Block-diagonal formulations

While previous variational 2-RDM calculations for electronic systems have employed the above formulation [20-31], the size of the largest block diagonal matrices in the 2-RDMs may be further reduced by using spin-adapted operators Ci in Eq. (9). Spin-adapted operators are defined to satisfy the following mathematical relations [54, 55] ... 

In this case, the variables for the primal SDP problem with free variables (Eq. (3)) and the dual SDP problem with equality constraints (Eq. (4)) are X,x) G S X IR and y G IR , respectively. Therefore the size of an SDP problem depends now on the size of each block-diagonal matrix of X, m, and s. We should also mention that the problem as represented by Eq. (4) is the preferred format for the dual SDP formulation of the variational calculation, which we present in the next section, too. 

The difficulty here is how to simultaneously constrain y and / y to be positive semidefinite. To formulate it as a primal SDP problem (Eq. (1)), we should express these two conditions as a positive semidefinite constraint over a single matrix let y be a block-diagonal matrix in which two symmetric matrices yj and y2 are arranged diagonally, and let us express the interrelation between these two matrices via linear constraints defined by the matrices Ap and the constants as in Eq. (1). That is. 

Table I (which can be deduced from Ref. [15]) shows the dimensions of the block-diagonal matrices of X and the number of linear equalities m in Eq. (1) relative to the number r of spin orbitals of a generic reference basis when employing the primal SDP formulation. It also considers conditions on oc electron number, total spin, and spin symmetries of the A-representability. In the table...

If we employ the dual SDP formulation and include the P, Q, G, Tl, and T2 conditions, the number of rows/columns of the largest block-diagonal matrices scale as 3r /16 again, while m scales as 3r" /64 and s as /A. 

The advantages of the dual SDP formulation are clear when comparing Tables I and II. First, notice that the sizes of the block-diagonal matrices are unchanged in both formulations. There is also an additional constraint = c in the dual SDP formulation, which is absent in the primal SDP formulation. Then, while the size m of equality constraint in the primal SDP formulation (see Eq. (1)) corresponds to the dimensions of the Q, G, Tl, and T2 matrices included in the formulation and scales as 25r /576, the dimension m of the variable vector y e R " in the dual SDP formulation (see Eq. (4)) corresponds to the dimension of the 2-RDM and scales merely as 3r" /64. The difference becomes more remarkable when more //-representabUity conditions are considered in these primal or dual SDP formulations. Computational implications when solving the SDP problems employing the primal and dual SDP formulations are discussed in Section V. 

We see from the way in which the effective rotational Hamiltonian is constructed that it is naturally expressed in terms of the angular momentum operator N. In the scientific literature, however, it is frequently written in terms of the vector R (which represents the rotational angular momentum of the nuclei) rather than N. While R = N — L occurs in the fundamental Hamiltonian (7.71), its use in the effective Hamiltonian is not satisfactory because R has matrix elements (due to L) which connect different electronic states and so is not block diagonal in the electronic states. In practice, authors who claim to be using R in their formulations usually ignore any matrix elements which they find inconvenient such as those of Lx and Ly. We shall return to this point in more detail later in this chapter. 

One of the central physical characteristics of the GMH model, the block diagonal form of the diabatic dipole-moment matrix, where a block includes all states charge-localized at a given site, may be compared with the picture emerging from alternative formulations of diabatic states, which (unlike the GMH model) do not involve any assumptions about dipole matrix elements, being based instead on the details of... 

There are various possible strategies for a formulation of this transformation [13], but for our purpose the FW transformation as a sequence of two transformations [49], is particularly recommended. Let us write the transformation operator Wpw as a product of two operators, such that the first one removes the non-diagonal blocks of D, while the second one reestablishes the normalization, such that the final transformation is unitary. 

Reconsider Problem 2.10, introducing a new basis in which Xi — Ofi + 3)/V2, X2 = X2, X3= (Xi X3)N2. Show that the matrix H is transformed to diagonal block form , the secular equations separating into two independent sets. Make a table to show how your basis functions and MOs behave under the symmetry operation of reflection across the plane bisecting the molecule. Express your results in matrix form, as in Problem 2.7. [Hint Express the new matrix elements ()f, h Xj) in terms of a and fi and formulate secular equations to determine the new mixing coefficients c,.]... 

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