Starting with Schr\"odinger's time-independent equation for
N indistinguishable particles, an exact equation is derived for the reduced density matrix of order
s, ${\ensuremath{\rho}}_{s}$. The equation contains terms dependent on ${\ensuremath{\rho}}_{s+1}$ and ${\ensuremath{\rho}}_{s+2}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\ensuremath{\rho}}_{2}$ and ${\ensuremath{\rho}}_{3}$ are particular functionals of ${\ensuremath{\rho}}_{1}$.
A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the
M occupied orbitals ${\ensuremath{\psi}}_{\ensuremath{\mu}}$ of an $N(\ensuremath{\le}M)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\ensuremath{\varrho}}=\ensuremath{\Sigma}{\ensuremath{\mu}=1}^{M}|{\ensuremath{\psi}}_{\ensuremath{\mu}}〉〈{\ensuremath{\psi}}_{\ensuremath{\mu}}|$. The condition placed on $\mathbf{\ensuremath{\varrho}}$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting
M=N, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\ensuremath{\varrho}}$. For
M>N there are three freedoms one may take with the fundamental condition on $\mathbf{\ensuremath{\varrho}}$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite
M>N, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of
N identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.
The hierarchy of the equations of motion for the reduced density matrices in quantum statistical mechanics is solved and the (cumulant) reduced density matrices at a time t are expressed in terms of those at an earlier time t0. Diagrams are introduced to express the results. With the aid of the technique of partial summations, the general term in the kinetic equation for the one-particle reduced density matrix or the generalized Boltzmann equation in quantum statistical mechanics is obtained. The equation is non-Markovian. A method of reducing the equation to Markovian is sketched.
We present a Hamiltonian method of constructing BBGKY-like hierarchies for quantum field theories. With suitable choices, our method creates a hierarchical system of evolution equations for the k-th order reduced density matrices. These equations can be closed at finite order using methods developed for the classical BBGKY hierarchy to give non-perturbative approximations for the full quantum equations of motion. Classical observables can then be numerically computed from these approximate equations, providing an analytically tractable method of modeling high-energy environments where quantum effects play a pronounced role.
The time-dependent Hartree-Fock equations are derived from a variational principle for the general N-body action of a Slater determinant of single-electron orbitals. The variational principle generalizes commonly used variational treatments based on reduced two-body actions. The self-consistent field equations are found to contain time-dependent corrections to the standard mean-field interactions. Their physical significance is discussed and a time-dependent phase shift to the Slater determinant is obtained that properly accounts for the total interaction energy in the mean-field approach.