2-positivity conditions

Harvest and connt the cells to be used for the experiment. For each antibody or experimental condition, label the tubes, one for the positive condition and one for the negative control see Note 3). 

Strength of positivity conditions Spin and spatial symmetry adaptation 1. Spin adaptation and S-representabiUty Open-shell molecules... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

Physically, the 3-positivity conditions restrict the probability distributions for three particles, two particles and one hole, one particle and two holes, and three holes to be nonnegative with respect to all unitary transformations of the one-particle basis set. These conditions have been examined in variational 2-RDM calculations on spin systems in the work of Erdahl and Jin [16], Mazziotti and Erdahl [17], and Hammond and Mazziotti [33], where they give highly accurate energies and 2-RDMs. 

Two different partial 3-positivity conditions have been proposed (i) the lifting conditions of Mazziotti [21, 33], and (ii) the T /T2 conditions of Erdahl [27, 34, 38]. The T1/T2 conditions have been implemented for molecules by Zhao et al. [27] and Mazziotti [34]. 

Because the addition of any two positive semidefinite matrices produces a positive semidefinite matrix, the four 3-positivity conditions [17] imply the following two less stringent constraints ... 

This dependence on ordering occurs because, unlike the set of operators and in the 3-positivity conditions, the operators do not include the set of single-particle excitation and deexcitation operators, that is,... 

The formal solution of Al-representability for the 2-RDM is developed in terms of a convex set of two-particle reduced Hamiltonian matrices. To complement the derivation of the positivity conditions from the metric matrices, we derive them from classes of these two-particle reduced Hamiltonian matrices. This interpretation allows us to demonstrate that the 2-positivity conditions are exact for certain classes of Hamiltonian operators for any interaction strength. In this section all of the ROMs are normalized to unity. Much of this discussion appeared originally in Refs. [21, 29]. 

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 ... 

Many methods in chemistry for the correlation energy are based on a form of perturbation theory, but the positivity conditions are quite different. Traditional perturbation theory performs accurately for all kinds of two-particle reduced Hamiltonians, which are close enough to a mean-field (Hartree-Fock) reference. There are a myriad of chemical systems, however, where the correlated wave-function (or 2-RDM) is not sufficiently close to a statistical mean field. Different from perturbation theory, the positivity conditions function by increasing the number of extreme two-particle Hamiltonians in which are employed as constraints upon the 2-RDM in Eq. (50) and, hence, they exactly treat a certain convex set of reduced Hamiltonians to all orders of perturbation theory. For the... 

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. 

Our discussion may readily be extended from 2-positivity to p-positivity. The class of Hamiltonians in Eq. (70) may be expanded by permitting the G, operators to be sums of products of p creation and/or annihilation operators for p > 2. If the p-RDM satisfies the p-positivity conditions, then expectation values of this expanded class of Hamiltonians with respect to the p-RDM will be nonnegative, and a variational RDM method for this class will yield exact energies. Geometrically, the convex set of 2-RDMs from p-positivity conditions for p > 2 is contained within the convex set of 2-RDMs from 2-positivity conditions. In general, the p-positivity conditions imply the (7-positivity conditions, where q < p. As a function of p, experience implies that, for Hamiltonians with two-body interactions, the positivity conditions converge rapidly to a computationally sufficient set of representability conditions [17]. 

The A -representability conditions on the 2-RDM can be systematically strengthened by adding some of the 3-positivity constraints to the 2-positivity conditions. For three molecules in valence double-zeta basis sets Table II shows... 

D. A. Mazziotti, Solution of the 1,3-contracted Schrodinger equation through positivity conditions on the 2-particle reduced density matrix. Phys. Rev. A 66, 062503 (2002). 

G. Gidofalvi and D. A. Mazziotti, Variational reduced-density-matrix theory strength of Hamiltonian-dependent positivity conditions. Chem. Phys. Lett. 398, 434 (2004). 

J. R. Hammond and D. A. Mazziotti, Variational two-electron reduced-density-matrix theory partial 3-positivity conditions for A-representability. Phys. Rev. A 71, 062503 (2005). 

The appropriate modification of the 2-RDM may be accomphshed by combining A-representability constraints, known as positivity conditions, with both the unitary and the cumulant decompositions of the 2-RDM. 

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