Universal method for computation of electrostatic potentials

A computational approach to determine electrostatic interaction and gravitational potentials by performing direct numerical integration is presented. The potential is expanded using finite-element functions of arbitrary order. The method does not involve any solutions of systems of linear equations. The potential is instead obtained as a sum of differential contributions. Thus, no boundary conditions for the potential are needed. It is computationally efficient and well suited for parallel computers, since the innermost loops constitute matrix multiplications and the outer ones can be used as parallel indices. Without using prescreening or other computational tricks to speed up the calculation, the algorithm scales as N4/3 where N denotes the grid size.     

Gaussian and finite-element Coulomb method for the fast evaluation of Coulomb integrals

The authors propose a new linear-scaling method for the fast evaluation of Coulomb integrals with Gaussian basis functions called the Gaussian and finite-element Coulomb (GFC) method. In this method, the Coulomb potential is expanded in a basis of mixed Gaussian and finite-element auxiliary functions that express the core and smooth Coulomb potentials, respectively. Coulomb integrals can be evaluated by three-center one-electron overlap integrals among two Gaussian basis functions and one mixed auxiliary function. Thus, the computational cost and scaling for large molecules are drastically reduced. Several applications to molecular systems show that the GFC method is more efficient than the analytical integration approach that requires four-center two-electron repulsion integrals. The GFC method realizes a near linear scaling for both one-dimensional alanine alpha-helix chains and three-dimensional diamond pieces.     

Efficient solution of Poisson's equation with free boundary conditions

Interpolating scaling functions give a faithful representation of a localized charge distribution by its values on a grid. For such charge distributions, using a fast Fourier method, we obtain highly accurate electrostatic potentials for free boundary conditions at the cost of O(N log N) operations, where N is the number of grid points. Thus, with our approach, free boundary conditions are treated as efficiently as the periodic conditions via plane wave methods.     

A new method to determine electrostatic potential around a macromolecule in solution from molecular wave functions

The three-dimensional reference interaction site model integral equation theory (3D-RISM) combined with the ab initio molecular orbital method (3D-RISM-SCF) is applied to a solvated macromolecular system. The solvation structure around a solute molecule is obtained from the 3D-RISM integral equation under the electrostatic potential of the solute molecule, calculated by the ab initio molecular orbital theory. The electrostatic potential should be calculated on each grid point in the three-dimensional real space. Therefore, the calculation of the electrostatic potential is the most time consuming part in this method. In this article, we propose a new procedure to save the computational cost for calculating the electrostatic potential and the solvated fock matrix. The strategy of this procedure is to evaluate the electrostatic potential and the solvated fock matrix in different ways, depending on the distance between solute and solvent. Inside the repulsive cores of solute atoms, it is possible to avoid the calculation of electrostatic potential and solvated Fock matrix by assuming the potential to be infinity. In the region sufficiently far from solute, they are evaluated classically by putting the effective point charge on each atom. In the intermediate region, the electrostatic potential is evaluated directly by integrating the molecular orbitals of the solute molecule. The electronic structure and the energy gradient of Methionine-Enkephalin and solvation structure are estimated by using this procedure in aqueous solution, and are compared with the results from other procedures. The results are compared also with those from the continuum model.     

Linear scaling calculation of maximally localized Wannier functions with atomic basis set

We have developed a linear scaling algorithm for calculating maximally localized Wannier functions (MLWFs) using atomic orbital basis. An O(N) ground state calculation is carried out to get the density matrix (DM). Through a projection of the DM onto atomic orbitals and a subsequent O(N) orthogonalization, we obtain initial orthogonal localized orbitals. These orbitals can be maximally localized in linear scaling by simple Jacobi sweeps. Our O(N) method is validated by applying it to water molecule and wurtzite ZnO. The linear scaling behavior of the new method is demonstrated by computing the MLWFs of boron nitride nanotubes.     

Efficient and accurate three-dimensional Poisson solver for surface problems

We present a method that gives highly accurate electrostatic potentials for systems where we have periodic boundary conditions in two spatial directions but free boundary conditions in the third direction. These boundary conditions are needed for all kinds of surface problems. Our method has an O(N log N) computational cost, where N is the number of grid points, with a very small prefactor. This Poisson solver is primarily intended for real space methods where the charge density and the potential are given on a uniform grid.     

A divide and conquer real-space approach for all-electron molecular electrostatic potentials and interaction energies

A computational scheme to perform accurate numerical calculations of electrostatic potentials and interaction energies for molecular systems has been developed and implemented. Molecular electron and energy densities are divided into overlapping atom-centered atomic contributions and a three-dimensional molecular remainder. The steep nuclear cusps are included in the atom-centered functions making the three-dimensional remainder smooth enough to be accurately represented with a tractable amount of grid points. The one-dimensional radial functions of the atom-centered contributions as well as the three-dimensional remainder are expanded using finite element functions. The electrostatic potential is calculated by integrating the Coulomb potential for each separate density contribution, using our tensorial finite element method for the three-dimensional remainder. We also provide algorithms to compute accurate electron-electron and electron-nuclear interactions numerically using the proposed partitioning. The methods have been tested on all-electron densities of 18 reasonable large molecules containing elements up to Zn. The accuracy of the calculated Coulomb interaction energies is in the range of 10(-3) to 10(-6) E(h) when using an equidistant grid with a step length of 0.05 a(0).     

Efficient implementation of three-dimensional reference interaction site model self-consistent-field method: application to solvatochromic shift calculations

The authors present an implementation of the three-dimensional reference interaction site model self-consistent-field (3D-RISM-SCF) method. First, they introduce a robust and efficient algorithm for solving the 3D-RISM equation. The algorithm is a hybrid of the Newton-Raphson and Picard methods. The Jacobian matrix is analytically expressed in a computationally useful form. Second, they discuss the solute-solvent electrostatic interaction. For the solute to solvent route, the electrostatic potential (ESP) map on a 3D grid is constructed directly from the electron density. The charge fitting procedure is not required to determine the ESP. For the solvent to solute route, the ESP acting on the solute molecule is derived from the solvent charge distribution obtained by solving the 3D-RISM equation. Matrix elements of the solute-solvent interaction are evaluated by the direct numerical integration. A remarkable reduction in the computational time is observed in both routes. Finally, the authors implement the first derivatives of the free energy with respect to the solute nuclear coordinates. They apply the present method to "solute" water and formaldehyde in aqueous solvent using the simple point charge model, and the results are compared with those from other methods: the six-dimensional molecular Ornstein-Zernike SCF, the one-dimensional site-site RISM-SCF, and the polarizable continuum model. The authors also calculate the solvatochromic shifts of acetone, benzonitrile, and nitrobenzene using the present method and compare them with the experimental and other theoretical results.     

Superlinear scaling in master-slave quantum chemical calculations using in-core storage of two-electron integrals

We describe the implementation of a parallel, in-core, integral-direct Hartree-Fock and density functional theory code for the efficient calculation of Hartree-Fock wave functions and density functional theory. The algorithm is based on a parallel master-slave algorithm, and the two-electron integrals calculated by a slave are stored in available local memory. To ensure the greatest computational savings, the master node keeps track of all integral batches stored on the different slaves. The code can reuse undifferentiated two-electron integrals both in the wave function optimization and in the evaluation of second-, third-, and fourth-order molecular properties. Superlinear scaling is achieved in a series of test examples, with speedups of up to 55 achieved for calculations run on medium-sized molecules on 16 processors with respect to the time used on a single processor.     

A particle-particle, particle-density algorithm for the calculation of electrostatic interactions of particles with slablike geometry

We present a fast and accurate method to calculate the electrostatic energy and forces of interacting particles with the boundary conditions appropriate to surfaces, i.e., periodic in the two directions parallel to the surface and free in the perpendicular direction. In the spirit of the Ewald method, the problem is divided into a short range and a long range part. The charge density responsible for the long range part is represented by plane waves in the periodic directions and by finite elements in the nonperiodic direction. Our method has computational complexity of O(N(g) log(N(g))) with a very small prefactor, where N(g) is the number of grid points.     

Similarity searching in files of three-dimensional chemical structures: Representation and searching of molecular electrostatic potentials using field-graphs

This paper reports a method for the identification of those molecules in a database of rigid 3D structures with molecular electrostatic potential (MEP) grids that are most similar to that of a user-defined target molecule. The most important features of an MEP grid are encoded in field-graphs, and a target molecule is matched against a database molecule by a comparison of the corresponding field-graphs. The matching is effected using a maximal common subgraph isomorphism algorithm, which provides an alignment of the target molecule's field- graph with those of each of the database molecules in turn. These alignments are used in the second stage of the search algorithm to calculate the intermolecular MEP similarities. Several different ways of generating field-graphs are evaluated, in terms of the effectiveness of the resulting similarity measures and of the associated computational costs. The most appropriate procedure has been implemented in an operational system that searches a corporate database, containing ca. 173,000 3D structures.     

Automated docking with grid-based energy evaluation

The ability to generate feasible binding orientations of a small molecule within a site of known structure is important for ligand design. We present a method that combines a rapid, geometric docking algorithm with the evaluation of molecular mechanics interaction energies. The computational costs of evaluation are minimal because we precalculate the receptor-dependent terms in the potential function at points on a three-dimensional grid. In four test cases where the components of crystallographically determined complexes are redocked, the “force field” score correctly identifies the family of orientations closest to the experimental binding geometry. Scoring functions that consider only steric factors or only electrostatic factors are less successful. The force field function will play an important role in our efforts to search databases for potential lead compounds.     

Efficient correlation-corrected vibrational self-consistent field computation of OH-stretch frequencies using a low-scaling algorithm

The authors present a new computational scheme to perform accurate and fast direct correlation-corrected vibrational self-consistent field (CC-VSCF) computations for a selected number of vibrational modes, which is aimed at predicting a few vibrations in large molecular systems. The method is based on a systematic selection of vibrational mode-mode coupling terms, leading to the direct ab initio construction of a sparse potential energy surface. The computational scaling of the CC-VSCF computation on the generated surface is then further reduced by using a screening procedure for the correlation-correction contributions. The proposed method is applied to the computation of the OH-stretch frequency of five aliphatic alcohols. The authors investigate the influence of different pseudopotential and all-electron basis sets on the quality of the correlated potential energy surfaces computed and on the OH-stretch frequencies calculated for each surface. With the help of these test systems, the authors show that their method offers a computational scaling that is two orders of magnitude lower than a standard CC-VSCF method and that it is of equal accuracy.     

Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Mo̸ller-Plesset perturbation theory

Many approximations have been developed to help deal with the O(N(4)) growth of the electron repulsion integral (ERI) tensor, where N is the number of one-electron basis functions used to represent the electronic wavefunction. Of these, the density fitting (DF) approximation is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to molecular size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decomposition to obtain a low-rank approximation to density fitting (tensor hypercontraction density fitting or THC-DF). This new approximation reduces the 4th-order ERI tensor to a product of five matrices, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling reduction for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N(4)) operations. This should be compared to the usual scaling behavior of O(N(5)) and O(N(6)) for MP2 and MP3, respectively. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and configuration interaction, promising significant gains in computational efficiency and storage reduction.     

Wavelet formulation of the polarizable continuum model

The first implementation of a wavelet discretization of the Integral Equation Formalism (IEF) for the Polarizable Continuum Model (PCM) is presented here. The method is based on the application of a general purpose wavelet solver on the cavity boundary to solve the integral equations of the IEF‐PCM problem. Wavelet methods provide attractive properties for the solution of the electrostatic problem at the cavity boundary: the system matrix is highly sparse and iterative solution schemes can be applied efficiently; the accuracy of the solver can be increased systematically and arbitrarily; for a given system, discretization error accuracy is achieved at a computational expense that scales linearly with the number of unknowns. The scaling of the computational time with the number of atoms N is formally quadratic but a N^1.5 scaling has been observed in practice. The current bottleneck is the evaluation of the potential integrals at the cavity boundary which scales linearly with the system size. To reduce this overhead, interpolation of the potential integrals on the cavity surface has been successfully used. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Quantitative comparison of molecular electrostatic potentials for structure-activity studies

An approach for representing, efficiently calculating and comparing discrete three-dimensional molecular electrostatic potentials using a quantitative similarity index (MEP-SI) based on a Carbo-type formalism is presented. A radial-type (MACRA) grid representation is described that provides more efficient storage of MEP information than a cubic grid of similar range, appropriate emphasis, and a convenient means for restricting the comparison of MEP functions to a local molecule region. The MACRA based MEP-SI formalism was used to evaluate the suitability of a variety of approximate methods for efficiently calculating the MEP for use in MEP-SI comparison of dissimilar molecules. The Mulliken charge method was found inadequate, while the method of potential-derived charges (PDCs), with additional charges for lone electron pairs included on sulfur, provided an efficient and sufficiently accurate representation of the MEP for this purpose. Convergence of the MEP-SI with respect to MACRA grid extent and mesh size was demonstrated; the effect of MEP error and different grid point emphasis in the MACRA versus the cubic grid results was investigated, and MEP-SI results were compared for different forms of the SI equation. The methodology proposed in this study provides an efficient and practical means for comparing MEP functions for two molecules and gives discriminating results for a sample series of molecular analogues consistent with expectations.     

Auxiliary functions for molecular integrals with Slater‐type orbitals. I. Translation methods

Many types of molecular integrals involving Slater functions can be expressed, with the ζ‐function method in terms of sets of one‐dimensional auxiliary integrals whose integrands contain two‐range functions. After reviewing the properties of these functions (including recurrence relations, derivatives, integral representations, and series expansions), we carry out a detailed study of the auxiliary integrals aimed to facilitate both the formal and computational applications of the ζ‐function method. The usefulness of this study in formal applications is illustrated with an example. The high performance in numerical applications is proved by the development of a very efficient program for the calculation of two‐center integrals with Slater functions corresponding to electrostatic potential, electric field, and electric field gradient. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

Ab initio molecular dynamics using hybrid density functionals

Ab initio molecular dynamics simulations with hybrid density functionals have so far found little application due to their computational cost. In this work, an implementation of the Hartree-Fock exchange is presented that is specifically targeted at ab initio molecular dynamics simulations of medium sized systems. We demonstrate that our implementation, which is available as part of the CP2K/Quickstep program, is robust and efficient. Several prescreening techniques lead to a linear scaling cost for integral evaluation and storage. Integral compression techniques allow for in-core calculations on systems containing several thousand basis functions. The massively parallel implementation respects integral symmetry and scales up to hundreds of CPUs using a dynamic load balancing scheme. A time-reversible multiple time step scheme, exploiting the difference in computational efficiency between hybrid and local functionals, brings further time savings. With extensive simulations of liquid water, we demonstrate the ability to perform, for several tens of picoseconds, ab initio molecular dynamics based on hybrid functionals of systems in the condensed phase containing a few thousand Gaussian basis functions.