Unrestricted Hartree-Fock Wave Function Research Articles

Analytic-Expansion Unrestricted Hartree-Fock Wave Function for theD5State of Fe

An accurate analytic-expansion unrestricted Hartree-Fock wave function is reported for the $^{5}D$ state of Fe. The computed values of the magnetic dipole hyperfine-interaction constants are compared with the experimental values obtained by Childs and Goodman and are in good agreement.

Unrestricted Hartree—Fock Calculations. II. Spin Properties of Pi-Electron Radicals

Unrestricted Hartree—Fock wavefunctions have been computed for a large number of conjugated-hydrocarbon pi-electron radicals. Pi-electron spin-density and charge-density functions have been computed with and without annihilation of the major contaminating spin multiplet in the wavefunction. Three different empirical relations have been used to relate our results to proton isotropic hyperfine splittings measured in ESR experiments. These are the simple McConnell expression; the Colpa and Bolton relation which includes a term depending on the excess charge density at the carbon atom; and that due to Giacometti, Nordia, and Pavan which introduces a contribution from the spin densities in the bonds between the carbon and its nearest neighbors. When the Pariser—Parr method with semiempirical electron-repulsion integrals is employed in the unrestricted Hartree—Fock calculation, then the experimental splittings are almost always bounded by those computed with spin densities before and after the annihilation of the major contaminating spin state, but spin densities after annihilation are closer to the experimental values derived with the empirical relations. This was found to be true for all the three relations used, but the simplest one gave slightly less satisfactory results. The general problem of the interpretation of spin densities obtained from an unrestricted Hartree—Fock wavefunction which is not an eigenstate of spin is discussed. It is found, however, that unrestricted computations with semiempirical integrals give small splittings of the underlying ``closed''shell orbitals, so that a single annihilation is a good approximation to projection and the spin densities after annihilation should be satisfactory from a theoretical point of view.

Unrestricted hartree fock wave functions and hyperfine coupling constants in aromatic radicals

It is shown that a simple expression, differing from the one currently used, must be adopted when calculating π electron spin densities from unrestricted H.F. wave functions in connection with the prediction of proton hyperfine couplings in aromatic free radicals. The proposed formula gives results which are almost identical to those one obtains after complete projection of the unwanted parts of the unrestricted function.

TheZ-expansion of unrestricted Hartree-Fock atomic orbitals

It is shown, in agreement with Stewart, that in the open-shell (i.e. unrestricted) Hartree-Fock wave function for the ground state 1S of helium-like atoms, the two orbitals u, v have terms in Z-1/2, Z-3/2,... as well as Z-1, Z-2,..., when expanded in a perturbation series in powers of Z-1. But standard perturbation theory methods are not able to fix the orbitals uniquely. Consideration of the appropriate Hartree-Fock equations for u and v show that, in terms of Z, u(r, Z1/2) = v(r, - Z1/2). If u is expanded as u(r, Z) = u0(r) + Z-1/2u1/2(r) + Z-1u1(r) +... an approximate expression for u1/2 is {2Zr + 3)/4 [radical]2}u0.