Effective Hamiltonians for Electronic Fine Structure and ...rwf.mit.edu/group/papers/rwf339.pdf · Effective Hamiltonians for Electronic Fine Structure and Polyatomic Vibrations Robert

Effective Hamiltonians for Electronic Fine Structureand Polyatomic Vibrations

Robert W. Field, Joshua H. Baraban, Samuel H. Lipoffand Annelise R. BeckDepartment of Chemistry, Massachusetts Institute of Technology, Cambridge, MA, USA

1 INTRODUCTION

1.1 Properties and Format of the Heff

The effective Hamiltonian, Heff, is an approximate rep-resentation of the exact Hamiltonian, Hexact, which is aninfinite-dimensional matrix. For all but a few exactly sol-uble toy problems, it is impossible to solve for the exacteigenvalues of an infinite-dimensional matrix. However, aneffective Hamiltonian is a finite-dimensional matrix andconsequently usable. The Heff is constructed to be capa-ble of reproducing, to measurement accuracy if needed, allof the energy levels and relative transition intensities thatare observable in a frequency-domain spectrum (Brown andCarrington 2003).

Each element of Heff is a sum over products of anexperimentally determinable molecular constant and an apriori known function of quantum numbers, for example

H(3Π, J )vΩ,vΩ ′ = Ev(3Π)δΩ,Ω ′

+ Bv[J (J + 1) − 2Ω2 + 1 + 2|Ω|]δΩ,Ω ′︸︷︷︸rotation

− 21/2Bv[J (J + 1) − ΩΩ ′]1/2δΩ,Ω ′±1︸︷︷︸spin−uncoupling

+ Av(|Ω| − 1)δΩ,Ω ′ + λv[2Ω2 − 4|Ω| + 2/3]δΩ,Ω ′︸︷︷︸spin–orbit and spin–spin fine structure

(1)

Handbook of High-resolution Spectroscopy. Edited by Martin Quackand Frederic Merkt. 2011 John Wiley & Sons, Ltd.ISBN: 978-0-470-74959-3.

where the molecular constants, Ev , Bv , Av , and λv , arethe energy, rotational constant, spin–orbit constant, andspin–spin constant of the vth vibrational level of a 3Π stateand v, J , and Ω are vibration, rotation, and body-frameprojection (Jz) quantum numbers (Lefebvre-Brion and Field2004, pages 94–98).

Another example, illustrated in Table 1, is an Heff

that describes anharmonic coupling between two harmonicvibrational modes (Lefebvre-Brion and Field 2004, pages689–692). This coupling is represented by a family ofvibrational matrices arranged along the main diagonal of theHeff. These matrices are needed to describe the vibrationalstates in the presence of an approximate ω1 ≈ 2ω2 1 : 2anharmonic resonance. Here, all the matrices are expressedin terms of the same set of molecular constants, harmonicfrequencies ω1 and ω2, diagonal anharmonicity constantsx11, x22, and x12 and a reduced interaction parameter,K1,22.

The values of the molecular constants that appear inan Heff are adjusted in a least-squares fit in which thedifferences between the eigenvalues of the Heff and theobserved energy levels are minimized. To think of theHeff merely as a convenient fit model is to grossly under-value the insights into dynamical intramolecular vibrationalredistribution (IVR) mechanisms, including all processesobservable in any imaginable time-domain experiment, thatalways are obtainable from a rationally constructed Heff.

Block-to-block quantum number scaling of the matrixelements of an Heff is an extremely valuable property.By fitting the parameters in an Heff against a subset ofobserved and assigned energy levels, it is possible to makeuseful energy and relative-intensity predictions about notyet observed or assigned transitions. This facilitates rapid

1462 Effective Hamiltonians for Electronic Fine Structure and Polyatomic Vibrations

Table 1 Heff for Fermi resonance polyads.

|0, 0〉 |0, 1〉 |0, 2〉 |1, 0〉 |0, 3〉 |1, 1〉 |0, 4〉 |1, 2〉 |2, 0〉 · · ·

〈0, 0| G(0, 0)

〈0, 1| G(0, 1)

〈0, 2| G(0, 2) 21/2K1,22

〈1, 0| 21/2K1,22 G(1, 0)

〈0, 3| G(0, 3) 61/2K1,22

〈1, 1| 61/2K1,22 G(1, 1)

〈0, 4| G(0, 4) 121/2K1,22 0〈1, 2| 121/2K1,22 G(1, 2) 41/2K1,22

〈2, 0| 0 41/2K1,22 G(2, 0)

.... . .

Heff(ω1, ω2, x11, x22, x12,K1,22, v1, v2) G(v1, v2) = ω1(v1 + 1/2) + x11(v1 + 1/2)2 + ω2(v2 + 1/2) + x22(v2 + 1/2)2 + x12(v1 + 1/2)(v2 + 1/2)

Diagonal elements: Heffv1,v2;v1v2

= G(v1, v2)

Off-diagonal elements: Heffv1,v2;v1+1v2−2 = K1,22[(v1 + 1)(v2)(v2 − 1)]1/2

Polyad matrices are enclosed in boxes. There are no nonzero matrix elements outside of the boxes (polyads) along the main diagonal of this Heff.

extensions of assignments and can direct experimentalattention to particularly revealing regions of the spectrum.Deviations from Heff predictions often convey importantinformation about previously unsampled interaction mech-anisms or, more prosaically, errors in the assignments uponwhich the current values of the parameters in the Heff arebased.

The Heff is a quantum mechanical object, because thevalues of the molecular constants contained in the Heff aredetermined by a fit to the energies of observed eigenstates.However, especially for vibrational problems, the Heisen-berg correspondence principle (Heisenberg 1925, Xiaoand Kellman 1989) enables replacement of the quantummechanical Heff by a classical mechanical Heff expressedin the action–angle representation. A classical mechani-cal Heff permits qualitative examination of the structure ofphase space. Of particular value is the emergence of qual-itatively new classes of intramolecular motions at “bifur-cations” (Kellman 1995) and a measure of the fractionof phase space that is classically chaotic (Jacobson et al.1999a).

In this article, we discuss methods for constructingan Heff, the small but essential differences between themolecular constants in the Heff vs. those in the Hexact,under what circumstances must an Heff model fail, whatkinds of failure are recoverable, and some recently pro-posed methods for extending the domain of validity of anHeff.

The exact Hamiltonian may be converted from differen-tial operator form, Hexact, to matrix form, Hexact, by choos-ing a complete set of basis functions, ϕi, and calculatingall integrals of the exact Hamiltonian operator between thesame or different states (diagonal and off-diagonal matrix

elements, respectively),

H exactij =

∫ϕ

i Hexactϕjdτ = ⟨

i|Hexact|j ⟩ . (2)

As the number of linearly independent functions ina complete basis set is generally infinite, Hexact is aninfinite-dimensional matrix. It is almost always impossibleto find the eigenvalues and eigenvectors of an infinite-dimensional matrix. Hexact may be reduced to a set ofsmaller dimensional matrices by exploiting the propertiesof operators that commute with Hexact. When an operatorA commutes with Hexact, i.e., [A, Hexact] = 0, then it ispossible to transform Hexact into block-diagonal form,where each block is associated with a different eigenvalue,ai , of A. If a set of several mutually commuting operatorsA, B, C, . . . (that each commute with the Hexact) exists,then the Hexact may be reduced to even smaller blocks alongthe diagonal, each associated with the set of rigorously goodquantum numbers, ai, bj , ck, . . . . Figure 1 schematicallyillustrates the reduction of Hexact to block-diagonal form.

Unfortunately, even after taking maximum advantage ofoperators that commute with Hexact, each block of Hexact isstill of infinite-dimension and the tools of linear algebracannot be used to solve for the eigenvalues and eigen-vectors of Hexact. However, it is almost always possibleto approximately factor each of the infinite-dimensionalblocks of Hexact into finite-dimensional sub-blocks alongthe main diagonal (“block-diagonalize”), of which only afew sub-blocks are relevant to the spectroscopic or dynam-ical problem under immediate study. These finite matricesare the effective Hamiltonian(s). The price one pays for thisenormous reduction in the dimension of Hexact is small andthe gain in describing the spectra and dynamics is great.

Effective Hamiltonians for Electronic Fine Structure and Polyatomic Vibrations 1463

(b)

(a)

Figure 1 (a) shows the exact block-diagonalization of Hexact

according to a set of operators that commute with the exactHamiltonian. (b) shows an expanded view of the lower rightsymmetry block. Nonzero matrix elements are indicated with dots,sized according to magnitude. Zero-order energies (on-diagonalmatrix elements) are marked with open circles, and are scaleddifferently than the off-diagonal elements. The boxes delineate apolyad-like structure, of the kind that is discussed in Section 4,for the vibrations of a three-mode system with both Fermi andDarling–Dennison resonances. The dynamics of such a systemare explored in Section 6.2.

Nevertheless, it is essential to understand both the priceand the methodology of this replacement of the Hexact byan Heff.

1.2 Local Versus Remote Perturbers: theImportance of the Mixing Angle

The construction of an Heff embodies decisions aboutthe relative importance of various terms in the Hexact

and the partitioning of state space into “local-perturber”and “remote-perturber” classes of basis states. The simplephysical criterion is resonance. Two basis states that arecoupled by an off-diagonal matrix element that is ofcomparable magnitude to their energy separation are inresonance. The key concept of mixing angle sufficesto guide the partitioning of state space, as embodiedin the organization of Heff into quasi-degenerate blocks,called polyads, which cannot be treated exclusively bynondegenerate perturbation theory.

First-order nondegenerate perturbation theory tells ushow to construct approximate eigenstates of the Heff,

|i〉 = |i〉0 +∑

j

′ [Hji

/(E

(0)i − E

(0)j

)]|j 〉0 (3)

where the superscripts 0 and (0) signify zero-order state(basis state) and zero-order energy (respectively, eigenstate

and eigenvalue of the zero-order Hamiltonian, H0), |i〉is called the “nominal” ith state because it has dominant|i〉0 basis state character, E

(0)i and E

(0)j are the energies

of the ith and j th basis states, Hji = 0⟨j |H′|i⟩0 (where

H′ ≡ H − H0), and∑

j′ indicates that the summation is

restricted to exclude j = i. The “mixing angle”, defined as

θ ij =[Hji

/(E

(0)i − E

(0)j

)], (4)

is the mixing coefficient that expresses the admixture ofthe j th basis state into the nominal ith statea. How a basisstate is treated in an Heff depends on whether the absolutevalue of its mixing angle with at least one of the localstates is larger or smaller than 1. If one mixing angleis large, the basis state is included in a quasi-degenerateblock, which must be diagonalized. If the mixing angles areall small, the basis state may be treated by nondegenerateperturbation theory. It is always the mixing angle, ratherthan the magnitude of the Hji coupling matrix element,that determines the “strength” of the coupling between theith and j th basis states and whether the j th basis state isexplicitly dealt with as a local-perturber or relegated toremote-perturber status and forgotten.

The reduction and truncation of the Hexact to an Heff arebased on the smallness of mixing angles. The vast majorityof mixing angles are ignorably small owing to the very

large(E

(0)i − E

(0)j

)term in the denominator. A mixing

angle can be large either accidentally or systematically.When an energy denominator is accidentally near zero andtherefore small relative to the Hji numerator, the Heff mustbe augmented by an additional local-perturber basis state.This is a necessary nuisance. Basis states that are connectedvia a large mixing angle are said to be quasi-degenerate.

When the couplings among a family of basis states arestrong, as manifested in consistently large mixing angles, allof these coupled states play essential roles in the appearanceof the spectrum and in the nature of the early-time dynam-ics. Sometimes there are simple rules, expressed as a func-tion of basis state quantum numbers, which specify boththe minimum membership of “local” basis states within oneblock of the Heff and the intrablock and interblock quantumnumber scaling of the matrix elements, H eff

j i . Examples ofthese membership and H eff

j i -scaling rules are discussed laterin the contexts of multiplet electronic states (Section 3.1)and vibrational polyads (Section 4.2).

It would be convenient if it were possible to reduce Hexact

to an Heff consisting entirely of 1 × 1 blocks along themain diagonal. Doing so would be equivalent to derivinga simple energy level formula in which all of the energylevels are obtained from a simple power series expressionin the quantum numbers, analogous to the Dunham formula


(Dunham 1932)

EvJ =∑,m

Y,m(v + 1/2)[J (J + 1)]m. (5)

This sort of reduction is very common for the rotation–vibration energy levels of electronic states in an extremelimit of one of the Hund’s coupling cases [in fact, thisreduction to simple formulas is a major reason whyspectroscopists are enamored of Hund’s cases, (see Watson1999)] and for most low-lying vibration–rotation levelsof spin and orbitally nondegenerate polyatomic molecules.However, there are many cases where reduction to simplepower series formulas is impossible or ill advised, asexemplified by the Heff matrices for a 3Π electronic state(Brown and Carrington 2003) and for ω1 ≈ 2ω2 vibrationalpolyads (Table 1). The transformation that reduces Hexact

to an Heff in fully 1 × 1 or ni × ni block-diagonal formis known as a Van Vleck or contact transformation (Brownand Carrington 2003, Papousek and Aliev 1982). Despiteappearances to the contrary, the algebra and the terminologyfor both the 1 × 1 and block-diagonal reductions derivefrom the same perturbation-theory methodology.

1.3 Structure of This Article

This article is organized as follows.An approximate block-diagonalization procedure for con-

structing the Heff is described in Section 2, both formallyand with a worked example. The block-diagonalizationtransformation of the Hexact effectively folds the effects ofremote-perturber basis states into the Heff that describes thelocal group of states sampled in the spectrum under study.Several crucial points regarding the incomplete correspon-dence between the Hexact and the Heff are emphasized atthe end of Section 2.

Section 3 contains a discussion of Heff models fordiatomic molecules, with particular emphasis on thespin–rotational structure (spin–orbit, spin–spin, spin–rotation, lambda doubling, and centrifugal distortion) of iso-lated electronic–vibrational states. The key point is thatdiatomic molecule fine structure is essentially a solvedproblem owing to the finite number of symmetries ofremote-perturber states. Difficulties associated with inter-actions among many vibrational levels of several electronicstates sometimes arise when building an Heff; however,these difficulties are quantitative rather than fundamental.Multistate fit models, based on inherent, a priori knowablerelationships among groups of basis states (e.g., Rydbergscaling rules, interaction matrix elements expressed as aproduct of calculable vibrational overlap and constant elec-tronic factors), are capable of representing even the most

complicated web of interactions. The key concept of deper-turbation is explained.

Section 4 begins with an introduction to the vibrations ofpolyatomic molecules, and describes the traditional anhar-monically coupled harmonic oscillator Heff model. Whenseveral vibrational modes are systematically resonant (e.g.,1 : 2 Fermi resonance, 2 : 2 Darling–Dennison resonance,and more complicated resonances involving three or moremodes), the natural extension of the anharmonically cou-pled harmonic oscillators model is the “polyad” Heff modelfor the interactions among the quasi-degenerate vibrationallevels. This model is based on harmonic oscillator matrixelement scaling and selection rules. The effects of sev-eral specific intermode interaction terms from the Hexact

on the Heff are derived and illustrated before present-ing the general “Van Vleck” or “contact” transformation.The vibrational Heff for linear ABA and ABBA moleculesprovides examples of doubly degenerate bending vibra-tions and the evolution from normal to local modes. Theinevitable failure of the anharmonically coupled harmonicoscillator Heff at high excitation energies is illustrated atthe end of Section 4.

Some promising ideas for extending the Heff to systemsuntreatable with the anharmonically coupled harmonicoscillator polyad model are presented in Section 5.

Several unconventional applications and extensions ofthe polyad Heff model are illustrated in Section 6. Theseinclude the analysis of the structure of classical mechani-cal action–angle phase space (qualitative classification ofvibrational eigenstates, bifurcations at which qualitativelynew stable large-amplitude motions emerge, the fraction ofphase space filled with chaotic trajectories), mechanismsand rates of intramolecular energy flow, visualizationsof dynamics, and barrier-related diagnostic patterns thatappear in what might naively be expected to be intractablycomplicated spectra.

Effective Hamiltonians are essential to molecular sci-ence. Their power, weaknesses, and future prospects arebriefly summarized in Section 7.

2 WHAT IS AN EFFECTIVEHAMILTONIAN?

2.1 Steps in the Construction of the Heff

Hexact is expressed in an infinite-dimensional state space.Heff is expressed in a finite-dimensional state space. Non-degenerate perturbation theory provides a framework fortruncating the infinite-dimensional Hexact. The truncationconsists of three steps: choosing the states that are theprimary subject or “target” of the present investigation;


identifying those states that are so strongly coupled to thetarget states that their effects on the target states cannotbe represented by nondegenerate perturbation theory; andexpressing the effects of the infinite number of remote statesas small corrections to the Heff.

The first step is the identification of the region of statespace that is principally relevant to the spectroscopic ordynamical problem one has chosen to investigate. Forexample, one might be interested in an electronic transi-tion, such as the CO b3Σ+−a3Π (vb = 2, va = 0) band(Rytel 1991, 1992). The relevant region of state space con-sists of the three multiplet components of the b3Σ+(vb =2) upper state and the six multiplet components of thea3Π(va = 0) lower state. One must add to these princi-pally relevant states the multiplet components of all ofthe other energetically nearby vibration–electronic statesthat are likely to be strongly coupled to either of thesetwo states. For example, the CO b3Σ+(v = 2) state isknown to tune through resonance with the v = 40–43 lev-els of the a′3Σ+ state for 0 < J < 21. Therefore, severalvibrational levels of the a′3Σ+ state must be added tothe b3Σ+(v = 2) block of Heff. These states are quasi-degenerate and must be treated by diagonalization of asuitable Heff. Alternatively, one might be interested inthe dynamics of highly excited vibrational levels of theacetylene electronic ground state (Herman 2007). Theprincipally relevant zero-order states are those that areaccessible via the chosen excitation scheme [e.g., over-tone pumping (Lehmann 1990) or Stimulated EmissionPumping (Kittrell et al. 1981)]. These accessible zero-order states are known as bright states. (“Brightness”is defined by the nature of the experimental excitationscheme by which one gains access to the states of inter-est. It is not an intrinsic quality.) Then the relevant regionof vibrational state space consists of a small numberof “bright” vibrational basis states and all of the ener-getically nearby vibrational states that are likely to bestrongly anharmonically or Coriolis-coupled to the brightstates. The group of strongly coupled vibrational statesis called a polyad (Lefebvre-Brion and Field 2004, pages689–692).

The second step is the partitioning of basis statesinto local and remote groups. The local states are thosethat are strongly coupled (mixing angle comparable toor larger than 1) directly or indirectly to one of thestates of primary interest. In this article, the local andremote basis states are labeled by right superscripts α

and φ, respectively. A crude finite-dimensional Heff-α isconstructed from the Hamiltonian matrix elements amongthe local basis states.

The third step is the correction of the crude Heff-α for allinteractions between the infinite number of basis states inthe remote φ group and the finite number of basis states in

the local α group. There are various formal procedures foraccomplishing this remote-perturber or out-of-block correc-tion of the crude Heff-α. These procedures are discussedin Sections 2.3 and 4.8. All of them give second-ordercorrection terms on the main diagonal of Heff-α , H

eff-α(2)iα,iα ,

equivalent to the second-order corrections to the energy thatone obtains from nondegenerate perturbation theory, E

(2)iα .

The second-order correction to the energy of the |iα〉 basisstate, due to interactions with all of the remote |mφ〉 basisstates, is

Heff-α(2)iα,iα ≡ E

(2)iα =

∑mφ

∣∣Hiα,mφ

∣∣2E

(0)iα − E

(0)

mφ

. (6)

Despite the infinite number of terms in this summation, thenumerical value of the second-order perturbation sum isusually small relative to all intra-α-block diagonal energydifferences, E

(0)iα − E

(0)jα .

This would complete the process of construction of Heff

if it were unnecessary to correct the off-diagonal elementsof the crude Heff−α for the effects of all of the remote |mφ〉basis states. The second-order correction to the off-diagonalα-block matrix element, H

eff-α(2)iα,jα , has the unsurprising form

(Lowdin 1951, Van Vleck 1951),

Heff-α(2)iα ,jα =

∑mφ

[Hiα,mφHmφ,jα

]12

[E

(0)iα + E

(0)jα

]− E

(0)

mφ

. (7)

The various schemes to correct for out-of-block effectsdiffer primarily in formalism. Subtle numerical differencesbetween these schemes appear only beyond second-orderperturbation theory.

2.2 Reduction of a 3 × 3 Hexact to a 2 × 2 Heff

The form of the second-order corrections to the crudeHeff-α is motivated here by the first-order nondegenerateperturbation-theory corrections to the wave function. Con-sider the simplest relevant state space, which consists oftwo α-block states, |1〉 and |2〉, and one φ-block state, |3〉.The 3 × 3 exact Hamiltonian is

|1〉0 |2〉0 |3〉0

Hexact =0〈1|0〈2|0〈3|

E + δ/2 V12 V13

V12 E − δ/2 V23

V13 V23 E3

.(8)

For simplicity, our Hexact is real, but this is not truegenerally. To reduce Hexact to a 2 × 2 Heff-α matrix, itis necessary to account for the effect of state |3〉0 on


the |1〉0, |2〉0 block of states via an approximateblock-diagonalization and truncation procedure. From first-order nondegenerate perturbation theory we have

|1〉 = |1〉0 + (V13/[E + δ/2 − E3])|3〉0 (9)

|2〉 = |2〉0 + (V23/[E − δ/2 − E3])|3〉0 (10)

|3〉 = |3〉0 − (V13/[E + δ/2 − E3])|1〉0

− (V23/[E − δ/2 − E3])|2〉0 (11)

and even though these three states are neither exactlynormalized nor mutually orthogonal (the normalization andorthogonality errors are of magnitude θ2 and thereforenegligibly small), it is possible to use them to define anapproximately unitary transformation (T † = T−1) of Hexact

that folds the effects of the remote state |3〉0 into the 2 × 2|1〉0, |2〉0 block of the Heff,

Hexact = T†HexactT (12)

where

T =

1 0 −V 13

[E+ δ2 −E3]

0 1 −V 23[E− δ

2 −E3]V13

[E+ δ2 −E3]

V23[E− δ

2 −E3]1

(13)

The resulting expression for the transformed Hexact

appears to be algebraically complicated, but the interpre-tation of each matrix element is simple and also essentialfor understanding the differences between the effective fitparameters in Heff and the true microscopic parameters inHexact. We now examine a few of the Hexact matrix elementsindividually.

The first diagonal matrix element is

H exact11 = E + δ/2︸︷︷︸

E(0)1

+ V 213/[E + δ/2 − E3]︸︷︷︸

E(2)1

+ E3V213/[E + δ/2 − E3]2︸︷︷︸

artifact

(14)

where the first two terms comprise E(0)1 , the third term is the

second-order correction to the energy, and the fourth term isan artifact that arises from the use of non-normalized statesin the transformation. The other diagonal matrix element inthe α block is

H exact22 = E − δ/2︸︷︷︸

E(0)2

+ V 223/[E − δ/2 − E3]︸︷︷︸

E(2)2

+ E3V223/[E−δ/2 − E3]2︸︷︷︸

artifact

. (15)

The off-diagonal matrix element in the 2 × 2 block is

Hexact12 = V12 + (1/2)V13V23

E + δ/2 − E3+ (1/2)V13V23

E − δ/2 − E3︸︷︷︸correction terms

− E3V13V23

[E + δ/2 − E3][E − δ/2 − E3]︸︷︷︸artifact

(16)

where the second and third terms are the expected correc-tion to the off-diagonal matrix element and the fourth termis an artifact. The second and third terms may be combined,following equation (7), thus

Hexact12

∼= V12 + V13V23

E − E3. (17)

The out-of-block off-diagonal matrix elements are

Hexact13 = V13

[V 2

13

(E + δ/2 − E3)2

+ V 223

(E + δ/2 − E3)(E − δ/2 − E3)

]

+ V12V23

E − δ/2 − E3(18)

Hexact23 = V23

[V 2

23

(E − δ/2 − E3)2

+ V 213

(E + δ/2 − E3)(E − δ/2 − E3)

]

+ V12V13

E + δ/2 − E3(19)

where the first term is the original out-of-block off-diagonal matrix element multiplied by the product of twomixing-angle factors and the second term is the intrablockoff-diagonal matrix element multiplied by one mixing-angle factor. (The residual interblock matrix elements,H exact

13 and H exact23 , contribute δE

(2)i = (

H exacti3

)2/(E − E3)

to the intrablock-diagonal energies.) The Hexact → Hexact

transformation reduces the neglected effects of interblockcoupling terms on intrablock energies by factors of θ4

and θ2. Thus, as far as energy levels are concerned, theerrors resulting from the replacement of the 3 × 3 Hexact

by the 2 × 2 Heff-α are negligible. However, the distortionsof fitted molecular constants and of computed transitionintensities may not be negligible (see Section 2.4).

To recapitulate, we start with Hexact for a state spaceconsisting of states |1〉0, |2〉0 and |3〉0. States |1〉0 and |2〉0

are classified as α-type and a crude Heff-α is constructed


by discarding the third row and column of Hexact. Animproved Heff-α is constructed by transforming Hexact →Hexact = T†HexactT using a T constructed following theprescription of nondegenerate perturbation theory for thefirst-order correction to the wavefunction. The matrixelements connecting the φ-block (remote) state |3〉0 to theα-block states |1〉0 and |2〉0 are reduced by mixing anglefactors, which are much smaller than 1. Now it is a muchbetter approximation to construct Heff-α by discarding thethird row and column of Hexact. Next, we generalize to aninfinite-dimensional Hexact. A cautionary note regarding theconstants that emerge from an Heff fit to an experimentalspectrum follows this generalization.

2.3 Formal Generalization of the Constructionof Effective Hamiltonians

We develop the framework for expressing the Heff in termsof Hexact in a few different but equivalent ways. All involvesome form of perturbation theory. The present discussionis based on the use of projection operators (Brown andCarrington 2003, see Section 7.2). [See Section 4.7 ofthis article for the contact transformation (Papousek andAliev 1982) and Section 5.1 for CVPT (McCoy and Sibert1991a)].

As is usual in perturbative methods, we partition ourexact Hamiltonian into a zero-order part and a perturbationterm

Hexact = H(0) + H′ = H(0) +jmax∑j=1

λj H(j) (20)

where λ is the standard order-sorting parameter. As indi-cated in equation (20), the perturbation term H′ can consistof many terms of differing order, but here we deal withonly jmax = 1, and therefore H′ = λH(1), for simplicity.The zero-order eigenfunctions are defined as follows:

H(0)|k〉0 = E(0)k |k〉0 (21)

where k denotes the set of quantum numbers associated withH(0). H(1) does not commute with H(0), and so k cannotrigorously be associated with the exact eigenfunctions ofHexact. The quantum numbers in the set k that remaingood, i, are used to block-diagonalize Hexact. Within a givensymmetry block, we can label both the zero-order and exacteigenfunctions by the set i, because this set correspondsto eigenvalues of operators that commute with Hexact. Theexact eigenstates are

Hexact|τ 〉 = Eτ |τ 〉 = (E(0)i + δEτ

i )|τ 〉 (22)

where τ is a unique label for each exact eigenstate,essentially its energy, but technically referred to as itseigenvalue rank. (It is pedagogically convenient, for thepurposes of this derivation, to be able to correlate the exacteigenstates with the zero-order states by labeling them withthe basis state index, i, but it is not necessary.)

We are interested in a subset of the eigenstates ofHexact that is defined by the nature of the spectrum beingexamined. We call this set of eigenstates the α subspace

|τα〉 =∑

i

cτi |i〉0 (23)

and we label both the zero-order and exact eigenfunctionswithin this local subspace with α, as |iα〉0 and |τα〉,respectively.

At this point, we concern ourselves with how to obtainthe exact eigenvalues Eτ = (E

(0)i + δEτ

i ) of the α stateswithout explicit reference to the remote φ states. This makesit possible practically to deal with the problem definedby equations (22) and (23), since the sum over all i inequation (23) includes an infinite number of states. [Tonotationally distinguish the remote states from the localstates, we label the remote states with m in φ, which isequivalent to i not in α, or symbolically, (m ∈ φ) ≡ (i α)]. Formally, we seek the effective Hamiltonian for the α

subspace, Heff-α , which satisfies the operator equation

Heff-α∑i∈α

dτi |i〉0 = (E

(0)i + δEτ

i )∑i∈α

dτi |i〉0. (24)

Note that this is equivalent to insisting that the matrixHeff-α has the same eigenvalues as Hexact, as discussed inSection 2.2 in the reduction of the 3 × 3 Hexact to a 2 × 2Heff. It is extremely important to realize that our objectivehere is only to obtain the exact eigenenergies, and not theexact eigenfunctions. The linear combination of zero-orderα states that yields an eigenenergy of Hexact under theoperation of Heff is not an exact eigenstate of Hexact, despitelinguistic and notational ambiguities. Misconceptions ofthis kind are a major source of confusion regarding effectiveHamiltonians, and are discussed in Section 2.4. The crucialpoint is that the construction of Heff enforces agreementwith Hexact only for the energy eigenvalues; since noeigenvalue equations are considered other than those ofthe Hamiltonian operator, the eigenstates of Heff will notproduce the same matrix elements that the exact eigenstatesof Hexact would when matrix elements of operators otherthan H are evaluated.

To accomplish our goal of determining the eigenenergiesof the exact eigenstates in the α subspace while using onlythe zero-order α states, we find it convenient to define the


following projection operators:

Pα =∑i∈α

|i〉0 0〈i| (25)

which has the effect Pα|τ 〉 =∑i∈α

cτi |i〉0, and

Qα = 1 − Pα =∑m∈φ

|m〉0 0〈m| =∑iα

|i〉0 0〈i|. (26)

Pα projects onto the zero-order α subspace, and Qα projectsonto the complementary subspace. Additionally, we alsoimagine the following operator U:

U∑i∈α

cτi |i〉0 = |τ 〉, (27)

which reverses the effect of Pα. The utility of U becomesapparent if we consider the product operator PαH(1)U actingon an eigenstate |τα〉 = ∑

i cτi |i〉0, which yields

PαH(1)U∑i∈α

cτi |i〉0 = δEτ

i

∑i∈α

cτi |i〉0. (28)

This result is precisely what we need to fulfill equation (24),as long as dτ

i = cτi for i and τ ∈ α; in other words, the

eigenfunctions of Heff-α are the projections of the exacteigenstates onto the α subspace. Therefore, we concludethat the effective Hamiltonian can be written as

Heff-α = H(0) + λPαH(1)U. (29)

Note that Heff-α is not merely a piece of Hexact. Rather,a part of Hexact is transformed in a special way so that,within the α subspace, Hexact and Heff-α yield the same setof eigenenergies.

What remains is to express U in terms of the knownquantities Pα , Qα , and H(1), first as a recursion relation,and subsequently as an expansion in powers of λH(1),a derivation that can be found in Brown and Carrington2003. Substituting the resulting expression for U intoequation (29), we arrive at our final result

Heff-α = H(0) + λPαH(1)Pα + λ2PαH(1) Qα

aH(1)Pα

+ λ3(

PαH(1) Qα

aH(1) Qα

aH(1)Pα

− PαH(1) Qα

a2H(1)PαH(1)Pα

)+ . . .

(30)

where a−1 is the operator inverse of the operator (E(0)iα −

H(0)) in the zero-order subspace complementary to α, suchthat for any integer r

Qα/ar =∑mφ

|mφ〉0 0〈mφ |/(E(0)iα − E

(0)

mφ )r . (31)

Note that the last term in equation (30) is not Hermitian.However, Heff-α may be put into Hermitian form bysymmetrizing the non-Hermitian terms. For a discussion ofthe Hermiticity of effective Hamiltonians and the projectionoperator method, (see Watson 2005).

Notice the appearance in equation (30) of the second-order correction to the energy

∑mφ

0〈iα|H(1)|mφ〉0 0〈mφ |H(1)|jα〉0

12

[E

(0)iα + E

(0)jα

]− E

(0)

mφ

(32)

in terms of order λ2 and higher, as seen in equation (7).Despite the reference to a remote state, the presence of theprojection operator Pα as the left- and right-most operatorin every term in Heff-α guarantees that Heff-α operates onlywithin the α subspace, as we originally specified.

The transformation of Hexact to Heff shown here is aformal procedure, used to derive and justify the form ofHeff. In practice, the constants that appear in Heff areempirically adjusted to obtain agreement with experimentalresults. This hybrid formal and fitting approach imparts apowerful combination of rigor and flexibility to the effectiveHamiltonian, but it is not without pitfalls (Brown andCarrington 2003).

2.4 Truth in Labeling

Every state in the infinite-member φ-block makes anindependent, additive contribution at both diagonal andoff-diagonal locations within the α-block effective Hamil-tonian, Heff-α .b Most of these individual contributions fromφ-states are negligibly small. However, the aggregate effectof these φ-block contributions to the α-block is pernicious.The “molecular constants”, derived by spectroscopists fromleast-squares adjustment of the Heff parameters until thedifferences between the eigenenergies of the Heff and theobserved energy levels are minimized, are not the samequantities computed ab initio by rigorous quantum chemi-cal methods, even though the same names are used by bothexperimentalists and theorists for these distinct quantities.

Out-of-block effects subtly corrupt the microscopicmeanings of molecular constants in an Heff. The fit param-eters are the microscopic molecular constants plus unspec-ified corrections that arise from the implicit transformationof Hexact into the form of the Heff-α fit model. Fits to spec-troscopic data often yield molecular constants determinedto an astonishing number of decimal places. This precisionis needed to satisfactorily fit the spectrum, but the effects


of out-of-block contributions to the fit parameters are oftenmuch larger than the uncertainties in high-level quantumchemical calculations of the microscopic molecular con-stants. It is naıve to think that experimentally determinedmolecular constants are the appropriate standard for check-ing the accuracy of computed molecular constants.

The gold standard is the raw spectroscopic data, notthe fitted molecular constants. It is sometimes possible, byapproximate calculation of the leading terms in the out-of-block contributions, to reduce the differences betweenfitted (experimentally determined) and computed (ab initio)molecular constants. However, the best way to eliminatethese hidden systematic errors would be to compute theexperimentally observed energy levels and to fit theselevels to the same Heff model used by the experimentalists.The computed energy levels would not, for example, bethe zero-order energy levels of a pure electronic state, afew locally interacting electronic states, or a large groupof strongly interacting vibrational states that constitutea polyad. They would be the same mixtures of zero-order states observed in the spectrum and described bythe Heff. The computed energy levels would have to begood approximations to the eigenstates of Hexact. It is sillyto expect that individual eigenenergies of Hexact can becomputed at an accuracy comparable to 0.1 cm−1. Instead,it is much more realistic to assess the quality of an ab initiocalculation by comparing computed differences between afew well-selected energy levels against the correspondingenergy differences between experimentally observed states(Lewerenz and Quack 1988).

It is important to avoid confusing the distinct goals ofexperiment and theory. An experimental fit model mustreproduce the observed energy levels to within measure-ment accuracy. A theoretical calculation must capture theessential features of structure and dynamics. Owing to out-of-block corrections, spectroscopically determined molec-ular constants are not equivalent to the true microscopicmolecular constants determined by a high-level ab ini-tio calculation. There is minimal microscopic reality, forexample, in a spectroscopically determined equilibriuminternuclear distance stated to an absolute accuracy of betterthan ±0.01 A.

2.5 Dynamics

One might also expect that an Heff that accurately accountsfor all of the observed energy levels would provide anexcellent account of the dynamics of the system subsequentto preparation in any imaginable initial state. However,truncation of the infinite-dimensional Hexact to a finite-dimensional Heff results in a multitude of subtle errors,principally the neglect of coupling terms between basisstates that are nominally zero.

Suppose, for example, that a specific zero-order state,ψ

(0)iα is prepared at t = 0 by a very short excitation pulse.

The prepared state is not an eigenstate, hence its observableproperties evolve in time. Small amounts of the preparedzero-order state are admixed into many local (α) and remote(φ) eigenstates,

Ψ (t = 0) = ψ(0)iα =

∑iα

aiαψiα +∑mφ

amφψmφ . (33)

The initially prepared noneigenstate evolves in time

Ψ (t) =∑iα

aiαψiα e−iEiα t/ +∑mφ

amφψmφ e−iEmφ t/ (34)

and the time-dependent probability of finding the system inthe initial state is

|Ψ (t = 0)Ψ (t)|2 =∑iα

|aiα |2 ∣∣ψiα

∣∣2 +∑mφ

∣∣amφ

∣∣2× ∣∣ψmφ

∣∣2 very small

+∑iα,i′α

2 cos

[(Eiα − Ei′α

)t

]Re

[aiαa

i′αψ(0)iα ψ

(0)

i′α]

− 2 sin[ωiα,i′α t

]Im

[aiαa

i′αψ

(0)iα ψ

(0)

i′α]large and slow

+∑

iα,mφ

2 cos

[ωiα,mφ t

]Re

[aiαa

mφψ

(0)iα ψ

(0)

mφ

]− 2 sin

[ωiα,mφ t

]Im

[aiαa

mφψ(0)iα ψ

(0)

mφ

]small and fast

+

∑mφ,m′φ

very small and mostly fast terms. (35)

In the Heff model, the Ψ (t = 0) does not contain out-of-block (φ) components and the intrablock (α) mixingcoefficients are subtly different than those in the Ψ (t = 0)

obtained from the Hexact ab initio calculation. The very fast(because of the large energy difference), small-amplitude(because of the small mixing angle) chatter in the Hexact

dynamics caused by remote perturbers may be viewed asnoise superimposed on the large-scale intrinsic dynamicscaused by a few strong resonances. Rightly or wrongly,this “noise” is removed when the Hexact is replaced by anHeff. An Heff, defined in terms of molecular constants thatare least-squares optimized to fit experimentally observedenergy level data (Lefebvre-Brion and Field 2004, pages248–261), always provides an excellent representationof the early-time, large-scale dynamics (Lefebvre-Brionand Field 2004, pages 684–700), but little is knownabout how rapidly such a representation runs off thetracks.


3 EXAMPLES OF Heff MODELS FORTHE ELECTRONIC STRUCTUREOF DIATOMIC MOLECULES

3.1 Heff for Electronic States of DiatomicMolecules

The essential ideas and procedures for setting up an Heff

model are best illustrated for the simplest possible non-atomic example: the electronic structure of a diatomicmolecule. Diatomic molecules are simple because rigorousselection rules limit the qualitative quantum number depen-dences contained in the Heff. Additionally, knowledge ofthe principal pathways by which local α-type states inter-act with remote φ-type states permits crude estimates of themagnitudes of parameters generated by the truncation ofHexact to Heff. The existence of only one nuclear displace-ment coordinate ensures that parameters in the Heff willexhibit calculable dependences on vibrational and rotationalquantum numbers. Finally, the evolution from very largeseparations between electronic states at low energy towardsmall separations at high energy provides opportunities toillustrate both locally reparable and globally irremediablebreakdowns of Heff models.

No discussion of the electronic structure of diatomicmolecules can begin without at least a passing refer-ence to Hund’s cases (Nikitin and Zare 1994, Watson1999, Lefebvre-Brion and Field 2004). There are threeimportant classes of interaction terms: nonsphericity ofthe electronic environment around each atomic nucleus,spin–orbit interaction of each open-shell electron, andmolecular rotation. Nonsphericity splits atomic L–S–Jstates into molecular Λ–S-Ω states, e.g., 3DJ → 3∆Ω ,3ΠΩ , 3Σ+

Ω . Spin–orbit lifts the Ω-degeneracy of a 2S+1Λ

state and causes ∆S = 0, ±1 and ∆Λ = 0, ±1 interactionsamong 2S+1Λ states. Rotation tries to change the body-frame quantization axis from the cylindrically symmetricinternuclear axis to the direction of the nuclear rotationangular momentum, R, causing ∆Λ = ±1 and ∆Σ = ±1interactions as it does so. Hund’s cases (a–e) classify thevarious possible relative strengths of these three interactions(Nikitin and Zare 1994). The following discussion is framedin the language of Hund’s case (a), in which the hierar-chy of coupling strengths is nonsphericity spin–orbit >

rotation.A 1Σ state is the simplest possible electronic state. The

Heff for each vibration–rotation level of a 1Σ state hasdimension 1. There is no fine structure. In contrast, a 3Π

state consists of six components with Ω = 2, 1, 0 and ±parity for each v, J . The Heff consists of two 3 × 3blocks (one for each parity) [See equation 1 and Table 4for simplified and full Heff, respectively (Brown and Merer

1979)]. There are spin–orbit, spin–spin, spin–rotation, andΛ-doubling contributions to the fine structure. The twomain reasons why the six components of a 3Π state aretreated as one rather than six independent electronic statesare as follows: (i) all six components have the same elec-tronic–vibrational and rotational energies, and (ii) the com-ponents interact with each other (via rotation) vastly morestrongly than with remote perturbers.

Suppose that we are interested in an electronic tran-sition between an isolated 1Σ+ state and an isolated3Π state, for example, the nominally forbidden COa3Π − X1Σ+ transition. We need to set up two effec-tive Hamiltonians, one for the X1Σ+ state and one forthe a3Π state. There are only five terms in Hexact respon-sible for interactions between our two groups of α-typestates and all of the remote φ-type states:

∑i,j 1/rij

(interelectronic repulsion), Hspin–orbit, and three parts ofHrotation (the L-uncoupling term, −B(R)J±L∓, the rota-tion–electronic term, +B(R)L±S∓, and the S-uncouplingterm, −B(R)J±S∓) (Lefebvre-Brion and Field 2004). Theselection rules [equations (36)–(40)] for each of these inter-action terms are very restrictive. We will see in Section 4that, for interactions among vibrational states of a poly-atomic molecule, there are many more plausible interactionterms and the selection rules for those vibrational interac-tion terms are less restrictive.

The selection rules for the diatomic molecule Heff are asfollows:∑

i,j

1/rij ∆S = ∆Λ = ∆Σ = ∆Ω = 0,

∆(spin–orbitals) ≤ 2 (36)

Hspin–orbit ∆S = 0,±1, ∆Λ = −∆Σ = ±1, 0,

∆Ω = 0, Σ± ↔ Σ∓,


B(R)J±L∓ ∆S = ∆Σ = 0, ∆Λ = ∆Ω = ±1,


B(R)L±S∓ ∆S = ∆Ω = 0, ∆Σ = −∆Λ = ±1,

Σ± ↔ Σ±, ∆(spin–orbitals) ≤ 2 (39)

B(R)J±S∓ ∆S = ∆Λ = 0, ∆Σ = −∆Ω = ±1,

Σ± ↔ Σ±, ∆(spin–orbitals) ≤ 1. (40)

∆(spin–orbitals) is a shorthand that specifies the numberof spin-orbitals (one-electron basis functions) by which theN -electron (Slater determinantal) ψ initial and ψfixed wavefunctions in

⟨ψ initial|H|ψfixed

⟩differ. The X1Σ+ state can

only interact with remote 1Σ+ states (via interelectronicrepulsion), 1Π states (via L-uncoupling), 3Σ−

0 states (viaspin–orbit), and 3Π0e states (via spin–orbit). These inter-actions add terms to the Heff that cannot be distinguished,


in the fit to spectroscopic data, from the true electronic-vibration energy and the true rotational constant, but theseadditional terms do not introduce qualitatively new finestructure. The interactions of the X1Σ+ state with all φ-type remote states are therefore ignored in the Heff.

The situation for the α-type a3Π state and the remoteφ-type states with which it interacts is much more compli-cated. Table 2 gives a complete list of all φ-type states withwhich an α-type 3Π state can interact.

There are, in principle, an infinite number of φ-type statesof each symmetry, but the interactions with all remote statesof the same symmetry are lumped into a single interac-tion parameter. Table 2 may seem like a very long list ofφ-type states, but the key point is that this list is complete,the selection rules are rigorous, and each αΩ → φΩ ′ →αΩ ′′ out-of-block interaction path generates an interactionparameter in the H eff-α

Ω,Ω ′′ matrix element that is multipliedby an a priori known function of the rotational quantumnumber, J , as shown in Table 3.

Each correction term in the Ωv, Ω ′′v matrix elementof Heff-α , H eff-α

Ωv,Ω ′′v , is implicitly defined by an infinitesummation over out-of-block φ-type states, which havespecified values of the S, Λ, Ω ′, and vS,Λ quantumnumbers,

H eff-αΩv,Ω ′′v =

∑φ,S,Λ,Ω ′,vS,Λ

HαΩv,φΩ ′vS,ΛHφΩ ′vS,Λ,αΩ ′′v

12

[E

(0)αΩv + E

(0)

αΩ ′′v

]− E

(0)

φΩ ′vS,Λ

.

(41)On the basis of their explicit J -dependences and Ω,Ω ′′

positions in the Heff-α matrix, all of the correction terms arecombined to yield the most general, algebraically compactHeff-α matrix (Brown and Carrington 2003). This reductionto a fit model is exemplified by the standard Heff model

for a 3Π state (Table 4), first proposed by Brown andMerer (1979) and derived following procedures describedby Brown and Carrington (2003). The 3Π matrix in Table4 is really two 3 × 3 matrices, one each for levels of e-and f -symmetry (Brown et al. 1975). Each of the Ω,Ω ′

locations in Heff contains several additive terms, writtenas a to-be-determined molecular constant times a knownfunction of quantum numbers. These additive terms comefrom matrix elements of the operators (Hrot, Hspin–orbit, etc.)from which Heff is constructed.

This Heff is expressed in terms of the following effectivemolecular constants, all of which contain contributions fromthe Hexact → Heff transformation and truncation: Ev , Av ,Bv , λv , γ v are the energy, spin–orbit constant, rotationalconstant, spin–spin constant, and spin–rotation constant forthe vth vibrational level of the 3Π state. Typically, the truemicroscopic spin–spin and spin–rotation values of λv andγ v are small relative to the “correction terms”.

Dv , ADv , and λDv are centrifugal distortion corrections toBv , Av , and λv (Zare et al. 1970, Veseth 1973). Dv and λDv

owe their existence to the explicit R−2 internuclear distancedependence of the rotational constant operator, B(R), andAv expresses the implicit R-dependence of the spin–orbitconstant, A(R).

ov , pv , and qv are lambda-doubling constants (Brownand Merer 1979). They describe the J , Ω dependenceof the splitting between the two-parity (or e, f -symmetryBrown et al. 1975) components of the 3Π state. All threelambda-doubling parameters are due to interactions with φ-block Σ states. The qv parameter comes from Hrot ⊗ Hrot

correction terms, as reflected by qv being multiplied byJ (J + 1) (the product of two ∆Ω = 1 matrix elements)

Table 2 Complete list of perturbers of a 3Π state.

φ-type state 3ΠΩ α-states affected ∆Ω selection rule interaction mechanism

1Σ+ Ω = 0e ∆Ω = 0 spin–orbit1Σ− Ω = 0f ∆Ω = 0 spin–orbit1Π Ω = 1 ∆Ω = 0 spin–orbit1∆ Ω = 2 ∆Ω = 0 spin–orbit3Σ+ Ω = 1e, (0, 1)f ∆Ω = 0 spin–orbit and rotation–electronic3Σ− Ω = 1f, (0, 1)e ∆Ω = 0 spin–orbit and rotation–electronic3Π Ω = 0, 1, 2 ∆Ω = 0 spin–orbit and interelectronic3∆ Ω = 1, 2 ∆Ω = 0 spin–orbit and rotation–electronic5Σ+ Ω = (1, 2)f, (0, 1, 2)e ∆Ω = 0 spin–orbit5Σ− Ω = (1, 2)e, (0, 1, 2)f ∆Ω = 0 spin–orbit5Π Ω = 0, 1, 2 ∆Ω = 0 spin–orbit5∆ Ω = 0, 1, 2 ∆Ω = 0 spin–orbit3Σ+ Ω = (0, 1, 2)f, (1, 2)e ∆Ω = ±1 L-uncoupling3Σ− Ω = (0, 1, 2)e, (1, 2)f ∆Ω = ±1 L-uncoupling3Π Ω = 0, 1, 2 ∆Ω = ±1 S-uncoupling3∆ Ω = 0, 1, 2 ∆Ω = ±1 L-uncoupling


Table 3 Relationships between Ω → Ω ′ → Ω ′′ second-order pathways and the J -dependence ofH eff

Ω,Ω ′′ .

αΩ → φΩ ′ → αΩ ′′path J -dependence of term in H eff-αΩ,Ω ′′

(∆Ω = 0) ⊗ (∆Ω = 0) J -independent

(∆Ω = 0) ⊗ (∆Ω = +1)[J (J + 1) − ΩΩ ′′]1/2 ≈ J

(∆Ω = +1) ⊗ (∆Ω = +1)[J (J + 1) − Ω Ω+Ω ′′

2

]1/2 [J (J + 1) − Ω+Ω ′′

2 Ω ′′]1/2 ≈ J (J + 1)

Table 4 Heff for a 3Π state (Brown and Merer 1979).∣∣∣3Π0e

f

⟩ ∣∣∣3Π1e

f

⟩ ∣∣∣3Π2e

f

⟩Ev − Av + (Bv − ADv + 2λDv/3)(x + 2) −(2x)1/2[Bv − γ v/2 − ADv/2 − λD/3 −[x(x − 2)]1/2(2Dv ± qv/2)⟨

3Π0e

f

∣∣∣ +2λv/3 − 2γ v − Dv(x2 + 6x + 2) ∓(pv + qv)/2 − 2Dv(x + 2)]

±(ov + pv + qv)

Ev − 4λv/3 + (Bv − 4λDv/3)(x + 2) −[2(x − 2)]1/2[Bv − γ v/2 + ADv/2⟨3Π1

e

f

∣∣∣ same as⟨3Π0

e

f|H|3Π1

e

f

⟩−2γ v − Dv(x

2 + 8x) ∓ qvx/2 −λDv/3 − 2Dvx]

Ev + Av⟨3Π2

ef

∣∣∣ same as⟨3Π0

ef|H|3Π2

ef

⟩same as

⟨3Π1

ef|H|3Π2

ef

⟩+(Bv + ADV

+ 2λDv/3)(x − 2)

+2λv/3 − Dv(x2 − 2x)

x ≡ J (J + 1).Reproduced from Brown and Merer 1979. Elsevier, 1979.

when it appears on-diagonal and in the Ω, Ω + 2 off-diagonal location, and by [J (J + 1)]1/2 (the product ofone ∆Ω = 1 and one ∆Ω = 0 matrix element) when itappears in the Ω, Ω + 1 off-diagonal location. The pv

parameter comes from Hrot ⊗ Hspin–orbit correction terms;its J -independent presence in the Ω = 0 diagonal locationexpresses the product of two ∆Ω = 0 matrix elements, andits [J (J + 1)]1/2 dependent presence in the Ω = 0,Ω = 1off-diagonal location expresses the product of one ∆Ω = 0and one ∆Ω = 1 matrix element. The ov parameter thatcomes from Hspin–orbit ⊗ Hspin–orbit is thus of ∆Ω = 0 ⊗∆Ω = 0 origin and is not multiplied by a J -dependentfactor.

3.2 Examples of Remote-perturber Originsof Terms in the 3Π Heff

The following three examples illustrate how interactionswith φ-type remote perturbers contribute to parameters inthe effective Hamiltonian for a 3Π state: (i) a 1Π statecontributes to λv; (ii) a 3∆ state contributes to γ v; and (iii)v′ = v vibrational levels of the α-type 3Π state generateDv (Zare et al. 1973).

3.2.1 Contribution of a Remote 1Π state to λv of 3Π

The spin–spin constant, λv , expresses the nonuniformity ofthe splittings between the Ω-components of the 3Π state

(see Table 4):

∆2,1 − ∆1,0 ≡ [E(3Π2, J = 0) − E(3Π1, J = 0)]

− [E(3Π1, J = 0) − E(3Π0, J = 0)]

= [A − 4B + 2λ] − [A − 2λ] = −4B + 4λ

(42)As is true for all spin–orbit interactions, the spin–orbitinteraction matrix element between 1Π and 3Π statesfollows the ∆Ω = 0 selection rule and is independent of J

as noted earlier (Table 3). Only the 3Π1 spin component isshifted

δE(2)(3Π1, v) =∑v′

⟨1Π, v′|Hspin–orbit|3Π1, v

⟩2E(0)(3Π1, v) − E(0)(1Π, v′)

≈ [A(3Π)]2

∆E(3Π1, 1Π)(43)

∆E(3Π2,3Π1) − ∆E(3Π1,

3Π0) = −4B − 2δE(2)(3Π1)

(44)thus

λv = −δE(2)(3Π1, v)/2 ≈ − A(3Π)2

2∆E(3Π, 1Π)

= A(3Π)2

2∆E(1Π, 3Π)> 0. (45)


The potential energy curves for two states that belong tothe same electronic configuration are nearly identical. Thus,to a very good approximation, only the v = v′ term in theequation (43) sum is nonzero and the sum over the v′ vibra-tional levels of the remote 1Π state collapses to a singlev′ = v term. In this term, the spin–orbit matrix elementand the energy denominator are approximately indepen-dent of v. Thus, as shown in equation (45), the contri-bution of a same-configuration 1Π state to the spin–spinconstant of the 3Π state is approximately [A(3Π)]2/

[2∆E(0)(3Π, 1Π)], the spin–orbit constant of the 3Π statesquared divided by twice the v = 0 to v = 0 energy dif-ference between the 1Π and 3Π states. However, the λv

constant in Table 4 contains HSO ⊗ HSO contributions frommany symmetries of φ-type states (see Table 3). The aggre-gate of all such HSO ⊗ HSO contributions is distributedbetween E(3Π, v) and λv .

3.2.2 The Contribution of 3∆ to γ v of 3Π

The spin–rotation constant appears in the ∆Ω = 1 off-diagonal matrix elements of the 3Π Heff (Table 4), forexample,

⟨3Π0, v|Heff|3Π1, v⟩ = [2J (J + 1)]1/2

2γ v (46)

The spin–rotation and centrifugal distortion of the spin–orbit constants, γ v and ADv , express corrections to theeffective rotational constants for the nominal Ω = 2, 1, 0spin components of a 3Π state,

∆Beff2,1 = [

BeffΩ=2 − Beff

Ω=1

] = 4Bv(Bv − γ v/2 + ADv/2)

Av(47)

∆Beff1,0 = [

BeffΩ=1 − Beff

Ω=0

] = 4Bv(Bv − γ v/2 − ADv/2)

Av

.

(48)

When γ v > 0, γ v is responsible for identical decreases, by2Bvγ v

Av, in both ∆Beff

2,1 and ∆Beff1,0. In contrast, ADv alters

∆Beff2,1 and ∆Beff

1,0 by opposite amounts.The φ-type 3∆ state contributes to the effective γ v

constant via the HSO ⊗ Hrot (L-uncoupling) path,⟨3Π0, v| − B(R)J+L−| 3∆1, v′⟩

×⟨

3∆1, v′∣∣∣∣A(R)

2L+S−

∣∣∣∣ 3Π1, v

⟩= −Bv,v′

[2J (J + 1)]1/2

2

⟨3Π |L−| 3∆⟩

× ⟨3∆|Av,v′L+| 3Π⟩

(49)

thus

γ v = −∑v′

Bv,v′⟨3Π |L−|3∆⟩ ⟨

3∆|Av,v′L+|3Π ⟩E(0)(3Π, v) − E(0)(3∆, v′)

. (50)

The γ v parameter appears in other locations in the 3Π

Heff in addition to the 3Π0, 3Π1 location. The same 3∆

contribution to γ v is obtained for each Ω,Ω ′ location inthe Heff. Only φ-type triplet states can make contributionsto γ v in the 3Π state because of the presence of the ∆S = 0Hrot term in the HSO ⊗ Hrot coupling path.

3.2.3 v′ = v Vibrational Levels of the α–type 3Π

State Generate Dv

The λv and γ v parameters embody contributions to theα-type 3Π state from φ-type remote perturbers. Any oper-ator that includes an R-dependent factor, either explicitly(B(R) in Hrot) or implicitly (HSO), generates “centrifugaldistortion” molecular constants (D, AD , λD, . . .) via vibra-tionally off-diagonal matrix elements within the α-type 3Π

state. The Dv parameter appears in every location in the3Π Heff (Table 4), for example,⟨3Π2, v = 0|Heff|3Π2, v = 0

⟩= −Dv=0[J (J + 1) − 2][J (J + 1)] (51)

The first step in obtaining equation (51) is to convert theR-dependence of B(R) into a form that is convenientfor evaluating vibrational matrix elements in a harmonicoscillator basis set,

X = R − Re (52)

where X is the displacement from equilibrium. For B(R)

and ωe in cm−1 units, expand B(R) in a power series

B(R) =[

4πcµ

]R−2 =

[

4πcµ

](X + Re)

−2

=[

4πcµ

]R−2

e

[1 − 2

(X

Re

)+ 3

(X

Re

)2

− . . .

]

= Be

[1 − 2

(X

Re

)+ 3

(X

Re

)2

− . . .

](53)

Evaluating the off-diagonal matrix element of X

〈v|X|v − 1〉 =[

4πcµω

]1/2

v1/2 (54)


the Hrot ⊗ Hrot Ω = 2, v → Ω = 2, v′ → Ω = 2, v cou-pling path yields a correction term⟨3Π2, v = 0|Heff|3Π2, v = 0

⟩=∑v′

⟨3Π2, v = 0|B(R)[J (J + 1) − 2]|3Π2, v

′⟩2E(0)(3Π, v = 0) − E(0)(3Π, v′)

(55)

and the Ω = 2, v → Ω = 1, v′ → Ω = 2, v coupling pathyields a similar coupling term,

∑v′

⟨3Π2, v = 0|B(R)[2J (J + 1) − 4]1/2|3Π1, v

′⟩2E(0)(3Π, v = 0) − E(0)(3Π, v′)

but with a [2J (J + 1) − 4]1/2 factor. When the termsfrom the two coupling paths are combined, the overallJ -dependence is

[J (J + 1) − 2]2 + [2J (J + 1) − 4]

= [J (J + 1) − 2][J (J + 1)].

Evaluating the vibrational matrix element of B(R) in theharmonic oscillator basis set and truncating the power seriesafter the X/Re term,⟨3Π2, v = 0|Heff|3Π2, v = 0

⟩= [J (J + 1) − 2][J (J + 1)]

B2e

ωe

⟨v = 0

∣∣∣∣−2X

Re

∣∣∣∣ v′ =1

⟩2

= −[J (J + 1) − 2][J (J + 1)]4B3

e

ω2e

, (56)

thus

D0 = 4B3e

ω2e

, (57)

which is the extremely useful Kratzer equation.

3.3 Multistate Interactions and Deperturbation

As can be seen from the examples above, the Heff for adiatomic molecule is much more than for a parametricallyparsimonious fit model. It can be an exquisitely sensitiveand specific detector/identifier of not quite remote per-turbers. Distortions of the splittings and effective rotationalconstants of the six components of a 3Π state often revealthe 2S+1Λ symmetry of a local perturber and whether thisperturber lies above or below the 3Π state. Violations ofthe Kratzer equation generally indicate the presence of a∆S = 0 perturber; a too large (small) Dv usually implies

that the perturber lies above (below) the α-type state ofinterest.

When the α-type state of interest is not isolated, itis straightforward to incorporate additional local-perturberstates into the Heff model. Each additional state increasesthe dimension of the Heff matrix. Masochism aside, there isno intrinsic limit to the (finite) number of basis states thatcan be incorporated into the Heff. However, it is importantto realize that there may be a physically sensible lower limitthat is larger than what would be required merely to achievea good Heff fit. A converged and physically sensible Heff

fit to a large group of interacting states yields deperturbedmolecular constants that are identical to what would havebeen obtained individually for each of the interacting statesif the 2S+1Λ states had been isolated. Suppose v = 0 and2 of a state of interest are truly isolated but v = 1 isaccidentally near-degenerate with a perturber. The fittedmolecular constants for v = 0 and 2 will be very similar toeach other, but the fitted constants for v = 1, erroneouslytreated as isolated, will be discrepant. However, whenthe local perturber(s) of v = 1 are incorporated into theHeff, deperturbed molecular constants similar to those forv = 0 and 2 will be obtained, confirming that the molecularconstants have some physical basis.

For example, the CO A1Π and a3Π states interactwith the a′3Σ+, e3Σ−, I1Σ−, d3∆, and D1∆ states. Bytreating all of these potentially interacting states (often twovibrational levels of several perturbers) using an expandedHeff model, fully deperturbed molecular constants for allseven electronic states were obtained (Field et al. 1972).The deperturbation model was used to compute the relativeintensities of all nominally forbidden CO (a′3Σ+, e3Σ−,I1Σ−, d3∆, D1∆) − X1Σ+ transitions, based on intensityborrowing from the A1Π−X1Σ+ transition, includingdestructive and constructive interference effects arisingfrom minuscule admixtures of two vibrational levels ofthe A1Π state (Rostas et al. 2000). A global modelthat accounts for all transition energies and intensitiesin a frequency-domain spectrum is certified as valid fordescribing all conceivable short-time dynamics.

3.4 Examples of Multistate Interactionsc

Four permanent gas diatomic molecules, H2, N2, O2, andCO were the subjects of disproportionate spectroscopicattention in the early days of electronic spectroscopy.Each molecule presented a problem of unexpected spectro-scopic complexity. The spectroscopic observations seemedincompatible with naıve molecular orbital-based expecta-tions. The resolutions of these paradoxes provided valuableinsights into the evil that lurks in the hearts of even thesmallest molecules.


The electronically excited 1Σ+g states of H2 are described

by a family of double-minimum potential energy curves.Each double-minimum curve is the adiabatic consequenceof a curve crossing between the diabatic H−(1s2)H+

ion-pair state and the 1Σ+g Rydberg series (nsσg)(σ g1s)

that converges to the H+2 X2Σ+

g electronic ground state(Wolniewicz and Dressler 1977, Lefebvre-Brion and Field2004, pages 161–168 and 384–387). The energy differ-ences between the lowest members of the ns Rydbergseries are sufficiently large (∆E(3s–2s) ∼ 15 000 cm−1,∆E(4s–3s) ∼ 5300 cm−1) that the ion-pair∼Rydberg inter-actions take the form of two-state interactions. The diabaticinterelectronic interaction energies are so large (much largerthan the vibrational spacing in either the Rydberg or ion-pair state) that the adiabatic (avoided crossing) basis setgives zero-order energy levels vastly closer to the observedpatterns in the spectrum than the diabatic (crossing curves)basis set. Nevertheless, there are many subtle additionaleffects (such as level shifts and intensity anomalies) inthe spectrum that imply local perturbations between lev-els in the inner and outer wells of each double-minimumstate.

Two lessons emerge. The more important one is thatnot all potential energy curves of diatomic molecules looklike the Morse potential. Less widely appreciated andvastly more devastating is that when a valence or ion-pairstate crosses Rydberg states at moderately high principalquantum number, the interactions can no longer be treatedas a series of isolated two-state interactions. One needsa multistate model based on (i) a scaling rule for theenergy spacings between members of a Rydberg series(2/n3) where is the Rydberg constant and n is theeffective principal quantum number, (ii) a scaling rule forthe interelectronic state matrix elements (∝ n−3/2), and(iii) the n-invariance of the Rydberg series potential energycurves (the potential curve of the ion core).

The 1Σ+u and 1Πu states of N2 are the lowest lying

states that have allowed transitions from the X1Σ+g elec-

tronic ground state. These 1Σ+u − X1Σ+

g and 1Πu − X1Σ+g

transitions lie far into the vacuum ultraviolet region ofthe spectrum, where the resolution of grating spectro-graphs was barely adequate to observe isotope shiftsand insufficient to resolve rotational structure. The spec-trum appeared as a random scatter of vibrational bandsthat could not easily be arranged into electronic states.The 1Πu states that are expected to appear in the spec-trum are three doubly excited valence states that belongto the (σ g2p)1(πu2p)3(πg2p)2 configuration (1Πu ⊗ 1Σ+

g ,3Πu ⊗ 3Σ−

g , and 1Πu ⊗ 1∆g) and the lowest membersof three Rydberg series npπu(σ g2p)−1 (converging to N+

2X2Σ+

g ) and nsσg(πu2p)−1 and ndσg(πu2p)−1 (convergingto the A2Πu state of N+

2 ). The three 1Πu valence states

have small ωe and Be, have similarly shaped potentialenergy curves, and have no oscillator strength for transi-tions from the X1Σ+

g ground state (because they differ bytwo orbital excitations from the ground state). The Ryd-berg states have large ωe and Be and have large oscillatorstrength for transitions from the ground state (npπu ←(σ g2p), nsσg ← (πu2p), or ndσg ← (πu2p) orbital exci-tations). The result is a relatively dense manifold of“dark” vibrational levels of the three valence 1Πu statesand strong local interactions with a “bright” Rydberg-state vibrational level. The interactions are not isolated2 × 2 valence–Rydberg interactions and must be treatedby a multistate Heff (Dressler 1969, Stahel et al. 1983).The physical content of this Heff includes three interelec-tronic interactions among the three 1Πu valence states[(1Πu(

1Σ+g ) ∼ 1Πu(

3Σ−g ), 1Πu(

1Σ+g ) ∼ 1Πu(

1∆g), and1Πu(

3Σ−g ) ∼ 1Πu(

1∆g)], three valence–Rydberg interac-tions with each of the three classes of Rydberg states(a total of nine valence–Rydberg interaction parameters),and many vibrational overlap integrals computed fromthe potential energy curves for each of the interact-ing electronic states. This spectrum of N2 was a clas-sic problem of spectroscopy solved by a heroic imple-mentation of building-block models. If such a multistatevalence–Rydberg interaction were to occur at sufficientlyhigh energy that the n + 1 ← n Rydberg electronic statespacing is comparable to the valence state vibrational spac-ings, it would be necessary to find a model that exploitsthe n-scaling of Rydberg energy spacings and matrixelements.

The CO b3Σ+ − a3Π “third positive” transition is promi-nent in discharges. It is complicated in its own right becauseit is a Hund’s case (b)–case (a) triplet–triplet transition,but has special complexity due to multiple perturbationsbetween the Rydberg (3sσ)(σ2p)−1b3Σ+ state and manyhigh vibrational levels of the valence (π 2p)3(π 2p) a′3Σ+

state (Rytel 1991, 1992). There is no corresponding pertur-bation in N2 because the Rydberg 3Σ+ state in N2 hasgerade symmetry and the valence 3Σ+ state has ungeradesymmetry. This is another example of multiple perturba-tions between one bright Rydberg state and many dark(because of poor Franck–Condon factors to the a3Π state)valence states. It is much simpler than the multistate inter-actions among 1Πu states in N2 because only two electronicstates are involved. However, it is experimentally diffi-cult because of the 27 rotational branches of a case (b)3Σ+ –case (a) 3Π electronic transition.

The Schumann–Runge B3Σ−u –X3Σ−

g bands of O2

(Lewis et al. 1996) shield the Earth’s surface from lethalλ < 200 nm solar radiation. It is the lowest energy allowedabsorption transition in O2. The B3Σ−

u state is perturbedby many 1,3,5Λu [(πu 2p)3 (πg 2p)3 and (πu 2p)4 (πg

2p)(σ u 2p)] valence states. Most of the interactions are


with the dissociation continua of lower-lying states. Thesestates leave their signatures on the B3Σ−

u state in a varietyof subtle features: erratic vibrational level shifts, variationsin the λ and γ constants that determine spin fine struc-ture, and v-, N -, and J -dependences of the homogeneouslinewidths. The complexity of the Heff for the O2 B3Σ−

u

state arises from the large number of perturbing electronicstates, most of which are difficult to characterize spectro-scopically because they are repulsive in the energy regionswhere they interact with the B3Σ−

u state.

3.5 The Approach to Disaster

The key question is not “how many interacting states istoo many for an Heff model?” but rather “in what situationmust the Heff approach fail irrecoverably?” and “what dowe do then?” It becomes impossible to partition state spaceinto a finite group of strongly interacting α-type states andan infinite group of remote-perturber φ-type states when theaverage spacing between interacting states becomes compa-rable to the largest interaction matrix elements among thesestates. This will always occur at high electronic excita-tion energy, where the density of electronic states increaseswithout limit. It does not occur for interactions betweenvibrational levels of two bound electronic states, ψ1 andψ2, near the dissociation limit of the ψ2 electronic state,because the number of bound vibrational states of the ψ2electronic state with non-negligible mixing angles with the

ψ1, v1 state,H1,v1;2,v2

E(0)(1,v1)−E(0)(2,v2), becomes large but remains

finite.There are several strategies for dealing with strong

interactions among many bound states. Any discussion ofstrategy must take into account that the experimental obser-vations will always be incomplete, owing to finite resolutionand dynamic range. Four classes of strategies are dis-cussed here: (i) continue adding states to the α-type groupof the Heff until all members of the strongly interactinggroup of states that are “of interest” are satisfactorily fitted(see Section 3.5.1); (ii) adopt a buffered-Heff strategy (seeSection 3.5.2); (iii) express the Heff in a different type ofelectronic–vibration basis set (e.g., diabatic vs. adiabatic)(see Section 3.5.3); (iv) find an alternative to an Heff model(see Section 3.5.4).

3.5.1 Increasing the Dimension of Heff

Suppose we are interested in the vibrational levels of twoelectronic states in the energy region centered at the energyof the avoided crossing between two adiabatic (ab initio)potential curves (see Section 3.5.3 below for a discus-sion of adiabatic vs. diabatic representations). An avoidedcrossing (Ex , Rx) is located at the internuclear distance

of closest vertical approach (Ex = 12 (V2(Rx) + V1(Rx)))

between two potential curves, 0 = ddR

(V2(R) − V1(R)),and the electronic matrix element is one-half this distanceof closest vertical approach, H electronic

1,2 (Rx) = 12 |V2(Rx) −

V1(Rx)|. This is a general result for the eigenvalues of thetwo-state problem

H =(

∆/2 V

V −∆/2

)(58)

E± = ±[(

∆

2

)2

+ V 2

]1/2

(59)

E+ − E− = 2V when ∆ = 0. (60)

In the diabatic (crossing curves) basis, the interactionmatrix elements between near-degenerate vibrational basisstates are the product of an electronic factor (evalu-ated at Rx) times a vibrational overlap factor, H1v1,2v2 =H electronic

1,2 (Rx) 〈v1|v2〉. The vibrational overlap factors arelargest near Ex and decrease rapidly as |E − Ex | increases(Lefebvre-Brion and Field 2004, pages 278–285).

If the electronic matrix element is large relative to thevibrational spacings within both electronic states, then itwill be necessary to include many vibrational levels ofboth electronic states in the Heff. How many? There is nosimple answer to this question. The answer depends onthe quality and extent of the spectroscopic data and thegoals of the analysis. The most modest goal is to determineexperimental values for Ex , Rx , and H electronic

1,2 (Rx). Themost ambitious goal is to fit all of the observed energylevels and relative transition intensities to measurementaccuracy. One simply increases the dimension of thevibrational basis until the fitted values of the quantitiesof interest become insensitive to further expansion of thebasis set.

It is essential that an Heff be constructed arounda model in which the number of adjustable parame-ters is minimal. Since many of the vibrational levelsexplicitly included in the Heff are not experimentallyobserved, each vibrational level must not be treated asif its energy and off-diagonal interaction matrix elementsare independent fit parameters. All of the vibrationallevels of each electronic state are derived from theirrespective V (R). The shapes of V1(R) and V2(R) enterinto the fit model in the form of a small number ofadjustable parameters. All of the perturbation matrix ele-ments are computed as the product of one fit parame-ter

[H electronic

1,2 (Rx)]

and a vibrational overlap factor com-puted from V1(R) and V2(R). The parametrically parsimo-nious form of this many-vibrational-state Heff model closelyresembles the vibrational polyad Heff models discussed inSection 4.


3.5.2 Buffered Heff

Suppose we are interested in the interactions between twoRydberg series converging to the same electronic–vibrationlevel of the ion core. Almost everything (e.g., energylevel differences, interaction matrix elements) in Rydbergstates is well described by scaling rules in the effec-tive principal quantum number, n (n−3 for energy dif-ferences or n

−3/21 n

−3/22 for matrix elements) (Mulliken

1964). When the interaction matrix element between near-degenerate members of two Rydberg series is comparable tothe energy below the ionization limit (IP − En = /n2),the number of coupled electronic states is infinite. Oneplausible solution is to create a five-block buffered-Heff

where blocks 1, 2, 4, and 5 correspond respectively tostates (n

1 − 2, n2 − 2), (n

1 − 1, n2 − 1), (n

1 + 1, n2 + 1),

and (n1 + 2, n

2 + 2) and block 3 corresponds to the statesof specific interest, (n

1, n2). The parameters in block 3 are

adjusted to give agreement with spectral data exclusivelyfrom the energy region of block 3, whereas all of the param-eters within and between blocks 1, 2, 4, and 5 and betweenblock 3 and blocks 1, 2, 4, and 5 are slaved to the fittedvalues within block 3. No spectral data directly samplingblocks 1, 2, 4, and 5 are included in the fit.

This buffered-Heff procedure is an attempt to make thebest of a bad situation. It might give useful results atmoderate values of n, but it must fail at high n. The onlyrecourse is to a completely different flavor of fit model, Themultichannel quantum defect theory (MQDT) (see Section3.5.4 below).

3.5.3 Express Heff in a Different Basis Set

It is often possible to express the Heff in any one ofseveral complete basis sets. Among these basis sets, onetypically finds pairs that correspond to opposite-limitingcases [e.g., diabatic vs. adiabatic (Lefebvre-Brion and Field2004, pages 90–94) or normal mode vs. local mode (Jacob-son et al. 1999b), see Section 4.8.2].

Any given problem can be approached in either of thetwo basis sets (e.g., diabatic vs. adiabatic) that correspondto opposite-limiting cases, provided that both basis sets arecomplete. The quality of the fit to the observed energy lev-els and the dynamics computed from the Heff expressed ineither of the two complete basis sets will be identical. How-ever, the initial difficulty in relating the spectroscopic obser-vations to the fit model can be prohibitive if the “wrong”model is used. For diatomic molecules, one must choosebetween a diabatic and an adiabatic representation for theelectronic–vibration states (and also make a choice amongHund’s coupling cases for the rotation–electronic states).

Spectroscopists and quantum chemists generally favor op-posite electronic–vibration limiting cases. Spectroscopists

tend to think in terms of potential energy curves that resem-ble a Morse potential, electronic wavefunctions that donot change rapidly with internuclear distance, and same-symmetry potential curves that cross; this is the diabaticlimit. Quantum chemists believe in the Born–Oppenheimerapproximation, and they compute potential energy curvesin the clamped-nuclei or adiabatic limit. Same-symmetryadiabatic potential curves do not cross, avoided crossingsresult in potential curves that do not resemble a Morse oscil-lator, and the electronic wavefunctions profoundly changecharacter as they traverse the region of an avoided crossing.

In the clamped-nuclei limit, the nuclear kinetic energyterm in the exact Hamiltonian is neglected to construct anelectronic–vibration basis set. This neglected term, whichinvolves first and second derivatives with respect to inter-nuclear distance, gives rise to off-diagonal (and diago-nal) matrix elements between electronic wavefunctions:∫

ψ1(R) d

dRψ2(R)dR. These matrix elements are gener-

ally small and slowly varying functions of R, except inthe region of an avoided crossing. The worst case in theadiabatic basis set is an extremely weakly avoided curvecrossing, which ensures that the electronic wavefunctionschange character within a very narrow range of internucleardistance. This is bad news because the off-diagonal matrixelements become large relative to the vibrational energylevel spacings within the two adiabatic electronic states. Itis extremely inconvenient to use the adiabatic basis set forstates in the Ex , Rx region near a weakly avoided curvecrossing because the dimension of the Heff is necessarilyenormous.

In the diabatic limit, the electronic Hamiltonian is notdiagonalized. The (unspecifiable) terms in the exact Hamil-tonian, which are implicitly neglected to permit potentialcurves to cross, appear in the Heff as off-diagonal elec-tronic matrix elements, H electronic

1,2 (Rx). A weakly avoidedadiabatic curve crossing, disastrous for an adiabatic-limitHeff, is meat-and-potatoes for a diabatic limit Heff calcu-lation because the off-diagonal electronic matrix element(equal to one-half the minimum vertical energy differencebetween adiabatic potential curves) is small relative to thevibrational spacings within both diabatic electronic states.In a diabatic limit calculation, only a few basis states mustbe treated as quasi-degenerate α-type states; the rest maybe dealt with as remote-perturber φ-type states. Conversely,a strongly avoided curve crossing is ideally suited for anadiabatic-limit Heff and disastrous for a diabatic limit Heff.

It is often necessary to make a choice between twoopposite-limit basis sets. However, sometimes the choiceis between two similarly appearing basis sets, one of whichis more convenient to use but is built on a fundamentallyflawed foundation. We present a very important example ofa model built on quicksand in the Section 4 discussion of the


polyad Heff model of anharmonic interactions among vibra-tional levels of polyatomic molecules. The anharmonicallycoupled harmonic oscillator Heff is exceptionally conve-nient owing to simple matrix element selection and scalingrules. But the harmonic oscillator basis is overly simplebecause it neglects the intrinsic diagonal anharmonicities ofthe vibrational modes, and, unlike the diabatic vs. adiabaticbasis sets for diatomic molecules, does not provide anexplicit recipe to compensate for this neglect. A coupledMorse oscillator basis set minimizes this deficiency, butat the cost of the loss of simplicity in both selection andscaling rules.

3.5.4 Multichannel Quantum Defect Theory(MQDT)

In MQDT (Ross 1991), the infinite number of electronicstates that belong to one Rydberg series

Enλv = IP − / n2λv (61)

(where IP is the ionization energy, n is the effective (non-integer) principal quantum number and is the Rydbergconstant) are treated as one “channel” and the totality ofinteractions between the two infinities of states associatedwith two Rydberg series is reduced to a two-channel prob-lem. This reduction of many implicitly related basis statesto one natural unit, a channel, is an elegant embodimentand extension of the idea of building a parametrically par-simonious model. Sometimes the traditional approach toa traditional problem, in this case, representing the statesobserved in a spectrum by an Heff model, becomes cum-bersome and physically opaque. Quantum defect theoryis a clamped-nuclei scattering-based approach to struc-ture and dynamics. An electron collides with the ion corein a specified initial rovibronic state. As the collisionprogresses, various electron–ion interaction mechanismsresult in the transfer of energy and angular momentumbetween electron and ion. The beauty of MQDT is that“electronic structure” is not viewed as a telephone direc-tory of potential energy curves and coupling matrix ele-ments. MQDT treats electronic structure as a coherentunit, and this permits the electron–ion inelastic scatter-ing to be viewed in terms of mechanisms rather than merenumbers.

A discussion of MQDT is beyond the scope of this arti-cle. The key point here is that a completely different rep-resentation of a familiar problem often yields both insightsand computational convenience. The effort of learningto speak and think in a new language is often richlyrewarded.

4 VIBRATIONS OF POLYATOMICMOLECULES

4.1 Vibrations: a Step into Deep Water

The derivations and formats of Heff models are quite differ-ent for the electronic fine structure of diatomic moleculesthan for the vibrational structure of polyatomic molecules.These differences reflect the substantially greater difficultyposed by vibrational anharmonic coupling.

The qualitative form of the Heff for a diatomic moleculeis an essentially solved problem because each electronicstate can only interact with a small number of symmetryclasses of remote-perturber states. Even though there arean infinite number of remote (φ-type) electronic–vibrationperturbers in each symmetry class, the correction termsto Heff are implicit summations over all φ-type statesin a given symmetry class. Although the approximatemagnitudes of most of the correction terms in the Heff

for a diatomic molecule may be crudely estimated, eithersemiempirically or ab initio, these correction terms aretreated by most spectroscopists merely as fit parameters.However, their magnitudes, signs, and vibrational leveldependences are often of considerable diagnostic value.

The situation with regard to the Heff for polyatomicmolecule vibrations could not be more different. Thepolyatomic molecule vibrational Heff is not a completelysolved, finite problem. The usual approach, anharmonicallyinteracting harmonic oscillator product basis states,

|v1v2 . . . v3N−6〉 =3N−6∏i=1

|vi〉 (62)

resulting in a polyad Heff, with a priori known matrix ele-ment selection and scaling rules, is very convenient, flex-ible, and powerful, but it is built on an inherently falsefoundation. The individual oscillators are not harmonic andon-diagonal corrections in the Heff for this anharmonicityare improperly ignored when computing off-diagonal inter-action matrix elements.

The anharmonically coupled harmonic oscillator app-roach works well at modest vibrational excitation, but itmust eventually fail at chemically relevant (> 10 000 cm−1)levels of vibrational excitation, as would be encounteredfor high-barrier unimolecular isomerization. Experimentaldata have only recently become available that will generateinsights into how the polyad Heff model begins to fail, if thefailure is gradual or catastrophic, and whether the inherentfallacy in the model can be fixed without sacrificing thesimplicity and convenience of anharmonically coupled har-monic oscillators (Ross 1991, Ishikawa et al. 1999). Is itpossible to construct a physically sound, yet parametrically


parsimonious, Heff model for molecular vibrations withoutkilling the harmonic oscillator golden goose?

A nonlinear N -atom molecule has 3N − 6 vibrationaldegrees of freedom. In the limit of infinitesimal dis-placements from equilibrium geometry (at equilibrium,all displacements are zero), it is possible to solve the3N coupled classical mechanical equations of motion todetermine the normal modes of vibration. Each normalmode i has a frequency ωi , and a set of three Carte-sian displacement amplitudes (aiJ eJ ) and phases (φiJ ) foreach atom. For the ith normal mode, all 3N displacementamplitudes,

Qi(t) =3N∑J=1

(qiJ (t) − qeJ )eJ =

3N∑J=1

δqiJ (t)eJ

=3N∑J=1

aiJ sin(2πcωit − φiJ )eJ (63)

oscillate sinusoidally at the same frequency ωi (in cm−1

units) about the equilibrium geometry Q0 (Q ≡ 0 at equi-librium),

Q0 =3N∑J=1

qeJ eJ . (64)

The potential energy surface may be expressed as apower series in the 3N − 6 normal-mode displacementcoordinates,

V (Q) = 1

2

3N−6∑i=1

kiiQ2i + 1

6

3N−6∑i,j,k

kijkQiQjQk

+ 1

24

3N−6∑i,j,k,l

kijklQiQjQkQl (65)

where the three- and four-index summations are unrestrictedand the kii , kijk and kijkl are partial derivatives of thepotential surface evaluated at the equilibrium geometry,Q = 0, for example,

kijk = ∂3V (Q)∂Qi∂Qj∂Qk

∣∣∣∣0

. (66)

All the nonzero terms in the power series expansion ofV (Q) are totally symmetric (the product of the irreduciblerepresentations of the displacement coordinates must con-tain the totally symmetric representation). For example,for formaldehyde, a four-atom C2v molecule, there are 6nonzero kii , 22 nonzero distinct kijk (it may be noted thatkiji = kiij = kjii are not distinct), and 45 nonzero distinct

kijkl .

One limitation of this traditional representation ofV (Q) is that it is a description localized infinitesimallyclose to the equilibrium geometry, Q = 0. It seems unrea-sonable to expect that derivatives of V (Q) evaluatedat Q = 0 will have much relevance to large ampli-tude, highly excited vibrational states. Another limitationis that an enormous number of derivatives might naıvelybe expected to contribute to the vibrational Heff. Thenumber of possibly relevant coupling terms and mech-anisms is unbounded, in sharp contrast to the situationfor diatomic molecules [We use V Q as shorthand forV (Q)].

4.2 Why is the Power Series Expansion ofV Q Useful?

This power series representation of V Q is widely usedfor three main reasons: (i) matrix elements of positiveinteger powers, ni , of displacement (Qi )

ni and momen-tum (Pi )

ni operators in the harmonic oscillator product|v1〉 |v2〉 . . . |v3N−6〉 basis set have simple ∆vi selectionrules and vi-dependent scaling rules; (ii) the magnitudesof matrix elements of

∏3N−6i=1 Qni

i operators are rapidlydecreasing functions of

∑i ni ; and (iii) resonance deter-

mines the basis states that must be treated as α-typequasi-degenerate as well as the operators that result in thestrongest interactions among this subset of quasi-degeneratebasis states.

4.2.1 Matrix Elements of (Qi )n and (Pi)

n in theHarmonic Oscillator Basis Set

The simplicity and convenience of the harmonic oscillatorbasis set is best appreciated when the (mass-weighted) Qi

and Pi operators are rewritten in dimensionless form, Qi

and Pi , which in turn are expressed as a sum of two simpleoperators, a†

i and ai , known as creation/annihilation orraising/lowering operators,

H(0)i = hcωi

2

[Q

2i + P

2i

]E

(0)i = hcωi(vi + 1/2) (67)

Qi =[

2πcωi

]1/2

Qi (units of length · mass1/2) (68)

Pi = [2πcωi]1/2Pi (units of momentum · mass−1/2)

(69)

Qi = 2−1/2[a†

i + ai

](70)

Pi = 2−1/2i[a†

i − ai

](71)


where

a†i |v1v2 . . . vi . . . v3N−6〉= (vi + 1)1/2 |v1v2 . . . vi + 1 . . . v3N−6〉 (72)

ai |v1v2 . . . vi . . . v3N−6〉= (vi)

1/2 |v1v2 . . . vi − 1 . . . v3N−6〉 (73)

and Ni = a†i ai is known as the number operator,

a†i ai |v1v2 . . . vi . . . v3N−6〉 = Ni |v1v2 . . . vi . . . v3N−6〉

= vi |v1v2 . . . vi . . . v3N−6〉 .

(74)

Qi and Pi are mass-weighted and it is not possible, asit is for a diatomic molecule, to rigorously write ωi =

12πc

(ki / µi)1/2, where µi would be a simple, coordinate-

independent, reduced mass. However, it is usually suffi-cient, for estimating key quantities, to express ωi in termsof diatomic molecule-like force constant and reduced massfactors.

Note that both Q i and Pi (and Qi and Pi) are operatorsthat have selection rules ∆vi = +1 and ∆vi = −1, whereasa†

i (ai ) has the usefully simpler selection rule ∆vi = +1(∆vi = −1),

〈v1v2 . . . vi + 1 . . . v3N−6| a†i |v1v2 . . . vi . . . v3N−6〉

= (vi + 1)1/2 (75)

〈v1v2 . . . vi − 1 . . . v3N−6| ai |v1v2 . . . vi . . . v3N−6〉= (vi)

1/2 (76)

The value of the nonzero matrix element of a†i or ai is

always the square root of the larger of the initial or finalstate vibrational quantum numbers. The advantage of thea†

i , ai notation is illustrated by matrix elements of Q2i

Q2i = (1/2)

[a†

i a†i + (2a†

i ai + 1) + aiai

](77)⟨

v1 + 2|a†i a†

i |vi

⟩= [(vi + 2)(vi + 1)]1/2⟨

vi |(2a†i ai + 1)|vi

⟩=2vi + 1 (78)

〈vi −2|aiai |vi〉= [vi(vi − 1)]1/2 (79)

where the three different (∆vi = 2, ∆vi = 0, ∆vi = −2)

nonzero matrix elements of Q2i are replaced by simple terms

involving a†i and ai , for which the selection rules and matrix

elements are easily remembered.The key advantages of the anharmonically coupled

harmonic oscillator product basis set are that all coupling

terms may be reduced to dimensionless operators, thenonzero matrix elements are sorted according to the overall∆v1,∆v2, . . . , ∆v3N−6 selection rule, and the matrixelements can be evaluated by inspection.

4.2.2 The Magnitudes of Matrix Elements of∏3N−6i=1 Qni

i Operators are Rapidly DecreasingFunctions of

∑i ni

The mass-weighted Qni

i is replaced by[

2πcωi

]ni/2Q

ni

i en

route to an expression written in terms of a†i and ai .

The mass-weighting of Qi can be removed, for order-of-magnitude calculations, by inserting a reduced mass factor,µi , into the scale factor. For ni = 1 and typical values ofµi ∼ 6 amu (e.g. a C2 molecule) and ωi ∼ 1000 cm−1, the[

2πcµiωi

]ni/2scale factor has the value of 7.5 × 10−2 A−1.

As a point of comparison, the distance between turningpoints for the v = 0 state is ∆Q(v = 0) = 2

1/2/(kµ)1/4 =0.14 A for this oscillator. For an operator of the form∏3N−6

i=1 Qni

i , the scale factor is of the order of (0.075)∑

i ni

A−∑i ni (i.e., 0.075 A when

∑i ni = 1).

Although we are mostly interested in intermode anhar-monic coupling terms such as kiij Q2

i Qj , the typical magni-tudes of the matrix elements of these intermode terms maybe illustrated by the magnitudes of intramode anharmoniccoupling terms of a Morse oscillator. For a diatomic Morseoscillator

V M(R)/hc = De [1 − e−β(R−Re)]2

(80)

β =[

4πcµωexe

]1/2

=[πcµω2

e

De

]1/2

(81)

(De = ω2e/4ωexe, ωe, and ωexe = [∆G(1/2) − ∆G(3/2)]/2

are, respectively, the dissociation energy, harmonic fre-quency, and the first anharmonic term, all in cm−1 units),the coupling terms associated with the various derivatives ofV (R), with respect to the non-mass-weighted displacementcoordinate, evaluated at Re(Q = 0), are

V M(Q) = 1

2

d2V M

dQ2

∣∣∣∣∣0

Q2 + 1

6

d3V M

dQ3

∣∣∣∣∣0

Q3

+ 1

24

d4V M

dQ4

∣∣∣∣∣0

Q4 + 1

120

d5V M

dQ5

∣∣∣∣∣0

Q5 (82)

V M(Q)/hc = De(βQ)2[1 − βQ + 7

12(βQ)2 − 1

4(βQ)3

](83)

βQ =[

2ωexe

ωe

]Q (84)


De(βQ)2 = ωe

2Q

2(85)

V M(Q)/hc = ωe

2Q

2

[1 −

(2ωexe

ωe

)1/2

Q

+ 7

12

(2ωexe

ωe

)Q

2 − 1

4

(2ωexe

ωe

)3/2

Q3

](86)

Since a typical value of 2ωexe

ωeis ∼ 1/50, the coeffi-

cient of each successively higher power of the dimension-less Q is smaller than the previous term by a factor of∼ (1/2)(1/50)1/2 . In general, the size and therefore theimportance of coupling terms decreases by 1 order of mag-nitude for each power of Q, roughly 0.1

∑i ni . Since

∑i ni

sets a limit on the sum of the changes in all of the vibra-tional quantum numbers in the harmonic oscillator productbasis set |v1v2 . . . v3N−6〉,

∆V ≡∑

i

|∆vi,max| =3N−6∑i=1

ni (87)

we have order-sorting according to ∆V .This is a very important result. Anharmonic coupling

terms for which∑

i ni = 3 are known as cubic,∑

i ni = 4as quartic,

∑i ni = 5 as quintic, etc. The off-diagonal

anharmonic matrix elements decrease roughly by a factorof 10 for each step in the order of the interaction,

∑i ni .

However, the spectroscopic and dynamical importanceof a coupling mechanism is embodied in the mixingangle, rather than the magnitude of the off-diagonal matrixelement by itself. Resonance, which occurs when an energydenominator is small relative to the matrix element in thenumerator, is more important than a large off-diagonalmatrix element.

4.2.3 Energy Resonance Selects the Subset ofLowest-order Coupling Terms that Act withinthe Quasi-degenerate Group of α-type States

Even for a molcule as small as formaldehyde, there arean enormous number of cubic and quartic coupling terms.However, only a few of these terms demand specialconsideration when the basis states are partitioned intoquasi-degenerate α-type states and remote φ-type states,according to the magnitude of the Hij coupling term relativeto the E

(0)i − E

(0)j energy difference. For example, when

ωi ≈ 2ωj (Fermi resonance) or ωi ≈ ωj + ωk , a quite smallcubic coupling term, ki,jj or ki,jk, will have disproportion-ate spectroscopic/dynamical importance relative to largerbut nonresonant coupling terms. (A comma in the force con-stant term is used to specify the resonant energy exchange

pattern: for example, k1,22 implies that one quantum frommode 1 is exchanged for two from mode 2.) Quartic reso-nances can have coupling terms of the forms k1,222, k11,22

(Darling–Dennison), k1,223, k1,234, etc.If an approximate energy resonance exists among the

normal-mode frequencies, ωi, the coupling term selectedby this energy resonance will make its presence feltat the lowest energy opportunity consistent with thecoupling mechanism. For example, for a 2 : 2 k11,22

Darling–Dennison resonance, the (v1, v2) states (0,0), (1,0),(0,1), and (1,1) will not exhibit any of the anomalies asso-ciated with a local perturbation. However, the (2,0) and(0,2) states will exhibit a classic two-state interaction. Asthe polyad quantum number, N = v1 + v2, increases, morepossibilities for interactions exist and the interaction matrixelements will be larger: for N = 3 there are (3, 0) ∼ (1, 2)

and (0,3) ∼ (2,1) 2 × 2 interactions; for N = 4 there isa 3 × 3 (4, 0) ∼ (2, 2) ∼ (0, 4) and a 2 × 2 (3, 1) ∼ (1, 3)

interaction. These groups of resonantly coupled states areknown as polyads, and the Heff for each polyad is known asa polyad Heff. Although the detailed forms of the different-dimensional polyad Heff matrices for each value of thepolyad quantum number, N , are not transparently related,all of the molecular constants in the (N + 1)th polyad maybe derived from those in the N th polyad.

Once a coupling term is known to cause experimen-tally observable interactions between systematically near-degenerate states, the existence of a spectroscopically anddynamically important resonance is established. It is pos-sible to deal with this resonance by exploiting very simpleharmonic oscillator matrix element selection rules and scal-ing rules. These scaling rules establish both membershipand the quantum number dependence within the associatedpolyads. This is one of the main advantages of the anhar-monically coupled harmonic oscillators basis set.

For example, membership is determined merely bythe polyad quantum number, N = v1 + v2: the N+ 1(v1, v2, v3) = (v1, N − v1, v3) states [(N, 0, v3), (N − 1,

1, v3), . . . (0, N, v3)] are coupled in same-v3 groups. Whenall the coupled modes do not belong to a totally symmetricirreducible representation of the molecular symmetry group(Cohen-Tannoudji et al. 1977), then the polyad Heff factorsinto uncoupled blocks along the diagonal, each associatedwith a different symmetry species.

Matrix element scaling is based on the properties ofthe a†

i and ai operators. For a 2 : 2 k11,22 resonancemechanism, the intrapolyad coupling matrix elements havethe form⟨

v1 − 2, v2 + 2|K11,22a1a1a†2a†

2|v1, v2

⟩= K11,22 [v1(v1 − 1)(v2 + 2)(v2 + 1)]1/2 (88)


⟨v1 − 2, N − v1 + 2|K11,22a1a1a†

2a†2|v1, N − v1

⟩= K11,22 [v1(v1 − 1)(N − v1 + 2)(N − v1 + 1)]1/2

(89)

and the same value of the K11,22 reduced coupling constantis expected to appear in polyads associated with all valuesof the polyad quantum number. (Upper case Ks are usedhere with dimensionless displacement coordinates.)

This discussion provides a glimpse of the power, con-venience, and transparency of the anharmonically cou-pled harmonic oscillators basis set and the polyad Heff

model constructed exploiting harmonic oscillator matrixelement selection and scaling rules. We show how toextend the polyad Heff to include more than one anhar-monic resonance mechanism and how to relate the molec-ular constants, in terms of which the polyad Heff isdefined, to partial derivatives of the ab initio V (Q)potential energy function evaluated at the equilibriumgeometry (Q = 0). But first it is instructive to ask thequestion, “When does observation of one local anhar-monic perturbation between two vibrational levels implythe existence of a family of polyads?” In other words,“Is it possible for a vibrational perturbation to be acci-dental or is a perturbation always a sign of somethingglobal?”

4.3 Global versus Accidental Resonances

The following is two-mode example. Suppose that twovibrational modes, ν1 and ν2, have similar frequencies.This is a common situation for two normal modes thatare symmetric and antisymmetric combinations of thestretching motions of two chemically identical bonds (forexample, modes 1 and 3 of HCCH: ν1 = 3372 cm−1, ν3 =3295 cm−1, or modes 1 and 5 of H2CO: ν1 = 2766 cm−1,

ν5 = 2843 cm−1 (Bunker and Jensen 1998).

θv1−2,v2+2;v1,v2 = K11,22(v1 − 1/2)(N − v1 + 3/2)

2(ω2 − ω1) − v1(4x11 + 4x22 − 4x12) + N(4x22 − 2x12) + 4(x11 + x22). (95)

Since anharmonic interactions can only occur betweenvibrational basis states of the same symmetry, the near-resonant interactions (i.e., those with mixing angle not smallrelative to 1) must have the form of trading two quanta ofmode 1 for two quanta of mode 2: |v1, v2〉∼|v1−2, v2 + 2〉.When v1 + v2 = N , the |v1 = N, v2 = 0〉, |N − 2, 2〉, . . . ,|0, N〉 basis states will comprise an N/2 + 1 (or N/2 +1/2) dimensional group of α-type near-degenerate states.

The vibrational energy levels and eigenstates are obtainedby diagonalizing a (N/2 + 1) by (N/2 + 1) Heff. Thisgroup of states is called a polyad and the Heff is the polyadmatrix.

Once two members of the polyad tune into resonance, asquantified by the mixing angle, θ ,∣∣θv1,v2;v1−2,v2+2

∣∣=∣∣∣∣ Hv1,v2;v1−2,v2+2

E(0)(v1, v2) − E(0)(v1 − 2, v2 + 2)

∣∣∣∣ ≥ 1 (90)

they can tune out of resonance only in an exceptionalsituation, as is shown here. The zero-order vibrationalenergy levels are

E(0)(v1, v2) = ω1v1 + x11v21 + ω2v2 + x22v

22 + x12v1v2

(91)and the energy difference between resonant levels is

E(0)(v1 − 2, v2 + 2) − E(0)(v1, v2) = 2(ω2 − ω1)

− (v1 − 1)(4x11) + (v2 + 1)(4x22) − 2x12(v2 − v1).

(92)

Replacing v2 by N − v1,

E(0)(v1 − 2, v2 + 2) − E(0)(v1, v2) = 2(ω2 − ω1)

− (v1 − 1)(4x11) + (N − v1 + 1)(4x22) − 2x12(N − 2v1)

(93)and writing the explicit v1, N dependence of the off-diagonal matrix element

Hv1−2,v2+2;v1,v2 ≈ K11,22(v1 − 1/2)(v2 + 3/2)

= K11,22(v1 − 1/2)(N − v1 + 3/2) (94)

we obtain a general expression for the v1, N dependenceof the mixing angle

The matrix element is largest in the middle of the polyad,where v1 − 1 = v2 + 1 = N/2, and there the mixing anglebecomes

θN/2+1,N/2−1;N/2−1,N/2+1

= K11,22(N/2 + 1/2)2

2(ω2 − ω1) + 2N(x22 − x11) + 4x12. (96)


The numerator of equation (96) increases approximately asN2, whereas the denominator increases approximately asN . Thus, as N = v1 + v2 increases, most of the mixingangles among the basis states within the polyad increase.

There is, however, an exceptional situation where thedenominator of equation (95) passes through 0 at low N

but is otherwise always large relative to the numerator evenas N increases. Let v1x,N be the value of v1 where thedenominator of equation (95) is zero for polyad N ,

v1x,N = 2(ω2 − ω1) + N(4x22 − 2x12) + (4x11 + 4x22)

4x11 + 4x22 − 4x12(97)

thus

θv1−2,v2+2;v1,v2 = K11,22(v1−1/2)(N−v1 + 3/2)

(v1x,N −v1)(4x11+4x22 − 4x12). (98)

Equations (97) and (98) tell us that if an accidentaldegeneracy is observed at low N because [2(ω2 − ω1) +4(x11 + x22)] and [4x22 − 2x12] have opposite signs, thenas N increases, the value of v1x,N rapidly tunes to anunattainable value (either negative or larger than N ), andthe mixing angle can no longer become accidentally large.

It is rare that a vibrational perturbation, observed at lowvibrational energy, does not replicate itself with increasingvirulence at higher vibrational excitation. The key to anescape from resonance is a small coupling matrix elementcombined with rapid detuning of the interacting levels.Detuning is especially rapid for 1 : 2 resonances, moreso when only one of the interacting modes is an RHhydride stretch (because the x anharmonicity constant ofRH stretches is generally much larger than for all othervibrational modes (e.g., x

ωfor ν1 and ν2 of HCCH are

−7.3 × 10−3 and −3.8 × 10−3) (Herman 2007).

4.4 Correction of the Vibrational Heff for Effectsof φ-type Perturbers

Once the important vibrational resonance is identified,the Heff is block-diagonalized, with one block for eachvalue of the polyad quantum number. Each block of theHeff is expressed in terms of diagonal harmonic ωi andanharmonic xij constants and one reduced off-diagonalcoupling constant, e.g., H11,22, H1,22, or H1,234. This Heff

must be corrected, by a Lowdin (1951), Van Vleck (1951) orcontact transformation (Papousek and Aliev 1982), for theeffects of interactions with all out-of-block φ-type states.Usually the out-of-block corrections are restricted to theeffects of all cubic and quartic coupling terms.

One result of this Van Vleck or contact transforma-tion (see Section 4.7) is that some new fit parameters areintroduced into the transformed Heff. However, the most

important effect is that correction terms are added to themicroscopic values of the diagonal ωi, xij fit parame-ters and the reduced off-diagonal coupling constants. Themolecular constants that define the Heff are no longer triv-ially related to partial derivatives of the potential energysurface, V Q, evaluated at Q = 0. As in the diatomiccase, many of the correction terms may be computed fromthe ab initio V Q to recover the (deperturbed) micro-scopic values of the fit parameters, provided that neglectof the infinite number of coupling terms of order higherthan quartic is justified. Computer programs exist, based onformulas derived by Mills (1972) and by Lehmann (1991),that exploit x –K relationships to derive ωi, xij and Ki,jkl

reduced coupling constants from the ab initio V Q, andother programs exist that implement the contact transfor-mation, again based on the ab initio V Q, to derive anoptimal Heff fit model (Sibert 1988).

4.5 Construction of a Multiresonance Heff

When, as is frequently the case, there is more than oneimportant vibrational resonance, all the important reso-nance coupling mechanisms may still be accommodatedwithin a single multiresonance Heff. In such cases, it isoften necessary to label each block of the Heff by morethan one polyad quantum number. A “resonance-vector”procedure (Lehmann 1991) is used to determine the lin-early independent “good” quantum numbers. If there are noresonances, then there are 3N − 6 good vibrational quan-tum numbers. One can imagine a 3N − 6 dimensional statespace in which each vibrational quantum number is associ-ated with one of 3N − 6 mutually orthogonal unit vectors.A k1,22Q1Q

22 resonance destroys the v1 and v2 quantum

numbers by causing mixing among systematically near-degenerate N = v1 + 2v2 zero-order levels. Each resonanceis associated with a state-space vector that points in a direc-tion determined by the coupling mechanism. For example,the k1,22 resonance is represented by the state-space vec-tor e1 + 2e2. One collects all of the important resonanceterms, constructs a resonance vector for each, and looksfor all of the remaining mutually orthogonal directions in3N − 6 dimensional state space that are orthogonal to all ofthe resonance vectors. These directions in state space cor-respond to the remaining good quantum numbers that arenot destroyed by the set of known anharmonic resonances.These are the polyad quantum numbers for a multiresonanceHeff (Kellman 1990).

For example, in the S0 state of acetylene, the normal-mode frequencies are in the approximate integer ratios,ω1(σ g) = 3397 cm−1 : ω2(σ g) = 1982 cm−1 : ω3(σ u) =3317 cm−1 : ω4(πg) = 609 cm−1 : ω5(πu) = 729 cm−1

≈ 5 : 3 : 5 : 1 : 1 (Kellman 1990). The resonance-vector


Table 5 The anharmonic resonances in HCCH expressed asresonance vectors.

Name Resonance vector(∆v1, ∆v2, ∆v3, ∆v4, ∆4, ∆v5, ∆5)

Stretch-DD (2, 0,−2, 0, 0, 0, 0) (DD = Darling–Dennison)Bend-DD I (0, 0, 0, 2, 0,−2, 0)

Bend-DD II (0, 0, 0, 2,±2,−2,∓2)

H3,245 (0,−1, 1,−1,±1,−1,∓1)

H1,255 (1,−1, 0, 0, 0,−2, 0)

H1,244 (1,−1, 0,−2, 0, 0, 0)

H14,35 (1, 0,−1, 1,∓1,−1,±1)

H34,15 (−1, 0, 1, 1,∓1,−1,±1)

-resonance (0, 0, 0, 0,±2, 0,∓2)

method tells us that all seven of the normal-mode quan-tum numbers (v1, v2, v3, v4, 4, v5, 5) are destroyed byanharmonic resonances. Table 5 provides a partial list ofresonances.

However, three polyad quantum numbers

Nresonance = 5v1 + 3v2 + 5v3 + v4 + v5

Nstretch = v1 + v2 + v3

total = 4 + 5 (99)

(where tot = 0, 1, 2, 3 correspond to σ , π, δ, φ) remaingood (Lehmann 1991). Each polyad is labeled by [Nres, Ns,

tot] as well as by rigorous symmetry labels (g, u as sub-scripts) and, for tot = 0 states, the reflection symmetrylabels + or − as superscripts.

4.6 The V Q ↔ Spectrum Inversion via the Heff

The analysis of anharmonic interactions presented in thissection is written in a style intended to mirror the treatmentof remote electronic perturbers in Sections 3.1 and 3.2.Consequently, it is more opaque and laboriously algebraicthan it ideally might be. We outline a more compactand useful implementation of the contact transformation inSection 4.7.

Let us now consider the polyatomic molecule vibrationalHeff in detail.

The Heff is a fit model expressed in terms of effectivemolecular constants ωi , xii , xij , yiii , etc., which appearon-diagonal in the Heff as

G(v1, v2, . . . v3N−6) =∑

i

ωi (vi + 1/2)

+∑i≥j

xij (vi +1/2)(vj + 1/2) + . . .

(100)

and interaction parameters, ki,jj (Fermi), kii,jj (Dar-ling–Dennison), etc. , which appear off-diagonal in theHeff as⟨

vi, vj , vt |ki,jjQiQ2j |vi ± 1, vj ± 0, 2, vt

⟩. (101)

V Q is expressed in terms of partial derivatives of V Qevaluated at equilibrium geometry Q = 0:

V Q =3N−6∑i=1

1

2

∂2V

∂Q2i

∣∣∣∣∣0

Q2i

+3N−6∑i,j,k

1

6

∂3V

∂Qi∂Qj∂Qk

∣∣∣∣0

QiQjQk + . . . . (102)

Each fit parameter, e.g., ωi , is actually a sum of amicroscopic term, e.g. ω

(0)i , which is directly related to

a partial derivative of V Q, and correction terms, e.g.,ω

(2)i , that originate in the transformation that approximately

block-diagonalizes Hexact into a set of finite-dimensionalHeff blocks.

The relationship between the partial derivatives of V Qand the microscopic molecular constants is illustrated herevia the following examples: (i) derivation of ω and x

in the energy level expression for the Morse oscillator,

GM(v) = ω(v + 1/2) + x(v + 1/2)2 from ∂2V M

∂Q2

∣∣∣0, ∂3V M

∂Q3

∣∣∣0,

and ∂4V M

∂Q4

∣∣∣0

for a Morse oscillator potential energy curve,

V M(Q) = De[1 − e−βQ]2; (ii) the unjustified neglect of thefirst-order nondegenerate perturbation theory corrections tothe harmonic oscillator basis states, |v〉0, because |vM〉 =|v〉0; (iii) the effects of remote-perturber states, introducedby a single intermode anharmonicity term, 1

2k122Q1Q22, on

the diagonal molecular constants ω1, ω2, x11, x22, x12 ofa two-mode molecule; (iv) the effects of remote-perturberstates, introduced by k12tQ1Q2Qt , on the effective diagonalmolecular constants ω1, ω2, x11, x22, x12 and the off-diagonal K11,22 perturbation term.

4.6.1 The Morse Oscillator

Nondegenerate perturbation theory recovers the exact vibra-tional energy levels of a Morse oscillator

GM(v) = ω(v + 1/2) + x(v + 1/2)2 (103)

where

x = − ω2

De.


The Morse potential is

V M(Q) = De[1 − e−βQ

]2(104)

β =[−4πcµx

]1/2

=[πcµω2

De

]1/2

. (105)

Expressing Q (not mass-weighted) in terms of the dimen-sionless Q

Q = Q

[

2πcµω

]1/2

(106)

and expanding V M as a power series in Q

V M(Q)/hc = ω

2Q2

[1 −

( ω

2De

)1/2Q + 7

12

( ω

2De

)Q2

− 1

4

( ω

2De

)3/2Q3 . . .

]. (107)

Replacing Q by a, a†, N

Q = 2−1/2(a + a†) (108)

N = a†a (109)

and exploiting the commutation rules [a, a†] = 1 and[a, N] = a to express Qn in the simplest form

Q2 = 1

2

(a2 + a†2) + (N + 1/2) (110)

Q3 = 2−3/2 [a3 + 3aN + 3a†(N + 1) + a†3] (111)

Q4 = 1

4

[a4 + 2a2(2N − 1) + 2a†2(2N + 3) + a†4

+ 6

(N + 1

2

)2

+ 3

2

]. (112)

Inserting Qn into the Hamiltonian

H/hc = ω

2P 2 + V (Q) = (ω/2)

[P 2 + Q2]

+ (V M/hc − (ω/2)Q2) (113)

= ω(N + 1/2)

− 1

8ω3/2De−1/2 [a3 + 3aN + 3a†(N + 1) + a†3]

+ 7

192(ω2/De)

[a4+ 2a2(2N − 1) + 6(N + 1/2)2

+ 3

2+ 2a†2(2N + 3) + a†4

](114)

The first term in equation (113) is exclusively diagonal (the∆v = ±2 matrix elements of the Q2 term from VM(Q) areexactly canceled by the (ω/2)P 2 kinetic energy term) andgives the zero-order harmonic oscillator energies

E(0)v /hc = ω(v + 1/2). (115)

The second term in equation (114) comes from Q3 and hasexclusively off-diagonal ∆v = ±3, ±1 matrix elements.The third term comes from Q4 and has both off-diagonal∆v = ±4, ±2 and ∆v = 0 diagonal matrix elements.Nondegenerate perturbation theory (the order of the per-turbation correction is given by right superscript (n)) gives

Ev = E(0)v + E(1)

v + E(2)v = H(0)

vv + ⟨v|H(1)|v⟩

+∑v′ =v

∣∣∣H(1)

vv′∣∣∣ 2

E(v)v − E

(0)

v′. (116)

The contribution to E(2)v from the Q3(cubic) term is

E(2)(cubic) = ω3

64De

[⟨v|a3|v + 3

⟩2−3ω

+⟨v|a†3|v − 3

⟩23ω

+ 3 〈v|aN|v + 1〉2

−ω+ 3

⟨v|a†(N + 1)|v − 1

⟩2ω

]

= − ω2

64De[30(v + 1/2)2 + 7/2] (117)

The contribution to E(1)v from the Q4 (quartic) term is

E(1)(quartic) = 7

64

ω2

De

[2(v + 1/2)2 + 1

2

]. (118)

Neglecting the Q4 contribution to E(2)v , we obtain the

expected v-dependent energy level expression for a Morseoscillator

Ev/hc = ω(v + 1/2) +(

42

192− 30

64

)ω2

De(v + 1/2)2

+ 7

128

ω2

De= ω(v + 1/2) − ω2

4De(v + 1/2)2

+ 7

128

ω2

De. (119)

The v-independent term, 21384

ω2

De , is small but does notagree with the exact energy level expression for a Morseoscillator. The contribution of the Q4 term to E(2)

v isexpected to be ∼ 100 times smaller than that for the Q3


term:

E(2)(quartic) = −(

7

192

)2ω3

De2[67(v + 1/2)

+ 68(v + 1/2)3]. (120)

These very small contributions to the ω and y [y is thecoefficient of (v + 1/2)3] parameters from E(2)(quartic) aremostly canceled by the E(1)(sextic) contribution.

The success of E(0)(harmonic) + E(1)(quartic) + E(2)

(cubic) in reproducing the Morse G(v) energy level expres-sion seems to imply that a harmonic oscillator basis set isadequate to deal with realistic anharmonicities. However,if we use perturbation theory to compute energies, then weshould also use perturbation theory to express corrections tothe wavefunctions, especially if we want to evaluate matrixelements of polyatomic molecule intermode coupling terms.

4.6.2 The Morse |vM〉 Expressed in Terms of theHarmonic |v〉0

First-order nondegenerate perturbation theory enables us toexpress Morse oscillator states as explicit linear combina-tions of harmonic oscillator states. Neglecting quintic andhigher order terms, we have

|vM〉 − |v〉0 =∑v′ =v

H(1)

vv′

E(0)v − E

(0)

v′|v′〉0 (121)

= 7

4 · 192

ω

De[v(v − 1)(v − 2)(v − 3)]1/2|v − 4〉0

− 1

38

( ω

De

)1/2[v(v − 1)(v − 2)]1/2|v − 3〉0

+ 7

192

ω

De(2v − 1)[v(v − 1)]1/2|v − 2〉0

− 3

8

( ω

De

)1/2(v − 2)v1/2|v − 1〉0

− 7

192

ω

De(2v + 3)[(v + 2)(v + 1)]1/2|v + 2〉0

+ 3

8

( ω

De

)1/2(v + 1)(v + 1)1/2|v + 1〉0

− 7

4 · 192

ω

De[(v+4)(v+3)(v+2)(v+1)]1/2|v+4〉0

+ 1

38

( ω

De

)1/2[(v + 3)(v + 2)(v + 1)]1/2|v + 3〉0

+ (1 − S2)1/2 |v〉0 (122)

where S2 in the coefficient of the |v〉0 term is the sum of thesquared mixing coefficients for all eight of the v′ = v terms.

Equation (122) is only valid when 124

(ωDe

)1/2v3/2 1.

For typical values of ω = 1000 cm−1 and De = 25 000cm−1, the ∆v = ±3,±1 mixing coefficients are >∼ 0.1when v > 5. This is bad news. Not only is the criticalapproximation of the anharmonically coupled harmonicoscillators approach, that |vM〉 = |v〉0, invalid but it is alsounreasonable to use perturbation theory to express |vM〉 =∑vM+4

vi=vM−4 αi |vi〉0 for vM > 5. (The vm − 4 to vm + 4 rangein the summation reflects the decision to exclude quinticand higher order terms from the expansion of V M(Q).)The low-v breakdown of the |vM〉 = |v〉0 approximationvitiates the harmonic oscillator scaling and selection rulesthat are the main advantages of the anharmonically coupledharmonic oscillators formulation of vibrational effectiveHamiltonians.

4.6.3 Effects of the k122Q1Q22 Intermode

Anharmonicity Term

Consider a two-mode molecule, where each mode is Morse-like. The diagonal matrix elements are given by

G(v1, v2) = ω01(v1 + 1/2) + x0

11(v1 + 1/2)2

+ ω02(v2 + 1/2) + x0

22(v2 + 1/2)2

+ x012(v1 + 1/2)(v2 + 1/2) (123)

where the values of ω0i and x0

ii are determined from the

∂2V

∂Q2i

∣∣∣∣0, ∂3V

∂Q3i

∣∣∣∣0, and ∂4V

∂Q4i

∣∣∣∣0

partial derivatives. The on-diag-

onal cross-anharmonicity term, x012(v1 + 1/2)(v2 + 1/2),

derives from diagonal matrix elements of the 14

∂4V

∂Q21∂Q2

2

∣∣∣∣0Q2

1

Q22 term

(k1122 ≡ ∂4V

∂Q21∂Q2

2

∣∣∣∣0

),

⟨v1v2|Q2

1Q22|v1v2

⟩ = 1

4〈v1v2|(2N1 + 1)(2N2 + 1)|v1v2〉

= (v1 + 1/2)(v2 + 1/2) (124)

x012 = 1

4k1122

[g11

2πcω1

] [g22

2πcω2

](125)

where gii is the reciprocal mass factor (analogous to 1/µ

in a single oscillator) in the 12 P†

i giiPi kinetic energy term.The lowest-order anharmonic terms in V Q that have

not already been treated [the effects of ∂3V

∂Q3i

∣∣∣∣0

are already

expressed in G(v1, v2)] are the intermode cubic 12k122Q1Q

22

and 12k112Q

21Q2 terms. A kijjQiQ

2j term is nonzero only


E(2)v1v2

/hc = K2

122

32

⟨v1, v2|a1a†2

2|v1 + 1, v2 − 2⟩2 −

⟨v1, v2|a†

1a22|v1 − 1, v2 + 2

⟩2−ω1 + 2ω2

+

⟨v1, v2|a†

1a†22|v1 − 1, v2 − 2

⟩2 − ⟨v1, v2|a1a2

2|v1 + 1, v2 + 2⟩2

ω1 + 2ω2

+

〈v1, v2|a1(2N2 + 1)|v1 + 1, v2〉2 −⟨v1, v2|a†

1(2N2 + 1)|v1 − 1, v2

⟩2−ω1

= K2122

32

[4v1v2 + 2v1 − v2

2 + v2

ω1 − 2ω2− 4v1v2 + 2v1 + 2v2

2 + 3v2 + 2

ω1 + 2ω2− (2v2 + 1)

ω1

]= K2

122

32

[ω1[2(v2 + 1/2)2 − 3/4] + ω2[16(v1 + 1/2)(v2 + 1/2)]

ω21 − 4ω2

2

− 4(v2 + 1/2)2

ω1

](130)

if the Qi displacement coordinate is totally symmetric.

(Mixed quadratic terms, such as kij ≡ ∂2V∂Qi∂Qj

∣∣∣0, are

rigorously zero owing to the definition of the normal-modedisplacement coordinates, Qi and Qj , and the requirement

that ∂V∂Qi

∣∣∣0

= 0 for all i.)

The present goal is to show how the k122Q1Q22 cubic

anharmonic coupling term affects all of the terms inG(v1, v2) as well as to demonstrate its contribution to the“effective” quartic K11,22 reduced interaction term,

1

2

∂3V

∂Q1∂Q22

∣∣∣∣0

Q1Q22 = 1

2k122Q1Q

22

= 1

2k122

[g11

2πcω1

]1/2 [g22

2πcω2

]Q1Q

22

(126)

= 1

2k122

[g11

2πcω1

]1/2 [g22

2πcω2

]× 2−3/2(a1 + a†

1)(a2 + a†2)

2 (127)

and absorbing all of the constants into the cubic K122

reduced interaction term (units of cm−1), we have

= 2−5/2K122(a1 + a†1)(a2 + a†

2)2 (128)

= 2−5/2K122[(a1a†22 + a†

1a22) + (a†

1a†22 + a1a2

2)

+ (2N2 + 1)(a1 + a†1)]. (129)

The a, a† operators are arranged in three groups, accordingto the selection rules ∆v1 = −2∆v2, ∆v1 = 2∆v2, and∆v2 = 0. This simplifies the algebra in evaluating E(2)

v1v2,

the second-order corrections to G(v1, v2),

Thus the corrections to the zero-order parameters due tothe 1

2k122Q1Q22 cubic anharmonicity are

ω1 = ω01 (131)

ω2 = ω02 (132)

x11 = x011 (133)

x22 = x022 + K2

122

32

[2ω1

ω21 − 4ω2

2

− 4

ω1

](134)

x12 = x012 + K2

122

32

[16ω2

ω21 − 4ω2

2

]. (135)

If ω1 ≈ 2ω2, then the4v1v2+2v1−v2

2+v2ω1−2ω2

term must be

excluded from E(2)v1v2

, all of the corrections to the molecularconstants must be recomputed, and it is necessary to set upa polyad Heff where membership in each quasi-degenerateblock of the polyad is determined by the polyad quantumnumber, N = 2v1 + v2. The N = 6 block of the polyad,for example, is a 4 × 4 Heff that describes the interactionsamong the (v1, v2) = (0, 6), (1, 4), (2, 2), and (3,0) basisstates:

Heff(N = 6)

=

G(0, 6)

(1516

)1/2K122 0 0(

1516

)1/2K122 G(1, 4)

(34

)1/2K122 0

0(

34

)1/2K122 G(2, 2)

(3

16

)1/2K122

0 0(

316

)1/2K122 G(3, 0)

,

(136)where the G(v1, v2) terms are corrected for all interactionswith out-of-block states,


G(v1, v2) = G(v1, v2) − K2122

32

[4(v1 + 1/2)(v2 + 1/2) + 2(v2 + 1/2)2 − (v2 + 1/2) + 1

ω1 + 2ω2+ 4(v2 + 1/2)2

ω1

]. (137)

The k122Q1Q22 term generates no out-of-block corrections

to any of the off-diagonal elements in the Heff(N = 6)

(Dubal and Quack 1984). However, if ω1 ≈ ω2, theninstead of the ω1 : 2ω2 polyad, it would be necessary toset up a 2ω1 : 2ω2 polyad, and there would be a correctionto the k11,22 Darling–Dennison

( 14k11,22Q

21Q

22

)interaction

term due to out-of-block effects from the 12k122Q1Q

22

term

K11,22 = K(0)11,22 − 2K2

122/ω2. (138)

It is important to remember that the zero-order “micro-scopic” molecular constants derived from partial deriva-tives of V Q evaluated at Q = 0 differ from the “effec-tive” fit parameters in the Heff. The differences betweenthe microscopic and effective parameters are due to addi-tive interactions via each of the intermode anharmoniccoupling terms. However, the form of these correctionsdepends critically on the presence of low-order resonancesbetween the harmonic frequencies, especially ω1 ≈ 2ω2 or2ω1 ≈ 2ω2.

4.6.4 Effects of the k12tQ1Q2Qt Term

The last remaining cubic term is the three-mode cubicanharmonic coupling term,

∂3V

∂Q1∂Q2∂Qt

∣∣∣∣0Q1Q2Qt = k12tQ1Q2Qt

= k12t2−3/2

[

2πc

]3/2 [g11g22gtt

ω1ω2ωt

]1/2

× (a1 + a†1)(a2 + a†

2)(at + a†t )

= 2−3/2K12t (a1 + a†1)(a2 + a†

2)(at + a†t )

= 2−3/2K12t (a1a2at + a†1a†

2a†t )

+ (a1a†2a†

t + a†1a2at ) + (a1a2a†

t + a†1a†

2at )

+ (a1a†2at + a†

1a2a†t ) (139)

which is nonzero if Γ Q1 ⊗ Γ Q2 ⊗ Γ Qt contains the totallysymmetric representation, Γ A1 . Equations 140–144 illus-trate two crucial algebraic strategies for combining thesecond-order contribution to E(2)

v1v2vt: pairwise combination

of (∆v1, ∆v2, ∆vt ) with (−∆v1, −∆v2, −∆vt ) terms andthe simplification of energy denominators using the ω1, ω2

sum (∑

12) and difference (∆12) terms.

The E(2) terms are first arranged into four pairs tosimplify the algebra in computing E(2)

v1v2vt,

E(2)v1v2vt

= K212t

8

[(⟨v1, v2, vt |a1a2at |v1 + 1, v2 + 1, vt + 1

⟩2−ω1 − ω2 − ωt

+⟨v1, v2, vt |a†

1a†2a†

t |v1 − 1, v2 − 1, vt − 1⟩2

ω1 + ω2 + ωt

)

+(⟨

v1, v2, vt |a1a†2a†

t |v1 + 1, v2 − 1, vt − 1⟩2

−ω1 + ω2 + ωt

+⟨v1, v2, vt |a†

1a2at |v1 − 1, v2 + 1, vt + 1⟩2

ω1 − ω2 − ωt

)

+(⟨

v1, v2, vt |a1a2a†t |v1 + 1, v2 + 1, vt − 1

⟩2−ω1 − ω2 + ωt

+⟨v1, v2, v3|a†

1a†2at |v1 − 1, v2 − 1, vt + 1

⟩2ω1 + ω2 − ωt

)

+(⟨

v1, v2, vt |a1a†2at |v1 + 1, v2 − 1, vt + 1

⟩2−ω1 + ω2 − ωt

+⟨v1, v2, vt |a†

1a2a†t |v1 − 1, v2 + 1, vt − 1

⟩2ω1 − ω2 + ωt

)](140)

= K212t

8

[v1v2vt − (v1 + 1)(v2 + 1)(vt + 1)

ω1 + ω2 + ωt

+ (v1 + 1)v2vt − v1(v2 + 1)(vt + 1)

−ω1 + ω2 + ωt

+ (v1 + 1)(v2 + 1)vt − v1v2(vt + 1)

ωt − ω1 − ω2

+ (v1 + 1)v2(vt + 1) − v1(v2 + 1)vt

−ω1 + ω2 + ωt

]= K2

12t

8

[−[v1v2 + v1vt + v2vt + v1 + v2 + vt + 1]

ω1 + ω2 + ωt

+ v2vt − v1v2 − v1vt − v1

−ω1 + ω2 + ωt

+ v1vt + v2vt + vt − v1v2

ωt − ω1 − ω2

+ v2vt + v1v2 + v2 − v1vt

−ω1 + ω2 − ωt

](141)


Algebraic reduction of equation (141) is facilitated bydefining

Σ12 = ω1 + ω2 (142)

∆12 = ω1 − ω2 (143)

E(2)v1v2vt

= K212t

8

[(Σ12)

[2(v1 + 1

2

) (vt + 1

2

) + 2(v2 + 1

2

) (vt + 1

2

)] − ωt

[2(v1 + 1

2

) (v2 + 1

2

) + 12

]ω2

t − (Σ12)2

+ (∆12)[2(v1 + 1

2

) (vt + 1

2

) − 2(v2 + 1

2

) (vt + 1

2

)] + ωt

[2(v1 + 1

2

) (v2 + 1

2

) − 12

](∆12)

2 − ω2t

](144)

The expression in equation (144) is valid except when oneof the denominators is near zero, ωt ≈ ω1 + ω2 or ωt ≈|ω1 − ω2|. This expression for E(2)

v1v2vtcontains corrections

to x1t , x2t , and x12,

x12 = x012

+ K212tωt

8

[2

(ω1 − ω2)2 − ω2t

+ 2

(ω1 + ω2)2 − ω2t

](145)

and similarly for x1t and x2t . In the special case whenω1 ≈ ω2, then the K11,22 coupling constant must be cor-rected for the out-of-block effects of K12t

K11,22 = K(0)11,22 − K2

12t

2ωt

(146)

where K(0)11,22 is the zero-order quartic coupling term.

So far, we have considered contributions to E(2) term byterm. Doing this is algebraically exhausting and physicallyopaque. A much better way is described in the followingsubsection, which deals with the contact transformation ina more abstract, compact, and computationally efficientway. We present this treatment here, rather than at thebeginning of the discussion of anharmonic couplings amongvibrational modes, because we believe we have motivatedthe urgent need for a general formalism.

4.7 The Contact Transformation for theVibrational Hamiltonian

The goal is to obtain an Heff that consists of finite-dimensional blocks along the main diagonal, yet isexpressed in terms of microscopic quantities that may,in principle, be computed ab initio. Owing to the enor-mous number of symmetry-allowed anharmonic interac-tion mechanisms, the derivation of the vibrational Heff byone or more contact transformations has the appearance

to humans of extraordinary algebraic complexity. Com-puters, programmed by humans, deal uncomplaininglywith this complexity. To develop some intuition, it isstill useful here to present a simplified version of theprocedure.

The formal derivation of the contact transformationis presented here in two stages, followed by a qualita-tive discussion of examples from the vibrational Hamil-tonian. First, a version of the contact transformation,expressed in terms of Hiα,mφ matrix elements betweenlocal (α-type) and remote (φ-type) states and energy

denominators,[E

(0)iα − E

(0)

mφ

], is derived. (Greek letters

denote groups of states, Roman letters index specificstates in each group.) This form is useful when the num-ber of out-of-block correction terms is small and order-sorting is uncomplicated. Second, the contact transfor-mation expressed in terms of explicitly definable con-tact transformation matrices, Sn, is derived. One veryimportant advantage of a formal treatment, expressed interms of operators and commutation rules for operatorsrather than in terms of sums of squared matrix ele-ments over energy denominators, is that manipulation ofthe operator expressions, in particular, the exploitation ofcommutation rules, results in vast simplifications of theformulas.

Before starting a formal discussion of the contact trans-formation, it is appropriate to comment on the partitioningof Hexact into terms organized according to their order ofimportance in the values of the exact energies.

Order-sorting is a topic that often inflames the passionsof experts, yet leaves bystanders amused at best. For theelectronic fine structure of diatomic molecules, it makessense to partition H into only H(0) (which implicitly definesa complete set of basis states) and a single perturbationterm, H(1), because the matrix elements of H(1) are readilydivided into intrablock and interblock terms. Energy differ-

ences∣∣∣E(0)

iα − E(0)

mφ

∣∣∣ between electronic states are generally

so much larger than H(1)

iα,mφ off-diagonal matrix elements(with the exception of occasional “accidental” degenera-cies) that once one decides the membership of the α-typelocally interacting group of states, it is better to diagonalizethe Heff exactly, rather than subject it to an order-by-orderperturbation theory algebraic solution.


The situation is profoundly different for polyatomicmolecule vibrations. The anharmonic matrix elements gen-erally decrease in order-of-magnitude steps,

H(quadratic) ≈ 10H(cubic) ≈ 100H(quartic) (147)

The quadratic terms are exclusively diagonal and give a setof harmonic frequencies ωi each of which contributes typ-ically 1000 cm−1 to the total vibrational energy. The cubicterms are exclusively off-diagonal, and therefore contributeto energy as [H(cubic)2/H(quadratic)] ≈ 10 cm−1. Thequartic terms make both diagonal, H(quartic) ≈ 10 cm−1,and off-diagonal, [H(quartic)2/H(quadratic)] ≈ 0.1 cm−1,contributions. Thus it makes sense to treat second-ordercubic and first-order quartic terms at the same order of per-turbation theory, to neglect quintic terms (10−3 cm−1), andto treat second-order quartic (0.1 cm−1) and first-order sex-tic terms (0.1 cm−1) together at the next higher order of per-turbation theory. Of course all of this order-sorting dependson the absence of resonance (small energy denominators).When a resonance occurs, a specific high-order term canmake a contribution to the energy comparable to those fromlower-order terms. Each resonance requires exclusion ofthis resonance from the nondegenerate perturbation-theorysummation and inclusion in a quasi-degenerate block of theHeff that must be numerically diagonalized.

Suppose we have a complete and exact vibrational Hexact.Hexact may be partitioned as

Hexact = H(0) + λH(1) + λ2H(2) + . . . (148)

where λ is an order-sorting book-keeping device (used toderive independent equations, one for each order of λ), andthe right superscript in parentheses connotes the relativeorder of magnitude, O(H(n))≈ 10−nO(H(0)), of the typicalmatrix element of H(n). If we choose H(0) to be a sumof individual normal-mode uncoupled harmonic oscillatorHamiltonians

H(0) =3N−6∑i=1

h(0)i = hc

2

3N−6∑i=1

ωi

(P

2i + Q

2i

)(149)

where Qi and Pi are the ith dimensionless displace-ment coordinate and conjugate linear momentum, then thenormal-mode product basis functions are eigenfunctions ofH(0),

H(0)

3N−6∏i=1

|vi〉0 =[hc

3N−6∑i=1

ωi(vi + 1/2)

]3N−6∏i=1

|vi〉0

= E(0)(v1, v2, . . . )

3N−6∏i=1

|vi〉0 (150)

and we seek the energy eigenvalues of Hexact. However,when Hexact is expressed in terms of matrix elements ofthe normal-mode product basis set, Hexact is an infinite-dimensional matrix.

The contact transformation (Papousek and Aliev 1982) isdesigned to factor Hexact into finite block-diagonal form, H,where the eigenvalues of H are identical to those of Hexact.A general unitary (T−1 = T†) transformation

T†HexactT = H (151)

has the form

T = eiλS (152)

where S is Hermitian (S† = S). Our goal is to derive theform of S on the basis of an order-sorting procedure,where terms in the equation to be solved are sorted into n

independent (one for each power of λ) equations accordingto the overall power of λ.

We begin by expanding all terms in equation (151)according to order,

Hexact = H(0) + λH(1) + λ2H(2) + . . . (153)

H = H(0) + λH(1) + λ2H(2) + . . . (154)

and expanding eiλS in a power series

T = eiλS = 1 + iλS − λ2

2S2 + . . . . (155)

Inserting equations (153) and (155) into the left-hand sideof equation (151)

H =(

1 − iλS − λ2

2S2 + . . .

) (H(0) + λH(1)

+ λ2H(2) + . . .) (

1 + iλS − λ2

2S2 + . . .

)(156)

which can be rearranged according to the powers of λ

H = λ0H(0) + λ1(H(1) − iSH(0) + iH(0)S)

+ λ2(

H(2) + iH(1)S − iSH(1) − 1

2S2H(0)

− 1

2H(0)S2

)+ . . . . (157)

We have an order-sorted expansion of H. These order-sorted terms may be combined with equation (154) to yield


the desired definitions of H(n) in terms of H(n) and S,thus

H(0) = H(0) (158)

H(1) = H(1) + i[H(0), S] (159)

H(2) = H(2) + SH(0)S+i[H(1), S]− 1

2(S2H(0) + H(0)S2).

(160)

Equation (158) implies that we should use the same basisfunctions for H as for H. We use equation (159), incombination with the nature of the block-diagonalizationthat we require, to derive all of the matrix elementsof S.

Suppose we want to divide the basis functions into theα-type group, in which we are interested, and the φ-typeremote group, in which we have no interest but must dealwith in a formally correct way. Consider three classes ofmatrix elements of H(1):

1. Within the α-block

⟨iα|H(1)|jα

⟩ = H(1)iα ,jα + i

∑kγ

(H

(0)iα,kγ Skγ ,jα − Siα,kγ H

(0)kγ ,jα

)(161)

where the kγ sum is over all states. This sum reduces to

= H(1)iα,jα + i

(E

(0)iα Siα,jα − E

(0)jα Siα,jα

)= H

(1)iα,jα + iSiα,jα

(E

(0)iα − E

(0)jα

)(162)

The φ-block and the α-φ-interblock matrix elements aresimilarly transformed.

2. Within the φ-block

⟨mφ |H(1)|nφ

⟩ = H(1)

mφ,nφ + iSmφ,nφ

(E

(0)

mφ − E(0)

nφ

)(163)

3. Between α- and φ-blocks

⟨iα|H(1)|nφ

⟩ = H(1)

iα ,nφ + i∑kγ

[H

(0)iα,kγ Skγ ,nφ − Siα,kγ H

(0)

kγ ,nφ

]= H

(1)

iα ,nφ + i[E

(0)iα Siα,nφ − Siα,nφE

(0)

nφ

]= H

(1)

iα ,nφ + iSiα,nφ

(E

(0)iα − E

(0)

nφ

)(164)

If we choose S such that

Siα,kγ = δikδαγ + [1 − δαγ ]iH(1)

iα,kγ

E(0)iα − E

(0)kγ

(165)

then ⟨iα|H(1)|jα

⟩ = ⟨iα|H(1)|jα

⟩(166)⟨

mφ|H(1)|nφ⟩ = ⟨

mφ |H(1)|nφ⟩

(167)⟨iα|H(1)|nφ

⟩ = 0 (168)

This choice of S leads to three very useful results:

H(1)(α−block) = H(1)(α−block) (169)

H(1)(φ−block) = H(1)(φ−block) (170)

and, most importantly,

H(1)(interblock) = 0 (171)

even though

H(1)(interblock) = 0 (172)

Thus T†HexactT makes no within-block changes to H(0)

and H(1), yet it causes H(1) to take block-diagonal form.This seems to be getting something for nothing. However,there is a price: information from H(0) and H(1) has beentransferred into a new and vastly more complicated formfor H(2).

We are interested in the intrablock matrix elements ofthe α-block of H(2). To examine them we exploit thecompleteness of the basis set, which requires summing overall φ states,

H(2)iα ,jα = H

(2)iα ,jα +

∑mφ

[Siα,kγ E

(0)kγ Skγ ,jα

+ i(H

(1)iα,kγ Skγ ,jα − Siα,kγ H

(1)kγ ,jα

)− 1

2

(Siα,kγ Skγ ,jαE

(0)jα + E

(0)iα Siα,kγ Skγ ,jα

)].

(173)

Inserting the equation (165) expression for Siα,kγ , we obtain

Hiα,jα = H(2)iα,jα +

∑kγ

[(E

(0)kγ − 1

2

(E

(0)jα + E

(0)iα

))

× −H(1)iα,kγ H

(1)kγ ,jα(

E(0)iα − E

(0)kγ

) (E

(0)kγ − E

(0)jα

)−

(H

(1)iα,kγ H

(1)kγ ,jα

E(0)kγ − E

(0)jα

− H(1)iα,kγ H

(1)kγ ,jα

E(0)iα − E

(0)kγ

)]


= H(2)iα,jα +

∑kγ

(E

(0)jα + E

(0)iα

)2

− E(0)kγ

× H

(1)iα,kγ H

(1)kγ ,jα(

E(0)iα − E

(0)kγ

) (E

(0)kγ − E

(0)jα

)

+ H(1)iα,kγ H

(1)kγ ,jα

[(E

(0)kγ − E

(0)jα

)−(E

(0)iα − E

(0)kγ

)][(

E(0)iα − E

(0)kγ

) (E

(0)kγ − E

(0)jα

)]

= H(2)iα,jα +

∑kγ

H(1)iα,kγ H

(1)kγ ,jα(

E(0)iα − E

(0)kγ

) (E

(0)kγ − E

(0)jα

)×[E

(0)jα + E

(0)iα

2− E

(0)kγ + 2E

(0)kγ −

(E

(0)jα + E0

iα

)](174)

Note that the energy denominator of equation (174), when

∣∣∣E(0)iα − E

(0)jα

∣∣∣ ∣∣∣∣∣∣(E

(0)iα + E

(0)jα

)2

− E(0)kγ

∣∣∣∣∣∣ (175)

is approximately equal to −[E

(0)kγ −

(E

(0)

iα+E

(0)

jα

)2

]2

, thus

H(2)iα,jα = H

(2)iα,jα +

∑kγ

H(1)iα ,kγ H

(1)kγ ,jα(

E(0)

iα+E

(0)

jα

2 − E(0)kγ

) . (176)

This is a wonderful result, as the diagonal elementsof H

(2)agree with what we expect from second-order

nondegenerate perturbation theory.It is possible to put part of the equation (160) expression

for H(2)

into a slightly more elegant form and thengeneralize to all orders of correction arising from thefirst contact transformation. First, equation (160) must besimplified,

SH(0)S − 1

2

(S2H(0) + H(0)S2) = SH(0)S

− 1

2

(S2H(0) − SH(0)S︸︷︷︸ + SH(0)S

+ H(0)S2 − SH(0)S︸︷︷︸ + SH(0)S)

(177)

where the terms with underbraces contain commutators,

= SH(0)S − 1

2(S

[S, H(0)

] + [H(0), S

]S + 2SH(0)S)

= −1

2

(S[S, H(0)

] − [S, H(0)

]S)

= −1

2

[S,[S, H(0)

]]. (178)

Now, it is possible to write the various orders of H in thecompact form,

H(1) = H(1) (179)

(based on the equation (165) definition of S ), and insertingequation (178) into equation (160), we have

H(2) = H(2) + i[S, H(1)

] − 1

2

[S,[S, H(0)

]](180)

H(3) = H(3) + i[S, H(2)

] − 1

2

[S,[S, H(1)

]]− i

6

[S,[S,[S, H(0)

]]](181)

H(n) = H(n) +n−1∑m=0

in−m

(n − m)!

[S,[S, . . .

[S, H(m)

]. . .

]].

(182)

Once the first contact transformation has been carried out,we have an H(1) in which the interblock matrix elementshave been eliminated. The effects of these interblock matrixelements of H(1) have been folded, as intrablock matrixelements, into H(2). But the H(2) part of H(2) still hasnonzero interblock matrix elements. A second contacttransformation, to create the doubly transformed H, denotedhere by ˜H, eliminates these interblock matrix elements ofH(2) and folds them into ˜H(3).

˜H = ei λ22 S2H e−i λ2

2 S2 = H(0) + λH(1) + λ2˜H2 + . . .

(183)˜H(0) = H(0) (184)˜H(1) = H(1) (185)˜H(2) = H(2) + i[H(0), S2

](186)˜H(3) = H(3) + i

[H(1), S2

](187)

(S2)iα,kγ = δikδαγ + (1 − δαγ

) iH (2)iα ,kγ

E(0)iα − E

(0)kγ

(188)


where the above form of S 2 is chosen by analogy withequation (165) to create an ˜H(2) in which all interblockmatrix elements are zero.

There is a much more elegant and powerful form of thecontact transformation, where S is expressed as a matrixrather than individually derived matrix elements (Papousekand Aliev 1982). If S 1 and S 2 are expressed as matrices,it is possible to use operator algebra to obtain all of thecontact-transformed H(n) and ˜H(m) matrices in maximallysimplified form.

Our goal is to determine H using an S matrix that causesH(1) = 0,

H = H(0) + H(1) + H(2) (189)

H(0) = H(0) (190)

H(1) = H(1) + i[S, H(0)

] = 0 (191)

H(2) = H(2) + i[S, H(1)

]− 1

2

[S,[S, H(0)

]](192)

but, since equation (191) requires [S,H(0)] = iH(1),

−1

2

[S,[S, H(0)

]] = −1

2

[S, iH(1)

]. (193)

Thus, inserting equation (193) into equation (192)

H(2) = H(2) + i

2

[S, H(1)

]. (194)

Finally, comparing equations (190), (191), and (194) toequation (189), we obtain

H = H(0) + H(2) + i

2

[S, H(1)

]. (195)

First, we must determine S :

S =∑n=1

Xn

[H(0),

[H(0),

[. . .

[H(0), H(1)

]]]](196)

the nth term in the sum contains n nested commutators,

S = X1[H(0), H(1)

]+X2[H(0),

[H(0), H(1)

]] + . . . (197)

where the set of Xn coefficients must be determined. Thenumber of terms in the equation (197) sum is limited bythe requirement that H(1), [H(0), H(1)], etc. all be linearlyindependent. The Xn coefficients are obtained by substi-tuting S from equation (197) into equation (191), settingthe sum of the coefficients of each linearly independentoperator to zero, and solving the set of linear equations forthe Xn.

The power of this procedure is illustrated for the cubicplus quartic coupled harmonic oscillator

H = H(0) + H(1) + H(0)

=[

1

2

3N−6∑k=1

ωk(P2k + Q

2k)

]︸︷︷︸

H(0)

+1

6

3N−6∑,m,n

KmnQQmQn

︸︷︷︸

H(1)

+ 1

24

3N−6∑k,,m,n

Kk,,m,nQkQQmQn +3∑

α=1

BαΠ2α

︸︷︷︸

H(2)

(198)

where the sums are unrestricted, Q and P are dimensionless,the three Bα are the Ae, Be, and Ce equilibrium (Q = 0)

rotational constants, and Πα is the vibrational angularmomentum.

Πα =3N−6∑k,

ζ αkQkP (199)

(ζ αk is a Coriolis interaction parameter), and all of the

diagonal elements of the H(1) cubic anharmonic couplingterm are zero. We require

0 = H(1) + i[S, H(0)

](200)

and obtain (see equation (16.1.3) of (Papousek and Aliev1982)

S = −1

6

∑,m,n

KmnΩ−1mn[2ωωmωnPPmPn

+ 3ωm(ω2 − ω2

m + ω2n)QQmQn] (201)

Ωmn = (ω + ωm + ωn)(−ω + ωm + ωn)

× (ω − ωm + ωn)(ω + ωm − ωn) (202)

(Ωmn is the global energy denominator that is generatedwhen all the additive terms with different energy denomi-nators are assembled into a single term) thus,

H = 1

2

∑k

ωk(P2k + Q

2k) + 1

24

∑k,,m,n

Kk,,m,nQkQQmQn

+∑α

BαΠ2α − 1

8

∑k,,m,n,r

KkmKknrΩ−1km

×[ωkωωm

(QnQr PPm + PPmQnQr + 2

3δmrδn

)

+ ωk(ω2 + ω2

m − ω2k)QQmQnQr

] . (203)


This is a remarkably compact and elegant result. It showshow H may be reduced to an Heff ≡ H harmonic plus quar-tic (KabcdQaQbQcQd ) form and how each of the Kabcd

quartic fit parameters is related to the microscopic quan-tities ωi, Kabc, Kabcd, and ζ α

ij , of which ωi andKabc, Kabcd may, in principle, be computed ab initiofrom the appropriate partial derivatives of V Q evalu-ated at Q = 0 and ζ α

ij calculated by vector analysis.Once Heff is expressed in terms of the effective quarticforce constants Kabcd, expressions defining all of theeffective ωi and xij parameters may be derived bynondegenerate perturbation theory. (See equations (8–12)from Lehmann 1991 and equations (43–49) from Mills1972.)

The first contact transformation reduces the vibra-tional Hamiltonian to a simple sum of individual modeharmonic oscillators and several pseudoquartic terms.These pseudoquartic terms combine true quartic terms

∂4V∂Qa∂Qb∂Qc∂Qd

∣∣∣0, second-order cubic terms, and vibra-

tional angular momentum terms. It is important to notesome qualitative patterns for the second-order cubic con-tributions to the pseudoquartic terms (Table 6) (seeequation (203)):

In the absence of resonances, only the Kaaaa, Kbbbb,and Kaabb terms contribute to the diagonal vibrationalconstants ωa , ωb, xaa , xab, xbb that determine the diagonalenergies

G (va, vb) = ωa (va + 1/2) + ωb (vb + 1/2)

+ xaa (va + 1/2)2 + xbb (vb + 1/2)2

+ xab (va + 1/2) (vb + 1/2) (204)

Table 6 Cubic anharmonic contributions to pseudoquartic forforce constant parameters.

Pseudoquartic term Cubic contributions

Kaaaa

∑k

KaakKaak

Kaaab

∑k

KaakKabk

Kaabb

∑k

KaakKbbk and∑

k

KabkKabk

Kaabc

∑k

KaakKkbc and∑

k

KabkKack

Kabcd

∑k

KabkKcdk,∑

k

KackKbdk

and∑

k

KadkKbck

where

xaa = 1

16Kaaaa − 1

16

∑k

Kaak

[8ω2

a − 3ω2k

ωk

(4ω2

a − ω2k

)] (205)

xab = 1

4Kaabb − 1

4

∑k

KaakKkbb

ωk

− 1

2

∑k

K2abkωk

(ω2

k − ω2a − ω2

b

)Ωabc

+[∑

α

Bα

(ζ

(α)a,b

)2][

ωa

ωb

+ ωb

ωa

](206)

Ωabc = (ωa + ωb + ωk)(ωa − ωb − ωk)

× (ωb − ωk − ωa)(ωk − ωa − ωb). (207)

Note the presence of a Coriolis contribution to xab butnot to xaa (Mills 1972). However, when one of the energydenominator factors in equations (205–207) is small, it isno longer possible to reduce the blocks along the maindiagonal of the vibrational Heff all the way down to1 × 1 form. It becomes necessary to exclude the specificcubic anharmonic terms with near-zero denominators fromthe second-order perturbation sums. When nondegenerateperturbation theory fails because of large mixing anglesamong a subset of basis states, the eigenvectors andenergies must be obtained by diagonalizing the block ofHeff that contains the quasi-degenerate basis states. Onemust set up a family of polyad matrices along the maindiagonal of Heff. Near degeneracies (resonances) amongthe classes of basis states connected by the pseudoquarticanharmonic terms (e.g., Kaabb, when ωa ≈ ωb) also requirepolyad treatment.

4.8 Effective Hamiltonians for Linear ABA andABBA Molecules

The problems associated with setting up the Heff for dou-bly degenerate bending vibrations and for describing thenormal-mode and local-mode limiting cases for moleculeswith two chemically identical bonds or benders are dis-cussed in this subsection.

Bending vibrations of linear molecules belong to doublydegenerate π irreducible representations. The usual choicefor basis functions is the two-dimensional isotropic har-monic oscillator. Matrix elements and selection rules in|N, 〉 and |nd, ng〉 forms of the two-dimensional isotropicharmonic oscillator basis set are discussed in Section 4.8.1.

Molecules with two chemically identical bonds (ABand BA) or benders (ABB and BBA) may be described


by either a normal-mode or a local-mode basis set.In the local-mode basis set, the power series expan-sion of V (Qleft, Qright) begins with a kRLQRQL term(L = left, R = right) and there is kinetic energy cou-pling, tRLPRPL, between the left and right local modes.(In section 4.8.2, the constants KRL and TRL are usedwith the dimensionless Q and P operators.) The rela-tionship between the parameters that define the Heff inthe normal- and local-mode basis sets are derived inSection 4.8.2.

4.8.1 Doubly Degenerate Bending Vibration

The Hamiltonian for a two-dimensional isotropic harmonicoscillator has the form (d = droite, g = gauche) (Cohen-Tannoudji et al. 1977, Jacobson et al. 1999b)

H/hc = ω[a†

dad + a†dag + 1

](208)

where a†d creates one quantum of vibration with one unit

of right circular angular momentum, and a†g creates one

quantum of vibration with one unit of left-circular angularmomentum,

a†d = 2−1/2 (a†

x + ia†y

), ad = 2−1/2 (ax − iay

)(209)

a†g = 2−1/2 (a†

x − ia†y

), ag = 2−1/2 (ax + iay

). (210)

The number operators are

Nd = a†dad (211)

Ng = a†gag (212)

and the vibrational angular momentum operator is

Lz =

(a†

dad − a†gag

)=

(Nd − Ng

)(213)

where Lz is equivalent to the difference between the numberoperators for right- and left-circular vibrational quanta.The only nonzero commutation rules among ad, a†

d, ag, a†g

are [ad, a†

d

]=[ag, a†

d

]= 1. (214)

The∣∣nd, ng

⟩basis functions are

∣∣nd, ng

⟩ = [(nd)!(ng)!]−1/2(a†

d)nd (a†

g)ng∣∣nd = 0, ng = 0

⟩(215)

or, expressed in terms of the total number of vibra-tional quanta, N = nd + ng , and the component of vibra-tional angular momentum along the molecular z-axis, = nd − ng ,

∣∣N = nd + ng, = nd − ng

⟩ = [(N+

2

)!

(N −

2

)!

]−1/2

× (a†

g

)N+2

(a†

g

)N−2 |0, 0〉 .

(216)

The |N, 〉 basis is widely used to describe the bendingvibrational levels of a linear molecule. For a given value ofN , there are N + 1 possible values of (changing in stepsof 2) over the range (nd + ng) ≥ ≥ −(nd + ng). In the|N, 〉 basis, the zero-order energies are

E(0)(N, )/hc = ω(N + 1) (217)

and the creation/annihilation operators have the expectedeffects on the

∣∣nd, ng

⟩basis functions,

ad

∣∣nd, ng

⟩ = n1/2d

∣∣nd − 1, ng

⟩(218)

a†d

∣∣nd, ng

⟩ = (nd + 1)1/2∣∣nd + 1, ng

⟩(219)

ag

∣∣nd, ng

⟩ = n1/2g

∣∣nd, ng − 1⟩

(220)

a†g

∣∣nd, ng

⟩ = (ng + 1)1/2∣∣nd, ng + 1

⟩. (221)

When the relationships between (nd, ng) and (N, ),

N = nd + ng (222)

= nd − ng (223)

nd = N +

2(224)

ng = N −

2(225)

are inserted into equations (218–221), we get the not-so-familiar effects of the creation/annihilation operators on the|N, 〉 basis functions

ad |N, 〉 =(

N +

2

)1/2

|N − 1, − 1〉 (226)

a†d |N, 〉 =

(N + + 2

2

)1/2

|N + 1, + 1〉 (227)

ag |N, 〉 =(

N −

2

)1/2

|N − 1, + 1〉 (228)

a†g |N, 〉 =

(N − + 2

2

)1/2

|N + 1, − 1〉 . (229)


We want to calculate matrix elements of the quadratic andquartic terms in the bending potential energy function

V (x, y)/hc = 1

2Kbend

(x2 + y2) + 1

4Kbb

(x2 + y2)2

(bb ≡ bend–bend) (230)

where

x = 2−1/2 (ax + a†x

) = 1

2

(ad + a†

d + ag + a†g

)(231)

y = 2−1/2 (ay + a†y

) = i

2

(ad − a†

d − ag + a†g

)(232)(

x2 + y2) = adag + a†da†

g + (N + 1) (233)(x2 + y2)2 = 3

2(N + 1)2 + 1

2− 2

2+ 2 (N + 1)

×(

adag + a†da†

g

)+ a2

da2g + a†2

d a†2g . (234)

Note that both (x2 + y2) and (x2 + y2)2 have ∆ = 0selection rules, but the quartic term partially lifts the -degeneracy of the −N ≤ ≤ N -components that belongto the same value of N . There is no bend–bend–bendcubic anharmonic term for a doubly degenerate bendingvibration, because such a term (πu ⊗ πu ⊗ πu = φu ⊕2πu) would not contain the totally symmetric irreduciblerepresentation (σ+

g ), and could only follow ∆ = ± 1, ± 3matrix element selection rules. The form for the quarticterm cannot be either x4 + y4 or x2y2 because both containterms of δg and γ g ( = 2 and 4) symmetry, unlike(x2 + y2)2, which is exclusively of σ+

g symmetry. Therelevant nonzero matrix elements of the terms involvingad, a†

d, ag, a†g are

⟨N, |adag|N + 2,

⟩ = 1

2[(N + + 2) (N − + 2)]1/2

(235)⟨N, |a†

da†g|N − 2,

⟩= 1

2[(N + ) (N − )]1/2 (236)

⟨N, |a2

da2g|N + 4,

⟩ = 1

2[(N + 4 + ) (N + 2 + )

× (N + 4 − ) (N + 2 − )]1/2

(237)⟨N, |a†2

d a†2g |N − 4,

⟩= 1

2[(N + − 2) (N + )

× (N − − 2) (N − )]1/2 .

(238)

These matrix elements are most readily derived using|nd, ng〉 rather than |N, 〉 labels for the basis functions.

Using these matrix elements it is possible to evaluatethe contributions of bend–stretch, bend–bend, and bend1 –bend2 anharmonic interactions to the molecular constants in

G(vi) =∑

s

ωs (vs + 1/2) +∑

t

ωt (vt + 1)

+∑s≥s′

xss′ (vs + 1/2) (vs′ + 1/2)

+∑s,t

xst (vs + 1/2) (vt + 1)

+∑t≥t ′

xtt ′ (vt + 1) (vt ′ + 1) +∑t≥t ′

gtt ′tt ′,

(239)

where t denotes a doubly degenerate (bending) mode ands denotes any singly degenerate mode. The relationshipsbetween derivatives of V (Q) evaluated at Q = 0 and xst ,xtt , xtt ′ , and gtt , given as equations (43–49) of Mills 1972,may be understood on the basis of the special treatmentrequired for doubly degenerate (bending) modes that hasbeen summarized in this subsection.

4.8.2 The transformation between Normal- andLocal-mode Basis Sets

The normal ↔ local-mode transformation has been dis-cussed by Lehmann (1983) for two identical, coupled one-dimensional anharmonic oscillators and by Jacobson et al.(1999b) for two identical coupled two-dimensional anhar-monic oscillators. The key pedagogical point is that, unlikethe normal-mode limit where there can be neither kineticenergy nor bilinear potential energy coupling between thenormal-mode basis states, there is both 1 : 1 kinetic energyand 1 : 1 potential energy coupling between the local-modebasis states.

The ABA molecule, treated in a local-mode basis set, isthe simplest example where it is necessary to deal explicitlywith kinetic energy coupling. Benign neglect of the kineticenergy term, based on experience with diatomic moleculeswhere the ∆v = ±2 off-diagonal matrix elements of theP

2and Q

2operators exactly cancel, is not justified. For

polyatomic molecules, particularly when the vibrationaldisplacements are not expressed as 3N Cartesian displace-ments of the N atoms, calculation of matrix elements of thekinetic energy operator is decidedly nontrivial (McCoy andSibert 1991b). Local bond stretches and curvilinear bend-ing displacements require careful analysis of kinetic energycoupling terms (McCoy and Sibert 1991a). The followingdiscussion of the stretching modes of an ABA molecule inthe normal- and local-mode limits illustrates the forms of


m3

m1

R R

f

m2

Figure 2 The mass and geometry dependence of the momentumcoupling term in the local-mode Hamiltonian.

HeffN and Heff

L and the transformation between the normal-and local-mode basis sets.

The simplest possible local-mode model for an ABAtriatomic molecule is expressed in terms of one local-mode harmonic frequency, ω0 = 1

2πc[k/µ]1/2, one 1 : 1

potential energy coupling term, kRL, and one 1 : 1 kineticenergy coupling term, which depends on known molec-ular geometry and atomic masses, and hence is notadjustable.

The classical mechanical local-mode HeffL is

HeffL = 1

2µ

[P 2

R + P 2L

] + k

2

[Q2

R + Q2L

]+ cos φ

m3PRPL + kRLQRQL. (240)

The masses and geometry are shown in Figure 2. Hereφ = π and cos φ = −1 for linear geometry and

µ = m1m3

m1 + m3= m2m3

m2 + m3(241)

µ

m3= m1

m1 + m3. (242)

HeffL may be rewritten as a quantum mechanical Heff

Lexpressed in terms of a†

R, aR , NR = a†RaR , a†

L, aL, andNL = a†

LaL,

HeffL = 1

2hcω0

[(2NR + 1) + (2NL + 1)

+(

kRL

k+ µ

m3cos φ

)(a†

RaL + aRa†L

)+(

kRL

k− µ

m3cos φ

)(a†

Ra†L + aRaL

)]. (243)

Note that the sum and difference of the potential andkinetic energy coupling terms appear, respectively, as thecoefficient of a “polyad-forming” (a†

RaL + aRa†L) term

and an out-of-polyad (a†Ra†

L + aRaL) term that results in

second-order corrections to the energy,

E(2)vRvL

/hc = −1

8ω0

[kRL

k− µ

m3cos φ

]2

(vR + vL + 1)

(244)

thus

HeffL /hc = ω′(NR + NL + 1) + 1

2ω0

(kRL

k+ µ

m3cos φ

)×(

a†RaL + aRa†

L

)(245)

where

ω′ = ω0

[1 − 1

8

(kRL

k− µ

m3cos φ

)2]

. (246)

This form of HeffL is overly simple because it depends on

only two freely adjustable parameters, k and kRL (µ, µ3and φ are determined by the atomic masses and moleculargeometry).

The stretching modes of an ABA triatomic moleculemay be represented by either a normal-mode or a local-mode Heff. We consider a mechanistically explicit, max-imally simple model, two local-mode Morse oscillators(ωM and xM) coupled by 1 : 1 kinetic energy and 1 : 1potential energy terms (KRLQRQL and TRLPRPL, whereKRL and TRL are reduced coupling constants, in cm−1

units).

Heff/hc = ωM [(NR + 1/2) + (NL + 1/2)

]+ xM [

(NR + 1/2)2 + (NL + 1/2)2]+ 1

2KRL

(aR + a†

R

) (aL + a†

L

)− 1

2TRL

(aR + a†

R

) (a†

L + aL

)(247)

= ωM [NR + NL + 1] + xM

2

[(NR + NL + 1)2

+ (NR − NL)2]+ 1

2(KRL + TRL)

(a†

RaL + aRa†L

)+ 1

2(KRL − TRL)

(aRaL + a†

Ra†L

). (248)

The (KRL + TRL) term is polyad forming in the local-mode basis set because (a†

RaL + aRa†L) couples the basis

states of systematically near-degenerate (vR, vL) and (vR ±1, vL ∓ 1) levels. The levels are near-degenerate becausethey belong to the same value of the polyad quantum


number, N = vR + vL. The (KRL − TRL) term results insecond-order corrections to the diagonal energies,

E(2)vRvL

/hc = 1

4(KRL − TRL)2

×[(vRvL)

2ωM− (vR + 1)(vL + 1)

2ωM

]= −1

8

(KRL − TRL)2

ωM[NR + NL + 1] . (249)

Thus the simplified form of Heff, expressed in the local-mode basis set, is

Heff/hc = ω′ [NR + NL + 1] + xM

2

[(NR + NL + 1)2

+ (NR − NL)2]+ 1

2(KRL + TRL)

(a†

RaL + aRa†L

)(250)

ω′ = ωM − (KRL − TRL)2

8ωM. (251)

Starting from this physical Heff model we derive relation-ships between the ωM, xM, KRL, and TRL parameters andthe values of the adjustable parameters in the two minimalfit models,

HeffN /hc = ωs (Ns + 1/2) + ωa (Na + 1/2)

+ xss (Ns + 1/2)2 + xaa (Na + 1/2)2

+ xsa (Ns + 1/2) (Na + 1/2)

+ (Kss,aa/4

) (a†2

s a2a + a2

s a†2a

)(252)

(s ≡ symmetric, a ≡ antisymmetric), where there are sixadjustable parametersωs, ωa, xss, xaa, xsa, Kss,aa, and

HeffL /hc = ω′ [NR + 1/2 + NL + 1/2

] + x[(NR + 1/2)2

+ (NL + 1/2)2]+ xRL (NR + 1/2) (NL + 1/2)

+ (HRL/hc)(

a†RaL + aRa†

L

)(253)

where there are 4 adjustable parameters ω′, x, xRL,

HRL.The Morse oscillator potential energy curve is V M(Q) =

De[1 − e−βQ]2. Its exact vibrational energy levels areGM(v) = ωM(N + 1/2) + xM(N + 1/2)2. The values ofωM and xM are expressed in terms of De and β inSection 4.6.1. The simplicity of the energy level expres-sion for the Morse oscillator combined with the qualita-tively reasonable form of V M(Q) justifies its incorporation

into models for polyatomic molecule stretching vibrations.However, despite the fact that we are treating Morse oscil-lator local modes, we are using harmonic oscillator a, a†

operators.By comparing terms in Heff and Heff

L , we find

ω′ = ωM − (KRL − TRL)2

8ωM(254)

x = xM (255)

xRL = 0 (256)

(HRL/hc) = 1

2(KRL + TRL). (257)

If we define a polyad quantum number N = NR + NL,

xM [(NR + 1/2)2 + (NL + 1/2)2] = xM

2

[(NR + NL + 1)2

+ (NR − NL)2] = xM

2

[(N + 1)2 + (NR − NL)2]

(258)

we see that xM

2 (NR − NL)2 is a polyad-breaking term thatattempts to counteract the polyad-forming 1

2 (KRL + TRL)

term.It is straightforward to transform Heff to the normal-mode

basis set by replacing the a†R , aR, a†

L, aL, NR , NL operatorsby

aR = 2−1/2(as + aa) (259)

aL = 2−1/2(as − aa) (260)

NR = 1

2

(a†

s + a†a

)(as + aa)

= 1

2

[Ns + Na + a†

s aa + asa†a

](261)

NL = 1

2

(a†

s − a†a

)(as − aa)

= 1

2

[Ns + Na − a†

s aa − asa†a

](262)

NR + NL = Ns + Na (263)

NR − NL = a†s aa + asa†

a (264)

a†RaL + aRa†

L = 1

2

[(a†

s + a†a

)(as − aa)

+ (as + aa)(a†

s − a†a

)]= 1

2[Ns − Na + (Ns + 1) − (Na + 1)]

= [Ns − Na] (265)


aRaL + a†Ra†

L = 1

2[(as + aa) (as − aa)

+ (a†

s + a†a

) (a†

s − a†a

)]= 1

2

[a2

s − a2a + a†2

s − a†2a

]. (266)

Inserting these equations into equation (250), we obtain

Heff/hc = ω′ [Ns + Na + 1] + xM

2

[(Ns + Na + 1)2

×(

a†22 a2

a + a2s a†2

a

)+ Ns (Na + 1)

+ (Ns + 1) Na]

+ 1

2(KRL + TRL) (Ns − Na) (267)

= ω′ [Ns + Na + 1]

+ xM

2

[4 (Ns + 1/2) (Na + 1/2)

+ (Ns + 1/2)2 + (Na + 1/2)2 − 1

2

]= ω′ [Ns + Na + 1]

+ xM

2

[4 (Ns + 1/2) (Na + 1/2)

+ (Ns + 1/2)2 + (Na + 1/2)2

− 1

2+ a†2

s a2a + a2

s a†2a

]+ 1

2(KRL + TRL) (Ns − Na) (268)

= (ω′ + λ

) (Ns + 1

2

)+ (

ω′ − λ) (

Na + 1

2

)+ xss (Ns + 1/2)2 + xaa (Na + 1/2)2

+ xsa (Ns + 1/2) (Na + 1/2)

+ (Kss,aa/4

) (a†2

s a2a + a2

s a†2a

)(269)

where

ωs = ω′ + λ (270)

ωa = ω′ − λ (271)

λ = 1

2(KRL + TRL) (272)

xss = xaa = 1

4xsa = xM/2 (273)

Kss,aa = 2xM. (274)

To identify the polyad-breaking and polyad-forming termsin Heff

L , we let the polyad quantum number be N = Ns + Na

and re-express the first two terms in HeffN /hc(

ω′ + λ)(Ns + 1/2) + (

ω′ − λ)(Na + 1/2)

= ω′ (N + 1) + 2λ (Ns − Na) (275)

thus (KRL + TRL)(Ns − Na) is polyad breaking andxM

2 (a†2s a2

a + a2s a†2

a ) is polyad forming. In HeffL ,

xM

2(NR − NL)2 = xM

2

(a†

s aa + asa†s

)2(276)

is polyad breaking, and

1

2(KRL + TRL)

(a†

RaL + aRa†L

)= 1

2(KRL + TRL)

× (Ns − Na) (277)

is polyad forming.It is significant that the polyad-forming and polyad-

breaking terms have exchanged roles in HeffN and Heff

L . Thisexchange of polyad-breaking and polyad-forming roles isfrequently encountered when the Heff is expressed in twoopposite-limit basis sets. The relative magnitudes of thetwo opposed parameters determine whether the energy levelpattern found in the spectrum resembles one or the other ofthe limiting cases.

We see, therefore, that making an appropriate matchbetween an Heff and a basis set is essential. The initialquality of the representation and the difficulty encounteredin expressing the Heff compactly depend on a close cor-respondence between the zero-order model and the actualbehavior of the system.

4.9 Failure is Guaranteed

Why must the anharmonically interacting harmonic oscil-lators Heff model eventually fail? The simple answer isthat, although a Morse oscillator is a much more realisticmodel than an anharmonic oscillator for a stretching mode,at high vibrational levels, where the outer turning point ofthe Morse oscillator is located at very large-R, extremelyhigh-v harmonic oscillator wavefunctions are required tocover the spatial region of the Morse oscillator outer turningpoint. For a Morse and a harmonic oscillator with identi-cal ωe, the outer turning point of the vth Morse vibrationallevel always lies vertically far below the outer turning pointof a (much higher-v) harmonic vibrational level, and theinner turning point of the vth Morse level lies verticallyabove the inner turning point of a (lower-v) harmonic level.These differences increase rapidly as the vibrational quan-tum number of the Morse oscillator increases. For a Morse


Q (Å)

Pot

entia

l (cm

−1 ×

103 )

1

(a) (b) (c) (d)

1.5 20

1

2

3

4

5

6

7

8

|⟨YHO|YMorse⟩|2 |⟨YHO|YMorse⟩|2 |⟨YHO|YMorse⟩|2

Har

mon

ic o

scill

ator

ene

rgy

(cm

−1 ×

103 )

0.45% De

0 0.5 10

2

4

6

8

10

12

14

1612.58% De

0 0.1 0.20

2

4

6

8

10

12

14

1625.48%De

0

2

4

6

8

10

12

14

16

0 0.05 0.1

Use of harmonic oscillator basis (YHO) to solve for morse oscillatorwavefunctions (YMorse)

Figure 3 It is a bad approximation to replace Morse oscillator wavefunctions by harmonic oscillator wavefunctions. (a) TheMorse anharmonic potential energy curve, the harmonic oscillator potential energy curve, and the v = 0, v = 14, and v = 30 Morsewavefunctions, located at Evib = 0.0045De, 0.1258De, and 0.2548De respectively. The dotted horizontal lines show the energies of theharmonic oscillator basis states. (b)–(d) The fractional characters of the harmonic oscillator wavefunctions (ωH

e = ωMe = 1000 cm−1)

in the vM = 0, 14, and 30 Morse wavefunctions. The energies on the harmonic potential at the inner and outer turning points of theMorse potential EH(QM

v− ) and EH(QMv+ ) are shown as horizontal lines on the vM = 0, 14, and 30 panels, respectively. The energy scale

is compressed in (b)–(d).

oscillator with De = 25 000 cm−1, ωe = 1000 cm−1, andωexe = 10 cm−1,

V M(Q) = hcDe[1 − e−βQ

]2(278)

and a harmonic oscillator with ωe = 1000 cm−1

V H(Q) = hcDe(βQ)2 (279)

the (noninteger-v) Morse vibrational “level” located atDe/2 = 12 500 cm−1 above the minimum of the potentialenergy curve has vM

1/2 = 14.14. The harmonic oscillatorvibrational levels that have the same inner or outer turningpoint (denoted by − and + subscripts, respectively) as

the vM1/2 = 14.14 level are vH

1/2− = 6.74 and vH1/2+ = 37.2.

Similarly, the Morse level at De/5 = 5000 cm−1 is vM1/5 =

4.78, which has the same inner and outer turning points asvH

1/5− = 2.94 and vH1/5+ = 8.26. This detuning of turning

points, especially rapid for the outer turning point wherean increasingly dominant fraction of the probability densityresides, ensures that the overlap integral between same-vMorse and harmonic wavefunctions, |〈vM|vH〉|, is smallerthan 1 and decreases rapidly as vM increases (See Figure 3).

The present discussion of harmonic and Morse oscil-lator models is restricted to polyatomic molecules. It isnever necessary to resort to such over-simple models for thevibration–rotation levels of diatomic molecules. The Ryd-berg–Klein–Rees (RKR) method (Le Roy 2004) provides


Ratio of morse diagonal and off -diagonal matrix elements

⟨vM a†viM ⟩

⟨vH a†vH − 1⟩

Quantum number v

Rat

io (

dim

ensi

onle

ss)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

viM = vM vi

M = vM − 2

viM = vM − 1

5 10 15 20 25 30 35 40

Figure 4 Comparison of ∆vM = +2, +1, and 0 a† matrix elements for Morse wavefunctions (ωMe = ωH

e = 1000 cm−1) to the onlynonzero harmonic oscillator matrix element,

⟨vH|a†|vH − 1

⟩ = (vH)1/2. Note that, up to 26% of De (vM = 31) only a 10% overestimateerror is made replacing

⟨vM|a†|vM − 1

⟩by (vM)1/2. However, the ∆vM = +2 and 0 matrix elements, which are zero for the harmonic

oscillator, become larger than 50% of the ∆vH = +1 value by vM = 14 for ∆vM = 0 and vM = 33 for ∆vM = +2.

accurate inversion of the experimentally observed vibra-tional [G(v)] and rotational [B(v)] constants to a potentialenergy curve, V (R). Accurate vibrational wavefunctions forone-dimensional potential curves are calculated by numer-ical integration of the vibrational Schrodinger equation.All vibrational integrals [overlap integrals

⟨v′|v′′⟩ and

expectation values of R-dependent quantities, e.g., Bv =〈v|B(R)|v〉] are evaluated numerically. The nonexistenceof an RKR procedure for polyatomic molecules makes itnecessary to exploit model potentials.

The use of harmonic oscillator rather than the morerealistic Morse oscillator vibrational-state functions as thebasis for vibrational scaling and selection rules must giveerroneous results. We are interested in matrix elements ofthe form

⟨vM|a†|vM − 1

⟩and

⟨vM|a†|vM

⟩and we naıvely

expect these matrix elements to have the harmonic oscil-lator values (vM)1/2 and 0, respectively. Exploiting thecompleteness of the Morse and harmonic oscillator basissets,

⟨vM|a†|vM − 1

⟩ = ∑vH,vH′

⟨vM|vH⟩ ⟨vH|a†|vH′⟩ ⟨

vH′|vM − 1⟩

=∑vH

⟨vM|vH⟩ (vH)1/2 ⟨vH − 1|vM − 1

⟩.

(280)

The Morse oscillator⟨vM|a†|vM − 1

⟩matrix element

becomes, as v increases, smaller than the value of the cor-responding harmonic oscillator

⟨vH|a†|vH − 1

⟩ = (vH)1/2

matrix element, as shown in Figure 4.In addition, the Morse oscillator matrix element⟨

vM|a†|vM⟩

(assumed, by analogy to the harmonic oscil-lator matrix element

⟨vH|a†|vH

⟩, to be zero) grows rapidly

even relative to⟨vH|a†|vH − 1

⟩, also shown in Figure 4.

Both the disagreement and the rate of divergence betweenthe harmonic and Morse oscillator basis sets increaserapidly with excitation energy. At some point, the errorsintroduced into the Heff constants become so large asto make the use of the anharmonically coupled har-monic oscillators model untenable. In the next section,we discuss ways to use the Heff in regions of phasespace where the harmonic oscillator is no longer a goodzero-order model. Canonical Van Vleck perturbation the-ory (CVPT) is a method that aims to retain the sim-plicity of harmonic oscillators by bypassing some ofthe difficulties mentioned earlier. A second, recent pro-posal by (Child et al. 2001) does away with the har-monic oscillator basis set by providing a semiclassicalmethod to easily obtain matrix elements for other zero-order models. We will refer to this second method hereas correspondence principle scaling (CPS) simply forconvenience.


5 IDEAS FOR IMPROVEMENTS

5.1 Canonical Van Vleck Perturbation Theory

CVPT transforms the Hexact into an Heff better suitedto treat highly excited vibrational states than the anhar-monically coupled harmonic oscillators Heff (McCoy andSibert 1991b). CVPT still makes exclusive use of har-monic oscillator raising and lowering operators; however,the method carries out multiple Van Vleck transformationsto obtain an Heff that deals explicitly with all of the nec-essary resonances, while avoiding divergent nondegenerateperturbation-theory terms. Two key points are that CVPTallows for any choice of coordinates (but often guides force-fully toward the most appropriate choice), and that thesolutions of the Heff derived by CVPT are expressed in amuch smaller basis set than the full dimensionality of Hexact.Despite the inherent simplicity of the harmonic oscillatoroperators, implementation of the CVPT procedure can bequite complicated owing to the high orders of perturbationtheory that spectroscopic accuracy requires for some sys-tems. Fortunately, a computer program VANVLK exists tocarry out all of the necessary transformations in operatorform (Sibert 1988).

An instructive overview of the utility and flexibility ofCVPT can be gained from considering the three formsof Heff that VANVLK is designed to construct. The first,almost degenerate perturbation theory, produces a block-diagonal Heff analogous to the polyad Heff. The advantagesof the CVPT Heff include both the identification of theimportant resonances directly from the Hexact and the auto-mated use of arbitrarily high orders of perturbation theorycompared to the second-order perturbation theory tradition-ally used. The second, the “Smax” method, uses a mixing-angle cutoff parameter, Smax, to control the terms includedin the S transformation matrix. Increasing Smax forces theHeff to take a more diagonal form, but at a rapidly increas-ing perturbation theory cost. The third CVPT Heff formuses a self-consistent basis, similar in spirit to the SCF(self-consistent field) approach used in electronic structurecalculations. Here the S matrix includes terms that consistof a product of number operators for all modes but one,multiplied by an off-diagonal operator in that single mode.Consequently, the basis functions of the CVPT SCF Heff

for a given mode contain implicitly the averaged effects ofall other modes, due to the number operators that accom-pany the off-diagonal operators of Heff in the transformationmatrix.

The broad array of options presented by CVPT isitself one of its advantages. By attempting to treat asystem using different approaches, it is possible to discernthe resonances that exist, and the ones that are mostimportant in determining the energies and dynamics. Since

CVPT is designed especially to deal with highly excitedvibrations, where the spectra and dynamics are likely to beextremely complex, this is the most welcome capability.

5.2 Correspondence Principle Scaling

The harmonic oscillator basis set is easy to use becauseof its simple matrix elements and strict selection rules foroperators expressed as integer powers of Q and P, but theanharmonically coupled harmonic oscillator model cannotaccurately represent many systems, especially at high exci-tation energy. Instead of attempting to correct or extendthe harmonic oscillator basis, a better option is to choosea completely new basis that is more physically reasonable(for instance, the Morse oscillator, which more accuratelydescribes a bond stretch). The physical unreasonablenessof the harmonic oscillator was discussed in Section 4.9.The matrix elements and selection rules in any nonhar-monic oscillator basis will be much more complicated thanthe matrix elements and selection rules in the harmonicoscillator basis, and, except in special cases, such as theMorse oscillator, the matrix elements cannot be evaluatedanalytically. However, this increased complexity may be anecessary price to pay for increased accuracy and abilityto describe highly excited states. In this section, a simplesemiclassical method, as outlined in Child et al. 2001, isdescribed, which can be used to obtain formulae for matrixelements in a basis set for any model potential energyfunction.

We introduce the semiclassical method via theWentzel–Kramers–Brillouin (WKB) approximation, whichgreatly simplifies the calculation of energy levels and wave-functions for an arbitrary potential. WKB theory assumesthat the potential, V (x), is varying slowly relative to thewavelength λ = h/p(x), where p(x) = [2µ(E − V (x))]1/2

is the classical mechanical momentum, and can be approx-imated as locally constant. The solutions for a constantpotential are simply plane waves with some amplitude A

and wavevector k. However, since the potential is not actu-ally constant, A and k vary with the position, and thewavefunction can be written in the form of a sine wave witha rapidly varying spatial frequency modulated by a slowlyvarying envelope function, A(x). Using a wavefunction ofthis form, the Schrodinger equation can be solved, with theresult

ψ(x) = C√p(x)

e±i∫

p(x)dx. (281)

Note that the probability density |ψ(x)|2 is inverselyproportional to the classical mechanical momentum. Thephysical justification for this result is that a particle withlarge momentum in a certain region between x and x + dx


will clearly spend a small amount of time in that region.Therefore, the probability of finding the particle in a regionwhere the momentum p(x) is large is expected to be smallMerzbacher (1998).

However, the WKB approximation is not valid near theturning points of the wavefunction, where the wavefunctionapproaches the classically forbidden E < V (x) region ofthe potential. Here, the momentum approaches zero andthe amplitude of the WKB wavefunction diverges. Tocreate a continuous wavefunction, the wavefunction in theclassical region is matched with the wavefunction in theclassically forbidden region. This matching leads to a setof boundary conditions and a quantization condition calledthe Bohr–Sommerfeld quantization rule, which provides anexpression for the integer quantum number n of a boundstate at energy E:

IE = (n + δ) = 1

2π

∮E

p(q)dq. (282)

(The value of δ depends on the dimension of the potential.For a one-dimensional potential, δ is 1/2.) The right-handside of equation (282) is the action at a certain energy,IE . The action is a classical mechanical quantity that isa constant of motion for a periodic orbit, and I and theconjugate variable angle, φ, are related to the momentumand position by a canonical transformation (Goldstein1980).

In quantum mechanics, wavefunctions are usually rep-resented as a function of one of the conjugate variables,position, q, and momentum, p. However, via the canonicaltransformation, the wavefunction can also be expressed interms of one of the conjugate variables, action, I , and angle,φ. This transformation is accomplished by first evaluatingequation (282) to obtain an expression for I in terms of E.To obtain an expression for the angle φ, I is substitutedfor E in equation (282); then the equation is differentiatedwith respect to I . The resulting integral is evaluated to giveφ. Details are given in (Child 1991).

From the WKB result, quantum mechanical matrix ele-ments can be evaluated by extending the semiclassicalrelationship between action and quantum number into aquantum mechanical “action operator”

I =(

−i∂

∂φ+ δ

) (283)

with eigenfunctions

ψn(φ) = 1√2π

einφ (284)

such that

Iψn(φ) = (n + δ) ψn(φ) = Inψn(φ). (285)

Now, matrix elements of any operator O that can beexpressed as a function of I and φ can be calculated asfollows:

〈n′|O|n〉 =∫ π

−π

ψ∗n′(φ)O(I , φ)ψn(φ)dφ

= 1

2π

∫ π

−π

O(I , φ)ei(n−n′)φdφ, (286)

where I is the average action, 12

[I (n) + I (n′)

]. Use of

the average action symmetrizes the resulting off-diagonalmatrix elements and ensures that O is Hermitian.

Equation (286) is part of the Heisenberg formulation ofthe correspondence principle (Heisenberg 1925):

〈n′|O|n〉 ≈ (n′|O|n) = 1

2π

∫ π

−π

O(I , φ)ei(n−n′)φdφ.

(287)Angle brackets are used to represent exact quantummechanical matrix elements, while the parentheses repre-sent semiclassical matrix elements or Fourier components,evaluated as Fourier integrals in action–angle space. Thequantity I is the average action, O is some operator, andO(I, φ) is that operator expressed as a function of actionand angle. The correspondence principle is most accuratewhen the wavefunctions more closely approximate classicalbehavior, that is, either when n and n′ are large or when(n − n′)/(n + n′) is small.

Use of the correspondence principle reduces the difficultproblem of calculating the exact matrix elements in somebasis set other than a harmonic oscillator basis into twosimpler parts: first, to transform the desired operator into(I, φ) variables, and second, to evaluate the Fourier inte-gral either analytically or numerically. The integrals can beevaluated analytically for the Morse oscillator and numer-ically for other model potentials [see Child et al. 2001]. Itis also useful to calculate the relationship between actionand scaled energy, in order to express the matrix elementsin terms of the scaled energy (often a more convenientquantity) rather than action.

The Morse oscillator provides a simple example. Exact,quantum mechanical matrix elements for this system can becalculated by evaluating integrals involving Laguerre poly-nomials (Sage 1978). For example, the exact expressionsfor the diagonal matrix elements of q are

〈n|q|n〉 = 1

alnγ + 1

a

n∑m=0

Cm

2[ψ(0)(m + 1) − ψ(0)(1)

]−ψ(0) (γ − n − m − 1)

(288)


where ψ(0)(z), the digamma function, is equal todlnΓ (z)/dz,

Cm = (γ − 2n − 1)n!Γ (γ − n − m − 1)

(n − m)!Γ (γ − n)(289)

and

γ = 2√

2µDe

β. (290)

De and β are constants of the Morse potential [seeequation (291)]. This exact expression is cumbersome, andits method of derivation is complicated. However, usingthe previously outlined semiclassical method, much simplerformulae for the matrix elements can be obtained. Afteran outline of the method, following Child et al. 2001, theresulting formulae for the matrix elements will be comparedto the exact results.

The Hamiltonian for the Morse oscillator is

H = 1

2µp2 + De

(1 − e−βq

)2(291)

The Bohr–Sommerfeld quantization expression [seeequation (282)] can be written as

I = 1

2π

∮p(q)dq

= 1

π

√De

∫ q+

q−

√2µ

[ε − (

1 − e−βq)2]dq = (n + δ)

(292)

where the scaled energy ε is defined as E/De. Afterobtaining this expression for I in terms of ε, the desiredoperator can be expressed in terms of an integral in φ withconstant ε.

To evaluate the Fourier integrals as in equation (292), the(q, p) variables must be transformed to (I, φ) variables.The transformation is as follows, with I replaced by ε forconvenience (see Child 1991 for derivation):

q = 1

βln

(1 + √

ε cos(φ)

1 − ε

)(293)

p = −√

2µDe

(√ε(1 − ε) sin(φ)

1 + √ε cos(φ)

). (294)

From these equations, the diagonal matrix elements of q

are given by

(n|q|n) = 1

βln

(1 + √

1 − ε

2(1 − ε)

). (295)

This expression is much simpler than, yet agrees with,the exact matrix elements as given by equation (288). Theformula for the off-diagonal matrix elements is also simple:

(n + j |q|n) = (−1)j+1 1

β

(1 − √

1 − ε√ε

)j

, j = 0. (296)

A similar approach can be used to solve for the matrixelements of any function of q and p by numerically evalu-ating the Fourier integrals. This is very useful for potentialsthat are not approximately modeled by a harmonic oscilla-tor—for instance, those with multiple minima or saddlepoints. For further examples of this method, (see Childet al. 2001), in which the matrix elements for “champagnebottle” and spherical pendulum potentials are numericallyevaluated.

This semiclassical approach to a problem provides analternative to using the harmonic oscillator basis set withmany correction terms. The utility of zeroth-order basisstates that more closely resemble the system can outweighthe convenience of the harmonic oscillator matrix elementscaling and selection rules.

6 UNCONVENTIONAL APPLICATIONSOF EFFECTIVE HAMILTONIANS

6.1 Classical Mechanical Heff from QuantumMechanical Heff

One of the most valuable benefits from a spectroscopi-cally determined Heff is also one of the most surprising.It is possible, following Heisenberg’s version of the cor-respondence principle (Heisenberg 1925), to replace thequantum mechanical Heff by a classical mechanical Heff

(Kellman 1995, Jacobson et al. 1999a). This Heff revealsthe structure of the phase space explored by the moleculeas a function of the rigorous (E, J , irreducible represen-tation of the molecular symmetry group) and approximate(polyad quantum numbers) constants of motion. Answersto important dynamical questions about the global struc-ture of phase space may be obtained from the Heff. Theseinclude, as a function of E and the conserved actions(treated as continuous rather than as discrete variables),the following: (i) What fraction of phase space is filledwith chaotic trajectories? (ii) How large are the phasespace regions that are filled with quasi-periodic trajec-tories surrounding the periodic trajectory to which theyare qualitatively related? and (iii) What are the loca-tions and natures of the bifurcations at which periodictrajectories of qualitatively new forms first appear? Theanswers to these questions could lead to rational use


of special states to gain spectroscopic access to barrier-proximal regions of a potential surface or to exert ratio-nal external control over large-amplitude intramolecularmotions.

The Heff derived from the Heff is significantly reduced indimension relative to the Hexact derived from Hexact becausethe existence of approximate constants of motion permitapproximate factorization of Hexact into Heff polyad blocks.An enormous number of nonzero off-diagonal elementsof Hexact are ignorable owing to the smallness of mixingangles. The Heff to Heff pathway both simplifies (dimensionreduction) and regularizes (elimination of “noise”, seeSection 2.5) the dynamics relative to what would begenerated by the full-dimensional classical mechanicalHexact.

The Heff is expressed as a sum of products of molecularconstants times number operators, Ni = a†

i ai , and prod-ucts of creation/annihilation operators, e.g., the (a†2

1 a23 +

a21a†2

3 ) Darling–Dennison resonance term. The Heisen-berg correspondence principle specifies a simple correspon-dence between quantum mechanical operators and classicalmechanical quantities:

Quantum Classical(Ni + 1

2

)= 1

2

(a†

i ai + aia†i

)Ii (297)

a†i I

1/2i eiφi (298)

ai I1/2i e−iφi . (299)

The conjugate variables in the classical mechanical Hamil-tonian, H are “action” (Ii) and “angle” (φi) in theaction–angle representation

∂H∂Ii

= dφi

dt= φi (300)

∂H∂φi

= −dI

dt= −I . (301)

If Ii is a constant of motion

Ii = 0 ⇒ ∂H∂φi

= 0 (302)

then φi does not appear in H and φi is linear in t ,

φi(t) = φi(0) + ∂H∂Ii

t. (303)

Suppose we have a linear, four-atom molecule such asacetylene in its S0 electronic state. There are seven vibra-tional quantum numbers (v1, v2, v3, v4, 4, v5, 5), wheremodes 4 and 5 are doubly degenerate bending vibrations.

The phase space associated with seven actions is 14-dimensional. It would be very difficult to examine trajec-tories in a 14-dimensional action–angle phase space. For-tunately, there are three approximate constants of motion,the polyad quantum numbers Nresonance, Nstretch, and total:

Nresonance = 5v1 + 3v2 + 5v3 + v4 + v5 (304)

Nstretch = v1 + v2 + v3 (305)

total = 4 + 5. (306)

In the special case when

Nstretch = 0 (a constant of motion) (307)

thus

Nresonance = v4 + v5 = Nbend, (308)

the state space associated with Nstretch = 0 consists oftwo doubly degenerate bending modes. But there are twoconstants of motion

Nbend = v4 + v5 → Ka (309)

total = 4 + 5 → Kb (310)

where Ka and Kb are the conserved actions. Thus atrajectory in the dynamically interesting part of the pure-bending phase space is described by the time dependenceof two nonconserved actions, Ja(t) and Jb(t), related tov4 − v5 and 4 − 5, and two conjugate angles, ψa(t)

and ψb(t) (following notation used by Jacobson et al.(1999a) and in Figure 5). Energy is also conserved, so thetrajectory lies on a three-dimensional energy shell in thefour-dimensional pure-bending phase space.

Poincare surfaces of section reveal the qualitative struc-tures (and the dependence of these structures on theconserved quantities, E,Nbend, and total) of the trajec-tories in the pure-bending phase space. The trajectoriesJa(t), ψa(t), Jb(t), ψb(t) are controlled by Heff(E,Ka,

Kb) (Heff is parametrically dependent on the constants ofmotion, Ka, Kb) via Hamilton’s equations of motion

∂Heff

∂Ja,b= ψa,b (311)

∂Heff

∂ψa,b= −Ja,b. (312)

A trajectory is launched at chosen t = 0 values of J 0b , ψ0

b,and at ψ0

a = 0 (superscript 0 denotes t = 0). Since thetrajectory lies on the three-dimensional energy shell, theinitial value of J 0

a is determined by the three linearly


0.2

0.1

0.0

(a)

(c) (d) (e)

(f) (g)

(i) (j)

(h)

(b)

−3 −2yb

yb

−1 0

Polyad [4,0], E = 2461 cm−1

Polyad [8,0], E = 5261 cm−1

1 2 3

5 420

−2−4

0

−5

2

0

−2

5

0

−5

−2 0

E = 13 860 cm−1

E = 14 660 cm−1 E = 15 060 cm−1 E = 15 460 cm−1

E = 14 060 cm−1 E = 14 160 cm−1

Local bend

2

yb yb

−2 0 2 0 2 4 6yb

0 2 4

Counter-rotation

6

yb

−2 0 2yb

−2 0 2

−3 −2yb

−1 0 1 2 3

J b

J bJ b J b

5

0

−5

J b

5

0

−5

J bJ b

J b

−0.1

−0.2

2

1

0

−1

−2

H1

H2

Line ofsight H–C C–H

Grey Color

Figure 5 (a), (b) Surfaces of section for the HCCH [Nb, l] = [4, 0] and [8, 0] polyads. The surface of section for the [4, 0] polyadshows that all of phase space is divided between cis-bend and trans-bend normal modes. The phase space structure for the [8, 0] polyadcontains large-scale chaos as well as at least two new qualitative behaviors. (c)–(j) Overview of the phase space and configurationspace dynamics associated with the HCCH [Nb = 22, l = 0] polyad. The plots (c)–(h) are surfaces of section for six energies withinthe polyad. Only simple structures are found near the (c) bottom (local bender) and (h) top (counter-rotator) of the polyad. Chaosdominates at energies near the middle of the polyad, but two classes of stable trajectories coexist with chaos. The plots (i), (j) showthe coordinate space motions of the left- and right-hand H-atoms that correspond to the two different periodic orbits in (f) (Jacobsonet al. 1999a; Jacobson and Field 2000a).

independent variables, J 0b , ψ0

b, ψ0a. The surface of section

is displayed on a planar cut through the three-dimensionalenergy shell. One is free to choose any location andorientation for this planar surface, but the Jb, ψb planelocated at ψa = 0 is convenient. Each time the trajectory

crosses through the dividing Jb, ψb plane in the dψa

dt> 0

direction, a point is plotted on the Jb, ψb plane. If thetrajectory is quasi-periodic, the points form a closed curveon the Jb, ψb plane. At the center of the region enclosedby the closed curve, at some set of values of the conservedquantities E,Ka, Kb, the closed curve shrinks to a singlepoint corresponding to the periodic orbit that controls thequalitative form of the set of surrounding quasi-periodicorbits. If the trajectory is chaotic, it generates a densearray of disconnected points on the surface of section. Theappearances of the quasi-periodic and chaotic regions onthe Poincare surface of section are strikingly different.

The surface of section can display a family of trajec-tories, each launched at a different initial value of thenonconserved J 0

a , ψ0a . The points on the Jb, ψb plane where

the different trajectories intersect the plane are color-codedaccording to the chosen initial value of J 0

a . In this way, tra-jectories spanning the full range of J 0

a values (constrainedby E,Ka,Kb) may be displayed on a single surface ofsection. There will be several nonintersecting curves cor-responding to each of the qualitatively distinct classes ofquasi-periodic trajectories that coexist at the chosen valuesof the conserved quantities, E,Ka,Kb.

The E,Ka, Kb dependence of the phase space structuremay be explored by varying each of the conserved quanti-ties in small increments (not limited by the integer values ofthe conserved actions or quantized energy levels) and com-puting a new surface of section at each set of E,Ka, Kbvalues. The regions associated with each class of quasi-periodic orbit may expand, contract, bifurcate, or vanish.One particularly interesting bifurcation is the replacementof normal-mode by local-mode trajectories. Surfaces ofsection reveal a rich structure of bifurcations as well as theincreasing dominance of chaos as E increases. Even whenthe phase space is dominated by chaos, a few quasi-periodic


regions remain. As long as the dJadψa area of a quasi-periodic region remains larger than Planck’s constant, h,there will be at least one regular, large-amplitude quantummechanical eigenstate embedded in a manifold of ergodiceigenstates.

Figure 5 shows an example of surfaces of section and theinsights they provide. The two left panels [(a), (b)] com-pare the phase space structures for the [Nbend, total] = [4, 0]and [8, 0] polyads. For the [4, 0] polyad all of phase spaceis divided between cis-bend and trans-bend normal-modequasi-periodic orbits. For the [8, 0] polyad, the areas ofthe cis-bend and trans-bend regions have decreased and atleast two qualitatively different classes of quasi-periodicorbits and some regions of chaos have emerged. The eightpanels on the right [(e)–(j)] show the phase space struc-ture associated with the [22, 0] polyad as a function ofE within the 1600 cm−1 energy range spanned by thepolyad eigenstates. The top and bottom energy regions ofthe polyad are dominated, respectively, by counter-rotatingand local-bending quasi-periodic orbits. The energy regionfrom 300 cm−1 above the bottom to 800 cm−1 below thetop of the polyad is dominated by chaos, but coexistingwith chaos are some regions of simple quasi-periodic orbitsthat are qualitatively distinct from the local benders andcounter-rotators.

Insights derived from the structure of polyad (Heff) phasespace direct attention to the existence of special classes oflarge-amplitude eigenstates of Hexact and suggest rationalstrategies for external control of intramolecular dynamics.It should be vastly more effective than brute force to designa control scheme around the selection and exploitation oflarge-amplitude eigenstates of a priori known qualitativeform.

6.2 Visualizations of Dynamics in State Space

The Heff model provides various ways to visualize the free-evolving dynamics of a molecule subsequent to a varietyof experimentally plausible short-pulse (“sudden”) initialexcitations. For a many-oscillator molecule with severalanharmonic interaction terms in the Heff, the dynamics con-tained in Ψ (t) are too complicated to understand withoutrecourse to one of many convenient reduced descriptions. Inthis section, we survey a variety of dynamics visualizationschemes that are based on knowledge of an experimentallydetermined Heff.

Two key questions must be answered: (i) What is the(idealized) nature of the initial preparation of the system?(ii) What are the various causes (e.g., a specific off-diagonalterm in Heff) and the qualitative patterns (the dominantpopulation flow pathways and flow rates in state space)of the dynamics?

We are going to use a simple bent ABA molecule withtwo off-diagonal coupling terms to illustrate the Heff-basedvisualizations of dynamics. The Heff is constructed with thediagonal energies described by

G(v1, v2, v3) = ω1(v1 + 1/2) + x11(v1 + 1/2)2

+ ω2(v2 + 1/2) + x22(v2 + 1/2)2

+ ω3(v3 + 1/2) + x33(v3 + 1/2)2

+ x13(v1 + 1/2)(v3 + 1/2) (313)

and off-diagonal elements described by⟨v1 + 2, v2, v3 − 2|Heff|v1, v2, v3

⟩= K11,33[(v1 + 2)(v1 + 1)(v3)(v3 − 1)]1/2 (314)⟨

v1 + 1, v2 − 2, v3|Heff|v1, v2, v3⟩

= K1,22[(v1 + 1)(v2)(v2 − 1)]1/2 (315)

where

ω1 = 950 cm−1

ω2 = 500 cm−1

ω3 = 1050 cm−1

x11 = −2 cm−1

x22 = 1 cm−1

x33 = −2 cm−1

x13 = −8 cm−1

K11,33 = −2 cm−1(Darling–Dennison)

K1,22 = 10 cm−1(Fermi)

The pair of anharmonic resonances in equations (314)and (315) destroys the three vibrational quantum numbers,v1, v2, v3, and replaces them by a polyad quantum number

N = 2v1 + v2 + 2v3, (316)

a rigorous symmetry label (either the A1 or B1 irreduciblerepresentation label for the C2v molecular symmetry group),and an energy rank label, τ , such that the highest energyeigenstate within each N,A1/B1 polyad is labeled τ = 1.We focus on the N = 20 polyad, where the A1 blockconsists of 36 basis states (all with even v3) and the B1

block consists of 30 basis states (odd v3).For dynamical qualities computed from Ψ (t) to evolve

in time, Ψ (0) must not be a single eigenstate of the Heff.


The concept of a “bright state” is extremely important here.Often, only one zero-order state is bright, owing to the one-electron form of an electronic transition moment operatorand to Franck–Condon or harmonic oscillator propensityrules for “sudden” transitions. It is frequently possible todesign a clever excitation scheme to select a particulardesired bright state.

Working in a direction opposite to the customary one, asingle basis state may be expanded as a linear combinationof eigenstates of the Heff,

Ψ i(0) = ψ(0)i =

∑j

aijψj (317)

where the set of N aij expansion coefficients is obtained

from the unitary transformation that diagonalizes the N ×N Heff

T†HeffT =

E1 0

E2. . .

0 EN

. (318)

The set of N aij that corresponds to the ith zero-order state

is the numbers that appear in the ith column of T†. Differentchoices of Ψ (0) result in different, but easily calculable,dynamics.

Suppose that one has chosen

Ψ i(0) = ψ(0)i (319)

as the initial preparation of the system. The dynamics of

Ψ i(t) =∑

j

T jie

−iEj t/ (320)

may be visualized in a variety of ways that reveal theprobability flow pathways (Where does the system start?Where does it go first? How fast? What is the driving forcefor a particular pathway?) as well as the terms in the Heff

responsible for each of the dynamical mechanisms.There are many convenient quantities for visualizing

dynamics in the state space ruled by an Heff (Jacobson andField 2000b) and (Lefebvre-Brion and Field 2004, pages692–700). These include

1. the number operator a†i ai = Ni ;

2. the autocorrelation function, 〈Ψ i(0)|Ψ i(t)〉, or thesurvival probability, |〈Ψ i(0)|Ψ i(t)〉|2;

3. the transfer probability,∣∣⟨Ψ i(0)|Ψ j(t)

⟩∣∣2, from theinitially prepared state i to some specified targetstate j ;

4. the expectation values of the resonance operator, e.g.,for a Darling–Dennison 2 : 2 anharmonic interaction,

HDD = K11,33

[a†2

1 a23 + a2

1a†23

](321)

5. the expectation values of the energy transfer rateoperator associated with the resonance operator inequation (321), e.g.

ODD = K11,33

[a†2

1 a23 − a2

1a†23

]. (322)

Once T† and the set of eigenenergies Eτ are known, the t-dependence of each dynamical quantity for any imaginableΨ (0) is easily calculated. Figure 6 illustrates many ofthese quantities for N = 20, A1 Ψ (0) = |0, 20, 0〉 [see alsoMarquardt and Quack 1991].

6.3 Key Patterns are Revealed inDouble-resonance Spectra

As the excitation energy increases, spectra becomeinexorably more congested and complex. The congestioncan be minimized by any of a wide variety of double-resonance schemes, but the complexity is dealt with throughthe informed use of Heff models. One tunable laser,the “PUMP”, by exciting a transition between relativelylow-energy states, selects one of a range of fully assignableintermediate states. A second tunable laser, the “PROBE”,is used to examine the energy levels in the target regionthat are accessible from the single, selected, and knownintermediate state.

This double-resonance scheme samples the rovibronicstates in the target region, exploiting both rigorousselection-rule restrictions on good quantum numbers (J ,parity, irreducible representation), and propensity rule-guided modifications of the relative intensities of groupsof related transitions.

The power of the combination of selection rules withdouble resonance to produce massively simplified andpartially assigned spectra is well understood. However,an Heff-based exploitation of propensity rules to uncoverdiagnostically significant patterns is neither understood norwidely exploited.

There are two kinds of diagnostic patterns: fine splittingsarising from tiny second-order constants (Λ-doubling, -doubling, spin fine structure, and hyperfine splittings)and large-scale intrapolyad relative intensity patterns. Theconcepts of “bright state” and “pluck” are needed to explainthe large-scale intrapolyad intensity patterns.

A polyad consists of a group of systematically near-degenerate anharmonically (and Coriolis) interacting zero-order states. Often, only one of these zero-order states is


Survival probability

0

0.5

1

Fermi energy

0

500

Darling−Dennison energy

Time (s × 10−12)

−5

0

5

Number operator (v2)

10

15

20


0

2

4


Time (s × 10−12)

0

0.05

0.1

Dynamical quantities for N = 20, Y0 = 0,20,0⟩

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5 0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

(a) (b)

(c) (d)

(e) (f)

Figure 6 Several dynamical quantities are plotted here for a three-mode system possessing both Fermi and Darling–Dennisonresonances, starting at t = 0 in the zero-order state |0, 20, 0〉. (a), (b) The two plots show the survival probability and the [closelyrelated, for this Ψ (0)] expectation value of the number operator for v2, 〈N2〉. (c)–(f) The four plots show the expectation values of thenumber operators for (d) v1 and (f) v3, and (c), (e) the expectation values of the corresponding resonance operator that moves energyinto each mode. Note that 2 〈N1〉 + 〈N2〉 + 2 〈N3〉 = 20 for all t , and that the expectation values of the number operator for v3 is 2orders of magnitude smaller than that for v1, since no resonance couples v2 and v3 directly. [Reproduced from Jacobson et al. (1999b). American Institute of Physics, 1999.]

bright because it is the terminus of a Franck–Condon-allowed transition from the chosen intermediate state. Aqualitative change in the pattern of intrapolyad relativeintensities occurs when a different (assignable) intermediatestate is chosen to select (via a Franck–Condon factor) adifferent bright state.

For example, in the acetylene S1←S0 electronic tran-sition, only the CC stretch and trans-bending vibrationsare Franck–Condon active. The polyad quantum num-bers for the S0 state (Nstretch, Nresonance, total) restrict thezero-order bright states that are accessed from any givenS1 intermediate vibrational pluck state. When any nν ′

2(CC stretch), mν′

3 (trans-bend), K ′a = 1 rovibrational level

(0, n, m, 0, 0, 0)1 of S1 is selected as the intermedi-ate state, only one zero-order vibrational level is brightin each [Nstretch, Nresonance, total = 0] S0 polyad: v′′

1 = 0,v′′

2 = Nstretch, v′′3 = 0, v′′

4 = Nresonance − 3Nstretch, ′′4 = 0,

v′′5 = 0, ′′

5 = 0. However, if the S1 vibrational pluckcontains v′

4 = 1 or v′6 = 1 in addition to v′

2 = n, v′3 =

m, then the bright states are (v′′1 , v′′

2 , v′′3 , v′′

4 , ′′4, v

′′5 , ′′

5) =(0, Nstretch, 0, Nresonance − 3Nstretch − 1, ±1, 1,∓1).

Since the Heff contains known values of all of theintrapolyad coupling terms, the relative intensities of tran-sitions into each eigenstate of the Heff for the [Nstretch,

Nresonance, total = 0] polyad are determined by the fractionalcharacter of the bright state(s) in each polyad eigenvector,

I (τ ) = | 〈τ |bright〉0 |2, (323)

where τ is the energy rank of the eigenstate within thepolyad and 〈τ |bright〉0 is the overlap between the 〈τ | eigen-bra and the |bright〉0 basis-ket. When there is more than onebright state in the polyad, there can be interference betweentransition amplitudes

I (τ ) =∣∣∣∣∣∑

i

ai

⟨τ |brighti

⟩0∣∣∣∣∣2

(324)

where ai is the excitation amplitude of the ith bright state.By choosing an S1 pluck state, one controls the intensity

pattern within a polyad. By selecting a different pluckstate, the intensity pattern changes in a way that permits


assignment of the dominant zero-order character of eacheigenstate, |τ 〉 (Jacobson and Field 2000b). Thus the Heff

guides the assignment of features observed in the spectrumor the design of an excitation scheme to externally controlthe intrapolyad dynamics.

The SEP spectrum of HCP Ishikawa et al. 1999 exem-plifies the use of sudden changes in molecular constants tosignal the approach to an isomerization barrier. The bendingvibrational levels |v, 〉 of a linear molecule form quasi-degenerate groups of v + 1 -levels, −v ≤ < v ( chang-ing in steps of 2). The most revealing levels are those with = 0, ±2 (even v) or = ±1 (odd v). The isomerization-sensitive parameters in the Heff are B, g, and γ , where g2

determines the spacings of same-v -components, and γ

expresses the parity splitting of = 0 levels. The g param-eter originates from the quartic anharmonic coupling term,kbend–bend(Q

2x + Q2

y)2 (see Section 4.8.1). The γ parameter

controls the strength of a second-order ∆ = 2 interaction

term,H2

ij

E(0)i

−E(0)j

, that arises from the off-diagonal Coriolis

matrix element⟨vbend, , vstretch|HROT|vbend − 1, ± 1, vstretch + 1

⟩∝ [J (J + 1) − ( ± 1)]1/2 . (325)

As the top of the HCP↔HPC barrier is approached, thelowest energy, lowest || member, |vstretch = 0, vbend =N〉0, of the 2 : 1 stretch–bend polyads, N = 2v1 + v2,decreases in energy rapidly relative to the other membersof the polyad. This decrease in the zero-order energy of|vstretch = 0, vbend = N〉0 occurs because the extreme bendzero-order state is most sensitive to the rapidly decreasingslope of the bending potential in the near-barrier region.Even though this |0, N〉0 state has fallen out of theenergy region spanned by other members of its polyad, itremains significantly admixed with the |1, N − 2〉0 state.The nominal |0, N〉 state (with its admixed |1, N − 2〉0

character) finds itself within the energy range of the N − 1polyad, which contains the |0, N − 1〉0 state. The characterof this |0, N − 1〉0 state is distributed throughout the N − 1polyad, but it is mostly localized near the low-energy edgeof the polyad. As the nominal |0, N〉 state falls deeperinto the N − 1 polyad, its admixed |1, N − 2〉0 characterinteracts, via the Qbend, Qstretch dependence of Hrot, withthe |0, N − 1〉0 character in the low-energy region of theN − 1 polyad. The strength of this ∆vstretch = −∆vbend =±1 interaction increases rapidly as the energy of the topof the barrier is approached. This interaction, expressed asa second-order correction to the N polyad Heff, results inrapid changes in both the effective rotational constant, Beff,and γ .

This sudden onset of changes in Beff and γ is an exampleof the rapid changes in resonance structure that must

always occur as a barrier region is approached (Jacobsonand Child 2001). The rapid evolution of the parametersin an Heff model, or the breakdown of the Heff matrixelement selection and scaling rules, will prove to be a usefulsignature of the barrier-proximal region of state space.Thus, even when the Heff model is dying, it whispers thename and weapon of its murderer.

Rapid evolution of electronic properties [e.g. hyperfinecoupling constants (Bechtel et al. 2008), electric dipolemoments measured by the Stark effect] also provides away to detect the special class of eigenstate that embodiesvery large-amplitude nuclear displacements from the equi-librium geometry. Recall the discussion in Section 6.1 ofhow the classical mechanical Heff (derived from the quan-tum mechanical Heff) is used to reveal the structure of phasespace. Often, quasi-periodic trajectories fill a minusculeregion of otherwise chaotic phase space. Electronic prop-erties serve as embedded reporters, distinguishing rare,large-amplitude eigenstates from vastly more numerous,profoundly mixed, ergodic eigenstates. It is possible that,guided by Heff models, large-amplitude eigenstates can belocated and used to map out chemically interesting regions(e.g., the minimum energy isomerization path) of a poten-tial energy surface in much the same way that vibrationalfundamentals are used to characterize the near-equilibriumregion.

6.4 Spaghetti Diagrams

The way we think about the structure and dynamics of thestates that appear in the spectrum is profoundly influencedby the basis set we use to express the Heff. When do thebasis state names cease to provide useful labels for theobserved eigenstates? Often Heff can be expressed in two(or more) different complete basis sets, such as normalmodes vs. local modes. The choice of basis set should makeno difference in the goodness of fit to the spectrum. If thebasis set is complete and the transformation from the infiniteHexact to the finite Heff is performed sensibly, the choice ofbasis cannot affect the quality of the Heff.

However, the choice of basis set can affect the quality ofcorrespondence between patterns in the spectrum and thezero-order energies. A poor choice of basis set might makeit more difficult to make initial correspondences betweenobserved and zero-order predicted energy levels, especiallywhen the resolution and intensity dynamic range of therecorded spectrum is insufficient to resolve and observe allof the energy levels.

The only universally good label in the absence ofrigorously good quantum numbers is eigenvalue rank, τ .An imperfect spectrum and a bad model ensure that errorswill be made in assigning eigenvalue rank to the subsetof observed levels. It is also possible that the number of


5800 5800

5600 5600

5400 5400

5200 5200

5000 5000

Inte

rnal

ene

rgy

(cm

−1)

Normal modebasis set(a)

(b)

Local modebasis setEigenstates

[8,0]g +

15 600 15 600

15 200 15 200

14 800 14 800

14 400 14 400

14 000 14 000

Inte

rnal

ene

rgy

(cm

−1)

[22,0]g +

Figure 7 Correlation diagrams for the normal- and local-mode basis sets for the (a) [8, 0]g+ and (b) [22, 0]g+ polyads. The lines in thediagram represent the state energies in the polyad in different limits. In the middle of the diagram are the eigenenergies, and at the farleft and right are the zero-order energies of the normal-mode and local-mode basis sets, respectively. At positions intermediate betweenthe eigenstate and basis state extremes, the energies are calculated by diagonalizing the effective Hamiltonian with the off-diagonalelements multiplied by a scaling factor between 0 (the unperturbed basis set limit) and 1 (the eigenstate limit). An eigenstate is likely tobe assignable in terms of quantum numbers associated with a given basis set if the line that passes from the eigenstate to a zero-orderbasis state does so with minimal deviations, or can be followed through avoided crossings. It is clear from this diagram that many moreeigenstates in the [22, 0]g+ polyad are assignable in the local-mode basis set than in the traditional normal-mode basis set, but that thenormal-mode basis set provides a better zero-order description of the eigenstates in the [8, 0]g+ polyad (Jacobson et al. 1999b).

independent, adjustable parameters required to fit all of theobserved levels to the measurement accuracy might dependon the choice of basis set.

Figure 7 is an example of a way to decide semi-globallyon the more appropriate basis set (Jacobson et al. 1999b).The eigenenergies of Heff for the pure-bend polyad(Nstretch = 0, Nresonance = Nbend, total = 0) levels of theacetylene S0 state are plotted in the middle of the figure. Atthe left and right sides, respectively, are the diagonal ener-gies in the normal- and local-mode basis sets. The linesconnecting the basis state energies at the left and right tothe eigenstate energies in the middle are computed by mul-tiplying all of the off-diagonal elements by λ and thenvarying λ continuously from 0 to 1, thereby uniformlyturning on all of the interactions between basis states. Sim-ilar diagrams are constructed at low and high energes. Thefigure shows that, at low energy, most of the connectingcurves between zero-order and eigenenergies are simple,almost straight lines with few signs of avoided crossings

for the normal-mode basis set but are riddled with avoidedcrossings for the local-mode basis set. At high energy, theopposite, mostly local-mode behavior is observed, but thereare energy regions within the polyad where the connectinglines on the normal-mode side of the diagram are moreregular than those on the local-mode side. This sort of“spaghetti diagram” gives a simple overview of the local-vs. normal-mode dominant characters of all of the polyadeigenstates.

7 SUMMARY AND CONCLUSIONS

The Heff is based on rigorous and automated proceduresfor truncating the Hexact. State space is partitioned into thelocal region, which is of current experimental or theoreticalinterest, and remote regions. This partitioning is basedon the smallness of mixing angles rather than couplingmatrix elements. One seeks an Heff, the eigenenergies of


which are equal to a subset of the eigenenergies of theHexact. However, the eigenvectors of the Heff and themolecular constants that define the Heff are not identicalto corresponding quantities from the Hexact. Caution isrecommended to avoid falling into the linguistic traps of“nominal states”, named after their dominant characterbasis state, and “perturbed” effective molecular constants,which are not identical to the ab initio computed pure(deperturbed) state properties with the same names.

The Hexact is truncated into the Heff by the VanVleck transformation or by a series of contact transforma-tions. These transformations combine ordinary Rayleigh–Schrodinger nondegenerate perturbation theory with quasi-degenerate perturbation theory, when necessitated by largemixing angles within a group of states (components of anS = 0 multiplet electronic state, an accidental near degen-eracy among group of states known as a perturbation, or asystematically near-degenerate group of vibrational levelsknown as a polyad ).

Replacement of an infinite Hexact by a finite Heff seemslike getting something for nothing. The implicit rather thanexplicit treatment of an infinite number of remote perturbersis a form of dynamical averaging analogous to a self-consistent field approximation. Filtering out an enormousnumber of very fast, very small-amplitude dynamical termsis justified if one is interested in a low-dimensional,mechanistic, nonstochastic representation of the early-timedynamics.

The electronic fine structure Heff is on firmer theoret-ical ground than the polyad Heff that describes anhar-monic interactions among a group of vibrational states.The reasons for this are simple. For electronic finestructure there are at most 2(2S + 1) components of anS = 0 multiplet electronic state and there are a finitenumber of electronic symmetries of remote perturbersthat can contribute to the Heff for an isolated multipletstate.

In contrast to the situation for electronic fine structure,anharmonic interactions among vibrational states are lim-ited in neither the number of possible interaction mech-anisms nor the number of systematically near-degenerateinteracting states. A harmonic oscillator product basis sethas traditionally been used to deal with anharmonic inter-actions. Even though each oscillator is formally treated asif it were anharmonic by the addition of anharmonicityconstants on the diagonal of the Heff, no anharmonicity cor-rection is made to the basis functions. The basis functionsare treated as harmonic oscillators to exploit exceptionallyconvenient harmonic oscillator matrix element selection andscaling rules. This anharmonically coupled harmonic oscil-lators model is fundamentally flawed. It must fail at highexcitation energy. This failure will invalidate the scalingof fit parameters from one polyad to the next. It will also

fail to describe the onset of non-negligible polyad-breakinginteractions between members of energetically overlappingpolyads.

The anharmonically interacting harmonic oscillator Heff

is of enormous value: for its convenience in constructingfit models, for the ease of describing the evolution betweennormal- and local-mode limiting behaviors, for insightfulquantum and classical mechanics based dynamics visualiza-tion and external control schemes, and for providing, in thespecific observable characteristics of its eventual failure ina conical-intersection or barrier-proximal region, informa-tion about the location and shape of a chemically interestingfeature of a potential energy surface. Two schemes, CVPTand CPS, offer extensions of the validity of the Heff modelwithout totally giving up the convenience of matrix elementselection and scaling rules, without committing the folly ofthrowing the baby out with the bathwater.

Molecules are, in effect, lower-dimensional entities thanimplied by the exact Hamiltonian. Even at high excita-tion, their spectra and unimolecular dynamics are far moreregular and localized than naıve “bag-of-atom” ergodicconsiderations would lead one to expect. The effectiveHamiltonian is a reduced-dimensional representation ofa full-dimensional reality. It is far more than a con-venient approximation awaiting replacement as enhancedcomputational capabilities become available. The effec-tive Hamiltonian takes us beyond molecular constants bytelling us how to visualize and understand the intramolec-ular dynamics that are encoded in frequency-domainspectra.

ACKNOWLEDGMENTS

Robert W. Field thanks the members of his researchgroup at MIT since 1974 for helping to formulate andclarify many of the ideas presented in this paper. He alsothanks the Department of Energy, the National ScienceFoundation, and the Air Force Office of Scientific Researchfor generous research grants in support of this curiosity-driven research. Annelise R. Beck acknowledges supportfrom the Barry M. Goldwater Scholarship Program. Thecontributions of Joshua H. Baraban and Samuel H. Lipoffto this material are based upon work supported by NationalScience Foundation Graduate Research Fellowships. JoshuaH. Baraban is also grateful for a David A. Johnson SummerGraduate Fellowship. Samuel H. Lipoff is also grateful fora Department of Homeland Security Graduate ResearchFellowship. We thank Adam Steeves and Tony Colombofor many valuable criticisms and insightful contributions,especially Figure 5. Peter Giunta prepared this manuscriptwith his usual artistry and exceptional attention to the finestdetails.


END NOTES

a.The mixing angle and the mixing coefficient are equivalentin the small-angle limit. The two terminologies seem tobe used interchangeably in the literature, but rigorously“mixing angle” = arctan “mixing coefficient”.b.At orders of perturbation theory beyond second order, thecontributions of different φ-block states to the α-block areentangled.c.This subsection contains a brief discussion of severaladvanced examples from the literature. The inexperiencedreader may wish to read subsection 3.5 first.

ABBREVIATIONS AND ACRONYMS

CVPT Canonical Van Vleck perturbation theoryCPS correspondence principle scalingIVR intramolecular vibrational redistributionRKR Rydberg–Klein–ReesSCF self-consistent fieldWKB Wentzel–Kramers–Brillouin

REFERENCES

Albert, S., Albert, K.K., Hollenstein, H., Tanner, C.M., andQuack, M. (2011) Fundamentals of rotation–vibration spectra,in Handbook of High-resolution Spectroscopy, Quack, M. andMerkt, F. (eds), John Wiley & Sons, Ltd., Chichester, UK.

Bechtel, H.A., Steeves, A.H., Wong, B.M., and Field, R.W.(2008) Evolution of chemical bonding during HCN HNCisomerization as revealed through nuclear quadrupole hyperfinestructure. Angewandte Chemie (International ed. in English),47, 2969–2972.

Breidung, J. and Thiel, W. (2011) Prediction of vibrational spec-tra from ab initio theory, in Handbook of High-resolution Spec-troscopy, Quack, M. and Merkt, F. (eds), John Wiley & Sons,Ltd., Chichester, UK.

Brown, J.M. and Carrington, A. (2003) Rotational Spectroscopyof Diatomic Molecules, Cambridge University Press, Cam-bridge, 29–33, 302–370.

Brown, J.M., Hougen, J.T., Huber, K.P., Johns, J.W.C.,Kopp, I., Lefebvre-Brion, H., Merer, A.J., Ramsay, D.A.,Rostas, J., and Zare, R.N. (1975) The labeling of parity doubletlevels in linear molecules. Journal of Molecular Spectroscopy,55, 500–503.

Brown, J.M. and Merer, A.J. (1979) Lambda-type doublingparameters for molecules in Π electronic states of triplet andhigher multiplicity. Journal of Molecular Spectroscopy, 74,488–494.

Bunker, P.R. and Jensen, P. (1998) Molecular Symmetry andSpectroscopy, NRC Research Press, Ottawa.

Carrington Jr., T. (2011) Using iterative methods to computevibrational spectra, in Handbook of High-resolution Spec-troscopy, Quack, M. and Merkt, F. (eds), John Wiley & Sons,Ltd., Chichester, UK.

Child, M.S. (1991) Semiclassical Mechanics with MolecularApplications, Oxford Press.

Child, M.S., Jacobson, M.P., and Cooper, C.D. (2001) Scalinglaws for strongly anharmonic vibrational matrix elements. Journalof Physical Chemistry A, 105, 10791–10799.

Cohen-Tannoudji, C., Diu, B., and Laloe, F. (1977) QuantumMechanics, Wiley Interscience, New York, pp. 727–741.

Dressler, K. (1969) The Lowest Valence and Rydberg States in theDipole-Allowed Absorption Spectrum of Nitrogen. A Survey ofTheir Interactions. Canadian Journal of Physics, 47, 547–561.

Dubal, H.-R. and Quack, M. (1984) Tridiagonal Fermi resonancestructure in the IR spectrum of the excited CH chromophore inCF3H. Journal of Chemical Physics, 81(9), 3779–3791.

Dunham, J.L. (1932) The energy levels of a rotating vibrator.Physical Review, 41, 721–731.

Field, R.W., Wicke, B.G., Simmons, J.D., and Tilford, S.G.(1972) Analysis of perturbations in the a3Π and A1Π statesof CO. Journal of Molecular Spectroscopy, 44, 383–399.

Goldstein, H. (1980) Classical Mechanics, Addison-Wesley, NewYork.

Heisenberg, W. (1925) Uber quantentheoretische Umdeutungkinematischer und mechanischer Beziehungen. Zeitschrift furPhysik, 33, 879–893.

Herman, M. (2007) The acetylene ground state saga. MolecularPhysics, 105, 2217–2241.

Herman, M. (2011) High-resolution infrared spectroscopy ofacetylene: theoretical background and research trends, in Hand-book of High-resolution Spectroscopy, Quack, M. and Merkt, F.(eds), John Wiley & Sons, Ltd., Chichester, UK.

Ishikawa, H., Field, R.W., Farantos, S.C., Joyeux, M.,Koput, J., Beck, C., and Schinke, R. (1999) HCP→CPH iso-merization: caught in the act! Annual Reviews of PhysicalChemistry, 50, 443–484.

Jacobson, M.P. and Child, M.S. (2001) Spectroscopic signaturesof bond-breaking internal rotation. II. rotation-vibration levelstructure and quantum monodromy in HCP. Journal of Chemi-cal Physics, 114, 262–275.

Jacobson, M. and Field, R.W. (2000a) Acetylene at the thresh-old of isomerization. Journal of Physical Chemistry, 104,3073–3086.

Jacobson, M.P. and Field, R.W. (2000b) Visualizing intramolec-ular vibrational redistribution: expectation values of resonanceoperators. Chemical Physics Letters, 320, 553–560.

Jacobson, M.P., Jung, C., Taylor, H.S., and Field, R.W.(1999a) State–by–state assignment of the bending spectrum ofacetylene at 15,000 cm−1: a case study of quantum–classicalcorrespondence. Journal of Chemical Physics, 111, 600–618.

Jacobson, M.P., Silbey, R.J., and Field, R.W. (1999b) Local modebehavior in the acetylene bending system. Journal of ChemicalPhysics, 110, 845–859.

Jungen, Ch. (2011) Elements of quantum defect theory, in Hand-book of High-resolution Spectroscopy, Quack, M. and Merkt, F.(eds), John Wiley & Sons, Ltd., Chichester, UK.


Kellman, M.E. (1990) Approximate constants of motion forvibrational spectra of many-oscillator systems with multi-ple anharmonic resonances. Journal of Chemical Physics, 93,6630–6635.

Kellman, M.E. (1995) Dynamical analysis of highly excited vibra-tional spectra: progress and prospects. In Molecular Dynamicsand Spectroscopy by Stimulated Emission Pumping, Dai, H.-L.and Field, R. (eds), World Scientific, pp. 943–997.

Kittrell, C., Abramson, E., Kinsey, J.L., McDonald, S., Reis-ner, D.E., Katayama, D., and Field, R.W. (1981) Selectivevibrational excitation by stimulated emission pumping. Journalof Chemical Physics, 75, 2056–2059.

Koppel, H., Cederbaum, L.S., and Mahapatra, S. (2011) Theoryof the Jahn–Teller effect, in Handbook of High-resolutionSpectroscopy, Quack, M. and Merkt, F. (eds), John Wiley &Sons, Ltd., Chichester, UK.

Le Roy, R.J. (2004) A Computer Program Implementing theFirst-Order RKR Method for Determining Diatomic MoleculePotential Energy Function . Chem. Phys. Res. Rep. CP-657R,University of Waterloo, http://leroy.uwaterloo.ca.

Lefebvre-Brion, H. and Field, R.W. (2004) The Spectra andDynamics of Diatomic Molecules, Elsevier.

Lehmann, K.K. (1983) On the relation of Child and Lawton’sharmonic coupled anharmonic-oscillator model and Darling-Dennison coupling. Journal of Chemical Physics, 79, 1098.

Lehmann, K.K. (1990) Where does overtone intensity come from?Journal of Chemical Physics, 93, 6140–6147.

Lehmann, K.K. (1991) Beyond the x−K relations: calculation of1−1 and 2−2 resonance constants with application to HCN andDCN. Molecular Physics, 66, 1129–1137.

Lewerenz, M. and Quack, M. (1988) Vibrational spectrum andpotential energy surface of the CH chromophore in CHD3.Journal of Chemical Physics, 88(9), 5408–5432.

Lewis, B.R., Dooley, P.M., England, J.P., Waring, K., Gibson,S.T., Baldwin, K.G.H., and Partridge, H. (1996) Observationof the second 3Πu valence state of O2. Physical Review A, 54,3923–3938.

Lowdin, P.O. (1951) Note on the quantum-mechanical perturba-tion theory. Journal of Chemical Physics, 19, 1396–1401.

Marquardt, R. and Quack, M. (1991) The wave packet motion andintramolecular vibrational redistribution in CHX3 moleculesunder infrared multiphoton excitation. Journal of ChemicalPhysics, 95(7), 4854–4876.

McCoy, A.B. and Sibert, E.L. (1991a) Perturbative calculations ofvibrational (J = 0) energy levels of linear molecules in normalcoordinate representations. Journal of Chemical Physics, 95,3476–3487.

McCoy, A.B. and Sibert, E.L. (1991b) Perturbative studies ofthe vibrations of polyatomic molecules using curvilinear coor-dinates. In Advances in Molecular Vibrations and CollisionDynamics, Bowman, J.M. and Ratner, M.A. (eds), JAI Press,pp. 255–282, Vol. 1A.

Merkt, F. and Quack, M. (2011) Molecular quantum mechanicsand molecular spectra, molecular symmetry, and interactionof matter with radiation, in Handbook of High-resolutionSpectroscopy, Quack, M. and Merkt, F. (eds), John Wiley &Sons, Ltd., Chichester, UK.

Merzbacher, E. (1998) Quantum Mechanics, John Wiley & Sons.

Mills, I.M. (1972) Vibration–rotation structure in asymmet-ric–and symmetric–top molecules. In Molecular SpectroscopyModern Research, Rao, K.N. and Matthews, C.W. (eds), Aca-demic, New York, pp. 115–140.

Mulliken, R.S. (1964) The Rydberg states of molecules. parts I-V.Journal of the American Chemical Society, 86, 3183–3197.

Nikitin, E.E. and Zare, R.N. (1994) Correlation diagrams forHund’s coupling cases in diatomic molecules with high rota-tional angular momentum. Molecular Physics, 82, 85–100.

Papousek, D. and Aliev, M.R. (1982) Molecular Vibrational-Rotational Spectra, Elsevier, Amsterdam.

Pratt, S.T. (2011) High-resolution valence-shell photoionization,in Handbook of High-resolution Spectroscopy, Quack, M. andMerkt, F. (eds), John Wiley & Sons, Ltd., Chichester, UK.

Ross, S.C. (1991) An MQDT Primer. AIP Conference Proceed-ings , New York, p. 225.

Rostas, F., Eidelsberg, M., Jolly, A., Lemaire, J.L., LeFloch, A.,and Rostas, J., (2000) Band oscillator strengths of the inter-system transitions of CO. Journal of Chemical Physics, 112,4591–4603.

Rytel, T. (1991) Analysis of the multiple perturbation arisingin the b3Σ+(v = 0) state of the CO molecule. Journal ofMolecular Spectroscopy, 145, 420–428.

Rytel, T. (1992) Analysis of the perturbed 2–2 band of the thirdpositive (b3Σ+− a3Πr ) system of CO. Journal of MolecularSpectroscopy, 151, 271–274.

Sage, M.L. (1978) Morse oscillator transition probabilities formolecular bond modes chemical physics. Chemical Physics, 35,375–380.

Schinke, R. (2011) Photodissociation dynamics of polyatomicmolecules: diffuse structures and nonadiabatic coupling, inHandbook of High-resolution Spectroscopy, Quack, M. andMerkt, F. (eds), John Wiley & Sons, Ltd., Chichester, UK.

Sibert, E.L. (1988) VANVLK: an algebraic manipulation programfor canonical Van Vleck perturbation theory. Computer PhysicsCommunications, 51, 149–160.

Stahel, D., Leoni, M., and Dressler, K. (1983) Nonadiabatic rep-resentations of the 1Σ+

u and 1Πu states of the N2 molecule.Journal of Chemical Physics, 79, 2541–2558.

Stohner, J. and Quack, M. (2011) Conventions, symbols, quan-tities, units and constants for high-resolution molecular spec-troscopy, in Handbook of High-resolution Spectroscopy,Quack, M. and Merkt, F. (eds), John Wiley & Sons, Ltd.,Chichester, UK.

Tew, D.P., Klopper, W., Bachorz, R.A., and Hattig, C. (2011) Abinitio theory for accurate spectroscopic constants and molecu-lar properties, in Handbook of High-resolution Spectroscopy,Quack, M. and Merkt, F. (eds), John Wiley & Sons, Ltd.,Chichester, UK.

Van Vleck, J.H. (1951) The coupling of angular momentumvectors in molecules. Reviews of Modern Physics, 23, 213–227.

Veseth, L. (1973) A direct approach for the reduction of diatomicspectra to molecular constants for the construction of RKRpotentials. Journal of Molecular Spectroscopy, 46, 37–66.

Watson, J.K.G. (1999) Rotation-electronic coupling in diatomicRydberg states. In The Role of Rydberg States in Spectroscopyand Photochemistry, Sandorfy, C. (ed), Kluwer, Netherlands,pp. 293–327.


Watson, J.K.G. (2005) Different forms of effective Hamiltonians.Molecular Physics, 103(24), 3283–3291.

Watson, J.K.G. (2011) Indeterminacies of fitting parameters inmolecular spectroscopy, in Handbook of High-resolution Spec-troscopy, Quack, M. and Merkt, F. (eds), John Wiley & Sons,Ltd., Chichester, UK.

Western, C.M. (2011) Introduction to modeling high-resolutionspectra, in Handbook of High-resolution Spectroscopy,Quack, M. and Merkt, F. (eds), John Wiley & Sons, Ltd.,Chichester, UK.

Wolniewicz, L. and Dressler, K. (1977) The EF and GK 1Σ+g

states of hydrogen: adiabatic calculation of vibronic statesin H2, HD, and D2. Journal of Molecular Spectroscopy, 67,416–439.

Worner, H.J. and Merkt, F. (2011) Fundamentals of electronicspectroscopy, in Handbook of High-resolution Spectroscopy,Quack, M. and Merkt, F. (eds), John Wiley & Sons, Ltd.,Chichester, UK.

Xiao, L. and Kellman, M.E. (1989) Unified semiclassical dynam-ics for molecular resonance spectra. Journal of ChemicalPhysics, 90, 6086–6098.

Zare, R.N., Schmeltekopf, A.L., Harrop, W.J., and Albritton,D.L. (1970) Modifications of the doublet energy formulae ofa diatomic molecule necessitated by the rotational stretching.Journal of Physics B, 3, 1677–1691.

Zare, R.N., Schmeltekopf, A.L., Harrop, W.J., and Albritton,D.L. (1973) A direct approach for the reduction of diatomicspectra to molecular constants for the construction of RKRpotentials. Journal of Molecular Spectroscopy, 46, 37–66.

RELATED ARTICLES

Albert et al. 2011: Fundamentals of Rotation–VibrationSpectra

Breidung and Thiel 2011: Prediction of Vibrational Spec-tra from Ab Initio TheoryCarrington 2011: Using Iterative Methods to ComputeVibrational SpectraHerman 2011: High-resolution Infrared Spectroscopyof Acetylene: Theoretical Background and ResearchTrendsJungen 2011: Elements of Quantum Defect TheoryKoppel et al. 2011: Theory of the Jahn–Teller EffectMerkt and Quack 2011: Molecular Quantum Mechan-ics and Molecular Spectra, Molecular Symmetry, andInteraction of Matter with RadiationPratt 2011: High-resolution Valence-shell Photoioniza-tionSchinke 2011: Photodissociation Dynamics of PolyatomicMolecules: Diffuse Structures and Nonadiabatic Cou-plingStohner and Quack 2011: Conventions, Symbols, Quanti-ties, Units and Constants for High-resolution MolecularSpectroscopyTew et al. 2011: Ab Initio Theory for Accurate Spectro-scopic Constants and Molecular PropertiesWatson 2011: Indeterminacies of Fitting Parameters inMolecular SpectroscopyWestern 2011: Introduction to Modeling High-resolutionSpectraWorner and Merkt 2011: Fundamentals of ElectronicSpectroscopy

Documents

Effective Hamiltonians for Electronic Fine Structure and ...rwf.mit.edu/group/papers/rwf339.pdf · Effective Hamiltonians for Electronic Fine Structure and Polyatomic Vibrations Robert