Wave functions Spin-orbit perturbed

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

Accordingly, the first-order spin-orbit perturbation of a triplet wave function may be written as a linear combination of unperturbed singlet, triplet, and quintet states with expansion coefficients defined in a similar way as those in Eq. [218]. 

The product symmetries of the excited a3Bi multiplet components are Ai, Ai, and f>2 The spin-orbit perturbed excited state wave functions are therefore given by... 

Gagliardi and Roos conducted a series of studies on actinide compounds. They follow a combined approach with DKH/AMFI Hamiltonians combined with CASSCF/CASPT2 for the energy calculation and an a posteriori added spin-orbit perturbation expanded in the space of nonrelativistic CSFs. This strategy aims to establish a balance of sufficiently accurate wave function and Hamiltonian approximations. Since the CASSCF wave function provides chemically reasonable but not highly accurate results (as witnessed, for instance, in the preceding section), it is combined with a quasi-relativistic Hamiltonian, namely the sc alar-relativistic DKH one-electron Hamiltonian. Additional effects — dynamic correlation and spin-orbit coupling — are then considered via perturbation theory. 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

Because of the spin-orbit selection rules, only triplet zeroth-order states contribute to the first-order perturbation correction of a singlet wave function. In Rayleigh-Schro dinger perturbation theory, the expansion coefficient a of a triplet zeroth-order state (3spin-orbit matrix element with the electronic ground state (in the numerator) and its energy difference with respect to the latter (in the denominator). 

The three multiplet components of an excited triplet state are degenerate in zeroth order. We have therefore, in principle, the freedom of choosing these in their spherical or Cartesian forms. On the other hand, the spin-orbit split triplet levels will transform according to the irreps of the molecular point group. For a smooth variation of the wave function gradient with respect to the perturbation parameter X, we employ Cartesian triplet spin functions also in the unperturbed case and express them as ket vectors ... 

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