Origin of warping in the E ‹ e Jahn-Teller problem: Quadratic vibronic coupling
versus anharmonicity and application to NaCl: Rh
2+
and triangular molecules
P. García-Fernández,
1
I. B. Bersuker,
2
J. A. Aramburu,
1
M. T. Barriuso,
3
and M. Moreno
1
1
Departamento de Ciencias de la Tierra y Física de la Materia Condensada, Universidad de Cantabria,
Avda. de los Castros s/n. 39005 Santander, Spain
2
Institute for Theoretical Chemistry, Department of Chemistry and Biochemistry, The University of Texas at Austin,
Austin, Texas 78712-1167, USA
3
Departamento de Física Moderna, Universidad de Cantabria, Avda. de los Castros s/n. 39005 Santander, Spain
Received 3 September 2004; revised manuscript received 11 March 2005; published 31 May 2005
A general method is suggested which allows one to separate the quadratic, cubic, and pseudo Jahn-Teller
contributions to the warping of the adiabatic potential energy surface APES of the E e problem employing
ab initio calculations. Numerical results were obtained for NaCl: Rh
2+
in a cluster approximation and triangular
molecules Na
3
,K
3
,Cu
3
,Ag
3
using density functional theory DFT and multireference second-order pertur-
bation theory CASPT2 methods. A largely unexpected result is that the contribution of cubic anharmonicity
to the energy barrier between the minima of the lower and upper branches of the APES is dominant in all the
systems and amounts for not less than 60% of the total. Another feature is that the three different contributions
mentioned above may have different signs, thus either enhancing or diminishing each other, affecting in
different ways the lower and upper branches of the APES. Other details of the numerical results are also
analyzed.
DOI: 10.1103/PhysRevB.71.184117 PACS numbers: 71.70.Ej, 71.23.An, 71.55.-i, 31.15.Ar
I. INTRODUCTION
Vibronic interactions of electronic states with low-
symmetry nuclear vibrations leading to Jahn-Teller JT or
pseudo JT effects play a key role in the theory of structure
and properties of matter.
1–8
In the realm of the JT effect,
particular attention has been focused on the E e problem
involving the interaction of a twofold degenerate electronic
E state with a twofold degenerate e vibration. The wide-
spread approach to JT effect problems was based on the as-
sumption that, since the nuclear displacements from the ref-
erence configuration are small, the main contribution to this
effect is due to the linear vibronic coupling terms, while the
quadratic coupling may be considered as a perturbation to
the solutions of the linear problem. If only the linear vibronic
coupling terms are taken into account in the E e problem
and vibrations are treated in the harmonic approximation, the
JT distortions are characterized by the adiabatic potential en-
ergy surface APES called the “Mexican Hat” Fig. 1a. It
has a continuum of equivalent minima forming a circular
trough. With the quadratic terms of vibronic coupling in-
cluded the APES becomes warped with alternating three
equivalent minima separated by three saddle points along the
bottom of the trough
9
Figs. 1b and 1c. This warping is
an essential feature of the APES of the E e problem affect-
ing all the main properties of the system.
1–9
In a further development it was shown that since the lin-
ear and quadratic terms are described by different and unre-
lated coupling constants, the quadratic terms are not neces-
sarily small as compared with the linear ones. Sufficiently
strong quadratic coupling changes drastically the topology of
the APES in JT systems and their properties. In particular, in
the E e problem large quadratic contributions produce three
additional conical intersections on the APES Ref. 10 that
under certain conditions change the symmetry and degen-
eracy of the ground state with all the consequences for the
observables.
11
In the T t
2
problem sufficiently strong qua-
dratic terms lead to additional seams of conical intersections
that change the ground state degeneracy too.
12
Important results obtained with strong quadratic coupling
terms raise the question of the role of higher order terms and
the convergence of the vibronic coupling approach taken in
consecutive approximations. To explore this issue we begin
with cubic terms of the vibronic coupling. Although it is well
known that cubic anharmonicity produces qualitatively the
same kind of warping of the APES in the E e problem see,
e.g., Refs. 13–16, there is not, up to now, a quantitative
evaluation of its significance in relation to the contribution of
quadratic terms. Additionally, it has also been suggested that
in transition metal systems the nd - n +1s mixing could also
contribute to the warping of the APES.
17–20
Note that anhar-
monicity and higher order contributions to the APES in JT
problems emerge not only from the higher order terms in the
FIG. 1. a APES shape for the linear JT effect, called the
“Mexican hat.” b “Tricorn” or warped “Mexican hat.” c Top
view of the “tricorn” showing the positions of three minima black
circles and saddle points black triangles.
PHYSICAL REVIEW B 71, 184117 2005
1098-0121/2005/7118/18411710/$23.00 ©2005 The American Physical Society 184117-1