Modified one-electron hamiltonian method in the MC SCF theory

For orbital optimization within the MC SCF theory a modification of the OEH method is proposed with the direction of descent determined according to the Fletcher–Reeves gradient method. The combined method developed on this basis ensures the convergence of the iterative process when the Hessian singularities occur. The convergence properties of the method proposed are studied by performing the ab initio water molecule calculations using two types of multiconfigurational wave functions.     

Cu as a one-electron atom: Molecular structure and dissociation energy of CuOH

The molecular structure and dissociation energy of CuOH have been investigated using non-empirical pseudopotentials. Two kinds of pseudopotential have been used, either including or not the 3d¹⁰ electrons of the Cu atom in the atomic core. At the SCF level and dealing explicitly with the 11 valence electrons of Cu, only about 60% of the experimental dissociation energy is recovered, while good agreement is found at the CI level. A difference of about 20 kcal/mol is found between the dissociation energies at the SCF level using the two kinds of pseudopotential. This difference is maintained at the CI level. Introduction of core-valence polarization and correlation effects overcomes the deficiency observed when only one electron is considered on the Cu atom.     

Scaled one-electron hamiltonian model for open-shell LCAO–MO–SCF calculations: Comparison with the restricted open-shell method of roothaan

The performance of a recently proposed scaled one-electron Hamiltonian (SOEH ) model is tested against parallel sets of restricted open-shell calculations by the method of Roothaan. It is found that the energy calculated by SOEH model, in general, lies slightly higher than the energy computed by the restricted open-shell method of Roothaan lending credibility to the application of variational argument to the scaled pseudoenergy functional (E^av) for deriving the SOEH model. The numerical stability of the converged SOEH energy with respect to changes in trial vectors indicates the reliability of the method. The SOEH model is shown to perform well in the calculation of geometries of radicals and ions. The convergence behavior of the SOEH model is compared with that of the restricted open-shell method of Roothaan.     

Chemistry and the near-sighted nature of the one-electron density matrix

Two different macrospopic pieces of copper have different external potentials and, because of the unique functional relationship between the electron density and the external potential as demanded by density functional theory, should possess different electron density distributions. Experimentally, however, an atom in the bulk exhibits the same electron density in both samples and they possess identical sets of intensive properties. Density functional theory does not account for the fundamental observation underlying the theory of atoms in molecules: that what are apparently identical distributions of charge can be observed for an atom or a grouping of atoms in systems with different external potentials and that these atoms contribute essentially identical amounts to the energies and all other properties of the systems in which they occur. It is shown that, unlike the external potential, the kinetic energy density and the potential energy density, defined by the virial of the Ehrenfest force acting on electron density, are short-range functions. As recorded in the first article on atoms in molecules, they exhibit a local dependence on the electron density that causes them to faithfully mimic the transferability of the atomic charge distributions from one system to another. The electron, the kinetic energy, and the virial densities are all determined directly by the one-electron density matrix, a function termed near-sighted by Professor Kohn. It is this near-sighted property of the one-matrix that underlies the working hypothesis of chemistry—that of a functional group exhibiting a characteristic set of properties. The observations obtained from the theory of atoms in molecules and the atomic theorems it determines demonstrate the existence of a local relationship between the electron density and all properties of a system. © 1995 John Wiley & Sons, Inc.     

Accurate electron density and one-electron properties for the beryllium atom

An accurate analytical electron density for the beryllium atom is obtained by using a fast and systematic method recently developed and tested for the neon atom. Asymptotic conditions both at the nucleus and at large distances are obeyed. A point-by-point comparison between our density and the one obtained from an almost “exact” configuration interaction wave function shows that differences are less than 0.5% for r between 0 and 5 bohrs and less than 1 % up to 9 bohrs. The accuracy of the density is also assessed by comparing results of density moments and x-ray scattering factors.     

The extended Koopmans' theorem Fock operator and the generalized overlap amplitude one-electron operator

The wave function of a system may be expanded in terms of eigenfunctions of the N −1 electron Hamiltonian times one-particle functions known as generalized overlap amplitudes (GOAS). The one-electron operator whose eigenfunctions are the GOAS is presented, without using an energy-dependent term as in the one-particle Green function or propagator approach. It is shown that this operator and the extended Koopmans' theorem (EKT) one-electron operator are of similar form, but perform complementary roles. The GOA operator begins with one-electron densities and total energies of N −1 electron states to generate the two-matrix and total energy of an N-electron state. The EKT operator begins with the two-matrix of an N-electron state to generate one-electron densities and ionization potentials (or approximations thereto) for N −1 electron states. However, whereas the EKT orbitals must be linearly independent, no such restriction applies to the GOAS. © 1996 John Wiley & Sons, Inc.     

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.     

Reduced density operators and the N-particle problem

An historical review of the theory of reduced density operators is given emphasizing the key role of the second order reduced matrix and related new concepts for understanding systems of many interacting fermions.     

Evaluation of one-electron integrals for arbitrary operators V(r) over Cartesian Gaussians: Application to inverse-square distance and Yukawa operators

A general procedure is presented for generating one-electron integrals over any arbitrary potential operator that is a function of radial distance only. The procedure outlines that for a nucleus centered at point C integrals over Cartesian Gaussians can be written as linear combinations of 1-D integrals. These Cartesian Gaussian functions are expressed in a compact form involving easily computed auxiliary functions. It is well known that integrals over the Coulomb operator can be expressed in terms of F_n(T) integrals, where By means of a substitution for F_n(T) by other simple functions, algorithms that form integrals over an arbitrary function can be generated. Formation of such integrals is accomplished with minor editing of existing code based on the McMurchie–Davidson formalism. Further, the method is applied using the inverse-square distance and Yukawa potential operators V(r) over Cartesian Gaussian functions. Thus, the proposed methodology covers a large class of one-electron integrals necessary for theoretical studies of molecular systems by ab initio calculations. Finally, by virtue of the procedure's recursive nature it provides us with an efficient scheme of computing the proposed class of one-electron integrals. © 1993 John Wiley & Sons, Inc.     

On the recognition of structural features: Some general problems illustrated by hamiltonian paths in polyhexes

Human and machine recognition skills are discussed, though not comprehensively reviewed, and some of the difficulties are illustrated by algorithms written to search for Hamiltonian paths in polyhexes. The most successful strategy for this is based upon the branching graph, a recently introduced graph-theoretical device which can aid the recognition of edges that arenot part of a Hamiltonian path. Another, more widely applicable approach that is interesting, although in this preliminary form only a little better than random methods, uses the metaphor of biological evolution, and tries to breed and grow paths subjected to natural selection.     

Stockholder projector analysis: a Hilbert-space partitioning of the molecular one-electron density matrix with orthogonal projectors

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.     

Effective operators and the Z-expansion method

The application of effective operators in the Z-expansion method is explored using hydrogenic functions. The relevant perturbation summations are evaluated using unit tensor operators and the results applied to the 1s²2s²2p² + 1s²2p⁴ complex of the carbon isoelectronic series.     

Pointwise bounds for the electron density and estimates of an expectation value for one-electron systems

Pointwise bounds for wavefunctions of a class of Schrödinger operators for one-electron systems are used to obtain pointwise bounds for the electron density, as well as estimates for the expectation value (r^?1).     

Assessment of a composite method based on selected density functional theory methods and complete basis set extrapolation formulas

Extrapolation formulas based on power and exponential formulas, as well as alternatives from a Taylor series, were tested and used with density functional theory (DFT) for the calculation of enthalpies of formation. The following four functionals were analyzed: B3LYP, BMK, M06-2X, and B2PLYP. Preliminary tests pointed to B2PLYP and B3LYP as the best and worst functionals, respectively. Taylor series expressions were as accurate as the power formulas and presented better performance than the exponential equation. The power formula (Equation (2)) and one of the simplest Taylor expressions (Equation (13)) were selected for the calculations with B3LYP and B2PLYP, with further empirical adjustments based on the higher level correction (HLC) and scaling of the experimental atomization energies used to calculate enthalpies of formation. HLC improved the B3LYP mean absolute error (MAE) from approximately 4.3 to 3.5 kcal mol⁻¹ using both extrapolation alternatives. For B2PLY, the MAEs were improved from 2.7 to 2.6 kcal mol⁻¹. Regarding the G3/05 test set, a significant improvement in the MAEs around 2.5 and 1.5 kcal mol⁻¹ were achieved using B3LYP and B2PLYP, respectively. The accuracy obtained from these empirical corrections was equivalent to other composite methods. The MAEs from B3LYP and B2PLYP may be suggested as ranges for the possible accuracy to be achieved by some DFT methods. The empirical corrections suggested in this work are improvements that may be considered to provide acceptable accuracy for enthalpies of formation and possibly other properties.     

Multiconfigurational expansions of density operators: equations of motion and their properties

 Multiconfigurational expansions of density operators which may be used in numerical treatments of the dynamics of closed and open quantum systems are introduced. The expansions of the density operators may be viewed as analogues of those used in the multiconfiguration time-dependent Hartree (MCTDH) method, which is a well-established and highly efficient method for propagating wavepackets in several dimensions. There is no unique multiconfigurational representation of a density operator and two sensible types of MCTDH-like expansions are studied. Equations of motion for these multiconfigurational expansions are presented by adopting the Dirac–Frenkel/McLachlan variational principle (or variants of thereof). Various properties of these sets of equations of motion are derived for closed and open system dynamics. The numerical and technical aspects of this approach have been recently discussed by us [(1999) J Chem Phys 111: 8759]. Here we discuss the formal aspects of the approach in a more general context. Received: 26 January 2000 / Accepted: 8 February 2000 / Published online: 12 May 2000     

Molecular dynamics simulation of the density and surface tension of water by particle-particle particle-mesh method

In this work, molecular dynamics simulation is performed to study the density and surface tension of water for a range of temperatures from 300 to 600 K. The extended simple point charge interaction potential for water is used. The particle-particle particle-mesh method, which automatically includes untruncated long-range terms, is used for the Lennard-Jones and the Coulombic terms. The results show that the long-range correction for the Lennard-Jones term is very important for the calculation of surface tension. It is found that the calculated density and surface tension of water fit well with experimental data for temperatures less than 500 K. Near the critical temperature, the simulation results are off from the experimental data.