Excited states of boron isoelectronic series from explicitly correlated wave functions

The ground state and some low-lying excited states arising from the 1s2 2s2p2 configuration of the boron isoelectronic series are studied starting from explicitly correlated multideterminant wave functions. One- and two-body densities in position space have been calculated and different expectation values such as , , , , , and , where r, r12, and R stand for the electron-nucleus, interelectronic, and two electron center of mass coordinates, respectively, have been obtained. The energetic ordering of the excited states and the fulfillment of the Hund's rules is analyzed systematically along the isoelectronic series in terms of the electron-electron and electron-nucleus potential energies. The effects of electronic correlations have been systematically studied by comparing the correlated results with the corresponding noncorrelated ones. All the calculations have been done by using the variational Monte Carlo method.     

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.     

The use of explicitly correlated,partially antisymmetric wave functions in atomic and molecular calculations

Trial wave functions, written as the sum of a configuration interaction expansion and an explicitly correlated term which is not antisymmetric, are proposed for use in calculating the electronic properties of atoms and molecules. A variational principle, modified to allow the use for such partially antisymmetric wave functions, is developed. It is shown that the consequences of partial antisymmetry on calculated expectation values can be estimated. The method avoids difficult three-electron integrals which arise in other theories.     

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.     

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.     

One- and two-body densities of carbon isoelectronic series in their low-lying multiplet states from explicitly correlated wave functions

The (3)P ground state and both the (1)D and (1)S excited states arising from the low-lying 1s(2)2s(2)2p(2) configuration of the carbon isoelectronic series are studied starting from explicitly correlated multiconfigurational wave functions. One- and two-body densities in position space have been calculated and different one- and two-body expectation values have been obtained. The effects of electronic correlations have been systematically studied. All the calculations have been done by means of variational Monte Carlo.     

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.     

On the use of explicitly correlated functions in variational computations for small molecules

The explicitly correlated wave functions used in variational molecular calculations are reviewed. Different types of such functions are considered. The state of art and future perspectives are briefly discussed. © 1994 John Wiley & Sons, Inc.     

Accurate calculations of intermolecular interaction energies using explicitly correlated wave functions

Explicitly correlated second-order M?ller-Plesset (MP2-F12) calculations of intermolecular interaction energies for the S22 benchmark set of Jurecka, Sponer, Cerny, and Hobza (Chem. Phys. Phys. Chem. 2006, 8, 1985) are presented and compared with standard MP2 results. The MP2 complete basis set limits are estimated using basis set extrapolation and augmented quadruple-zeta and quintuple-zeta basis sets. Already with augmented double-zeta basis sets the MP2-F12 interaction energies are found to be closer to the complete basis set limits than standard MP2 calculations with augmented quintuple-zeta basis sets. Various possible approximations in the MP2-F12 method are systematically tested. Best results are obtained with localized orbitals and the diagonal MP2-F12/C(D) ansatz. Hybrid approximations, in which some contributions of the auxiliary basis set are neglected and which considerably reduce the computational cost, have a negligible effect on the interaction energies. Also the orbital-invariant fixed-amplitude approximation of Ten-no leads to only slightly less accurate results. Preliminary results for the neon and benzene dimers, obtained with the recently proposed CCSD(T)-F12a approximation, indicate that the CCSD(T) basis set limits can also be very closely approached using augmented triple-zeta basis sets.     

Parallel algorithm for calculating ro-vibrational states of diatomic molecules

In this article, we develop and implement an algorithm for calculating the rovbrational states of diatomic molecules optimized for multiple instructions multiple data computers of distributed memory. The method is based upon the p-version of the finite element method and has been implemented on an INTEL iPSC/2 machine with 16 processors. © 1994 by John Wiley & Sons, Inc.     

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.     

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.     

An efficient algorithm for calculating whole-profile functions in crystal structure solution from powder diffraction data

A method for calculation of the whole-profile powder diffraction R-factor R_wp, adapted specifically for use in direct-space structure solution, is shown to be faster by a factor of approximately 20 than standard methods for calculating R_wp.     

Optimized gaussian bond functions for CO

Optimum bond function parameters of ξ_1s = 1.12 and ξ_2p = 0.70 placed at 0.44 of the bond distance from the oxygen atom are reported for the CO molecule. Using these parameters, the total ground-state energy is lower than that obtained by Neumann and Moskowitz using two sets of 3d type polarization functions on each atomic center with exponents of 0.5 and 1.5. The one-electron properties, however, are slightly inferior to those calculated using the 3d functions.     

SF-[2]R12: a spin-adapted explicitly correlated method applicable to arbitrary electronic states

The [2](R12) method [M. Torheyden and E. F. Valeev, J. Chem. Phys. 131, 171103 (2009)] is an explicitly correlated perturbative correction that can greatly reduce the basis set error of an arbitrary electronic structure method for which the two-electron density matrix is available. Here we present a spin-adapted variant (denoted as SF-[2](R12)) that is formulated completely in terms of spin-free quantities. A spin-free cumulant decomposition and multi-reference generalized Brillouin condition are used to avoid three-particle reduced density matrix completely. The computational complexity of SF-[2](R12) is proportional to the sixth power of the system size and is comparable to the cost of the single-reference MP2-R12 method. The SF-[2](R12) method is shown to decrease greatly the basis set error of multi-configurational wave functions.     

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.     

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.     

Analytic first derivatives for explicitly correlated,multicenter, Gaussian geminals

Variational calculations utilizing the analytic gradient of explicitly correlated Gaussian molecular integrals are presented for the ground state of the hydrogen molecule. Preliminary results serve to motivate the need for general formulas for analytic first derivatives of molecular integrals involving multicenter, explicitly correlated Gaussian geminals with respect to Gaussian exponents and coordinates of the orbital centers. Explicit formulas for analytic first derivatives of Gaussian functions containing correlation factors of the form exp(-βr_ij²) are derived and discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 991–999, 1997     

An explicitly correlated Mukherjee's state specific coupled cluster method: development and pilot applications

This paper reports development of the explicitly correlated variant of Mukherjee's state specific multireference coupled cluster method (MkCC-F12). The current implementation is restricted to conventional single and double excitations and to pseudo-double excitations related to the Slater Type Geminal (STG) correlation factor using the SP ansatz. The performance of the MkCCSD-F12 was tested on calculations of singlet methylene, dissociation curve of the fluorine molecule, and the BeH(2) insertion pathway. As expected, the results of the newly developed method reconfirm the significantly faster convergence with respect to the basis set limit compared to the traditional expansion in Slater determinants. Results prove that treating the correlation factor separately for each reference is appropriate.