Wave functions, approximate correct zeroth-order

In the first-order approximation to NMR spectra when H° is taken as the first two terms in (8.78), the nuclear-spin energy levels are, in general, degenerate. Show for the A2X2 case (where A and X are protons) that the functions (8.79) are the correct zeroth-order wave functions for the perturbation H where H is the last term in (8.78). 

Before constructing approximate molecular orbitals for other HJ states, we consider how the trial function (13.57) can be improved. From the viewpoint of perturbation theory, (13.57) is the correct zeroth-order wave function. We know that the perturbation of molecule formation will mix in other hydrogen-atom states besides Is. Dickinson in 1933 used a trial function with some 2po character mixed in (since the ground state of HJ is a tr state, it would be wrong to mix in 2p i functions) he took... 

In Table II we also compare our total variational energies with the energies obtained by Wolniewicz. In his calculations Wolniewicz employed an approach wherein the zeroth order the adiabatic approximation for the wave function was used (i.e., the wave function is a product of the ground-state electronic wave function and a vibrational wave function) and he calculated the nonadiabatic effects as corrections [107, 108]. In general the agreement between our results... 

After this step one may already identify the paramagnetic and diamagnetic terms as arising from the pp and pn first order orbital corrections to the zeroth order wave function. The contribution is then further approximated by neglecting all off-diagonal terms in the pn-pn part of the Hessian... 

Table III. The MRCI and SOCI results are not in as good agreement with the FCI as the MRCI-hQ or SOCI+Q results. The +Q correction does not overshoot as it did for N2 and CH2, in part because of the larger number of electrons correlated here. As noted above, when only six electrons are correlated, the MRCI accounts for such a high percentage of the correlation that the +Q correction overestimates the missing correlation. For more than six electrons, or for cases where the zeroth-order wave function used is less satisfactory than was the CASSCF for N2 and CH2, the +Q correction may become a better approximation. This is especially true where quantities involving large differential correlation effects, such as electron affinities, are sought [7]. Thus the +Q correction substantially improves the agreement with FCI for the electron affinity of fluorine [8], even if large CASSCF active spaces and SOCI wave functions are employed.

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