T. L. Gilbert

Single‐Configuration Wavefunctions and Potential Curves for Low‐Lying States of He2+, Ne2+, Ar2+, F2−, Cl2− and the Ground State of Cl2

Wavefunctions, orbital energies, and potential curves for He2+, Ne2+, Ar2+, F2−, and Cl2− have been calculated in the molecular‐orbital, self‐consistent‐field approximation over a range Re ≲ R < ∞ for the ground state and those excited states which dissociate into an atom and an ion in their ground states. The ground state potential curve for Cl2 has been calculated for R ∼ Re. Dissociation energies and other parameters obtained from the calculated potential curves are compared with corresponding parameters obtained from elastic differential scattering measurements and resonant electron capture in noble gases, measurements of afterglow line profiles in dissociative recombination radiation in neon and argon, optical absorption spectra of VK centers, and endoergic charge transfer studies of halogen ions and molecules. A formal analysis and discussion of the sources of correlation error in the calculated potential curves and a discussion of the expansion errors are also given.

Soft‐Sphere Model for Closed‐Shell Atoms and Ions

Using recent spectroscopic data for the alkali halide monomers, it is shown that approximate additivity rules of the form Rij = Ri + Rj and ρij = ρi + ρj hold for the radii and hardness parameters in the Born–Mayer repulsive potential, Uij(R) = fρijexp[(Rij − R) / ρij].

Relation between charge and force parameters of closed-shell atoms and ions

Combining rules for parameters which characterize the short‐range repulsive forces between closed‐shell atoms and ions are used to determine the radius and softness for a number of alkali and halide ions and rare‐gas atoms from scattering and spectroscopic data. A corresponding set of charge radii and softnesses are determined from atomic charge densities calculated in both the relativistic and nonrelativistic Hartree–Fock approximations. These characteristic atomic parameters are compared, and it is found that, for species with the same charge belonging to the same column of the periodic table, the relations between charge and force radii and between charge and force softnesses are very nearly linear. The radial distance dependence of the softness is examined, and some areas where further work is needed to follow up this empirical study are indicated.

Multiconfiguration self‐consistent‐field theory for localized orbitals. II. Overlap constraints, Lagrangian multipliers, and the screened interaction field

The general, standard, and localized forms of the multiconfiguration self‐consistent‐field orbital equations, with and without overlap constraints, are derived and compared. The equations are very similar in form to the corresponding single configuration (Hartree‐Fock) equations. The constrained equations may be formulated in terms of a single Hermitian orbital operator with a Lagrangian multiplier matrix which is always Hermitian. The equations for orbitals localized on an embedded fragment (an atomic shell, bond, atom, molecular fragment, or other subsystem that is part of a larger system) differ from the orbital equations for an isolated fragment only by the addition of a screened interaction field. Dropping the orbital overlap constraints introduces an additional condition on the Lagrangian multipliers that must be satisfied in order for the energy to be stationary with respect to variations in the overlap matrix.

Maximum‐Overlap Directed‐Hybrid Orbitals

A previous method for constructing best hybrid orbitals based on the principle of maximum overlap by Murrell is evaluated. A simpler and more general procedure is outlined. Use is made of the Carlson-Keller theorem which requires that positive roots be used and eliminates ambiguity. (P.C.H.)

SCF potential curves for AlH and AlH+ in the attractive and repulsive regions

Self‐consistent‐field calculations have been carried out for the systems AlH and AlH+ in their ground and excited states. Distances considered range from separated atoms and ions to 0.1 bohr, or ΔE ?2700 eV. Additional calculations were performed to determine the effect of core and valence electrons at small internuclear separations. A comparison was made of the ab initio potential curve obtained for the 1Σ+ state with results obtained from the Thomas–Fermi statistical model.

Spline representation. I. Linear spline bases for atomic calculations

The use of a basis of cardinal splines for atomic calculations with the expansion method is proposed. The basis has the property that the coefficients Cp in the orbital expansion φ (r) = Σp=−1N+1 Cpχp(r) are given by C−1=φ′(r0), Cp=φ(rp), p=0,1, ···, N, and CN+1=φ′(rN), where r0, r1, ···,rN is the mesh of knots on which the spline is defined. The basis functions χp(r) are continuous with continuous first and second derivatives, and may readily be calculated from the expansion χp(r) = Σk=1N Σs=03 rsdk (r)Sksp, where dk(r)=1 if rk−1 ≤ r<rk and 0 otherwise and Sksp is a matrix of coefficients which are uniquely determined by the mesh alone and may be constructed by simple algebraic operations, the most complex being the inversion of a single (N+1)×(N+1) matrix. Atomic Hartree‐Fock matrix elements are given by simple polynomial expressions. The number of two‐electron integrals increases only as the square of the number of basis functions and can be stored in factored form, so that the storage requirements for...

Spline bases for atomic calculations

The use of spline bases has been proposed for ab initio electronic structure calculations. They offer the apparent advantage that no nonlinear parmameters occur, so that exponent optimization is unnecessary, and the number of two‐electron integrals in SCF calculations increases only as n2 rather than n4 where n is the number of basis functions. The results of test calculations on the helium atom are reported in this note. Analysis of the computer time required to achieve accuracy comparable to other methods indicates that spline bases are not suitable for SCF calculations. (AIP)

The spline representation: II. Shell orbitals and exponential spline bases

The cardinal spline expansion method for atomic calculations, introduced in a preceding article, is extended from linear to exponential splines. These are obtained by replacing the spline basis functions χp(r) by χp(u), where u = e−ζr. An orbital multiplying factor rn−1e−ζnlr is introduced for each shell in order to improve convergence. The Hartree−Fock equations for closed−shell atoms are transformed to equations for nodeless orbitals localized in atomic shells (shell orbitals), and the construction of shell orbitals for atoms embedded in molecules is discussed briefly. Explicit expressions for the one−center matrix elements are derived. A major advantage of the linear spline representation, viz., that the number of two−electron integrals increases only as the square of the basis size and the off−diagonal integrals can be factored, is retained. A comparison is made of the relative advantages and disadvantages of linear and exponential spline representations with regard to probable storage requirements, c...

Selection of self-consistent-field orbitals

A procedure introduced by Davidson for selecting orbital bases on the occupied and virtual Hartree‐Fock manifolds is shown to be equivalent to a procedure introduced by Adams.