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International Journal of Quantum Chemistry

Integral–geometrical consideration of density matrices

Boris P. Zapol

Published Dec 5, 1995• 2 citations
Publisher

Abstract

Abstract The ensemble N‐representability problem for the k‐th order reduced density matrix (k‐RDM) as well as the problem of reconstruction of the N‐particle system density matrices (N‐DM) from a given k‐RDM are studied. The spatial parts of the k‐RDM expansion in terms of spin tensorial operators Θ are represented using particular values (at specially chosen ) of the Radon transform of the N‐DM spatial parts (or their sums) 𝒟Nλ(x′ | x″) (here, is a d‐plane in the n‐space ℝn of x = (x′, x″)), with n = 6N, d = 3 (N − k), x′ ≡ (r′1, ⃛, r′N), x′ ≡ (r1″, ⃛, r″N ()). In this way, the problem is reduced to investigation of the properties of the functions . For a normalizable N – DM, it is proved that are bounded functions. The properties of implied by the N‐DM permutational symmetry, Hermiticity, and positive definiteness are found. A formal procedure of reconstruction of all N‐DM corresponding to a given k‐RDM is proposed. © 1995 John Wiley & Sons, Inc.

DOI
10.1002/qua.560560511

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The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).
The traces of the p-order reduced density matrices (p-RDM) split into independent contributions associated to the subsets of p-electron eigenstates of the Ŝ2 and Ŝz operators. Here, we report the partial traces for the blocks of the low-order RDMs corresponding to pure spin states of an N-electron system. A systematic method for calculating those of higher order RDMs is described and some useful relations are also given. All these relations which must be fulfilled independently by a RDM can be considered as N- and S-representability conditions © 1997 John Wiley & Sons, Inc.
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Abstract The spin structure of the first order reduced density matrix (RDM‐1) for an arbitrary many‐electron state with zero z‐projection of the total spin is examined. It is well known that for the state ΨS0(r1σ1,…,rN σN), which is an eigenstate of operators S2^{\widehat {S^2}}S2 and S^z{\widehat S}_zSz​ with quantum numbers S and M = 0, the matrix elements for spins α and β are equal for any r and r′: ρ (r|r′) = ρ (r|r′). In the present article, it is shown that the same is true for any state ΦM = 0(r1σ1,…,rN,σN) with indefinite total spin if in the expansion ΦM = 0 = ∑S DS ΨS0 only spins S with the same parity are present. To prove the statement, it is shown that the wave function ΨS0 acquires the phase factor (−1)N/2−S when all spin functions α(σi) are changed for β(σi) and vice versa. In the developed proof, the Hamiltonian was not used at all and it was not even assumed that the wave function ΨS0 is an eigenfunction of some Hamiltonian. Therefore the obtained result is valid for the stationary and non‐stationary states, ground and excited states, with and without homogeneous magnetic field imposed, exact and approximate wave functions. From the result obtained it follows, in particular, that for the stationary state to be spin‐polarized (ρ (r|r) ≠ ρ (r|r)) it is necessary for the Hamiltonian to mix states with different parity spins. The consequences from the proved statement for the antiferromagnetic state are discussed. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008