Weinhold-Wilson constraints

The (2, 2) constraints are associated with the first three Weinhold-Wilson constraints [23]. This is clear when we rewrite the inequalities in terms of the... 

These inequalities are the fourth and fifth Weinhold-Wilson constraints. These (2, 3) inequalities have a probabilistic interpretation very similar to the (2, 2) inequalities. Returning to the fundamental number operators form of the (3, 3) inequalities (cf. Eq. (45)), we see that Eq. (48) was generated from... 

In general, it is difficult to derive the Q, R) conditions directly. An exception occurs for the R, K) constraints, which have an especially simple form based on the positive semidefinite Hamiltonian in Eq. (29). Fortunately, the Q, R) conditions (Q < K) are easily derived from the (/ , R) conditions [26]. In Section III.D we used this result to derive the Weinhold-Wilson constraints on the diagonal elements of the 2-matrix [23]. (The Weinhold-Wilson constraints are identical to the (2, 3) conditions.)... 

Constraints on the diagonal element of the density matrix can be useful in the context of the density matrix optimization problem, Eq. (8). As Weinhold and Wilson [23] stressed, the A-representability constraints on the diagonal elements of the density matrix have conceptually appealing probabilistic interpretations this is not true for most of the other known A-representability constraints. 

The second and third inequalities in Eq. (43) are the same, except for a permutation of the indices.) Weinhold and Wilson use the fact that p, represents the probability of observing an electron in orbital i and py represents the probability that orbitals i and j are both occupied to show that each of these constraints has a straightforward probabilistic interpretation. 

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