Journal of Molecular Spectroscopy 216, 15–23 (2002)doi:10.1006/jmsp.2002.8675

Nonadiabatic Predissociation Studies of the (1-3)3Πg System of Al2by Means of a Complex Rotated Finite Element Method

Stefan Andersson and Nils Elander

Stockholm Center for Physics, Astronomy, and Biotechnology, Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden

E-mail: stan@physto.se

Received December 4, 2001; in revised form July 16, 2002

A general numerical Runge–Kutta–Fehlberg based diabatization procedure for electronic states in diatomics was applied tothe adiabatic (1-3)3�g system of Al2 in order to obtain a strictly diabatic basis. Using an exterior complex rotated finite elementmethod, adiabatic Born–Oppenheimer (BO) as well as diabatic rovibronic term energy values and predissociation widths for the(2)3�g; (v, N ) = (0 − 50, 0 − 25) and (3)3�g; (v, N ) = (0 − 17, 0 − 25) levels were computed. Comparing rotationless BO anddiabatic energies, differences between 10 and 25 cm−1 are found for the (2)3�g levels while the (3)3�g levels display an almostconstant shift ∼12 cm−1. From the widths, the nonradiative lifetime for each rovibronic level was calculated. Based on existingrotationless radiative lifetimes, an estimation of an upper limit of about 50 ns was used to determine a number of rovibronic(2, 3)3�g levels which may be experimentally observed. C© 2002 Elsevier Science (USA)

I. INTRODUCTION

Predissociation mechanisms play a very important role in thedestruction and creation of molecules since they in many caseshave much larger cross sections than the corresponding directmechanisms. It is therefore of great interest to study these pro-cesses in detail, experimentally as well as to model them the-oretically. In diatomics one frequently has two or more closelying, interacting potential energy curves (PECs), of the samespin and symmetry, correlating with different dissociation lim-its. According to the von Neumann–Wigner noncrossing rule(1), the adiabatic PECs are not allowed to cross each other.The Born–Oppenheimer (BO) separation of the nuclear andelectronic wavefunctions is not appropriate in the region ofclosely avoided intersections, where nonadiabatic effects arelikely to appear. Physically, nonadiabatic couplings betweenstates, of which at least one belongs to the continuous spec-trum, are of particular interest because there is now the possi-bility of radiationless transitions from the vibronically boundstate to the repulsive state, resulting in predissociation of themolecule.

Predissociative levels may be viewed as resonances, char-acterized by complex eigenvalues, Eres = E − i�/2, having fi-nite lifetimes, τ , in correspondence with their widths, � = 1/τ .A well known approach when studying nonadiabatic effects isto form a rigorous, or approximately, rigorous diabatic basisthrough an orthogonal transformation from the adiabatic repre-sentation. This often simplifies numerical treatments, especiallyin the case of strongly avoided crossings.

The Al2 dimer has been the subject of considerable interestthe past 10 years, both theoretically (2–7) and experimentally

15

(8–12). Among several spectroscopic problems in Al2, one ofthe most interesting is the predissociation mechanism in the(1-3) 3�g manifold. This three-state system, illustrated in Fig. 1(13, 14), consists of one repulsive ((1)3�g) and two attractive((2, 3)3�g) electronic states. Each pair of these states is coupledto each other through nonadiabatic derivative couplings (13).This implies a possibility that predissociation processes mayoccur.

Theoretical studies of nonadiabatic effects in the (1-3) 3�g

system of Al2 have been carried out by Han et. al. (13), usingmulticonfiguration self-consistent field interaction wavefunc-tions, and by Han and Yarkony (14) by means of a nonper-turbative computational approach. Vibrational levels as well asradiative and nonradiative lifetimes, for a number of (2, 3) 3�g

vibrational levels, were reported in these studies. The conclusionwas that the nonradiative transitions are the dominating decaymechanism in this three-state system.

We here apply a general diabatization procedure to the(1-3)3�g system of Al2, using the adiabatic PECs and non-adiabatic derivative couplings from Han et al. (13). The result-ing diabatic representation is used to form a coupled-channelSchrodinger problem which is numerically solved by means ofa complex rotated, one-dimensional finite element method.

Our theoretical approach is described in Section 2, wherethe adiabatic and diabatic basis state description of the coupled-channel Schrodinger problem is outlined in Sections 2.1 and 2.2.Our numerical approach, based on the Runge–Kutta–Fehlbergtechnique and a one-dimensional finite element method, is dis-cussed in Sections 3.1 and 3.2. Our calculations are presented inSection 3.3. In Section 4 our results are presented and discussed.Section 5 summarizes this paper.

0022-2852/02 $35.00C© 2002 Elsevier Science (USA)

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16 ANDERSSON AND ELANDER

FIG. 1. Adiabatic (solid lines) and diabatic (dashed lines) potential energycurves, Uαα(R) as in Eqs. [9] and [18], for the (1-3)3�g three-state approxima-tion of Al2.

II. THEORY

All formulas and equations in this section are expressed inatomic units if nothing else is stated.

II.1. The Adiabatic Formulation

Consider a diatomic molecule with nuclei masses MA andMB , and where nuclear and electronic coordinates are denotedby R and r, respectively. The electronic Hamiltonian, He, includ-ing the kinetic energy operator, Te, and total potential energy,V (R, r), together with a real set of electronic wavefunctions,φβ(R, r), form a fixed nuclei eigenvalue problem,

Heφβ(R, r) = Uβ(R)φβ(R, r). [1]

Using completeness of {φβ(R, r)} for an expansion of the totalwavefunction,

Ψ(R, r) =∞∑

β=1

χβ(R)φβ(R, r), [2]

with the nuclear wavefunctions, χβ(R), as coeficients, the rovi-bronic eigenvalue problem is formed as

∞∑β=1

〈α|H − Ev J |β〉χβ(R) = 0; α = 1, 2, . . . , [3]

where the notation |φγ 〉 = |γ 〉; γ = α, β is used. Here H =(1/2M)∇2

R + He, M = MA MB/(MA + MB). From Eq. [1],and the orthonormality 〈α|β〉 = δαβ , the rovibrational problem

C© 2002 Elsevier

described by Eq. [3] reduces to

− 1

2M

[∇2

Rχα(R) +∞∑

β=1

{2Fαβ(R) · ∇R + Gαβ(R)χβ(R)}]

+∞∑

β=1

Uαβ(R)χβ(R) = Ev J χα(R). [4]

Here

Uαβ(R) = 〈α|He|β〉, [5]

are elements of the electronic potential energy matrix, and

Fαβ(R) = 〈α|∇R|β〉 = −Fβα(R), [6]

Gαβ(R) = 〈α|∇2R|β〉 = Gβα(R), [7]

couple adiabatic electronic states (20).Truncating the sums in Eq. [4] by n PECs, and using the BO-

adiabatic (a) formulation (1) for well separated eigenvalues, aproblem of practical use is formed,

∇2Rχ (a)

α (R) +n∑

β=1

{2F (a)

αβ (R) · ∇R + G(a)αβ (R)χ (a)

β (R)}

+ Vαα(R)χ (a)α (R) = 0, [8]

where

Vαα(R) = −2M[U (a)

αα (R) − Ev J]. [9]

II.2. The Diabatic Formulation

Solving the adiabatic problem defined by Eq. [8], integrationshave to be carried out on a set of second order differential equa-tions (21, 22) also containing first derivatives. Furthermore, thederivative coupling elements Fαβ(R) and Gαβ(R) are often vary-ing fast. The adiabatic basis may also be inconvenient for treatingsome perturbation situations (23). The diabatic (d) basis offersan alternative representation of a coupled-channel Schrodingerproblem. It is obtained via

φ(d)α (R, r) =

n∑β=1

φ(a)β (R, r)Tβα(R), [10]

where α = 1, 2, . . . , n and Tβα(R) are elements of the n × nadiabatic to diabatic transformation (ADT) matrix T(R). In orderto preserve the total wavefunction, Eq. [2] implies that

Ψ(R, r) =n∑

β=1

φ(d)β (R, r)χ (d)

β (R), [11]

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NONADIABATIC PREDISSOCIATION STUDIES 17

where

χ (d)α (R, r) =

n∑β=1

Tαβ(R)χ (a)β (R, r). [12]

From Eqs. [12] and [8] a new matrix SE is formed,

T∇2Rχ

(d) + 2[∇RT + F(a)T] · ∇Rχ(d) + [VT + K]χ(d) = 0,

[13]

where

K = ∇2RT + 2F(a)∇RT + G(a)T. [14]

Equation [12], together with the requirement

∇RT + F(a)T = 0, [15]

implies that T is orthogonal and that K vanishes (20). Assuming

limR→∞

F(R) = 0 ⇒ limR→∞

T(R) = I, [16]

where I is the identity matrix, we obtain a boundary condition toEq. [15]. This implies that we, solving Eq. [15] for the condition[16], have unique solutions of the diabatization problem for anynumber of coupled equations [4]. Thus, the diabatic coupled SEtakes the form

∇2Rχ

(d) + Wχ(d) = 0, [17]

where W = T−1VT and, using Eq. [9], consequently

U(d) = T−1U(a)T. [18]

Using index notation in Eq. [17] we obtain

∇2Rχ (d)

α (R) +n∑

β=1

Wαβ(R)χ (d)β (R) = 0, [19]

where Wαβ(R) = Tαβ(R)Vαβ(R)Tβα(R) and Vαβ(R) are definedby Eq. [9]. The rotational contribution to U(d)(R) is described interms of quantum numbers for the total angular momentum J ,and the electronic angular momentum �,

U (d)αβ (R) = U (d)

αβ (R, J, �)

= U (d)αβ (R, 0, 0) + δαβ

2M R2{J (J + 1) − �2}. [20]

II.3. Exterior Complex Scaling

Eigenvalue problems containing bound levels and resonancesmay be solved by using complex scaling (CS) methods (26,27, 29, 31, 32). The eigenenergy and the width of a level is

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then a solution of a non-Hermitian complex symmetric analyticcontinuation of a nondilated Hermitian Schrodinger problem.In the original uniform CS, suggested by Balslev and Combes(26, 27), the radial coordinates are scaled as

R → Reiφ with 0 ≤ φ < φcrit, [21]

where φ is a rotation angle, and φcrit is a limiting angle dependingon the potential in H = H0 + V(R) (28). The Hamiltonian andthe wavefunction are transformed to new operators

Hφ = e−2iφ H0 + Vφ while Ψφ = ei φ

2 Ψ(Reiφ). [22]

The operator Hφ has the same real eigenvalues as H (26, 27, 29,31, 32) while the pure continuum eigenenergies are rotated inthe complex plane by an angle −2φ. In addition, Hφ uncoversa new class of eigenvalues, the resonances. The uniform CSrequires V(R) to be analytic such that V(Reiφ) can be obtained.When the analytic form of V(R) is unknown we cannot apply theuniform CS method. Therefore the idea of exterior CS (ECS) wassuggested (30, 33, 34)

R →{

R, R ≤ Rs

Rs + (R − Rs)eiφ, R > Rs .[23]

Here Rs is the exterior scaling radius up to which the potentialsare real while the outer part, R > Rs , is analytically continued.The ECS requirement is that only V(R) for R > Rs needs to be afunction that can be continued analytically. The potential in theinner region R ≤ Rs may in fact be described only by discretenumerical values. The accuracy of the calculation then dependson the density of that point grid.

III. NUMERICAL APPROACH

III.1. Computational Diabatization

Equations [15] and [16] give us a formal tool and a compu-tational possibility to uniquely obtain a set of diabatic PECs foran arbitrary adiabatic PEC matrix U(a)(R).

Using a matrix version of the Runge–Kutta–Fehlberg(RKF45) method (39–41), we are able to propagate T(R) inthe interval from Rmax to Rmin. The RKF45 method has a builtin error estimate which allows us to control the accuracy of ourcalculated ADT matrix T(R) to only depend on the word lengthof the used computer code.

III.2. A One-Dimensional Finite Element Method

The basic idea of the finite element method (FEM) is to dis-cretize the solution region into a finite number of subregions. Thetotal trial wavefunction is expanded in a finite element basis,

Ψφ(R) =∑

i j

ci j fi j (R), [24]

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18 ANDERSSON AND ELANDER

where fi j (R) denotes the j th basis function localized in elementi and it is here assumed that ECS has been used. The complexvalued, φ-dependent expansion coefficients, ci j , are defined ineach element and restricted through continuity boundary condi-tions (16) for Ψφ(R) over element boundaries. The local basisfunctions, fi j , are nonzero only inside a given element i ,

fi j (R) ≡ 0 for R /∈ [Ri−1, Ri ], i = 1, . . ., K . [25]

The Rayleigh–Ritz variational principle (37) provides an esti-mate to the complex resonance energies, εk , which are obtainedas eigenvalues of 〈Ψφ|Hφ|Ψφ〉, and they are evaluated solvingfinite-dimensional problems of the form

(Hφ − εk Sφ)ck = 0, [26]

where

(H )i j,k� = 〈i j |Hφ|k�〉,[27]

(S)i j,k� = 〈i j |k�〉,

with the notation | fi j 〉 = |i j〉, and

εk = Ek − i�k/2. [28]

The H i j,k� and Si j,k� elements are identically zero for i �= k,and therefore the global matrices Hφ and Sφ become bandedand relatively sparse (16).

In practice, the number of elements needed, when solving one-dimensional eigenvalue problems for diatomics, varies with theproblem: the size of the global space, the localization of thewavefunctions, etc. A starting grid should contain ∼20–30 el-ements and ∼8–10 local basis functions within each element.These two parameters are then slowly increased until conver-gence is reached. The finite element basis functions, fi j , maybe of any kind but polynomials, exponential functions, trigono-metric functions, or products of these are commonly used.

III.3. Calculations

The reduced mass µ = 13.4907703 a.u. for Al2 was obtainedfrom Huber and Herzberg (15). The initial data, in the form ofadiabatic PECs and nonadiabatic derivative coupling elementsfor the (1-3)3�g system, were obtained from Han et al. (13)and are here represented as tension splines (42). The ADT wascarried out by applying the RKF45 method on the system ofcoupled Eqs. [15]. The resulting 3 × 3 ADT matrix T(R) wasthen used in Eq. [17] in order to obtain the diabatic potentialmatrix U(d)(R) containing the diagonal diabatic electronic po-tential energies, and the off-diagonal nonadiabatic interactionpotentials.

Rovibronic eigenenergies and widths were obtained from theADT result by means of a complex rotated FEM method. TheBO, single-channel problem is formulated by the uncoupled

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TABLE 1Finite Element Calculation Parameters Including Number of

Finite Element Intervals (NFEM) and Number of Basis Functionsper FEM-Interval (Nbas)

NFEM Nbas Ri Rs R f φ

130 13 2.30 10.00 15.0 20.0

Note. The innermost mesh point (Ri ), exterior scaling point (Rs ), and outer-most mesh point (R f ), all given in a0. The standard dilation angle (φ) is givenin degrees.

SE [8] while the corresponding diabatic, coupled-channel prob-lem is given by Eq. [17]. The accuracy in an ECS FEM compu-tation depends on the density of the point grid used in the regionbefore the ECS point Rs , as was pointed out in Section 2.3.Therefore, a total number of about 1200 spline points was usedin all our calculations.

Although the energy, E , and the width, � = 1/τ , are indepen-dent of the ECS angle, φ, the resonances become φ-dependent innumerical approximations. Then E and � are defined by meansof the complex variational principle (37),

d E

dφ

∣∣∣∣φR

= 0 andd�

dφ

∣∣∣∣φ I

= 0, [29]

where the two optimal angles φR and φI converge to one singleangle as the numerical accuracy is increased.

The theory and numerical methods described in Sections 2and 3 were implemented in a Fortran 77 code (16) and used tocompute adiabatic and diabatic rovibronic term values, E(v, N ),as well as predissociation widths, �(v, N ), and nonradiative life-times, τnr (v, N ), for the (2, 3)3�g states in the (1-3)3�g mani-fold of Al2. The step length in our diabatization calculation was1 · 10−4a0 yielding an error estimate ∼2 · 10−11 cm−1. Parame-ters used in the FEM calculations are given in Table 1.

Following Scrinzi and Elander (16) the convergence of thecomputed complex eigenvalues (εk in Eq. [28]), using thenumber of elements NFEM and the degree of the polynomialp = Nbas − 1, is such that |εtrue − εcalc|/εcalc < 10−12. Thus, weconsider levels with widths less than 10−8 cm−1 as bound states.

We have neglected the spin–orbit coupling terms because nosuch data, consistent with the PECs used here, are available. Thisimplies that we, as well as Han et al. (13, 14), use the case (b) ro-tational quantum number, N , i.e., {J (J + 1) − �2} → {N (N +1)} in Eq. [20].

IV. RESULTS AND DISCUSSION

In the following sections the rovibronic energy levels andwidths will be denoted as E(v(i), N ) and �(v(i), N ), respectively,where N is the rotational quantum number and v(i) are the vi-brational quantum numbers for the (2)3�g (i = 2) and (3)3�g

(i = 3) levels. For each (v(i), N ) level both adiabatic (BO) and

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NONADIABATIC PREDISSOCIATION STUDIES 19

diabatic results were obtained. In the rotationless case, levelswere obtained for (v(2), N ) = (0 − 70) and (v(3), N ) = (0 − 17)while for the rovibronic case the (v(2), N ) = (0 − 50, 0 − 25) and(v(3), N ) = (0 − 17, 0 − 25) levels were calculated.

IV.1. Diabatization Results

Before presenting our main results it might be of interest todiscuss and compare our ADT results with previous calculations.The RKF45 routine was tested on exponentially growing and de-creasing problems with various exponents. The differences be-tween analytically obtained solutions and our numerical resultswere found to agree with the error estimates of our numeri-cal code (see Section 3.3). In Figs. 1 and 2 the present diabaticPECs and the nonadiabatic interaction potentials Uαβ(R), for the(1-3)3�g states of Al2 are displayed. Comparing these resultswith the corresponding plots of Han et al. (13), we found anexcellent agreement in the region 3.5 ≤ R ≤ 7.5a0. Outside thisinterval there is no comparable data available.

IV.2. Rovibronic Energy Levels

The (2, 3)3�g energy levels were identified by studying thecorresponding total wavefunctions, i.e., by counting the numberof nodes of the single-channel total wavefunction in the BO case,and the second and third components of the coupled-channeltotal wavefunction in the diabatic case. For close lying dia-batic E(v(2), 0) and E(v(3), 0) levels this was further supportedby studying the linear N (N + 1) dependence for each E(v(i),0 − 25) term series. In Fig. 3, this is illustrated for the overlap-ping region (v(2), N ) = (27 − 50, 0 − 25) and (v(3), N ) = (0 − 9;0 − 25) i.e., in the energy region ∼(28600–32600) cm−1.

Comparing the rotationless BO vibrational energy levels,EBO(v(i), 0), with the corresponding diabatic ones, E(v(i), 0),the latter ones are shifted to higher energies for all studied lev-els. These shifts originate mainly from the derivative coupling

FIG. 2. Interaction potentials, Uαβ as in Eq. [18], for the (1-3)3�g three-state approximation of Al .

2

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FIG. 3. The N (N + 1) dependence of the calculated diabatic rovibronictermvalues, E(v(i), N ), where � denotes the (v(2), N ) = (26 − 50, 0 − 25) levelsand + denotes the (v(3), N ) = (0 − 9, 0 − 25) levels.

elements Fαβ(R), α, β ∈ 1, 2, 3. In Fig. 4, the energy differences,EBO(v(i), 0) − E(v(i), 0), are plotted as functions of v(i). For the(2)3�g levels, the energy differences are strongly oscillating be-tween 10 and 25 cm−1 while an almost constant shift ∼12 cm−1

is found for the (3)3�g levels. This behavior continues also forhigher vibrational levels not displayed here.

The presented E(v(2), 0) and E(v(3), 0) levels agree quitewell with the results reported by Han et al. (13) and Hanand Yarkony (14). In Table 2, a selected number of energiesfor rotationless vibrational levels are presented. Using the no-tations in this table we find that E(v(2), 0)a − E(v(2), 0)b,c ≤±1 cm−1 except for v(2) = 7 − 9 for which E(v(2) = 7 − 9, 0)a −E(v(2) = 7 − 9, 0)c ∼ 2 − 3 cm−1. A similar comparison for the

FIG. 4. The energy shifts, E(v(i), 0) − EBO(v(i), 0), between the diabaticthree-state approximation and the Born–Oppenheimer (BO) formulation wherei = 2 and i = 3 levels are denoted by � and +, respectively.

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20 ANDERSSON AND ELANDER

TABLE 2Term Energy Values, E(v (i), N) [cm−1], Nonradiative, τnr (v (i), N)[µs, ns, ps], and Radiative Lifetimes, τr (v (i), N)[ns], for a Selected

Number of (v (i), N) levels of the (1-3)3Πg manifold of Al2

i (v(i), N ) E(v(i), N )a τnr (v(i), N )a E(v(i), N )b τnr (v(i), N )b E(v(i), N )c τnr (v(i), N )c τr (v(i), N )c

2 (0, 0) 23949.26 4.32 ps 23949.3 2.29 ps 23948.2 0.83 ps 796 ns2 (1, 0) 24389.33 15.40 24389.6 168.98 24390.2 2.33 3362 (2, 0) 24671.83 4.26 24671.4 3.98 24672.1 2.78 1912 (3, 0) 24760.89 2.14 24761.4 2.24 24761.5 1.67 3522 (4, 0) 24905.14 1.52 24905.4 1.38 24905.5 1.44 3192 (5, 0) 25057.99 1.03 25058.7 0.84 25057.8 1.01 2792 (6, 0) 25217.77 0.88 25217.9 0.66 25216.7 0.75 2332 (7, 0) 25383.46 0.94 25384.1 0.61 25381.3 0.56 1872 (8, 0) 25552.58 1.29 25552.8 0.68 25549.7 0.39 1512 (9, 0) 25723.77 2.44 25724.2 0.94 25720.4 0.20 1252 (11, 0) 26069.31 18.316 ns2 (11, 2) 26070.22 31.0012 (11, 3) 26071.13 63.4372 (11, 4) 26072.35 349.2582 (11, 5) 26073.87 750.0002 (24, 0) 28328.17 0.1262 (24, 15) 28360.33 476.884 µs

3 (0, 0) 28661.73 0.016 ns 28661.6 6.37 ns 28661.2 0.006 ns 34.83 (1, 0) 29150.34 0.787 29150.1 0.02 29149.6 0.062 35.43 (2, 0) 29618.94 1.977 29618.5 0.34 29618.2 1.512 35.23 (3, 0) 30070.26 0.823 30069.7 0.32 30069.5 0.474 35.13 (3, 20) 30144.31 0.169 µs3 (3, 21) 30151.71 2.0703 (4, 0) 30505.94 0.337 ns 30505.3 0.13 30505.1 0.078 35.23 (5, 0) 30927.31 0.921 30926.7 0.28 30926.3 0.024 35.33 (6, 0) 31335.00 1.633 31334.1 1.54 31333.7 0.048 35.63 (6, 20) 31407.30 243.5313 (7, 0) 31729.86 0.161 31728.9 0.42 31728.5 0.027 35.93 (8, 0) 32112.87 7.4993 (8, 3) 32114.89 60.4123 (8, 4) 32116.24 2.656 µs3 (12, 0) 33534.31 19.08 ps3 (12, 13) 33563.59 12.428 µs3 (15, 0) 34485.78 6.656 ns3 (15, 11) 34506.16 61.8363 (15, 12) 34509.88 151.2113 (15, 13) 34513.91 984.3003 (15, 14) 34518.25 2.213 µs3 (15, 15) 34522.90 164.605 ns3 (15, 16) 34527.86 51.767

Note. The index i = 2, 3 denotes the (2, 3)3�g vibrational levels. The index a denotes results from the present calculations while the indices b and c denoteresults reported in Refs. (13, 14).

v(3) rotationless levels shows that E(v(2), 0)a − E(v(2), 0)b,c ≤1 cm−1, i.e., that all energies calculated here lie above the re-sults reported in Refs. (13, 14). To summarize, the comparedrotationless vibrational energies, obtained by three different ap-proaches, all agree within (1–3) cm−1.

The fact that the (2)3�g and (3)3�g states both have rovi-bronic levels with energies between 28600 and 32600 cm−1

implies that strong homogeneous perturbations may exist in thisregion. Therefore the rovibronic E(v(2), 0 − 25) and E(v(3), 0 −−25) energies are displayed as functions of N (N + 1) in Fig. 4.Taking a closer look in this figure, two sets of parallel lines ap-

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pear where each set corresponds to one of the (2, 3)3�g states.This, together with the wavefunction analysis discussed above,was used to identify each E(v(i), 0 − 25); i = 2, 3, term series.

IV.3. Predissociation Widths

The (2, 3)3�g PECs are in the adiabatic as well as in the di-abatic description coupled to each other when determining therovibrational energy levels, in particular those located in theoverlap region (28600–32600) cm−1. These levels are furtherembedded in the continuum of the (1)3�g repulsive state.

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NONADIABATIC PREDISSOCIATION STUDIES 21

FIG. 5. The N (N + 1) dependence of the logarithm for calculated widths,�(v(2), N ), of some (2)3�g levels.

This suggests that the (v(2), N ) and (v(3), N ) levels most likelyare short-lived. By studying the widths we indeed see that(0.30 · 10−3 ≤ �(v(i), 0) ≤ 30) cm−1, i.e., that all rotationless(v(i), 0); i = 2, 3, levels are more or less predissociated.

The variation of �(v(i), N ) with N within a v(i) term seriesmay reflect electronic perturbations (18). Therefore we havestudied the N (N + 1) dependence for log10�(v(2), 0 − 25) andlog10�(v(3), 0 − 25). A few chosen results for the (v(2), N ) and(v(3), N ) levels are displayed in Figs. 5 and 6, respectively.

Starting with the lower (v(2), N ) levels (not dispalyed here),interactions with the (v(3), N ) levels are almost negligible. Forthe lowest level, log10�(v(2) = 0, N ) is essentially constant,i.e., log10�(v(2) = 0, N ) ≈ log10 �(v(2) = 0, N + 1). The nextfive (v(2), N ) levels show an almost linear dependence where

FIG. 6. The N (N + 1) dependence of the logarithm for calculated widths,�(v(3), N ), of some (3)3�g levels.

C© 2002 Elsevier

log10�(v(2) =1−5, N ) > log10 �(v(2) = 1 − 5, N + 1) but whenincreasing v(2) = 1 one step at a time this negative inclina-tion is getting smaller and smaller such that log10�(v(2) =6 − 7, N ) ≈ log10 �(v(2) = 6 − 7, N + 1) and log10�(v(2) = 8 −10, N ) < log10 �(v(2) = 8 − 10, N + 1); i.e., we now have apositive inclination. For v(2) = 11 the linear dependence isbroken. Furthermore, as is clearly seen in Fig. 5, log10�(v(2) =11, N<5)> log10 �(v(2) =11, 5)< log10 �(v(2) = 11, N>5); i.e.,we have a dip with a distinct minimum for N = 5 where�(v(2) = 11, 5) = 0.78 · 10−5 cm−1.

For (v(2), N ) = (12 − 23, 0 − 25) the linear dependence ap-pears again and, as was noted before, the inclination changesfrom negative to almost constant for (v(2), N ) = (15, 0 − 25),and continues to positive when increasing v(3) further. The(v(2), N ) = (24, 0 − 25) levels show a similar behavior as wasseen for (v(2), N ) = (11, 0 − 25) and from Fig. 5 a minimum of�(v(2), N ) = (24, 15) = 1.12 · 108 cm−1 is found for N = 15. Inthe region where (v(2), N ) = (25 − 41, 0 − 25), a similar behav-ior as was seen for (v(2), N ) = (12 − 23, 0 − 25) appears butwhere signs of nonlinearity for (v(2), N ) = (39 − 41, 0 − 25)are seen. The nonlinear dependence for v(2) = 42 is plotted inFig. 5 where a minimum of �(v(2) = 42, 23) = 0.36 · 10−3 cm−1

is seen for N = 23. For (v(2), N ) = (43 − 50, 0 − 25) the linearpattern described above appears again but with the exceptionthat log10�(v(2) = 43, N > 20) behaves slightly nonlinearly.

Except for v(2) = 11, 24, 41, and 42, the (v(2), N ) levels appearto be relatively unperturbed. This is not the case for the studied(v(3), N ) levels which, except for a few cases, all show a moreor less nonlinear dependence for log10�(v(3), N ) as functionsof N (N + 1). This is clearly seen in Fig. 6 where the widthsfor v(3) = 3, 6, 8, 12, and 15 all have a minimum for someN , with values in the interval (0.42 · 10−6 ≤ �(v(3), N ) ≤ 0.22 ·10−4 cm−1). For v(3) = 1, 6, and 16 the width has a max-imum for some N . For the remaining results displayed inFig. 6, we have that log10�(v(3)

A , N ) < log10 �(v(3)A , N + 1) and

that log10�(v(3)B , N ) > log10 �(v(3)

B , N + 1), where v(3)A = [2, 9 −

11, 13 − 14, 17] and v(3)B = [4 − 5, 7].

In order to check if the irregularities of log10�(v(i), N ) versusN imply similar irregularities in the corresponding E(v(i), N )series the local B-value, defined as

B(v, N ) = E(v, N ) − E(v, 0)

N (N + 1), [30]

was calculated for all vibrational term series displaying non-smooth logarithmic width dependence. The variation of B(v, N )within all so studied term series was smaller than 1 promille.

To summarize this subsection we note that the rotationlessvibrational levels studied here predissociate. The linear depen-dence described above thus indicates that the correspondingrovibronic levels also predissociate. However, for a consider-able number of (v(i), N ) levels, clear signs of spectroscopicperturbations were seen in the form of a nonlinear behaviorof log10�(v(i), N ). At the present time we do not have a simple

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22 ANDERSSON AND ELANDER

explanation for these perturbations. In the regions around eachminimum the widths are relatively small, implying that theselevels are more long-lived than expected from studies of the cor-responding rotationless levels. The next subsection is focused onthat question.

IV.4. Nonradiative Lifetimes

Using the calculated predissociation widths, �(v(i), N ), thecorresponding nonradiative lifetimes, τnr (v(i), N ), for transi-tions between the (2, 3)3�g rovibronic levels and the (1)3�g

continuum, were obtained. Comparing the (2, 3)3�g rotation-less lifetimes, τnr (v(i), 0), presented in Table 2, one sees thatτnr (v(3), 0) are in general about 100–1000 times longer thanτnr (v(2), 0), particularly in the overlapping region (28600–32600) cm−1. This is most likely a consequence of the fact thatF13(R) � F12(R) ∼ F23(R) (13).

By converting the predissociation widths to nonradiative life-times we were able to compare the present results with theFermi–Wentzel Golden Rule (44–46) results of Han et al. (13)and the more elaborate Golden Rule formalism in Han andYarkony (14). These comparisons are relevant since it is believedthat Golden Rule-type treatments are appropriate for narrow res-onances.

Results, originating from the three different approaches, arecollected in Table 2. Using the notations in this table we findthat τnr (v(2), 0)a ≤ 2 × τnr (v(2), 0)b for most of the levels withthe exception that τnr (v(2) = 1, 0)b ≈ 11 × τnr (v(2), 0)a . A simi-lar comparison with the results of Han and Yarkony (14) showsthat τnr (v(2) = 2 − 7, 0)a ≈ 1.5 × τnr (v(2) = 2 − 7, 0)c while theremaining levels with v(2) = 0, 1, 8, and 9 have lifetimes forwhich τnr (v(2) = 0 − 1; 8 − 9, 0)a ∼ (3 − 12) × τnr (v(2) = 0 −1; 8 − 9, 0)c.

For the (v(3), 0) levels compared here the agreement isworse than for the previous (v(2), 0) levels. From Table 2 wehave that τnr (v(3) = 6, 0)a ≈ τnr (v(3) = 6, 0)b while τnr (v(2) =1 − 5, 0)a ∼ (3 − 40) × τnr (v(2) = 1 − 5, 0)b. For v(3) = 0 and 7we have the opposite situation and in particular we note thatτnr (v(2) = 0, 0)b ≈ 400 × τnr (v(2) = 0, 0)a . Continuing this com-parison with the results of Han and Yarkony (14) we have thatτnr (v(3) = 0 − 7, 0)a ∼ (1.5 − 40) × τnr (v(2) = 0 − 7, 0)c; i.e.,the agreement is bad and strongly varying with the (v(3), 0)levels.

Even if the discussion above clearly shows that there areconsiderable differences between the here presented nonradia-tive lifetimes and the results reported by Han and Yarkony(14) and Han et al. (14), we note that the agreement be-tween their nonradiative lifetimes is rather poor. In Table 2we find that τnr (v(2) = 2 − 7, 0)b ≈ τnr (v(2) = 2 − 7, 0)c andτnr (v(3) = 3 − 5; 7, 0)b ≈ τnr (v(3) = 3 − 5; 7, 0)c while the restof the levels have lifetime differences of variable size. For ex-ample, we find that τnr (v(2) = 1, 0)b ≈ 72 × τnr (v(2) = 1, 0)c

and τnr (v(2) = 1, 0)b ≈ 1061 × τnr (v(3) = 0, 0)c. Thus, the threeapproaches compared here disagree considerably with respect

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to rotationless nonradiative lifetimes. However, according toScrinzi and Elander (16) the numerical approach used in thepresent paper is quickly convergent and extremely stable. Thisimplies that the nonradiative lifetimes presented here are reli-able with respect to the PECs and nonadiabatic couplings thatwere used as initial data.

IV.5. Spectroscopically Measurable Rovibronic Levels

In the previous subsection we reported a number of rovibroniclevels belonging to the (2)3�g as well as (3)3�g states whichhave quite narrow widths. This makes us believe that they maybe spectroscopically observable assuming the used PECs ma-trix (U(R)) correctly is able to model the (2, 3)3�g states ofthe Al2 molecule. Han et al. (13) and Han and Yarkony (14)have computed the radiative lifetimes, τr (v(i), 0), for some rota-tionless levels. We here assume that τr (v(i), N ) varies smoothlywith N within a rovibronic termseries (23, 43), i.e., τr (v(i), N ) ≈τr (v(i), 0). Using that the total lifetime of a level, τ (v(i), N ), isrelated to τnr (v(i), N ) and τr (v(i), N ) as

τ = τnr + τr

τnrτr, [31]

we are able to estimate transitions originating in these upper(v(i), N ) levels which may be observed experimentally.

The radiative lifetimes for some (2, 3)3�g rotationless (v(i), 0)levels, calculated by Han and Yarkony (14), are listed inTable 2 as well as displayed in Fig. 7. Here τr (v(2) = 0 − 9, 0)decrease from 796 ns (v(2) = 0) to 125 ns (v(2) = 9) whileτr (v(3) = 0 − 7, 0) are all around 35 ns. Based on this we as-sume that for higher (v(i), N ) levels τr (v(2), N ) ≈ 100 ± 50 nswhile τr (v(3), N ) ≈ 35 ns.

If we now consider levels for which τr (v(i), N ) ≈ τnr (v(i), 0)we find that only (v(2), N ) levels with �(v(2), N ) ≤ 5 · 10−5 cm−1

FIG. 7. The v(i), i = 1, 2, dependence of the rotationless radiative lifetimes,τr (v(i), 0), for the (2)3�g (with �) and (3)3�g (with +) levels reported inRef. (14).

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NONADIABATIC PREDISSOCIATION STUDIES 23

or log10�(v(2), N ) ≤ −4 may be observable. Following this wesuggest that the levels below the horizontal dashed line in Fig. 5,i.e., (v(2), N ) = (11, 2 − 5) and (24, 15), may be experimentallymeasurable.

For the (v(3), N ) levels we find an upper limit where�(v(3), N ) ≈ 5 · 10−5 cm−1 or that levels for which log10�

(v(2), N ) ≤ −4 may be observable. According to Fig. 6 thisincludes all levels below the horizontal dashed line, i.e.,(v(3), N ) = (3, 20 − 21), (6, 20), (8, 3 − 4), (12, 13), and (15,

11 − 16).Thus, concludingly we find that in spite of the fact that the

(2, 3)3�g PECs are strongly affected by the repulsive (1)3�g

state we are able to find a few rovibronic levels that may beobserved in a spectroscopic study. Even though the positions,and particularly the widths of the computed rovibronic levels,are sensitive to the potential energy matrix (U(R)) we in thisstudy give evidence that observable levels indeed may exist.

V. SUMMARY

We have in this work applied a general matrix Runge–Kutta–Fehlberg diabatization procedure on the (1-3)3�g manifold ofAl2, using adiabatic electronic potential energy curves (PECs)and nonadiabatic coupling elements originally obtained by Hanet al. (13). Rotationless BO vibrational as well as diabatic rovi-bronic levels, including energies and widths, were computed forthe (2, 3)3�g states. From the widths, the nonradiative lifetimeswere computed for each rovibronic (2, 3)3�g level. Compar-ing the rotationless BO and diabatic vibrational energies, dif-ferences within (10–25) cm−1 were found for the (2)3�g levelswhile the (3)3�g energies have an almost constant shift of about12 cm−1. Most of the diabatic rotationless levels were found toagree within ±1 cm−1 when compared with the theoretical re-sults reported by Han et al. (13) and Han and Yarkony (14). Fora similar comparison between the nonradiative lifetimes lessgood agreements were found. Using the rotationless radiativelifetimes reported by Han and Yarkony (14) we estimated a limitof the nonradiative lifetimes for which the (2, 3)3�g rovibroniclevels may be experimentally observable. From this we sug-gested a number of such levels, having nonradiative lifetimeswithin the interval (0.052–476.884) µs.

ACKNOWLEDGMENTS

This work was supported by a grant from the Swedish National ScienceFoundation (NFR) and by Stockholm University.

REFERENCES

1. B. H. Bransden and C. J. Jochain, “Physics of Atoms and Molecules.”Longman, New York, 1983.

2. H. Basch, W. J. Stevens, and M. Krauss, Chem. Phys. Lett. 109, 212 (1984).

C© 2002 Elsevier

3. C. W. Bauschlicher, S. R. Langhoff, P. R.Taylor, and S. P. Walch, J. Chem.Phys. 86, 7007 (1987).

4. S. R. Langhoff and C. W. Bauschlicher, J. Chem. Phys. 92, 1879 (1990).5. C. W. Bauschlicher and S. R. Langhoff, Chem. Rev. 91, 701 (1991).6. C. W. Bauschlicher and S. R. Langhoff, Science 254, 394 (1991).7. K. K. Sunil and K. D. Jordan, J. Phys. Chem. 92, 2774 (1988).8. M. F. Cai, T. P. Dzugan, and V. E. Bondybey, Chem. Phys. Lett. 155, 430

(1989).9. M. F. Cai, C. C. Carter, T. A. Miller, and V. E. Bondybey, Chem. Phys. 155,

233 (1991).10. Z. Fu, G. W. Lemire, G. A. Bishea, and M. D. Morse, J. Chem. Phys. 93,

8420 (1990).11. M. A. Douglas, R. H. Hauge, and J. L. Margrave, J. Phys. Chem. 87, 2945

(1983).12. H. Abe and D. M. Kolb, Ber. Bunsenges Phys. Chem. 87, 523 (1983).13. S. Han et al., J. Chem. Phys. 102, 1955 (1995).14. S. Han and D. R. Yarkony, J. Chem. Phys. 103, 7336 (1995).15. K. P. Huber and G. Herzberg, “Molecular Spectra and Molecular Structure.

IV. Constants of Diatomic Molecules.” Litton Educ. Publ. Inc., New York,1979.

16. A. Scrinzi and N. Elander, J. Chem. Phys. 98, 3866 (1993).17. M. R. Manaa and D. R. Yarkony, J. Chem. Phys. 100, 8204 (1994).18. C. Carlsund-Levin, “Numerical Studies of Resonances using Complex

Scaling.” Doct. Diss., Theor. Phys., Dept. of Phys. R. Inst. of Tech.,Stockholm, 2000.

19. M. Born and J. R. Oppenheimer, Ann. Phys. (Leipzig) 84, 457 (1927).20. M. Baer, Chem. Phys. Lett. 35, 112 (1975).21. R. G. Gordon, J. Chem. Phys. 51, 14 (1969).22. R. de Vogelaere, J. Res. Natl. Bur. Std. 54, 119 (1954).23. H. Lefebvre-Brion and R. W. Field, “Perturbations in the Spectra of Di-

atomic Molecules.” Academic Press, Orlando, 1986.24. C. A. Mead and D. G. Truhlar, J. Chem. Phys. 77, 6090 (1982).25. S. Han and D. R. Yarkony, Mol. Phys. 88, 53 (1996).26. J. Aquilar and J. M. Combes, Commun. Math. Phys. 22, 269 (1971).27. E. Balslev and J. M. Combes, Commun. Math. Phys. 22, 280 (1971).28. M. Rittby, N. Elander, and E. Brandas, Mol. Phys. 45, 553 (1982).29. B. Simon, Ann. Math. 97, 247 (1973).30. B. Simon, Phys. Lett. A 71, 211 (1979).31. C. van Winter, J. Math. Appl. 47, 368 (1974).32. C. van Winter, J. Math. Appl. 18, 633 (1974).33. C. A. Nicolaides and D. R. Beck, Phys. Lett. A 65, 11 (1978).34. B. Gyarmati and T. Vertse, Nucl. Phys. A 160, 523 (1971).35. P. B. Kurasov, A. Scrinzi, and N. Elander, Phys. Rev. A 49, 5095

(1994).36. B. Helffer, in “Resonances, Lecture Notes in Physics” (E. Brandas and

N. Elander, Eds.), Vol. 325. Springer-Verlag, Berlin, 1989.37. N. Moiseyev, P. R. Certain, and F. Weinhold, Mol. Phys. 336, 1613 (1978);

47, 585 (1982).38. N. Elander et al., Physica Scripta 20, 631 (1979).39. E. Fehlberg, “Low-order Classical Runge-Kutta Formulas With Step Size

Control.” NASA, TR R-315.40. L. F. Shampine, H. A. Watts, and S. Davenport, “Solving Nonstiff Ordinary

Differential Equations—The State of the Art.” Sandia Labaratories Report,SAND75-0182.

41. L. Baker, “C Tools for Scientists and Engineer.” McGraw–Hill, HamburgNew York, 1989.

42. A. K. Cline, Comm. ACM 17, 218 (1974).43. N. Elander, J. Oddershede, and N. H. F. Beebe, Ap. J. 216, 165 (1977).44. G. Wentzel, Phys. Z. 29, 321 (1928).45. O. K. Rice, Phys. Rev. 33, 748 (1929).46. O. K. Rice, Phys. Rev. 34, 1451 (1929).

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