Hamiltonian positive semidefinite

When N = p, the set B simply contains the p-particle reduced Hamiltonians, which are positive semidefinite, but when N = p + 1, because the lifting process raises the lowest eigenvalue of the reduced Hamiltonian, the set also contains p-particle reduced Hamiltonians that are lifted to positive semidefinite matrices. Consequently, the number of Wrepresentability constraints must increase with N, that is, B C B. To constrain the p-RDMs, we do not actually need to consider all pB in B, but only the members of the convex set B, which are extreme A member of a convex set is extreme if and only if it cannot be expressed as a positively weighted ensemble of other members of the set (i.e., the extreme points of a square are the four corners while every point on the boundary of a circle is extreme). These extreme constraints form a necessary and sufficient set of A-representability conditions for the p-RDM [18, 41, 42], which we can formally express as... 

Any arbitrary one-particle reduced Hamiltonian shifted by its A-particle ground-state energy must be expressible by the extreme HamUtonian elements in the convex set As we showed in Eq. (52), keeping the 1-RDM positive semide-finite is equivalent to applying the Al-representability constraints in Eq. (50) for the class of extreme positive semidefinite which may be parameterized by... 

Because is the A -particle energy and not the lowest eigenvalue of K, some of the eigenvalues of C will be negative, and this portion of the reduced Hamiltonian cannot be represented by the positive semidefinite Hamiltonians in Eq. (56). 

The three complementary representations of the reduced Hamiltonian offer a framework for understanding the D-, the Q-, and the G-positivity conditions for the 2-RDM. Each positivity condition, like the conditions in the one-particle case, correspond to including a different class of two-particle reduced Hamiltonians in the A-representability constraints of Eq. (50). The positivity of arises from employing all positive semidefinite in Eq. (50) while the Q- and the G-conditions arise from positive semidefinite and B, respectively. To understand these positivity conditions in the particle (or D-matrix) representation, we define the D-form of the reduced Hamiltonian in terms of the Q- and the G-representations ... 

If the G-matrix is positive semidefinite, then the above expectation value of the G-matrix with respect to the vector of expansion coefficients must be nonnegative. Similar analysis applies to G, operators expressible with the D- or Q-matrix or any combination of D, Q, and G. Therefore variationally minimizing the ground-state energy of n (H Egl) operator, consistent with Eq. (70), as a function of the 2-positive 2-RDM cannot produce an energy less than zero. For this class of Hamiltonians, we conclude, the 2-positivity conditions on the 2-RDM are sufficient to compute the exact ground-state A-particle energy on the two-particle space. 

For the next step, we show how we consider the N-representability conditions for the 1-RDM y for a system with N particles that is all of its eigenvalues should be between zero and one [17]. In other words, this condition is equivalent to saying that y and / — y are positive semidefinite, where / is the identity matrix. Assuming that H is the one-body Hamiltonian, we have... 

In order to get significant results, the initial data must be formed by a set of clearly non-A -representable second-order matrices, which would generate upon contraction a closely ensemble A -representable 1-RDM. It therefore seemed reasonable to choose as initial data the approximate 2-RDMs built by application of the independent pair model within the framework of the spin-adapted reduced Hamiltonian (SRH) theory [37 5]. This choice is adequate because these matrices, which are positive semidefinite, Hermitian, and antisymmetric with respect to the permutation of two row/column indices, are not A -representable, since the 2-HRDMs derived from them are not positive semidefinite. Moreover, the 1-RDMs derived from these 2-RDMs, although positive semidefinite, are neither ensemble A -representable nor 5-representable. That is, the correction of the N- and 5-representability defects of these sets of matrices (approximated 2-RDM, 2-HRDM, and 1-RDM) is a suitable test for the two purification procedures. Attention has been focused only on correcting the N- and 5-representability of the a S-block of these matrices, since the I-MZ purification procedure deals with a different decomposition of this block. 

Positive semidefinite Hamiltonians can be constructed by taking an operator that depends on g creation/annihilation operators and multiplying the operator... 

Since it is obviously impossible to require that Tr[Pg AfFg] > 0 for every choice of Pq, one imposes this constraint only for a few operators. Moreover, because one needs to be able to prove that the operators are positive semidefinite, the operators that are selected for use as constraints are typically much simpler than a molecular Hamiltonian. This is unfortunate, because if one could ensure that Tr[Hg Fg] > Egs Hff) for the Hamiltonian of interest, then the computational procedure would be exact. Future research in V-representability might focus on developing constraints based on molecular considerations. 

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.)... 

In Eq. (7.5), (x) and S(x) are the coarse-grained energy and entropy functions, respectively. The antisymmetric operator L defines a generalized Poisson bracket A, = L that possesses the same properties as the classical Poisson bracket described above. The last term in Eq. (7.5) is new compared to Hamiltonian dynamics (7.4) and describes dissipative, irreversible phenomena. The friction matrix M is symmetric and positive, semidefinite. Together with the degeneracy... 

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