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Physical Review A

Effective-potential theory for time-dependent many-electron wave functions

T. Kato, Kaoru Yamanouchi

Published Aug 10, 2018• 4 citations
Publisher

Abstract

We derive a variation equation for a time-dependent effective potential that is local both in time and in space and show that this potential generates and propagates spin orbitals with which a many-particle, time-dependent, multiconfigurational wave function $\mathrm{\ensuremath{\Psi}}(t)$ is constructed. Within this approach, the wave function and the effective potential are determined simultaneously in a self-consistent manner both in time-independent and in time-dependent cases. We also derive an equation that determines a real-valued effective potential and show that the equation establishes a relation among the first-order and second-order reduced density matrices and the electron repulsion integrals in the spin-orbital representation. By introducing the first-order density equation, we show that the variation equation for the effective potential can be simplified for an exact wave function. We also show that we can derive an effective potential with which a ground-state wave function that fulfills the Brillouin-Brueckner condition is constructed and that we can derive the effective potential proposed by Slater, with which a ground-state wave function represented by the multiconfiguration expansion is calculated, when an additional constraint is imposed on the varitation equation.

DOI
10.1103/physreva.98.023405

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We derive a variation equation for a time-dependent effective potential that is local both in time and in space and show that this potential generates and propagates spin orbitals with which a many-particle, time-dependent, multiconfigurational wave function \ensuremathΨ(t)\mathrm{\ensuremath{\Psi}}(t)\ensuremathΨ(t) is constructed. Within this approach, the wave function and the effective potential are determined simultaneously in a self-consistent manner both in time-independent and in time-dependent cases. We also derive an equation that determines a real-valued effective potential and show that the equation establishes a relation among the first-order and second-order reduced density matrices and the electron repulsion integrals in the spin-orbital representation. By introducing the first-order density equation, we show that the variation equation for the effective potential can be simplified for an exact wave function. We also show that we can derive an effective potential with which a ground-state wave function that fulfills the Brillouin-Brueckner condition is constructed and that we can derive the effective potential proposed by Slater, with which a ground-state wave function represented by the multiconfiguration expansion is calculated, when an additional constraint is imposed on the varitation equation.
Many-electron theory is treated here with much attention focused on the synthesis of the density functional and density matrix theory. This synthesis shows that the exchange–correlation potential, normally obtained as the functional derivative with respect to the electron density of an assumed approximate exchange–correlation energy, can be written exactly in terms of the “near-diagonal” behavior of the fully interacting first-order density matrix and the diagonal of the second-order density matrix, thereby bypassing the functional derivative. Also involved, since the exchange–correlation potential is an “artifact” of reducing the many-electron Schrödinger equation to one-body Slater–Kohn–Sham equations, are the one-body orbitals derived from these latter equations, not just the density thereby obtained. This situation has then motivated a discussion of what constraints are placed on the low-order density matrices of atoms and molecules, described by an assumed spin-independent Hamiltonian, by the requirement that these matrices are constructed from an N-electron wave function which is an eigenfunction of the spin operators S2 and Sz. Some briefer discussion is added on the optimally convergent determinantal expansion of such an N-electron wave function, the relation between Brueckner and Löwdin natural orbitals being commented on as a by-product. In this latter context, it is noted that the exchange-only potential of density functional theory can be represented as a sum of two line integrals, each of which is path-dependent, the sum being, of course, path-independent. But in addition to knowledge of Slater–Kohn–Sham orbitals appearing as in the Harbola–Sahni exchange-only potential (which is one of the path-dependent terms), there is a kinetic correction. This involves Löwdin natural orbitals and their occupation numbers, but only now in the weak coupling limit of the electron–electron interaction, as emphasized in the kinetic “correction” to the Harbola–Sahni formula exhibited by Levy and the writer. Some directions for further work are briefly set out. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 163–176, 1999
We derive a multiconfigurational time-dependent Hartree theory for systems with particle conversion. In such systems particles of one kind can convert to another kind and the total number of particles varies in time. The theory thus extends the scope of the available and successful multiconfigurational time-dependent Hartree methods---which were solely formulated for and applied to systems with a fixed number of particles---to a broader class of physical systems and problems. As a guiding example we treat explicitly a system where bosonic atoms can combine to form bosonic molecules and vice versa. In the theory for particle conversion, the time-dependent many-particle wave function is written as a sum of configurations made of a different number of particles, and assembled from sets of atomic and molecular orbitals. Both the expansion coefficients and the orbitals forming the configurations are time-dependent quantities that are fully determined according to the Dirac-Frenkel time-dependent variational principle. By employing the Lagrangian formulation of the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the atomic and molecular orbitals and one for the expansion coefficients. The first set is comprised of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with coefficients forming a time-dependent Hermitian matrix. Particular attention is paid to the reduced density matrices of the many-particle wave function that appear in the theory and enter the equations of motion. There are two kinds of reduced density matrices: particle-conserving reduced density matrices which directly only couple configurations with the same number of atoms and molecules, and particle nonconserving reduced density matrices which couple configurations with a different number of atoms and molecules. Closed-form and compact equations of motion are derived for contact as well as general two-body interactions, and their properties are analyzed and discussed.
By minimizing the difference between the left- and the right-hand sides of the many-body time-dependent Schr\"odinger equation with the Slater-determinant wave function, we derive a nonadiabatic and self-interaction-free time-dependent single-particle effective potential, which is the generalization to the time-dependent case of the so-called localized Hartree-Fock potential. The new potential can be efficiently used within the framework of the time-dependent density-functional theory as we demonstrate by the evaluation of the wave vector and frequency-dependent exchange kernel ${f}_{x}^{h}(q,\ensuremath{\omega})$ of the homogeneous electron gas. This is found to be nonsingular and causal, and it satisfies the positiveness of the dissipation, in contrast to the earlier known kernel from the first-order perturbation theory, which makes our fxh{f}_{x}^{h}fxh​ promising for applications.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.