Spin-orbit operators functions

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

In particular, within the orbital model of electronic structure (which is developed more systematically in Section 6), one can not construct trial wavefunctions which are simple spin-orbital products (i.e., an orbital multiplied by an a or (3 spin function for each electron) such as 1 sa 1 s(32sa2s(32pia2poa. Such spin-orbital product functions must be made permutationally antisymmetric if the N-electron trial function is to be properly antisymmetric. This can be accomplished for any such product wavefunction by applying the following antisymmetrizer operator ... 

Spin-orbit operator and the part of the spin-orbit operator that acts on the spatial part of the wave function. 

The spin-orbit operator LS given in Eq. (67) is expressed in terms of the individual electron-orbital and spin-momentum operators rather than the total momentum operators L and S. It can be shown (/, 5) that when evaluating integrals involving only LS functions of the same configuration, ls can be replaced by... 

Symmetry dictates that the representations of the direct product of the factors in the integral (3 /T Hso 1 l/s2) transform under the group operations according to the totally symmetric representation, Aj. The spin part of the Hso spin-orbit operator converts triplet spin to singlet spin wavefunctions and singlet functions to triplet wavefunctions. As such, the spin function does not have a bearing on the symmetry properties of Hso- Rather, the control is embedded in the orbital part. The components of the orbital angular momentum, (Lx, Ly, and Lz) of Hso have symmetry properties of rotations about the x, y, and z symmetry axes, Rx, Ry, and Rz. Thus, from Table 2.1, the possible symmetry... 

For a matrix element of the spin-orbit operator to be different from zero, it is in principle sufficient that the direct products of the irreps of space and spin functions on both sides of the matrix element be equal. Thus,... 

The spatial parts of the a, b XA states can couple to X 3Bi via the y component of the spin-orbit operator. The operator Sy couples the singlet spin function So (Ai) to the Bi triplet function. 

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. 

Second, it is a good approximation for an ion like HeAr+ to assume that the spin-orbit coupling operator is the same as that for the free Ar+ ion, L S, where f is the atomic spin orbit coupling constant. If the basis functions are confined to those arising from the 2P3/2 and 2Pi/2 states, the spin-orbit operator is also diagonal in a case (e) basis... 

Fig. 6. Average relativistic effective core potential and relativistic effective core potential energy curves for two states of Bi2. HF, Hartree-Fock GVB(pp), eight-configuration perfect-pairing generalized valence bond FVCI, full-valence Cl based on the GVB(pp) wave functions FV7R, full-valence Cl plus single and double promotions to virtual MOs relative to seven-dominant configurations. (The FVCI and FV7R calculations include the REP-based spin-orbit operator.)...

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

Thus this model maps density over atoms rather than spatial coordinates. If overlap is included some other definition of charge density such as Mulliken s17 may be employed. Eq. (30) and (31) are then used with this wave function to calculate the hyperfine constants as a function of the pn s. If symmetry is high enough, there will be enough hyperfine constants to determine all the p s, otherwise additional approximations may be necessary. For transition metal complexes, where spin-orbit effects are appreciable, it is necessary to include admixtures of excited-state configurations that are mixed with the ground state by the spin-orbit operator. To determine the extent of admixture, we must know the value of the spin-orbit constant X and the energy of the excited states. 

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