A time-dependent functional formalism of molecular positions is developed in order to study non- equilibrium problems. It is based on the Liouvillean quantum field dynamics and is different from the density-functional framework. It provides a direct self-consistent approach to molecular dynamics of a system which is initially in thermodynamic equilibrium and is subjected to time-dependent external force.
A time-dependent functional formalism of molecular positions is developed in order to study non- equilibrium problems. It is based on the Liouvillean quantum field dynamics and is different from the density-functional framework. It provides a direct self-consistent approach to molecular dynamics of a system which is initially in thermodynamic equilibrium and is subjected to time-dependent external force.
We develop a nonequilibrium time-dependent functional theory of coupled interacting fields in terms of the Liouvillian quantum field dynamics. In this theory the expectation values of the appropriate field operators are treated as the functional variables. Thus, the formalism proposed here goes beyond the existing time-dependent density-functional method, in that we now have a mixed-state multivariable dynamical theory. A large class of condensed matter problems, with the given initial condition that the system be in thermodynamic equilibrium and subjected to time-dependent external forces, can now be examined. In particular, we formulate the self-consistent coupled equations for the electron, ion, and the electromagnetic fields. The above formalism provides a firm physical and mathematical basis for time-dependent dynamical simulation procedures such as those envisaged in the Car-Parrinello approach by replacing the fictitious Lagrangian by a precise action principle. The theory involves a certain functional representing the mutual interactions and correlations among these fields. In parallel with this, we also generalize the Baym \ensuremath{\Phi}-derivable method in many-body theory to investigate the class of problems mentioned above. This involves a functional of the exact one-particle Green functions. We compare the two methods and suggest the use of the latter in the former scheme as a way of going beyond the simple mean-field schemes. We illustrate the utility of the framework by citing a variety of illustrative examples.
Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg, Kohn, and Sham [4,5] in the mid-1960s. According to this theory, the fundamental physical information about a many-body system is provided by single-particle densities in a three-dimensional space, which are obtained variationally within a time-independent framework. When compared with other previous formalisms, DFT presents two clear advantages: (1) it is able to treat many-body problems in a sufficiently accurate way and (2) it is computationally simple. This explains why it is one of the most widely used theories to deal with electronic structure-the electronic ground-state energy as a function of the position of the atomic nuclei determines molecular structures and solids, providing at the same time the forces acting on the atomic nuclei when they are not at their equilibrium positions. At present, DFT is used routinely to solve many problems in gas phase and condensed matter. Furthermore, it has made possible the development of accurate molecular dynamics schemes in which the forces are evaluated quantum mechanically ‘‘on the fly.’’ Nonetheless, DFT is a fundamental tool provided the systems studied are relatively large; for small systems, standard methods based on the use of the wave function render quite accurate results [6]. Moreover, it is also worth stressing that all practical applications of DFT rely on essentially uncontrolled approximations [7] (e.g., the local density approximation [4,5], the local spin-density approximation, or generalized gradient approximations [8]), and therefore the validity of DFT is conditioned to its capability to provide fairly good values of the experimental data.
Classical molecular dynamics (MD) provides a powerful and widely used approach to determining thermodynamic properties by integrating the classical equations of motion of a system of atoms. Time-Dependent Density Functional Theory (TDDFT) provides a powerful and increasingly useful approach to integrating the quantum equations of motion for a system of electrons. TDDFT efficiently captures the unitary evolution of a many-electron state by mapping the system into a fictitious non-interacting system. In analogy to MD, one could imagine obtaining the thermodynamic properties of an electronic system from a TDDFT simulation in which the electrons are excited from their ground state by a time-dependent potential and then allowed to evolve freely in time while statistical data are captured from periodic snapshots of the system. For a variety of systems (e.g., many metals), the electrons reach an effective state of internal equilibrium due to electron-electron interactions on a time scale that is short compared to electron-phonon equilibration. During the initial time-evolution of such systems following electronic excitation, electron-phonon interactions should be negligible, and therefore, TDDFT should successfully capture the internal thermalization of the electrons. However, it is unclear how TDDFT represents the resulting thermal state. In particular, the thermal state is usually represented in quantum statistical mechanics as a mixed state, while the occupations of the TDDFT wavefunctions are fixed by the initial state in TDDFT. We work to address this puzzle by (A) reformulating quantum statistical mechanics so that thermodynamic expectations can be obtained as an unweighted average over a set of many-body pure states and (B) constructing a family of non-interacting (single determinant) TDDFT states that approximate the required many-body states for the canonical ensemble.
In this paper we propose an ab initio method to solve quantum many-body problems of molecular dynamics where both electronic and nuclear degrees are represented by ensembles of trajectories and guiding waves in physical space. Both electrons and nuclei can be treated quantum mechanically where the guiding waves obey a set of coupled Schrodinger equations (quantum-quantum description) or, alternatively, coupled Schrodinger-Newtonian equations are solved for the quantum-classical approximation. The method takes into account local and nonlocal quantum correlation effects in a self-consistent manner. The general formalism is applied to one- and two-dimensional hydrogen molecules subjected to a strong ultrashort optical pulse. Comparison is made with the results from the "exact" Ehrenfest molecular dynamics for the molecular ionization and for the evolution of the internuclear distance as the molecule dissociates.
Abstract Chapter 14 covers the statistical mechanics of driven quantum systems and the non-equilibrium ensembles associated with them. The topic is presented both from the state-vector and density matrix perspectives. The chapter begins with a discussion of driving forces resulting from light-matter interactions and their treatment via time-dependent perturbation theory. The Fermi Golden Rule is derived, and it is shown how to incorporate the Golden Rule into the calculation of frequency spectra within the canonical ensemble, which then connects to canonical quantum time correlation functions. The formalism is derived again from the quantum Liouville equation. The discussion then turns to the challenge of computing quantum time correlation functions and approximate techniques based on the Feynman path integral of Chapter 12. These schemes include centroid and ring-polymer molecular dynamics as well as open-path integral methods