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The Journal of Chemical Physics

A time-dependent variational principle and the time-dependent Hartree approximation in hydrodynamical form

James Lill, M. I. Haftel, G. H. Herling

Published May 1, 1989• 1 citation
Publisher

Abstract

A quantum mechanical time-dependent variational principle is generalized using the classical theory of fluids to obtain a variational principle suitable for the fluid dynamical description of mixed state quantum mechanics. A newly derived set of moment equations, in both standard and renormalized form, can be derived with the aid of this principle through minimization of the error in expressing the total derivative of the Wigner function. Coupled systems are studied in the time-dependent Hartree (TDH) approximation using a novel variational principle, and the renormalization procedure used earlier in the examination of single particle dynamics is extended to the TDH analysis. Use of a local Maxwellian ansatz for each particle results in a particularly simple ‘‘two-fluid’’ theory, the TDH/LM approximation, which does not violate the standard and renormalized energy conservation theorems derived earlier for the single particle equations. The fluid dynamical TDH/LM approximation is shown to possess a simple semiclassical interpretation.

DOI
10.1063/1.456590

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A quantum mechanical time-dependent variational principle is generalized using the classical theory of fluids to obtain a variational principle suitable for the fluid dynamical description of mixed state quantum mechanics. A newly derived set of moment equations, in both standard and renormalized form, can be derived with the aid of this principle through minimization of the error in expressing the total derivative of the Wigner function. Coupled systems are studied in the time-dependent Hartree (TDH) approximation using a novel variational principle, and the renormalization procedure used earlier in the examination of single particle dynamics is extended to the TDH analysis. Use of a local Maxwellian ansatz for each particle results in a particularly simple ‘‘two-fluid’’ theory, the TDH/LM approximation, which does not violate the standard and renormalized energy conservation theorems derived earlier for the single particle equations. The fluid dynamical TDH/LM approximation is shown to possess a simple semiclassical interpretation.
A newly derive set of quantum fluid dynamical equations appropriate for the description of mixed state dynamics is presented. Based essentially on moments of the Wigner function, the theory presented here uses an expansion of the pseudodistribution function in conjunction with a renormalization technique to obtain a semiclassical approximation for the dynamics of the probability, probability current, and energy densities. Particularly simple equations, ideal fluid dynamics with a nonlocal potential, result from using a local Maxwellian ansatz for the Wigner function. Further, the transformed potential may be easily computed from the fields being convected. Analogies with classical kinetic theory and fluid dynamics are exploited whenever possible.
We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan [Mol. Phys. 8, 39–44 (1964)], minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer and Saraceno [Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics Vol. 140 (Springer, Berlin, 1981)], imposes the stationarity of an action functional. We characterize both principles in terms of metric and symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.
For an N-electron system, a connection is explored between density-functional theory and quantum fluid dynamics, through a dynamical extension of the former. First, we prove the Hohenberg–Kohn theorem for a time-dependent harmonic perturbation under conditions which guarantee the existence of the corresponding steady (or quasiperiodic) states of the system. The corresponding one-particle time-dependent Schrödinger equation is then variationally derived starting from a fluid-dynamical Lagrangian density. The subsequent fluid-dynamical interpretation preserves the ’’particle’’ description of the system in the sense that the N-electron fluid has N components each of which is an independent-particle Schrödinger fluid characterized by a density function ρj and an irrotational velocity field uj, j = 1,⋅⋅⋅,N. However, the mean velocity u of the fluid is not irrotational, in general. The force densities and the stress tensor occurring in the Navier–Stokes equation are physically interpreted. The present work is another step towards the interpretation of physicochemical phenomena in three-dimensional space.
We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration interaction wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules.
We discuss the particle method in quantum mechanics which provides an exact scheme to calculate the time-dependent wavefunction from a single-valued continuum of trajectories where two spacetime points are linked by at most a single orbit. A natural language for the theory is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture and the particle theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle. The Euler-Lagrange equations imply a fourth-order nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates using as input the initial wavefunction. The admissible solutions are those consistent with quasi-potential flow. The effect of the superposition principle is represented via a nonclassical force on each particle. The wavefunction is computed via the standard map between the Lagrangian coordinates and the Eulerian fields, which provides the analogue in this model of Huygens principle in wave mechanics. The method is illustrated by calculating the time-dependence of a free Gaussian wavefunction. The Eulerian and Lagrangian pictures are complementary descriptions of a quantum process in that they have associated Hamiltonian formulations that are connected by a canonical transformation. The de Broglie-Bohm interpretation, which employs the same set of trajectories, should not be conflated with the Lagrangian version of the hydrodynamic interpretation. The theory implies that the mathematical results of the de Broglie-Bohm model may be regarded as statements about quantum mechanics itself rather than about its interpretation.
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.