Density functionals for coulomb systems

This paper has three aims: (i) To discuss some of the mathematical connections between N-particle wave functions ψ and their single-particle densities ρ (x). (ii) To establish some of the mathematical underpinnings of “universal density functional” theory for the ground state energy as begun by Hohenberg and Kohn. We show that the HK functional is not defined for all ρ and we present several ways around this difficulty. Several less obvious problems remain, however. (iii) Since the functional mentioned above is not computable, we review examples of explicit functionals that have the virtue of yielding rigorous bounds to the energy.     

Information entropies of many-electron systems

The Boltzmann–Shannon (BS ) information entropy S_ρ = ∫ ρ(r)log ρ(r)dr measures the spread or extent of the one-electron density ρ(r), which is the basic variable of the density function theory of the many electron systems. This quantity cannot be analytically computed, not even for simple quantum mechanical systems such as, e.g., the harmonic oscillator (HO ) and the hydrogen atom (HA ) in arbitrary excited states. Here, we first review (i) the present knowledge and open problems in the analytical determination of the BS entropies for the HO and HA systems in both position and momentum spaces and (ii) the known rigorous lower and upper bounds to the position and momentum BS entropies of many-electron systems in terms of the radial expectation values in the corresponding space. Then, we find general inequalities which relate the BS entropies and various density functionals. Particular cases of these results are rigorous relationships of the BS entropies and some relevant density functionals (e.g., the Thomas–Fermi kinetic energy, the Dirac–Slater exchange energy, the average electron density) for finite many-electron systems. © 1995 John Wiley & Sons, Inc.     

Dissipative many-electron dynamics of ionizing systems

In this paper, we perform many-electron dynamics using the time-dependent configuration-interaction method in its reduced density matrix formulation (ρ-TDCI). Dissipation is treated implicitly using the Lindblad formalism. To include the effect of ionization on the state-resolved dynamics, we extend a recently introduced heuristic model for ionizing states to the ρ-TDCI method, which leads to a reduced density matrix evolution that is not norm-preserving. We apply the new method to the laser-driven excitation of H(2) in a strongly dissipative environment, for which the state-resolve lifetimes are tuned to a few femtoseconds, typical for dynamics of adsorbate at metallic surfaces. Further testing is made on the laser-induced intramolecular charge transfer in a quinone derivative as a model for a molecular switch. A modified scheme to treat ionizing states is proposed to reduce the computational burden associated with the density matrix propagation, and it is thoroughly tested and compared to the results obtained with the former model. The new approach scales favorably (～N(2)) with the number of configurations N used to represent the reduced density matrix in the ρ-TDCI method, as compared to a N(3) scaling for the model in its original form.     

Correlation effects in externally perturbed many-electron systems

Different methods for the calculation of the electron correlation contribution to atomic and molecular properties are analyzed and evaluated. The methods based on the self-consistent solution of the external perturbation problem are shown to offer several formal and computational advantages. The analysis of the correlation perturbation series for properties of many-electron systems indicates the importance of the appropriate treatment of unlinked diagrammatic contributions. In particular, the standard limited configuration interaction scheme based on single and double substitutions in the reference function may significantly suffer from the erratic treatment of unlinked clusters and needs to be corrected appropriately. The basis set choice for the calculation of highly accurate values of properties is also discussed. In order to circumvent the dimensionality problem the use of basis sets with explicit dependence on the external perturbation strength is recommended and methods for their choice and optimization are presented. A particular attention is paid to the many-body perturbation theory involving singly and doubly substituted intermediate states and based on the coupled Hartree–Fock solution for the one-electron perturbation problem. Different computational aspects of this method are discussed and compared with other techniques currently in use.     

Search for a density-based alternative quantum mechanics of many-electron systems

There are three reasons for seeking an alternative density-based quantum mechanics of many-electron systems, incorporating both interpretive and basic quantum mechanical aspects: (i) failure of popularad hoc chemical concepts underab initio scrutiny; (ii) failure ofab initio calculations to provide simple concepts; and (iii) highly attractive concepts and pictures generated by the electron density in three-dimensional space. At present the three interlinked pillars for such a density mechanics (in contrast to wave mechanics) are: (a) density functional theory; (b) quantum fluid dynamics; and (c) property densities in three-dimensional space. This article describes several studies dealing with these aspects. Although a density mechanics may well be an impossible ideal to realize, the search for it is indeed rejuvenating the whole of quantum chemistry.     

A graphical approach to permutation group representations for many-electron systems

A simple procedure is presented for obtaining the standard Young tableaux for the representation [(N/2) + S,(N/2) − S] of the permutation group ℒ_N for an N-electron system in spin state S directly from the spin branching diagram. We redefine the coordinate axes of the branching diagram to obtain a graph in terms of the partitions of the two-rowed Young diagram and define walks in this graph which yield directly the first rows of the allowed standard Young tableaux spanning a given representation when suitable weights have been assigned to the nodes in the graph. The allowed states are in a lexically ordered form and permit going easily from an index to an array and vice versa. © 1996 John Wiley & Sons, Inc.     

A spin-dependent unitary group approach to many-electron systems

In this article we derive a segment-level formula for the matrix elements of the U(2n) generators in a basis symmetry adapted to the subgroup U(n) × U(2) (i.e., spin-orbit basis), for the representations appropriate to many-electron systems. This enables the direct evaluation of the matrix elements of spin-dependent Hamiltonians.     

Quantum chemistry of many-electron systems: high-priority aspects

The review generalizes the studies devoted to the development of a new quantum chemistry method representing an alternative to the Hartree–Fock approximation. Based on the hypothesis of prohibition of equipotential surfaces, which clarifies the physical sense of the Pauli exclusion principle, and taking account of the condition for antisymmetrical wave function of the triplet state (3S) of He atom, the Hartree–Fock approximation is inappropriate for a priori determination of the nodal surfaces of many-electron wave functions (MWFs) for the test systems traditionally used in quantum chemistry, namely, excited triplet state of H₂ molecule and the ground electronic states of Li atom and LiH molecule. The nodal surfaces of the wave functions corresponding to the minimum basis set of Slater orbitals in the Hartree–Fock approximation are constructed and analyzed. An alternative to the Hartree–Fock approximation is provided by the MWF quantum chemical method being developed by the authors. In the MWF method, the nodal surfaces for H₂(³Σ _u ^v ) and Li(²S) are specified a priori. Some aspects of geometric interpretation of the Pauli exclusion principle are discussed. Unlike the MWF method, the Hartree–Fock approximation is unsuitable for taking account of the dependence of the MWF nodal surfaces on the nuclear charges and on correlation effects related to the motion of electrons with antiparallel spins because such nodal surfaces are predefined by the mathematical properties of Slater determinants rather than by physically clear and more practically valuable algebraic products of electrostatic potential differences.     

Use of green's coulomb function in perturbation theory for atoms

Green's Coulomb function for negative values of energy is applied to the calculation of sums and integrals arising in second-order field form of atomic perturbation theory.     

Phase associated with the single-particle density of many-electron systems

In excited states of atoms and molecules, as well as in time-dependent situations, the one-electron density no longer suffices to completely characterize the electronic state; in addition, one now requires information about the electronic phase or the current density. We show that, for a stationary electronic state, the continuity equation of quantum fluid dynamics represents a differential equation for the electronic phase, which must be solved subject to certain periodicity conditions. These periodicity conditions arise from the nodal topology of the wave function and give rise to quantized vortices of current. The consequences of writing an electronic “wave function” for a many-electron system directly in terms of the single-particle density and phase have been investigated. We have shown that such a procedure leads to the appearance of an “internal magnetic vector potential.” We also establish the connection between the electronic phase and the geometrical (“Berry”) phase accompanying the adiabatic transport of a quantal system around a closed loop in parameter space. This leads to a generalization of the current density concept and allows us to discuss the geometrical phase in terms of the circulation of this current in parameter space.     

The symmetrized basis function method for SU(2) and its application to the total spin states of many-electron systems

In this paper, the symmetrized basis function method is extended and used for the classification of the total spin states of many-electron systems. The reduction matrix, which is expressed as a series of products and direct products of matrices is derived. It is very advantageous that the method can be completed by computer. As an example, the reduction matrix for a 5-electron system calculated by IBM PC computer and the classification of the total spin states is presented.