Time‐dependent density operator in the random impulse approximation: I. Connections with the “generalized sudden approximation” for photoionization and electron impact ionization; and II. Relevance to reduced density operators, particle–hole matrices, and propagators for the initial state
Abstract A density operator analysis is made of the particle‐ejection (e.g., photoionization) problem in the “generalized sudden approximation” for the case in which the ejected particle is detected but its dynamical properties are not observed. It is shown that the resultant statistical density operator may be derived from that obtained in a particle‐conserving Hamiltonian‐evolution linear‐response model with a random impulse perturbation, and that the physical applicability of the latter model embraces the range of applicability of the former. The initial value of the statistical density operator for the above particle‐ejection problem is shown to be the N – (1)‐particle reduced density operator of the initial state; and the initial value of the statistical density operator for the excitations in the random impulse model is shown to be the N‐particle Hermitian operator whose matrix representation is the G matrix of Garrod and Percus. The significance of the eigenvalue spectrum of these operators to the excitation properties of the system is discussed, especially for the random impulse model, where a large eigenvalue of the G matrix can signal strong preferential excitation to its corresponding particle‐hole collective state, even for a random perturbation. Extensions of these ideas to excitations from states with a large eigenvalue of the two‐particle reduced density operator (e.g., superconducting states) are mentioned. The The applicability of these density matrices to the description of the excitation spectrum due to a well‐defined perturbation is discussed. The relationship of these time‐dependent density operators to the one‐particle propagator and the particle–hole propagator (polarization propagator) is established.
Abstract A density operator analysis is made of the particle‐ejection (e.g., photoionization) problem in the “generalized sudden approximation” for the case in which the ejected particle is detected but its dynamical properties are not observed. It is shown that the resultant statistical density operator may be derived from that obtained in a particle‐conserving Hamiltonian‐evolution linear‐response model with a random impulse perturbation, and that the physical applicability of the latter model embraces the range of applicability of the former. The initial value of the statistical density operator for the above particle‐ejection problem is shown to be the N – (1)‐particle reduced density operator of the initial state; and the initial value of the statistical density operator for the excitations in the random impulse model is shown to be the N‐particle Hermitian operator whose matrix representation is the G matrix of Garrod and Percus. The significance of the eigenvalue spectrum of these operators to the excitation properties of the system is discussed, especially for the random impulse model, where a large eigenvalue of the G matrix can signal strong preferential excitation to its corresponding particle‐hole collective state, even for a random perturbation. Extensions of these ideas to excitations from states with a large eigenvalue of the two‐particle reduced density operator (e.g., superconducting states) are mentioned. The The applicability of these density matrices to the description of the excitation spectrum due to a well‐defined perturbation is discussed. The relationship of these time‐dependent density operators to the one‐particle propagator and the particle–hole propagator (polarization propagator) is established.
A detailed account of the density matrix formalism and the resolvent operator theory is given. The statistical mixture of quantum states is described in terms of the density matrix operator, a concept which in an elegant manner allows one to obtain the ensemble average of an observable [a Hermitian operator, which eigenspectrum is complete]. On the basis of an integral equation for the evolution operator, described in the Interaction Representation, the Liouville equation, describing the dynamical evolution of the density matrix operator, is established. A comprehensive analysis of thermal equilibrium in electron systems is presented. For a description of the dynamical time evolution of a system in contact with a heat bath, a reservoir in thermal equilibrium, the den- sity matrix operator approach is shown to be extremely convenient. The Kubo formalism for the time-dependent density matrix is discussed in its most general form. The so-called resolvent propagator formalism is set up starting from a discussion of the analytical part of a Dirac delta function. Having established integral equations for the retarded and advanced scalar propagator operators, the resolvent propagator concept is introduced and analyzed. The resolvent of the Hamiltonian is of particular importance in physics, and the singularities of this resolvent (poles and branch cuts) are identified. We finish by a study of correlated systems in terms of the reduced density matrix operator concept.
In this work, we analyze the Born, Bogoliubov, Green, Kirkwood, and Yvon (BBGKY) hierarchy of equations for describing the full time evolution of a many-body fermionic system in terms of its reduced density matrices (at all orders). We provide an exhaustive study of the challenges and open problems linked to the truncation of such a hierarchy of equations to make them practically applicable. We restrict our analysis to the coupled evolution of the one- and two-body reduced density matrices, where higher-order correlation effects are embodied into the approximation used to close the equations. We prove that within this approach, the number of electrons and total energy are conserved, regardless of the employed approximation. Further, we demonstrate that although most of the truncation schemes available in the literature give acceptable ground-state energy, when applied to describe driven electron dynamics, they exhibit undesirable and unphysical behavior, e.g., violation and even divergence in local electronic density, both in weakly and strongly correlated regimes. We illustrate and analyze these problems within the few-site Hubbard model. The model can be solved exactly and provides a unique reference for our detailed study of electron dynamics for different values of interaction, different initial conditions, and the large set of approximations considered here. Moreover, we study the role of compatibility between two hierarchical equations and positive semidefiniteness of reduced density matrices in the instability of electron dynamics. We show that even if the used approximation holds the compatibility, electron dynamics can still diverge when positive definitiveness is violated. We propose some partial solutions of such problems and point out the main paths for future work in order to make this approach applicable for the description of the correlated electron dynamics in complex systems.
In our analysis of the dynamic one-electron difference-density matrices arising from time-dependent many-body states, we have observed two very distinct modes for the migration of an excitation through a system. In one case, coherences in the density matrix give rise to single particle and hole states that appear to migrate as independent units. Though interacting, they are independent in the sense that the single-electron state that describes an individual particle or hole is everywhere entangled with the same state of the electrons that have been integrated/traced out. These pseudo-wavefunctions carry nontrivial phase information (nonzero momentum), which is consistent with their observed direction of travel. The other case is dominated by a transition that can be thought of as the closing of a particle--hole pair in one location, entangled with the opening of such a pair in another location. In this case, the spatially-dependent phase of a given particle or hole state lies very near the real axis, reminiscent of time-independent molecular orbitals.
A general theory of long range correlations in the ground state of a closed shell nucleus is outlined, using a number of ideas introduced or adumbrated, but not always successfully developed, in previous work of the authors and others. These include the following: (i) Any operator can be formally expressed as a series in particle-hole (p-h) excitation and destruction operators which should converge rapidly for a suitable choice of single-particle basis at the closed shells. By means of these relations and with the help of sum rules all observables can be computed without the explicit use of wave functions. (ii) For a fixed choice of single-particle basis, the ground state energy can be computed from a variational principle in which in lowest nontrivial approximation, in consequence of (i), the parameters are amplitudes of the same type as occur in the random phase approximation (RPA). The precise relation to RPA is fully detailed. (iii) The single particle basis chosen is the natural orbital basis of Lowdin, which presents itself as the obvious choice both for formulation of the dynamics and for the algorithm of solution. (iv) Though the theory deals with pairs of fermion operators, particular attention is paid that all Pauli principle restrictions are satisfied. (v) It is verified that in the perturbation limit, known results are reproduced. The method is generally applicable to any system for which the correlations can be viewed as particle-hole excitations with respect to a reference Slater determinant.
In the classical work by Irving and Zwanzig [Irving J.H. and Zwanzig R.W., J. Chem. Phys. 19 (1951), 1173-1180 ] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. This work derives the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the semi-classical limit of the quantum observables in the conservation laws, also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit is to: (i) approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a non linear eigenvalue problem, (ii) define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.