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).

Algorithms and working expressions for a grid-based fast multipole method (GB-FMM) have been developed and implemented. The computational domain is divided into cubic subdomains, organized in a hierarchical tree. The contribution to the electrostatic interaction energies from pairs of neighboring subdomains is computed using numerical integration, whereas the contributions from further apart subdomains are obtained using multipole expansions. The multipole moments of the subdomains are obtained by numerical integration. Linear scaling is achieved by translating and summing the multipoles according to the tree structure, such that each subdomain interacts with a number of subdomains that are almost independent of the size of the system. To compute electrostatic interaction energies of neighboring subdomains, we employ an algorithm which performs efficiently on general purpose graphics processing units (GPGPU). Calculations using one CPU for the FMM part and 20 GPGPUs consisting of tens of thousands of execution threads for the numerical integration algorithm show the scalability and parallel performance of the scheme. For calculations on systems consisting of Gaussian functions (α = 1) distributed as fullerenes from C20 to C720, the total computation time and relative accuracy (ppb) are independent of the system size.

The direct approach to gravitation and electrostatics (DAGE) algorithm is an accurate, efficient, and flexible method for calculating electrostatic potentials. In this paper, we show that the algorithm can be easily extended to consider systems with many different kinds of periodicities, such as crystal lattices, surfaces, or wires. The accuracy and performance are nearly the same for periodic and aperiodic systems. The electrostatic potential for semiperiodic systems, namely defects in crystal lattices, can be obtained by combining periodic and aperiodic calculations. The method has been applied to an ionic model system mimicking NaCl, and to a corresponding covalent model system.

A new implementation of vibrational coupled-cluster (VCC) theory is presented, where all amplitude tensors are represented in the canonical polyadic (CP) format. The CP-VCC algorithm solves the non-linear VCC equations without ever constructing the amplitudes or error vectors in full dimension but still formally includes the full parameter space of the VCC[n] model in question resulting in the same vibrational energies as the conventional method. In a previous publication, we have described the non-linear-equation solver for CP-VCC calculations. In this work, we discuss the general algorithm for evaluating VCC error vectors in CP format including the rank-reduction methods used during the summation of the many terms in the VCC amplitude equations. Benchmark calculations for studying the computational scaling and memory usage of the CP-VCC algorithm are performed on a set of molecules including thiadiazole and an array of polycyclic aromatic hydrocarbons. The results show that the reduced scaling and memory requirements of the CP-VCC algorithm allows for performing high-order VCC calculations on systems with up to 66 vibrational modes (anthracene), which indeed are not possible using the conventional VCC method. This paves the way for obtaining highly accurate vibrational spectra and properties of larger molecules.

We present an integration scheme for optimizing the orbitals in numerical electronic structure calculations on general molecules. The orbital optimization is performed by integrating the Helmholtz kernel in the double bubble and cube basis, where bubbles represent the steep part of the functions in the vicinity of the nuclei, whereas the remaining cube part is expanded on an equidistant three-dimensional grid. The bubbles' part is treated by using one-center expansions of the Helmholtz kernel in spherical harmonics multiplied with modified spherical Bessel functions of the first and second kinds. The angular part of the bubble functions can be integrated analytically, whereas the radial part is integrated numerically. The cube part is integrated using a similar method as we previously implemented for numerically integrating two-electron potentials. The behavior of the integrand of the auxiliary dimension introduced by the integral transformation of the Helmholtz kernel has also been investigated. The correctness of the implementation has been checked by performing Hartree-Fock self-consistent-field calculations on H, HO, and CO. The obtained energies are compared with reference values in the literature showing that an accuracy of 10 to 10 E can be obtained with our approach.