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The computational approach to the Hirshfeld [Theor. Chim. Acta 44, 129 (1977)] atom in a molecule is critically investigated, and several difficulties are highlighted. It is shown that these difficulties are mitigated by an alternative, iterative version, of the Hirshfeld partitioning procedure. The iterative scheme ensures that the Hirshfeld definition represents a mathematically proper information entropy, allows the Hirshfeld approach to be used for charged molecules, eliminates arbitrariness in the choice of the promolecule, and increases the magnitudes of the charges. The resulting “Hirshfeld-I charges” correlate well with electrostatic potential derived atomic charges. [ABSTRACT FROM AUTHOR]

Classical dynamics can be described with Newton’s equation of motion or, totally equivalently, using the Hamilton-Jacobi equation. Here, the possibility of using the Hamilton-Jacobi equation to describe chemical reaction dynamics is explored. This requires an efficient computational approach for constructing the physically and chemically relevant solutions to the Hamilton-Jacobi equation; here we solve Hamilton-Jacobi equations on a Cartesian grid using Sethian’s fast marching method [J. A. Sethian, Proc. Natl. Acad. Sci. USA 93, 1591 (1996)]. Using this method, we can—starting from an arbitrary initial conformation—find reaction paths that minimize the action or the time. The method is demonstrated by computing the mechanism for two different systems: a model system with four different stationary configurations and the H+H2→H2+H reaction. Least-time paths (termed brachistochrones in classical mechanics) seem to be a suitable chioce for the reaction coordinate, allowing one to determine the key intermediates and final product of a chemical reaction. For conservative systems the Hamilton-Jacobi equation does not depend on the time, so this approach may be useful for simulating systems where important motions occur on a variety of different time scales. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Ground state eigenvectors of the reduced Bardeen–Cooper–Schrieffer Hamiltonian are employed as a wavefunction Ansatz to model strong electron correlation in quantum chemistry. This wavefunction is a product of weakly interacting pairs of electrons. While other geminal wavefunctions may only be employed in a projected Schrödinger equation, the present approach may be solved variationally with polynomial cost. The resulting wavefunctions are used to compute expectation values of Coulomb Hamiltonians, and we present results for atoms and dissociation curves that are in agreement with doubly occupied configuration interaction data. The present approach will serve as the starting point for a many-body theory of pairs, much as Hartree–Fock is the starting point for weakly correlated electrons. [ABSTRACT FROM AUTHOR]

In this brief report, we show that the three different chemical hardness definitions developed in the framework of the temperature-dependent density functional theory—namely, the electronic, the thermodynamic, and the Helmholtz hardnesses—imply both the hard and soft acids and bases (HSAB) principle and the maximum hardness (MH) principle. These hardnesses are identified as the second derivative of a thermodynamic state function and avoid the somewhat arbitrary approach, based on the parabolic interpolation of the energy versus electron number, that is normally used to justify these principles. This not only leads to a more mathematically sound justification of the HSAB and MH principles in the low-temperature limit but also establishes that the HSAB and the MH principles hold at any temperature of chemical relevance. [ABSTRACT FROM AUTHOR]

Using results from atomic spectroscopy, we show that there are two types of flat-planes conditions. The first type of flat-planes condition occurs when the energy as a function of the number of electrons of each spin, Nα and Nβ, has a derivative discontinuity on a line segment where the number of electrons, Nα + Nβ, is an integer. The second type of flat-planes condition occurs when the energy has a derivative discontinuity on a line segment where the spin polarization, Nα - Nβ, is an integer, but does not have a discontinuity associated with an integer number of electrons. Type 2 flat planes are rare-we observed just 15 type 2 flat-planes conditions out of the 4884 cases we tested-but their mere existence has implications for the design of exchange-correlation energy density functionals. To facilitate the development of functionals that have the correct behavior with respect to both fractional number of electrons and fractional spin polarization, we present a dataset for the chromium atom and its ions that can be used to test new functionals. [ABSTRACT FROM AUTHOR]

We present a new, non-variational orbital-optimization scheme for the antisymmetric product of one-reference orbital geminal wave function. Our approach is motivated by the observation that an orbital-optimized seniority-zero configuration interaction (CI) expansion yields similar results to an orbital-optimized seniority-zero-plus-two CI expansion [L. Bytautas, T. M. Henderson, C. A. Jimenez-Hoyos, J. K. Ellis, and G. E. Scuseria, J. Chem. Phys. 135, 044119 (2011)]. A numerical analysis is performed for the C2 and LiF molecules, for the CH2 singlet diradical as well as for the symmetric stretching of hypothetical (linear) hydrogen chains. For these test cases, the proposed orbital-optimization protocol yields similar results to its variational orbital optimization counterpart, but prevents symmetry-breaking of molecular orbitals in most cases. [ABSTRACT FROM AUTHOR]

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities. (The same type of relationship holds also for the electron-electron repulsion energy.) This extends a known property of the Levy-Valone Ensemble Constrained-Search and the Lieb Legendre-Transform refomulations of the Hohenberg-Kohn functional to the individual components of the functional. Moreover, we observe that the kinetic and electron-repulsion results also apply to densities with fractional electron number (even if there are no degeneracies), and we close with an analogous point-wise property involving the external potential. Examples where different degenerate states have different kinetic energy and electron-nuclear attraction energy are given; consequently, individual components of the ground state electronic energy can change abruptly when the molecular geometry changes. These discontinuities are predicted to be ubiquitous at conical intersections, complicating the development of universally applicable density-functional approximations. [ABSTRACT FROM AUTHOR]

A previously introduced partitioning of the molecular one-electron density matrix over atoms and bonds [D. Vanfleteren et al., J. Chem. Phys. 133, 231103 (2010)] is investigated in detail. Orthogonal projection operators are used to define atomic subspaces, as in Natural Population Analysis. The orthogonal projection operators are constructed with a recursive scheme. These operators are chemically relevant and obey a stockholder principle, familiar from the Hirshfeld-I partitioning of the electron density. The stockholder principle is extended to density matrices, where the orthogonal projectors are considered to be atomic fractions of the summed contributions. All calculations are performed as matrix manipulations in one-electron Hilbert space. Mathematical proofs and numerical evidence concerning this recursive scheme are provided in the present paper. The advantages associated with the use of these stockholder projection operators are examined with respect to covalent bond orders, bond polarization, and transferability. [ABSTRACT FROM AUTHOR]

Despite the importance of non-singlet molecules in chemistry, most variational second order density matrix calculations have focused on singlet states. Ensuring that a second order density matrix is derivable from a proper N-electron spin state is a difficult problem because the second order density matrix only describes one- and two-particle interactions. In pursuit of a consistent description of spin in second order density matrix theory, we propose and evaluate two main approaches: we consider constraints derived from a pure spin state and from an ensemble of spin states. This paper makes a comparative assessment of the different approaches by applying them to potential energy surfaces for different spin states of the oxygen and carbon dimer. We observe two major shortcomings of the applied spin constraints: they are not size consistent and they do not reproduce the degeneracy of the different states in a spin multiplet. First of all, the spin constraints are less strong when applied to a dissociated molecule than when they are applied to the dissociation products separately. Although they impose correct spin expectation values on the dissociated molecule, the dissociation products do not have correct spin expectation values. Secondly, both under 'pure spin state conditions' and under 'ensemble spin state' conditions is the energy a convex function of the spin projection. Potential energy surfaces for different spin projections of the same spin state may give a completely different picture of the molecule's bonding. The maximal spin projection always gives the most strongly constrained energy, but is also significantly more expensive to compute than a spin-averaged ensemble. In the dissociation limit, both the problem of nondegeneracy of equivalent spin projections, size-inconsistency and unphysical dissociation can be corrected by means of subspace energy constraints. [ABSTRACT FROM AUTHOR]

The eigenvectors of the electronic stress tensor can be used to identify where new bond paths form in a chemical reaction. In cases where the eigenvectors of the stress tensor are not available, the gradient-expansion-approximation suggests using the eigenvalues of the second derivative tensor of the electron density instead; this approximation can be made quantitatively accurate by scaling and shifting the second-derivative tensor, but it has a weaker physical basis and less predictive power for chemical reactivity than the stress tensor. These tools provide an extension of the quantum theory of atoms and molecules from the characterization of molecular electronic structure to the prediction of chemical reactivity. [ABSTRACT FROM AUTHOR]

Variational second order density matrix theory under 'two-positivity' constraints tends to dissociate molecules into unphysical fractionally charged products with too low energies. We aim to construct a qualitatively correct potential energy surface for F3- by applying subspace energy constraints on mono- and diatomic subspaces of the molecular basis space. Monoatomic subspace constraints do not guarantee correct dissociation: the constraints are thus geometry dependent. Furthermore, the number of subspace constraints needed for correct dissociation does not grow linearly with the number of atoms. The subspace constraints do impose correct chemical properties in the dissociation limit and size-consistency, but the structure of the resulting second order density matrix method does not exactly correspond to a system of noninteracting units. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A double-index atomic partitioning of the molecular first-order density matrix is proposed. Contributions diagonal in the atomic indices correspond to atomic density matrices, whereas off-diagonal contributions carry information about the bonds. The resulting matrices have good localization properties, in contrast to single-index atomic partitioning schemes of the molecular density matrix. It is shown that the electron density assigned to individual atoms, when derived from the density matrix partitioning, can be made consistent with well-known partitions of the electron density over atom in the molecule basins, either with sharp or with fuzzy boundaries. The method is applied to a test set of about 50 molecules, representative for various types of chemical binding. A close correlation is observed between the trace of the bond matrices and the shared electron density index. [ABSTRACT FROM AUTHOR]

A variational optimization of the second-order density matrix under the P-, Q-, and G-conditions was carried out for a set of diatomic 14-electron molecules, including N2, O22+, NO+, CO, and CN-. The dissociation of these molecules is studied by analyzing several chemical properties (dipole moments, population analysis, and bond indices) up to the dissociation limit (10 and 20 Å). Serious chemical flaws are observed for the heteronuclear diatomics in the dissociation limit. A careful examination of the chemical properties reveals that the origin of the dissociation problem lies in the flawed description of fractionally occupied species under the P-, Q-, and G-conditions. A novel constraint is introduced that imposes the correct dissociation and enforces size consistency. The effect of this constraint is illustrated with calculations on NO+, CO, CN-, N2, and O22+. [ABSTRACT FROM AUTHOR]

A previous study of diatomic molecules revealed that variational second-order density matrix theory has serious problems in the dissociation limit when the N-representability is imposed at the level of the usual two-index (P,Q,G) or even three-index (T1,T2) conditions [H. Van Aggelen et al., Phys. Chem. Chem. Phys. 11, 5558 (2009)]. Heteronuclear molecules tend to dissociate into fractionally charged atoms. In this paper we introduce a general class of N-representability conditions, called subsystem constraints, and show that they cure the dissociation problem at little additional computational cost. As a numerical example the singlet potential energy surface of Be B+ is studied. The extension to polyatomic molecules, where more subsystem choices can be identified, is also discussed. [ABSTRACT FROM AUTHOR]

Relationships between third-order reactivity indicators in the closed system [N, v(r)], open system [μ, v(r)], and density [ρ(r)] pictures are derived. Our method of derivation unifies and extends known results. Among the relationships is a link between the third-order response of the energy to changes in the density and the quadratic response of the density to changes in external potential. This provides a link between hyperpolarizability and the system’s sensitivity to changes in electron density. The dual descriptor is a unifying feature of many of the formulas we derive. [ABSTRACT FROM AUTHOR]

The number of the potential energy calculations required by the quadratic string method (QSM), and the fast marching method (FMM) is significantly reduced by using Shepard interpolation, with a moving least squares to fit the higher-order derivatives of the potential. The derivatives of the potential are fitted up to fifth order. With an error estimate for the interpolated values, this moving least squares enhanced Shepard interpolation scheme drastically reduces the number of potential energy calculations in FMM, often by up 80%. Fitting up through the highest order tested here (fifth order) gave the best results for all grid spacings. For QSM, using enhanced Shepard interpolation gave slightly better results than using the usual second order approximate, damped Broyden-Fletcher-Goldfarb-Shanno updated Hessian to approximate the surface. To test these methods we examined two analytic potentials, the rotational dihedral potential of alanine dipeptide and the SN2 reaction of methyl chloride with fluoride. [ABSTRACT FROM AUTHOR]

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory. [ABSTRACT FROM AUTHOR]

We have implemented the sweep algorithm for the variational optimization of SU(2) x U(1) (spin and particle number) invariant matrix product states (MPS) for general spin and particle number invariant fermionic Hamiltonians. This class includes non-relativistic quantum chemical systems within the Born-Oppenheimer approximation. High-accuracy ab initio finite field results of the longitudinal static polarizabilities and second hyperpolarizabilities of one-dimensional hydrogen chains are presented. This allows to assess the performance of other quantum chemical methods. For small basis sets, MPS calculations in the saturation regime of the optical response properties can be performed. These results are extrapolated to the thermodynamic limit. [ABSTRACT FROM AUTHOR]