On the eigenfunctions of many‐particle systems in quantum mechanics

J. Chem. Theory Comput.

2014

We discuss molecular orbital basis sets that contain both Gaussian and polynomial (ramp) functions. We show that, by modeling ramp−Gaussian products as sums of ramps, all of the required one-and two-electron integrals can be computed quickly and accurately. To illustrate our approach, we construct R-31+G, a mixed ramp−Gaussian basis in which the core basis functions of the 6-31+G basis are replaced by ramps. By performing self-consistent Hartree−Fock calculations, we show that the thermochemical predictions of R-31+G and 6-31+G are similar but the former has the potential to be significantly faster.

Section: Basis Functionsmentioning

confidence: 99%

Mixed Ramp–Gaussian Basis Sets

McKemmish

Gilbert

J. Chem. Theory Comput.

2014

“…Because all of the integrals that arise in Hartree−Fock and traditional post-Hartree−Fock calculations can be reduced 12−27 to elementary functions and error functions, 28 Gaussians are ubiquitous in molecular calculations and have even challenged plane waves in solidstate calculations. 29,30 However, from a theoretical viewpoint, Gaussians are suboptimal for two reasons: they lack a cusp 31 at r = 0, and they decay too fast 32 at large r. The complementary nature of these deficiencies becomes clear if an exponential function (the exact wave function for a hydrogen atom) and a Gaussian (the exact wave function for a harmonic oscillator) are superimposed as shown in Figure 1. The logarithmic transformation that converts the exponential into a straight line converts the Gaussian into a sigmoidal curve that is flat at both ends of the domain.…”

Section: Introductionmentioning

confidence: 99%

Gaussian Expansions of Orbitals

McKemmish

J. Chem. Theory Comput.

2012

Using numerical calculations and analytic theory, we examine the convergence behavior of Gaussian expansions of several model orbitals. By following the approach of Kutzelnigg, we find that the errors in the energies of the optimal n-term even-tempered Gaussian expansions of s-type, p-type, and d-type exponential orbitals are ε n s ∼ exp(−π(3n) 1/2 ), ε n p ∼ exp(−π(5n) 1/2 ), and ε n d ∼ exp(−π(7n) 1/2 ), respectively. We show that such "root-exponential" convergence patterns are a consequence of the orbital cusps at r = 0, rather than the over-rapid decay of Gaussians at large r. We find that even-tempered expansions of the cuspless Lorentzian orbital also exhibit root-exponential convergence but that this is a consequence of its fat tail.

“…Those in the first class, which include configuration interaction, Møller-Plesset perturbation theory and coupled cluster theory, 9 are based on the mathematical observation that an improved wavefunction can be formed by linearly combining eigenfunctions of the HF Hamiltonian. Although in the limit these methods provide exact results, they are intrinsically inefficient because they have to approximate the derivative discontinuities in the exact wavefunction 10 by sums of smooth functions. As a consequence, their computational costs become prohibitive even for quite small systems.…”

Section: Introductionmentioning

confidence: 99%

A family of intracules, a conjecture and the electron correlation problem

Phys. Chem. Chem. Phys.

Crittenden

O’Neill

et al. 2006

We present a radical approach to the calculation of electron correlation energies. Unlike conventional methods based on Hartree-Fock or density functional theory, it is based on the twoelectron phase-space information in the Omega intracule, a three-dimensional function derived from the Wigner distribution. Our formula for the correlation energy is isomorphic to the Hartree-Fock energy expression but requires a new type of four-index integral. Preliminary results, obtained using a model that is based on the known correlation energies of small atoms, are encouraging.