Harrison Shull

SUPERPOSITION OF CONFIGURATIONS AND NATURAL SPIN ORBITALS. APPLICATIONS TO THE He PROBLEM.

The method of superposition of configurations is examined in its application to the helium atom in two cases: a 21×21 matrix including all configurations up to 〈6s〉2, and a 20×20 matrix with all configurations up to the 4‐quantum level, including angular terms. A new radial limit is established at −2.87900±0.00003, and this is used to discuss the convergence of such expansions in Legendie functions. The variation with scale factor is discussed in detail. The wave functions are analyzed in terms of natural spin orbitals (NSOs), which seem to have many advantages. The first NSO bears a striking resemblance to the Hartree‐Fock function, and the first two together provide a close approximation to the solution of the extended Hartree‐Fock equations with different orbitals for different electrons. An energy of −2.877924 is obtained for the best (u, v) function found. An analysis of the results suggests that inner orbitals may be better represented by pure exponentials than by Hartree‐Fock orbitals whenever additional correlational degrees of freedom are permitted. Expressed in approximate NSO form, the wave function is almost invariant to choice of basis set, provided that the latter is reasonably chosen. In particular, the necessity of including continuum terms along with the discrete hydrogen‐like set is demonstrated.

Electronic correlation energy in 3- and 4-electron atoms☆

The electronic correlation energy in 3- and 4-electron atomic systems is compared to previously well established correlation energies in 2-electron atoms. It is shown that the distribution of correlation energy in the K shell of these atoms between radial and angular correlation parallels that of the 2-electron system very closely. It is found, however, that the correlation in the L shell of the Be ground state is almost purely angular correlation energy. There is negligible correlation energy associated with K-L interaction. Analysis of the Z dependence of the correlation energy of 4 electron atoms shows a term linear in Z. It is suggested that this term arises from degeneracies existing in the limit of infinite Z, and a tabulation of states expected to have this property is given. The analysis suggests a convenient scheme for constructing a semiempirical method for estimating atomic energies rather accurately. It is pointed out that a similar analysis for molecules in terms of the internuclear parameters suggests there may be inherent difficulties in constructing such a scheme for the molecular case. (auth)

Molecular Calculations. I. LCAO MO Self‐Consistent Field Treatment of the Ground State of H2O

A theoretical study of the ground state of the water molecule utilizing the self‐consistent field molecular orbital method in the linear‐combination‐of‐atomic‐orbitals approximation is reported. All 10 electrons have been explicitly included, and all integrals have been retained in the calculations. Results are listed for six different values of the bond angle from 90 to 180 degrees, but only for the experimentally observed OH distance. The effect of inner‐shell outer‐shell mixing is investigated and shown to have a significant effect upon the results. The electronic configuration is discussed both from the viewpoint of molecular and equivalent orbitals. The computed dipole moment is 1.51 D. The SCF ionization potentials agree well with the experimental values, their order of assignment being quite definitely established. The calculated total electronic energy is in error by about 0.75%. This energy changes by only 0.13% for a change in bond angle from 90° to 180°, however, and consequently, the treatment...

Correlation Splitting in Helium‐Like Ions

The correlation effects in the ground state of the He‐like ions are investigated by using the idea that the two electrons may occupy different orbitals in space. By using a scale factor η and a splitting parameter v, a radial function [open phi][r] is split into two different orbitals, [open phi][η(1+v)r] and [open phi][η(1—v)r], following Hylleraas [E. A. Hylleraas, Z. Physik 54, 347 (1929)], and Eckart [C. Eckart, Phys. Rev. 36, 878 (1930)]. It is shown that by using orbitals obtained by splitting a simple exponential function, a surprisingly large amount of the radial correlation energy may be taken into account, whereas an attempt to split the atomic self‐consistent field function for He led to a negative result. The lowest triplet state is also treated by using the same approach.

Natural Spin Orbital Analysis of Hydrogen Molecule Wave Functions

Approximate wave functions for the hydrogen molecule ground state previously available in the literature are analyzed quantitatively into approximate natural spin orbital functions with particular attention to the corresponding occupation numbers. The analysis demonstrates the very great similarity of all such trial wave functions, and especially the largely molecular orbital nature of the wave function. In addition it shows a close relationship between the molecular orbital and valence bond functions, and the importance for allowing for angular correlation of the electrons by including terms dependent upon the azimuthal coordinate. The analysis particularly demonstrates that approximate natural spin orbital occupation numbers are nearly invariant under a wide variety of choices of basis functions, and therefore are particularly suitable for comparison of different approximate functions and for discussion of their respective properties.

The Chemical Bond in Molecular Quantum Mechanics

It is postulated that a properly antisymmetrized product function over geminals (electron‐pair wave functions) is adequate for discussion of the principal chemical properties of molecules. By application of the virial theorem it is shown that such a wave function has both of the properties essential to the bond‐energy concept; namely (a) the energy of a molecule is the sum of the energies of its individual bonds and (b) the bond energies are invariant from one molecule to another. Within the framework of this approximation, bond energies become identified in magnitude with the kinetic energies associated with the respective geminals. The concepts are sufficiently general to include both localized and nonlocalized bonds, unshared electron pairs, odd electrons, and states of various multiplicities.

Floating Wave Functions for H2+ and H2

Floating wave functions using ordinary 1s orbitals are reexamined for both H2+ and H2. Results previously reported in the literature are found to be in error. The newly obtained energies are disappointingly high. It is conjectured that these relatively poor energies are largely the result of the inability of this kind of approximation to give a good representation of the discontinuities in the wave function in the vicinity of the nuclei.

Single‐Center Expansions for One‐Electron Systems. H2+

The expansion of the H2+ wave function in terms of the set of associated Laguerre functions of order (2l+2) based on a single point in space, the molecular midpoint, is studied by the usual variational approach. In particular, the convergence of the expansion is studied as a function of the number and type of basis functions used, the respective scaling parameters, and the internuclear distance. A peculiar behavior with respect to the scaling parameter, in which, for example, for certain optimum choices of the scale parameter for one group of basis functions no improvement is observed upon the addition of one more such function, is proved to occur generally for these functions with one‐electron Hamiltonians. This behavior emphasizes the necessity for treating such scale factors as mathematical parameters rather than as constants to be chosen by physical intuition. This single‐center expansion is concluded to be only slowly convergent and therefore, the authors feel, likely to have relatively limited use f...

Single‐Center Wave Function for the Hydrogen Molecule

The ground state of the hydrogen molecule is studied using an expansion based on a single center, the molecular midpoint, with basis orbitals constructed from associated Laguerre functions with a single orbital exponent. The convergence of the expansion is studied by systematic addition of terms and is found to be slow. The best wave functions attained have energies of —1.15086 (38 axially symmetric terms), and —1.16141 (44 terms). The results are shown to be very similar to those obtained by using Slater orbitals with nonintegral principal quantum number both from the energy increments observed and from a natural spin orbital occupation number analysis. It is concluded that the slow convergence probably results from failure to represent adequately the singularities at the nuclei, and that further use of single‐center expansions in diatomic problems (except at very small internuclear distances) seems unprofitable, irrespective of what set of orbitals is used as a basis.

Nature of the Two‐Electron Chemical Bond. VII. Multicenter Bonds and H3+

The nature of the chemical bonding in H3+ has been studied by means of a natural spin–orbital analysis of a previously calculated wavefunction. H3+ is shown to resemble closely its united‐atom analog, Li+, and the Hartree–Fock energy of H3+ is estimated to be −1.301 hartrees. Also, electron‐density plots are presented in order to clarify the physical picture of the bonding in H3+, and suggestions are made as to how further calculations might proceed most efficiently.