Spin-orbit diagonal matrix elements

EOMCCSD(T) calculations, while facilitating the open-shell implementation of the CR-EOMCCSD(T) method employing the restricted open-shell Hartree-Fock (ROHE) orbitals [59]. Indeed, the use of spin-orbital energy differences (ca -f -f Cc — — ej — e ) instead of the complete form of the diagonal matrix elements of involving triply excited determinants to... 

It is interesting to emphasize that the diagonal matrix elements of the spin-own-orbit interaction vanish for half-filled shells. 

All the off-diagonal matrix elements of the spin-orbit coupling in the >, Tl> [ basis are thus reduced by the factor y, and we use the experimentally observed quenching to calculate Ej j and the corresponding geometrical distortion (14). In the Cs2NaYClg host lattice the total spread of the four spin-orbit components of T2 is 32 cm whereas crystal field theory without considering a Jahn-Teller effect predicts a total spread of approximately 107 cm-. ... 

The interaction is time-even (K = 1). Hence (from the time-reversal selection rules) within a manifold with Q = g, K G [gX Q QKq an the requirement that QKQ = +1 the diagonal matrix elements will vanish if the interaction is not a totally symmetric scalar in spin and orbital angular momenta. 

Here, grs is a parameter that is quantified either from experimental data, or is calculated by an ab initio method as one-half of the singlet-triplet excitation energy gap of the r—s bond. In terms of the qualitative theory in Chapter 3, grs is therefore identical to the key quantity —2(3 5 - This empirical quantity incorporates the effect of the ionic components of the bond, albeit in an implicit way. (c) The Hamiltonian matrix element between two determinants differing by one spin permutation between orbitals r and s is equal to grs. Only close neighbor grs elements are taken into account all other off-diagonal matrix elements are set to zero. An example of a Hamiltonian matrix is illustrated in Scheme 8.1 for 1,3-butadiene. 

In the present treatment, we retain essentially all the diagonal matrix elements of X these are the first-order contributions to the effective electronic Hamiltonian. There are many possible off-diagonal matrix elements but we shall consider only those due to the terms in Xrot and X o here since these are the largest and provide readily observable effects. The appropriate part of the rotational Hamiltonian is —2hcB(R)(NxLx + NyLy). The matrix elements of this operator are comparatively sparse because they are subject to the selection rules AA = 1, A,Y=0 and AF=0. The spin-orbit coupling term, on the other hand, has a much more extensive set of matrix elements allowed... 

In order to appreciate this point more clearly, we confine our attention to the contributions to 3Qff produced by perturbations from the spin-orbit coupling 3Q0 and the electronic Coriolis mixing 30-ot- If we represent an off-diagonal matrix element of the former by (L S) and the latter by (N L), we can describe some examples of these higher order terms, as shown in table 7.1. The third-rank terms appear only in states of quartet or higher multiplicity and the fourth-rank terms in states of quintet (or higher) multiplicity. With the important exception of transition metal compounds, the vast majority of electronic states encountered in practice have triplet multiplicity or lower. 

Here, co represents the Euler angles (orbital Zeeman interaction, we see that it has off-diagonal matrix elements which link electronic states with A A = 0, 1, as well as purely diagonal elements. It is clearly desirable to remove the effect of these matrix elements by a suitable perturbative transformation to achieve an effective Zeeman Hamiltonian which acts only within the spin-rotational levels of a given electronic state rj, A, v), in the same way as the zero-field effective Hamiltonian in equation (7.183). 

We now use the results we have obtained to calculate the energy levels in a magnetic field, determine the field values for the allowed electric dipole transitions, and compare the results with the experimental spectrum [56]. It is already clear that in the case (b) basis set we shall have to take note of the extensive mixing of different rotational levels by the AN = 2 off-diagonal matrix elements of the spin-spin interaction. In SO the spin spin parameter X is comparable with the rotational constant B0, and, as we shall see, in heavier molecules like SeO, X is so much larger than B0, because of spin orbit coupling, that a case (a) basis is more appropriate. 

For off-diagonal matrix elements we have (from the orthogonality of the one-electron spin-orbitals)... 

The energies of the magnetic interaction of orbital and spin angular momenta are —A and +A/2 in the atomic ion, while they are —A/2, 0, and +A/2 in the molecular ion. Here, A is a spin-orbit coupling constant in the ground state of atomic ion. An additional off-diagonal matrix element arises in the molecular ion from the microscopic spin-orbit operator... 

The absolute sign of an off-diagonal matrix element cannot be determined, since it depends arbitrarily on the chosen phase for the determinantal wave function, namely, on the order of the spin-orbitals. However, the relative sign of two off-diagonal matrix elements can often be determined experimentally. Thus, care must be taken to define the phases of the wavefunctions consistently. 

For diagonal matrix elements (diagonal in all quantum numbers including S), we can define A, the spin-orbit coupling constant as... 

Note that the signs of the off-diagonal matrix elements in Eq. (3.4.37b) depend on the phase choice (i.e., the selected standard order of the spin-orbitals). However, the overall signs of the matrix elements in Eq. (3.4.37b) do not affect the sign or magnitude of the As values which are derived from the square of these matrix elements. 

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