Reduction of the N-Particle Variational Problem

Abstract

A variational method is presented which is applicable to N-particle boson or fermion systems with two-body interactions. For these systems the energy may be expressed in terms of the two-particle density matrix: Γ(1, 2 | 1′, 2′)=(Ψ |a2+a1+a1′a2′| Ψ). In order to have the variational equation: δE/δΓ = 0 yield the correct ground-state density matrix one must restrict Γ to the set of density matrices which are actually derivable from N-particle boson (or fermion) systems. Subsidiary conditions are presented which are necessary and sufficient to insure that Γ is so derivable. These conditions are of a form which render them unsuited for practical application. However the following necessary (but not sufficient) conditions are shown by some applications to yield good results: It is proven that if Γ(1, 2 | 1′, 2′) and γ(1 | 1′) are the two-particle and one-particle density matrices of an N-particle system [normalized by tr Γ = N(N − 1) and trγ = N] then the associated operator: G(1,2 | 1′,2′)=δ(1−1′)γ(2 | 2′)+σ Γ(1′,2 | 1,2′)−γ(2 | 1)γ(1′ | 2′) is a nonnegative operator. [Here σ is + 1 or − 1 for bosons or fermions respectively.]

Spin Symmetry Requirements in Density Functional Theory: The Proper Way to Predict Magnetic Coupling Constants in Molecules and Solids

“…. ; r n s n ) in a known basis set and to find the expansion coefficients using the variational method and with all necessary and sufficient constraints (spin and space symmetries) to prevent the variational collapse [41]; this is to avoid the convergence to a mathematical solution with no physical meaning. This is the basis of the so-called Full Configuration Interaction (FCI) method which provides the most accurate possible solution [40].…”

A primal–dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix

“…Early attempts, however, produced unrealistic results [4] and it was soon realized [5] that non-trivial constraints are needed to ensure that the 2DM is derivable from a physical wave function. These constraints were called N -representability conditions by Coleman [6], and Garrod and Percus [7] derived two such conditions, the so-called Q and G conditions, which can be expressed as matrixpositivity constraints. With these constraints there were some attempts, some of which quite successful, to solve this problem numerically in the 1970s [8][9][10][11].…”

The electronic ground-state energy problem: A new reduced density matrix approach

“…[7]) were also proposed in order to precise the accuracy of the N-representability conditions in specific cases. In practice, only approximate RDM minimization problems, in which only a few necessary N-representability conditions are imposed (see the geometric constraints of [8], or the so-called P,Q,G conditions [9,10]), can be considered. The first numerical studies relying on this strategy gave encouraging results [11].…”