Harris J. Silverstone

Piecewise polynomial configuration interaction natural orbital study of 1 s2 helium

We report here an analysis of extensive configuration interaction (CI) wave functions for the 1s2 ground state of the helium atom using piecewise polynomial basis functions. Large numbers of natural radial orbitals (NROs) with l ranging from 0 to 11 have been treated accurately and analyzed systematically. The contribution of each NRO to the total energy is found to follow the formula ΔE∼−0.42(l+1/2) × (n−1/2)−6a.u., where n is the principal quantum number, and the expansion coefficient of the NRO configuration is found to follow the formula, c∼−0.35(l+1/2)2/3 (n−1/2)−4. (The constants for l=0 are a little different). From an examination of the tails of the NROs, we are able to suggest an l‐dependent universal asymptotic formula, rβ−l−1+δe−ζr, where β is a constant, where δ=δl0, and where ζ2/2 is the ionization potential. The nodes of the NROs are also found to behave in a systematic way that yields valuable information on the choice of basis functions. So‐called ’’L’’‐limit energies EL, more accurate tha...

Modified Perturbation Theory for Atoms and Molecules Based on a Hartree–Fock φ0

The improvement of Hartree–Fock wavefunctions by perturbation theory is hampered by slow convergence with the usual choice of H0 =  ∑ i = 1N F(i), where F denotes the one‐electron Hartree–Fock Hamiltonian. The slow convergence is usually attributed to the unphysical nature of the excited states of F, which do not describe electrons moving in the field of the nuclei, shielded by N‐1 electrons. It is shown how to redefine F so that the zeroth‐order wavefunction still remains the Hartree–Fock wavefunction but so that the excited states of F correspond to an electron moving in the field of the nuclei screened by N‐1 electrons. The redefined F thus results in a more appropriate H0. Calculations of some second‐order polarization and semi‐internal correlation energies in first‐row atoms are given to illustrate the use of the redefined F.

On the Evaluation of Two‐Center Overlap and Coulomb Integrals with Noninteger‐n Slater‐Type Orbitals

Some formulas are developed which facilitate computation of two‐center overlap and Coulomb integrals with noninteger principal‐quantum‐number (n) Slater‐type orbitals by the Fourier‐transform convolution technique. A conceptually simple, closed‐form analytical expression for the overlap and Coulomb integrals with integer‐n Slater‐type orbitals is also given.

Piecewise polynomial electronic wavefunctions

Piecewise polynomials are examined as basis functions for electronic wavefunctions. The spline function method is a special case, which is shown to be less accurate, for a fixed set of mesh points, than a method based directly on Hermite’s interpolation formula. The determination of a suitable mesh is discussed both inductively and deductively, and a logarithmic formula for the 1s orbital of helium is ’’derived.’’ The accuracy is shown to depend on the number of points N+1 and on the polynomial order 2s+1, approximately according to the formula, δE∼N−4s−2, for appropriate meshes. A striking result is the possibility for systematically increasing the accuracy of the energy by systematically increasing the number of points, without encountering linear dependence problems, is demonstrated by calculations on the helium atom. With a 16‐point theoretically derived mesh, and with seventh order polynomials, we obtain a Hartree–Fock energy for helium of −2.8616799956122 a.u.

Expansion about an Arbitrary Point of Three‐Dimensional Functions Involving Spherical Harmonics by the Fourier‐Transform Convolution Theorem

Expansion of ψ(r)=ψ(r)YLM(θ,φ) in terms of spherical harmonics and radial functions, whose coordinates are measured from an arbitrary point in space, is obtained by use of the Fourier‐transform convolution theorem. For a specific ψ(r), two integrals most be evaluated to determine the expansion explicitly: (1) the radial part ψ(k) of the Fourier transform of ψ(r); and (2) an integral of ψ(k) with spherical Bessel functions. The examples of noninteger‐n and integer‐n Slater‐type orbitals are worked out by contour integration.

Unified Treatment of Two‐Center Overlap, Coulomb, and Kinetic‐Energy Integrals

A single analytical formula is derived which gives the two‐center overlap, Coulomb, nuclear‐attraction, and kinetic‐energy integrals as special cases. There are no difficulties when orbital exponents are nearly equal. Simplification of the derivative operators, which are characteristic of the approach, is discussed in detail. A computational scheme is outlined, and computing times are given.

Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

The theory of generalized functions and Fourier transforms is used to derive the Laplace‐type expansion for r12nYlm(θ12, φ12). This approach leads naturally to a general formula for the Dirac delta‐function terms which occur when n≤−3 and n − l is odd.

Analytical Evaluation of Multicenter Integrals of r12−1 with Slater‐Type Atomic Orbitals. V. Four‐Center Integrals by Fourier‐Transform Method

The four‐center integral of r12−1 with Slater‐type atomic orbitals is evaluated analytically. The Fourier‐transform convolution theorem is used to express the integral as an infinite sum in which the internuclear angles appear in spherical harmonics, and the internuclear distances in integrals over spherical Bessel functions and exponential‐type integrals. These “radial” integrals are evaluated as convergent infinite expansions by contour integration techniques. The formulas are valid for general values of the n, l, m, ζ parameters of the orbitals and for general nonzero values of the internuclear distance vectors.

Simulation of the EMR Spectra of High-Spin Iron in Proteins

Very detailed information about the energy levels and orientations of d orbitals in heme proteins has been obtained by combining EMR studies of paramagnetic samples with other structural information from X-ray crystallography and optical studies. As a result, the chemistry of heme enzymes can be discussed in detail. While the aim of this chapter is to review progress in bringing the chemistry of mononuclear iron centers in nonheme proteins to a similar level of knowledge, our understanding of line shape analysis for high-spin iron is dominated by the vast literature on heme samples. We begin this introduction with some of the history of EMR spectroscopy of methemoglobin and metmyoglobin.

Analytical Evaluation of Multicenter Integrals of r12−1 with Slater‐Type Atomic Orbitals. III. (2–2)‐Type Three‐Center Integrals

The general three-center one-electron nuclear-attraction integral with integer-n Slater-type orbitals is evaluated analytically by letting the orbital exponent of a 1s orbital in the analytical formula for two-electron three-center electron-repulsion integrals tend to infinity. The result is an infinite sum in which the internuclear angles appear in spherical harmonics, and the internuclear distances appear in modified spherical Bessel functions and exponential-type integrals.