The divide-expand-consolidate family of coupled cluster methods: numerical illustrations using second order Møller-Plesset perturbation theory

Previously, we have introduced the linear scaling coupled cluster (CC) divide-expand-consolidate (DEC) method, using an occupied space partitioning of the standard correlation energy. In this article, we show that the correlation energy may alternatively be expressed using a virtual space partitioning, and that the Lagrangian correlation energy may be partitioned using elements from both the occupied and virtual partitioning schemes. The partitionings of the correlation energy leads to atomic site and pair interaction energies which are term-wise invariant with respect to an orthogonal transformation among the occupied or the virtual orbitals. Evaluating the atomic site and pair interaction energies using local orbitals leads to a linear scaling algorithm and a distinction between Coulomb hole and dispersion energy contributions to the correlation energy. Further, a detailed error analysis is performed illustrating the error control imposed on all components of the energy by the chosen energy threshold. This error control is ultimately used to show how to reduce the computational cost for evaluating dispersion energy contributions in DEC.     

Fast density matrix-based partitioning of the energy over the atoms in a molecule consistent with the Hirshfeld-I partitioning of the electron density

For the Hirshfeld-I atom in the molecule (AIM) model, associated single-atom energies and interaction energies at the Hartree-Fock level are efficiently determined in one-electron Hilbert space. In contrast to most other approaches, the energy terms are fully consistent with the partitioning of the underlying one-electron density matrix (1DM). Starting from the Hirshfeld-I AIM model for the electron density, the molecular 1DM is partitioned with a previously introduced double-atom scheme (Vanfleteren et al., J Chem Phys 2010, 132, 164111). Single-atom density matrices are constructed from the atomic and bond contributions of the double-atom scheme. As the Hartree-Fock energy can be expressed solely in terms of the 1DM, the partitioning of the latter over the AIM naturally leads to a corresponding partitioning of the Hartree-Fock energy. When the size of the molecule or the molecular basis set does not grow too large, the method shows considerable computational advantages compared with other approaches that require cumbersome numerical integration of the molecular energy integrals weighted by atomic weight functions.     

Interatomic interactions in momentum space. Momentum density and kinetic energy in chemical bonding

On the basis of the virial theorem for a uniform scaling process of a polyatomic system, the total energy and its gradient are quantitatively related with the behavior of the electron density in momentum space through the kinetic energy of the system. For attractive and repulsive interactions, the behavior of the momentum density distribution and its effect on the stabilization energy and the interatomic force are examined. Some guiding principles are deduced for their interrelation. The results are used to clarify the role of kinetic energy in chemical bonding. Possible energy partitioning in this approach is also mentioned.     

Atoms-in-molecules in momentum space: a Hirshfeld partitioning of electron momentum densities

A direct application of the Hirshfeld atomic partitioning (HAP) scheme is implemented for molecular electron momentum densities (EMDs). The momentum density contributions of individual atoms in diverse molecular systems are analyzed along with their topographical features and the kinetic energies of the atomic partitions. The proposed p-space HAP-based charge scheme does seem to possess the desirable attributes expected of any atoms in molecules partitioning. In addition to this, the main strength of the p-space HAP is the exact knowledge of the kinetic energy functional and the inherent ease in computing the kinetic energy. The charges derived from HAP in momentum space are found to match chemical intuition and the generally known chemical characteristics such as electronegativity, etc.     

Energy partitioning schemes

The paper gives an overview, generalization and systematization of the different energy decomposition schemes we have devised in the last few years by using both the 3-D analysis (the atoms are represented by different parts of the physical space) and the Hilbert space analysis in terms of the basis orbitals assigned to the individual atoms. The so called "atomic decomposition of identity" provides us the most general formalism for analyzing different physical quantities in terms of individual atoms or pairs of atoms. (The "atomic decomposition of identity" means that we present the identity operator as a sum of operators assigned to the individual atoms.) By proper definitions of the atomic operators, both Hilbert-space and the different 3-D decomposition schemes can be treated on an equal footing. Several different but closely related energy decomposition schemes have been proposed for the Hilbert space analysis. They differ by exact or approximate treatment of the three- and four-center integrals and by considering the kinetic energy as a part of the atomic Hamiltonian or as having genuine two-center components, too. (Also, some finite basis correction terms may be treated in different manners.) The exact schemes are obtained by using the "atomic decomposition of identity". In the approximate schemes a projective integral approximation is also introduced, thus the energy components contain only one- and two-center integrals. The diatomic energy contributions have also been decomposed into terms of different physical nature (electrostatic, exchange etc.) The 3-D analysis may be performed either in terms of disjunct atomic domains, as in the case of the AIM formalism, or by using the so called "fuzzy atoms" which do not have sharp boundaries but exhibit a continuous transition from one to another. The different schemes give different numbers, but each is capable of reflecting the most important intramolecular interactions as well as the secondary ones--e.g. intramolecular interactions of type C-H[...]O.     

The elusive atomic rationale for DNA base pair stability

A systematic analysis of the electrostatic interaction between 27 natural DNA base pairs was carried out, based on ab initio correlated wave functions and the topology of the electron density. Using high rank multipole moments we show that the atomic partitioning of the interaction energy contains many substantial contributions between distant atoms. Profiles of cumulative energy versus internuclear distance show large fluctuations and provide an electrostatic fingerprint of the partitioning of interaction energy in a complex. A quantified comparison between each pair of energy profiles, one for each base pair, makes clear that there is no correlation between the total base pair interaction energy and the shape of the profile. In other words, base pairs with similar interaction energy are not stable for the same reasons in terms of atomic partitioning. In summary, simple rules to rationalize the pattern of energetic stability of naturally occurring base pairs in terms of subsets of atoms are elusive. Our work cautions against inappropriate use of Jorgensen's secondary interaction hypothesis.     

Correlated one-particle method: numerical results

In a previous paper a correlated one-particle method was formulated, where the effective Hamiltonian was composed of the Fock operator and a correlation potential. The objective was to define a correlated one-particle theory that would give all properties that can be obtained from a one-particle theory. The Fock-space coupled-cluster method was used to construct the infinite-order correlation potential, which yields correct ionization potentials (IP's) and electron affinities (EA's) as the negative of the eigenvalues. The model, however, was largely independent of orbital choice. To exploit the degree of freedom of improving the orbitals, the Brillouin-Brueckner condition is imposed, which leads to an effective Brueckner Hamiltonian. To assess its numerical properties, the effective Brueckner Hamiltonian is approximated through second order in perturbation. Its eigenvalues are the negative of IP's and EA's correct through second order, and its eigenfunctions are second-order Brueckner orbitals. We also give expressions for its energy and density matrix. Different partitioning schemes of the Hamiltonian are used and the intruder state problem is discussed. The results for ionization potentials, electron affinities, dipole moments, energies, and potential curves are given for some sample molecules.     

Partitioning schemes for the 2-matrix and the pair density

Several schemes are discussed for partitioning the second-order reduced density matrix Γ into two parts, Γ⁰ and Γ′. The Γ⁰s are based on the independent particle model and the Γ′s are corrections due to electron correlation. The difficulties of choosing a Γ⁰ that will serve as a suitable reference point for studying electron correlation are discussed. In order to compare alternative partitioning schemes, an atomic wave function for the ¹S ground state of the Be atom in the configuration-interaction approximation was selected. A fifty-two configuration wave function was computed and contour graphs were made of the total pair density Γ(1 2) and of the “correlation pair density” Γ′(1 2) for several choices of the reference Γ⁰.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

The role of quantum-mechanical interference and quasi-classical effects in conjugated hydrocarbons

The nature of the chemical bond in conjugated hydrocarbons is analyzed through the generalized product function energy partitioning (GPF-EP) method, which allows the calculation of the quantum-mechanical interference and quasi-classical contributions to the energy. The method is applied to investigate the differences between the chemical bonding in conjugated and non-conjugated hydrocarbon isomers and to evaluate the contribution from the energy components to the stabilization of the molecules. It is shown that in all cases quantum-mechanical interference has the effect of concentrating π electron density between the two carbon atoms directly involved in the (C-C)π bonds. For the conjugated isomers, this effect is accompanied by a substantial reduction of electron density in the π space of the neighbouring (C-C)σ bond. On the other hand, quasi-classical effects are shown to be responsible for the extra stabilization of the conjugated isomers, in which a decrease of the π space kinetic reference energy seems to play an important role. Finally, it is shown that the polarization of p-like orbitals in compounds with alternating single and double bonds ultimately increases electron density in the π space of the neighbouring (C-C)σ bond. Therefore, quasi-classical effects, rather than covalent ones, seem to be responsible for several properties of conjugated molecules, such as thermodynamic stability, planarity and (C-C)σ bond shortening. The shortcomings of the delocalization concept are discussed.     

Sigma-pi energy separation in homodesmotic reactions.

A well-established quantity for specifying the aromaticity or antiaromaticity of cyclic conjugated molecules is the so-called aromatic stabilization energy (ASE), which can be derived-either experimentally or theoretically-from appropriate homodesmotic reactions. To gain further insight into the origin of aromaticity, several schemes have been devised to partition ASE into nuclear and electronic as well as sigma and pi contributions, some of which have resulted in contradictory statements about the driving force of aromatic stabilization. Currently, these contradictions have not been resolved and have resulted in a confusing distinction between two different types of aromaticity: extrinsic and intrinsic aromaticity. By investigating different homodesmotic reactions we show that, in contrast to ASE itself, the individual contributions that enter the ASE can strongly depend on the type of reaction. Caution is therefore advised if conclusions or physical interpretations are derived from the individual components. The contradictions result from the fact that some reactions suffer from an imbalance in the number of interaction terms at the two sides of the reaction equation. The concept of isointeractional reactions is introduced and results in the elimination of the imbalance. For these reactions, the contradictions disappear and the distinction between intrinsic and extrinsic aromaticity becomes unnecessary. As far as the sigma-pi partitioning is concerned, several schemes proposed in the literature are compared. Contradictory results are obtained depending on the partitioning scheme and reaction used. In this context, it is demonstrated that for the partitioning of the electron-electron interaction, the scheme introduced by Jug and K?ster is the one that is most theoretically grounded.     

An orbital localization criterion based on the topological analysis of the electron localization function at correlated level

This work describes a procedure for localizing orbitals based on the topological analysis of the electron localization function at correlated level. The decomposition of the overlap matrix according to the partitioning of the three dimensional physical space into basins provided by that function allows us to define a localization index to be maximized using isopycnic orbital transformations. The localization algorithm has been computationally implemented and its efficiency tested on selected molecular systems at equilibrium, stretched, and twisted geometries. We report results which allow to analyze the influence of the correlated and uncorrelated treatments on the orbital localization.     

Partitioning scheme for theab initio SCF energy

As a tool for the interpretation ofab initio SCF calculations, an energy partitioning scheme is presented. When performed within an orthogonalized basis, the scheme allows the deduction of well transferable, almost basis independent two-center terms which characterize bond strengths and non-bonded interactions. The results for a large number of molecules are given. The construction of an orthogonal minimal basis (OMBA) from arbitrary basis sets as a generalization of the symmetrical orthogonalization is described. The transferability of Fock matrix elements is discussed. The energy partitioning quantities are related to the corresponding terms obtained with the semiempirical schemes CNDO and MINDO/3.     

Three-dimensional numerical integration for electronic structure calculations

Two three-dimensional numerical schemes are presented for molecular integrands such as matrix alements of one-electron operators occuring in the Fock operator and expectation values of one-electron operators describing molecular properties. The schemes are based on a judicious partitioning of space so that product-Gauss integration rules can be used in each region. Convergence with the number of integration points is such that very high accuracy (8–10 digits) may be obtained with obtained with a modest number of points. The use of point group symmetry to reduce the required number of points is discussed. Examples are given for overlap, nuclear potential, and electric field gradient integrals.     

Atomic decomposition of identity: General formalism for population analysis and energy decomposition

The concept of “atomic resolution of identity” has been introduced, leading to a very simple general formalism for generating different decomposition schemes of molecular quantities. Thus, different population analysis and energy partitioning schemes can be treated as special cases of a common framework. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Atomic shell structure based on inhomogeneity measures of the electron density

A measure of electron density inhomogeneity is proposed based on the distance of function values to the average value within a chosen region. The choice of the examined regions follows the approach of restricted space partitioning with fixed inhomogeneity as the restriction. The integration of electron density over the regions of the partitioning yields discrete distribution of charges, which is analyzed. The resulting functionals depend on the particular definition of distance. Two possibilities are selected for the distance definition, and the functionals are applied to examine the shell structure of the atoms Li to Xe.     

In silico partitioning and transmembrane insertion of hydrophobic peptides under equilibrium conditions

Nascent transmembrane (TM) polypeptide segments are recognized and inserted into the lipid bilayer by the cellular translocon machinery. The recognition rules, described by a biological hydrophobicity scale, correlate strongly with physical hydrophobicity scales that describe the free energy of insertion of TM helices from water. However, the exact relationship between the physical and biological scales is unknown, because solubility problems limit our ability to measure experimentally the direct partitioning of hydrophobic peptides across lipid membranes. Here we use microsecond molecular dynamics (MD) simulations in which monomeric polyleucine segments of different lengths are allowed to partition spontaneously into and out of lipid bilayers. This approach directly reveals all states populated at equilibrium. For the hydrophobic peptides studied here, only surface-bound and transmembrane-inserted helices are found. The free energy of insertion is directly obtained from the relative occupancy of these states. A water-soluble state was not observed, consistent with the general insolubility of hydrophobic peptides. The approach further allows determination of the partitioning pathways and kinetics. Surprisingly, the transfer free energy appears to be independent of temperature, which implies that surface-to-bilayer peptide insertion is a zero-entropy process. We find that the partitioning free energy of the polyleucine segments correlates strongly with values from translocon experiments but reveals a systematic shift favoring shorter peptides, suggesting that translocon-to-bilayer partitioning is not equivalent but related to spontaneous surface-to-bilayer partitioning.     

On the localization of defects

The range in ordinary space of a defect in a solid, polymer, or a large molecule is an important parameter in many physical and chemical situations like, e.g., substitutional and interstitial impurities in ionic crystals, deep and shallow energy levels in semiconductors, core holes in systems studied by photoelectron spectroscopy, and hydrogen in metals. Both the electronic structure and the lattice of the host molecule or solid are normally influenced by the defect, and it is desirable to treat, as far as possible, these two aspects together. An explicit procedure for investigating the degree of localization is proposed that is based on the direct calculation of Wannier functions and partitioning technique.     

Interactions between phosphate and water in solution: a natural bond orbital based analysis in a QM/MM framework

The natural bond orbital (NBO) and natural energy decomposition analysis (NEDA) calculations are used to analyze the interaction between mono-methyl phosphate-ester (MMP) and its solvation environment in a combined quantum mechanical/molecular mechanical (QM/MM) framework. The solute-solvent configurations are generated using a specific parametrization of the self-consistent-charge density functional tight-binding (SCC-DFTB) model for the MMP and TIP3P for water. The NBO and NEDA calculations are done with several QM/MM partitioning schemes with HF/6-31+G** as the QM level. Regardless of the size of the QM region, a notable amount of charge transfer is observed between MMP and the neighboring water molecules and the charge-transfer interactions are, in the NEDA framework, as important as the electric (electrostatic and polarization) components. This work illustrates that NBO based analyses are effective tools for probing intermolecular interactions in condensed phase systems.