The problem considered is that of the diagonal N -representability of a p th-order reduced density matrix, p ≥ 2 p \geq 2 , for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. M = N + p M = N + p ; Case II. M > N + p M > N + p ; Case III. M > N + p M > N + p . Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of
N antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order
k has two sets of indices of each
k variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order
N over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete
k-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly
N spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order
M, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.
The
N-representability problem for the first- and second-order density matrices is considered when
N is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for
N-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the
N-representability problem for the second-order density matrix can be solved are considered.
Given a function w completely antisymmetric in n variables, there may exist a set of n functions of one variable such that the given function is a Slater determinant in the latter. The first problem considered is that of obtaining a criterion for this to be the case for a given function. This problem is solved by considering the function w as a mapping of the space of functions in n − 1 variables onto the space of functions of one variable. A necessary and sufficient condition for the initial function to be a Slater determinant is then shown to be that the image space be n dimensional. This criterion is converted into practical algorithms which can be employed for the determination. The application of one of these yields the theorem that an arbitrary linear combination of the n + 1 Slater determinants in n variables formed from n + 1 one-variable functions can always be written as a single Slater determinant. It is further proved that if the image space of the mapping is m(>n) dimensional, the original function can be expressed as a linear combination of m!/(m − n)!n! Slater determinants in n variables formed from m one-variable functions. Playing an important role in the analysis is the product of the mapping described above by its adjoint (the product is simply related to Dirac's density matrix for a quantum mechanical system of identical particles) as well as the eigenvectors and eigenvalues of this Hermitian positive semidefinite mapping. The latter form a basis for a systematic approximation procedure for representing a given function by a single Slater determinant or by sums of Slater determinants formed from a particular number of one-variable functions, which yields results obtained previously by Löwdin. Problems of simultaneous approximation of sets of antisymmetric functions and possible physical applications to many-fermion systems are briefly discussed.
For a quantum-mechanical system of
N identical fermions, the
N-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric
N-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles,
N, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be
N-representable is that its natural spin-orbital occupation numbers be equal in pairs.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of
p particles for each value of
p. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the
N-representability problem is given, and two sufficient conditions and one necessary condition for
N representability are given in terms of distances and norms.