Notes on: Molecular Physics

2004

Prof. W. Ubachs

Vrije Universiteit Amsterdam

1. Introduction

1.1 TextbooksThere are a number a textbooks to be recommended for those who wish to study molecularspectroscopy; the best ones are:

1)The series of books by Gerhard HerzbergMolecular Spectra and Molecular StructureI. Spectra of Diatomic MoleculesII. Infrared and Raman Spectroscopy of Polyatomic MoleculesIII. Electronic Spectra of Polyatomic Molecules

2)Peter F. BernathSpectra of Atoms and Molecules

3)Philip R. Bunker and Per JensenMolecular Symmetry and Spectroscopy, 2nd edition

4)Hlne Lefebvre-Brion and Robert W. FieldPerturbations in the Spectra of Diatomic Molecules

A recent very good book is that of:5) John Brown and Alan CarringtonRotational Spectroscopy of Diatomic Molecules

1.2 Some general remarks on the spectra of moleculesMolecules are different from atoms:- Apart from electronic transitions, always associated with the spectra of atoms, also purelyvibrational or rotational transitions can occur. These transitions are related to radiation bymultipole moments, similar to the case of atoms. While in atoms a redistribution of the elec-tronic charge occurs in a molecule the transition can occur through a permanent dipole mo-ment related to the charges of the nuclei.- Superimposed on the spectral lines related to electronic transitions, there is always a rovi-brational structure, that makes the molecular spectra much richer. In the case of polyatomicsthree different moments of inertia give rise to rotational spectra, in diatomics only a single ro-tational component. Each molecule has 3n-6 vibrational degrees of freedom where n is thenumber of atoms.- Atoms can ionize and ionization continua are continuous quantum states that need to be con-sidered. In molecules, in addition, there are continuum states associated with the dissociationof the molecule. Bound states can couple, through some interaction, to the continua as a resultof which they (pre)-dissociate.

- 5 -

1.3 Some examples of Molecular SpectraThe first spectrum is that of iodine vapour. It shows resolved vibrational bands, recorded bythe classical photographic technique, in the so-called B30u+ - X1g+ system observed inabsorption; the light features signify intense absorption. The discrete lines are the resolvedvibrations in the excited state going over to the dissociative continuum at point C. Leftwardof point C the spectrum looks like a continuum but this is an effect of the poor resolution.This spectrum demonstrates that indeed absorption is possible (in this case strong) to thecontinuum quantum state.

Later the absorption spectrum was reinvestigated by Fourier-transform spectroscopy result-ing in the important iodine-atlas covering the range 500-800 nm. There is several lines ineach cm-1 interval and the numbers are well-documented and often used as a reference forwavelength calibration. Note that the resolution is determined by two effects: (1) Dopplerbroadening and (2) unresolved hyperfine structure. The figure shows only a small part of theiodine atlas of Gerstenkorn and Luc.

- 6 -

The hyperfine structure can be resolved when Doppler-free laser spectroscopic techniques areinvoked. The following spectrum is recorded with saturation spectroscopy. A single rotationalline of a certain band is shown to consist of 21 hyperfine components. These are related to theangular momentum of the two I=5/2 nuclei in the I2 molecule.

Usually molecular spectra appear as regular progressions of lines. In the vibrational bands ofdiatomic molecules the rotational lines are in first order at equal separation. If a quantum stateis perturbed that may be clearly visible in the spectrum. This is demonstrated in the spectra oftwo bands of the SiO molecule in the H1+- X1+ system. The upper spectrum pertains to the(0,0) band and is unperturbed; the lower one of the (1,0) band clearly shown perturbation of therotational structure.

-3000 -2000 -1000Frequency, MHz

R17(16-1)*

- 7 -

2. Energy levels in molecules; the quantum structure2.1. The Born-Oppenheimer approximationThe Hamiltonian for a system of nuclei and electrons can be written as:

where the summation i refers to the electrons and A to the nuclei. The first term on the rightcorresponds to the kinetic energy of the electrons, the second term to the kinetic energy ofthe nuclei and the third term to the Coulomb energy, due to the electrostatic attraction andrepulsion between the electrons and nuclei. The potential energy term is equal to:

The negative terms represent attraction, while the positive terms represent Coulomb-repul-sion. Note that a treatment with this Hamiltonian gives a non-relativistic description of themolecule, in which also all spin-effects have been ignored.

Now assume that the wave function of the entire molecular system is separable and canbe written as:

where el represents the electronic wave function and nuc the wave function of the nuclearmotion. In this description it is assumed that the electronic wave function can be calculatedfor a particular nuclear distance R. Then:

The Born-Oppenheimer approximation now entails that the derivative of the electronic wavefunction with respect to the nuclear coordinates is small, so is negligibly small. Inwords this means that the nuclei can be considered stationary, and the electrons adapt theirpositions instantaneously to the potential field of the nuclei. The justification for this origi-nates in the fact that the mass of the electrons is several thousand times smaller than the massof the nuclei. Indeed the BO-approximation is the least appropriate for the light H2-mole-cule.

If we insert the separable wave function in the wave equation:

then it follows:

H "2

2m------- i2

i "

2

2M A-----------A

2

A V R r( , )+=

V R r( , ) Z Ae2

4pi0rAi-------------------

A i, Z AZBe

2

4pi0 RA RB-----------------------------------

A B> e

2

4pi0rij-----------------

i j>+ +=

mol ri RA( , ) el ri R;( )nuc R( )=

i2elnuc nuci

2el=

A2 elnuc elA

2 nuc 2 Ael( ) Anuc( ) nucA2 el+ +=

Ael

H E=

- 8 -

The wave equation for the electronic part can be written separately and solved:

for each value of R. The resulting electronic energy can then be inserted in the wave equationdescribing the nuclear motion:

We have now in a certain sense two separate problems related to two wave equations. The firstrelates to the electronic part, where the goal is to find the electronic wave functionand an energy . This energy is related to the electronic structure of the molecule analo-gously to that of atoms. Note that here we deal with an (infinite) series of energy levels, a groundstate and excited states, dependent on the configurations of all electrons. By searching the eigenvalues of the electronic wave equation for each value of R we find a function for the electronicenergy, rather than a single value.

Solution of the nuclear part then gives the eigen functions and eigen energies:

In the BO-approximation the nuclei are treated as being infinitely heavy. As a consequence thepossible isotopic species (HCl and DCl) have the same potential in the BO-picture. Also all cou-plings between electronic and rotational motion is neglected (e.g. -doubling).

2.2. Potential energy curves

The electrostatic repulsion between the positively charged nuclei:

is a function of the internuclear distance(s) just as the electronic energy. These two terms canbe taken together in a single function representing the potential energy of the nuclear motion:

Hmol nuc"

2

2m------- i2

i e

2

4pi0rij-----------------

i j> Z Ae

2

4pi0rAi-------------------

A i,+

el +=

elZ AZBe

2

4pi0 RA RB-----------------------------------

A B> "

2

2M A-----------A

2

A

nuc+ Etotalmol=

"2

2m------- i2

i e

2

4pi0rij-----------------

i j> Z Ae

2

4pi0rAi-------------------

A i,+

el ri R;( ) Eel R( )el ri R;( )=

"2

2M A-----------A

2

A Z AZBe

2

4pi0 RA RB-----------------------------------

A B>+

nuc R( ) Eel R( )nuc R( )+ Etotalnuc R( )=

el ri R;( )Eel R( )

nuc R( )

Enuc Etotal Eel R( ) Evib Erot+= =

V N R( )Z AZBe

2

4pi0 RA RB-----------------------------------

A B>=

V R( ) V nuc R( ) Eel R( )+=

- 9 -

In the case of a diatom the vector-character can be removed; there is only a single internu-clear distance between two atomic nuclei.

In the figure below a few potential energy curves are displayed, for ground and excitedstates. Note that:

- at small internuclear separation the energy is always large, due to thee dominant role ofthe nuclear repulsion

- it is not always so that de electronic ground state corresponds to a bound state- electronically excited states can be bound.

Electronic transitions can take place, just as in the atom, if the electronic configuration in themolecule changes. In that case there is a transition form one potential energy curve in themolecule to another potential energy curve. Such a transition is accompanied by absorptionor emission of radiation; it does not make a difference whether or not the state is bound. Thebinding (chemical binding) refers to the motion of the nuclei.

2.3. Rotational motion in a diatomic moleculeStaring point is de wave equation for the nuclear motion in de Born-Oppenheimer approxi-mation:

where, just as in the case of the hydrogen atom the problem is transferred to one of a reducedmass. Note that represents now the reduced mass of the nuclear motion:

Before searching for solutions it is interesting to consider the similarity between this waveequation and that of the hydrogen atom. If a 1/R potential is inserted then the solutions (ei-genvalues and eigenfunctions) of the hydrogen atom would follow. Only the wave function

has a different meaning: it represents the motion of the nuclei in a diatomic mole-

RR

VV

"2

2------R V R( )+ nuc R( ) Enuc R( )=

M AMB

M A MB+-----------------------=

nuc R( )

- 10 -

cule. In general we do not know the precise form of the potential function V(R) and also it isnot infinitely deep as in the hydrogen atom.

Analogously to the treatment of the hydrogen atom we can proceed by writing the Laplacianin spherical coordinates:

Now a vector-operator N can be defined with the properties of an angular momentum:

The Laplacian can then be written as:

The Hamiltonian can then be reduced to:

Because this potential is only a function of internuclear separation R, the only operator with an-gular dependence is the angular momentum N2, analogously to L2 in the hydrogen atom. Theangular dependent part can again be separated and we know the solutions:

The eigenfunctions for the separated angular part are thus represented by the well-known spher-ical harmonics:

and the wave function for the molecular Hamiltonian:

Inserting this function gives us an equation for the radial part:

R

1R2------

R R2 R

1R2 sin----------------- sin

1R2 sin----------------- 22

+ +=

N x i" y z

zx

i" sin

cot cos

+ = =N y i" z x

x

z

i" cos( ) cot sin

+ = =

N z i" x y y

x

i"

= =

R

1R2------

R R2 R

N2

"2R2

------------=

"2

2R2-------------

R R2 R

12R2-------------N2

V R( )+ + nuc R( ) Enuc R( )=

N2

N M,| "2N N 1+( ) N M,| = with N 0 1 2 3 etc, , , ,=N z N M,| "M N M,| = with M N N 1 N,+,=

N M,| Y NM ,( )=

nuc R( ) R( )Y NM ,( )=

- 11 -

Now the wave equation has no partial derivatives, only one variable R is left.

2.4. The rigid rotor

Now assume that the molecule consists of two atoms rigidly connected to each other. Thatmeans that the internuclear separation remains constant, e.g. at a value Re. Since the zeropoint of a potential energy can be arbitrarily chosen we choose V(Re)=0. The wave equationreduces to:

The eigenvalues follow immediately:

where B is defined as the rotational constant. Hence a ladder of rotational energy levels ap-pears in a diatom. Note that the separation between the levels is not constant, but increaseswith the rotational quantum number N.For an HCl molecule the internuclear separation is Re=0.129 nm; this follows from the anal-ysis of energy levels. Deduce that the rotational constants 10.34 cm-1.

This analysis gives also the isotopic scaling for the rotational levels of an isotope:

2.5 The elastic rotor; centrifugal distortion

In an elastic rotor R is no longer constant but increases with increasing amount of rotationas a result of centrifugal forces. This effect is known as centrifugal distortion. An estimateof this effect can be obtained from a simple classical picture. As the molecule stretches the

"2

2R2-------------

Rdd R2 Rd

d N N 1+( ) V R( )+ + R( ) EvN R( )=

12Re

2---------------N2

nuc Re( ) Erotnuc Re( )=

EN"

2

2Re2---------------N N 1+( ) BN N 1+( )= =

N

B 1---

- 12 -

centrifugal force Fc is, at some new equilibrium distance Re, balanced by the elastic bindingforce Fe, which is harmonic. The centripetal and elastic forces are:

By equating Fc=Fe and by assuming it follows:

The expression for the rotational energy including the centrifugal effect is obtained from:

Now use Re for the above equations and expanding the first term of the energy expression itfollows:

The quantum mechanical Hamiltonian is obtained by replacing N by the quantum mechanicaloperator . It is clear that the spherical harmonics YNM() are also solutions of that Hamilto-nian. the result for the rotational energy can be expressed as:

where:

is the centrifugal distortion constant. This constant is quite small, e.g. 5.32 x 10-4 cm-1 in HCl,but its effect can be quite large for high rotational angular momentum states (N4 dependence).Selection rules for the elastic rotor are the same as for the rigid rotor (see later).

2.6. Vibrational motion in a non-rotating diatomic moleculeIf we set the angular momentum N equal to 0 in the Schrdinger equation for the radial part andintroduce a function Q(R) with than a somewhat simpler expression results:

This equation cannot be solved straightforwardly because the exact shape of the potential V(R)is not known. For bound states of a molecule the potential function can be approximated with aquadratic function. Particularly near the bottom of the potential well that approximation is valid(see figure).

Fc 2Re=

N2

Re3-------------- Fe k Re Re( )=

Re Re

Re ReN2

kRe3-------------=

E N2

2Re2-----------------

12--- Re Re( )+=

E N2

2Re2------------- 1

N2

kRe4------------+

2 12---

N4

2kRe6---------------+

N2

2Re2-------------

N4

22kRe6------------------ +=

N

EN BN N 1+( ) DN2 N 1+( )2=

D4Be

3

e2---------=

R( ) Q R( ) R="

2

2------ R22

dd

V R( )+ Q R( ) EvibQ R( )=

- 13 -

Near the minimum R=Re a Taylor-expansion can be made, where we use = R - Re:

and:

Here again the zero for the potential energy can be chosen at Re. The first derivative is 0 atthe minimum and k is the spring constant of the vibrational motion. The wave equation re-duces to the known problem of the 1-dimensional quantum mechanical harmonic oscillator:

The solutions for the eigenfunctions are known:

where Hv are the Hermite polynomials; de energy eigenvalues are:

with the quantum number v that runs over values v=0,1,2,3.

From this we learn that the vibrational levels in a molecule are equidistant and that there isa contribution form a zero point vibration. The averaged internuclear distance can be calcu-lated for each vibrational quantum state with . These expectation values are plottedin the figure. Note that at high vibrational quantum numbers the largest density is at the clas-sical turning points of the oscillator.

Re

V(R)

V R( ) V Re( ) RddV

Re 12--- R2

2

dd V

Re

2+ + +=

V Re( ) 0= RddV

Re0=

R2

2

dd V

Re

k=

"2

2------ 22

dd

12---k

2+ Q ( ) EvibQ ( )=

Qv ( )2 v 2 1 4

v!pi1 4------------------------

12---

2exp Hv ( )= with

e"

----------= ek---=

Evib "e v12---+ =

Qv ( ) 2

- 14 -

The isotopic scaling for the vibrational constant is

Note also that the zero point vibrational energy is different for the isotopes.

2.7. Anharmonicity in the vibrational motion

The anharmonic vibrator can be represented with a potential function:

On the basis of energies and wave functions of the harmonic oscillator, that can be used as afirst approximation, quantum mechanical perturbation theory can be applied to find energy lev-els for the anharmonic oscillator (with parameters k and k):

In the usual spectroscopic practice an expansion is written (in cm-1),

with e, exe, eye and eze to be considered as spectroscopic constants, that can be deter-mined from experiment.Note that for the anharmonic oscillator the separation between vibrational levels is no longerconstant. In the figure below the potential and the vibrational levels for the H2-molecule areshown.

e1

-------

V ( ) 12---k2 k'3 k''4+ +=

Evib "e v12---+

154------

k'2

"e----------

"e----------

3v

2v

1130------+ + O k''( )+=

G v( ) e v 12---+ exe v12---+

2 eye v

12---+

3eze v

12---+

4+ + +=

- 15 -

H2 has 14 bound vibrational levels. The shaded area above the dissociation limit contains acontinuum of states. The molecule can occupy this continuum state! For D2 there are 17bound vibrational states.

A potential energy function that often resembles the shape of bound electronic state poten-tials is the Morse Potential defined as:

where the three parameters can be adjusted to the true potential for a certain molecule. Onecan verify that this potential is not so good at . By solving the Schrdinger equationwith this potential one can derive the spectroscopic constants:

The energies of the rovibrational levels then follow via the equation:

Another procedure that is often used for representing the rovibrational energy levels withina certain electronic state of a molecule is that of Dunham, first proposed in 1932:

In this procedure the parameters Ykl are fit to the experimentally determined energy levels;the parameters are to be considered a mathematical representation, rather than constants with

V R( ) De 1 ea R Re( )

[ ]2=

r

ea

2pi------2De

---------= xe"e4De----------= e 6

xeBe3

e------------- 6

Be2

e------= Be

"2

2Re2-------------=

EvN e v12---+ xee v

12---+

2 BeN N 1+( ) DeN2 N 1+( )2 e v 12---+ N N 1+( )+ +=

EvN Y kl v12---+

kN l N 1+( )l

k l,=

- 16 -

a physical meaning. nevertheless a relation can be established between the Ykl and the molecularparameters Be, De, etc. In approximation it holds:

2.8. Energy levels in a diatomic molecule: electronic, vibrational and rotationalIn a molecule there are electronic energy levels, just as in an atom, determined by the configu-ration of orbitals. Superimposed on that electronic structure there exists a structure of vibration-al and rotational levels as depicted in the figure.

Transitions between levels can occur, e.g. via electric dipole transitions, accompanied by ab-sorption or emission of photons. Just as in the case of atoms there exist selection rules that de-termine which transitions are allowed.

2.9. The RKR-procedureThe question is if there exists a procedure to derive a potential energy curve form the measure-ments on the energy levels for a certain electronic state. Such a procedure, which is the inverseof a Schrdinger equation does exist and is called the RKR-procedure, after Rydberg, Klein andRees.

Y 10 e Y 01 Be Y 20 exe Y 11 e Y 30 eye

- 17 -

3. Transitions between quantum states

3.1. Radiative transitions in moleculesIn a simple picture a molecule acts in the same way upon incident electromagnetic radiationas an atom. The multipole components of the electromagnetic field interacts with the chargedistribution in the system. Again the most prominent effect is the electric dipole transition.In a molecule with transitions in the infrared and even far-infrared the electric dipole approx-imation is even more valid, since it depends on the inequality. The wavelength of the ra-diation is much longer than the size of the molecule d:

In the dipole approximation a dipole moment interacts with the electric field vector:

In a quantum mechanical description radiative transitions are treated with a "transition mo-ment" Mif defined as:

This matrix element is related to the strength of a transition through the Einstein coefficientfor absorption is:

Very generally the Wigner-Eckart theorem can be used to make some predictions on al-lowed transitions and selection rules. The dipole operator is an -vector, so a tensor of rank1. If the wave functions have somehow a dependence on a radial part and an angular part thetheorem shows how to separate these parts:

In the description the tensor of rank 1 q can take the values 0, -1 and +1; this correspondswith x, y, and z directions of the vector. In all cases the Wigner-3j symbol has a value une-qual to 0, if J=0, -1 and +1. This is a general selection rule following if J is an angular mo-mentum:

The rule M=0 only holds for q=0, so if the polarisation is along the projection of the fieldaxis.

2pi------d 1

H int E er E= =

M fi f E i =

B fi ( )pie

2

30"2-------------- f E i

2=

r

JM rq1( ) 'J'M' 1( )J M J 1 J'

M q M'J r 1( ) 'J' =

J J' J 1 0 1, ,= =J J' 0= = forbidden

M M' M 1 0 1, ,= =

- 18 -

3.1. Two kinds of dipole moments: atoms and moleculesIn atoms there is no dipole moment. Nevertheless radiative transitions can occur via a transitiondipole moment; this can be understood as a reorientation or relocation of electrons in the systemas a result of a radiative transition. Molecules are different; they can have a permanent dipolemoment as well. The dipole moment can be written as:

Where e and N refer to the electrons and the nuclei. In fact dipole moments can also be createdby the motion of the nuclei, particularly through the vibrational motion, giving rise to:

where the first term is the electronic transition dipole, similar to the one in atoms, the second isthe permanent or rotating dipole moment and the third is the vibrating dipole moment.

3.2. The Franck-Condon principleHere we investigate if there is a selection rule for vibrational quantum numbers in electronictransitions in a diatom. If we neglect rotation the wave function can be written as:

The transition matrix element for an electronic dipole transition between states and is:

Note that on the left side within the integral there appears a complex conjugated function. Thedipole moment contains an electronic part and a nuclear part (see above). Insertion yields:

If two different electronic states el and el are concerned then the second term cancels, be-cause electronic states are orthogonal. Note: it is the second term that gives rise to pure vibra-tional transitions (also pure rotational transitions) within an electronic state of the molecule.Here we are interested in electronic transitions. We write the electronic transition moment:

In first approximation this can be considered independent of internuclear distance R. This is theFranck-Condon approximation, or the Franck-Condon principle. As a result the transition ma-

e N+ erii

eZ ARAA+= =

0 Rdd Re

12--- R2

2

dd

2+ + +=

mol ri RA( , ) el ri R;( )vib R( )=

if ''' d=

if 'el'vib e N+( )''el''vib r Rdd= ='ele''el rd( )'vib''vib Rd= 'el''el r 'vibN''vibd Rd+

Me R( ) 'ele''el rd=

- 19 -

trix element of an electronic transition is then:

The intensity of a transition is proportional to the square of the transition matrix element,hence:

So the Franck-Condon principle gives usa selection rule for vibrational quantum numbers inelectronic transitions. The intensity is equal to the overlap integral of the vibrational wavefunction of ground and excited states. This overlap integral is called the Frank-Condon fac-tor. It is not a strict selection rule forbidding transitions!

3.3. Vibrational transitions: infrared spectraIn the analysis of FC-factors the second term in the expression for the dipole matrix elementwas not further considered. This term:

reduces, in case of a single electronic state (the first integral equals 1 because of orthogonal-ity) it can be written as:

where the first term represents the permanent dipole moment of the molecule. In higher or-der approximation in a vibrating molecule induced dipole moments play a role, but these aregenerally weaker.

if Me R( ) 'vib''vib Rd=

I if2

v' v'' | 2

if 'el''el r 'vibN''vibd Rd=

v' |vib v''| v' | a b2+ +( ) v''| =

- 20 -

An important consequence is that in a homonuclear molecule there exists no dipole moment,, so there is no vibrational or infrared spectrum!

If we proceed with the approximation of a harmonic oscillator then we can use the knownwave functions Qv() to calculate intensities in transitions between states with quantum num-bers vk and vn:

form which a selection rule follows for purely vibrational transitions:

In case of an anharmonic oscillator, or in case of an induced dipole moment so-called overtonetransitions occur. Then:

These overtone transitions are generally weaker by a factor of 100 than the fundamental infraredbands.

Note that vibrational transitions are not transitions involving a simple change of vibrationalquantum number. In vibrational transitions the selection rules for the rotational or angular partmust be satisfied (see below).

3.4. Rotational transitionsInduced by the permanent dipole moment radiative transitions can occur for which the electron-ic as well as the vibrational quantum numbers are not affected. The transition moment for a tran-sition between states and can be written as:

where the states represent wave functions:

The projection of the dipole moment onto the electric field vector (the quantization axis) can bewritten in vector form (in spherical coordinates) in the space-fixed coordinate frame:

The fact that the vector can be expressed in terms of a simple spherical harmonic function Y1m

vib 0=

n | k| Qn ( )Qk ( ) d "------- n2---k n 1, n 1+2------------k n 1+,+= =

v v' v 1= =

v 1 2 etc, ,=

NM| NM| M fi NM E NM =

NM| NM ( ) eR ( )Y NM ( )= =

0 cossin sinsin

cos0Y 1m=

- 21 -

allows for a simple calculation of the transition moment integral:

This gives only a non-zero result if:

So rotational transitions have to obey these selection rules. The same holds for the vibration-al transitions.

3.5. Rotation spectraThe energy expression for rotational energy levels, including centrifugal distortion, is:

Here we adopt the usual convention that ground state levels are denoted with N" and excitedstate levels with N. The subscript refers to the vibrational quantum number of the state.Then we can express rotational transition between ground and excited states as:

Assume N=N"+1 for absorption:

If the centrifugal absorption is neglected and an equally spaced sequence of lines is found:

The centrifugal distortion causes the slight deviation from equally separated lines.

Note that in a pure rotation spectrum there are only absorbing transitions for which N=N-N"= 1, so in the R-branch (see below).

M fi 0 Y NM cossin sinsin

cosY NM d

Y NMY 1mY NM d0pi02pi=

14pi------ 2N 1+( )3 2N 1+( )

N 1 N0 0 0

N 1 NM m M

=

N N N 1= =M M M 0 1,= =

Fv BvN N 1+( ) DvN2 N 1+( )2=

F N( ) F N( )=BvN N 1+( ) DvN2 N 1+( )2 BvN N 1+( ) DvN2 N 1+( )2[ ]( )=

abs Bv N 1+( ) N 2+( ) N N 1+( )[ ] Dv N 1+( )2 N 2+( )2 N2 N 1+( )2[ ]=2B N 1+( ) 4D N 1+( )3=

abs N( ) abs N 1( ) 2B=

- 22 -

3.6. Rovibrational spectra

Now the term values, or the energies, are defined as:

For transitions one finds the transition energies:

Here is the so-called band origin, the rotationless transition. Note thatthere is no line at this origin. So:

Now the different branches of a transition can be defined. The R-branch relates to transition forwhich N=1. Note that this definition means that the rotational quantum number of the excitedstate is always higher by 1 quantum, irrespective of the fact that the transition can relate to ab-sorption or emission. With neglect of the centrifugal distortion one finds the transitions in theR-branch:

Similarly transitions in the P-branch, defined as N=-1 transitions, can be calculated, again withneglect of centrifugal distortion:

Now the spacing between the lines is roughly 2B; more precisely:

where the statement on the right holds if Bv < Bv". Hence the spacing in the P-branch is largerin the usual case that the rotational constant in the ground state is larger. There is a pile up oflines in the R-branch that can eventually lead to the formation of a bandhead, i.e. the pointwhere a reversal occurs.An energy level diagram for rovibrational transitions is shown in the following figure. Wherethe spacing between lines is 2B the spacing between the R(0) and P(1) lines is 4B. Hence thereis a band gap at the origin.

T G ( ) F N( )+=F N( ) BvN N 1+( ) DvN2 N 1+( )2=

G ( ) e 12---+ exe 12---+

2=

( ) F N( ) F N( ) G ( ) G ( )+=

0 G ( ) G ( )=

( ) 0 F N( ) F N( )+=

R 0 B N 1+( ) N 2+( ) BN N 1+( )+=0 2B 3B B( )N B B( )N2+ + +=

P 0 B B+( )N B B( )N2+=

R N 1+( ) R N( ) 3B B so 2B

3.7. Rovibronic spectraIf there are two different electronic states involved rovibronic transitions can occur, i.e. tran-sitions where the electronic configuration, the vibrational as well as the rotational quantumnumbers change. Transitions between a lower electronic state A and a higher excited stateB as in the following scheme can take place:

Possible transitions between the lower and excited state have to obey the selection rules, in-cluding the Franck-Condon principle. Transitions can be calculated:

0123

0123

0

R0R1

R2

P1P2

P3

"

'

R-branch P-branch

N"

N

TA

TB

Tterm valueof excited state

T"term valueof lower state

- 24 -

Again R and P branches can be defined in the same way as for vibrational transitions with tran-sition energies:

But now the constants have a slightly different meaning: 0 is the band origin including theelectronic and vibrational energies, and the rotational constants Bv and Bv" pertain to electron-ically excited and lower states. If now we substitute:

Then we obtain an equation that is fulfilled by the lines in the R branch as well as in the P-branch:

This is a quadratic function in m; if we assume that B < B", as is usually the case, then:

a parabola results that represents the energy representations of R and P branches. Such a parab-ola is called a Fortrat diagram or a Fortrat Parabola. The figure shows one for a single rovi-bronic band in the CN radical at 388.3 nm.

T T=T T B G v( ) F N( )+=

T T A G v( ) F N( )+=

R 0 2B 3B B( )N B B( )N2+ + +=P 0 B B+( )N B B( )N2+=

m N 1+= for R branchm N= for P branch

0 B B+( )m B B( )m2+ +=

0 m m2+=

- 25 -

Note that there is no line for m=0; this implies that again there is a band gap. From such fig-ures we can deduce that there always is a bandhead formation, either in the R-branch or inthe P-branch. In the case of CN in the spectrum above the bandhead forms in the P-branch.The bandhead can easily be calculated, assuming that it is in the P-branch:

It follows that the bandhead is formed at:

3.8. Population distributionIf line intensities in bands are to be calculated the population distribution over quantumstates has to be accounted for. From statistical thermodynamics a partition function followsfor population of states at certain energies under the condition of thermodynamic equilibri-um. In case of Maxwell-Boltzmann statistics the probability P(v) of finding a molecule inquantum state with vibrational quantum number v is:

When filling in the vibrational energy it follows:

where N is the Zustandssumme, and kT is expressed in cm-1. As often in statistical physics(ergodic theorem) P(v) can be interpreted as a probability or a distribution. As an exampleP(v) is plotted as a function of v in the following figure.

At each temperature the ratio of molecules in the first excited state over those in the ground

NddP B B+( ) 2N B B( )+ 0= =

N B B+B B------------------=

P v( ) eE v( )( ) kT( )

eE v( )( ) kT( )

v-------------------------------------=

P v( ) 1N----e

e v12---+

kT---------------------------exe v

32---+

kT-----------------------------+

=

- 26 -

state can be calculated. P(v=1)/P(v=0) is listed in the Table for several molecules for 300K and1000K.

In case of the distribution over rotational states the degeneracy of the rotational states needs tobe considered. Every state has (2J+1) substates . Hence the partition function becomes:

In the figure the rotational population distribution of the HCl molecule is plotted. Note that itdoes not peak at J=0. The peak value is temperature dependent and can be found through setting:

J| JM|

P J( ) 2J 1+( )eErot kT( )

2J 1+( )e Erot kT( )J-----------------------------------------------------

1N rot---------- 2J 1+( )e BJ J 1+( ) DJ

2 J 1+( )2+= =

Jdd P J( ) 0=

Herzberg fig 59

- 27 -

4. High vibrational levels in the WKB-approximation

4.1. The Wentzel Kramers Brillouin approximationThe Wentzel Kramers Brillouin (WKB) approximation is a tool to solve the one-dimensionalSchrdinger equation. Both, wave functions and energy levels can be determined for a given po-tential. Before computers were in common use, this approximation belonged to the standard topicsin basic quantum mechanic courses, but after the 1960s this approximation received less and lessattention. Recently, however, the WKB approximation gained interest again, due to developmentsin the field of cold atoms. The WKB is wielded to determine the binding energy of the upper vibra-tional levels in diatomic molecules which is important for the value of the scattering length for s-wave collisions; an important parameter for experiments with cold atoms.In molecular physics this approximation can aid in the investigation of vibrational levels close toa dissociation limit and can give tunneling rate constants in the case of, for instance, auto-ionizationor pre-dissociation.Besides this, the WKB approximation is widely applicable for a lot of quantum mechanical prob-lems, provided a potential curve is known, and can help to interpret physical phenomena in somedetail. Therefore, this approximation is presented in this chapter.

This and the following three paragraphs are based on the book: Quantum Mechanics, by E. Merz-bacher (1964).

The derivation starts with the one-dimensional Schrdinger equation. The variable is chosen to beR rather than x, as this is the symbol for the internuclear distance in diatomic molecules.

with as the reduced mass.In the case that V = constant, the equation is easily solved and the solutions are

with

In the following only k will be used, whether E > V or E < V. In the latter case a purely positiveimaginary number is assumed for k.If V = V(R) the solutions can be written in the form

The problem is now to obtain the function u(R). Substitute this wave function and also

d2dR2----------

2"

2------ E V[ ]+ 0=

e ikR= for E V>

e R= for E V