The main effects of JT vibronic couplings are due to the special dynamics of the nuclear configuration that follows from the JT instability. The energy levels and wavefunctions describing these effects are solutions of the system of coupled equations (2.6) in Section 2.1. They were obtained for the most important JT problems formulated in Chapters 3 and 4, and are discussed in this chapter.

Weak vibronic coupling, perturbation theory

Calculation of the energy spectrum and wavefunctions of a JT or PJT molecule as solutions of the coupled equations (2.6) is a very complicated problem which cannot be solved in a general form, without simplifications, for arbitrary systems. However, as in similar quantum-mechanical situations, analytical solutions for some limiting cases in combination with exact numerical solutions of some particular cases yield the general trends and provide understanding of the origin and mechanism of the phenomenon as a whole.

For vibronic problems the limiting cases of weak and strong vibronic coupling with relatively small and large vibronic coupling constants, respectively, can be solved analytically. A quantitative criterion of weak and strong coupling can be defined by comparing the JT stabilization energy with the zero-point energy of -fold degenerate vibrations. Denote. Then, if, the vibronic coupling will be regarded as weak, and if λΓ ≫ 1, the coupling is strong; λΓ is the dimensionless vibronic coupling constant.

General: JT vibronic coupling effects in geometry and reactivity

This chapter is devoted to applications of the JTE, PJTE, and RTE theory to solve the problems of origin of geometry and spectra, as well as other properties of molecular systems. With regard to geometry, as mentioned in the introduction, increased computer power allows sufficiently accurate calculations of APES of small to moderate-sized molecular systems to determine their absolute minima, which are assumed to define the molecular geometry. In many cases and for many electronic states, ground and excited, the APES are rather complicated: there may be several equivalent minima which do not correspond to the high-symmetry configuration that one may expect from a general, classical point of view. Instead, complicated dynamics due to the multi-minimum APES, anharmonicity, conical intersections, and lines of conical intersections may occur. Obviously, molecular properties other than geometry, including spectra and reactivity, are strongly dependent on the APES too.

On the other hand, as follows from the previous chapters, the JT vibronic effects are the only sources of instability of high-symmetry configurations. Therefore the vibronic effects may serve as a fundamental basis for understanding (rationalizing) the results of quantum-chemical computations (as well as results obtained by other methods). In addition, for large systems ab initio calculations with geometry optimization are impossible or impractical, leaving the perturbational vibronic approach as the only source of information on bulk and local instabilities.

This chapter is to formulate in general the different “traditional” situations that occur due to the lack of minimum of the APES in systems with degenerate electronic terms, as predicted by the JT theorem (Section 2.5). They are labeled as JT problems. Each of them is characterized by the type of degenerate electronic term Γ and the JT-active vibrational modes γi (denoted as the Γ ⊗ (γ1 + γ2 + ···) problem), and has a specific APES. The latter is most important to the solution of the vibronic coupling equations (2.6), which yield the energies and wavefunctions considered in Chapter 5. But the APES has also independent value, allowing qualitative and semiquantitative evaluation of important properties of the system. Other (less traditional) situations including PJT, RT, and product JT problems are formulated in Chapter 4.

Basic formulation. The simplest (E ⊗ b1) and E ⊗ (b1 + b2) problems

The JT problems refer to polyatomic (molecular and crystalline) systems with degenerate electronic states, for which the JT theorem is operative. As follows from group-theoretical considerations, degenerate states may occur in systems with high symmetry that have at least one rotational or rotoflection axis of the order n > 2. The higher the symmetry, the more and higher-order degenerate terms are possible. Systems with axial symmetry have twofold degenerate E terms, cubic systems have E and threefold degenerate T terms (with spin–orbital interaction included they may have also fourfold degenerate G terms), while icosahedral systems have E, T, G, and fivefold degenerate H terms (see the corresponding point groups in the appendix).

Definitions

Symmetry operations leave a set of objects in indistinguishable configurations which are said to be equivalent. A set of symmetry operators always contains at least one element, the identity operator E. When operating with E the final configuration is not only indistinguishable from the initial one, it is identical to it. A proper rotation, or simply rotation, is effected by the operator R(φn), which means “carry out a rotation of configuration space with respect to fixed axes through an angle φ about an axis along some unit vector n.” The range of φ is − π < φ ≤ π. Configuration space is the three-dimensional (3-D) space ℛ3 of real vectors in which physical objects such as atoms, molecules, and crystals may be represented. Points in configuration space are described with respect to a system of three space-fixed right-handed orthonormal axes x, y, z, which are collinear with OX, OY, OZ (Figure 2.1(a)). (A right-handed system of axes means that a right-handed screw advancing from the origin along OX would rotate OY into OZ, or advancing along OY would rotate OZ into OX, or along OZ would rotate OX into OY.) The convention in which the axes x, y, z remain fixed, while the whole of configuration space is rotated with respect to fixed axes, is called the active representation.

Crystallographic magnetic point groups

Because the neutron has a magnetic moment, neutron diffraction can reveal not only the spatial distribution of the atoms in a crystal but also the orientation of the spin magnetic moments. Three main kinds of magnetic order can be distinguished. In ferromagnetic crystals (e.g. Fe, Ni, Co) the spin magnetic moments are aligned parallel to a particular direction. In antiferromagnetically ordered crystals, such as MnO, the spins on adjacent Mn atoms are antiparallel, so there is no net magnetic moment. In ferrimagnetic crystals (ferrites, garnets) the antiparallel spins on two sublattices are of unequal magnitude so that there is a net magnetic moment. In classical electromagnetism a magnetic moment is associated with a current, and consequently time reversal results in a reversal of magnetic moments. Therefore the point groups G of magnetic crystals include complementary operators ΘR, where Θ is the time-reversal operator introduced in Chapter 13. The thirty-two crystallographic point groups, which were derived in Chapter 2, do not involve any complementary operators. In such crystals (designated as type I) the orientation of all spins is invariant under all R ∈ G. In Shubnikov's (1964) description of the point groups, in which a positive spin is referred to as “black” and a negative spin as “white,” so that the time-reversal operator Θ induces a “color change,” these groups would be singly colored, either black or white.