Reduced matrix elements interaction

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

The reduced matrix element of T2(C) can be treated in the usual manner but we shall not pursue this rather messy analysis any further, particularly since the dipolar interaction in CsF is very small ... 

As was the case with the dipolar interaction, the quadrupole interaction has matrix elements which connect the ground state with excited states we neglect these by putting e/ = 0. The reduced matrix element of T2(g) was evaluated in (9.14), so with these replacements, including the neglect of matrix elements off-diagonal in N, we obtain the simplified result ... 

The reduced matrix element of T2(A, S2) was previously evaluated in appendix 8.3, and to be consistent with the form (9.99) of the spin-spin interaction we take the q = 0... 

Another useful reduced matrix element is for the interaction of electrons whose states involve spin—orbit coupling. 

It follows from the preceding results that the electro-optical properties of molecules in degenerate electronic states should have unusual temperature dependence, which is absent in the case of nondegenerate states. Even for nondipolar degenerate electronic states (e.g., for states in which the reduced matrix elements of the dipole moment are zero) for certain relationships between the vibronic constant and the temperature, there may be a quadratic dependence of the Kerr effect on p, similar to that observed in the case of molecules that are simultaneously anisotropic polarizable and possess a proper dipole moment. The nonlinear dependence on p under consideration is due exclusively to the vibronic interaction that redetermines the vibronic spectrum and leads to different polarizability in different vibronic states. This dependence on p has to be distinguished from that which arises due to the nonzero value of the dipole moment in the initial ground electronic state (e.g., as in the case of the E term in molecules with D3h symmetry). The two sources of the... 

However, numerical estimates of the effect of frequency dependence of the coefficients of depolarization based on the calculation of molecular cross sections Gj(na>, Aw) are rather difficult. But it can be shown that in the cases when different Gj magnitudes are nonzero by symmetry selection rules (neglecting the spin-orbital interaction), they are of the same order of magnitude, since they are determined by the same energy denominators and reduced matrix elements of the operator of the dipole moment. 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

Since the term labels of electron configurations are fixed, the evaluation of the reduced matrix elements L5 V(11) LS) is helpful (Table 8.44). This is of great utility since now one can readily see which terms are connected (interact) through the one-electron spin-orbit operator. However, the matrix elements between the /-components need the evaluation of one 6/-symbol. 

The decoupling formula for the reduced matrix element yields (a) the first type (pair-interaction) reduced matrix elements, which are straightforward... 

Pair-interaction reduced matrix elements of the biquadratic exchange for a general triad3... 

For ring compounds we need to add the interaction —based on the reduced matrix element R yN(r,s-,i,S)-The chain compounds include several special cases [13] ... 

The contents of the large square brackets must thus represent the reduced matrix element in accordance with (52). Since the operator does not contain a spin part, we may use (48) and interpret the interaction as a tensor product with a unit scalar operating in spin space. The real space operator is, however, not yet in the form of a spherical tensor operator, but using (46) we arrive at... 

The second equality emphasizes the relationship between the quantities C(K, K ) and the reduced matrix element. In contrast to the previous section, the interaction operator operates in both spin and real space, where the real space tensors emerge as (r) from a plane-wave expansion of the exponential. The application of eq. (46) allows us to make use of eq. (57) and we obtain... 

Substitution (101) appears to be the simplest way in which spin-dependent phenomena can be incorporated into the theory. The operator (100) commutes with S and is thus consistent with effects produced by such spin-independent interactions as the Coulomb and crystal-field terms in the Hamiltonian. Moreover, the effect of (101) turns out to be equivalent to adding to each reduced matrix element of F " a part (proportional to c,) that involves the reduced matrix element of the double tensor where = s (Judd 1977b). Such double tensors are straightforward to evaluate furthermore the proportionality = /4 holds for all terms of... 

The trace summation in this equation is identified as a scalar interaction constant, which is represented by the reduced matrix element. 

Using the transfer model, we can also express the reduced matrix elements for the e a2 channel. Even though there is no overlap between these orbitals, they do give rise to a transfer-term intensity. Orbital interaction does indeed delocalize the e t2g) orbitals over the ligands. The dipole operators, centred on the complex origin, will then couple the e f ) and a2 f) ligand-centred orbitals. Hence, we write ... 

For the ss interaction, the reduced matrix elements between the core states comprise an odd symmetrical double tensor (u + k odd) and are therefore the same for (nl) n l and (n/) n /. The soo interaction is composed of two parts The first, comprising the factors k and s behaves under conjugation like the spin-orbit interaction. It is represented by an even symmetrical double tensor (u + k even) and therefore changes sign the second part, comprising ( and s is represented by an odd symmetrical operator, and therefore remains invariant under conjugation. 

For a half filled shell the reduced matrix elements of with even k satisfy the selection rule Av = 2 therefore the triple product (1.148) also satisfies the selection rule Av = 2 (whereas the electrostatic energy matrices satisfy the selection rule Av = 0, 4). Hence, all diagonal elements of Hi vanish, and all second-order effects are well represented by two-electron effective interactions. 

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