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     

Perimetric scale-shape coordinates for triatomic molecules

Perimetric nuclear coordinates of a triatomic molecule treat all three nuclei equivalently and are not subject to the triangle conditions. Through an appropriate orthogonal transformation they can be separated into one scale coordinate, viz., the circumference, and two shape coordinates, which are determined by the angles. The parameter space of the shape coordinates is an equilateral triangle. The basic formulas are given and the relationship between points in coordinate space and molecular shapes are elucidated. Received: 10 January 1996 / Accepted: 2 January 1997     

Symmetry-adapted integrals over many-electron basis functions and operators

We consider integrals over symmetry-adapted basis functions that involve the coordinates of more than one electron. We focus on basis functions that can be written as products of one-electron functions and (say) a two-electron function. We show first that the two-electron parts of the basis functions can be absorbed into the operator, resulting in an integral over only one-electron basis functions, but a more complicated many-electron operator. We then prove a general formula for expressing such integrals in terms of symmetry-distinct integrals only. Received: 16 June 2000 / Accepted: 10 July 2000 / Published online: 19 January 2001     

Kohn-Sham potentials corresponding to Slater and Gaussian basis set densities

A new method based on linear response theory is proposed for the determination of the Kohn-Sham potential corresponding to a given electron density. The method is very precise and affords a comparison between Kohn-Sham potentials calculated from correlated reference densities expressed in Slater-(STO) and Gaussian-type orbitals (GTO). In the latter case the KS potential exhibits large oscillations that are not present in the exact potential. These oscillations are related to similar oscillations in the local error function δ _i (r)=(−ɛ _i )ϕ _i (r) when SCF orbitals (either Kohn-Sham or Hartree-Fock) are expressed in terms of Gaussian basis functions. Even when using very large Gaussian basis sets, the oscillations are such that extreme care has to be exercised in order to distinguish genuine characteristics of the KS potential, such as intershell peaks in atoms, from the spurious oscillations. For a density expressed in GTOs, the Laplacian of the density will exhibit similar spurious oscillations. A previously proposed iterative local updating method for generating the Kohn-Sham potential is evaluated by comparison with the present accurate scheme. For a density expressed in GTOs, it is found to yield a smooth “average” potential after a limited number of cycles. The oscillations that are peculiar to the GTO density are constructed in a slow process requiring very many cycles. Received: 24 February 1997 / Accepted: 18 June 1997     

Implementation of gradient formulas for correlated gaussians: He, ∞He,Ps2, 9Be,and ∞Be test results

New formulas in the basis of explicitly correlated Gaussian basis functions, derived in a previous article using powerful matrix calculus, are implemented and applied to find variational upper bounds for nonrelativistic ground states of ⁴He, ^∞He, Ps₂, ⁹Be, and ^∞Be. Analytic gradients of the energy are included to speed optimization of the exponential variational parameters. Five different nonlinear optimization subroutines (algorithms) are compared: TN, truncated Newton; DUMING, quasi-Newton; DUMIDH, modified Newton; DUMCGG, conjugate gradient; and POWELL, direction set (nongradient). The new analytic gradient formulas are found to significantly accelerate optimizations that require gradients. We found that the truncated Newton algorithm out-performs the other optimizers for the selected test cases. Computer timings and energy bounds are reported. © 1997 John Wiley & Sons, Inc.     

Hydrogen-type orbitals in terms of Gaussian functions

We express the hydrogen-type orbitals as integral transforms of Gaussian functions. The topographic features of the analytical expressions for the weight functions provide ways for the design of basis sets. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 143–150, 1997     

A systematic ab initio study of the water dimer in hierarchies of basis sets and correlation models

A systematic, high-level ab initio investigation of the water dimer has been performed. The oxygen-oxygen bond distance has been estimated to be around 2.90 ?, about 0.05 ? shorter than the experimentally estimated distance, challenging the accuracy of the latter. The interaction energy has been obtained at −5.0±0.1 kcal/mol, which compares favourably with the experimentally estimated value of −5.4±0.7 kcal/mol. The importance of employing basis sets that include diffuse functions in correlated calculations on hydrogen-bonded systems is confirmed. In correlated calculations on the water dimer and the hydrogen fluoride dimer, the counterpoise-corrected interaction energies converge considerably slower towards the basis set limit than do the uncorrected energies, provided that the correlation-consistent basis sets are augmented with diffuse functions. Received: 12 February 1997 / Accepted: 5 June 1997     

Basis-set dependence of nuclear spin-spin coupling constants

The convergence of NMR indirect spin-spin coupling constants with the extension of the basis set is analyzed, based on calculations carried out at the multiconfigurational self-consistent-field level for the HF and H₂O systems. For the dominant and difficult Fermi-contact contribution, the standard correlation-consistent basis sets of electronic-structure theory are not suitable, lacking flexibility in the core region. Improved but not satisfactory convergence of the couplings is observed when decontracting the s functions of the correlation-consistent cc-pVXZ basis sets for 2≤X≤6. Next, by systematically extending these basis sets with tight s functions, new sets are obtained that are sufficiently flexible for accurate calculations of indirect nuclear spin-spin couplings, without sacrificing the smooth convergence behavior of the correlation-consistent basis sets. Received: 22 September 1997 / Accepted: 30 December 1997     

Master formulas for two‐ and three‐center one‐electron integrals involving Cartesian GTO,STO, and BTO

As a first application of the shift operators method we derive master formulas for the two‐ and three‐center one‐electron integrals involving Gaussians, Slater, and Bessel basis functions. All these formulas have a common structure consisting in linear combinations of polynomials of differences of nuclear coordinates. Whereas the polynomials are independent of the type (GTO, BTO, or STO) of basis functions, the coefficients depend on both the class of integral (overlap, kinetic energy, nuclear attraction) and the type of basis functions. We present the general expression of polynomials and coefficients as well as the recurrence relations for both the polynomials and the whole integrals. Finally, we remark on the formal and computational advantages of this approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 83–93, 2000     

Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials

We present auxilliary basis sets for the atoms H to At – excluding the Lanthanides – optimized for an efficient treatment of molecular electronic Coulomb interactions. For atoms beyond Kr our approach is based on effective core potentials to describe core electrons. The approximate representation of the electron density in terms of the auxilliary basis has virtually no effect on computed structures and affects the energy by less than 10⁻⁴ a.u. per atom. Efficiency is demonstrated in applications for molecules with up to 300 atoms and 2500 basis functions. Received: 17 December 1996 / Accepted: 8 May 1997     

Contracted Gaussian-type basis functions revisited II. Atoms Na through Ar

We report five minimal-type contracted Gaussian-type function (CGTF) basis sets of the second-row atoms, Na – Ar, for molecular applications. Three of the present CGTF sets are revised versions of those given by Huzinaga and co-workers and the other two are newly developed for more accurate calculations. Practical utility and improved reliability of the present basis sets, augmented by polarization functions, are confirmed by test calculations on the P atom and P₂ molecule both at the self-consistent field (SCF) and configuration interaction (CI) levels. Received: 10 February 1997 / Accepted: 23 April 1997     

Extracting parameters for base-pair level models of DNA from molecular dynamics simulations

 A method is described to extract a complete set of sequence-dependent energy parameters for a rigid base-pair model of DNA from molecular dynamics (MD) simulations. The method is properly consistent with equilibrium statistical mechanics and leads to effective inertia parameters for the base-pair units as well as stacking and stiffness parameters for the base-pair junctions. We give explicit formulas that yield a complete set of base-pair model parameters in terms of equilibrium averages that can be estimated from a time series generated in an MD simulation. The expressions to be averaged depend strongly both on the choice of coordinates used to describe rigid-body orientations and on the choice of strain measures at each junction. Received: 12 July 2000 / Accepted: 5 January 2001 / Published online: 3 May 2001     

New recurrence relations for the rapid evaluation of electron repulsion integrals based on the accompanying coordinate expansion formula

We present an algorithm for the rapid computation of electron repulsion integrals (ERIs) over Gaussian basis functions based on the accompanying coordinate expansion (ACE) formula. The present algorithm uses equations termed angular momentum reduced expressions and introduces two types of recurrence relations to ACE formulas. Numerical efficiencies are assessed for (p pmid R:p p) and (sp spmid R:sp sp) ERIs by using the floating-point operation count. The algorithm is suitable for calculating ERIs for the same exponents but different angular momentum functions, such as L shells and derivatives of ERIs. The present algorithm is also capable of calculating ERIs with highly contracted Gaussian basis functions.     

Calculation of infrared absorption spectra using Gaussian variational wave packets

Formulas are developed for calculation of infrared absorption spectra with the help of Gaussian variational wave packets. The excitation is viewed as occurring to a linear combination of basis states corresponding to a Gaussian multiplied by Hermite polynomials of coordinates. Application of the formulas is examplified on HDO isolated in a D2O matrix. A simple approximation is proposed for the fundamental transitions.     

Four‐center integrals for Gaussian and exponential functions

The shift operator technique is used for deriving, in a unified manner, the master formulas for the four‐center repulsion integrals involving Gaussian (GTO), Slater (STO), and Bessel (BTO) basis functions. Moreover, for the two classes of exponential‐type functions (ETO), i.e., STO and BTO, we give the expressions corresponding to both the Gauss and Fourier transforms. From the comparison of the master formulas of GTO and ETO, we conclude that ETO can perform more efficiently than GTO, and we remark the points where the effort must be focused to carry out this possibility. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 16–28, 2001     

Analytical formulas for the magnetic field produced by a spin or a paramagnetic current density in the case of Gaussian basis functions

We present a derivation of simple formulas for the evaluation at any point of space of the magnetic field produced by a spin or a paramagnetic orbital current when Cartesian Gaussian basis functions are used, as is often the case in quantum chemistry. These formulas can be useful to plot the magnetic field vector density obtained from ab initio calculations or from a density operator fitted on experimental data. The magnetic field density is the observable probed in polarized neutron diffraction (PND) experiment, for it is, in fact, with this quantity that the neutron spins interact and not with the spin or magnetization density. The formulas make extensive use of the confluent hypergeometric function. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 11–15, 2001     

Digital erosion in the evaluation of molecular integrals

 In the course of checking our work on the symbolic calculation of molecular integrals over Slater orbitals, we obtained some results in substantial disagreement with two recent articles that describe numerical schemes. We believe that these schemes suffer from digital erosion, possibly because recurrence formulas were used outside their regions of stability. Our results were obtained using the ζ-function method, which expands the orbital on one atom onto the other, and integrates in polar coordinates. They were checked using elliptic coordinates. Both sets of calculations were performed symbolically. We summarize these calculations and discuss the impact of symbolic calculation on the accuracy of molecular computations. Received: 31 August 2001 / Accepted: 29 October 2001 / Published online: 8 April 2002     

Maxwell–Cartesian spherical harmonics in multipole potentials and atomic orbitals

 The nature of the Maxwell–Cartesian spherical harmonics S ⁽ⁿ⁾ _K and their relation to tesseral harmonics Y _nm is examined with the help of “tricorn arrays” that display the components of a totally symmetric Cartesian tensor of any rank in a systematic way. The arrays show the symmetries of the Maxwell–Cartesian harmonic tensors with respect to permutation of axes, the traceless properties of the tensors, the linearly independent subsets, the nonorthogonal subsets, and the subsets whose linear combinations produce the tesseral harmonics. The two families of harmonics are related by their connection with the gradients of 1/r, and explicit formulas for the transformation coefficients are derived. The rotational transformation of S ⁽ⁿ⁾ _K functions is described by a relatively simple Cartesian tensor method. The utility of the Maxwell–Cartesian harmonics in the theory of multipole potentials, where these functions originated in the work of Maxwell, is illustrated with some newer applications which employ a detracer exchange theorem and make use of the partial linear independence of the functions. The properties of atomic orbitals whose angular part is described by Maxwell–Cartesian harmonics are explored, including their angular momenta, adherence to an Uns?ld-type spherical symmetry relation, and potential for eliminating an angular momentum “contamination” problem in Cartesian Gaussian basis sets. Received: 9 July 2001 / Accepted: 7 September 2001 / Published online: 19 December 2001     

Generator coordinate Hartree–Fock method applied to the choice of a contracted Gaussian basis for the second-row atoms

The generator coordinate Hartree–Fock (GCHF) method is employed as a criterion for the selection of a 18s12p Gaussian basis for the atoms Na–Ar. The role of the weight functions in the assessment of the numerical integration range of the GCHF equations is shown. The extended basis is then contracted to (10s6p) by a standard procedure and in combination with the previously contracted (7s5p) Gaussian basis for the atoms Li–Ne is enriched with polarization functions. This basis is tested for AlF, SiO, PN, BCl, and P₂. The properties of interest were HF total energies, MP2 dipolar moments, bond distances, and dissociation energies. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 927–934, 1997     

Treatment of multibody interactions in molecular simulations of systems with general bond networks

One of the most formidable difficulties in the computer programming of molecular simulations is the sometimes complicated bookkeeping required for keeping track of internal coordinates and their derivatives. A completely general method for keeping track of stretch (two-body), bend (three-body), and torsion, wag, and other four-body interactions for ANY bond network is presented. Computer code using this method for calculating internal coordinates and their derivatives can be used for completely different types of bond networks, no matter how complex, with little or no modification. The method is designed to incorporate recent improved formulas for calculating internal coordinates and their derivatives to ensure the most optimal calculation sequence. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1513–1522, 1997