Jahn-Teller Systems

VOLUME 83, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1999

Multiconical Intersections and Nondegenerate Ground State in E ≠ e Jahn-Teller Systems

Hiroyasu Koizumi* and Isaac B. BersukerDepartment of Chemistry and Biochemistry, The University of Texas, Austin, Texas 78712

(Received 18 June 1999)

We have investigated the E ≠ e Jahn-Teller problem with large quadratic coupling. Our resultsreverse the paradigm that, in E ≠ e Jahn-Teller systems, the ground vibronic state should be the sameas that of the initial doubly degenerate electronic E term. The E vibronic state is the ground statefor small quadratic coupling when the dominant tunneling path between the potential surface minimaencircles one central conical intersection, whereas for large quadratic coupling it goes around fourconical intersections and the ground state becomes nondegenerate.

PACS numbers: 71.70.Ej, 03.65.Bz, 31.30.Gs, 82.90.+ j

There is a widespread belief that the ground state sym-metry of any vibronic system is the same as that of the ini-tial degenerate electronic state. The assumed ground statesymmetry significantly affects the interpretation of the ob-served properties of the system. In particular, this assump-tion lies in the base of the theory of vibronic reductionfactors widely used to derive the ground state propertiesfrom those of the degenerate electronic state [1,2]. There-fore, the problem of whether the ground state symmetryis that of the initial degenerate electronic state or not is amatter of fundamental importance. The concern about thispoint has been increased after the finding that for some pa-rameter values the ground vibronic state in the Jahn-TellerH ≠ h problem is nondegenerate [3], which shattered theparadigm that the vibronic ground state should have thesame degeneracy and symmetry as the initial degenerateelectronic state.

In this Letter, we revisit the E ≠ e Jahn-Teller sys-tem with strong vibronic coupling including quadratic andfourth order terms, and examine whether the nondegener-ate A symmetry ground state exists or not. Contrary to theexperience-based belief, we found that the ground state ofthis system may be nondegenerate, of A type. It existsin cases where the quadratic vibronic coupling constant issufficiently large.

In the case of the strong vibronic coupling, the tun-neling between the minima of the adiabatic potential ofJahn-Teller systems was first studied by Bersuker [4]. Hestarted from the deep minima of the adiabatic potentialof the ground state and diagonalized the full Hamilton-ian using the displaced harmonic oscillators as a basis set.The ground state of the system was found in this way tobe nondegenerate with the twofold degenerate E tunnel-ing level higher in energy. This result was questionedafter the work of O’Brien [5], in which numerical calcu-lations starting from the linear E ≠ e problem and treat-ing quadratic vibronic coupling as a perturbation yield thedoubly degenerate E level as the ground term. The subse-quent works on the E ≠ e problem followed this schemeand resulted in the E ground state, contributing thus to theformation of the above-mentioned paradigm.

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The double degeneracy of the vibronic ground state wasfirst obtained by Longuet-Higgins et al. [6]. In the case ofstrong vibronic coupling with deep potential minima it canbe associated with the Berry phase [5,7,8] arising fromthe conical intersection of the adiabatic potential surface.O’Brien attached a phase factor to the wave functionin order to satisfy the sign-change boundary condition[5]. Polinger obtained the E symmetry ground stateconsidering a closed loop around the conical intersectionat the center and using the WKB connection formulafor the tunneling by imposing an antiperiodic boundarycondition [9]. It is also shown by Ham that the sign-change boundary condition required by the Berry phasegives the doubly degenerate ground state [8].

However, in the case of sufficiently strong quadraticcoupling, three additional conical intersections on the in-terminimum barriers come close to the one at the center.Then, two types of important tunneling paths that penetratethe interminimum barrier are possible: one goes throughthe saddle point between the conical intersections at thecenter and at the barrier, and the other goes through thesaddle point outside of both of them. The dominant tun-neling path will be the one that yields the larger tunnelingrate between the two. It is plausible that the two types oftunneling paths, mentioned above, form parts of two dif-ferent stationary semiclassical paths for the path integralrepresentation of the resolvent [10], where one of themencircles the conical intersection at the center only, andthe other goes around all four. As shown by Zwanzigerand Grant, the electronic wave function does not changesign after the circular transportation along a closed loopthat goes outside of all four conical intersections [11]. Byemploying the same type of argument as that of Polinger[9] for this closed path, it can be easily shown that theground state of this system is nondegenerate with A sym-metry. Then we can assume that the change of the groundstate symmetry and degeneracy from A to E is correlatedwith the change of the dominant tunneling path. We willshow below that this is indeed the case and the changeof the ground state degeneracy (the A-E level intersec-tion) is directly correlated to the change of the dominant

© 1999 The American Physical Society 3009


tunneling path which is estimated by the one-dimensionalWKB tunneling rate.

The model Hamiltonian for doubly degenerate elec-tronic states �ja�, jb�� of the E representation anddoubly degenerate normal vibrational coordinatesQ1 � r cosu, Q2 � r sinu of the E representation isgiven by

H � H0 1 H1 1 H2 , (1)where

H0 �X

e�a,bh2DHOje� �ej , (2)

where h2DHO the Hamiltonian for the two-dimensionalisotropic harmonic oscillator,

h2DHO � 212

≠2

≠r2 21

2r

≠

≠r2

12r2

≠2

≠u2 112

r2,

(3)H1 is the vibronic interaction Hamiltonian given by

H1 �

µkre2iu 1

12

gr2ei2u

∂ja� �bj 1 H.c. , (4)

and H2 is the fourth order term

H2 �X

e�a,b

fr4je� �ej . (5)

This term is included in order to have a bound groundstate for a large quadratic coupling constant g $ 1 [12].

The adiabatic potential surfaces are obtained as

U6 �12

r2 1 fr4 6

sk2r2 1 kgr3 cos3u 1

14

g2r4 ,

(6)

with double-valued adiabatic electronic wave functionsjw1� � �e2ia�2ja� 1 eia�2jb��

p2 and jw2� �

�e2ia�2ja� 2 eia�2jb��p

2, for U1 and U2, respectively,where a � tan21��k sinu 2 0.5gr sin2u��k cosu 1

0.5gr cos2u�.We diagonalize the Hamiltonian (1) numerically

using the basis set je� ≠ �r, ujy, m�, e � a, b,y � 1, 2, . . . , m � y 2 1, y 2 3, . . . , 2y 1 1, where

�r, ujy, m� � �21�n

µn!e2r2

p�n 1 jmj�!

∂1�2

rjmjLjmjn �r2�e2imu ,

(7)

with n � �y 2 jmj 2 1��2. The basis functions�r, ujy, m� are eigenfunctions for the two-dimensionalharmonic oscillator Hamiltonian h2DHO with eigen-values Eym � y. Nonzero matrix elements for the linearvibronic coupling term re2iu are given by [6,13]

�y, mjre2iujy 1 1, m 1 1� �p

�y 1 m 1 1��2 ,

�y, mjre2iujy 2 1, m 1 1� �p

�y 2 m 2 1��2 ,

(8)

and matrix elements for reiu , r2e2i2u , r2ei2u , and r4

can be calculated using Eq. (8).

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Let us start with the f � 0 case. In this case, thequadratic coupling parameter g must be smaller than 1in order to have the U2 value bounded from below. InFig. 1(a), vibronic energies for the lowest eight statesof the k � 0 and the f � 0 case are shown [14]. Atg � 0.918 the crossing of the lowest E and the lowestA energy levels occurs and the nondegenerate A statebecomes the lowest for g . 0.918. Even when k fi 0(a small linear coupling term is turned on), the A statebecomes the ground state at large g values. With k � 0.1,the ground state becomes an A state when g is larger than0.922 [Fig. (b)]. If the quadratic coupling parameter g islarger than 1, the fourth order terms must be included. InFig. 1(c), the case k � 0.5 and f � 0.5 is shown. Thecrossing of the lowest E and A states occurs at g � 4.50.

Now let us consider the situation where the vibroniccoupling is very strong, and the ground state nucleardynamics is considered to be performed on the singlepotential surface with deep minima. In this case, theenergy difference between the lowest A and E statesis considered as due to the tunneling splitting [4,9].The relevant potential energy surface is given by V �U2 1 HBH, where HBH is the Born-Huang term [15] (orcentrifugal energy [2]) given by

HBH �12

2Xi�1

ø≠w2

≠Qi

Ç≠w2

≠Qi

¿�

18

�=a�2, (9)

where

=a �2kg sin3uer 1 �2k2r21 2 k2r 2 kg cos3u�eu

2k2 112g2r2 1 2kgr cos3u

.

(10)

The potential energy surfaces U6 exhibit a conicalintersection at the center r � 0 and three periph-eral ones at �Q1, Q2� � �2kg21 cos�321�2n 1 1�p,2kg21 sin�321�2n 1 1�p�, n � 0, 1, 2 (Fig. 2) [16]. Ateach conical intersection �Q1c, Q2c�, HBH provides a

0.9 0.91 0.92 0.93 0.94 0.95

0.4

0.6

1

g

E

4 4.2 4.4 4.6 4.8 5

0

1

2

3

g

E

0.9 0.91 0.92 0.93 0.94 0.950.4

0.6

0.8

1

1.2

g

E

(a) (b) (c)

ω

ω2 ω2 ω2

ω ω

FIG. 1. Energies for the lowest eight vibronic states versusthe quadratic coupling constant g. Solid and dotted lines de-note nondegenerate and doubly degenerate states, respectively.(a) k � 0, f � 0; (b) k � 0.1, f � 0; (c) k � 0.5, f � 0.5.


FIG. 2. Potential surface of the E ≠ e system for k � 0.9,g � 2, and f � 0.05. (a) Cross section of the potentialsurfaces V (solid line) and U6 (dotted line) at Q2 � 0. C1 andC2 indicate conical intersections, while P1 and P2 indicate twosaddle points on V . (b) Contour plot of the potential surfaceV . “�” and “1” indicate a conical intersection and a minimum,respectively. Two steepest decent tunneling paths P1 and P2are shown by thick lines.

diverging term proportional to ��Q1 2 Q1c�2 1 �Q2 2

Q2c�221 r22c because the adiabatic wave function

depends on the circular angle around the conical intersec-tion uc tan21��Q2 2 Q2c��Q1 2 Q1c�, which givesrise to the contribution proportional to jr21

c ≠jw2��≠ucj2.

Thus, the access of the nuclear motion to conical intersec-tions is prohibited. A fictitious magnetic field that givesrise to the appearance of the Berry phase and causes thesign change of the wave function exists at each conicalintersection.

Two steepest decent paths that connect nearby minimathrough two different saddle points, one between theconical intersections at the center and at the barrier �P1�and the other outside both of them �P2� for the valuesk � 0.9, g � 2.0, and f � 0.05, are shown in Fig. 2. Asis seen from Fig. 2(a), HBH gives a large contribution toenergy near the conical intersections and makes the barrierfor the path P1 significantly higher than that for the pathP2. As a result, the barrier for P1 is high and thin, whilethat for P2 is characterized as low and thick.

The absolute values of the energy difference betweenthe lowest A and E states for three values g � 2.0, 1.9, 1.8and f � 0.05 by varying k from 0.5 to 1 are shown inFig. 3(a). The triply degenerate ground state intersectionpoints are obtained at k � 0.82, 0.68, and 0.54 for g �2.0, 1.9, and 1.8, respectively. If k is larger than thisvalue, the ground state is E, and if it is smaller the groundstate is A symmetry.

We estimate the tunneling rate for P1 and P2 using theWKB method given by

exp�2Si� � exp

µ2

ZPi

p2�V 2 E� dq

∂, i � 1, 2 .

(11)

In Fig. 3(b), j exp�S1 2 S2� 2 1j and the correlationof the g versus k values at the intersection points

FIG. 3. (a) The absolute values of the energy differencejEA 2 EEj between the lowest A and E levels, and (b) theratios of the tunneling rates via P1 and P2 paths (plotted asj exp�S1 2 S2� 2 1j) as functions of k for f � 0.05 and threevalues of g � 2.0, 1.9, and 1.8. Calculated points are indicatedby “1” and polynomially interpolated curves for g � 2.0, 1.9,and 1.8 cases are depicted by solid, dotted, and dash-dottedlines, respectively, in (b). Inset: The g versus k values at thecrossing points EA � EE (solid line) and at S1 � S2 (dottedline).

EA � EE with the corresponding values at S1 � S2are shown. A qualitatively good correlation betweenthe crossing points of energy levels EA � EE and byequitunneling rate points via the two paths S1 � S2 isobtained [Fig. 3(b), inset]. Note that the present estimateof the tunneling rate based on the one-dimensional WKBis not very accurate. A more accurate estimate includingthe multidimensionality of the tunneling [17] may yield aquantitatively better correlation. This strong correlationbetween the change of the degeneracy of the ground stateand the change of the dominant tunneling path suggeststhat the former is caused by the latter through the changeof boundary condition on the wave function, which givesus a clear explanation of the origin of the energy levelE-A crossover as due to the presence of two alternativetunneling passes which yield two Berry phases, p and4p, respectively.

The change of ground state symmetry and degeneracyas a function of intermolecular interactions seems to beof general importance. For instance, it follows fromthe crystal field theory that the ground state symmetrychanges with the strength of the crystal (or ligand) field;the phenomenon is known as spin crossover and it hassignificant implications in magnetic materials [18]. Theenergy level crossover discussed above may be regardedas a part of this general understanding.

In summary, we confirmed the existence of the nonde-generate vibronic ground state in the E ≠ e Jahn-Tellersystems first suggested in [4]. It exists in cases where thequadratic coupling constant is sufficiently large [19]. Wehave also elucidated the mechanism of the E-A energylevel crossover by demonstrating the correlation betweenthe change of the degeneracy of the tunneling-split groundstate and the change of the dominant tunneling path:one goes through the saddle point between the conical

3011


intersections at the center and at the barrier, the other goesthrough the saddle point outside of both of them.

The authors acknowledge fruitful discussion withDr. V. Z. Polinger and Dr. H. Köppel. H. K. is supportedin part by the travel grant from Hyogo prefectural gov-ernment in Japan.

*On leave of absence from Faculty of Science, HimejiInstitute of Technology, Kamigori, Ako-gun, Hyogo678-1297, Japan.

[1] R. Englman, The Jahn-Teller Effect in Molecules andCrystals (Wiley, New York, 1972).

[2] I. B. Bersuker and V. Z. Polinger, Vibronic Interactions inMolecules and Crystals (Springer-Verlag, Berlin, 1989).

[3] M. C. P. Moate, C. M. O’Brien, J. L. Dunn, C. A. Bates,Y. M. Liu, and V. Z. Polinger, Phys. Rev. Lett. 77, 4362(1996).

[4] I. B. Bersuker, Opt. Spectrosc. 11, 319 (1961); Zh. Eksp.Teor. Fiz. 43, 1315 (1962) [Sov. Phys. JETP 16, 933(1963)].

[5] M. C. M. O’Brien, Proc. R. Soc. London A 281, 323(1964).

[6] H. C. Longuet-Higgins, U. Öpik, M. H. L. Pryce, and R. A.Sack, Proc. R. Soc. London A 244, 45 (1958).

[7] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).[8] F. S. Ham, Phys. Rev. Lett. 58, 725 (1987).[9] V. Z. Polinger, Sov. Phys. Solid State 16, 1676 (1975).

[10] See, for example, J. W. Negele and H. Orland, QuantumMany-Particle Systems (Perseus Books, Reading, MA,1988), Sect. 7.4.

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[11] J. W. Zwanziger and E. R. Grant, J. Chem. Phys. 87, 2954(1987).

[12] The units of energy, time, length, and mass-weightedlength (for normal coordinates) in the present Letter aregiven by h̄v, v21, �h̄�mv�1�2, and �h̄�v�1�2, respec-tively, where h̄v and m are the energy quantum and massfor the two dimensional harmonic oscillator, respectively.

[13] H. Koizumi and S. Sugano, J. Chem. Phys. 101, 4903(1994); 103, 7651(E) (1995).

[14] The lowest A state for the k � 0 and f � 0 casehas the energy 2

p1 2 g with the eigenfunction FA �

�1�p

2� �ja� ≠ eiu 2 jb� ≠ e2iu�r�r, uj2, 0�.[15] M. Born and K. Huang, Dynamical Theory of Crystal

Lattices (Oxford University, London, 1954), App. VIII.[16] For k fi 0 and g � 0 values, three peripheral conical

intersections are considered to exist at limg!02kg21 �`, while for the k � 0 and g fi 0 all four conicalintersections merge into one Renner–Teller-type crossing.This latter situation is actually found at near regulartetrahedral configurations in the H4 system [H. Koizumiet al. (unpublished)].

[17] See, for example, H. Ushiyama and K. Takatsuka,J. Chem. Phys. 106, 7023 (1997), and references therein.

[18] See, for example, I. B. Bersuker, Electronic Structureand Properties of Transition Metal Compounds (Wiley,New York, 1996).

[19] Using a variational method, Bersuker observed the E-Aenergy level crossing for large quadratic coupling in 1972,but to avoid publication of a possible inaccuracy only thedata for smaller quadratic couplings before the crossing,were published [I. B. Bersuker, Cord. Chem. Rev. 14, 357(1975), see p. 390].

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Jahn-Teller Systems