Cooperative JT effect

The JT origin of structural phase transitions in crystals with JT centers was realized at very early stages of the study of the JT effect. It is obvious that the interaction of JT local distortions in crystals, as mentioned above, under certain conditions, at temperatures T0 in equation (1) may lead to their ordering resulting in crystal structures of lower symmetry (cooperative JT effect for review see Ref. [8]). 

With a lattice of coupled molecules, one also has to consider the possibility of a cooperative JT effect. This could move the system away from the molecular description. To facilitate hopping between the molecules, it might be advantageous to fix the phase of the distortion from site to site, which conflicts with the dynamic distortion [18]. If band properties dominate the behavior of the system, they could impose a periodic arrangement of distortions, which would possibly open a gap in the electronic structure and turn the system into a band insulator. Thus, it is hard to predict theoretically which limit, band or molecular, might be the most relevant in a given crystalline phase. 

An increase in the extent of valence d electron localization is expected for smaller principal quantum numbers and as one moves to the right in a period because of a contraction in the size of the d orbitals. For example, with compounds of the late 3d metals, a mixture of 4s bands, and more-or-less localized 3d atomic orbitals may coexist, in which case, it becomes possible for cubic crystal fields to split the degenerate d orbitals and give rise to a localized JT distortion (e.g. a single octahedra), or small polaron in physics terminology. High concentrations of JT ions, where the polyhedra share stmctural elements, are subject to a cooperative JT effect, which can cause distortion to a lower crystalline symmetry. 

Structural phase transitions induced by JT interactions (cooperative JT effect) are perhaps the most striking structural manifestations. Owing to the importance of such an effect in solid state physics and chemistry many review articles are available ... 

We shall not discuss this subject further, because till now no example of cooperative JT effect has been reported in biomolecules. 

The key feature distinguishing the OOA from the cooperative JT effect is the way the chemical bonding effects are included. This will be an important part of the present study. In Sect. 2, we present a simple qualitative description of the cooperative JT effect. First, in Sect. 2.1, we demonstrate the chemical nature of the JT instability. In Sect. 2.2, the cooperative JT effect is presented as interplay of short-range chemical bonding effects with the long-range intercell elastic coupling. To reveal the most important differences of the two approaches from one another, in Sect. 3 we present a simplified version of the OOA. In Sect. 4, we discuss some additional effects that are closely related to JT instability in crystals. 

Both the cooperative JT effect in its traditional form and the OOA has a reach list of applications discussed in many related publications. Numbered in thousands, they all cannot be reviewed in one paper. Therefore, in no way the present work can be considered as a comprehensive review of either one of these two approaches. Fortunately, some JT crystals, mostly perovskites, were considered by different authors in both ways, applying the OOA and the traditional approach. This provides an opportunity to compare their results using as example a relatively short list of JT crystals. Regarding the cooperative JT effect, an updated review is provided by Kaplan in the present book [6]. For the OOA, its most important details can be found in the fundamental review by Kugel and Khomskii [9]. Recent results were reviewed by Khomskii [10] and Khaliullin [11]. 

In the traditional theory of the cooperative JT effect, its significant part is one-center JT problem in a low-symmetry mean field (see the last paragraph of Sect. 2.2). In particular, it includes the eigenvalue problem for the Hamiltonian, similar to (7), operating in an infinite manifold of vibrational one-center states. Compared to this relatively complex step, in the OOA, the mean-field approximation is much simpler. In the OOA, one has to solve just a finite-size matrix (2 x 2 in this case) or, for other JT cases, a somewhat larger matrix but finite anyway. In the theory of the cooperative JT effect, this important advantage of the OOA allows to proceed farther than... 

Parameters (18) include vibronic coupling constant, V, and phonon band-structure factors, (/, y K j, A). For a particular crystal, finding these factors is a laborious problem of crystal lattice dynamics. Instead, in the OOA, 7y are used as free parameters of the theory. Still consistent with the fundamental theory of the JT effect, in this form the OOA is not directly derived from the theory. In other words, in the theory of cooperative JT effect, the OOA is a phenomenological approach. 

In the OOA, one of its basic assumptions is reducing the intercell correlation to symmetry equivalent intersite orbital exchange coupling. This assumption simplifies the physical picture of the cooperative JT effect. Ligands and all the respective... 

The third lucky case is the so-called T g gg problem in a cubic system. The JT instability of the orbital triplet term (the T term) is due to its linear coupling to twofold degenerate vibrations, Qe and Qe, shown in Fig. 3b and c [1,2]. As it is similar to the tetragonal E g b g case, for this cooperative JT effect we do not go into its details. Just note that there are three minimum points in this case and there are three matrix positions on the main diagonal allowing the respective simultaneous coordinate shift to these three vertices. 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

Due to its intrinsic complexity arising from the orbital multiplicity, the researches in this field have been almost exclusively concerned with the JT effect in rather simple systems like molecules, small clusters, and a single JT impurity center in solids in which itinerant electrons do not play an important role [2,3]. Even if the JT crystals, in which an infinite number of such JT centers occupy regular positions in a lattice, are considered in the context of the cooperative JT effect, relevant electrons in the system have usually been assumed to be localized [4]. 

---