Finite Basis Set Calculations Research Articles

A path integral ground state approach for asymmetric top rotors with nuclear spin symmetry: Application to water chains

A path integral ground state (PIGS) approach for the simulation of asymmetric top rotors is presented. The method is based on Monte Carlo sampling of angular degrees of freedom. A symmetry-adapted rotational density matrix is used to account for nuclear spin statistics. To illustrate the method, ground-state properties of collections of para-water molecules confined to a one-dimensional lattice are computed. Those include energetic and structural observables. An advantage of the PIGS method is that expectation values can be obtained directly since the square of the wavefunction is sampled during a simulation. To benchmark the method, ground state energies and orientational distributions are computed using exact diagonalization for a single para-water molecule in an external field using a finite basis of symmetric top eigenfunctions. Benchmark results are also provided for N = 2 para-water molecules pinned to lattice sites at various distances to sample the crossover from hydrogen bonding to the dipole-dipole interaction regime. Excellent agreement between the PIGS results and the finite basis set calculations is observed. A thorough analysis of the convergence in terms of the imaginary time propagation length and systematic Trotter error is performed. The PIGS approach is then applied to a chain of N = 11 water molecules, and an equation of state is constructed in terms of the intermolecular separation. Ordering effects are also studied, and a transition between hydrogen bonding to dipole-dipole alignment is observed. The method is scalable and can also be applied in higher dimensions.

The same number of optimized parameters scheme for determining intermolecular interaction energies.

We propose the Same Number Of Optimized Parameters (SNOOP) scheme as an alternative to the counterpoise method for treating basis set superposition errors in calculations of intermolecular interaction energies. The key point of the SNOOP scheme is to enforce that the number of optimized wave function parameters for the noninteracting system is the same as for the interacting system. This ensures a delicate balance between the quality of the monomer and dimer finite basis set calculations. We compare the SNOOP scheme to the uncorrected and counterpoise schemes theoretically as well as numerically. Numerical results for second-order Møller-Plesset perturbation theory (MP2) and coupled-cluster with single, double, and approximate triple excitations (CCSD(T)) show that the SNOOP scheme in general outperforms the uncorrected and counterpoise approaches. Furthermore, we show that SNOOP interaction energies calculated using a given basis set are of similar quality as those determined by basis set extrapolation of counterpoise-corrected results obtained at a similar computational cost.

Elimination of spurious solutions of the Dirac equation using the Lanczos algorithm

Spurious solutions of the Dirac equation which arise and also occur in finite basis-set calculations can easily be identified and eliminated using the Lanczos algorithm. Numerically convergent eigensolutions of the Dirac Hamiltonian for a relativistic electron in an external central Coulomb-like potential are obtained to demonstrate the method.

Self-consistent effective local potentials

An effective local potential (ELP) is a multiplicative operator whose deviation from a given nonlocal potential has the smallest variance evaluated with a prescribed single-determinant wave function. ELPs are useful in density functional theory as alternatives to optimized effective potentials (OEPs) because they do not require special treatment in finite basis set calculations as OEPs do. We generalize the idea of variance-minimizing potentials by introducing the concept of a self-consistent ELP (SCELP), a local potential whose deviation from its nonlocal counterpart has the smallest variance in terms of its own Kohn-Sham orbitals. A semi-analytical method for computing SCELPs is presented. The OEP, ELP, and SCELP techniques are applied to the exact-exchange-only Kohn-Sham problem and are found to produce similar results for many-electron atoms.

Comparison of the polarizabilities and hyperpolarizabilities obtained from finite basis set and finite difference Hartree–Fock calculations for diatomic molecules: III. The ground states of N2, CO and BF**This paper is dedicated to Victor R Saunders, on his official retirement from Daresbury Laboratory.

We investigate the accuracy with which the electric dipole polarizability, αzz, and the hyperpolarizability, βzzz, can be calculated by using the algebraic approximation, i.e. finite basis set expansions, and by means of the finite difference method in calculations for the ground states of the 14 electron systems N2, CO and BF within the Hartree–Fock model at their respective experimental equilibrium geometries. For a well-chosen grid, the finite difference technique can provide Hartree–Fock energy and dipole moment expectation values approaching machine precision which can be used to assess the accuracy of corresponding calculations carried out within the algebraic approximation. The finite field approximation is used to determine polarizabilities and hyperpolarizabilities from finite difference Hartree–Fock dipole moment expectation values. The results are compared with finite basis set calculations of the corresponding quantities which are carried out analytically using coupled perturbed Hartree–Fock theory. For the N2 molecule, the Hartree–Fock polarizability is found to be 14.9512 au within the finite basis set approximation and 14.945 au within the finite difference approach. For the CO molecule, the corresponding results are 14.4668 au and 14.4668 au, whilst for the BF molecule the values are 16.6450 au and 16.6450 au, respectively. The Hartree–Fock hyperpolarizability of the CO molecule is found to be 31.4081 au and 31.411 au within the finite basis set and finite difference approximations, respectively. The corresponding hyperpolarizability values for the BF molecule are 63.9687 au and 63.969 au, respectively.

A new approach for the two‐electron cumulant in natural orbital functional theory

AbstractThe cumulant expansion gives rise to a useful decomposition of the two‐matrix D in which the pair correlated matrix (cumulant) is disconnected from the antisymmetric product of the one‐matrix Γ. A new explicit antisymmetric approach for the two‐particle cumulant matrix in terms of two symmetric matrices, Δ and Λ, as functionals of the occupation numbers is proposed for singlet ground states of closed‐shell systems. It produces a natural orbital functional that reduces to the exact expression for the total energy in two‐electron systems. The functional form of matrix Λ is readily generalized to any system with an even number of electrons. The diagonal elements of Δ equal the square of the occupation numbers, and the N‐representability positivity necessary conditions of the two‐matrix impose several bounds on the off‐diagonal elements of matrix Δ. The well‐known mean value theorem and the partial sum rule obtained for the off‐diagonal elements of Δ provide a prescription for deriving a practical functional. In particular, when the mean values {J} of the Coulomb interactions {Jij} for a given orbital i taking over all orbitals j ≠ i are assumed to be equal {Kii/2}, a functional close to self‐interaction‐corrected GU functional is obtained, but the two‐matrix fermionic antisymmetric holds. An additional term for the matrix elements of Λ between HF occupied orbitals is proposed to ensure a correct description of the occupation numbers for the lowest occupied levels. The functional is tested in fully variational finite basis set calculations of 57 molecules. It gives reasonable molecular energies at the equilibrium geometries. The calculated values of dipole moments are in good agreement with the available experimental data. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Estimating the Hartree—Fock limit from finite basis set calculations

It is demonstrated that the polarization-consistent basis sets, which are optimized for density functional methods, are also suitable for Hartree–Fock calculations, and can be used for estimating the Hartree–Fock basis set limit to within a few micro-hartree accuracy. Various two- and three-point extrapolation schemes are tested and exponential functions are found to be superior compared to functions depending on the inverse power of the highest angular momentum function in the basis set. Total energies can be improved by roughly an order of magnitude, but atomization energies are only marginally improved by extrapolation.

Ab initio study of magnetochiral birefringence

The magnetically induced axial birefringence of six closed-shell chiral molecules (methyloxirane, C3H6O, fluoro- and methylcyclopropanone, C3H3OF and C4H6O, carvone, C10H14O, limonene, C10H16, and proline, C5H9NO2) is determined at the Hartree–Fock wave-function level by evaluating the frequency dependent quadratic response functions entering the molecular property expression, according to Barron and Vrbancich [Mol. Phys. 51, 715 (1984)]. Both the magnetic dipole and the electric quadrupole contributions are taken into account and their relative importance is discussed. A proof of the origin independence of the magnetochiral birefringence is presented for the exact wave function and the dependence on the origin is investigated in finite basis set calculations at the Hartree–Fock level. For carvone, limonene, and proline the results are compared with recent experimental data obtained by two different experimental groups, which are in disagreement with respect to the magnitude of the magnetochiral effect. A parallel study of the natural optical rotation shows that in the three larger molecules the optical rotatory strengths are strongly affected by changes in conformations. Nonetheless the magnetochiral birefringence computed for various different conformers—although varying remarkably—is much smaller in absolute value than experimentally observed. The disagreement—of more than three orders of magnitude—between some experimental data and theory appears to be hard to reconcile and to attribute entirely to limitations in the computational approach.

Assessment of simple exchange-correlation energy functionals of the one-particle density matrix

An improved density matrix functional (DMF) combining the properties of the “corrected Hartree” (CH) and “corrected Hartree–Fock” (CHF) approximations is proposed. Functionals of the CH/CHF type and the closely related natural orbital functional of Goedecker and Umrigar (GU) are tested in fully variational finite basis set calculations of light atoms, the lowest energy singlet methylene, and, for the first time, potential energy curves of diatomic molecules. Although CH/CHF-style DMFs may give reasonable energies for atoms and molecules near equilibrium geometries, they predict unrealistically shallow minima in the potential energy curves for diatomic molecules with more than two electrons. The calculated CH and CHF molecular dissociation curves exhibit the same patterns of over- and under-correlations as the corresponding correlation energy plots for the homogeneous electron gas undergoing a transition from high to low densities. In contrast, the GU functional yields not only accurate atomic and molecular energies but also plausible dissociation curves. The reasons behind the observed performance are analyzed.

The Dirac equation in the algebraic approximation: VIII. Comparison of finite basis set and finite element molecular Dirac–Hartree–Fock calculations for the H2, LiH, and BH ground states

AbstractUsing basis sets of distributed s‐type Gaussian functions with positions and exponents optimized so as to support Hartree–Fock total energies with an accuracy approaching the sub‐μHartree level, Dirac–Hartree–Fock–Coulomb calculations are reported for the ground states of the H2, LiH, and BH molecules at their respective equilibrium geometries. The calculated energies are compared with those obtained by Kullie et al. using the finite element technique for the Dirac–Hartree–Fock–Coulomb problem. Finite basis set calculations using the Dirac–Hartree–Fock–Breit approximation are also reported and compared with a first‐order treatment of the frequency‐independent Breit interaction. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

Exchange-correlation potentials

We describe our implementation of the Zhao, Morrison, and Parr method [Phys. Rev. A 50, 2138 (1994)] for the calculation of molecular exchange-correlation potentials from high-level ab initio densities. The use of conventional Gaussian basis sets demands careful consideration of the value of the Lagrange multiplier associated with the constraint that reproduces the input density. Although formally infinite, we demonstrate that a finite value should be used in finite basis set calculations. The potential has been determined for Ne, HF, N2, H2O, and N2(1.5re), and compared with popular analytic potentials. We have then examined how well the Zhao, Morrison, Parr potential can be represented using a computational neural network. Assuming vxc=vxc(ρ), we incorporate the neural network into a regular Kohn–Sham procedure [Phys. Rev. A 140, 1133 (1965)] with encouraging results. The extension of this method to include density derivatives is briefly outlined.

Density functional-time-dependent local density approximation calculations of autoionization resonances in noble gases

Absolute photoionization cross section profiles and asymmetry parameters of Ne, Ar and Kr have been calculated at the time-dependent local density approximation level. We employed a very accurate B-spline finite basis set and the modified Sternheimer approach, which is a first-order perturbative scheme particularly suited to finite basis set calculations. The gradient-dependent van Leeuwen and Baerends (VLB) exchange-correlation potential has been used, since it has the correct Coulombic behaviour at large distances which is a necessary condition for the existence of the Rydberg states. A simple modification of the boundary conditions considerably improves the number of terms of the Rydberg series. The resonance parameters obtained by fitting the calculated Fano profiles with a suitable analytic expression are compared with the experimental parameters.

Orthogonality constraints in finite basis set calculations

The orthogonality-constrained one-body eigenvalue problem is formulated for complete, finite-dimensional spaces in terms of projection operators that partition the space into two orthogonal subspaces, one containing the constraints and one containing the desired solutions. We derive a Hermitian operator, associated with the Hamiltonian, having eigenvectors that are correspondingly partitioned, each subset of the eigenvectors providing a basis for one of the subspaces. This Hermitian operator advantageously replaces a non-Hermitian operator that was proposed in recent work of this group. The non-zero spectrum of the two operators is demonstrably the same, and the occurrence of null eigenvalues is clarified through characterization of the associated eigenvectors. The deviation from zero of these eigenvalues (apart from numerical inaccuracies) is related directly to the normalization of the constraint vectors. The formalism is extended to the case of non-orthonormal primitive basis sets, and a numerical application is carried out using B-spline functions.

Absorbing boundary conditions by the partial integration exterior scaling method

The partial integration exterior scaling (PIES) method is developed. This method enables one to insert absorbing boundary conditions in finite basis set calculations in a rigorous way. The results are parameter independent when the basis set approaches completeness. Illustrative numerical examples are given for the calculations of resonance positions and lifetimes and of cumulative reaction probabilities.

Indirect nuclear spin-spin coupling constants within the coupled multiconfiguration hartree-fock approximation

The multiconfigurational coupled Hartree-Fock expression for the Fermi contact contribution to the indirect nuclear spin-spin coupling constants is derived Numerical calculations on H2 and FH indicate that the nuclear spin-spin coupling is very difficult to describe in finite basis set calculations.

A new solution for the gauge origin problem

A variation scheme for the determination of the optimal gauge origin in the finite basis set calculations of second-order molecular magnetic properties, is presented. The best gauge origin is defined in terms of the smallest possible violation of the completeness relation for a pair of hermitean operators relevant for a given second-order magnetic property. In contrast to some previous proposals, the present scheme implicitly minimizes the basis set error of the computed second-order energy and the best gauge origin may not coincide with the electronic centroid. The latter only maximizes the diamagnetic contribution to the magnetic susceptibility tensor. The proposed method is illustrated by the calculation of the proton and fluorine shielding constant for the HF molecule.