Completeness criteria for explicitly correlated Gaussian geminal bases of axial symmetry

The completeness criteria for the basis set of explicitly correlated Gaussian-type geminals adapted to C_∞v symmetry are given. Specifically, we show that any pair function of Σ⁺ symmetry can be expanded in terms of products involving two spherical Gaussian orbitals located on the internuclear axis and a Gaussian correlating factor with a positive exponent. Pair functions corresponding to other irreducible representations of C_∞v can be expressed as linear combinations of products of a σ⁺ function and an angular factor depending on the azimuthal angles. The minimal set of the angular factors needed for completeness is given. These factors are relevant also for other explicitly correlated bases. © 1997 John Wiley & Sons, Inc.     

Analytical gradients for Singer's multicenter n‐electron explicitly correlated Gaussians

Analytical gradients for Singer's basis of n‐electron multicenter explicitly correlated Gaussian functions are derived and implemented to variationally optimize the energy and wave function of molecular systems within the Born–Oppenheimer approximation. Wave functions are optimized with respect to (½n(n+1)+3n) nonlinear variational parameters and one linear coefficient per term in the basis set. Preliminary results for the ground states of H₃⁺ and H₃ suggest that the method can be more flexible and can achieve lower energies than previously reported calculations. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 151–159, 2001     

Combining explicitly correlated R12 and Gaussian geminal electronic structure theories

Explicitly correlated R12 methods using a single short-range correlation factor (also known as F12 methods) have dramatically smaller basis set errors compared to the standard wave function counterparts, even when used with small basis sets. Correlations on several length scales, however, may not be described efficiently with one correlation factor. Here the authors explore a more general MP2-R12 method in which each electron pair uses a set of (contracted) Gaussian-type geminals (GTGs) with fixed exponents, whose coefficients are optimized linearly. The following features distinguish the current method from related explicitly correlated approaches published in the literature: (1) only two-electron integrals are needed, (2) the only approximations are the resolution of the identity and the generalized Brillouin condition, (3) only linear parameters are optimized, and (4) an arbitrary number of (non-)contracted GTGs can appear. The present method using only three GTGs and a double-zeta quality basis computed valence correlation energies for a set of 20 small molecules only 2.2% removed from the basis set limit. The average basis set error reduces to 1.2% using a near-complete set of seven GTGs with the double-zeta basis set. The conventional MP2 energies computed with much larger quadruple, quintuple, and sextuple basis sets all had larger average errors: 4.6%, 2.4%, and 1.5%, respectively. The new method compares well to the published MP2-R12 method using a single Slater-type geminal (STG) correlation factor. For example, the average basis set error in the absolute MP2-R12 energy obtained with the exp(-r12) correlation factor is 1.7%. Correlation contribution to atomization energies evaluated with the present method and with the STG-based method only required a double-zeta basis set to exceed the precision of the conventional sextuple-zeta result. The new method is shown to always be numerically stable if linear dependencies are removed from the two-particle basis and the zeroth-order Hamiltonian matrix is made positive definite.     

Matrix elements of N-particle explicitly correlated Gaussian basis functions with complex exponential parameters

In this work we present analytical expressions for Hamiltonian matrix elements with spherically symmetric, explicitly correlated Gaussian basis functions with complex exponential parameters for an arbitrary number of particles. The expressions are derived using the formalism of matrix differential calculus. In addition, we present expressions for the energy gradient that includes derivatives of the Hamiltonian integrals with respect to the exponential parameters. The gradient is used in the variational optimization of the parameters. All the expressions are presented in the matrix form suitable for both numerical implementation and theoretical analysis. The energy and gradient formulas have been programmed and used to calculate ground and excited states of the He atom using an approach that does not involve the Born-Oppenheimer approximation.     

Application of explicitly correlated Gaussian functions for calculations of the ground state of the beryllium atom

The electronic energy of atoms and molecules may be evaluated accurately by the use of wave functions where the interelectronic distances are explicitly present. In particular, explicitly correlated Gaussian-type functions make these types of calculations feasible and computationally tractable even for more extended systems. The resulting multielectron integrals may be reduced to standard one- and two-electron integrals that are readily evaluated. Initial calculations have been made for the Be atom where all four electrons were correlated at the same time. The preliminary results show that accurate results may be obtained. © 1993 John Wiley & Sons, Inc.     

Benchmark calculations for He2 and LiH molecules using explicitly correlated Gaussian functions

Explicitly correlated Gaussian (ECG) functions with carefully optimized non-linear parameters are used to calculate the electronic energies of He₂⁺ and LiH at their equilibrium internuclear distances. The obtained variational upper bounds (−4.99464392 and −8.070538 hartree, respectively) are the lowest reported to date. By extrapolating results obtained with various expansion lengths, the estimations of the Born–Oppenheimer limits are made.     

Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

The matrix differential calculus is applied for the first time to a quantum chemical problem via new matrix derivations of integral formulas and gradients for Hamiltonian matrix elements in a basis of correlated Gaussian functions. Requisite mathematical background material on Kronecker products, Hadamard products, the vec and vech operators, linear structures, and matrix differential calculus is presented. New matrix forms for the kinetic and potential energy operators are presented. Integrals for overlap, kinetic energy, and potential energy matrix elements are derived in matrix form using matrix calculus. The gradient of the energy functional with respect to the correlated Gaussian exponent matrices is derived. Burdensome summation notation is entirely replaced with a compact matrix notation that is both theoretically and computationally insightful. © 1996 John Wiley & Sons, Inc.     

Simple coupled-cluster singles and doubles method with perturbative inclusion of triples and explicitly correlated geminals: The CCSD(T)R12 model

To approach the complete basis set limit of the "gold-standard" coupled-cluster singles and doubles plus perturbative triples [CCSD(T)] method, we extend the recently proposed perturbative explicitly correlated coupled-cluster singles and doubles method, CCSD(2)(R12) [E. F. Valeev, Phys. Chem. Chem. Phys. 8, 106 (2008)], to account for the effect of connected three-electron correlations. The natural choice of the zeroth-order Hamiltonian produces a perturbation expansion with rigorously separable second-order energy corrections due to the explicitly correlated geminals and conventional triple and higher excitations. The resulting CCSD(T)(R12) energy is defined as a sum of the standard CCSD(T) energy and an amplitude-dependent geminal correction. The method is technically very simple: Its implementation requires no modification of the standard CCSD(T) program and the formal cost of the geminal correction is small. We investigate the performance of the open-shell version of the CCSD(T)(R12) method as a possible replacement of the standard complete-basis-set CCSD(T) energies in the high accuracy extrapolated ab initio thermochemistry model of Stanton et al. [J. Chem. Phys. 121, 11599 (2004)]. Correlation contributions to the heat of formation computed with the new method in an aug-cc-pCVXZ basis set have mean absolute basis set errors of 2.8 and 1.0 kJmol when X is T and Q, respectively. The corresponding errors of the standard CCSD(T) method are 9.1, 4.0, and 2.1 kJmol when X=T, Q, and 5. Simple two-point basis set extrapolations of standard CCSD(T) energies perform better than the explicitly correlated method for absolute correlation energies and atomization energies, but no such advantage found when computing heats of formation. A simple Schwenke-type two-point extrapolation of the CCSD(T)(R12)aug-cc-pCVXZ energies with X=T,Q yields the most accurate heats of formation found in this work, in error on average by 0.5 kJmol and at most by 1.7 kJmol.     

New correlation factors for explicitly correlated electronic wave functions

We have investigated the correlation factors exp(-zetar12), r12 exp(-zetar12), erfc(zetar12), and r12 erfc(zetar12) in place of the linear-r12 term for use in explicitly correlated electronic-structure methods. The accuracy obtained with all of these correlation factors is significantly greater than that obtained with the plain correlation factor r12. Polarization functions that are more diffuse than those of standard basis sets give even better results. The correlation factor exp(-zetar12) is very close to the optimum correlation factor for helium and outperforms the others.     

Newton–Raphson optimization of the explicitly correlated Gaussian functions for the ground state of the beryllium atom

Explicitly correlated Gaussian functions have been used in variational calculations on the ground state of the beryllium atom. In such calculations on systems with more electrons, it becomes imminent and essential to develop effective strategies for optimizing the parameters involved in the basis functions. The theory of analytical first and second derivatives of the variational functional with respect to the Gaussian exponents and its computational implementation in conjunction with the Newton–Raphson optimization technique is described. Some numerical results are presented to illustrate the performance of the method. © 1995 John Wiley & Sons, Inc.     

Matrix elements for Ĵ2 and Ĵz operators over explicitly correlated Cartesian Gaussian functions

General formalism for evaluation of multiparticle integrals involving J?² and J?_z operators over explicitly correlated Cartesian Gaussian functions is presented. The integrals are expressed in terms of the general overlap integrals. An explicitly correlated Cartesian Gaussian function is a product of spherical orbital Gaussian functions, powers of the Cartesian coordinates of the particle, and exponential Gaussian factors, which depend on interparticular distances. This development is relevant to both adiabatic and nonadiabatic calculations of energy and properties of multiparticle systems. © 1995 John Wiley & Sons, Inc.     

Multicenter and multiparticle integrals for explicitly correlated cartesian gaussian-type functions

An analytical derivation of multicenter and multiparticle integrals for explicitly correlated Cartesian Gaussian-type cluster functions is demonstrated. The evaluation method is based on the application of raising operators that transform spherical cluster Gaussian functions into Cartesian Gaussian functions.     

Energy and energy gradient matrix elements with N-particle explicitly correlated complex Gaussian basis functions with L=1

In this work we consider explicitly correlated complex Gaussian basis functions for expanding the wave function of an N-particle system with the L=1 total orbital angular momentum. We derive analytical expressions for various matrix elements with these basis functions including the overlap, kinetic energy, and potential energy (Coulomb interaction) matrix elements, as well as matrix elements of other quantities. The derivatives of the overlap, kinetic, and potential energy integrals with respect to the Gaussian exponential parameters are also derived and used to calculate the energy gradient. All the derivations are performed using the formalism of the matrix differential calculus that facilitates a way of expressing the integrals in an elegant matrix form, which is convenient for the theoretical analysis and the computer implementation. The new method is tested in calculations of two systems: the lowest P state of the beryllium atom and the bound P state of the positronium molecule (with the negative parity). Both calculations yielded new, lowest-to-date, variational upper bounds, while the number of basis functions used was significantly smaller than in previous studies. It was possible to accomplish this due to the use of the analytic energy gradient in the minimization of the variational energy.     

Accurate explicitly correlated wave functions for two electrons in a square

An explicitly correlated linear-r(12) variational method is developed for a system of two electrons confined to a two-dimensional square well with infinite walls. The wave function is written as an expansion in products of non-negative integer powers of the relative and center-of-mass electronic coordinates and powers of r(12) restricted to 0 and 1. This form indirectly includes higher powers of the interelectronic distance and exhibits a much faster convergence than a similar expansion without r(12)-dependent terms. The method is implemented using high-precision floating-point arithmetic. Ground-state total energies are reported with at least 12 accurate significant figures for squares with sides from 1 to 50 bohrs. The method can be used "as is" for excited states and for two-dimensional rectangular wells. We also show that wave functions for two electrons in a square and in a rectangle have a higher symmetry than can be accounted for by the point group of the system.     

Newton–Raphson optimization of the explicitly correlated Gaussian functions for calculations of the ground state of the helium atom

Explicitly correlated Gaussian functions have been used in variational calculations on the ground state of the helium atom. The major problem of this application, as well as in other applications of the explicitly correlated Gaussian functions to compute electronic energies of atoms and molecules, is the optimization of the nonlinear parameters involved in the variational wave function. An effective Newton–Raphson optimization procedure is proposed based on analytic first and second derivatives of the variational functional with respect to the Gaussian exponents. The algorithm of the method and its computational implementation is described. The application of the method to the helium atom shows that the Newton–Raphson procedure leads to a good convergence of the optimization process. © 1994 by John Wiley & Sons, Inc.     

Analytic energy second derivatives for general correlated wavefunctions,including a solution of the first-order coupled-perturbed configuration-interaction equations

The first theoretical method for the analytic determination of energy second derivatives for configuration-interaction wave-functions is presented. Several test cases are reported, the largest being an 8385 configuration formaldehyde wavefunction.     

An algorithm for calculating atomic D states with explicitly correlated gaussian functions

An algorithm for the variational calculation of atomic D states employing n-electron explicitly correlated gaussians is developed and implemented. The algorithm includes formulas for the first derivatives of the hamiltonian and overlap matrix elements determined with respect to the gaussian nonlinear exponential parameters. The derivatives are used to form the energy gradient which is employed in the variational energy minimization. The algorithm is tested in the calculations of the two lowest D states of the lithium and beryllium atoms. For the lowest D state of Li the present result is lower than the best previously reported result.     

Systematically convergent basis sets for explicitly correlated wavefunctions: the atoms H, He, B-Ne, and Al-Ar

Correlation consistent basis sets have been optimized for use with explicitly correlated F12 methods. The new sets, denoted cc-pVnZ-F12 (n=D,T,Q), are similar in size and construction to the standard aug-cc-pVnZ and aug-cc-pV(n+d)Z basis sets, but the new sets are shown in the present work to yield much improved convergence toward the complete basis set limit in MP2-F12/3C calculations on several small molecules involving elements of both the first and second row. For molecules containing only first row atoms, the smallest cc-pVDZ-F12 basis set consistently recovers nearly 99% of the MP2 valence correlation energy when combined with the MP2-F12/3C method. The convergence with basis set for molecules containing second row atoms is slower, but the new DZ basis set still recovers 97%-99% of the frozen core MP2 correlation energy. The accuracy of the new basis sets for relative energetics is demonstrated in benchmark calculations on a set of 15 chemical reactions.     

The performance of explicitly correlated methods for the computation of anharmonic vibrational frequencies

Anharmonic vibrational frequencies for closed-shell molecules computed with CCSD(T)-F12b/aug-cc-pVTZ differ from significantly more costly composite energy methods by a mean absolute error (MAE) of 7.5 cm⁻¹ per fundamental frequency. Comparison to a few available gas phase experimental modes, however, actually lowers the MAE to 6.0 cm⁻¹. Open-shell molecules have an MAE of nearly a factor of six greater. Hence, open-shell molecular anharmonic frequencies cannot be as well-described with only explicitly correlated coupled cluster theory as their closed-shell brethren. As a result, the use of quartic force fields and vibrational perturbation theory can be opened to molecules with six or more atoms, whereas previously such computations were limited to molecules of five or fewer atoms. This will certainly assist in studies of more chemically interesting species, especially for atmospheric and interstellar infrared spectroscopic characterization.