Cartesian expressions for electric multipole moment operators

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensor technique

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model.

New insights in chemical reactivity from quantum chemical topology

Based on the quantum chemical topology of the modified electron localization function ELF_x , an efficient and robust mechanistic methodology designed to identify the favorable reaction pathway between two reactants is proposed. We first recall and reshape how the supermolecular interaction energy can be evaluated from only three distinct terms, namely the intermolecular coulomb energy, the intermolecular exchange-correlation energy and the intramolecular energies of reactants. Thereafter, we show that the reactivity between the reactants is driven by the first-order variation in the coulomb intermolecular energy defined in terms of the response to changes in the number of electrons. Illustrative examples with the formation of the dative bond B-N involved in the BH₃ NH₃ molecule and the typical formation of the hydrogen bond in the canonical water dimer are presented. For these selected systems, our approach unveils a noticeable mimicking of E_dual onto the DFT intermolecular interaction energy surface calculated between the both reactants. An automated reaction-path algorithm aimed to determine the most favorable relative orientations when the two molecules approach each other is also outlined.

The bonding picture in hypervalent XF3 (X = Cl, Br, I, At) fluorides revisited with quantum chemical topology

Hypervalent XF₃ (X = Cl, Br, I, At) fluorides exhibit T-shaped C_2V equilibrium structures with the heavier of them, AtF₃ , also revealing an almost isoenergetic planar D_3h structure. Factors explaining this behavior based on simple "chemical intuition" are currently missing. In this work, we combine non-relativistic (ClF₃ ), scalar-relativistic and two-component (X = Br - At) density functional theory calculations, and bonding analyses based on the electron localization function and the quantum theory of atoms in molecules. Typical signatures of charge-shift bonding have been identified at the bent T-shaped structures of ClF₃ and BrF₃ , while the bonds of the other structures exhibit a dominant ionic character. With the aim of explaining the D_3h structure of AtF₃ , we extend the multipole expansion analysis to the framework of two-component single-reference calculations. This methodological advance enables us to rationalize the relative stability of the T-shaped C_2v and the planar D_3h structures: the Coulomb repulsions between the two lone-pairs of the central atom and between each lone-pair and each fluorine ligand are found significantly larger at the D_3h structures than at the C_2v ones for X = Cl - I, but not with X = At. This comes with the increasing stabilization, along the XF₃ series, of the planar D_3h structure with respect to the global T-shaped C_2v minima. Hence, we show that the careful use of principles that are at the heart of the valence shell electron pair repulsion model provides reasonable justifications for stable planar D_3h structures in AX₃ E₂ systems. © 2017 Wiley Periodicals, Inc.

An Integral-Direct Linear-Scaling Second-Order Møller-Plesset Approach

An integral-direct, iteration-free, linear-scaling, local second-order Møller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calculations as well as for the efficient evaluation of the perturbative terms of double-hybrid density functionals. The method is based on a fragmentation approximation: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calculated directly invoking the density fitting approximation; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calculations show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an average (maximum) error below 1 kJ/mol (4 kJ/mol). Our benchmark calculations demonstrate that the new method enables MP2 calculations for molecules with more than 2300 atoms and 26000 basis functions on a single processor.

HPAM: Hirshfeld Partitioned Atomic Multipoles

An implementation of the Hirshfeld (HD) and Hirshfeld-Iterated (HD-I) atomic charge density partitioning schemes is described. Atomic charges and atomic multipoles are calculated from the HD and HD-I atomic charge densities for arbitrary atomic multipole rank l(max) on molecules of arbitrary shape and size. The HD and HD-I atomic charges/multipoles are tested by comparing molecular multipole moments and the electrostatic potential (ESP) surrounding a molecule with their reference ab initio values. In general, the HD-I atomic charges/multipoles are found to better reproduce ab initio electrostatic properties over HD atomic charges/multipoles. A systematic increase in precision for reproducing ab initio electrostatic properties is demonstrated by increasing the atomic multipole rank from l(max) = 0 (atomic charges) to l(max) = 4 (atomic hexadecapoles). Both HD and HD-I atomic multipoles up to rank l(max) are shown to exactly reproduce ab initio molecular multipole moments of rank L for L ≤ l(max). In addition, molecular dipole moments calculated by HD, HD-I, and ChelpG atomic charges only (l(max) = 0) are compared with reference ab initio values. Significant errors in reproducing ab initio molecular dipole moments are found if only HD or HD-I atomic charges used.

A finite field method for calculating molecular polarizability tensors for arbitrary multipole rank

A finite field method for calculating spherical tensor molecular polarizability tensors α(lm;l'm') = ∂Δ(lm)/∂ϕ(l'm')* by numerical derivatives of induced molecular multipole Δ(lm) with respect to gradients of electrostatic potential ϕ(l'm')* is described for arbitrary multipole ranks l and l'. Interconversion formulae for transforming multipole moments and polarizability tensors between spherical and traceless Cartesian tensor conventions are derived. As an example, molecular polarizability tensors up to the hexadecapole-hexadecapole level are calculated for water using the following ab initio methods: Hartree-Fock (HF), Becke three-parameter Lee-Yang-Parr exchange-correlation functional (B3LYP), Møller-Plesset perturbation theory up to second order (MP2), and Coupled Cluster theory with single and double excitations (CCSD). In addition, intermolecular electrostatic and polarization energies calculated by molecular multipoles and polarizability tensors are compared with ab initio reference values calculated by the Reduced Variation Space method for several randomly oriented small molecule dimers separated by a large distance. It is discussed how higher order molecular polarizability tensors can be used as a tool for testing and developing new polarization models for future force fields.

Physical nature of ethidium and proflavine interactions with nucleic acid bases in the intercalation plane

On the basis of the crystallographic structures of three nucleic acid intercalation complexes involving ethidium and proflavine, we have analyzed the interaction energies between intercalator chromophores and their four nearest bases, using a hybrid variation-perturbation method at the second-order Møller-Plesset theory level (MP2) with a 6-31G(d,p) basis set. A total MP2 interaction energy minimum precisely reproduces the crystallographic position of the ethidium chromophore in the intercalation plane between UA/AU bases. The electrostatic component constitutes the same fraction of the total energy for all three studied structures. The multipole electrostatic interaction energy, calculated from cumulative atomic multipole moments (CAMMs), was found to converge only after including components above the fifth order. CAMM interaction surfaces, calculated on grids in the intercalation planes of these structures, reasonably reproduce the alignment of intercalators in crystal structures; they exhibit additional minima in the direction of the DNA grooves, however, which also need to be examined at higher theory levels if no crystallographic data are given.

Multipole-based integral estimates for the rigorous description of distance dependence in two-electron integrals

We derive multipole-based integral estimates (MBIE) as rigorous and tight upper bounds to four-center two-electron integrals in order to account for the 1/R distance decay between the charge distributions, which is missing in the Schwarz screening commonly used in ab initio methods. Our screening criteria are valid for all angular momenta and can be formulated for any order of multipoles. We have found the expansion limited to dipoles to be sufficiently tight for estimating the integrals in Hartree-Fock and density-functional theories, while the screening effort is negligible. For, e.g., a DNA fragment with 1052 atoms and 10,674 basis functions (6-31G*) the exchange part is faster by a factor of 2.1 as compared to the Schwarz screening both within our linear exchange scheme, whereas a smaller factor of 1.3 is gained for the Coulomb part within the continuous fast multipole method. Most importantly, our new MBIE screening is perfectly suited to exploit the strong distance decay of electron-correlation effects of at least 1/R4 in atomic-orbital-based formulations of correlation methods.