Accurate Hartree-Fock Research Articles

Electronic Structure of Diatomic Molecules. III. A. Hartree—Fock Wavefunctions and Energy Quantities for N2(X1Σg+) and N2+(X2Σg+, A2Πu, B2Σu+) Molecular Ions

The problem of the convergence of a sequence of Hartree—Fock—Roothaan wavefunctions and energy values to the true Hartree—Fock results is examined for N2(X1Σg+). This critical study is based on a hierarchy of Hartree—Fock—Roothaan wavefunctions which differ in the size and composition of the expansion basis set in terms of STO symmetry orbitals. The concluding basis set gives a total Hartree—Fock energy of −108.9956 hartree and Re(HF)=2.0132 bohr for N2(X1Σg+). Results are also presented from direct calculations for three states of the N2+ molecular ion (X2Σg+, A2Πu, B2Σu+) which are also thought to be very close approximations to the true Hartree—Fock values. The results give EHF=−108.4079, −108.4320, and −108.2702 hartree and Re(HF)=2.0385, 2.134, and 1.934 bohr for the X2Σg+, A2Πu, and B2Σu+ states of N2+, respectively. Extensive calculations for various R values establish that the X2Σg+ and A2Πu states are reversed in order relative to experiment, a short-coming ascribed to the Hartree—Fock approximation.

Electric Dipole Polarizability of Atoms by the Hartree—Fock Method. I. Theory for Closed-Shell Systems

A method for the calculation of the dipole polarizabilities of closed-shell atomic systems is presented. This method involves the direct calculation of the Hartree—Fock wavefunction of the atom in the presence of the perturbing field. The orbitals are expressed as linear combinations of Slater-type functions. A large number of carefully chosen and optimized basis functions are used so as to assure a good fit to the true Hartree—Fock wavefunction. The polarizability is then calculated from the limiting value of α=p/F for F=0, where F is the electric field strength, and p is the induced dipole moment.

Orbital Theories of Electronic Structure. II. Molecularly Invariant Orbitals

We express mathematically the question: Do orbitals for localized groups in molecules exist such that they are accurately the same in all molecules containing the same localized electron group? The advantage of giving the question mathematical form is that then the search for such molecularly invariant orbitals requires that accurate Hartree—Fock calculations be carried out only on a set of reference systems. Constructing the Hartree—Fock wavefunction of a molecule from the localized orbitals of subgroups in the reference systems, one can then test the accuracy to which the subgroup orbitals are molecularly invariant orbitals. Based on our quantitative approach, we present arguments and suggest conditions which favor the existence of molecularly invariant orbitals. The conditions favoring their existence are as follows. (a) The subgroup orbitals in general should not be orthogonal to the orbitals of any other subgroup in a molecule. (b) The subgroup orbitals should be well localized, in the sense that they are small in the neighborhood of the nuclei of the other subsystems. (c) The nuclear separations in the molecule from which the localized orbitals are taken, and the nuclear separations in the molecule to which the orbitals are transferred, should be identical. (d) If two molecules differ only through the substitution of one ion for another, the most favorable case will be that in which the two ions have the same net charge. (e) The differences between the eigenvalues of the orbitals of a subsystem in one molecule and those of the same subsystem in another molecule decrease at least as rapidly as the fourth power of the reciprocal of the separation of the subsystem from the rest of the system, for increasing separation.

Simple Basis Set for Molecular Wavefunctions Containing First- and Second-Row Atoms

The self-consistent field functions for the ground state of the first- and second-row atoms (from He to Ar) are computed with a basis set in which two Slater-type orbitals (STO's) are chosen for each atomic orbital. The reported STO's have carefully optimized orbital exponents. The total energy is not far from the accurate Hartree—Fock energy given by Clementi, Roothaan, and Yoshimine for the first-row atoms and unpublished data for the second-row atoms. The obtained basis sets have sufficient flexibility to be a most useful starting set for molecular computations, as noted by Richardson. With the addition of 3d and 4f functions, the reported atomic basis sets provide a molecular basis set which duplicate quantitatively most of the chemical information derivable by the more extended basis set needed to obtain accurate Hartree—Fock molecular functions.

Calculation of Molecular Properties with Limited-Basis Hartree—Fock Functions

The use of the limited-basis (Roothaan) approximation to Hartree—Fock functions and its implications for the accuracy of calculations on one-electron operators are discussed. Only for the true Hartree—Fock function does Brillouin's theorem guarantee second-order errors for expectation values of these operators. The mean square deviation of one Roothaan function from a second with an incremented basis set provides a measure of how closely the true Hartree—Fock function is being approached. Expectation values of one-electron operators reflect this as well, but it is shown that the improvement or lowering of energy on addition of functions to the basis is much less suitable.

Atomic scattering factors for the lithium- and beryllium-isoelectronic sequences from accurate analytic Hartree-Fock wave functions

Atomic scattering factors for the lithium- and beryllium-isoelectronic sequences from accurate analytic Hartree-Fock wave functions

Exchange Polarization Effects in Hyperfine Structure

Exchange polarization of core electrons by outer unpaired electrons has been calculated for 10 different atomic configurations of Li, Na, K, F, Cl, Be, B, and N in the unrestricted Hartree-Fock (UHF) approximation. Numerical integration techniques were used and accurate conventional Hartree-Fock (HF) wave functions were also obtained for these configurations. The theory of atomic hyperfine structure in the UHF approximation is developed and the HF and UHF calculated values of the hyperfine coupling constants are compared with available experimental data. The importance of core polarization in solid state problems is briefly mentioned with particular attention to color centers. Finally, unsuccessful attempts to calculate core polarization by perturbation expansion methods are discussed.

LCAO Wave Functions for Hydrogen Fluoride with Hartree-Fock Atomic Orbitals

Accurate Hartree-Fock fluorine functions and an exponential Slater 1s hydrogen orbital have been used as basis functions in a conventional SCF LCAO—MO treatment of the HF molecule at five internuclear distances. Comparison with Hartree-Fock calculations for the isoelectronic F— and Ne systems gives a qualitative indication that a rather close approximation to the true molecular Hartree-Fock solution for HF has been achieved. Configuration interaction has been included with the restriction that the 1σ orbital remains filled. Molecular energies, dipole moments, and other molecular quantities are evaluated and compared with experimental results and with other theoretical work.