Spin-adapted reduced hamiltonian

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. 

C. Valdemoro, Theory and practice of the spin adapted reduced Hamiltonian, in Density Matrices and Density Functionals (R. Erdahl and V. Smith, eds.). Proceedings of the A. J. Coleman Symposium, Kingston, Ontario, 1985, Reidel, Dordrecht, 1987. 

C. Valdemoro, Spin-adapted reduced Hamiltonian. 1 Elementary excitations. Phys. Rev. A 31, 2114 (1985). 

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. 

From the begining of the development of the RDM theory the need to render N-and 5-representable a 2-RDM obtained by an approximative method was patent. The development first of the spin-adapted reduced Hamiltonian methodology and, more recently, that of the second-order contracted Schrodinger equation rendered the solution of this problem urgent. The purification strategies... 

C. Valdemoro, M. P. de Lara-Castells, R. Bochicchio, and E. Perez-Romero, A relevant space within the spin-adapted reduced Hamiltonian theory. 1. Study of the BH molecule. Int. J. Quantum Chem. 65, 97 (1997). 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

So, to use the spin permutation technique we constructed the symmetry adapted lattice Hamiltonian in a compact operator form and essentially reduced the dimensionality of the corresponding eigenvalue problem. The effects of tpp 0 and the additional superexchange of copper holes are considered in [48]. 

Obviously, the sfss technique is not bounded to be applied only in AIMP calculations or in other valence-only calculations, but it can be used with any relativistic Hamiltonian which can be separated in spin-free and spin-dependent parts [48]. Being a very simple procedure, it is an effective means for the inclusion of dynamic correlation and size consistency in spin-orbit Cl calculations with any choice of Cl basis, such as determinants, double-group adapted configuration state functions, or spin-free Cl functions. In the latter case [46], the technique reduces to changing the diagonal elements of the spin-orbit Cl matrix. 

One can take advantage of any spin or spatial symmetry in the Hamiltonian by symmetry adapting the metric matrices and thereby reducing the size of the 2-RDM to be optimized [16]. For the ladder model, we transform the RDMs to bonding and antibonding spaces and then Fourier transform to take advantage of the translational symmetry. We consider linear combination of creation and annihilation operators to form two disjoint one-electron subspaces... 

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