Guanghua Liu

Badger's rule revisited

Numerical experiments demonstrate that the accuracy of stretching force constants ke provided by Badgers rule is unlikely to be substantially improved either by modification of the functional dependence on the equilibrium bond length Re or the inclusion of bond parameters related to electron density. These results, based upon both the experimental and QCISD/6-311++G(3d2f, 3p2d) values of Re and ke, imply that most of the universal characteristics of the bond strength vs. bond length dependence are accounted for by Badgers rule, the more detailed features being unexplainable by first-order response properties such as electron density.

Ionization potentials and electron affinities from the extended Koopmans’ theorem applied to energy-derivative density matrices: The EKTMPn and EKTQCISD methods

The extended Koopmans’ theorem (EKT) is combined with the energy-derivative formalism for the one- and two-particle reduced density matrices. Such a combination produces a versatile methodology for calculations of ionization potentials and electron affinities that, being applicable to any level of electron correlation treatment, is more general than the equation-of-motion (EOM) formalism. General expressions for the Feynman–Dyson amplitude, the pole strength, and the one-particle reduced density matrix of the hole state are derived. Like the electron propagator theory (EPT), the present approach provides a one-electron description of the electron attachment–detachment processes that is advantageous from the interpretive point of view. Numerical tests show that EKT calculations are capable of affording ionization potentials with accuracy comparable to that of the EPT methods but at a substantially lower computational cost.

Electron intracule densities and Coulomb holes from energy-derivative two-electron reduced density matrices

Application of the energy-derivative formalism to two-electron reduced density matrices produces a robust approach to the approximate evaluation of electron intracule densities I(R) and Coulomb holes in atoms and molecules. The versatility of this approach, which makes routine calculations of correlated I(R) feasible at any level of electronic structure theory, is demonstrated by results of selected MP2 calculations. The MP2/(20s10p10d) values of I(0) are within 10% of their “exact” counterparts in systems such as H−, He, Li+, Be2+, Li, and Be. Quantitative reproduction of the exact I(R) is found to be contingent upon the inclusion of Gaussian primitives with high angular momenta in the basis sets.

Topology of electron-electron interactions in atoms and molecules. I. The Hartree-Fock approximation

Topologies of the electron intracule and extracule densities, I(R) and E(R), are analyzed. These topologies are found to be inherently more complex than those of the one‐electron density. The main topological features of I(R) and E(R) are already present in the densities calculated within the Hartree–Fock (HF) approximation. Results of test calculations on several planar systems show that the positions and properties of attractors in I(R) and E(R) are predicted with a surprising fidelity by a naive independent‐atom model, making it possible to index distinct types of electron pairs present in atoms and molecules. In general, each pair of atoms in a given molecule has the potential of producing a pair of attractors in I(R). At the HF level of theory, all the atoms collectively furnish a single attractor in I(R) at R=0, but this topological pattern is bound to change upon the inclusion of electron correlation. The attractors in E(R) stem from both individual atoms and atomic pairs. In addition, attractors t...

Topology of electron-electron interactions in atoms and molecules. II. The correlation cage

The concept of the correlation cage provides new insights into electron–electron interactions in atoms and molecules. The cage constitutes the domain in the space of interelectron distance vectors R within which correlation effects are substantial. Its shape and size are entirely determined by the topological properties of the electron intracule density I(R), thus avoiding any references to ill-defined “uncorrelated” quantities. Integration of observables related to I(R) over the correlation cage affords quantitative measures of electron correlation. The number of strongly correlated electron pairs Mcorr[I], their electron–electron repulsion energy Wcorr[I], and the cage volume Vcorr[I] that characterizes the spatial extent of electron correlation are functionals of I(R). The ratio κ[I] of I(0)Vcorr[I] and Mcorr[I], which measures the strength of short-range correlation effects, is small for systems such as H− and closer to one for those with weaker correlation effects.

FAST EVALUATION OF ELECTRON INTRACULE AND EXTRACULE DENSITIES ON LARGE GRIDS OF POINTS

A new approach to fast evaluation of the electron intracule and extracule densities on large grids of points is described. Substantial (50‐ to 100‐fold) speed ups over the conventional algorithms are attained through the use of precomputed intermediates in the grid‐dependent phase of calculations. These intermediates are evaluated only once in a grid‐invariant procedure that employs efficient two‐stage integral screening to reduce computational effort. In addition to delivering high performance, the new algorithm facilitates calculations of analytical gradients and Hessians of the intracule and extracule electron densities. For regular grids with shared components of Cartesian coordinates, the present method allows the factorization of the primitive quartet contributions that makes the cost of calculations proportional to the cubic root of the number of grid points.

The concerted trimerization of ethyne to benzene revisited

Abstract CCSD(T)/6-311G ∗∗ //QCISD/6-311G ∗∗ calculations on the concerted [2+2+2] trimerization of ethyne to benzene yield Δ H trim o (HCCH)=−140.2 kcal/mol and Δ H act o (HCCH)=53.1 kcal/mol. The corresponding transition state (TS) possesses C 2 symmetry, although both the planar D 3h and nonplanar D 3 structures are negligibly higher in energy, indicating extreme flatness of the potential energy hypersurface along the distortion paths. The analogous trimerizations of HCCCl and ClCCCl are predicted to be considerably more exothermic. As the respective TSs cannot be located and the planar pseudo-TSs that possess several imaginary vibrational frequencies are associated with high reaction barriers, the concerted mechanism can be ruled out for these reactions.

Orbital hardness matrix and Fukui indices, their direct self-consistent-field calculations, and a derivation of localized Kohn–Sham orbitals

Formulas governing fixed orbital hardnesses and their relation to the hardness kernel are derived. It is shown how the orbital hardness matrix and its inverse matrix, the orbital softness matrix, may thus be directly calculated, and then the total chemical hardness, softness, and electronegativity of a molecular species. These quantities are calculated for the molecule HCN, using Dirac exchange and von Barth–Hedin correlation in the local density form of Kohn–Sham theory. The result complies with the frontier orbital theory. As quantitative indicators of orbital reactivity, the frontier orbital softness and Fukui indices generally have larger values than inner electron orbitals. The relation of orbital hardness matrix elements to the two-electron orbital integrals in a typical molecular orbital calculation is discussed, and it is demonstrated that diagonalization of the orbital hardness matrix leads to orbitals more localized than conventional Kohn–Sham orbitals.

Topology of electron–electron interactions in atoms and molecules. III. Morphology of electron intracule density in two 1Σg+ states of the hydrogen molecule

The differences in electronic structures of two 1Σg+ states of the hydrogen molecule are vividly reflected in their intracule densities I(r). The ground-state wave function of H2 is associated with two distinct topologies of I(r) (one of which pertains to the united atom limit), whereas no fewer than 11 unequivalent sets of critical entities are found for I(r) of the EF state that involves multiple electronic configurations. These sets and the catastrophes that interrelate them, which arise from conflicts between topological features of I(r) pertinent to different configurations, are characterized in detail. The usefulness of topological analysis of I(r) in the detection and characterization of various types of electron correlation is demonstrated.

Electrostatic interaction energies from a generalized Gaussian quadrature

Abstract When used in conjunction with a properly chosen curvilinear coordinate system, a generalized Gaussian quadrature affords a new versatile approach to the computation of energies of electrostatic interaction between two non-overlapping, continuous charge distributions. Compared with the conventional discretization of distributions, this approach offers a vast reduction in computational effort without a concomitant loss of accuracy. Although potentially slower than the fast multipole method, the present formalism is more general and less prone to discretization errors due to finite grid sizes. Results of test calculations on the energies of classical electrostatic interaction between AIMs demonstrate the power of the new method.