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Physical Review A

Addendum to “Time-dependent variational principle with constraints for parametrized wave functions”

Katsuhisa Ohta

Published Apr 26, 2006• 2 citations
Publisher

Abstract

The sensitivity analysis for the time-dependent variational principle (TDVP) with constraints for parametrized wave functions [K. Ohta, Phys. Rev. A 70, 022503 (2004)] is investigated to assess the geodesic deviation caused by external parameters. The constraints for the sensitivity functions, inherited from the TDVP, are dealt with their consistency conditions. As an example of the sensitivity analysis, the geodesic deviation in the neighborhood of stationary states is investigated for general wave functions.

DOI
10.1103/physreva.73.044502

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The sensitivity analysis for the time-dependent variational principle (TDVP) with constraints for parametrized wave functions [K. Ohta, Phys. Rev. A 70, 022503 (2004)] is investigated to assess the geodesic deviation caused by external parameters. The constraints for the sensitivity functions, inherited from the TDVP, are dealt with their consistency conditions. As an example of the sensitivity analysis, the geodesic deviation in the neighborhood of stationary states is investigated for general wave functions.
The time-dependent variational principle in the stationary action principle form is formulated with constraint conditions for parametrized wave functions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical mechanics based on the commutability of operators. If the local bases in the parametrized wave functions construct a complete basis set, the classification of constraints with the commutator becomes equivalent to that with the complex generalized Poisson brackets (CGPB). However, in approximate wave functions with limited variational parameters, the equivalence between the CGPB and the commutator of operators is lost. The first-class constraints and the constants of motion classified with the commutator should be reconsidered as the pseudo-second-class constraints in the approximate wave functions.
Abstract In this work, we consider two conditions required for the nonsingularity of constraints in the time‐dependent variational principle (TDVP) for parametrized wave functions. One is the regularity condition which assures the static nonsingularity of the constraint surface. The other condition is the second‐class condition of constraints which assures the dynamic nonsingularity of the constraint surface with a symplectic metric. For analytic wave functions for complex TDVP‐parameters, the regularity and the second‐class conditions become equivalent. The second‐class condition for expectation values is reduced to the noncommutability of the corresponding quantum operators. The symplectic singularity of the equation of motion of TDVP is also shown to be a local breakdown of the second‐class condition in an extended canonical phase‐space. © 2012 Wiley Periodicals, Inc.
Methods of quantum nuclear wave-function dynamics have become very efficient in simulating large isolated systems using the time-dependent variational principle (TDVP). However, a straightforward extension of the TDVP to the density matrix framework gives rise to methods that do not conserve the energy in the isolated system limit and the total system population for open systems where only energy exchange with environment is allowed. These problems arise when the system density is in a mixed state and is simulated using an incomplete basis. Thus, the basis set incompleteness, which is inevitable in practical calculations, creates artificial channels for energy and population dissipation. To overcome this unphysical behavior, we have introduced a constrained Lagrangian formulation of TDVP applied to a non-stochastic open system Schrödinger equation [L. Joubert-Doriol, I. G. Ryabinkin, and A. F. Izmaylov, J. Chem. Phys. 141, 234112 (2014)]. While our formulation can be applied to any variational ansatz for the system density matrix, derivation of working equations and numerical assessment is done within the variational multiconfiguration Gaussian approach for a two-dimensional linear vibronic coupling model system interacting with a harmonic bath.