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Molecular Physics

On a variational principle for the time-dependent Schrödinger equation

Roger F. Prat

Published Jan 1, 1967• 2 citations
Publisher

Abstract

Discussion on McLachlan's (1) formulation of a variational method for the time-dependent Schrödinger equation as compared to Frenkel's one. It appears that McLaclan's will not give better results.
DOI
10.1080/00268976700101001

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The Journal of Physical Chemistry A•Apr 6, 2025•5 citations
A minimum principle is described which has von Neumann’s equation of motion of the statistical operator as its ultimate solution. When the statistical operator is initially restricted to be a projection of trace unity, the minimum principle has Schrödinger’s time−dependent equation as its ultimate solution. As a result, the present minimum principle is shown to be a generalization of one first introduced by McLachlan, with a modification of the Dirac−Frenkel variational principle as a necessary consequence. In physical terms, the quantity that is minimized is shown to be the mean−squared deviation of an effective time−dependent Hamiltonian, which can be attributed to a time−dependent statistical operator initially chosen to describe a system, from the actual time−dependent Hamiltonian of the system. Examples of approximate solutions to von Neumann’s equation and Schrödinger’s equation that are obtained by use of the minimum principle confirm this identification.
Real-time propagation methods for chemistry and physics are invariably formulated using variational techniques. The time-dependent bivariational principle (TD-BIVP) is known to be the proper framework for coupled-cluster type methods, and is here studied from a differential geometric point of view. It is demonstrated how two distinct classical Hamilton's equations of motion arise from considering the real and imaginary parts of the action integral. This in turn leads to two distinct bivariational principles for real bivariational approximation submanifolds. Conservation laws and Poisson brackets are introduced, completing the analogy with classical mechanics. Furthermore, the time-dependent univariational principles (the time-dependent variational principle, the McLachlan principle, and the Dirac-Frenkel principle) are reconstructed using the TD-BIVP and a bivariational submanifold on product form. An overview of established real-time propagation methods is given in the context of our formulation of the TD-BIVP, namely time-dependent traditional coupled-cluster theory, orbital-adaptive coupled-cluster theory, time-dependent orthogonal optimized coupled-cluster theory, Brueckner coupled-cluster theory, and equation-of-motion coupled cluster theory.