Slater hull

C. Necessary Conditions from the Slater Hull (R, R) Conditions... 

E. Sufficiency of the Slater Hull Constraints for Diagonal Elements of Density Matrices... 

E. Constraints on Off-Diagonal Elements from Unitary Transformation of the Slater Hull Constraints... 

The similarities and differences between methods based on the electron density, electron-pair density, and reduced density matrices have recently been reviewed [5]. This chapter is not intended as a comprehensive review, but as a focused consideration of A-representability constraints that are applicable to diagonal elements of reduced density matrices. Such constraints are useful both to researchers working with the Q-density and to researchers working with g-electron reduced density matrices, and so we shall attempt to review these constraints in a way that is accessible to both audiences. Our focus is on inequalities that arise from the Slater hull because the Slater hull provides an exhaustive list of A-representability conditions for the diagonal elements of the density matrix, Although the Slater hull constraints... 

The Slater hull constraints are not directly applicable to existing approaches to pair-density functional theory because they are formulated in the orbital representation. Toward the conclusion of this chapter, we will also address A-representability constraints that are applicable when the spatial representation of the pair density is used. 

A Q-rmirix is in the Slater hull if it satisfies the (Q, K) constraints, where K is the number of spin orbitals in the basis set. If Tq is in the Slater hull, it satisfies... 

Henceforth, we will use the statements Vq is in the Slater hull, Tg satisfies the Slater hull conditions, and Tg satisfies the Q, K) conditions interchangeably. 

Necessary conditions based on the Slater hull have been pursued by Kummer, McRae and Davidson, Yoseloff and Kuhn, and Deza and Laurent, among others [24-26, 29, 42-44]. 

Even for a relatively small system, obtaining the Q, R) conditions is computationally challenging [25]. For example, using the 2-matrix to describe the beryllium atom in a minimal basis would require the (2,10) conditions for a four-electron system. In this case, the Slater hull is a polyhedron with on the order of ten billion facets. Only small R is interesting for computational applications. 

As previously discussed, when developing necessary conditions for N-representability, it is useful to consider the subset of Hamiltonians whose ground-state energy is zero. Applying this idea to the Slater hull, the following Hamiltonians arise as important constraints ... 

Figure 2. Pictorial representation of how one defines the Slater hull using hyperplanes. The vertices of the tetrahedron correspond to Slater determinants. The base of the Slater hull, which is shaded in this figure, lies in the hyperplane defined by Tr[n n/jF ] = 0. Moving clockwise, starting at the front of the tetrahedron, the sides of the Slater hull are contained in the hyperplanes...

In fact, even the full set of (R, K) conditions is insufficient. Since the full set of Slater hull constraints is not sufficient to ensure the A-representability of the /f-matrix, there exist some F that satisfy the Slater hull constraints,... 

The Slater hull constraints represent the entire family of A -representabUity constraints that can be expressed using only the diagonal elements of the reduced density matrix [25, 43]. That is, the complete set of (g, K) conditions is necessary and sufficient to ensure the A -representability of the g-density. 

Pi is an A -representable g-density if and only if it satisfies all of the Slater hull constraints. That is, is A -representable if and only if it... 

Although every g-density in the Slater hull is A-representable, not every g-matrix that satisfies the Slater hull constraints is A-representable. That is, if Pii...iQ =, g ,j satisfies the (g, K) conditions, some choices for the off-... 

Because the Slater hull constraints are insufficient to ensure A -representability, it is important to find additional methods for constraining the off-diagonal elements of the density matrix. Obtaining constraints that supersede the Slater hull requires considering Hamiltonians with a more general form than polynomials of number operators. As discussed in Section UFA, matters are especially simple if the Hamiltonian has nonnegative eigenvalues, because then the necessary conditions for A-representability take the form... 

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