-
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:
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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
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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)
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114101-3 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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
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114101-4 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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.
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114101-5 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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
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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).
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114101-7 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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.
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114101-8 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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 (, ).
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114101-9 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
+ 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
)
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114101-10 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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.
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114101-11 L. V. Poluyanov and W. Domcke J. Chem. Phys. 137,
114101 (2012)
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)
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