A modification of Frenkel's time-dependent variational principle (TDVP) determines the complex phase of the wave function: the real and imaginary parts give the phase prescribed by the Schrodinger equation and the correct normalization, respectively. They are given by the hermitized and anti-hermitized components of the frequency operator. The quantum-mechanical TDVP is transcribed into a classical action principle where the hermitized and anti-hermitized components of the frequency operator are shown to be related to the velocity potential and stream function, respectively, of the complex potential method of two-dimensional hydrodynamic flow. The product of metric coefficients that appears in the Poisson brackets is the derivative of the circulation; the conservation of circulation, angular momentum, and particle number are all equivalent.
A modification of Frenkel's time-dependent variational principle (TDVP) determines the complex phase of the wave function: the real and imaginary parts give the phase prescribed by the Schrodinger equation and the correct normalization, respectively. They are given by the hermitized and anti-hermitized components of the frequency operator. The quantum-mechanical TDVP is transcribed into a classical action principle where the hermitized and anti-hermitized components of the frequency operator are shown to be related to the velocity potential and stream function, respectively, of the complex potential method of two-dimensional hydrodynamic flow. The product of metric coefficients that appears in the Poisson brackets is the derivative of the circulation; the conservation of circulation, angular momentum, and particle number are all equivalent.
Abstract Time-dependent variational principle (TDVP) provides powerful methods in solving the time-dependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum–classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314). We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for ‘energy related dynamics’ and the other for ‘dynamics of wave-flow’. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.
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.
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.
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. The latter is new, and can in principle be used to develop novel propagation methods. Conservation laws and Poisson brackets are introduced, completing the analogy with classical mechanics. 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, and equation-of-motion coupled cluster theory.
Abstract A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.
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.