Reduced Hamiltonian matrix

As in the previous section, by connected we mean all terms that scale linearly with N. Wedge products of cumulant RDMs can scale linearly if and only if they are connected by the indices of a matrix that scales linearly with N transvec-tion). In the previous section we only considered the indices of the one-particle identity matrix in the contraction (or number) operator. In the CSE we have the two-particle reduced Hamiltonian matrix, which is defined in Eqs. (2) and (3). Even though the one-electron part of scales as N, the division by A — 1 in Eq. (3) causes it to scale linearly with N. Hence, from our definition of connected, which only requires the matrix to scale linearly with N, the transvection... 

In order to regulate the convergence of the iterative process by damping or accelerating its rate according to convenience, the basic procedure has been further refined [88]. This is achieved by substituting the numerical reduced Hamiltonian matrix by a new one... 

For calculating the reduced Hamiltonian matrix element B, we need the diagonal matrix elements llRj, and Hfp, and the overlap between R and P ... 

For printing purposes, only the reduced Hamiltonian matrix, <[J,K-K+ ett J, KJK+y, is shown in Fig. III.11, and the reader should keep in mind that the matrix elements indicated by the letter Z are diagonal in M while the matrix elements indicated with the letter S are off diagonal in M with M = M 1. 

Fig. III. 11. Pattern of nonvanishing elements of the reduced Hamiltonian matrix. en for ethyleneoxide. The symmetry species of the rotational states...

The Hamiltonian matrix factorizes into blocks for basis functions having connnon values of F and rrip. This reduces the numerical work involved in diagonalizing the matrix. 

Section II discusses the real wave packet propagation method we have found useful for the description of several three- and four-atom problems. As with many other wave packet or time-dependent quantum mechanical methods, as well as iterative diagonalization procedures for time-independent problems, repeated actions of a Hamiltonian matrix on a vector represent the major computational bottleneck of the method. Section III discusses relevant issues concerning the efficient numerical representation of the wave packet and the action of the Hamiltonian matrix on a vector in four-atom dynamics problems. Similar considerations apply to problems with fewer or more atoms. Problems involving four or more atoms can be computationally very taxing. Modern (parallel) computer architectures can be exploited to reduce the physical time to solution and Section IV discusses some parallel algorithms we have developed. Section V presents our concluding remarks. 

Since the reduced Hamiltonian in this case has a single isolated critical point, the transformed monodromy matrix, in the y basis, becomes... 

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

After K steps of recursion, the Lanczos vectors can be arranged into an N x K matrix Q, which reduces the Hamiltonian matrix to a tridiagonal form ... 

Because many physical systems possess certain types of symmetry, its adaptation has become an important issue in theoretical studies of molecules. For example, symmetry facilitates the assignment of energy levels and determines selection rules in optical transitions. In direct diagonalization, symmetry adaptation, often performed on a symmetrized basis, significantly reduces the numerical costs in diagonalizing the Hamiltonian matrix because the resulting block-diagonal structure of the Hamiltonian matrix allows for the separate... 

Molecules, in general, have some nontrivial symmetry which simplifies mathematical analysis of the vibrational spectrum. Even when this is not the case, the number of atoms is often sufficiently small that brute force numerical solution using a digital computer provides the information wanted. Of course, crystals have translational symmetry between unit cells, and other elements of symmetry within a unit cell. For such a periodic structure the Hamiltonian matrix has a recurrent pattern, so the problem of calculating its eigenvectors and eigenvalues can be reduced to one associated with a much smaller matrix (i.e. much smaller than 3N X 3N where N is the number of atoms in the crystal). 

Hamiltonians involving more than two electron interactions. I shall use this to illustrate the general case of arbitrary p. The second-order reduced density matrix (2-RDM) of a pure state ij/, a function of four particles, is defined as follows ... 

Direct minimization of the energy as a functional of the p-RDM may be achieved if the p-particle density matrix is restricted to the set of Al-represen-table p-matrices, that is, p-matrices that derive from the contraction of at least one A-particle density matrix. The collection of ensemble Al-representable p-RDMs forms a convex set, which we denote as P. To define P, we first consider the convex set of p-particle reduced Hamiltonians, which are... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

Hamiltonian may be lifted by Eq. (48) to an (r — A )-hole Hamiltonian, which shares the same ground-state as the A -particle Hamiltonian. A similar lifting may be extended to the G-form of the reduced Hamiltonian, but the procedure is slightly more subtle since the G-matrix combines the particle and the hole perspectives. 

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 ... 

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

The matrix K is the reduced Hamiltonian [22, 25] and has the same symmetry properties as the two-electron matrix that is. 

Nakatsuji [37] in 1976 first proved that with the assumption of N-representability [3] a 2-RDM and a 4-RDM will satisfy the CSE if and only if they correspond to an A-particle wavefunction that satishes the corresponding Schrodinger equation. Just as the Schrodinger equation describes the relationship between the iV-particle Hamiltonian and its wavefunction (or density matrix D), the CSE connects the two-particle reduced Hamiltonian and the 2-RDM. However, because the CSE depends on not only the 2-RDM but also the 3- and 4-RDMs, it cannot be solved for the 2-RDM without additional constraints. Two additional types of constraints are required (i) formulas for building the 3- and 4-RDMs from the 2-RDM by a process known as reconstruction, and (ii) constraints on the A-representability of the 2-RDM, which are applied in a process known as purification. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

The unitary decomposition may be applied to any Hermitian, antisymmetric two-particle matrix including the 2-RDM, the two-hole RDM, and the two-particle reduced Hamiltonian. The decomposition is also readily generalized to treat p-particle matrices [80-82]. The trial 2-RDM to be purified may be written... 

J. Karwowski, W. Duch, and C. Valdemoro, Matrix-elements of a spin-adapted reduced Hamiltonian. Phys. Rev. A 33, 2254 (1986). 

As laid out in the previous section, the quest for necessary conditions for A-representabihty reduces to a quest for Hamiltonians whose ground-state energy is known. As shown in Eq. (15), the Hamiltonian can then be shifted so that its ground-state energy is zero, so that a necessary condition for the A-representability of the g-matrix can be written in terms of the reduced Hamiltonian... 

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