Wave functions corrections

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

For gas-phase molecules the assumption of electronic adiabaticity leads to the usual Bom-Oppenheimer approximation, in which the electronic wave function is optimized for fixed nuclei. For solutes, the situation is more complicated because there are two types of heavy-body motion, the solute nuclear coordinates, which are treated mechanically, and the solvent, which is treated statistically. The SCRF procedures correspond to optimizing the electronic wave function in the presence of fixed solute nuclei and for a statistical distribution of solvent coordinates, which in turn are in equilibrium with the average electronic structure. The treatment of the solvent as a dielectric material by the laws of classical electrostatics and the treatment of the electronic charge distribution of the solute by the square of its wave function correctly embodies the result of... 

Calculation of the nonlogarithmic polarization operator contribution is quite straightforward. One simply has to calculate two terms given by ordinary perturbation theory, one is the matrix element of the radiatively corrected external magnetic field, and another is the matrix element of the radiatively corrected external Coulomb field between wave functions corrected by the external magnetic field (see Fig. 9.13). The first calculation of the respective matrix elements was performed in [34]. Later a number of inaccuracies in [34] were uncovered [22, 23, 40, 43, 44, 45] and the correct result for the nonlogarithmic contribution of order a Za) EF to HFS is given by... 

It should be apparent that the expressions for the wave functions after interaction [equations (3.38) and (3.39)] are equivalent to the Rayleigh-Schrodinger perturbation theory (RSPT) result for the perturbed wave function correct to first order [equation (A.109)]. Similarly, the parallel between the MO energies [equations (3.33) and (3.34)] and the RSPT energy correct to second order [equation (A. 110)] is obvious. The missing first-order correction emphasizes the correspondence of the first-order corrected wave function and the second-order corrected energy. Note that equations (3.33), (3.34), (3.38), and (3.39) are valid under the same conditions required for the application of perturbation theory, namely that the perturbation be weak compared to energy differences. 

In summary, the wave function correct to first order and the energy correct to second order are... 

With the first-order eigenvalue and wave function corrections in hand, we can carry out... 

Substitution of the expansions (1.197) and (1.198) into (1.195) followed by expansion of the first-order wave-function correction according to... 

With the first-order eigenvalue and wave function corrections in hand, we can carry out analogous operations to determine the second-order corrections, then the third-order, etc. The algebra is tedious, and we simply note the results for the eigenvalue corrections, namely... 

To examine how the Kramers doublets are affected by a decrease of the ligand field anisotropy it is instructive to consider Wi (or Wi) as a wave function correct to first order, with the terms in dxz and dxy as the first order corrections (106). The spin-orbit coupling operator for a one-particle system is... 

As an example of the connection between perturbation theory wave function corrections and polarizability, we now calculate the linear polarizability, ax. The states are corrected to first order in H. Since the polarization operator (Zx) is field independent, polarization terms linear in the electric field arise from products of the unperturbed states and their first-order corrections from the dipole operator. The corrected states are [12]... 

G 2) means that the polarization expression (eq 8b) generates terms of order 2 as well. These contribute to / , along with terms that are generated from matrix elements arising from second order wave function corrections absent from eq 8 a. 

We have added an ellipse in the 3 and 4 terms to emphasize that further corrections to terms of these orders arise from wave function corrections of higher-order than explicitly written in eq 10a. 

Quantum mechanical calculations of 33S nuclear quadrupole coupling constants are not an easy matter (not only for the 33S nucleus, but for all quadrupolar nuclei). Indeed, the electric field gradient is a typical core property, and it is difficult to find wave functions correctly describing the electronic distribution in close proximity to the nucleus. Moreover, in the case of 33S, the real importance of the Sternheimer shielding contribution has not been completely assessed, and in any case the Sternheimer effect is difficult to calculate. 

Fig. 1. The QED contributions of order a/it) to the bound-electron gj factor depicted as Feynman diagrams. Double lines indicate bound fermions, wavy bnes indicate photons. The interaction with the magnetic field is denoted by a triangle. Diagram (a) is also termed SE, ve (self-energy vertex correction), diagrams (c) and (e) SE, wf (self-energy wave-function correction), diagram (b) VP, pot (vacuum-polarization potential correction), and diagrams (d) and (f) VP, wf (vacuum-polarization wave-function correction)...

The decomposition of the irreducible part of the self-energy wave-function correction term is depicted in Fig. 2. The divergent terms are these with zero and one interaction in the binding potential present, below referred to as zero-potential term and one-potential term , respectively. The charge divergences cancel between both terms. In addition, a mass counter term dm has to be subtracted to obtain proper mass renormalization similar to the case of the free self energy [47] (for our schemes see also [44]). The zero- and one-potential... 

To obtain a similar formula from Eq. (267) for the wave function corrections it is convenient to introduce the Green function for the operator K defined by... 

Thus G is an inverse of K for all such functions which have no component along the unperturbed ground state. From Eq. (266) the wave function corrections are of this type so we can apply G to Eq. (267) to obtain the recursive formula... 

Equations (270) and (274) provide a practical scheme for the systematic calculation of the energy and wave function corrections to large order. It should be noted that in order to obtain the fcth-order correction to the energy it is necessary to calculate the first k — 1 corrections to the wave function. 

Up to this point we are still dealing with undetermined quantities, energy and wave function corrections at each order. The first-order equation is one equation with two ImEnowns. Since the solutions to the unperturbed Schrodinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrodinger perturbation theory, and the equation in (4.32) becomes... 

Fig. 21. The vacuum-polarization-like Feynman diagrams for the hyperfine-structure splitting, (a) and (b) are the wave-function corrections and (c) is the magnetic-loop modification. The cross x signifies the interaction with the nuclear magnetization distribution.

The only difference between the first-order self energy correction and this wave function correction term is the appearance of (5a) instead of a). Thus, from this expression we can conclude that in the case of the first-order self energy the correction term vanishes. For the screening case only the first term vanishes and we are left with... 

---