Communication: Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

A double-atom partitioning of the molecular one-electron density matrix is used to describe atoms and bonds. All calculations are performed in Hilbert space. The concept of atomic weight functions (familiar from Hirshfeld analysis of the electron density) is extended to atomic weight matrices. These are constructed to be orthogonal projection operators on atomic subspaces, which has significant advantages in the interpretation of the bond contributions. In close analogy to the iterative Hirshfeld procedure, self-consistency is built in at the level of atomic charges and occupancies. The method is applied to a test set of about 67 molecules, representing various types of chemical binding. A close correlation is observed between the atomic charges and the Hirshfeld-I atomic charges.     

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.     

Second quantization for systems with a constant number of particles in the Dirac notation

The relation between the completeness condition for an appropriate one-particle basis set and the occupation number representation (second quantization) is shown for the time-independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one-electron density matrix and with the projector on the Fermi sea in the one-electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one-electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.     

Atoms-in-molecules from the Stockholder Partition of the Molecular Two-electron Distribution

The effective one-electron distributions of bonded atoms obtained from the “stockholder” partition of the molecular two-electron density are reported. These two-electron stockholder (S) atoms are compared with their one-electron analogs represented by the corresponding Hirshfeld (H) one-electron stockholder pieces of the molecular electron density. The influence of the exchange (Fermi) and Coulomb correlation between electrons on the resultant shapes of bonded atoms is investigated The vertical (for the fixed molecular electron density) and horizontal (involving the electron density displacement) correlation influences on the two-electron stockholder atoms are examined. The two sets of bonded stockholder atoms in the near-dissociation bond-elongated diatomics are compared for different approximations of the electron correlation effects. The cluster components in atomic resolution of the S-partitioning scheme are investigated for illustrative homonuclear and heteronuclear diatomics: H₂, LiH, HF, LiF, and N₂. This framework facilitates an understanding of the origins of the observed differences between the S and H variants of Atoms-in-Molecules. With the exception of hydrogen atoms, especially in light molecules, the two sets of bonded atoms were found to be practically identical. For H₂ and LiH the S atoms were shown to exhibit a distinctly higher degree of the bonding character, compared to their H analogs. The main electron correlation effects have been found to be well represented already at the exchange-only level, e.g., in the unrestricted Hartree–Fock (UHF) theory. An inclusion of the extra vertical Coulomb correlation exerts a marginal moderating influence on the ionic/covalent composition of the chemical bond already predicted by the UHF approximation, in the direction of a slightly more covalent (less ionic) bond character. The horizontal shifts of the molecular density due to Coulomb correlation, relative to the UHF reference, often act in the opposite direction.     

Covalent bond orders and atomic anisotropies from iterated stockholder atoms

Iterated stockholder atoms are produced by dividing molecular electron densities into sums of overlapping, near-spherical atomic densities. It is shown that there exists a good correlation between the overlap of the densities of two atoms and the order of the covalent bond between the atoms (as given by simple valence rules). Furthermore, iterated stockholder atoms minimise a functional of the charge density, and this functional can be expressed as a sum of atomic contributions, which are related to the deviation of the atomic densities from spherical symmetry. Since iterated stockholder atoms can be obtained uniquely from the electron density, this work gives an orbital-free method for predicting bond orders and atomic anisotropies from experimental or theoretical charge density data.     

Infinite-order quasirelativistic density functional method based on the exact matrix quasirelativistic theory

The exact one-electron matrix quasirelativistic theory [Kutzelnigg and Liu, J. Chem. Phys. 123, 241102 (2005)] is extended to the effective one-particle Kohn-Sham scheme of density functional theory. Several variants of the resultant theory are discussed. Although they are in principle equivalent, consideration of computational efficiency strongly favors the one (F(+)) in which the effective potential remains untransformed. Further combined with the atomic approximation for the matrix X relating the small and large components of the Dirac spinors as well as a simple ansatz for correcting the two-electron picture change errors, a very elegant, accurate, and efficient infinite-order quasirelativistic approach is obtained, which is far simpler than all existing quasirelativistic theories and must hence be regarded as a breakthrough in relativistic quantum chemistry. In passing, it is also shown that the Dirac-Kohn-Sham scheme can be made as efficient as two-component approaches without compromising the accuracy. To demonstrate the performance of the new methods, atomic calculations on Hg and E117 are first carried out. The spectroscopic constants (bond length, vibrational frequency, and dissociation energy) of E117(2) are then reported. All the results are in excellent agreement with those of the Dirac-Kohn-Sham calculations.     

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.     

Toward a physical understanding of electron-sharing two-center bonds. I. General aspects

In 1916, Lewis and Kossel laid the empirical ground for the electronic theory of valence, whose quantum theoretical foundation was uncovered only slowly. We can now base the classification of the various traditional chemical bond types in a threefold manner on the one- and two-electron terms of the quantum-physical Hamiltonian (kinetic, atomic core attraction, electron repulsion). Bond formation is explained by splitting up the real process into two physical steps: (i) interaction of undeformed atoms and (ii) relaxation of this nonstationary system. We aim at a flexible bond energy partitioning scheme that can avoid cancellation of large terms of opposite sign. The driving force of covalent bonding is a lowering of the quantum kinetic energy density by sharing. The driving force of heteropolar bonding is a lowering of potential energy density by charge rearrangement in the valence shell. Although both mechanisms are quantum mechanical in nature, we can easily visualize them, since they are of one-electron type. They are however tempered by two-electron correlations. The richness of chemistry, owing to the diversity of atomic cores and valence shells, becomes intuitively understandable with the help of effective core pseudopotentials for the valence shells. Common conceptual difficulties in understanding chemical bonds arise from quantum kinematic aspects as well as from paradoxical though classical relaxation phenomena. On this conceptual basis, a dozen different bond types in diatomic molecules will be analyzed in the following article. We can therefore examine common features as well as specific differences of various bonding mechanisms.     

Orthogonal natural atomic orbitals form an appropriate one-electron basis for expanding CASSCF wave functions into localized bonding schemes and their weights

Localized bonding schemes and their weights have been obtained for the pi-electron system of nitrone by expanding complete active space self-consistent field wave functions into a set of Slater determinants composed of orthogonal natural atomic orbitals (NAOs) of Weinhold and Landis (Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, 2005). Thus, the derived bonding schemes are close to orthogonal valence bond structures. The calculated sequence of bonding scheme weights accords with the sequence of genuine resonance structure weights derived previously by Ohanessian and Hiberty (Chem Phys Lett 1987, 137, 437), who employed nonorthogonal atomic orbitals. This accord supports the notion that NAOs form an appropriate orthogonal one-electron basis for expanding complete active space self-consistent field wave functions into meaningful bonding schemes and their weights.     

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.     

Molecular structural formulas as one-electron density and hamiltonian operators: the VIF method extended

The valency interaction formula (VIF) method is given a broader and more general interpretation in which these simple molecular structural formulas implicitly include all overlaps between valence atomic orbitals even for interactions not drawn in the VIF picture. This applies for VIF pictures as one-electron Hamiltonian operators as well as VIF pictures as one-electron density operators that constitute a new implementation of the VIF method simpler in its application and more accurate in its results than previous approaches. A procedure for estimating elements of the effective charge density-bond order matrix, Pmunu, from electron configurations in atoms is presented, and it is shown how these lead to loop and line constants in the VIF picture. From these structural formulas, one finds the number of singly, doubly, and unoccupied molecular orbitals, as well as the number of molecular orbitals with energy lower, equal, and higher than -1/2Eh, the negative of the hydrogen atom's ionization energy. The VIF results for water are in qualitative agreement with MP2/6311++G3df3pd, MO energy levels where the simple VIF for water presented in the earlier literature does not agree with computed energy levels. The method presented here gives the simplest accurate VIF pictures for hydrocarbons. It is shown how VIF can be used to predict thermal barriers to chemical reactions. Insertion of singlet carbene into H2 is given as an example. VIF pictures as one-electron density operators describe the ground-state multiplicities of B2, N2, and O2 molecules and as one-electron Hamiltonian operators give the correct electronegativity trend across period two. Previous implementations of VIF do not indicate singly occupied molecular orbitals directly from the pictorial VIF rules for these examples. The direct comparison between structural formulas that represent electron density and those that represent energy is supported by comparison of a simple electronegativity scale, chiD=N/n2, with well-known electronegativity scales of Pauling, Mulliken, and Allen. This scale comes from the method used to calculate Pmumu for sp3 hybridized period-two elements and is comparable to electronegativity because it has the same form as <1/r> for hydrogenic orbitals. It therefore provides a physical basis for the representation of one electron density and Hamiltonian operators by the same VIF picture.     

Iterative Atomic Charge Partitioning of Valence Electron Density

We propose an atomic charge partitioning scheme, iterative adjusted charge partitioning (I-ACP), belonging to the stockholder family and based on partitioning of the valence molecular electron density. The method uses a Slater-type weighting factor c_Ar^2n–2exp(–α_Ar), where α_A is a fixed parameter and c_A is determined iteratively. The parameters α_A were fitted for 17 main-group elements. The I-ACP scheme is shown to produce consistent, chemically meaningful atomic charges. Several stockholder-type charge-partitioning are compared. Extensive numerical tests demonstrate that in most cases, I-ACP surpasses most other methods by reproducing more accurately molecular dipole moments. © 2018 Wiley Periodicals, Inc.     

The distribution of electronic charge in the ground state of cubic boron nitride

The distribution of electronic charge in cubic boron nitride is investigated using the bond orbital wave functions recently calculated by Coulson and Doggett. Plots of the one-electron density function, in the (110) plane, are found to be insensitive to the choice of atomic basis functions, in contradistinction to the previously calculated effective atomic charges. A number of structure amplitudes are also calculated for each of the bond orbital wave functions.     

An ab initio study of crystal field effects,part 3: Solid- and gas-phase geometry of formamide,modeling the changes in a peptide group due to hydrogen bonds

A model of the solid state of formamide is constructed by optimizing a central molecule in an electrostatic field of the proper symmetry. Attention is paid to the way the electrostatic charges are obtained. Point charges obtained from a Mulliken population analysis yield a final set of atomic charges in the central molecule that agree reasonably well with those obtained experimentally after ak-refinement of formamide. Point charges from a so-called stockholder partitioning agree slightly less. Furthermore, the simple crystal field adaptation of standard ab initio methods reproduces within experimental limits the differences in C=O and C-N lengths, observed between the gas-phase and the solid state geometry. Again, a Mulliken field agrees slightly better than a stockholder field, but the difference in performance is statistically insignificant. In a survey of 221 high-quality single-crystal x-ray determinations of compounds containing the peptide group N-C=O, we found evidence supporting quantitatively the conclusion that the increase of C=O and the decrease of C-N bond length in the gas-to-solid transition is dominated by the effects of hydrogen bonding. It was shown that the C=O bond lengthens by about 0.011 å per H-bond it accepts, while the N-C bond diminishes by about 0.015 å per H-bond it donates.Part 2, see Ref. [5].     

On numerical solution of the integral Hartree-Fock equations for molecules by the method of MO LCAO in the momentum space

To develop a numerical solution of mentioned equations the method of factorized projection of integral operator kernel is applied. All matrix elements of the method are calculated analytically, being expressed in terms of two types of standard integrals: the overlap integrals and one-electron Coulomb integrals. To calculate the integrals we used the O(4)-symmetry of hydrogen-like atomic orbitals as well as operational technique of differentiation with respect to scalar and vector parameters.     

A definition for the covalent and ionic bond index in a molecule

Formulae for hermitian operators representing covalent, ionic, and total bond indices are derived. The eigenstates of these operators come in pairs, and can be considered as bonding, anti-bonding and lone-pair orbitals. The form of these operators is derived by generalising the rule that the bond order be defined as the net number of bonding electron pairs. The percentage of covalency and ionicity of a chemical bond may be obtained, and bond indices can also be defined between groups of atoms. The calculation of the bond indices depends only on the electron density operator, and certain projection operators used to represent each atom in the molecule. Bond indices are presented for a series of first and second row hydrides and fluorides, hydrocarbons, a metal complex, a Diels–Alder reaction and a dissociative reaction. In general the agreement between the bond indices is in accord with chemical intuition. The bond indices are shown to be stable to basis set expansion.     

A reference‐free stockholder partitioning method based on the force on electrons

We argue that when one divides a molecular property into atom‐in‐a‐molecule contributions, one should perform the division based on the property density of the quantity being partitioned. This is opposition to the normal approach, where the electron density is given a privileged role in defining the properties of atoms‐in‐a‐molecule. Because partitioning each molecular property based on its own property density is inconvenient, we design a reference‐free approach that does not (directly) refer atomic property densities. Specifically, we propose a stockholder partitioning method based on relative influence of a molecule's atomic nuclei on the electrons at a given point in space. The resulting method does not depend on an “arbitrary” choice of reference atoms and it has some favorable properties, including the fact that all of the electron density at an atomic nucleus is assigned to that nucleus and the fact all the atoms in a molecule decay at a uniform asymptotic rate. Unfortunately, the resulting model is not easily applied to spatially degenerate ground states. Furthermore, the practical realizations of this strategy that we tried here gave disappointing numerical results. © 2017 Wiley Periodicals, Inc.     

An analysis of dynamic linear response properties at RPA level

A separation of the dynamic linear response is developed, which distinguishes between the one- and two-electron contributions to the molecular response, by partitioning the RPA equation. The derivation of the partitioning is given in both an RPA , equation of motion, type approach and using the alternative, but equivalent, density matrix method. Three physically distinct contributions are obtained, called the direct, interaction, and back contributions. The direct term is composed entirely of one-electron effects, while the interaction and back terms account for the electron-interaction contributions to the response. Results for the dynamic dipole polarizability suggest that while the one-electron contribution is dominant in the zero-frequency limit, the two-electron contribution becomes increasingly important as the frequency of the perturbation increases. This implies that approximation of the linear response by only one-electron contributions is acceptable for the static case, but is less relevant for the dynamic case. The ramifications of this observation, for the scaling of sum-over-states-type calculations of large molecular systems, is briefly discussed, as is the application of our partitioning method to the higher polarizabilities. © 1995 John Wiley & Sons, Inc.     

Comparative study of the Hartree–Fock and the Hartree–Fock–Slater method

The Hartree–Fock method (standard Roothaan closed-shell HF –LCAO theory) and the Hartree–Fock–Slater method (restricted HFS –LCAO –DV method developed by Baerends and Ros) have been compared with emphasis on the respective one-electron equations and on the matrix elements of the respective Fock operators. Using the same STO basis in the two cases, the matrix elements of the Fock operators and of their separate one-electron, Coulomb, and exchange contributions have been calculated for the same orbitals and density of the ground state of the diatomic molecule ZnO. The effects of methodical (exchange potential) and numerical (DV method, density fit) differences between the HF and HFS methods on the various matrix elements have been analyzed. As expected the methodical effect prevails and is responsible for the higher (less negative) values of the matrix elements of the HFS Fock operator compared to those of the HF Fock operator. Numerical effects are observable also and are caused by the difference in integration procedures (DV method), not by the density fit.