New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

An application of double exponential formula to radial quadrature grid in density functional calculation

We report an application of the double exponential formula to the numerical integration of the radial electron distribution function for atomic and diatomic molecular systems with a quadrature grid. Three types of mapping transformation in the double exponential formula are introduced into the radial quadrature scheme to generate new radial grids. The double exponential grids are examined for the electron-counting integrals of He, Ne, Ar, and Kr atoms which include occupied orbitals up to the 4p shell. The performance of radial grid is compared for the double exponential formula and the formulas proposed in earlier studies. We mainly focus our attention on the behavior of accuracy by the quadrature estimation for each radial grid with varying the mapping parameter and the number of grid points. The convergence behavior of the radial grids with high accuracy for atomic system are also examined for the electron-counting integrals of LiH, NaH, KH, Li₂, Na₂, K₂, HF, HCl, HBr, F₂, Cl₂, Br₂, LiF, NaCl, KBr, [ScH]⁺, [MnH]⁺, and [CuH]⁺ molecules. The results reveal that fast convergence of the integrated values to the exact value is achieved by applying the double exponential formula. It is demonstrated that the double exponential grids show similar or higher accuracies than the other grids particularly for the Kr atom, Br₂ molecule, alkali metal hydrides, alkali metal halogenides, and transition metal hydride cations, suggesting that the double exponential transformations have potential ability to improve the reliability and efficiency of the numerical integration for energy functionals.     

A reexamination of virial coefficients of the Lennard-Jones fluid

The fourth-order virial coefficients have been calculated exactly to five decimal places for pure fluids of the Lennard-Jones potential at many points in the phase diagram. The calculations were performed through direct evaluation of the integrals, or diagrams, which make up the density expansion of the radial distribution function: included were the standard fast Fourier transform method of evaluating the simply connected diagrams and the evaluation of the bridge diagram for the fourth order in density by expansion in Legendre polynomials. The polynomial-order dependence of the bridge diagram calculation and the range dependence of the simply connected diagrams of the fourth order are found to have more significance than was thought from previous studies, especially in the low-temperature range. This result was confirmed by direct evaluation of the diagrams which construct the virial coefficients, as given by Rowlinson, Barker, and coworkers. This calculation confirmed that numerical convergence has not been achieved at the precision levels previously reported in the literature. These differences, though minor at higher temperatures, can be seen to be more significant at the lower temperature ranges. Received: 31 July 2000 / Accepted: 18 September 2000 / Published online: 21 December 2000     

Half-numerical evaluation of pseudopotential integrals

A half-numeric algorithm for the evaluation of effective core potential integrals over Cartesian Gaussian functions is described. Local and semilocal integrals are separated into two-dimensional angular and one-dimensional radial integrals. The angular integrals are evaluated analytically using a general approach that has no limitation for the l-quantum number. The radial integrals are calculated by an adaptive one-dimensional numerical quadrature. For the semilocal radial part a pretabulation scheme is used. This pretabulation simplifies the handling of radial integrals, makes their calculation much faster, and allows their easy reuse for different integrals within a given shell combination. The implementation of this new algorithm is described and its performance is analyzed.     

Density functional calculations of the pseudorotational flexibility of tetrahydrofuran

The flexibility of the five-membered ring in tetrahydrofuran was investigated using quantum mechanical methods involving density functional, Hartree-Fock, and many-body perturbation theory (MP2, MP4) calculations. We found that motion along the pseudorotational path of tetrahydrofuran is nearly free. The 0.1 kcal/mol energy barrier for pseudorotation, calculated at the highest MP4(SDQ)/6-311++G(2d,p)//MP2/6-311++G(2d,p) level of theory, agrees well with the experimental value of 0.16±0.03 kcal/mol. Similar results were obtained with the S-VWN, B3-LYP and B-LYP density functional calculations using the 6-31G(d) set of atomic orbitals. Also the density functional dipole moments and geometries were in good agreement with both the MP2 and experimental benchmarks. However, all density functional methods that utilized the default integration grid in the Gaussian 94 program were found to provide false stationary points of the C ₁ symmetry near the pseudorotational angle of 100°. These stationary points disappeared when a denser spherical-product grid was used. Overall, the hybrid B3-LYP functional was found to be the most promising quantum mechanical method for the modeling of biomolecules containing the furanose ring. Received: 17 June 1997 / Accepted: 20 November 1997     

The computation of radial integrals with nonclassical quadratures for quantum chemistry and other applications

The computation of radial integrals on the semi-infinite axis is an important computationally intensive feature of quantum chemistry computer codes with additional applications to physics and engineering. There have been numerous algorithms proposed to evaluate these integrals efficiently. Many of these approaches involve the transformation of the semi-infinite axis, \(r \in [0,\infty )\), to the finite interval, \(x \in [-1,1]\), and the use of the Gauss–Legendre or Gauss–Chebyshev quadratures to evaluate the integrals. These mappings redistribute the quadrature points in many different ways. The approach in this paper is to compute the radial integrals with the Gauss–Maxwell nonclassical quadrature defined by the weight function \(w(r)=r^2e^{-r^2}\) appropriate for the semi-infinite interval and to also use scaling of the quadrature points. We carry out numerical experiments with simple model radial integrands and compare with the results of previous workers.     

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     

Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm

Orthogonal polynomials of a discrete variable have been widely investigated as fundamental tools of numerical analysis. This work aims to propose the extension of their use to quantum mechanical problems. By exploiting both their connection with coupling and recoupling coefficients of angular momentum theory and their asymptotic relationships (semiclassical limit) with spherical and hyperspherical harmonics, a discretization procedure, the hyperquantization algorithm, has been developed and applied to the study of anisotropic interactions and of reactive scattering. One of the most appealing features of this method turns out to be a drastic reduction of memory requirements and computing time for extensive dynamical calculations. Examples of the application of this technique to stereodirected dynamics via an exact representation for the S matrix as well as to the characterization of molecular beam polarization are also illustrated. Received: 17 September 1999 / Accepted: 3 February 2000 / Published online: 5 June 2000     

Calculation of three-center electric and magnetic multipole moment integrals using translation formulas for Slater-type orbitals

 By the use of translation formulas for the expansion of Slater-type orbitals (STOs) in terms of STOs at a new origin, three-center electric and magnetic multipole moment integrals are expressed in terms of two-center multipole moment integrals for the evaluation of which closed analytical formulas are used. The convergence of the series is tested by calculating concrete cases. Computer results with an accuracy of 10⁻⁷ are obtained for 2^ν– pole electric and magnetic multipole moment integrals for 1≤ν≤5 and for arbitrary values of screening constants of atomic orbitals and internuclear distances. Received: 28 October 1999 / Accepted: 15 February 2000 / Published online: 5 June 2000     

Grid-free DFT implementation of local and gradient-corrected XC functionals

Following an approach to density functional theory calculations based on the matrix representation of operators, we implemented a scheme as an alternative to traditional grid-based methods. These techniques allow integrals over exchange-correlation operators to be evaluated through matrix manipulations. Both local and gradient-corrected functionals can be treated in a similar way. After deriving all the required expressions, selected examples with various functionals are given. Received: 7 March 1998 / Accepted: 21 May 1998 / Published on line: 6 August 1998     

Numerical integration of atomic electron density with double exponential formula for density functional calculation

In the preceding study, we reported an application of the double exponential formula to the radial quadrature grid for numerical integration of the radial electron distribution function. Three-type new radial grids with the double exponential transformation were introduced. The performance of radial grids was compared between the double exponential grids and the grids proposed in earlier studies by applying to the electron-counting integrals of noble gas atoms and diatomic molecules including alkali metals, halogens, and transition metals. It was confirmed that the change in accuracy of the quadrature approximation depending on atomic or molecular species is not significant for the double exponential integration schemes rather than the other integration schemes. In the present study, we further investigate the accuracy of the double exponential formula for the electron-counting integrals of all the atoms from H to Kr in the periodic table to elucidate the stable performance of the double exponential radial grids. The electron densities of the atoms are calculated with the Gauss-type orbital basis functions at the B3LYP level. The quadrature accuracy and convergence behavior of numerical integration are compared among the double exponential formula and the formulas proposed by Treutler et al. and by Mura et al. The results reveal that the double exponential radial grids remarkably improve the convergence rate toward high accuracy compared with the previous radial grids, particularly for heavy elements in the 4th period, without fine tuning of the radial grids for each atom.     

Linear scaling computation of the Fock matrix. III. Formation of the exchange matrix with permutational symmetry

 A direct comparison is made between two recently proposed methods for linear scaling computation of the Hartree–Fock exchange matrix to investigate the importance of exploiting two-electron integral permutational symmetry. Calculations on three-dimensional water clusters and graphitic sheets with different basis sets and levels of accuracy are presented to identify specific cases where permutational symmetry may or may not be useful. We conclude that a reduction in integrals via permutational symmetry does not necessarily translate into a reduction in computation times. For large insulating systems and weakly contracted basis sets the advantage of permutational symmetry is found to be negligible, while for noninsulating systems and highly contracted basis sets a fourfold speedup is approached. Received: 8 October 1999 / Accepted: 3 January 2000 / Published online: 21 June 2000     

Second-order Møller–Plesset perturbation theory with terms linear in the interelectronic coordinates and exact evaluation of three-electron integrals

 The second-order correlation energy of M?ller–Plesset perturbation theory is computed for the neon atom using a wave function that depends explicitly on the interelectronic coordinates (MP2-R12). The resolution-of-identity (RI) approximation, which is invoked in the standard formulation of MP2-R12 theory, is largely avoided by rigorously computing the necessary three-electron integrals. The basis-set limit for the second-order correlation energy is reached to within 0.1 mE _h. A comparison with the conventional RI-based MP2-R12 method shows that only three-electron integrals over s and p orbitals need to be computed exactly, indicating that the RI approximation can be safely used for integrals involving orbitals of higher angular momentum. Received: 9 May 2001 / Accepted: 31 October 2001 / Published online: 9 January 2002     

A fast algorithm to compute atomic charges based on the topology of the electron density

This work proposes a novel algorithm to compute atomic charges as defined by the theory of “atoms in molecules” (AIM). Using the divergence theorem it is possible to express the 3D volume integral over an atomic basin purely in terms of 2D surface integrals. Hence, it can be proven that an atomic charge is equal to the flux of the electric field of the whole molecule through the atom's complete boundary. This boundary consists of the interatomic surfaces and the so-called outeratomic surface, which is the open side of the atom. When fine-tuned the algorithm can generate atomic charges in the order of minutes without introducing any approximations. Moreover, the problem of the geometrical cusp occurring in atomic basins and that of multiple intersections is also eliminated. The computational overhead of computing the electric field (which is analytical) is compensated by the gain in computing time by eliminating one dimension of quadrature. The proposed algorithm opens an avenue to invalidate the oft-quoted drawback that AIM charges are computationally expensive. We explain the details of the implementation in MORPHY01 and illustrate the novel algorithm with a few examples. Received: 1 June 2000 / Accepted: 4 October 2000 / Published online: 23 January 2001     

Formulas and numerical table for the radial part of overlap integrals with the same screening parameters of Slater-type orbitals

 The numerical properties of the radial part of overlap integrals with the same screening parameters in the form of polynomials in p = ξR over Slater-type orbitals have been studied and obtained by using three different methods. For that purpose, the characteristics of auxiliary functions were used first, then Fourier transform convolution theorem, and recurrence relations for the basic coefficients of A ^s _{n l λ, n ′ l ′λ} were used. The calculations of the radial part of overlap integrals with the same screening parameters were made in the range 1 ≤ n ≤ 75, 1 ≤ n′ ≤ 75, and 10⁻⁶ ≤ p. Received: 18 January 2001 / Accepted: 5 April 2001 / Published online: 27 June 2001     

Excited-State Electronic Structure with Configuration Interaction Singles and Tamm-Dancoff Time-Dependent Density Functional Theory on Graphical Processing Units

Excited-state calculations are implemented in a development version of the GPU-based TeraChem software package using the configuration interaction singles (CIS) and adiabatic linear response Tamm-Dancoff time-dependent density functional theory (TDA-TDDFT) methods. The speedup of the CIS and TDDFT methods using GPU-based electron repulsion integrals and density functional quadrature integration allows full ab initio excited-state calculations on molecules of unprecedented size. CIS/6-31G and TD-BLYP/6-31G benchmark timings are presented for a range of systems, including four generations of oligothiophene dendrimers, photoactive yellow protein (PYP), and the PYP chromophore solvated with 900 quantum mechanical water molecules. The effects of double and single precision integration are discussed, and mixed precision GPU integration is shown to give extremely good numerical accuracy for both CIS and TDDFT excitation energies (excitation energies within 0.0005 eV of extended double precision CPU results).     

Applications of fourier transforms in molecular orbital theory. Calculation of optical properties and tables of two-center one-electron integrals

Fourier transform methods initiated by Geller and Harris are applied to the calculation of optical properties of molecules. Tables of one-electron two-center integrals needed for the accurate computation of molecular absorption and optical activity are calculated by the Fourier transform method. A general theorem is derived which allows the angular part of the integrals to be treated by means of projection operators. The radial parts of the integrals are treated by the methods of Harris. The results are obtained in a simple closed form which avoids the usual transformation to local coordinates. The two-center integrals evaluated include matrix elements of the momentum operator, the dipole moment operator, the tensor operator , the quadrupole moment operator, and the angular momentum operator. These are evaluated between 1s, 2s, and 2p Slater-type atomic orbitals located on different atoms. The results are expressed as functions of the Slater exponents and of the relative coordinates of the two atoms.     

Symmetrical “nonproduct” quadrature rules for a fast calculation of multicenter integrals

A complete method of numerical integration, designed especially for density functional theory, is presented. We first refer to already known methods and then present a new development of the angular integration. A set of symmetrical quadrature rules which is equivalent to the popular Lebedev scheme has been developed for any arbitrary point group. In case of octahedral symmetry our method turns out to be exactly identical to Lebedev's. These formulas integrate exactly spherical harmonics of the highest possible order with, most probably, the least possible number of grid points. Nevertheless a rigorous mathematical proof of this statement has not yet been found. Examples of quadrature rules for noncubic point groups (not covered by Lebedev's grid), e.g., the icosahedral, pentagonal, or hexagonal ones are given. The application of this method to the resolution of the Poisson's equation is also presented. © 1997 John Wiley & Sons, Inc.     

The occurrence of electron transfer in aromatic nitration: dynamical aspects

Electron transfer (ET) from toluene to the nitronium ion in the region of van der Waals intermolecular distances has been investigated by a quantum dynamical analysis performed on potential-energy surfaces computed at the ab initio multireference configuration interaction level. The results show that ET is very fast, occurring on a timescale of a few picoseconds. This has important implications for the mechanism of aromatic nitration: the ET path can compete efficiently with the direct attack of the nitronium ion to the aromatic substrate to yield the Wheland intermediate and that could explain some unsettled points in the mechanism of aromatic nitration. Received: 16 September 1999 / Accepted: 3 February 2000 / Published online: 12 May 2000