Relativistic T × T and T × E Jahn–Teller coupling in tetrahedral systems

Chemical Physics 374 (2010) 86–93

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Relativistic T  T and T  E Jahn–Teller coupling in tetrahedral systems Leonid V. Poluyanov a, Wolfgang Domcke b,* a b

Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 142432, Russia Department of Chemistry, Technische Universität München, Lichtenbergstrasse 4, D-85747 Garching, Germany

a r t i c l e

i n f o

Article history: Received 22 February 2010 In final form 18 June 2010 Available online 23 June 2010 This article is dedicated to Horst Köppel on the occasion of his 60th birthday. Keywords: Jahn–Teller effect Spin-orbit coupling Relativistic effects

a b s t r a c t It is shown that orbitally triply degenerate spin-doublet states (2T2) in tetrahedral systems exhibit, in addition to the well-known electrostatic Jahn–Teller effects, Jahn–Teller couplings which are of relativistic origin, that is, they arise from the spin-orbit operator. The linear relativistic Jahn–Teller Hamiltonian of a 2T2 state involving vibrational modes of E and T2 symmetry is derived with group-theoretical methods. The 2T2  (E + T2) Jahn–Teller Hamiltonian is transformed to a SO-adapted basis in which the zerothorder spin-orbit-coupling matrix is diagonal. For the 4  4 sub-matrices corresponding to the G3/2 irreducible representation of the spin double group T 0d , the adiabatic potential-energy surfaces and the geometric phases of the adiabatic electronic wave functions are given in analytic form. Ó 2010 Published by Elsevier B.V.

1. Introduction As is well known, tetrahedral, cubic and icosahedral systems possess doubly and triply orbitally degenerate electronic states as well as doubly and triply degenerate vibrational modes [1–4]. In addition, open-shell electronic states may exhibit spin-degeneracy (spin doublets, triplets, quartets, etc.). In open-shell electronic systems containing second-row or heavier elements, the spin-orbit (SO) interaction becomes relevant. In general, the SO coupling lifts the orbital degeneracy as well as the spin degeneracy. In systems of high symmetry, such as tetrahedral or cubic systems, however, the lifting of the electronic degeneracies by the SO coupling is incomplete. For example, the four-fold degeneracy of a non-relativistic 2E state is retained and a six-fold degenerate non-relativistic 2T state is split into two-fold and four-fold degenerate levels [1–4]. In addition, time-reversal symmetry is preserved by the SO interaction [5]. As a consequence, the adiabatic electronic potential surfaces as well as the vibronic energy levels must be at least doubly degenerate in systems with an odd number of electrons (Kramers degeneracy) [5]. The interplay of Jahn–Teller coupling [1–4] and SO coupling [6] has extensively been analyzed in the vast literature on the spectroscopy of impurity centers in crystals. In these studies, the description of the SO coupling has been based on empirical single-center SO operators of the type

HSO ¼ kL  S;

* Corresponding author. Fax: +49 89 289 13622. E-mail address: [email protected] (W. Domcke). 0301-0104/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.chemphys.2010.06.025

ð1Þ

where k is a phenomenological SO coupling constant and L and S are the orbital and spin angular momenta of the impurity atom [7]. While the empirical ansatz (1) may be appropriate for a partially occupied inner shell of an impurity atom in a rigid crystal environment, a more accurate description of SO coupling may be necessary for molecules, atomic clusters and multi-center transition-metal complexes. In recent work, we have developed the description of SO coupling effects in 2E states of tetrahedral systems on the basis of the microscopic Breit-Pauli (BP) SO operator, which results from the reduction of the Dirac equation to two-component form in the so-called Pauli approximation [8]. It has been shown that the four-fold degeneracy of a non-relativistic 2E state is lifted in first order in the nuclear displacements of T2 symmetry [9]. This implies the existence of a linear 2E  T2 JT effect which is of purely relativistic origin, that is, it arises exclusively from the SO-coupling operator. This JT effect is not predicted by the time-honored JT selection rules, since the latter have been derived for the spin-free (electrostatic) Hamiltonian [10]. The 4  4 vibronic-coupling Hamiltonian of a 2E state interacting with a vibrational mode of T2 symmetry has been derived in Ref. [9] and its adiabatic eigenvalues and eigenfunctions have been discussed. For small displacements from the tetrahedral reference geometry, the adiabatic potential-energy surfaces represent a ‘‘Mexican Hat” in four-dimensional space (the energy as a function of three nuclear coordinates) [9]. The adiabatic electronic eigenfunctions carry a nontrivial geometric phase [11] which depends on the orientation of the plane of the integration contour [9]. In the present paper, we elaborate the description of JT-coupling effects arising from the BP operator in orbitally triply

L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93

degenerate states (T2) of doublet spin multiplicity in tetrahedral symmetry. It will be shown that the microscopic SO operator gives rise to linear JT coupling terms involving both E-type and T2-type vibrational modes. These novel JT couplings are of purely relativistic origin and should not be confused with the well-established electrostatic JT couplings in T2 electronic states [1–4]. 2. Symmetry selection rules Orbitally degenerate states (E, T1, T2) in tetrahedral systems (point group Td) are subject to the JT effect. The spin-free JT selection rules are [10] 2

½E  ¼ A þ E;

ð2aÞ

½T 21;2  ¼ A þ E þ T 2 ;

ð2bÞ

where [C2] denotes the symmetrized square of the degenerate irreducible representation C [12]. The vibrational modes of a fouratomic tetrahedral system (X4) transform as A, E and T2. In electronic states of T1 or T2 symmetry, the E mode as well as the T2 mode are thus JT active (in first order). In electronic states of E symmetry, only the E mode is JT active according to the selection rule (2a). In the Pauli approximation [6,8], the electronic Hamiltonian can be written as the sum of the electrostatic Hamiltonian Hes and the BP SO operator HSO. Hes may be chosen, for example, as the generalized Fock operator of multi-configuration self-consistent-field (MCSCF) theory. For a single unpaired electron in the field of four identical nuclei, the BP operator reads [6]

HSO ¼

ig e b2e qS

4 X 1  ðrk  rÞ; 3 r k¼1 k

ð3Þ

where q is the effective nuclear charge of the four equivalent atomic cores,

S¼

 1 irx þ jry þ krz ; 2

ð4Þ

rx, ry, rz are the Pauli spin matrices, eh ; 2me c

be ¼

is the Bohr magneton, ge = 2.0023 is the g-factor of the electron, and i, j, k are the cartesian unit vectors. The rk, rk in Eq. (3) are defined as

rk ¼ r  R k ;

ð5aÞ

r k ¼ jrk j;

ð5bÞ

where r is the radius vector of the single unpaired electron and Rk, k = 1, 2, 3, 4, denotes the radius vectors from the origin to the four corners of the tetrahedron. For the analysis of the symmetry properties, it is convenient to write the BP operator (3) in determinal form [9]

HSO

  rx  1 2 ¼ ig e be  Ux  @ 2  @x



ry rz  

Uy Uz ; @ @y

@ @z

 

ð6aÞ

where

U¼

4 X q r k¼1 k

ð6bÞ

@U : @x

ð6cÞ

and

Ux ¼

87

Eq. (6) reveals that the SO operator is completely determined by the electrostatic potential U(r). The simple one-electron expressions (3) and (6) for the BP operator are suitable for the analysis of the symmetry properties of the SO operator. For actual calculations of relativistic JT coupling parameters in many-electron systems, the full microscopic BP Hamiltonian [13] should be employed, see e.g., Ref. [14]. The symmetry group of the electronic Hamiltonian including electron spin and SO coupling is the tetrahedral spin double group T 0d [12]. T 0d can be constructed by determining the complete set of generalized symmetry operators of spin-12 systems

Z n ¼ C n U yn ;

ð7Þ

where the Cn are the spatial symmetry operations of the point group Td and the Un are SU2 matrices acting on the spin functions a, b. The complete set of the 48 symmetry operations of T 0d is given in Appendix A [15]. We shall be concerned with tetrahedral systems with a single unpaired electron. The spin functions a, b transform according to the E1/2 irreducible representation of T 0d [16]. The treatment of electronic states of higher spin multiplicity, e.g., quartets or sextets, is a straightforward extension of the present analysis, since the degenerate representations of these spin states can be decomposed into the irreducible representations with the character table of T 0d :2 T 1 , 2 T2 and 2E states transform in T 0d symmetry as follows: 2

T 1 ¼ T 1  E1=2 ¼ G3=2 þ E1=2 ;

ð8aÞ

2

T 2 ¼ T 2  E1=2 ¼ G3=2 þ E5=2 ;

ð8bÞ

2

E ¼ E  E1=2 ¼ G3=2 ;

ð8cÞ

where E1/2, E5/2, G3/2 are the three two-valued representations of T 0d : E1=2 and E5/2 are two-dimensional, while G3/2 is a four-dimensional irreducible representation [12,16]. According to Eqs. (8a), (8b), the 2T2 (2T1) non-relativistic electronic manifold splits into G3/2 and E5/2(E1/2) irreducible representations. The six-fold degeneracy of a 2T2(2T1) state is thus partially lifted by the SO coupling, resulting in a four-fold degenerate G3/2 level and a two-fold degenerate E5/2(E1/2) level. The four-fold degeneracy of a non-relativistic 2E state, on the other hand, is not lifted by the SO coupling at the tetrahedral reference geometry [3,4]. As a consequence of time-reversal symmetry, the two-fold degeneracy of the E1/2 and E5/2 levels cannot be lifted by any intra-molecular interaction [5], while the energy levels of the G3/2 manifold can split into two two-fold degenerate levels. The JT selection rules in the T 0d spin double group are [17]

E1=2  G3=2 ¼ E þ T 1 þ T 2 ;

ð9aÞ

E5=2  G3=2 ¼ E þ T 1 þ T 2 ; n o G23=2 ¼ A1 þ E þ T 2 :

ð9bÞ ð9cÞ

Here C1  C2 denotes the direct product of C1 and C2 and {C2} denotes the antisymmetrized square of C [12]. Eq. (9c) reveals that the four-fold degeneracy of the G3/2 manifold can be lifted in first order by vibrational modes of E or T2 symmetry. For a 2E state, which transforms as G3/2 in the T 0d group, the T2 mode is not JT active in the non-relativistic limit, see Eq. (2a). It follows that the JT activity of the T2 mode according to Eq. (9c) must arise from the SO operator. The corresponding 2E  T2 JT Hamiltonian has been derived in Ref. [9]. On the other hand, a detailed analysis of the symmetry properties of the individual terms of the Taylor expansion of the BP operator reveals that the first-order relativistic 2E  E coupling term vanishes [9]. In a 2E(G3/2) state, the E mode is thus JT active through the electrostatic forces, while the T2 mode is JT active through the relativistic forces. For 2T2 and 2T1 states, the selection rule (9c) predicts that both the E mode and the T2 mode are JT active in first order in the G3/2

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manifold. In addition, the E and T2 modes can couple the G3/2 and E5/2 (for 2T2) or E1/2 (for 2T1) electronic manifolds via pseudo-JTtype interactions. A detailed analysis of the symmetry properties of the Taylor expansion of the BP operator (see Section 3.1.2 below) shows that both E and T2 modes are relativistically JT active in firstorder. Thus both electrostatic forces (Eq. (2b)) as well as relativistic forces (Eqs. (9a)–(9c)) contribute to the lifting of the electronic degeneracy of 2T1,2 electronic states.

3.1.1. The electrostatic vibronic matrix Symmetry-adapted atomic displacements transforming as x, y and z in the Td point group are

ð10aÞ ð10bÞ ð10cÞ

where

ð11Þ

Dimensionless normal coordinates of a suitable nondegenerate reference state can be obtained by multiplication with appropriate scaling factors which depend on the atomic masses and the harmonic force constants of the reference state [18]. Molecular orbitals transforming according to the T2 representation of Td can be constructed as linear combinations of atomic p orbitals on the four atoms

/ðkÞ x ¼ xk f ðr k Þ;

ð12aÞ

/ðkÞ y

ð12bÞ

/ðkÞ z ¼ zk f ðr k Þ;

k ¼ 1; 2; 3; 4;

ð12cÞ

where rk is defined in Eq. (5b) and f(r) is an exponential function (for Slater-type orbitals) or a Gaussian function (for Gaussian-type orbitals). The molecular orbitals of T2 symmetry are

 1  ð1Þ ð3Þ ð4Þ /x þ /ð2Þ ; x þ /x þ /x 2   1 ð1Þ ð3Þ ð4Þ / þ /ð2Þ ; wy ðrÞ ¼ y þ /y þ /y 2 y   1 ð1Þ ð4Þ / þ /zð2Þ þ /ð3Þ : wz ðrÞ ¼ z þ /z 2 z

wx ðrÞ ¼

ð13aÞ ð13bÞ ð13cÞ

The electrostatic Hamiltonian Hes is expanded at the reference geometry in powers of the T2 normal coordinates Qx, Qy, Qz up to second order

Hes ¼ H

ð0Þ

þ

Hxð1Þ Q x

þ

Hð1Þ y Qy

þ

Hð1Þ z Qz

1 1 þ Hð2Þ Q 2 þ Hð2Þ Q 2 2 xx x 2 yy y

1 ð2Þ ð2Þ þ Hð2Þ Q 2 þ Hð2Þ xy Q x Q y þ H yz Q y Q z þ H zx Q z Q x ; 2 zz z

ð15bÞ ð15cÞ

H(0) is invariant under the transformations of Td. The triples ð2Þ ð2Þ and Hxy ; Hð2Þ yz ; Hzx

transform as T2. The triple

forms a reducible representation of Td which can be decomposed into A1 and E irreducible representations. Second-order polynomials transforming as A1 and E are given by

The spin-free Hamiltonian for T2  T2 JT coupling up to second order in T2 normal modes in tetrahedral symmetry is well established and can be found in many reviews and monographs [1–4]. In the following subsection, we briefly sketch the derivation of the spin-free T2  T2 JT Hamiltonian to set the stage for the derivation of the SO-induced 2T2  T2 Hamiltonian in Section 3.1.2.

¼ yk f ðrk Þ;

ð15aÞ

ð2Þ ð2Þ Hxx ; Hyy ; Hð2Þ zz

3.1. The 2T2  T2 Jahn–Teller Hamiltonian

Rkm ¼ jRk  Rm j:

Hð0Þ ¼ Hes ð0Þ;  @Hes Hxð1Þ ¼ ;  @Q x 0 @Hes ð2Þ ¼ ; etc: Hxy @Q x @Q y 0 Hxð1Þ ; Hyð1Þ ; Hð1Þ z

3. Derivation of the vibronic-coupling Hamiltonians

1 Sx ¼ pffiffiffi ðR12  R34 Þ; 2 1 p Sy ¼ ffiffiffi ðR14  R23 Þ; 2 1 Sz ¼ pffiffiffi ðR13  R24 Þ; 2

where

ð14Þ

 1  PA1 ðQ Þ ¼ pffiffiffi Q 2x þ Q 2y þ Q 2z ; 3  1  2 PEa ðQ Þ ¼ pffiffiffi Q z  Q 2y ; 2  1  2 PEb ðQ Þ ¼ pffiffiffi 2Q x  Q 2y  Q 2z : 6

ð16aÞ ð16bÞ ð16cÞ

The vibronic matrix of the T2 state is obtained by the calculation of the matrix elements of Hes in the diabatic electronic basis wx, wy, wz of Eq. (13). Since the individual terms of the Hamiltonian as well as the electronic basis functions transform according to irreducible representations of Td, it is straightforward to determine the nonvanishing matrix elements: only those parts of the integrand which transform totally symmetric give a contribution to the integral. Including, finally, the nuclear kinetic-energy operator TN, which is diagonal in the diabatic electronic basis, the T2  T2 JT Hamiltonian is

 1 ð2Þ Hes ðT 2  T 2 Þ ¼ T N þ xR2 13 þ Hð1Þ es ðT 2  T 2 Þ þ Hes ðT 2  T 2 Þ; 2 ð17aÞ 0 1 0 Qz Qy ð1Þ ð17bÞ Hes ðT 2  T 2 Þ ¼ a@ Q z 0 Q x A; Qy Qx 0 ð2Þ Hes ðT 2  T 2 Þ 0 1 BQ x Q y BQ x Q z Að2Q 2x  Q 2y  Q 2z Þ B C 2 2 2 ¼@ BQ x Q y Að2Q y  Q x  Q z Þ BQ y Q z A; BQ x Q z

BQ y Q z

Að2Q 2z  Q 2x  Q 2y Þ ð17cÞ

where

R¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Q 2x þ Q 2y þ Q 2z ;

ð17dÞ

13 denotes the 3  3 unit matrix, x is the harmonic vibrational frequency of the T2 mode, a is the first-order electrostatic JT coupling constant and A, B are second-order electrostatic JT coupling constants. Discussions of the adiabatic electronic potential-energy surfaces (the eigenvalues of Hes(T2  T2)  TN13) can be found in the literature [1–4]. 3.1.2. The spin-orbit vibronic matrix HSO of Eq. (3) is expanded in a Taylor series in Qx, Qy, Qz in analogy to Eq. (14). Assuming that the SO coupling is weak compared to the electrostatic interactions, the expansion is terminated after the first order:

  ð0Þ ð1Þ ð1Þ ð1Þ HSO 2 T 2  T 2 ¼ h þ hx Q x þ hy Q y þ hz Q z ; ð0Þ

x

y

z

rx þ h ry þ h rz ; y z ¼ rx þ hx ry þ hx rz ; y z ¼ rx þ hy ry þ hy rz ; ð1Þ x y z hz ¼ hz rx þ hz rz þ hz rz ; h

ð1Þ hx ð1Þ hy

¼h

x hx x hy

ð18aÞ ð18bÞ ð18cÞ ð18dÞ ð18eÞ

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L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93

where

wþx ¼ wx ðrÞa;

ð24aÞ

 @U @ @U @ x  ; h ¼ ig e b2e q @y @z @z @y  @U @ @U @ y ; h ¼ ig e b2e q  @z @x @x @z  @U @ @U @ z h ¼ ig e b2e q  ; @x @y @y @x

wþy ¼ wy ðrÞa;

ð24bÞ

ð19aÞ ð19bÞ ð19cÞ

x

x

hx ¼

@h ; @Q x

ð20aÞ

wþz ¼ wz ðrÞa;

ð24cÞ

wz ¼ wz ðrÞb;

ð24dÞ

wy ¼ wy ðrÞb;

ð24eÞ

wx ¼ wx ðrÞb:

ð24fÞ

The calculation of the matrix elements of the SO Hamiltonian (23) with the basis functions (24) is greatly simplified by the existence of time-reversal symmetry. The time-reversal operator s is defined as [5]

x

x

hy ¼

@h ; @Q y @h ; etc: @Q z

b denotes the operation of complex conjugation. The timewhere cc reversal operator acts on the spin-orbitals (24) as follows

ð21Þ

First-order polynomials in Qx, Qy, Qz which transform according to the irreducible representations of T 0d are:

1 PA2 ðQ ; ~ rÞ ¼ pffiffiffi ðQ x rx þ Q y ry þ Q z rz Þ; 3

ð22aÞ

1 PEa ðQ ; ~ rÞ ¼ pffiffiffi ð2Q x rx  Q y ry  Q z rz Þ; 6

ð22bÞ

1 rÞ ¼ pffiffiffi ðQ y ry  Q z rz Þ; PEb ðQ ; ~ 2

ð22cÞ

1 rÞ ¼ pffiffiffi ðQ z ry þ Q y rz Þ; PT 1x ðQ ; ~ 2

ð22dÞ

1 rÞ ¼ pffiffiffi ðQ x rz þ Q z rx Þ; PT 1y ðQ ; ~ 2

ð22eÞ

1 rÞ ¼ pffiffiffi ðQ y rx þ Q x ry Þ; PT 1z ðQ ; ~ 2

ð22fÞ

1 rÞ ¼ pffiffiffi ðQ z ry  Q y rz Þ; PT 2x ðQ ; ~ 2

ð22gÞ

1 rÞ ¼ pffiffiffi ðQ x rz  Q z rx Þ; PT 2y ðQ ; ~ 2

ð22hÞ

1 rÞ ¼ pffiffiffi ðQ y rx  Q x ry Þ: PT 2z ðQ ; ~ 2

ð22iÞ



HSO T 2  T 2 ¼ h

x

0

b cc;

ð25Þ

b b b swþx ¼ wx cc; swþy ¼ wy cc; swþz ¼ wz cc;  þ  þ  b b b swz ¼ wz cc; swy ¼ wy cc; swx ¼ wþx cc; where it is assumed that the spatial orbitals (13) are real-valued. The representation of the operator s is thus the 6  6 matrix

0 B B B B s¼B B B B @

0 0 0 0 0 1

0

0

0 0

1

1

0 1 0C C C 0 0 1 0 0C C: 0 1 0 0 0 C C C 1 0 0 0 0 A 0

0

0

0

0 0

ð26Þ

0

It is straightforward to show that s2 = 16, as is required for an oddelectron system. The requirement

ð27Þ

leads to the vanishing of six of the matrix elements of HSO and requires others to be equal or equal up to a minus sign. Using the expansion of HSO in symmetry-adapted polynomials and the symmetry properties of the spin-orbital basis, it is straightforward to determine the nonvanishing elements of the 2T2  T2 vibronic matrix. The result is

    ð0Þ  ð1Þ  HSO 2 T 2  T 2 ¼ HSO 2 T 2 þ HSO 2 T 2  T 2 ; 1 0 0 i 0 1 0 0 B i 0 0 i 0 0C C B C B C B   0 0 0 0 i 1 ð0Þ C; HSO 2 T 2 ¼ DB C B 1 i 0 0 0 0 C B C B @ 0 0 i 0 0 iA 0

rx þ hy ry þ hz rz þ hA2 PA2 þ hEa PEa

0

0 0

B 0 B B B iQ x  ð1Þ 2 HSO T 2  T 2 ¼ aB B iQ B z B @ Q 0

þ hEb PEb þ hT 1x PT 1x þ hT 1y PT 1y þ hT 1z PT 1z þ hT 2x PT 2x þ hT 2y PT 2y þ hT 2z PT 2z ;

1

½HSO ; s ¼ 0;

The Taylor expansion of Eq. (18) can thus be rewritten as an expansion in symmetry-adapted polynomials

2



ð20cÞ

and U is given by Eq. (6b). Making use of the explicit symmetry operators of T 0d given in ð1Þ ð1Þ ð1Þ Appendix A, it can be shown that the operator triple hx ; hy ; hz as well as rx, ry, rz transform according to the irreducible reprex y z sentation T1 of T 0d . The nine operators hx ; hx ; . . . ; hz transform according to the direct product representation

T 1  T 2 ¼ A2 þ E þ T 1 þ T 2 :

0 1

b ¼ s ¼ iry cc

x

x

hz ¼



ð20bÞ

1

0 0

i 0 iQ x iQ z

ð28aÞ

ð28bÞ

Qþ

0

iQ y

Qz

0

iQ y

0

0

Q z

Qz 0

0 Q z

0 iQ y

iQ y 0

Q 

iQ z

iQ x

0

0

1

Q þ C C C iQ z C C; iQ x C C C 0 A 0 ð28cÞ

ð23Þ

where hA2 transforms as A2, the pair hEa ; hEb transforms as E, the triple hT 1x , hT 1y ; hT 1z transforms as T1, and the triple hT 2x ; hT 2y ; hT 2z transforms as T2 in T 0d . A non-relativistic spin-orbital basis of a 2T2 state is given by the spatial orbitals of Eq. (13), multiplied by spin functions a, b:

where

Q  ¼ Q x  iQ y

ð28dÞ

and D and a are real constants which represent the zeroth-order SO splitting of the 2T2 state and the first-order relativistic 2T2  T2 JT coupling, respectively.

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L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93

The detailed analysis of this and the preceding sub-section confirms that the T2 mode is JT active both in the electrostatic Hamiltonian (Eq. (17b)) as well as via the SO operator (Eq. (28c)). The electrostatic (a) and relativistic (a) coupling constants are real, but can be positive or negative. There may thus be constructive or destructive interference of the electrostatic and relativistic JT couplings.

   e ð1Þ G3=2  T 2 ¼ T N þ 1 xR2  D 14 H 2 0 0 Q þ 0 B Q 0 Q z B ~ þ ia1 @ 0 0 Qz Q z 0 Q þ

1 Qz 0 C C: Q A 0

ð35Þ

The eigenvalues of the vibronic matrix (35) are 3.1.3. Spin-orbit adapted basis and adiabatic potential-energy surfaces  ð0Þ  The hermitian matrix HSO 2 T 2 of Eq. (28b) can be transformed to diagonal form by a unitary transformation U

    e ð0Þ 2 T 2 ¼ U y Hð0Þ 2 T 2 U: H SO SO

ð29Þ

  e ð0Þ 2 T 2 has degenerate eigenvalues, Since the transformed matrix H SO the unitary matrix U is not unique. A suitable choice of U is given in Appendix B, Eq. (B.1). In the transformed basis (see Eq. (B.2) in Appendix B)

~ ¼ U y w; w

ð30Þ

where w is the vector of electronic basis functions defined in Eq. (24), the SO vibronic matrix takes the form

  e ð0Þ 2 T 2 ¼ diagðD; D; D; D; 2D; 2DÞ: H SO



e H T2  T2 ¼ U  T 2 ÞU; es 2  2  ð1Þ ð1Þ y e T2  T2 ¼ U H T 2  T 2 U; H SO

y

ð1Þ Hes ðT 2 SO

ð32aÞ ð32bÞ

yields

        e ð0Þ 2 T 2 þ H e ð1Þ 2 T 2  T 2 þ H e ð1Þ 2 T 2  T 2 e ð1Þ 2 T 2  T 2 ¼ H H es SO 1 0 ~1 Q þ ~1 Q z ~2 Q  ia 0 ia 0 ia D C B B ia ~ ~1 Q z ~2 Q  2 piffiffi a ~ Q C D ia 0  piffiffi3 a B 1Q  3 2 z C C B C B 0 ~1 Q z ~1 Q  ~2 Q þ ia D ia ia 0 C B C; B ¼B ~ 2 i C ffiffi ffiffi p p ~ ~ ~ i a Q 0 i a Q  D i Q  Q a a 1 z 1 þ B 3 2 z 3 2 þC C B C B 0 piffiffi a ~2 Q  i p2ffiffi a ~2 Q þ ~2 Q z ia 2D 0 C B 3 3 A @ piffiffi a ~2 Q þ 2  piffiffi a ~2 Q z ~2 Q  0 0 2 D ia 3 3 ð33Þ where

1 ~1 ¼ pffiffiffi ða þ 2aÞ; a 3 1 ~2 ¼ pffiffiffi ða  aÞ: a 2

ð34aÞ ð34bÞ

~1 ; a ~2 It is seen that the effective first-order JT coupling parameters a have an electrostatic (a) and a relativistic (a) contribution. The second-order electrostatic vibronic matrix, Eq. (17c), can be transformed to the SO-adapted basis analogously. For sufficiently large SO splitting D, the G3/2 manifold (with eigenvalue D) can be considered to be approximately decoupled from the E5/2 manifold (with eigenvalue 2D). In this approximation, the 2T2  T2 JT Hamiltonian is reduced to a 4  4 matrix

ð36aÞ ð36bÞ

~1 is given by Eq. (34a). The adiawhere R defined in Eq. (17d) and a batic potentials (36) are doubly degenerate (Kramers degeneracy) and represent a ‘‘Mexican Hat” in four-dimensional space. A convenient choice of the normalized and orthogonalized e ð1Þ  T N 14 are the columns of the matrix eigenvectors of H

0

1 i/ 1 B B e sin h U ¼ pffiffiffi B 0 2@ cos h

ð31Þ

In agreement with the group-theoretical result (8b), the zeroth-order SO coupling splits the 6-fold degenerate 2T2 manifold into a doubly degenerate manifold (E5/2) and a quadruply degenerate manifold (G3/2). Rewriting the first-order electrostatic vibronic matrix (17b) in the spin-orbital basis (24) (which means doubling of the 3  3 mað1Þ trix to a 6  6 matrix) and transformation of Hð1Þ es þ HSO to the SO adapted basis

 ð1Þ 2

1 ~1 jR; V 1;2 ¼ D þ xR2  ja 2 1 ~1 jR; V 3;4 ¼ D þ xR2 þ ja 2

0 i cos h

1 e

i/

sin h

i

0

iei/ sin h

 cos h

0

1

i cos h C C C; i A

ð37Þ

ei/ sin h

where h and / are the polar and azimuthal angles in Qx, Qy, Qz space. The single-valued form (37) of the adiabatic eigenvectors is suitable for the calculation of the geometric phase of the adiabatic electronic wave functions. The standard definition of the geometric phase (Berry phase) is [11]

cn ðCÞ ¼ i

I

dQ huðQ ÞjrQ uðQ Þi;

ð38Þ

C

where u(Q) is a single-valued adiabatic wave function which depends parametrically on the nuclear coordinate vector Q. The integral is to be evaluated over a closed path in parameter space which encloses the origin Q = 0. To evaluate the contour integral in Eq. (38), we consider an arbitrary plane through the origin

aQ x þ bQ y þ cQ z ¼ 0; where a, b, c determine the orientation of the plane in the threedimensional normal-coordinate space. The results are

pc ffi: c1 ¼ c3 ¼ c2 ¼ c4 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 þ b þ c2

ð39Þ

It is seen that the adiabatic electronic wave functions carry a nontrivial geometric phase. The geometric phases cn depend on the orientation of the plane spanned by the integration contour, but are independent of the radius of the loop of integration (for linear JT coupling). The wave functions within a Kramers doublet, (c1, c2) and (c3, c4), exhibit geometric phases of opposite sign. The geometric phases (39) are the same as found for the G3/2  T2 Hamiltonian which arises from relativistic vibronic coupling in a 2E state of tetrahedral systems [9]. It is noteworthy that the geometric phases spoil the equivalence of the x, y, z coordinates, which holds in the spin-less case. As Eq. (39) shows, the z-axis is distinguished from the x and y-axes. The origin of this inequivalence is the quantization of the spin with respect to the z-axis. While the SO contribution to the energy, given by the scalar product of orbital and spin angular momenta, is invariant with respect to interchanges of x, y, z, the electronic wave function is not. Therefore, the lifting of the spatial invariance appears in the geometric phases, while the adiabatic PE surfaces are spatially invariant (cf. Eq. (36)). The analysis presented here for a 2T2 state can be performed in a completely analogous manner for a 2T1 state [19]. The results are

91

L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93 ð0Þ 2

similar, but certain matrix elements in HSO have a different sign.



ð1Þ 2

T 1 and HSO

T1  T2



3.2. The T2  E Jahn–Teller Hamiltonian 3.2.1. The electrostatic vibronic matrix Symmetry-adapted coordinates transforming according to the E representation in Td are given by the following linear combinations of the edges Rkm of the tetrahedron

1 ðR13 þ R24  R14  R23 Þ; 2 1 Sb ¼ pffiffiffi ð2R12 þ 2R43  R13  R24  R14  R23 Þ: 2 3

Sa ¼

ð40aÞ

ð45eÞ

rÞ ¼ qþ rz ; PT 2z ðq; ~

ð45fÞ

ð41Þ

1 ð1Þ ð2Þ Hes ðT 2  EÞ ¼ T N þ xq2 13 þ Hes ðT 2  EÞ þ Hes ðT 2  EÞ; ð42aÞ 2 0 1 0 0 qb B C ð42bÞ qþ 0 A; Hð1Þ es ðT 2  EÞ ¼ c @ 0 0 0 q 0

pffiffiffi 3 1 qa  q : 2 2 b

ð45gÞ

With these definitions, we can rewrite the Taylor expansion (43) in terms of symmetry-adapted polynomials

  x y z HSO 2 T 2  E ¼ h rx þ h ry þ h rz þ hT 2x qa rx þ hT 2y q ry

x



0

ð45dÞ

PT 2y ðq; ~ rÞ ¼ q ry ;

ð45cÞ

þ hT 2z qþ rz þ hT 1x qb rx þ hT 1y qþ ry  hT 1z q rz :

ð0Þ

pffiffiffi 3q a q b

rÞ ¼ q rz ; PT 1z ðq; ~ PT 2x ðq; ~ rÞ ¼ qa rx ;

q ¼

H0 and Hð2Þ þ transform according to the A1 representation, while the ð1Þ ð2Þ ð2Þ pairs Hð1Þ a ; Hb and Hab ; H transform according to the E representation of Td. Matrix elements of Hes(T2  E) with the symmetry-adapted electronic basis functions (13) can be straightforwardly evaluated as described in Subsection 3.1.1. It is found that all off-diagonal matrix elements vanish. The resulting T2  E Hamiltonian is

 q2a Þ 

ð45aÞ ð45bÞ

where

  1 ð0Þ ð1Þ Hes ðT 2  EÞ ¼ H0 þ Hað1Þ qa þ Hb qb þ Hþð2Þ q2a þ q2b 2   1 ð2Þ þ Hab qa qb þ Hð2Þ q2a  q2b þ    2

1 ðq2b 2

PT 1x ðq; ~ rÞ ¼ qb rx ; PT 1y ðq; ~ rÞ ¼ qþ ry ;

ð40bÞ

Dimensionless normal coordinates qa, qb can be obtained by multiplication with an appropriate scaling factor [18]. The Taylor expansion of Hes up to second order in qa, qb can be written as

0

ð44Þ

First-order polynomials in qa, qb which transform as T1, T2 are

2

Hð2Þ es ðT 2  EÞ 0 q2a  q2b B ¼ CB @ 0 0

E  T1 ¼ T1 þ T2:

1

C C; A pffiffiffi 1 2 2 ðq  q Þ þ 3 q q a b a b 2 0

ð42cÞ where

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ q2a þ q2b ; pffiffiffi 3 1 q ¼ q  q 2 a 2 b

ð42dÞ ð42eÞ

and c(C) is the first-order (second-order) electrostatic T2  E JT coupling constant. Discussions of the adiabatic electronic potential-energy surfaces of the Hamiltonian (42) can be found in the literature [1–4].

y

ð46Þ

z

Here the triples h , h , h and hT 1x ; hT 1y ; hT 1z transform as T1, while the triple hT 2x ; hT 2y ; hT 2z transforms as T2. The calculation of matrix elements of HSO(2T  E) of Eq. (46) with the symmetry-adapted basis functions (24) is further simplified by the commutation of HSO with the time-reversal operator (26). The result is

0

0 B iq B  B B 0  ð1Þ 2 HSO T 2  E ¼ cB B q B þ B @ 0 0

iq

0

qþ

0

0

0

iqb

0

0

0

0

iqb

iqb

0

0

0

0

iqb

0

0

0

qþ

0

iq

0

1

0 C C C qþ C C; 0 C C C iq A

ð47Þ

0

where c is the first-order relativistic coupling constant of the E mode in a 2T2 electronic state. The E mode is thus JT active in first-order through electrostatic forces (Eq. (42b)) as well as through relativistic forces (Eq. (47)). 3.2.3. Spin-orbit adapted basis and adiabatic potential-energy surfaces Applying the unitary transformation U given in Eq. (B.1) in ð1Þ 2 Appendix B to Hð1Þ T 2  E , we obtain the vibronic es ðT 2  EÞ and H SO matrix in the SO-adapted basis:

     e ð1Þ 2 T 2  E ¼ U y Hð0Þ þ Hð1Þ ðT 2  EÞ þ Hð1Þ 2 T 2  E U H es SO SO 1 0 ~c1 qþ D þ ~c1 q 0 0 ~c2 q 0 B ~c2 qþ C 0 D  ~c1 q ~c1 qþ 0 0 C B C B þ ~c1 q ~c1 q ~c2 q C B 0   D þ 0 0 C; ¼B B ~c qþ 0 0 D  ~c1 q ~c2 qþ 0 C 1 C B C B @ ~c2 q ~c2 qþ 0 0 2D 0 A ~c2 qþ ~c2 q 0 0 0 2D ð48Þ

3.2.2. The spin-orbit vibronic matrix The Taylor expansion of the SO operator up to first order in qa, qb reads

HSO

2

 x y z x y z T 2  E ¼ h rx þ h ry þ h rz þ ðha rx þ ha ry þ ha rz Þqa x

y

z

þ ðhb rx þ hb ry þ hb rz Þqb þ   

ð43Þ

Making use of Appendix A, it can be shown that the operators hx, hy, hz and the Pauli matrices rx, ry, rz transform according to the T1 representation of the spin double group T 0d . The six operators n hm ; n ¼ x; y; z; m ¼ a; b, as well as the operator products rnqm transform according to the direct product representation

where

c  c; 2 1 ~c2 ¼ pffiffiffi ðc þ cÞ: 2

~c1 ¼

ð49aÞ ð49bÞ

The second-order part of the electrostatic vibronic matrix, Eq. (42c), can analogously be transformed to the SO-adapted basis. If D is sufficiently large, the E5/2 manifold can be approximately decoupled from the G3/2 manifold. The 4  4 JT Hamiltonian of the G3/2 manifold is

92

L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93

0

q 0 0  B 0 q qþ 1 B  e ð1Þ ðG3=2  EÞ ¼ T N þ xq2  D 14 þ ~c1 B H @ 0 qþ q 2 qþ

0

0

1 qþ 0 C C C; 0 A q ð50Þ

±

where q± and q are defined in Eqs. (42e) and (45g), respectively. The adiabatic potential-energy surfaces of the Hamiltonian (50) are

1 V 1;2 ¼ D þ xq2  j~c1 jq; 2 1 V 3;4 ¼ D þ xq2 þ j~c1 jq; 2

ð51aÞ ð51bÞ

where q is defined in Eq. (42d). Eq. (51) represents a (doubly-degenerate) ‘‘Mexican Hat” in three-dimensional space (the energy as a function of two nuclear coordinates). The analysis presented here for a 2T2 state can be performed in an analogous manner for a 2T1 state [19]. The eigenvector matrix of the vibronic matrix (50) can be written as

0 pffiffiffi pffiffiffi        1 3 sin /2 cos /2 cos /2  3 sin /2 B pffiffiffi pffiffiffi /        C C 3 sin /2  3 sin /2 cos / 1B C B cos U ¼ B pffiffiffi 2/ pffiffiffi 2/ C; /  /  C 2 B 3 sin 3 sin  cos cos A @ 2 2 2 2 / pffiffiffi / pffiffiffi /  / 3 sin 2 3 sin 2 cos 2  cos 2 ð52Þ where / is the azimuthal angle in the qa, qb plane. Using the definition (41), it can easily be shown that all four geometric phases are zero. A Hamiltonian for G3/2  (T2 + E) JT coupling has been given by Moffitt and Thorson for octahedral systems in the limit of strong SO coupling [20]. While the adiabatic electronic eigenvalues of the G3/2  T2 and G3/2  E vibronic matrices of Moffitt and Thorson are the same as Eqs. (36) and (51), respectively, the adiabatic electronic wave functions, the nonadiabatic couplings and therefore the vibronic spectra are not identical. The geometric phases are different as well: while the Moffitt–Thorson vibronic matrix yields geometric phases of ±p in all cases, the present Hamiltonians yield nontrivial geometric phases which depend on the orientation of the integration contour in the G3/2  T2 case, while the geometric phases are zero in the G3/2  E case. 4. Discussion and conclusions We have presented a systematic analysis of JT coupling by T2 and E vibrational modes in a 2T2 electronic state in tetrahedral systems. In addition to the well-established electrostatic (spin-free) JT couplings, the JT couplings arising from the microscopic SO operator have been determined. To the best of our knowledge, the corresponding relativistic JT coupling matrices, in particular Eqs. (33) and (48), are new in JT theory. For 2T2 and 2T1 states in tetrahedral symmetry, the electrostatic and the relativistic JT couplings obey the same selection rules, that is, both E and T2 vibrational modes are JT active in first order. The zeroth-order SO matrix elements in 2T2 and 2T1 states can be transformed to diagonal form, which defines the SO-adapted basis. In the latter, the electrostatic and the SO-induced vibronic 6  6 matrices have the same structure. It is therefore possible to define relativistically corrected JT coupling constants. The relativistic forces can either enhance or weaken the electrostatic forces. The analysis described in the present work can be extended to electronic states of higher spin multiplicity, i.e. MT1,2 states, M = 4, 6 . . . High-spin states are often encountered in transition-

metal complexes, for example. The spin functions of even multiplicity for M > 2 are reducible in the T 0d group. After the reduction to irreducible representations of T 0d , the SO vibronic matrices can be constructed as discussed in Sections 3.1.2 and 3.2.2. It should be emphasized that the additive action of electrostatic and relativistic JT forces found in 2T2 and 2T1 states of tetrahedral systems is the exception rather than the rule. In 2E states of tetrahedral systems, for example, the normal mode of E symmetry is JT active at the electrostatic level, while the T2 normal mode is JT active through the SO operator [9]. In several of the molecular point groups with degenerate irreducible representations, the relativistic JT forces are complementary to the electrostatic JT forces. The JT selection rules of 1937 are therefore insufficient for the prediction of the distortion of degenerate states in systems containing heavy elements. The relativistic JT coupling parameters discussed in the present work have been calculated in Ref. [14] for tetrahedral cluster cations of the elements of the fifth main group of the periodic table, þ þ þ Pþ 4 ; As4 ; Sb4 and Bi4 . As expected, the absolute magnitude of the JT coupling parameters arising from the SO operator increases strongly with the nuclear charge. For heavier elements, the relativistic JT forces can be of the same order of magnitude as the electrostatic JT forces [14]. There exists a rich variety of essentially untouched vibroniccoupling phenomena in molecules, clusters and crystals containing heavy elements. The vibronic-coupling parameters arising from the microscopic SO operator can be computed with existing quantum chemical programs [14]. The novel phenomena arising from SO vibronic coupling can thus be explored in a systematic manner, independent of the existence of experimental spectroscopic data. The recording of vibronically resolved electronic spectra or vibrational spectra of isolated molecules and clusters containing heavy atoms is notoriously difficult due to the generally very low volatility of such compounds. The theoretical predictions can identify selected species which exhibit particularly spectacular electronic or vibrational spectra and therefore should be worthwhile objects of future experimental investigations. Appendix A. The symmetry operators of the tetrahedral spin double group T 0d We need to find the complete set of symmetry operators Zn for which

Z n HSO Z 1 n ¼ H SO ; where HSO is given by Eq. (6a). For of the symmetry operators is

Z n ¼ C n U yn ;

ðA:1Þ spin-12

systems, the general form

ðA:2Þ

where Cn is an operation of the spatial symmetry group Td and Un is a SU2 matrix acting on the spin functions a, b. For a given Cn, the solution of Eq. (A.1) defines the corresponding SU2 matrix Un. By straightforward calculations one can verify that the 24 symmetry operators given in Table A.1 fulfill Eq. (A.1). This set T is not closed with respect to multiplication. To obtain a group of symmetry operators, we need to include the additional element R = I. The direct-product set

T 0d ¼ T  ðI; RÞ;

ðA:3Þ

is a group of order 48. It is the so-called double group of tetrahedral spin 12 systems [5,12]. The full symmetry group of the SO operator of tetrahedral systems includes the time-reversal operator s. This group of order 96 is given by

L.V. Poluyanov, W. Domcke / Chemical Physics 374 (2010) 86–93

G96 ¼ T 0d  ðI; sÞ:

ðA:4Þ

Given the symmetry operators of Table A.1, it is straightforward to construct the group multiplication table and to determine the classes and the irreducible representations of T 0d . Appendix B. The SO-adapted basis of a 2T2 state ð0Þ 2

A suitable matrix U, which transforms HSO form, is

T2



to diagonal

Table A.1 The set T of symmetry operators of the SO operator in tetrahedral symmetry. Abbreviations: m ¼ p1ffiffi2 ; e ¼ expð i4pÞ. The super/subscripts a, b, c, d refer to the four corners of the tetrahedron. 1. I (identical transformation)  0 1 x 2. Z x2 ¼ iC 2 1 0  0 i y 3. Z y2 ¼ iC 2 i 0  1 0 z 4. Z z2 ¼ iC 2 0 1  m m 5. Z a3 ¼ eþ C a3 im im  m m 6. Z b3 ¼ e C b3 im im  m m 7. Z c3 ¼ eþ C c3 im im  m m 8. Z d3 ¼ e C d3 im im  þ me me 2 a2 9. Z a3 ¼ iC 3 meþ me  2 me meþ b2 10. Z b3 ¼ iC 3 me meþ  meþ me 2 c2 11. Z c3 ¼ iC 3 þ  m e me   2 me meþ d2 12. Z d3 ¼ iC 3  me meþ  im m 13. Z ab ¼ rab m i m  0 e 14. Z ac ¼ rac þ e 0  im im 15. Z ad ¼ rad im im  im im 16. Z bc ¼ rbc im im  0 eþ 17. Z bd ¼ rbd  e 0  im m 18. Z cd ¼ rcd m im  m im 19. Z x4 ¼ iSx4 im m  m m 20. Z y4 ¼ iSy4 m m  þ e 0 21. Z z4 ¼ iSz4 0 e  3 3 m im 22. Z x4 ¼ iSx4 im m  3 3 m m 23. Z y4 ¼ iSy4 m m   3 3 e 0 24. Z z4 ¼ iSz4 0 eþ

1 p1ffiffi 0 i p1ffiffi2 0 0 i p1ffiffi3 6 C B p1ffiffi p1ffiffi B 3 0 i p1ffiffi6 0 0 C 2 C B C B 0 0  p1ffiffi2 i p1ffiffi6 C i p1ffiffi3 B 0 C B ffiffi q C; U¼B 2 B i p1ffiffi 0 0 0 0 C C B 3 3 B qffiffi C C B 2C B 0 p1ffiffi 0 0 0 3A @ 3 p1ffiffi 0 0 i p1ffiffi6 0 i p1ffiffi3 2

93

0

ðB:1Þ

The SO-adapted basis functions of a 2T2 state, defined by Eq. (30), are

  ~ 1 ¼ p1ffiffiffi wþ þ iwþ þ 2w ; w x y z 6   1 ~ 2 ¼ pffiffiffi iw þ w ; w y x 2  1  þ ~ w3 ¼ pffiffiffi 2wz þ iwy  wx ; 6  1  þ ~ w4 ¼ pffiffiffi iwx  iwþy ; 2  1  þ ~ w5 ¼ pffiffiffi wx þ iwþy  wz ; 3   1 ~ 6 ¼ pffiffiffi wþ  iw þ w ; w z y x 3

ðB:2aÞ ðB:2bÞ ðB:2cÞ ðB:2dÞ ðB:2eÞ ðB:2fÞ

~1  w ~ 4 transform as G3=2 ; w ~ 5; w ~ 6 transform as E5/2 in T 0 . w d References [1] M.D. Sturge, Solid State Phys. 20 (1967) 91. [2] R. Englman, The Jahn–Teller Effect, Wiley, New York, 1972. [3] I.B. Bersuker, V.Z. Polinger, Vibronic Interactions in Molecules and Crystals, Springer, Heidelberg, 1989. [4] I.B. Bersuker, The Jahn–Teller Effect, Cambridge University Press, Cambridge, 2006. [5] E. Wigner, Group Theory, Academic Press, New York, 1959. [6] H.A. Bethe, E.E. Salpeter, Quantum Mechanics for One- and Two-Electron Atoms, Springer, Berlin, 1957. [7] J.H. van Vleck, Rev. Mod. Phys. 23 (1941) 213. [8] A. Wolf, M. Reiher, B.A. Heß, in: Relativistic Electronic-Structure Theory Part I. Fundamentals, Elsevier, Amsterdam, 2002 (Chapter 11). [9] L.V. Poluyanov, W. Domcke, J. Chem. Phys. 129 (2008) 224102. [10] H.A. Jahn, E. Teller, Proc. Roy. Soc. (Lond.) A 161 (1937) 220. [11] M.V. Berry, Proc. Roy. Soc. (Lond.) A 392 (1984) 45. [12] M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley, Reading, 1962. [13] B.A. Heß, C.M. Marian, in: P. Jensen, P.R. Bunker (Eds.), Computational Molecular Spectroscopy, Wiley, New York, 2000, p. 152. [14] D. Opalka, M. Segado, L.V. Poluyanov, W. Domcke, Phys. Rev. A 81 (2010) 042501. [15] In ref. [9], p. 3, it has erroneously been stated that HSO lacks invariance with respect to generalized reflections and improper rotations. The set of operators given in ref. [9] is the rotational subgroup of the tetrahedral spin-double group T 0d . The complete set of operators of T 0d is given in Appendix A. [16] We use the symbols E1/2, E5/2, G3/2 of Landau and Lifshitz and Hamermesh [12] for the representations of T 0d . In the solid-state literature, these representations are denoted as C6, C7, C8. [17] H.A. Jahn, Proc. Roy. Soc. (Lond.) A 164 (1938) 117. [18] E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. [19] L.V. Poluyanov, W. Domcke, unpublished, 2010. [20] W. Moffitt, W. Thorson, Phys. Rev. 108 (1957) 1251.