Spin-coupled wave function determination

The ESR hyperfine coupling is determined by triplet perturbations. Thus, in principle one should use an unrestricted wave function to describe the reference state. However, it is also possible to use a spin-restricted wave function (Fernandez et al. 1992) and take into account the triplet nature of the perturbation in the definition of the response. Within such a (e.g., SCF or MCSCF) restricted-unrestricted approach, first-order properties are given as the sum of the usual expectation value term and a response correction that takes into account the change of the wave function induced by the perturbation (of the type (0 H° 0)). This restricted-unrestricted approach has also been extended to restricted Kohn-Sham density functional theory (Rinkevicius et al. 2004). 

The C-H spin couplings (Jen) have been dealt with in numerous studies, either by determinations on samples with natural abundance (122, 168, 224, 231, 257, 262, 263) or on samples specifically enriched in the 2-, 4-, or 5-positions (113) (Table 1-39). This last work confirmed some earlier measurements and permitted the determination for the first time of JcH 3nd coupling constants. The coupling, between a proton and the carbon atom to which it is bonded, can be calculated (264) with summation rule of Malinovsky (265,266), which does not distinguish between the 4- and 5-positions, and by use of CNDO/2 molecular wave functions the numerical values thus - obtained are much too low, but their order agrees with experiment. The same is true for Jch nd couplings. 

The spin—orbit coupling constants f4f and fsd, which represent the relativistic spin—orbit interaction in the 4f and 5d shells, also determined by means of the radial wave functions Rrd of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23... 

It should be pointed out that a somewhat different expression has been given for the Knight shift [32] and used in the analysis of PbTe data that in addition to the g factor contains a factor A. The factor A corresponds to the I PF(0) I2 probability above except that it can be either positive or negative, depending upon which component of the Kramers-doublet wave function has s-character, as determined by the symmetry of the relevant states and the mixing of wavefunctions due to spin-orbit coupling. 

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 ... 

The product of bond wave functions in Equation 3.8, involves so-called perfect pairing, whereby we take the Lewis structure of the molecule, represent each bond by a HL bond, and finally express the full wave function as a product of all these pair-bond wave functions. As a rule, such a perfectly paired polyelectronic VB wave function having n bond pairs will be described by 2" determinants, displaying all the possible 2x2 spin permutations between the orbitals that are singlet coupled. 

In this wave function, the last determinant has a pure triplet AY moiety as before in R. However, now the first two terms are also a pure triplet AY (with Ms = 0). As such, by using the orthogonalization procedure, we generated a spectroscopic promoted state, where X is coupled to a triplet AY to a total doublet spin state the spin pairing is 50% between X and A and 50% between X and Y. The corresponding G expression becomes then ... 

So far we know the selection rules for spin-orbit coupling. Further, given a reduced matrix element (RME), we are able to calculate the matrix elements (MEs) of all multiplet components by means of the WET. What remains to be done is thus to compute RMEs. Technical procedures how this can be achieved for Cl wave functions are presented in the later section on Computational Aspects. Regarding symmetry, often a complication arises in this step Cl wave functions are usually determined only for a single spin component, mostly Ms = S. The Ms quantum numbers determine the component of the spin tensor operator for which the spin matrix element (S selection rules dictated by the spatial part of the ME. 

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... 

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