Eigenvector matrix

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

This is a quantity which can be easily constructed given a set of molecular orbitals (the coefficients C ) and a precalculated set of atomic orbital integrals. At this point, the Hartree-Fock equations have been reduced to a matrix eigenvector problem, FC = SCe, but not in a computationally convenient form. Following the analysis leading to equation 84, we first define the transformed Fock matrix as... 

The second order electronic density matrix can be constructed from the Hamiltonian matrix eigenvector given in eq(5) above to give... 

This can be seen to have a macroscopically large eigenvalue signifying a condensate of electron pairs. All yj (Mt) low-lying states have the same density matrix eigenvector and eigenvalue. 

Consequently the thermally averaged second order reduced electronic density matrix eigenvector is... 

E.R. Davidson. Matrix Eigenvector Methods in Methods in Computational Molecular Physics (ed. G.H.F. Dierksen and S. Wilson) Reidel, Dordrecht, 1983. 

The computational procedure involves obtaining the matrix representation of the symmetry operators of the (2n + 2) site chain in the direct product basis. The matrix representation of both J and P for the new sites in the Fock space is known from their definitions. Similarly, the matrix representations of the operators J and P for the left (right) part of the system at the first iteration are also known in the basis of the corresponding Fock space states. These are then transformed to the density matrix eigenvectors bcisis. The matrix representation of the symmetry operators of the full system in the direct product space are obtained as the direct product of the corresponding matrices ... 

Figure 1 Optical gap (in unit of t) as a function of inverse chain length for Hubbard chains with U = 4.0t and U = 6.0t m corresponds to the number of density matrix eigenvectors retained in the DMRG procedure.

A. Graovac, D. PlavSid, M. Kaufman, T. Pisanski, and E. C. Kirby, Application of the adjacency matrix eigenvectors method to geometry determination of toroidal carbon molecules, J. Chem. Phys. 113 (2000) 1925-1931. 

Finally, the calculation of the square matrix eigenvector is composed by the relative weights of the elements considered. However, before using the weights, a consistency test should be performed, as indicated by Saaty (1983) and Alonso (2006). 

Matrix.eigenvalueO and Matrix.eigenvector() - Code for eigenvalue and eigenvector problems. 

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