Towards a “Chemical” Hamiltonian

An analysis of the LCAO Hamiltonian is performed in terms of a “mixed” formulation of the second quantization for nonorthogonal orbitals, compressing the different interactions to one- and two-center terms as far as possible by performing appropriate projections. For this purpose an operator of atomic charge is also introduced, the expectation values of which are the Mulliken gross atomic populations on the individual atoms. The LCAO Hamiltonian is decomposed into terms having different physical meaning and significance: (i) sum of effective atomic Hamiltonians; (ii) the electrostatic interactions in the point-charge approximation; (iii) the electrostatic effects connected with the deviation of the actual charge distribution from the pointlike one; (iv) two-center overlap effects; (v) finite basis (“counterpoise”) correction terms related to the individual atoms; and (vi) similar finite basis correction terms with respect to the two-center interactions. Only terms of types (i) to (iv), containing no three- or four-center integrals, are considered as having physical significance. Based on the analysis of the Hamiltonian, an energy partitioning scheme is developed, and explicit expressions are given for one- and two-center (and basis extension) components of the SCF energy. The approach is also applied to the problem of intermolecular interactions, and an explicit formula is given permitting calculation of the “counterpoise” part of the supermolecule energy by properly taking into account that it depends not only on the extension of the basis, but also on the occupation of the additional orbitals in the intervening molecule—a factor completely overlooked in the usual scheme of calculations.     

Simulations of high-dielectric Stockmayer fluids in hyperspherical geometry

The static dielectric properties of Stockmayer fluids are investigated in the hyperspherical geometry, S(3). Different methods of obtaining the static dielectric constant ε(r) are compared. Tested methods include the evaluation of the Kirkwood factor, fluctuations of the total dipole moment, and a two-center potential correlation formula to obtain the dielectric constant through effective interactions. With no coupling to the "surrounding," the different methods give consistent estimates of the dielectric constant. Adding a coupling to the surrounding gives large size dependencies and the two-center potential correlation formula breaks down. For low dipole moments, there is a good agreement in the dielectric constant with previous studies.     

Diatomic interactions in momentum space. The effect of polarization and floating functions

The method of momentum density for interatomic interactions is used to investigate the pictures and roles of the polarization and floating functions in momentum (p-) space. Referring to the previous results from the minimal LCAO (Finkelstein-Horowitz) momentum density, we quantitatively discuss the effect of these functions for the bonding process in the ground state of H ₂ ⁺ system. The essence of the polarization and floating effects is found to be a modulation of the oscillation in the two-center part of the momentum density. The polarization function introduces a term with a phase and the floating function enlarges the period of the oscillation. An increased migration of the density from the one-center to the two-center part is also important. As a result, both the functions contribute to emphasize the contraction and expansion of momentum density observed previously. However, the floating function disturbs the density distribution in high momentum region, reflecting the destruction of cusps in position (r-) space. We point out an error in the pioneer work of Duncanson.     

Orbital interpretation of kinetic energy density and a direct space comparison of chemical bonding in tetrahedral network solids

We present how the kinetic energy density (KED) can be interpreted on the basis of the orbital interactions within the Kohn-Sham theory and propose how to utilize a direct space function in chemical bonding analysis, the relative entropy density (RED), which is constructed from the KED, the Thomas-Fermi KED (TF-KED), and the electron density. From the detailed analysis of the KED of wave functions and the TF-KED from the free electron model, it is shown that the RED can reveal the nodal properties of individual wave functions and provide a variationally meaningful way of accumulating chemical bonding information from the wave functions, hence allowing quantitative bonding analysis in direct space. To substantiate the proposal, the RED function has been tested on the tetrahedral network solids, including the group 14 elements and the III-V binary compounds with the zinc blende structure. The direct space maps of the RED quantitatively reflect the trend in metallicity and the polarity of their two-center, two-electron bonds in terms of the absolute values of the RED, the location of the minimum values, and the behavior of the deformation from the spherical symmetry of the atomic RED.     

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.     

Local response dispersion method. II. Generalized multicenter interactions

Recently introduced local response dispersion method [T. Sato and H. Nakai, J. Chem. Phys. 131, 224104 (2009)], which is a first-principles alternative to empirical dispersion corrections in density functional theory, is implemented with generalized multicenter interactions involving both atomic and atomic pair polarizabilities. The generalization improves the asymptote of intermolecular interactions, reducing the mean absolute percentage error from about 30% to 6% in the molecular C(6) coefficients of more than 1000 dimers, compared to experimental values. The method is also applied to calculations of potential energy curves of molecules in the S22 database [P. Jure?ka et al., Phys. Chem. Chem. Phys. 8, 1985 (2006)]. The calculated potential energy curves are in a good agreement with reliable benchmarks recently published by Molnar et al. [J. Chem. Phys. 131, 065102 (2009)]. These improvements are achieved at the price of increasing complexity in the implementation, but without losing the computational efficiency of the previous two-center (atom-atom) formulation. A set of different truncations of two-center and three- or four-center interactions is shown to be optimal in the cost-performance balance.     

Two-electron integrations in the quantum theory of atoms in molecules

A method to compute two-electron integrals over arbitrary regions of space is introduced and particularized to the basins appearing in the quantum theory of atoms in molecules. The procedure generalizes the conventional multipolar approach to account for overlapping densities. We show that the approach is always convergent and computationally efficient, scaling as N(4) in the worst, two-center case. Several numerical results supporting our claims are also presented.     

A semiempirical model for the two-center repulsion integrals in the NDDO approximation

A self-consistent formalism is proposed for the two-center electron repulsion integrals in the NDDO approximation, based on their expansion in terms of multipole-multipole interactions and free from adjustable parameters.     

Development of the cyclic cluster model formalism for Kohn-Sham auxiliary density functional theory methods

The development of the cyclic cluster model (CCM) formalism for Kohn-Sham auxiliary density functional theory (KS-ADFT) methods is presented. The CCM is a direct space approach for the calculation of perfect and defective systems under periodic boundary conditions. Translational symmetry is introduced in the CCM by integral weighting. A consistent weighting scheme for all two-center and three-center interactions appearing in the KS-ADFT method is presented. For the first time, an approach for the numerical integration of the exchange-correlation potential within the cyclic cluster formalism is derived. The presented KS-ADFT CCM implementation was applied to covalent periodic systems. The results of cyclic and molecular cluster model (MCM) calculations for trans-polyacetylene, graphene, and diamond are discussed as examples for systems periodic in one, two, and three dimensions, respectively. All structures were optimized. It is shown that the CCM results represent the results of MCM calculations in the limit of infinite molecular clusters. By analyzing the electronic structure, we demonstrate that the symmetry of the corresponding periodic systems is retained in CCM calculations. The obtained geometric and electronic structures are compared with available data from the literature.     

A theoretical study on the regioselectivity of 1,3-dipolar cycloadditions using DFT-based reactivity indexes

The regioselectivity for a series of four 1,3-dipolar cycloaddition reactions has been studied using global and local reactivity indexes. The results of the theoretical analysis suggest that for asynchronous cycloadditions associated to polar processes, the regioselectivity is consistently explained by the most favorable two-center interactions between the highest nucleophilic and electrophilic sites of the reagents.     

Expansion Formulae for Two-center Integer and Noninteger <Emphasis Type="Italic">n</Emphasis> STO Charge Densities and their Use in Evaluation of Multi-center Integrals

Using complete orthonormal sets of Ψ^α-exponential type orbitals (Ψ^α-ETOs, α =1, 0, −1, −2, ...) introduced by the author, the series expansion formulae are derived for the two-center integer and noninteger n STO (ISTO and NISTO) charge densities in terms of integer n STOs at a third center. The expansion coefficients occurring in these relations are presented through the two-center overlap integrals between STOs with integer and noninteger principal quantum numbers. The general formulae obtained for the STO charge densities are utilized for the evaluation of two-center Coulomb and hybrid integrals of NISTOs appearing in the Hartee–Fock–Roothaan approximation. The final results are expressed in terms of both the overlap integrals and the one-center basic integrals over integer n STOs. It should be noted that the result for the multi-center multielectron integrals with two-center noninteger n STO charge densities presented in this paper were not appeared in our past publications.     

Efficient multipole model and linear scaling of NDDO-based methods

Fast growth of computational costs with that of the system's size is a bottleneck for the applications of traditional methods of quantum chemistry to polyatomic molecular systems. This problem is addressed by the development of linear (or almost linear) scaling methods. In the semiempirical domain, it is typically achieved by a series of approximations to the self-consistent field (SCF) solution. By contrast, we propose a route to linear scalability by modifying the trial wave function itself. Our approach is based on variationally determined strictly local one-electron states and a geminal representation of chemical bonds and lone pairs. A serious obstacle previously faced on this route were the numerous transformations of the two-center repulsion integrals characteristic for the neglect of diatomic differential overlap (NDDO) methods. We pass it by replacing the fictitious charge configurations usual for the NDDO scheme by atomic multipoles interacting through semiempirical potentials. It ensures invariance of these integrals and improves the computational efficiency of the whole method. We discuss possible schemes for evaluating the integrals as well as their numerical values. The method proposed is implemented for the most popular modified neglect of diatomic overlap (MNDO), Austin model 1 (AM1), and PM3 parametrization schemes of the NDDO family. Our calculations involving well-justified cutoff procedures for molecular interactions unequivocally show that the proposed scheme provides almost linear scaling of computational costs with the system's size. The numerical results on molecular properties certify that our method is superior with respect to its SCF-based ancestors.     

Synthesis, structure, and bonding of BaTl(3): an unusual competition between cluster and classical bonding in the thallium layers.

The new BaTl(3) compound has been synthesized and characterized by physical property measurements and electronic structure calculations. Its structure (Cmcm) is a new intermediate in the Ni(3)Sn family (P6(3)/mmc), and consists of thallium layers formed from two-center bond formation between the parallel chains of face-sharing octahedral clusters. The valence electron concentration (VEC) of the thallium layers is consistent with their two-dimensional nature, in comparison with those in other AX(3)-type compounds with one- or three-dimensional anionic networks with the same building blocks and different VECs. The unique geometric features of the anionic thallium layers bring on an unusual competition between inter- and intracluster bonds. Detailed studies of the energetics of BaTl(3) reveal for the first time the important role of cation-anion interactions in the bonding competition in such an anionic substructure.     

New translation method for STOs and its application to calculation of two-center two-electron integrals

The new translation method for Slater-type orbitals (STOs) previously tested in the case of the overlap integral is extended to the calculation of two-center two-electron molecular integrals. The method is based on the exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. Fair uniform convergence and stability under wide changes in molecular parameters are obtained for all studied two-center hybrid, Coulomb, and exchange repulsion integrals. Ten-digit accuracy in the final numerical results is achieved through multiple precision arithmetic calculation of common angular coefficients and Gaussian numerical integration of some of the analytical formulas resulting for the radial integrals. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 91–100, 2000     

Modified all-valence INDO/spd method for ground and excited state properties of isolated molecules and molecular complexes

A new semiempirical all-valence method, GRINDOL (Ghost and Rydberg INDO ), based on the INDO approximation, is described. Linderberg–Seamans relation (extended to the d and Rydberg orbitals) for the resonance integrals and a new semitheoretical expression for the core-core repulsion term and energy correction including basis-set superposition error (intermolecular as well as intramolecular) has been applied. The proposed method enables calculation of ground and excited state properties. The ground state results (including intermolecular interactions) as well as the spectral properties are in reasonable agreement with relevant experimental (or ab initio) studies for isolated molecules, molecular complexes, and transition metal compounds. The method contains only one adjustable parameter, all two-center integrals and terms are only basis-set dependent. The one-center integrals are evaluated from the respective atomic terms.     

Two-center Dirac equation with separable potentials

It is shown that the two-center Dirac equation with a separable potential is analytically solvable. As a numerical example we construct a two-center separable potential whose energy eigenvalues resemble the two-center nuclear Coulomb potential.     

Analysis on solvated molecules with a new energy partitioning scheme for intra- and intermolecular interactions

A new partitioning scheme for the total energy of molecules is presented. In the scheme, the Hartree-Fock total energy of a molecular system is represented as the sum of one- and two-center terms exactly. The present method provides physically reasonable behavior for a wide range of interactions, and intermolecular interaction is treated equivalently with intramolecular interaction. The method is applied to analysis on the inter- and intramolecular interactions of molecular complexes both in gas phase and in aqueous solution. The results strongly indicate that the present method is a powerful tool to understand not only the bonding nature of molecules but also interaction between molecules.     

Boron,carbon, and aluminum supertetrahedral graphane analogues

The electronic and spatial structures of carbon, boron, and aluminum supertetrahedrane models of graphane have been studied by means of density functional theory methods in the supermolecular approximation (B3LYP/6-311G(df,2p)) and with imposing periodic boundary conditions (PBEPBE/6-311G (d,p), HSEH1PBE/6-311G (d,p)). Calculations predict that pure boron and aluminum structures are narrow-gap semiconductors. For supertetrahedral carbon graphane, calculations predict properties intermediate between the semiconductor and insulator properties. All bonds in the carbon system are two-center two-electron (2с–2е), while for the boron system, intratetrahedrane bonds are three-center two-electron (3с–2е), and intertetrahedrane bonds are common two-center two-electron bonds (2с–2е).     

Reaction electronic flux: a new concept to get insights into reaction mechanisms. Study of model symmetric nucleophilic substitutions

The present work is focused on studying chlorine and fluorine identity SN2 substitutions on a methyl center, within the framework of the newly introduced reaction electronic flux J(xi), that allows one to identify charge transfer and polarization mechanisms that take place along the reaction coordinate. The main results concern the discovery of different charge transfer mechanism, despite both reactions have the same energetic pattern with simultaneous bond breaking and formation. It turns out that the chlorine substitution is mainly driven by polarization effects and characterized by through bond interactions while intermolecular charge transfer dominates the fluorine exchange reaction, that is characterized by through space interactions.