Origin of warping in theE⊗eJahn-Teller problem: Quadratic vibronic coupling versus anharmonicity and application toNaCl:Rh2+

Abstract

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 ͑APES͒ 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 ͑DFT͒ and multireference second-order perturbation theory ͑CASPT2͒ 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.

Origin of Small Barriers in Jahn–Teller Systems: Quantifying the Role of 3d–4s Hybridization in the Model System NaCl:Ni+

“…In a previous work 11 analyzing the different factors contributing to the barrier, B, we showed that when the E g term of an octahedral d 9 complex splits because of distortion in the Q θ coordinate, the energy surfaces of the resulting A 1g and B 1g electronic states can be written as …”

“…This mechanism of energy lowering is sometimes termed the pseudo-JT effect. 11,12 If we only consider the excited states arising from the e g 2 4s 1 configuration for NaCl:Ni + , the difference between V PJT (A 1g ) and V PJT (B 1g ) can be understood by looking at Figure 3, taking into account that the 4s orbital can only be coupled to the 3z 2 − r 2 one through the W 1θ (r) operator. As in the B 1g state there are two electrons in the a 1g *(∼3z 2 −r 2 ) orbital and only one in the A 1g state ( Figure 3) the energy gain due to this mechanism for B 1g is essentially twice that for A 1g .…”

“…Accordingly, a MX 6 complex (M = d 9 or d 7 ion) embedded in an insulating cubic lattice would prefer an elongated over a compressed geometry provided that it is elastically decoupled from the rest of the lattice where it is embedded. 11,21 By contrast, other authors 12,22,23 have ascribed the origin of B to the quadratic JT term involving operators that depend quadratically on Q θ and Q ε coordinates. However, a previous quantitative study performed on several JT systems points out that the quadratic JT term plays a minor role while the vibronic admixture of e g * orbitals (∼3z 2 − r 2 ; ∼x 2 − y 2 ) with other ones due to the linear JT term but treated in secondorder perturbations yields an important contribution 11 to B.…”

“…More precisely, in NaCl:Rh 2+ , about 40% of the barrier has been shown to come from the vibronic admixture of the a 1g *(∼3z 2 -r 2 ) orbital with the mainly 5s level of Rh 2+ , although the dominant contribution does arise from the anharmonicity. 11 Moreover, because the RhCl 6 4− complex in NaCl is elastically decoupled from the rest of the lattice, both contributions have been shown 11,13,14 to favor an elongated geometry, making B = 511 cm −1 . These results explain our interest in Ni + systems because the 3d−4s separation is only 2 eV for the free ion while values close to 10 eV correspond to the 4d−5s separation for free Ag 2+ or Rh 2+ ions.…”

“…In that section, particular attention is paid to exploring the energy, E(A 1g ) and E(B 1g ), of both A 1g and B 1g electronic states as a function of the Q θ coordinate. Seeking to clear out the mechanism responsible for the barrier, B, in NaCl:Ni + , such dependencies are analyzed following the procedure 11 employed in an analysis of cluster calculations on NaCl:Rh 2+ . For the sake of completeness, a recall of that procedure is previously given in section 3.…”