Jahn-Teller, Pseudo-Jahn-Teller, And Spin-Orbit Coupling Hamiltonian of a d Electron in an...

Jahn-Teller, pseudo-Jahn-Teller, and spin-orbit coupling Hamiltonian of a delectron in an octahedral environmentLeonid V. Poluyanov and Wolfgang Domcke Citation: J. Chem. Phys. 137, 114101 (2012); doi: 10.1063/1.4751439 View online: http://dx.doi.org/10.1063/1.4751439 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i11 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

Downloaded 20 Sep 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

http://jcp.aip.org/?ver=pdfcovhttp://aipadvances.aip.org/resource/1/aaidbi/v2/i1?&section=special-topic-physics-of-cancer&page=1http://jcp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Leonid V. Poluyanov&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://jcp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Wolfgang Domcke&possible1zone=author&alias=&displayid=AIP&ver=pdfcovhttp://jcp.aip.org/?ver=pdfcovhttp://link.aip.org/link/doi/10.1063/1.4751439?ver=pdfcovhttp://jcp.aip.org/resource/1/JCPSA6/v137/i11?ver=pdfcovhttp://www.aip.org/?ver=pdfcovhttp://jcp.aip.org/?ver=pdfcovhttp://jcp.aip.org/about/about_the_journal?ver=pdfcovhttp://jcp.aip.org/features/most_downloaded?ver=pdfcovhttp://jcp.aip.org/authors?ver=pdfcov

THE JOURNAL OF CHEMICAL PHYSICS 137, 114101 (2012)

Jahn-Teller, pseudo-Jahn-Teller, and spin-orbit coupling Hamiltonianof a d electron in an octahedral environment

Leonid V. Poluyanov1 and Wolfgang Domcke21Institute of Chemical Physics, Academy of Sciences, Chernogolovka, Moscow 142432, Russia2Department of Chemistry, Technische Universitt Mnchen, D-85747 Garching, Germany

(Received 16 May 2012; accepted 24 August 2012; published online 17 September 2012)

Starting from the model of a single d-electron in an octahedral crystal environment, the Hamiltonianfor linear and quadratic Jahn-Teller (JT) coupling and zeroth order as well as linear spin-orbit (SO)coupling in the 2T2g + 2Eg electronic multiplet is derived. The SO coupling is described by themicroscopic Breit-Pauli operator. The 10 10 Hamiltonian matrices are explicitly given for alllinear and quadratic electrostatic couplings and all linear SO-induced couplings. It is shown that the2T2g manifold exhibits, in addition to the well-known electrostatic JT effects, linear JT couplingswhich are of relativistic origin, that is, they arise from the SO operator. While only the eg mode is JT-active in the 2Eg state in the nonrelativistic approximation, the t2g mode becomes JT-active throughthe SO operator. Both electrostatic as well as relativistic forces contribute to the 2T2g 2Eg pseudo-JT coupling via the t2g mode. The relevance of these analytic results for the static and dynamic JTeffects in octahedral complexes containing heavy elements is discussed. 2012 American Instituteof Physics. [http://dx.doi.org/10.1063/1.4751439]

I. INTRODUCTION

The Jahn-Teller (JT) effect is an extremely widespreadphenomenon in molecular and solid-state spectroscopy.16

It is reflected, for example, in the absorption and photolu-minescence spectra of transition-metal or rare-earth ions incrystal lattices1, 712 or in ultrafast radiationless transitionsand photochemical reactions in organometallic coordinationcomplexes.1316

The interplay of the splitting of degenerate electronicstates by linear and quadratic JT coupling of electrostatic ori-gin and by spin-orbit (SO) coupling is ubiquitous in solid-state spectroscopy. In the vast theoretical literature on thisfield, the JT coupling is described by a Taylor expansion ofthe spin-free electronic Hamiltonian up to second order in therelevant vibrational displacement coordinates. The SO cou-pling, on the other hand, is approximated in zeroth order, thatis, by its value at the high-symmetry reference geometry. Inmost cases, the SO coupling is represented by an effectivesingle-center atomic SO operator which, as such, is indepen-dent of the nuclear coordinates.16 This level of description(spin-free electronic potentials up to second order, SO cou-pling in zeroth order) may be termed the standard model ofJT theory.

In the present work, we address the treatment of SO-coupling effects beyond the standard model for cubic andoctahedral systems. The analysis will be based on the mi-croscopic Breit-Pauli SO operator17 and all SO-inducedvibronic-coupling terms will systematically be included up tofirst order in the normal-mode displacements. It will be shownthat there exist JT as well as pseudo-JT (PJT) couplings whicharise from the SO operator and thus are of relativistic origin.Since the symmetry selection rules in the relativistically gen-eralized cubic and octahedral spin double groups (O ,O h) aredifferent from the selection rules in the spin-less case (point

groups O, Oh), there arise novel SO-induced JT and PJT cou-plings which partly are complementary to the electrostatic JTcouplings.

Octahedral coordination is particularly common in solidsand in organometallic compounds. The simplest example ofsuch systems is an octahedrally coordinated transition-metalion with a single electron in the d-shell, such as the Ti3+

ion in sapphire (Al2O3),10, 18 which is of particular interest

as a highly versatile lasing material.19, 20 As is well known,the fivefold degenerate d orbital splits into a threefold degen-erate orbital of T2g symmetry and a twofold degenerate or-bital of Eg symmetry.21 The JT-active modes are of t2g andeg symmetry.16 In addition to the JT couplings within the de-generate T2g and Eg electronic manifolds, the T2g and Eg statescan interact via the t2g normal mode (PJT coupling). Inclusionof spin doubles the minimal electronic basis from the five spa-tial orbitals of a d level to ten spin-orbitals. The JT and PJTvibronic Hamiltonians are thus given by 10 10 matrices.

The JT and SO coupling effects in transition-metal andrare-earth compounds are not only responsible for complexelectronic spectra and ultrafast photophysical dynamics (seeRefs. 1416, 22, 23 for recent experimental studies andRefs. 2427 for theoretical studies), but also for distortions ofground-state equilibrium structures from the expected tetra-hedral or octahedral shapes (the so-called static JT effect16).In most cases, the electrostatic (spin-free) JT selection rules28

and spin-free electronic-structure calculations were employedfor the analysis of the static JT effect in tetrahedral and octa-hedral systems.2933 Only relatively recently, due to the avail-ability of broadly applicable relativistic electronic-structurecodes, the effects of strong SO couplings on the equilibriumgeometries of heavy-element compounds have been eluci-dated for a number of examples.3437 The expected quench-ing of the static JT distortion38 by large SO splittings was

0021-9606/2012/137(11)/114101/11/$30.00 2012 American Institute of Physics137, 114101-1


http://dx.doi.org/10.1063/1.4751439http://dx.doi.org/10.1063/1.4751439http://dx.doi.org/10.1063/1.4751439

114101-2 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137, 114101 (2012)

observed. It was shown that SO splittings can be of the sameorder of magnitude as crystal-field splittings for the hexa-halides of the heaviest transition-metal elements.37

In the present work, we develop a systematic theory ofJT, PJT, and SO coupling in octahedral systems which is notlimited to the assumption of weak SO-coupling effects. Theeigenvalues of the 10 10 JT matrix define a tenfold adia-batic potential-energy surface as a function of the t2g and egnormal coordinates. This model is suitable for a systematicfitting of ab initio calculated energy data and thus allows thequantitative extraction of JT couplings and SO couplings, in-cluding also SO-induced JT couplings, from ab initio data.

II. ELECTRONIC HAMILTONIAN, ELECTRONIC BASISFUNCTIONS, VIBRATIONAL NORMAL MODES AND JTSELECTION RULES

In this work, we are concerned with XY6 complexesof octahedral symmetry (point group Oh), where X is atransition-metal atom or ion and Y is a main-group atom orion. We consider the case of a single electron in the d-shell ofthe central atom (or, equivalently, a d9 configuration), whileY is a closed-shell atom or ion.

For the purpose of symmetry analysis, the electrostatic(spin-free) Hamiltonian of the single electron in an octahedralcrystal environment can be written as (in atomic units)

Hes = 122 (r) (1a)

(r) = q0r

+6

k=1

q1

rk, (1b)

where q0 and q1 are effective charges of the atomic centers Xand Y, respectively, and

r = |r| (1c)

rk = |r Rk| , (1d)where r is the radius vector of the electron and the Rk , k= 1. . . 6, denote the radius vectors from the origin to the sixcorners of the octahedron. The X atom is at the origin of thecoordinate system.

The Breit-Pauli SO coupling operator for this systemreads17

HSO = ige2e S[q0

r3(r ) + q1

6k=1

1

r3k(rk )

], (2a)

where

S = 12 (ix + jy + kz), (2b)

x, y, z are the Pauli spin matrices, e is the Bohr magne-ton, ge is the g-factor of the electron, and i, j, k are the Carte-sian unit vectors. For the analysis of the symmetry properties,it is useful to write the Breit-Pauli operator in determinal form

HSO = 12 ige2e

x y z

x y zx

y

z

, (3a)

where

x = x

, etc. (3b)

Equation (3a) reveals that the SO coupling operator is com-pletely determined by the electrostatic potential (r). Since(r) depends on the nuclear coordinates, so does HSO.

The symmetry group of the SO operator (2) is O h, theoctahedral spin double group. The elements of O h are of theform

Zn = CnU n, (4)where the Cn are the 48 spatial symmetry operations of thepoint group Oh and the Un are unitary 2 2 matrices actingon the spin functions , . To close the group, Zn has tobe included for each Zn of Eq. (4), which doubles the grouporder to 96.39 The explicit form of the symmetry operators Znis given in Appendix A.

In addition, HSO is time-reversal invariant. The time-reversal operator for a single unpaired electron is the anti-unitary operator (up to an arbitrary phase factor)39

= iycc =(

0 11 0

)cc, (5)

where cc denotes the operation of complex conjugation. Thefull symmetry group of HSO of Eq. (2) is thus

G = O h (E, ), (6)where E is the identity operator. The order of the group G is192. The operations of O h commute with .

As is well known, the fivefold degenerate d-orbital onthe transition-metal atom splits into a threefold degenerate or-bital of T2g symmetry and a twofold degenerate orbital of Egsymmetry.21 The former can be written as

x = yzf (r)y = xzf (r)z = xyf (r)

T2g, (7)

where x, y, z are the coordinates of the electron with respectto the center of the octahedron and f(r) is an exponential orGaussian radial function. The orbitals of Eg symmetry can bewritten as

a = 16 (2z2 x2 y2)f (r)b = 12 (x2 y2)f (r)

}Eg. (8)

Including electron spin, the electronic basis set is given by theten spin orbitals{x,y,z,z,y,x,a,b,b,a

}.

(9)These basis functions define a ten-dimensional double-valuedreducible representation

(T2g + Eg) Eg1/2, (10a)which can be decomposed into irreducible representationsof O h

(T2g + Eg) Eg1/2 = Eg5/2 + 2Gg3/2. (10b)



6

C'

C

7

5

2

1

3

4

8

OA'

A

B

B'

x

y

z

FIG. 1. Enumeration of the corners of the octahedron (A, B, C, A, B, C)and the corners of the enclosing cube (1. . . 8).

Here Eg1/2 and Eg5/2 are two-dimensional double-valued ir-reducible representations, while Gg3/2 is a four-dimensionaldouble-valued irreducible representation of O h.

40 One of thetwo Gg3/2 manifolds corresponds to the 2Eg state; the otherGg3/2 manifold is an irreducible component of the 2T2g elec-tronic state.

The 15 vibrational normal modes of the centered octahe-dron are of the species

= a1g + eg + t2g + 2t1u + t2u. (11)

The definitions of the symmetry coordinates and pictorialrepresentations can be found, for example, in Table 2.2 andFig. 2.3 of Ref. 6, respectively. Designating the central atomas O and the atoms at the corners of the octahedron as A, B,C, C, B, A, see Fig. 1, symmetry-adapted linear combina-tions of t2g and eg symmetry are

Sx = 12 (rBC + rBC rBC rBC)Sy = 12 (rAC + rAC rAC rAC)Sz = 12 (rBA + rBA rBA rBA)

t2g, (12)

sa = 123 (2rOC + 2rOC rOA rOA rOB rOB)sb = 12 (rOA + rOA rOB rOB)

eg,

(13)

where rAB, etc., are displacements of interatomic distancesfrom the octahedral reference geometry. Normal coordinatesQx, Qy, Qz of t2g symmetry and qa, qb of eg symmetry areobtained by multiplication of the symmetry coordinates withappropriate mass-dependent conversion factors.41

The well-known spin-free JT selection rules for T2g andEg electronic states in cubic/octahedral symmetry are28

[Eg]2 = A1g + Eg, (14a)

[T2g]2 = A1g + Eg + T2g, (14b)

where []2 denotes the symmetrized square of the irreduciblerepresentation .39 In electronic states of Eg symmetry, thevibrational mode of eg symmetry is JT-active in first order,while in electronic states of T2g symmetry, the eg mode aswell as the t2g mode are active in first order according toEq. (14). The selection rule for EgT2g PJT coupling is

Eg T2g = T1g + T2g. (15)The t2g mode is thus PJT-active in first order.

The JT selection rules relevant for the d 1 configurationin the spin double group O h are

42{G2g3/2

} = A1g + Eg + T2g, (16a)Gg3/2 Eg5/2 = Eg + T1g + T2g. (16b)

Here {2} denotes the antisymmetrized square of the irre-ducible representation .39 According to Eq. (16a), the four-fold degeneracy of a Gg3/2 level is lifted by t2g and eg modes,and according to Eq. (16b), Gg3/2 and Eg5/2 levels interact viathe t2g and eg modes. JT splitting of the twofold degenerateEg5/2 level is excluded by time-reversal symmetry, which re-quires at least twofold degeneracy of levels for an odd numberof electrons (Kramers degeneracy).

It should be noted that in the spin-free case only the egmode is JT-active in the Eg electronic state (Eq. (14a)), whileboth eg and t2g modes are active in the 2E (Gg3/2) electronicstate (Eq. (16a)). This implies that the JT-activity of the t2gmode must arise from the SO operator. The same applies intetrahedral symmetry.43, 44

III. (2T2g + 2Eg) (t2g + eg) JT AND PJT HAMILTONIANA. Taylor expansion of the electronic Hamiltonian andcalculation of matrix elements

In accordance with the standard model of JT coupling,we expand the electrostatic Hamiltonian up to second orderin the t2g and eg normal-mode displacements. Assuming thatSO coupling is somewhat weaker than the electrostatic inter-actions, we terminate the Taylor expansion of the SO operatorafter the first order. The Hamiltonian thus has the structure

H = Hes + HSO, (17)

Hes = H (0)es + H (1)es,Q + H (1)es,q + H (2)es,Q + H (2)es,q + H (2)es,Qq,(18)

HSO = H (0)SO + H (1)SO,Q + H (1)SO,q . (19)Here, H (0)es is the electrostatic Hamiltonian of Eq. (1) at the oc-tahedral reference geometry. Likewise, H (0)SO is the SO opera-tor of Eq. (2) at the reference geometry. The first- and second-order expansion terms of the electrostatic Hamiltonian can be



written as

H(1)es,Q = A1(t2g)Qx + A2(t2g)Qy + A3(t2g)Qz, (20)

H (1)es,q = B1(eg)qa + B2(eg)qb, (21)

H(2)es,Q = 13 C(a1g)

(Q2x +Q2y +Q2z

)+ 1

6D1(eg)

(2Q2z Q2x Q2y

)+ 1

2D2(eg)

(Q2x Q2y

)+ E1(t2g)QyQz + E2(t2g)QxQz + E3(t2g)QxQy,

(22)

H (2)es,q = 12 F (a1g)(q2a + q2b

)+ 12G1(eg)

(q2a q2b

)

2G2(eg)qaqb, (23)

H(2)es,Qq =

3

2 H1(t1g)Qxq+ +

32 H2(t1g)Qyq + H3(t1g)Qzqb

32 I1(t1g)Qxq

+

32 I2(t1g)Qyq

+ + I3(t1g)Qzqa.(24)

H(0)SO is written as

H(0)SO = hx(t1g)x + hy(t1g)y + hz(t1g)z. (25)

The first-order expansion terms of the SO operator take theform

H(1)SO,Q = 13 a(a2g)(Qxx +Qyy +Qzz)

+ 16b1(eg)(2Qxx Qyy Qzz)

+ 12b2(eg)(Qyy Qzz)

+ 12c1(t1g)(Qzy +Qyz)

+ 12c2(t1g)(Qxz +Qzx)

+ 12c3(t1g)(Qyx +Qxy)

+ 12d1(t2g)(Qzy Qyz)

+ 12d2(t2g)(Qxz Qzx)

+ 12d3(t2g)(Qyx Qxy), (26)

H(1)SO,q = e1(t1g)qx + e2(t1g)q+y + e3(t1g)qaz

+ f1(t2g)q+x f2(t2g)qy + f3(t2g)qbz, (27)where

q = 12 (

3qa qb), (28a)

q = 12 (qa

3qb). (28b)

In Eqs. (20)(27), the symmetries of the electronic opera-tors A . . . I and a . . . f , h, which are the coefficients of thenormal-mode expansion, are indicated in parentheses. Sincethe Pauli spin matrices have been factored out, the expan-sion coefficients in Eqs. (26) and (27) transform accordingto single-valued irreducible representations of O h. Given thetransformation properties of the electronic operators, the cal-culation of matrix elements with the electronic basis functionsof the d-orbital (Eq. (9)) is straightforward. Note that the ex-pansions ((20)(28)) also are useful for the construction ofJT/PJT vibronic matrices for p-orbitals or f-orbitals in octahe-dral systems.

The labor of calculating the matrix elements of theHamiltonian is significantly reduced by the hermiticity of theHamiltonian and by time-reversal symmetry. In the basis setof Eq. (9), the time-reversal operator has the representationgiven by the 10 10 matrix T in Appendix A. The require-ment that the Hamiltonian matrices commute with T leads tothe vanishing of many of the matrix elements and requiresothers to be equal, or equal up to a minus sign.

The matrix elements of the electronic Hamiltonian de-fined by Eqs. (17)(28) form a 10 10 hermitean matrix,which we write as

H = Hes +HSO, (29a)

Hes = H (0)es +H (1)es +H (2)es,Q +H (2)es,q +H (2)es,Qq, (29b)

HSO = H (0)SO +H (1)SO,Q +H (1)SO,q , (29c)

where H (1)es contains the linear JT and PJT coupling terms ofboth t2g and eg modes. For convenience, the totally symmetricquadratic part of the potentials is included in H (0)es , that is,

H (0)es =(ET + 122T Q2 + 122T q2

)16

(EE + 122EQ2 + 122Eq2)14, (30)where ET and EE are the vertical energies of the 2T2g and 2Egstates, 16 and 14 are 6 6 and 4 4 unit matrices, respec-tively, T, E and T, E are the harmonic vibrational fre-quencies of the t2g and eg modes in the 2T2g and 2Eg states,and

Q2 = Q2x +Q2y +Q2z, (31a)

q2 = q2a + q2b . (31b)The electrostatic vibronic matrices H (1)es , H

(2)es,Q, H

(2)es,q , and

H(2)es,Qq are given in Figs. 25. The vibronic matrices H

(0)SO ,

H(1)SO,Q resulting from the expansion of the SO operator are

given in Figs. 6 and 7. The vibronic matrices in Figs. 27 arethe main results of the present work. We have verified that theHamiltonian matrices in Figs. 27 commute with the symme-try operators of O h as well as with the time-reversal operatorgiven in Appendix A.



cq aQz aQy 0 0 0 dQx

3dQx

aQz cq+ aQx 0 0 0 dQy

3dQy

aQy aQx cqa 0 0 0 2dQz 0 0 0

0 0 0 cqa aQx aQy 0 0 0 2dQz

0 0 0 aQx cq+ aQz 0 0

3dQy dQy

0 0 0 aQy aQz cq 0 0

3dQx dQx

dQx dQy 2dQz 0 0 0 bqa bqb

3dQx

3dQy 0 0 0 0 bqb bqa

0 0

0 0

0 0

0 0

0 0 0 0

3dQy

3dQx 0 0 bqa bqb

0 0 0 2dQz dQy dQx 0 0 bqb bqa

FIG. 2. Representation of the first-order electrostatic JT Hamiltonian, H (1)es,Q +H (1)es,q , in the spin-orbital basis (9). a, b, c, d are real parameters representinglinear T2g t2g coupling (a), T2g eg coupling (c), Eg eg coupling (b) and T2g Eg PJT coupling by the t2g mode (d).

B. 2T2g (t2g + eg) JT Hamiltonian

In this and Subsection III C, we assume that thecrystal-field splitting is large compared to other parame-ters of the system, such that the 2T2g and 2Eg electronic

states can approximately be considered as decoupled. TheJT Hamiltonian of the 2T2g state is given by the upper-left 6 6 block of the matrices in Figs. 27. Omitting,for brevity and clarity, the quadratic JT coupling terms, wehave

H[2T2g (t2g + eg)] = (ET + 122TQ2 + 122T q2)16

+

cq i+ aQz aQy iQx iQz Q+ 0i+ aQz cq+ aQx + iQy i+ Qz 0 Q+aQy + iQx aQx iQy cqa 0 i Qz + iQz+ iQz i+ Qz 0 cqa aQx + iQy aQy iQx

Q 0 i Qz aQx iQy cq+ i+ aQz0 Q iQz aQy + iQx i+ aQz cq

,

(32)

where q is defined in Eq. (28a),

Q = Qx iQy, (33)and a, c, , are real parameters. The parameter a is thelinear electrostatic T2g t2g JT coupling constant, c is thelinear electrostatic T2g eg JT coupling constant, repre-sents the zeroth-order SO splitting of the 2T2g state and isthe linear relativistic T2g t2g JT coupling constant. It is seenthat the t2g mode is JT-active via electrostatic (parameter a)as well as relativistic (parameter ) forces. The eg mode, onthe other hand, is JT-active only via the electrostatic forces(parameter c).

The structure of the zeroth-order SO contribution (thatis, the upper left 6 6 block in Fig. 6) is the same as foundfor a 2T2 state derived from p-type orbitals in tetrahedral

symmetry.44 This nondiagonal matrix can be transformed todiagonal form by a unitary transformation U which defines aSO-adapted electronic basis. A suitable choice of U has beenfound in Ref. 44. In the transformed basis, the zeroth-orderSO matrix of a 2T2g state takes the form

H(0)SO

(2T2g

) = diag(,,,, 2, 2), (34)in agreement with the group-theoretical result (10b). The 2T2 (t2 + e) Hamiltonian matrix in the SO-adapted basis isgiven in Ref. 44 (Eqs. (33) and (48)).

It should be noted that in tetrahedral systems the 2T2 e JT coupling has electrostatic as well as relativistic con-tributions (parameters c and ). In octahedral and cubic sys-tems, on the other hand, the linear relativistic 2T2g eg cou-pling parameter is zero by symmetry. The requirement that


114101-6L.V.P

oluyanovand

W.D

omcke

J.Chem

.Phys.137,114101

(2012)

FIG. 3. Representation of the second-order electrostatic JT Hamiltonian, H (2)es,Q, of the t2g mode in the spin-orbital basis (9). A, B, C, D are real parameters representing quadratic T2g t2g coupling (A, B), quadratic T2g eg coupling (C) and quadratic T2g Eg PJT coupling by the t2g mode (D).

FIG. 4. Representation of the second-order electrostatic JT Hamiltonian, H (2)es,q , of the eg mode in the spin-orbital basis (9). A and C are real parameters representing quadratic T2g t2g coupling (A) and quadratic Eg eg coupling (C).



the 2T2g e vibronic matrix must commute with the sym-metry operators of O h (Appendix A) enforces the vanishingof the coupling parameter . This is another example whichreveals that the symmetry selection rules for the SO operatordiffer from those for the electrostatic Hamiltonian in a subtlemanner.

C. 2Eg (t2g + eg) JT Hamiltonian

The JT Hamiltonian of the 2Eg state is given by the lowerright 4 4 block of the matrices in Figs. 27. Omitting, forbrevity and clarity, the quadratic JT coupling terms, we have

H[

2E2g (t2g + eg)] = (EE + 122EQ2 + 122Eq2)14

+

bqa bqb + iQz iQ 0bqb iQz bqa 0 iQ

iQ+ 0 bqa bqb + iQz0 iQ+ bqb iQz bqa

.

(35)

Note that there is no zeroth-order SO splitting in the 2Egstate. In accordance with the JT selection rules discussed inSec. II, the eg mode is JT active through electrostatic forces(coupling constant b), while the t2g mode is JT-active throughrelativistic forces (coupling constant ), as in tetrahedralsystems.43

It is straightforward to determine the adiabatic potential-energy functions as eigenvalues of the matrix (35)

V1 = V2 = EE + 122EQ2 + 122Eq2 b2q2 + 2Q2

V3 = V4 = EE + 122EQ2 + 122Eq2 +b2q2 + 2Q2.

(36)

The energy functions (36) represent an elliptic Mexican hatin six-dimensional space (the energy as a function of five nu-clear coordinates). Since the adiabatic eigenvectors also aregiven in analytic form, it is straightforward to calculate thegeometry phases of the adiabatic wave functions as a contourintegral45 as discussed in Ref. 43. The adiabatic wave func-tions carry nontrivial geometric phases which depend on theorientation of the plane of integration in the five-dimensionalnuclear coordinate space.

D. (2T2g + 2Eg) (t2g + eg) PJT couplingOmitting, for brevity, second-order terms, the 6 4 block

which couples the 6 6 2T2g matrix with the 4 42 Eg matrixreads

HT2g,Eg

=

dQx + iQy

3dQx + i+Qy i + Qz i

3 + +QzdQy iQx

3dQy + i+Qx + iQz

3 i+Qz

2dQz 2i 2Q+ 2Q2Q+ 2Q 2i 2dQz

3 i+Qz + iQz

3dQy i+Qx dQy + iQxi3 +Qz i Qz

3dQx i+Qy dQx iQy

,

(37)

where

=

3, (38a)

=

3 . (38b)It is seen from Eq. (37) that there exists a zeroth-order rela-tivistic coupling of the 2T2g and 2Eg states (parameter ), afirst-order electrostatic coupling by the t2g mode (parameter

d), as well as first-order relativistic couplings by the t2g mode(parameters , ). The eg mode does not contribute to the 2T2g 2Eg PJT coupling.

The nonrelativistic 2T2g 2Eg PJT coupling is deter-mined by the single parameter d in Eq. (37). Our result is inagreement with the linear PJT coupling Hamiltonian given byStoneham and Lannoo for cubic and tetrahedral systems.46

The relativistic linear 2T2g 2Eg PJT coupling by the t2gmode (parameters , ) is a new result.



0 EQzqa3

2EQyq

+ 0 0 0 H(17)Qq H

(18)Qq 0 0

EQzqa 0 32 EQxq 0 0 0 H(27)Qq H

(28)Qq 0 0

32EQyq

+ 32EQxq

0 0 0 0 FQzqa GQzqb 0 0

0 0 0 0

32EQxq

32EQyq

+ 0 0 GQzqb FQzqa

0 0 0 32EQxq

0 EQzqa 0 0 H(28)Qq H

(27)Qq

0 0 0 32EQyq

+ EQzqa 0 0 0 H(18)Qq H

(17)Qq

H(17)Qq H

(27)Qq FQzqa 0 0 0 0 0 0 0

H(18)Qq H

(28)Qq GQzqb 0 0 0 0 0 0 0

0 0 0 GQzqb H(28)Qq H

(18)Qq 0 0 0 0

0 0 0 FQzqa H(27)Qq H

(17)Qq 0 0 0 0

FIG. 5. Representation of the electrostatic JT Hamiltonian bilinear in the t2g and eg modes, H(2)es,Qq , in the spin-orbital basis (9). E, F, G are real parameters

representing bilinear T2g (t2g + eg) JT coupling (E) and bilinear T2g Eg PJT coupling (F, G). See Appendix B for the definition of H (17)Qq , H (27)Qq , H (18)Qq ,H

(28)Qq .

0 i 0 0 0 0 0 i i3

i 0 0 i 0 0 0 0 3

0 0 0 0 i 0 2i

i 0 0 0 0 0 0 2i 0

0 0 i 0 0 i 3

0 0 0 i 0 i3 i

0 0 0 0 3 i

3 0 0

0 0 2i 0 i 0 0

i 0 2i 0 0 0 0

i3

3 0 0 0 0 0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

FIG. 6. Representation of the zeroth-order SO coupling Hamiltonian, H (0)SO ,in the spin-orbital basis (9). and are real parameters.

IV. CONCLUSIONS

We have presented a systematic derivation of the JT, PJT,and SO coupling Hamiltonian arising from a singly occupiedd-orbital in an octahedral crystal environment. The electro-static electronic Hamiltonian has been expanded up to sec-ond order in t2g and eg normal-mode displacements. The SOcoupling Hamiltonian has been expanded up to first order inthese modes. While the expansion of the electrostatic poten-tial up to second order corresponds to the standard model ofJT theory,16 the expansion of the SO coupling operator up tofirst order extends the theory beyond the standard model, inwhich SO coupling is considered as an atomic property whichis independent of the nuclear coordinates.16

It has been shown that there exist JT and PJT forceswhich are of relativistic origin, arising from the SO operator.In some cases, e.g., for the 2T2g t2g JT effect, the electro-static and relativistic forces act additively, resulting in con-structive or destructive interference of electrostatic and rela-tivistic JT couplings. In other cases, e.g., for the 2E2g (t2g

0 0 iQx iQz Q+ 0 iQy i+Qy Qz +Qz

0 0 iQy Qz 0 Q+ iQx i+Qx iQz i+Qz

iQx iQy 0 0 Qz iQz 0 0 2Q+ 2Q

iQz Qz 0 0 iQy iQx 2Q+ 2Q 0 0

Q 0 Qz iQy 0 0 i+Qz iQz i+Qx iQx

0 Q iQz iQx 0 0 +Qz Qz i+Qy iQy

iQy iQx 0 2Q i+Qz +Qz 0 iQz iQ 0

i+Qy i+Qx 0 2Q+ iQz Qz iQz 0 0 iQ

Qz iQz 2Q 0 i+Qx i+Qy iQ+ 0 0 iQz

+Qz i+Qz 2Q+ 0 iQx iQy 0 iQ+ iQz 0

FIG. 7. Representation of the first-order SO coupling Hamiltonian of the t2g mode, H(1)SO,Q, in the spin-orbital basis (9). , , , are real parameters representing

relativistic linear T2g t2g coupling (), E2g t2g coupling () and T2g Eg PJT coupling (, ).



+ eg) JT effect, the electrostatic and relativistic forces arecomplementary. In the 2Eg state, the eg mode is JT-activethrough electrostatic forces, while the t2g mode is JT-activethrough relativistic forces. The 2T2g and 2Eg states interact inzeroth order through the SO operator and in first order throughthe t2g mode. The PJT coupling via the t2g mode has elec-trostatic and relativistic origins. The electrostatic JT couplingterms derived in the present work agree with the JT textbooksfor the terms linear and quadratic in the t2g and eg modes. Forthe mixed coupling matrix H (2)es,Qq (Fig. 5), we could not findliterature results. The first-order relativistic JT/PJT couplingmatrix H (1)SO,Q (Fig. 7) also is new in JT theory.

While the analysis has been performed for the model ofan electron in a d-orbital at the central atom of an octahedralcomplex, the derived JT and PJT matrices should be gener-ally valid for d-orbitals in systems of cubic symmetry (pointgroups O, Oh), such as cubic X8 systems or centered cubicXY8 systems.

The magnitude of the relativistic JT and PJT coupling pa-rameters scales such as the SO splittings, that is, with Z2 in thevalence shell of heavy elements.47 For the first-row transition-metal elements, the relativistic JT forces are expected to beweak compared to the electrostatic forces. For the third-rowtransition metals, on the other hand, not only the zeroth-orderSO couplings can be large, but also the SO-induced JT cou-plings may be of the same order of magnitude as the electro-static JT couplings.

It has recently been stated that Today, no one has postu-lated a relativistic JT theorem; therefore, molecular geometrydistortions due to the dynamic JT effect are a consequenceof nonrelativistic treatments.37 This statement is supersededby our previous results for tetrahedral systems43, 44 and thepresent results for cubic and octahedral systems. A JT theoryis now available which systematically includes SO-couplingeffects up to first order in normal-mode displacementsfrom the reference geometry. If considered necessary, theexpansion of the BP operator (Eq. (19)) could be extended toinclude second-order terms.

To avoid the need of an explicit diabatization of the abinitio electronic wave functions,48 it is usually convenient todetermine the electrostatic and relativistic JT couplings andSO-splitting parameters by the fitting of the eigenvalues of theJT model Hamiltonian to adiabatic ab initio energy data (di-abatization by ansatz). For such applications, it is essentialto know which gradients and second derivatives of the elec-trostatic and SO energies are zero by symmetry and thus arenot adjustable parameters in the fitting procedure.

To the knowledge of the authors, no attempts have beenmade so far towards the ab initio determination of geometricdistortions in transition-metal or rare-earth compounds whicharise from purely relativistic forces and are thus not predictedby the time-honored electrostatic JT selection rules.28 Sincethe zeroth-order SO splitting of 2E states is zero by symmetryin tetrahedral and octahedral systems, these states are generi-cally unstable with respect to relativistic distortions in normalmodes of t2 symmetry, cf. Eq. (16a). The existence of suchSO-induced 2E t2 JT distortions has recently been demon-strated by relativistic multi-reference self-consistent-field cal-culations for the 2E ground states of the tetrahedral clustercations of the elements of the fifth main group (P+4 , As

+4 , Sb

+4 ,

Bi+4 ).49, 50

APPENDIX A: THE SYMMETRY OPERATORSOF THE SPIN DOUBLE GROUP Oh

The corners of the cube enclosing the octahedron areenumerated 1. . . 8, see Fig. 1. Twofold and fourfold rotationsaround the coordinate axes are denoted C2, k, C4, k, k = x, y,z. Threefold rotations around axes through opposite cornerson the enclosing cube are denoted C3, ij, i, j = 1. . . 8. Twofoldrotations around axes through opposite edges of the enclosingcube are denoted C2, ij, kl, i, j, k, l = 1. . . 8. The unit operationis denoted as E, the inversion as I. In this notation, the 24 es-sential symmetry operators of the vibronic Hamiltonian are

Z1 = E(

1 00 1

)Z2,x = iC2,x

(0 11 0

)

Z2,y = iC2,y(

0 ii 0

)Z2,z = iC2,z

(1 00 1

)

Z3,18 = eC3,18( i i

)Z3,27 = eC3,27

( i i

)

Z3,36 = e+C3,36( i i

)Z3,45 = e+C3,45

(

i i)

Z23,18 = e+C23,18(

i i)

Z23,27 = e+C23,27(

i i

)

Z23,36 = eC23,36(

i i)

Z23,45 = eC23,45(

i i

)

Z2,12,78 = C2,12,78(i

i

)Z2,43,56 = C2,43,56

(i i

)



Z2,23,76 = C2,23,76(

0 ee+ 0

)Z2,14,58 = C2,14,58

(0 e+

e 0)

Z2,16,38 = C2,16,38(

i ii i

)Z2,25,38 = C2,25,38

(i ii i

)

Z4,x = C4,x(

ii

)Z4,y = C4,y

(

)

Z4,z = C4,z(

e 00 e+

)Z34,x = C34,x

( ii

)

Z34,y = C34,y(

)

Z34,z = C34,z(

e+ 00 e

)

where

= 12

e = ei/4.24 additional symmetry operators are given by the product ofthe above operators with the inversion operator:

Z24+n = IZn, n = 1 . . . 24.

To close the group, the operator Zn has to be in-cluded for each Zn. This defines the 96 operators of thegroup Oh. Note that the point-group symmetry operators actboth on the electronic basis states as well as on the nuclearcoordinates.

In the spin-orbital basis of Eq. (9), the electronic parts ofthe above symmetry operators are given by 10 10 matrices.Z3, 27, for example, reads

Z3,27 = e

0 1 0 0 1 0 0 0 0 00 0 1 1 0 0 0 0 0 01 0 0 0 0 1 0 0 0 0i 0 0 0 0 i 0 0 0 00 0 i i 0 0 0 0 0 0

0 i 0 0 i 0 0 0 0 00 0 0 0 0 0 12

3

2

32

12

0 0 0 0 0 0

32 12 12

3

2

0 0 0 0 0 0 i

32 i2 i2 i

3

2

0 0 0 0 0 0 i2 i

32

i

32 i2

C(n)3,27.

The superscript n in the above equation indicates that thispoint-group symmetry operator acts on the nuclear degrees offreedom.

The 23 other essential symmetry operator matrices areconstructed analogously. The JT+PJT Hamiltonian matrixmust commute with all 24 essential symmetry operators andthe inversion operator I. The commutation with the remainingsymmetry operators follows trivially. In addition, the Hamil-tonian matrix must commute with the time-reversal operator.

In the basis of Eq. (9), it is given by

T =

0 0 0 0 0 1 0 0 0 0

0 0 0 0 1 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 1 0 0 0

cc.



APPENDIX B: ABBREVIATIONS IN FIG. 5

The abbreviations in Fig. 5 are defined as follows:

H(17)Qq =

( 34Gq+ +

3

4 Fq)Qx, (B1)

H(27)Qq =

(34Gq

3

4 Fq+)Qy, (B2)

H(18)Qq =

(

3

4 Gq+ 34Fq)Qx, (B3)

H(28)Qq =

(

3

4 Gq 34Fq+)Qy. (B4)

1M. D. Sturge, Solid State Phys. 20, 91 (1967).2R. Englman, The Jahn-Teller Effect (Wiley, New York, 1972).3I. B. Bersuker and V. Z. Polinger, Vibronic Interactions in Molecules andCrystals (Springer, Heidelberg, 1989).

4M. C. M. OBrien and C. C. Chancey, Am. J. Phys. 61, 688 (1993).5I. B. Bersuker, Chem. Rev. 101, 1067 (2001).6I. B. Bersuker, The Jahn-Teller Effect (Cambridge University Press, Cam-bridge, 2006).

7S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of Transition MetalIons (Academic, New York, 1970).

8D. Reinen and C. Friebel, Structure and Bonding (Springer, Berlin, 1979),Vol. 37.

9D. Reinen and M. Atanasov, Magn. Reson. Rev. 15, 167 (1991).10M. Grinberg, A. Mandelis, and K. Fjeldsted, Phys. Rev. B 48, 5922 (1993).11F. Rodriguez and F. Aguado, J. Chem. Phys. 118, 10867 (2003).12M. N. Sanz-Ortiz and F. Rodriguez, J. Chem. Phys. 131, 124512 (2009).13A. Juris, V. Balzani, F. Barigelletti, S. Campagna, P. Belser, and A. von

Zelewsky, Coord. Chem. Rev. 84, 85 (1988).14A. Hauser, Top. Curr. Chem. 234, 155 (2004).15E. A. Juban, A. L. Smeigh, J. E. Monat, and J. K. McCusker, Coord. Chem.

Rev. 250, 1783 (2006).16A. Cannizzo, F. van Mourik, W. Gawelda, G. Zgrablic, C. Bressler, and

M. Chergui, Coord. Chem. Rev. 254, 2677 (2010).17H. A. Bethe and E. E. Salpeter, Quantum Mechanics for One- and Two-

Electron Atoms (Springer, Berlin, 1957).18S. Garcia-Revilla, F. Rodriguez, R. Valiente, and M. Pollnau, J. Phys. Con-

dens. Matter 14, 447 (2002).19P. Moulton, Optics News 8, 9 (1982).

20P. Albers, E. Stark, and G. Huber, J. Opt. Soc. Am. B 3, 134 (1986).21W. Moffitt and C. J. Ballhausen, Annu. Rev. Phys. Chem. 7, 107 (1956).22T. Hofbeck and H. Yersin, Inorg. Chem. 49, 9290 (2010).23J. N. Schrauben, K. L. Dillman, W. F. Beck, and J. K. McCusker, Chem.

Sci. 1, 405 (2010).24C. Daniel, Top. Curr. Chem. 241, 119 (2004).25R. G. McKinlay, J. M. Zurek, and M. J. Paterson, Adv. Inorg. Chem. 62,

351 (2010).26J.-L. Heully, F. Alary, and M. Boggio-Pasqua, J. Chem. Phys. 131, 184308

(2009).27A. Y. Freidzon, A. V. Scherbinin, A. A. Bogaturyants, and M. V. Alfimov,

J. Phys. Chem. A 115, 4565 (2011).28H. A. Jahn and E. Teller, Proc. R. Soc. London, Ser. A 161, 220 (1937).29A. Ceulemans, D. Beyens, and L. G. Vanquickenborne, J. Am. Chem. Soc.

106, 5824 (1984).30L. G. Vanquickenborne, A. E. Vinckier, and K. Pierloot, Inorg. Chem. 35,

1305 (1996).31R. Bruyndonckx, C. Daul, P. T. Manoharan, and E. Deiss, Inorg. Chem. 36,

4251 (1997).32M. Hargittai, Chem. Rev. 100, 2233 (2000).33D. Reinen, M. Atanasov, and P. Khler, J. Mol. Struct. 838, 151 (2007).34L. Alvarez-Thon, J. David, R. Arriata-Perez, and K. Seppelt, Phys. Rev. A

77, 034502 (2008).35J. David, P. Fuentealba, and A. Restrepo, Chem. Phys. Lett. 457, 42

(2008).36M. J. Molski and K. Seppelt, Dalton Trans. 2009, 3379.37J. David, D. Guerra, and A. Restrepo, Inorg. Chem. 50, 1480 (2011).38F. S. Ham, Phys. Rev. A 138, 1727 (1965).39M. Hamermesh, Group Theory and Its Application to Physical Problems

(Addison-Wesley, Reading, 1962).40We use the symbols E1/2, E5/2, G3/2 of Landau and Lifshitz and

Hamermesh39 for the representations of O. In the solid-state literature,these representations are denoted as 6, 7, 8.

41E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations(McGraw-Hill, New York, 1955).

42H. A. Jahn, Proc. R. Soc. London, Ser. A 164, 117 (1938).43L. V. Poluyanov and W. Domcke, J. Chem. Phys. 129, 224102 (2008).44L. V. Poluyanov and W. Domcke, Chem. Phys. 374, 86 (2010).45M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).46A. M. Stoneham and M. Lannoo, J. Phys. Chem. Solids 30, 1769 (1969).47P. Pyykk, Annu. Rev. Phys. Chem. 63, 45 (2012).48H. Kppel, in Conical Intersections: Electronic Structure, Dynamics and

Spectroscopy, edited by W. Domcke, D. R. Yarkony, and H. Kppel (WorldScientific, Singapore, 2004), Chap. 4.

49D. Opalka, M. Segado, L. V. Poluyanov, and W. Domcke, Phys. Rev. A 81,042501 (2010).

50D. Opalka, L. V. Poluyanov, and W. Domcke, J. Chem. Phys. 135, 104108(2011).


http://dx.doi.org/10.1016/S0081-1947(08)60218-0http://dx.doi.org/10.1119/1.17197http://dx.doi.org/10.1021/cr0004411http://dx.doi.org/10.1103/PhysRevB.48.5922http://dx.doi.org/10.1063/1.1569847http://dx.doi.org/10.1063/1.3223459http://dx.doi.org/10.1016/0010-8545(88)80032-8http://dx.doi.org/10.1007/b93641http://dx.doi.org/10.1016/j.ccr.2006.02.010http://dx.doi.org/10.1016/j.ccr.2006.02.010http://dx.doi.org/10.1016/j.ccr.2009.12.007http://dx.doi.org/10.1088/0953-8984/14/3/313http://dx.doi.org/10.1088/0953-8984/14/3/313http://dx.doi.org/10.1364/ON.8.6.000009http://dx.doi.org/10.1364/JOSAB.3.000134http://dx.doi.org/10.1146/annurev.pc.07.100156.000543http://dx.doi.org/10.1021/ic100872whttp://dx.doi.org/10.1039/c0sc00262chttp://dx.doi.org/10.1039/c0sc00262chttp://dx.doi.org/10.1007/b83770http://dx.doi.org/10.1016/S0898-8838(10)62009-0http://dx.doi.org/10.1063/1.3254196http://dx.doi.org/10.1021/jp111303ahttp://dx.doi.org/10.1098/rspa.1937.0142http://dx.doi.org/10.1021/ja00332a012http://dx.doi.org/10.1021/ic941381ihttp://dx.doi.org/10.1021/ic961220+http://dx.doi.org/10.1021/cr970115uhttp://dx.doi.org/10.1016/j.molstruc.2007.01.049http://dx.doi.org/10.1103/PhysRevA.77.034502http://dx.doi.org/10.1016/j.cplett.2008.04.003http://dx.doi.org/10.1039/b821121chttp://dx.doi.org/10.1021/ic102048jhttp://dx.doi.org/10.1103/PhysRev.138.A1727http://dx.doi.org/10.1098/rspa.1938.0008http://dx.doi.org/10.1063/1.3035189http://dx.doi.org/10.1016/j.chemphys.2010.06.025http://dx.doi.org/10.1098/rspa.1984.0023http://dx.doi.org/10.1016/0022-3697(69)90245-5http://dx.doi.org/10.1146/annurev-physchem-032511-143755http://dx.doi.org/10.1103/PhysRevA.81.042501http://dx.doi.org/10.1063/1.3629779

Jahn-Teller, Pseudo-Jahn-Teller, And Spin-Orbit Coupling Hamiltonian of a d Electron in an...

Documents

Transcript of Jahn-Teller, Pseudo-Jahn-Teller, And Spin-Orbit Coupling Hamiltonian of a d Electron in an...