On the Beebe-Linderberg two-electron integral approximation

The Beebe-Linderberg two-electron integral approximation, which is generated by a Cholesky decomposition of the two-electron integral matrix ([μν|λσ]), is slightly modified. On the basis of test calculations, two key questions concerning this approximation are discussed: The numerical rank of the two-electron integral matrix and the relationship between the integral threshold and electronic properties. The numerical results presented in this work suggest that the modified Beebe-Linderberg approximation might be considered as an alternative to effective core potential methods.     

An interaction site model integral equation study of molecular fluids explicitly considering the molecular orientation

We implemented an interaction site model integral equation for rigid molecules based on a density-functional theory where the molecular orientation is explicitly considered. In this implementation of the integral equation, multiple integral of the degree of freedom of the molecular orientation is performed using efficient quadrature methods, so that the site-site pair correlation functions are evaluated exactly in the limit of low density. We apply this method to Cl(2), HCl, and H(2)O molecular fluids that have been investigated by several integral equation studies using various models. The site-site pair correlation functions obtained from the integral equation are in good agreement with the one from a simulation of these molecules. Rotational invariant coefficients, which characterize the microscopic structure of molecular fluids, are determined from the integral equation and the simulation in order to investigate the accuracy of the integral equation.     

Improving the convergence of closed and open path integral molecular dynamics via higher order Trotter factorization schemes

Higher order factorization schemes are developed for path integral molecular dynamics in order to improve the convergence of estimators for physical observables as a function of the Trotter number. The methods are based on the Takahashi-Imada and Susuki decompositions of the Boltzmann operator. The methods introduced improve the averages of the estimators by using the classical forces needed to carry out the dynamics to construct a posteriori weighting factors for standard path integral molecular dynamics. The new approaches are straightforward to implement in existing path integral codes and carry no significant overhead. The Suzuki higher order factorization was also used to improve the end-to-end distance estimator in open path integral molecular dynamics. The new schemes are tested in various model systems, including an ab initio path integral molecular dynamics calculation on the hydrogen molecule and a quantum water model. The proposed algorithms have potential utility for reducing the cost of path integral molecular dynamics calculations of bulk systems.     

Hybrid quantum/classical path integral approach for simulation of hydrogen transfer reactions in enzymes

A hybrid quantum/classical path integral Monte Carlo (QC-PIMC) method for calculating the quantum free energy barrier for hydrogen transfer reactions in condensed phases is presented. In this approach, the classical potential of mean force along a collective reaction coordinate is calculated using umbrella sampling techniques in conjunction with molecular dynamics trajectories propagated according to a mapping potential. The quantum contribution is determined for each configuration along the classical trajectory with path integral Monte Carlo calculations in which the beads move according to an effective mapping potential. This type of path integral calculation does not utilize the centroid constraint and can lead to more efficient sampling of the relevant region of conformational space than free-particle path integral sampling. The QC-PIMC method is computationally practical for large systems because the path integral sampling for the quantum nuclei is performed separately from the classical molecular dynamics sampling of the entire system. The utility of the QC-PIMC method is illustrated by an application to hydride transfer in the enzyme dihydrofolate reductase. A comparison of this method to the quantized classical path and grid-based methods for this system is presented.     

A New Method of Calculating Multicenter Matrix Elements in MO LCAO Theory Using an Exponential Type Basis

Using integral representation of the product of reduced Bessel functions (RBF) specified on different centers and a new generalized integral identity for RBF one can prove that the 4-center integral of Coulomb repulsion in an exponential type AO basis may be expressed as a three-dimensional integral over the volume of a cube with an edge 1. A new method of calculating the multicenter matrix elements of quantum chemistry in an exponential AO basis is suggested based on this representation. Numerical calculations of a number of multicenter integrals using this algorithm illustrate the efficiency of the method.     

A boundary element method for molecular electrostatics with electrolyte effects

A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.     

Integral enthalpy of mixing of the liquid ternary Au–Cu–Sn system

The integral enthalpy of mixing of the ternary Au–Cu–Sn has been determined with a Calvet type calorimeter at 6 different cross sections at 1273 K. The substitutional solution model of Redlich–Kister–Muggianu was used for a least square fit of the experimental data in order to get an analytical expression for the integral enthalpy of mixing. The ternary extrapolation models of Kohler, Muggianu and Toop were used to calculate the integral enthalpy of mixing and to compare measured and extrapolated values. Additional calculations of the integral enthalpy of mixing using the Chou model have been performed. With the calculated data, the iso-enthalpy lines have been determined using the Redlich–Kister–Muggianu model. A comparison of the data has been made.     

Kohn–Sham potentials by an inverse Kohn–Sham equation and accuracy assessment by virial theorem

In the recent study, the authors have proposed an integral equation for solving the inverse Kohn–Sham problem. In the present paper, the integral equation is numerically solved for one-dimensional model of a He atom and an H₂ molecule in the electronic ground states. For this purpose, we propose an iterative solution algorithm avoiding the inversion of the kernel of the integral equation. To quantify the numerical accuracy of the calculated exchange-correlation potentials, we evaluate the exchange and correlation energies based on the virial theorem as well as the reproduction of the exact ground-state electronic energy. The results demonstrate that the numerical solutions of our integral equation for the inverse Kohn–Sham problem are accurate enough in reproducing the Kohn–Sham potential and in satisfying the virial theorem.     

Complex Franck—Condon overlap integral in nonradiative decays

Considering the nuclear coordinate (Q) dependence of the electronic energy denominator appearing in the virbonic coupling matrix element, a complex Franck—Condon overlap integral which is needed in order to evaluate the nonradiative decay rate constant not only in the weak coupling but also in the strong coupling case is derived. The real part of the overlap integral plays an important role in the weak coupling case. The imaginary part is originated from the potential energy surface crossing regions and, consequently, contributes to the nonradiative decay rate constant in the strong coupling case. When the Q-dependence of the electronic energy denominator is neglected, the complex overlap integral leads to the results ontained by using the usual Herzberg—Teller expansion method. It is shown that the complex integral is expressed by the optical Franck—Condon overlap integral multiplied by a correction factor when the nonradiative decay from the vibrationless state is considered.     

Efficient calculation of vibrational magnetic dipole transition moments and rotational strengths

A priori prediction of vibrational magnetic dipole transition moments and vibrational strengths requires the calculation of the overlap integral of the derivatives of the electronic wavefunction with respect to nuclear displacement and an external magnetic field. The efficient calculation of this integral, using coupled Hartree-Fock theory, is described.     

Determination of the fiber-size distribution function in polydisperse dielectric fibrous materials

A simple method has been proposed for determining the average fiber length in unit volume of a polydisperse fibrous material as depending on fiber diameter. The method consists in the measurement of light attenuation as a function of the distance from an examined sample. The method entails comparison of the measured energy fluxes that reach a detector before and after scattering by an examined fibrous material sample and uses an integral relation that expresses the intensity of light transmitted through a random medium via the electric-field correlation function. Formulas have been found for the electric-field correlation function after the passage through a layer of a polydisperse fibrous material with random arrangement and orientation of fibers. The obtained correlation function enables one to derive an integral equation that expresses the logarithmic ratio of the energy fluxes reaching the radiation detector before and after the passage through a scattering medium via the fiber size-distribution function. Solution of this integral equation makes it possible to determine the fiber-size distribution function from the light attenuation measured as depending on the distance from the point of observation. Experiments have been carried out for several fibrous filters and relevant calculations have been presented. The results of the solution of the integral equation agree with the data obtained by other experimental methods and with visual processing of electron micrographs.     

Kinetic analysis of solid‐state reactions: Precision of the activation energy calculated by integral methods

The integral methods are extensively used for the kinetic analysis of solid‐state reactions. As the Arrhenius integral function [p(x)] does not have an exact analytical solution, different approximated equations have been proposed in the literature for performing the kinetic analysis of experimental integral data. Since the first approximation of Van Krevelen, a large number of equations have been proposed with the objective of increasing the precision in the determination of the Arrhenius integral, as checked from the standard deviation of the approximated function with regard to the real exact value of the integral. However, the main application of these equations is the determination of the kinetic parameters, in particular activation energies, and not the computation of the Arrhenius integral. A systematic analysis of the errors involved in the determination of the activation energy from these integral methods is still missing. A comparative study of the precision of the activation energy as a function of x and T computed from the different integral methods has been carried out. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 658–666, 2005     

Path integral Monte Carlo method for ab initio calculation

The Feynman path integral method is applied to the many-electron problem. We first give new closure relations in terms of ordinary complex and real numbers, which could be derived from an arbitrary complete set of state vectors. Then, in the path integral form, the partition function of the system and the ensemble average of energy are explicitly expressed in terms of these closure relations. It is impossible to evaluate the path integral by direct numerical integrations because of its huge amount of integration variables. Therefore, we develop an algorithm by the Monte Carlo method with constraints corresponding to the normalization condition of states to calculate the required integral. Finally, the ensemble average of energy for the hydrogen molecule is explicitly evaluated by the quantum Monte Carlo method and results are compared with the result obtained by the ordinary full configuration interaction (CI) method. © 1996 John Wiley & Sons, Inc.     

Evaluation of retention time for temperature-programmed gas chromatography by use of a closed-form integral

A closed-form integral for the Arrhenius exponential was obtained by using an approximate relation instead of the usual one in a linear temperature program. The closed-form integral can be applied to predict the retention time very well both for a packed column and for a capillary column, in constant the velocity mode.     

Calculation of quantum mechanical free energy at low temperature

The path integral method is used to calculate the quantum mechanical free energy at low temperature. Based on the variational harmonic reference system and implemented by the partial averaging technique, the path integral can be cast into the form of a classical configurational integral with the original potential replaced by an effective one. We compared this approach with other related methods and found that it gave better results than the others considered in this paper. Furthermore, the multidimensional implementation of this method is discussed. Received: 15 September 1997 / Accepted: 1 October 1997     

A path-integral Langevin equation treatment of low-temperature doped helium clusters

We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)] sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of He(N)-CO(2) clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], an open-source molecular simulation package.     

A path integral influence functional for excess electron in fluids: Density-functional formulation

In this paper, we propose a path integral influence functional from a solvent to determine a self-correlation function of a quantum particle in classical simple fluid. It is shown that the influence functional is related to a grand potential functional of the pure solvent under a three-dimensional external field arising from a classical isomorphic polymer, on which the quantum particle is mapped. The influence functional can be calculated from the self-correlation function, the solute-solvent and the solvent-solvent pair correlation function. The obtained equation of the self-correlation function is applied to an excess electron problem in fluid helium. The Fourier path-integral Monte Carlo method is employed to perform the path integral of the electron. The solute-solvent pair correlation function is estimated from a reference interaction site model integral equation. These results obtained form our proposed influence functional and from that proposed by Chandler, Singh, and Richardson are compared with those provided by a path integral Monte Carlo simulation with the explicit helium solvent.     

Attainable accuracy in the numerical integration of the Thomas—Fermi kinetic-energy functional using a modeled electron density

In the Hohenberg and Kohn formation of the density-functional theory of an electronic system, the basic variable is the electron (number) density. This quantity, however, is not known. For this reason, in an actual calculation, one has to resort to an approximate electron (number) density in order to evaluate the integral occurring in the Hohenberg and Kohn density functional framework. This poses the question: what is the accuracy beyond which one cannot penetrate in the numerical evaluation of the integrals? The present work attempts to provide an answer to this question by considering the Ne atom as an example and using the simplest energy-density functional, namely the Thomas-Fermi functional. In this functional, composed of three terms, there is only one term, the kinetic-energy functional, that has to be evaluated numerically. The evaluation of this integral is done by modeling the electron (number) density of the Ne atom and resorting to Simpson's compound rule. Following this, an error bound for the integral is established. This is the central result of this paper.     

Libcint: An efficient general integral library for Gaussian basis functions

An efficient integral library Libcint was designed to automatically implement general integrals for Gaussian‐type scalar and spinor basis functions. The library is able to evaluate arbitrary integral expressions on top of p, r and σ operators with one‐electron overlap and nuclear attraction, two‐electron Coulomb and Gaunt operators for segmented contracted and/or generated contracted basis in Cartesian, spherical or spinor form. Using a symbolic algebra tool, new integrals are derived and translated to C code programmatically. The generated integrals can be used in various types of molecular properties. To demonstrate the capability of the integral library, we computed the analytical gradients and NMR shielding constants at both nonrelativistic and 4‐component relativistic Hartree–Fock level in this work. Due to the use of kinetically balanced basis and gauge including atomic orbitals, the relativistic analytical gradients and shielding constants requires the integral library to handle the fifth‐order electron repulsion integral derivatives. The generality of the integral library is achieved without losing efficiency. On the modern multi‐CPU platform, Libcint can easily reach the overall throughput being many times of the I/O bandwidth. On a 20‐core node, we are able to achieve an average output 8.3 GB/s for C₆₀ molecule with cc‐pVTZ basis. © 2015 Wiley Periodicals, Inc.     

Two-center molecular repulsion integrals over slater functions

For the general two-electron two-center integral over Slater functions, use of the Neumann expansion for the electron-electron interaction term yields the standard auxiliary functions. These are expanded and integrated explicitly by two independent methods. The resulting simple analytic formula for the total integral is completely general, requiring only the Slater function quantum numbers and exponents and the internuclear separation. Hence all two-electron hydrid, coulomb, exchange, and one-center integrals are considered. The efficiency of calculation of this expression is compared with those of other methods, indicating an order of magnitude improvement in speed over recursion for the exchange integral.