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Chemical Physics 322 (2006) 405410The relativistic E E JahnTeller effect revisited

Wolfgang Domcke a,*, Sabyashachi Mishra a, Leonid V. Poluyanov b

a Department of Chemistry, Technical University of Munich, D-85747 Garching, Germanyb Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 14232, Russia

Received 31 May 2005; accepted 9 September 2005Available online 11 November 2005Abstract

The vibronic Hamiltonian describing linear JahnTeller coupling as well as spinorbit coupling in systems of trigonal symmetry isderived in the diabatic spinorbital representation, employing the microscopic (BreitPauli) spinorbit coupling operator in the sin-gle-electron approximation. The analysis generalizes previous treatments based on more phenomenological models of spinorbit cou-pling. It is shown that an alternative time-reversal symmetry adapted diabatic representation exists, in which the 4 4 JahnTellerspinorbit vibronic Hamiltonian is block-diagonal. The adiabatic electronic wave functions are obtained in explicit form. The topologicalphases (Berrys phases) of all four adiabatic electronic basis states are calculated. As found previously for the special case of systems withD3h symmetry, the topological phases depend on the radius of the loop of integration. 2005 Elsevier B.V. All rights reserved.

Keywords: JahnTeller effect; Spinorbit coupling; Geometric phase1. Introduction

The interplay of JahnTeller (JT) and spinorbit (SO)interactions in degenerate electronic states of molecularsystems has extensively been studied since decades, begin-ning with a paper by Jahn in 1938 [1]. The most commoncase is the E E JT effect, where a doubly degeneratevibrational mode lifts the degeneracy of a doubly degener-ate electronic state in first order [26]. It has been shown bymany authors that the JT coupling and the SO couplingtend to quench each other, see, e.g. [6,8].

A particular interesting aspect of JT systems (or, moregenerally, of conical intersections) is the nontrivial topologi-cal phase (also referred to as Berrys phase) carried by theadiabatic electronic wave functions [911]. The first analysisof the effect of SO coupling on the topological phase of a JTsystem has been given by Stone within a phenomenologicalmodel of the JT and SO interactions of a single electronin the field of three nuclei arranged in an equilateral triangle0301-0104/$ - see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.chemphys.2005.09.009

* Corresponding author. Fax: +49 89 289 13622.E-mail addresses: domcke@ch.tum.de, chemphys@ch.tum.de (W.

Domcke).[12]. Mead [13] and Yarkony and collaborators [1418] haveextended this analysis to the more general case of conicalintersections. Matsika and Yarkony and Han and Yarkonyhave given a comprehensive analysis of conical intersectionswith inclusion of SO coupling in the general case of no sym-metry [17,18]. The most detailed analysis of geometric phasesin the special case of E E JTSO systems has been given byKoizumi and Sugano [19] as well as Schon and Koppel [20].

It seems that the explicit analysis of geometric phaseeffects in E E JTSO systems has so far been restrictedto systems which exhibit an additional (horizontal) planeof symmetry of the nuclear frame, such as systems withthree identical nuclei forming an equilateral triangle[12,13,19,20]. The significance of such a plane of symmetryfor the existence of conical intersections and geometricphases has first been pointed out by Mead [13,21] andYarkony [1418]. While in general five conditions mustbe fulfilled to have an exact degeneracy of adiabatic termsin SO-coupled systems, the number of conditions reducesto three when a plane of symmetry exists [1421]. The planeof symmetry simplifies the JTSO problem, since certainmatrix elements of the SO operator with nonrelativisticelectronic wave functions vanish in this case [13].

mailto:domcke@ch.tum.demailto:chemphys@ch.tum.de

406 W. Domcke et al. / Chemical Physics 322 (2006) 405410It has been shown that the E E JTSO problem,which in general involves four coupled electronic states,decouples into two identical two-state subproblems ifthe plane of symmetry exists [1921]. The Kramersdegeneracy of the energy levels of odd-electron systems,which in general is the consequence of time-reversal sym-metry of the Hamiltonian, reduces to spin degeneracy inthis special case, that is, the z-axis projections of the spinof the unpaired electron are good quantum numbers[19,20].

In the absence of a plane of symmetry, the electronicangular momentum and spin projections are not con-served, and Kramers degeneracy is nontrivial in the sensethat it arises solely from time-reversal symmetry. It hasbeen pointed out by Mead [22] that interesting geometricphase factors associated with non-Abelian gauge groupsare expected in this case.

In the present work, we consider the E E JT effect withinclusion of SO coupling in molecular systems with a singlethreefold symmetry axis, which do not possess a horizontalplane of symmetry. Examples are four-atomic radicals orradical cations with C3v symmetry, such as CH3S, CH3I

+

or PF3 . We restrict ourselves to linear JT coupling (the ef-fects of quadratic JT coupling on the geometric phases hasbeen analyzed, for example, in [20]). Unlike the previousinvestigations, where simplified models of the SO interac-tion of the type

HSO ALzSz; 1

where Lz, Sz are the z-axis projections of the orbital andspin angular momentum and A is a phenomenologicalconstant, have been assumed, we base our analysis onthe microscopic BreitPauli SO operator in the sin-gle-electron approximation [23,24]. We shall derive the4 4 JTSO Hamiltonian matrix in a time-reversal sym-metry adapted (in short, T-adapted) basis of nonrelativ-istic diabatic spinorbital electronic states. It will beshown that alternative T-adapted diabatic basis statescan be introduced, which render the JTSO Hamiltonianblock-diagonal. In this new basis, the electronic angularmomentum and spin projections are no longer goodquantum numbers. The adiabatic electronic wave func-tions of the 4 4 JTSO problem and their geometricphase factors will be obtained in closed analytic form.2. The vibronic Hamiltonian in the diabatic representation

Let us consider, as an example of an odd-electronmolecular system with a threefold symmetry axis, a pyra-midal molecule with C3v symmetry of the nuclear frame.We describe the unpaired electron in an effective single-par-ticle picture. The three identical nuclei are arranged in thex, y plane. The fourth atom is located above or below theorigin of the x, y plane.

The electronic Hamiltonian consists of an electrostaticand a SO partHel Hes HSO; 2

Hes 1

2r2

X3n1

Qrn

Q4r4

; 3

HSO Axrx Ayry Azrz. 4

Here Q(=Q1 = Q2 = Q3) and Q4 are effective nuclearcharges. The rn, n = 1. . .4, are the electronnuclei distances

rn x Xn2 y Y n2 z2h i1

2

; n 1; 2; 3

r4 x2 y2 z Zn2h i1

2

;

5

and the rx, ry, rz are the Pauli spin matrices. The Ax, Ay,Az are differential operators in electronic coordinate space.Their explicit form is given in Appendix A. The operatorHSO in Eq. (4) is the well-known BreitPauli operatorfor a single electron moving in the field of four nuclear cen-ters [23,24].

The electronic Hamiltonian specified by Eqs. (2)(4)commutes with the operator C3 of the the C3v point groupas well as with the time-reversal operator,

T 0 11 0

c.c; 6

where c.c denotes the operation of complex conjugation,

Hel;C3 0 7Hel; T 0. 8

Note that the vertical symmetry planes (3rv) are symmetryelements of the electrostatic Hamiltonian, but not of theSO operator (HSO is antisymmetric with respect to rv).

When an electron moving in the field of three identicalnuclei is considered, as in [12,13,19], the electrostatic Ham-iltonian commutes with the reflection operation with re-spect to the molecular plane (rh). As can be seen fromthe explicit expressions given in Appendix A, Ax and Ayare antisymmetric with respect to rh, while Az is symmetric.As a result, the matrix elements of Ax and Ay with a givenorbital of Hes vanish by symmetry. Only matrix elementsof Az are nonzero. For this reason, the four-atomic modelconsidered here, which lacks the symmetry element rh, ismore general than the previously considered models ofthree identical nuclei [12,13,19].

Let us next introduce atom-centered basis functions

vn v~rn; n 1; 2; 3; 4;~rn ~r ~Rn; n 1; 2; 3;~r4 ~r ~R4;~R

T

n Xn; Y n; 0; n 1; 2; 3;~R

T

4 0; 0; Z4.

9

To take advantage of the C3v symmetry of the electrostaticHamiltonian, we introduce the symmetry-adapted linearcombinations

z

W. Domcke et al. / Chemical Physics 322 (2006) 405410 407wE1 6122v~r1 v~r2 v~r3;

wE2 212v~r2 v~r3;

w1A 312v~r1 v~r3;

w2A v~r4.

10

wE1 and wE2 span the Hilbert space of a degenerate elec-tronic state of E symmetry. In the following, only this 2Estate is considered, assuming that is sufficiently separatedin energy from the other electronic states.

Following Longuet-Higgins [25], we introduce complex-valued electronic basis functions

w 212wE1 iwE2 11

312 v~r1 e2pi3 v~r2 e

2pi3 v~r3

h iw 2

12wE1 iwE2

312 v~r1 e2pi3 v~r2 e

2pi3 v~r3

h i.

These functions transform as follows under C3

C3w e2pi3 w

C3w e2pi3 w.

12

Including the spin function of the electron, we have thefollowing nonrelativistic basis states

w1 wjai;w2 wjai;w3 wjbi;w4 wjbi;

13

where a (b) denotes the spin projection 12 1

2 of the un-

paired electron. These electronic basis functions are T-adapted in the sense of Mead [21]

Tw1 w4 Tw4 w1;Tw2 w3 Tw3 w2.

14

Consider next symmetry-adapted nuclear displacementsfrom the C3v equilibrium geometry. The bond-stretchingcoordinates, for example, are given by

dRE1 6122dR1 dR2 dR3;

dRE2 212dR2 dR3;

dRA 312dR1 dR2 dR3.

15

As is well known, normal modes are obtained by the diag-onalization of the F and G matrices [26]. According to theJT theorem [27], only the normal modes of E symmetry arerelevant for the vibronic coupling within the 2E electronicstate. We consider here a single E mode with dimensionlessnormal coordinates Qx, Qy. It is convenient to introducethe complex-valued combinations

Q Qx iQy qeiu; 16

where q and u are polar coordinates in the QE1 , QE2 plane.The transformation properties of Q under C3 areC3Q e2pi3 Q

C3Q e2pi3 Q.

17

The vibronic Hamiltonian is obtained by expanding Helof Eqs. (2)(4) in a Taylor series up to first order in Q andevaluating matrix elements with the diabatic basis states(Eq. (13)), taking account of the symmetry properties ofEqs. (7), (8), (12), (14) and (17). The result is

HelHesHSO; 18

Hes

0 jqeiu 0 0

jqeiu 0 0 0

0 0 0 jqeiu

0 0 jqeiu 0

0BBB@

1CCCA; 19

HSO

Dz 0 Dx iDy 00 Dz 0 Dx iDyDx iDy 0 Dz 00 Dx iDy 0 Dz

0BBB@

1CCCA. 20

The energy E0 of the doubly degenerate electronic state atthe reference geometry has been set to zero. The (real)parameter j in Eq. (19) is the well-known coupling con-stant of the (nonrelativistic) linear E E JT effect [27].The parameters Dx, Dy, Dz are matrix elements of the SOoperator and are real numbers. Their definition is givenin Appendix A. They are independent of the nuclear coor-dinates (q, u) up to first order in q. Note that the zeros inthe cross-diagonal of Eq. (20) are a consequence of thetime-reversal invariance of the T-adapted electronic basisfunctions (Eq. (14)). It should also be noted that theBreitPauli SO operator is not diagonal in the nonrelativ-istic diabatic basis (13). Including only its diagonal ele-ments [28] is an approximation.

Including the harmonic part of the potential-energy sur-face with vibrational frequency hx (which arises from thespin-paired electrons not explicitly considered here) andthe nuclear kinetic-energy operator, we have the final vib-ronic Hamiltonian

H TN h2xq2

1Hes HSO. 21

Here 1 denotes the four-dimensional unit matrix and TN isgiven by

TN hx2

o2

oq2 1

qo

oq 1

q2o2

ou2

. 22

As mentioned above, the SO matrix elements Dx, Dy vanishif Hes possesses a horizontal plane of symmetry. The SOcoupling is then characterized by the single parameter Azand the vibronic Hamiltonian is block diagonal.

Hes HSO

Dz jqeiu 0 0

jqeiu Dz 0 00 0 Dz jqeiu

0 0 jqeiu D

0BBB@

1CCCA. 23

408 W. Domcke et al. / Chemical Physics 322 (2006) 405410In this special case, considered previously in [68,12,19,20], for example, the spin projections are goodquantum numbers; the upper-left (lower-right) block ofEq. (23) corresponds to the spin projections 1

2 1

2.

3. Transformation of the vibronic Hamiltonian to block-diagonal form

At the reference geometry (q = 0), the electrostatic partHes of the vibronic potential-energy matrix vanishes. Theremaining matrix HSO of Eq. (20) can be diagonalized,by the following constant (that is, independent of q andu) unitary transformation matrix

S

0 0 eiv2 sin c e

iv2 cos c

eiv2 sin c e

iv2 cos c 0 0

0 0 eiv2 cos c e

iv2 sin c

eiv2 cos c e

iv2 sin c 0 0

0BBBB@

1CCCCA; 24

where

cos c 1ffiffiffi2

p 1 DzD

12

sin c 1ffiffiffi2

p 1 DzD

12

v argDx iDy

25

and

D2 D2x D2y D2z . 26The transformed Hamiltonian reads

SHS TN h2xq2

1

D 0 jqeiu 0

0 D 0 jqeiu

jqeiu 0 D 00 jqeiu 0 D

0BBB@

1CCCA.

27Reordering rows and columns, we obtain the Hamilto-

nian in block-diagonal forms (see also [11], Appendix C)

H0 TN h2

xq2

1

D jqeiu 0 0

jqeiu D 0 00 0 D jqeiu

0 0 jqeiu D

0BBB@

1CCCA.

28Note that the electronic basis states corresponding to thevibronic Hamiltonian matrix (28) are diabatic states, sincethe unitary transformation S is independent of q and u.They are related to the original spinorbital diabatic elec-tronic basis states wk (Eq. (13)) as follows:

/1/2/3/4

0BBB@

1CCCA

eiv2 cosc 0 e

iv2 sinc 0

0 eiv2 cosc 0 e

iv2 sinc

eiv2 sinc 0 eiv2 cosc 00 eiv2 sinc 0 eiv2 cosc

0BBBB@

1CCCCA

w1w2w3w4

0BBB@

1CCCA.

29Eq. (29) reveals that the new diabatic basis states involvelinear combinations of the original basis states (Eq. (13))with different orbital angular momentum and spin projec-tions. Thus, neither the electronic angular momentum pro-jections (K = 1) nor the spin projections (ms = 1/2) aregood quantum numbers in the basis which block-diagonal-izes the vibronic Hamiltonian. The block-diagonal struc-ture is a consequence of the time-reversal symmetry ofthe Hamiltonian.

The basis states /k are T-adapted in the same manner asthe original basis states

T/1 /4 T/4 /1;T/2 /3 T/3 /2.

30

Note that the operator T does not mix the pairs (/1, /4)and (/2, /3). The formal substitution D ! D interchangesthe pairs (/1, /4) and (/2, /3).

The relativistic linear E E JT Hamiltonian (28) has thesame form as that derived previously for trigonal moleculeswith a horizontal symmetry plane [19,20]. The SO parame-ter Dz of the latter case is replaced by the parameter D de-fined in Eq. (26).

4. Adiabatic potential-energy surfaces and adiabatic

electronic wave functions

It is trivial to diagonalize the Hamiltonian (28) in thefixed-nuclei limit (TN ! 0). The adiabatic terms are

U 1;4q h2xq2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2q2 D2

q;

U 2;3q h2xq2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2q2 D2

q.

31

Note that the adiabatic terms are determined by the singleparameter D defined in the Eq. (26), rather than by the indi-vidual matrix elements Dx, Dy, Dz. Eq. (31) represents tworotationally symmetric hyperboloids in QE1 , QE2 space.The degeneracy of the nonrelativistic eigenvalue at q = 0is lifted by the SO splitting 2D. Each adiabatic term is dou-bly degenerate (Kramers degeneracy).

The unitary matrix which diagonalizes the upper-leftblock of the potential matrix reads

W eiu2ig1 cos h e

iu2ig2 sin h

eiu2ig1 sin h eiu2ig2 cos h

!; 32

where

cos h jqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2q2 D2

p D

2 q2j2

r

sin h U 1;4 Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2q2 D2

p D

2 q2j2

r < 0 33

and g1, g2 are arbitrary phases. For the lower-right block ofthe potential matrix, the unitary transformation matrix is

W. Domcke et al. / Chemical Physics 322 (2006) 405410 409W eiu2ig3 cos h eiu2ig4 sin heiu2ig3 sin h eiu2ig4 cos h

!; 34

where g3, g4 again are arbitrary phases. The adiabatic elec-tronic functions are thus given in the following form

/ad1 eig1 eiu2 cos h/1 e

iu2 sin h/2

;

/ad2 eig2 eiu2 sin h/1 e

iu2 cos h/2

;

/ad3 eig3 eiu2 cos h/3 e

iu2 sin h/4

;

/ad4 eig4 eiu2 sin h/3 e

iu2 cos h/4

.

35

These adiabatic electronic wave functions have to be T-adapted, that is, we require

T/ad1 /ad4 T/

ad4 /

ad1 ;

T/ad2 /ad3 T/

ad3 /

ad2 .

36

The adiabatic functions /ad1 , /ad4 belong to the upper adi-

abatic potential-energy sheet U1,4(q) and are transformedinto each other by time reversal. The functions /ad2 , /

ad3

are likewise associated with the lower potential-energysheet U2,3(q). The conditions (36) determine the phase an-gles g1 g4, with the result

/ad1 eiu2 cos h/1 e

iu2 sin h/2

;

/ad2 i eiu2 sin h/1 e

iu2 cos h/2

;

/ad3 i eiu2 cos h/3 e

iu2 sin h/4

;

/ad4 eiu2 sin h/3 e

iu2 cos h/4

.

37

The adiabatic electronic wave functions (37) are double-valued functions of the angular coordinate u (because ofthe phase angle e

iu2 ). They can alternatively be written as

single-valued functions

~/ad

1 eiu cos h/1 sin h/2

;

~/ad

2 i eiu sin h/1 cos h/2

;

~/ad

3 i cos h/3 eiu sin h/4

;

~/ad

4 sin h/3 eiu cos h/4

.

38

The dependence of the /adk on the nuclear coordinates q,u gives rise to nondiagonal matrix elements of TN and thusto nonadiabatic couplings.

The single-valued form (38) of the adiabatic wave func-tion is most suitable for the calculation of the topologicalphase according to the expression [29]

cnC iIc

d~Rhn~RjrRn~Ri; 39

where n~R is a single-valued adiabatic wave functionwhich depends parametrically on the nuclear coordinate~R. The integral is to be evaluated over a closed path inparameter space q, u which encloses the origin q = 0.Let us illustrate the calculation of the topological phasefor the wave function ~/

ad

1 (Eq. (38)). Integrating over a cir-cle of radius R around the origin, we have

c1 iZ 2p0

Rdu eiu cosh; sin h 1

Ro

ou

eiu cos h

sin h

qR

40

c1 p 1Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2j2 D2p

!. 41

The topological phases of the other three adiabatic statesare

c2 p 1Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2j2 D2p

!; 42

c3 c2; 43c4 c1. 44

The two components of an adiabatic electronic Kra-mers doublet thus have geometric phases of oppositesigns. As found previously for planar triatomic systems[12,13,19,20], the geometric phases depend on the radiusof the integration contour when SO coupling is included.In the nonrelativistic case (D = 0), on the other hand, wehave c = p, independent of R. In the limit jR D, thatis, for a large radius of the integration contour, Eq. (41)yields c1 = c2 p, c3 = c4 p like in the nonrelativisticcase. In the opposite limit jR D, we find c4 = c1 2pand c2 = c3 0, that is, the absence of a nontrivial geo-metric phase. This result indicates that the whole area en-closed by the integration contour contributes to thegeometric phase in the general case, while in the nonrela-tivistic case the source of the geometric phase is exclu-sively the conical intersection at q = 0 [13].5. Conclusions

We have derived the Hamiltonian of the linear E E JTeffect in trigonal systems with the inclusion of SO coupling,employing the microscopic (BreitPauli) SO operator in thesingle-electron approximation. This analysis generalizesprevious derivations for planar triatomic systems, wherethe existence of a plane of symmetry eliminates some ofthe SO coupling elements.

When the usual diabatic spin orbitals are employed aselectronic basis functions, the vibronic Hamiltonian is a4 4 matrix, that is, four electronic basis states are coupledby the vibronic and SO interactions. We have shown thatthe linear JTSO Hamiltonian can be transformed toblock-diagonal form by a constant (geometry-independent)unitary transformation.

The electronic angular momentum and spin projectionsare no longer good quantum numbers in this transformedbasis. These basis states, like the original spin orbitals,are T-adapted, that is, they form two pairs which aremapped onto each other by the time-reversal operator.

410 W. Domcke et al. / Chemical Physics 322 (2006) 405410The adiabatic electronic states, which diagonalize theJTSOHamiltonian in the fixed-nuclei approximation, havebeen obtained in explicit form. It has been shown that theycan be made T-adapted like the diabatic electronic basisstates. Each of the two sheets of the adiabatic potential-energysurfaces is doublydegenerate.This is a (nontrivial) con-sequence of the time-reversal symmetry of the Hamiltonian.

The geometric phases of the adiabatic electronic wavefunctions have been calculated for the four-state system.The geometric phases of the two adiabatic electronic stateswithin a Kramers doublet have opposite signs. If the radiusof the integration contour encircling the origin of the nuclearcoordinate phase is sufficiently large, the geometric phases ofthe JTSO effect are p, like in the nonrelativistic case. Inthe opposite limit, the geometric phases tend towards zeroor 2p. The latter results agree with those obtained previouslyfor planar triatomic systems [12,13,19,20].

Acknowledgments

This work has been supported by the Deutsche Fors-chungsgemeinschaft by a research grant as well as by a vis-itor grant for L.V.P.

Appendix A

The differential operators Ax, Ay, Az areAx igb2X3n1

Qr3n

yno

oz z o

oy

igb2 Q4

r34yo

oz z4

o

oy

;

A1

Ay igb2X3n1

Qr3n

zo

ox xn

o

oz

igb2 Q4

r34z4

o

ox x o

oz

;

Az igb2X3n1

Qr3n

xno

oy yn

o

ox

igb2 Q4

r34xo

oy y o

ox

;

where

xn x Xn;yn y Y n;

rn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2n y2n z2

q; n 1; 2; 3;

z4 z Z4;

r4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2 z24

q.

A2Xn, Yn, Zn are coordinates of the nuclei, g 2.0023 is g-fac-tor of the electron and b eh

2mecis the Bohr magneton.

The expressions for Dx, Dy, Dz are given below

Dx 1

2

ZwAxw d

3r 12

ZwAxw d

3r;

Dy 1

2

ZwAyw d

3r 12

ZwAyw d

3r;

Dz 1

2

ZwAzw d

3r 12

ZwAzw d

3r.

A3References

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The relativistic E times E Jahn - Teller effect revisitedIntroductionThe vibronic Hamiltonian in the diabatic representationTransformation of the vibronic Hamiltonian to block-diagonal formAdiabatic potential-energy surfaces and adiabatic electronic wave functionsConclusionsAcknowledgmentsAppendix AReferences