Um estudo da autoconsistência da matriz Y como condição para N-representabilidade da estrutura eletrônica de átomos e moléculas

Abstract

We are interested in confirrning the self consistency of writing the density rnatrix of order one, y, when written in terrns of the Natural Transition Orbitals, as well as being a reduction of the rnatrix of order two; that is: i) Y (x/x)¹=EiXi(x)nixxi(x')=2/N-1 (xx2(x'x/2) Ax2 The Natural Transition Orbitals, xi' are eigenvectors of the operator P, ii) r Xi = Ei Xi F is the non-hermition operator which represents the effective interaction between the electrons, when correlation effects are included, as well as the kinetic energy and nuclear interaction terrns. The E1s are the eigenvectors of the F operator and are interpreted as the ionization energies of the spin orbitals Xi. The ni's are occupation numbers, whits values between 0 and 1 iii) 0<ni<1 At the sarne time we require the matrix of arder two, r, to be herrnitian, antisyrnetric and of trace N(N-1)/2, iv) tr r = N(1-1)/2 and its diagonal elements restricted to values bétween O and 1/2. During all of our researches, in which we utilized various minimization programs, we observed that the above conditions were insufficient to guarantee N-representability of the matrix r. ln addition, we utilized an approximation for the second - arder densi ty rnatrix, as being an antisymmetric prcduct of density matrices of first order. As a result of this approximation we obtained an operator P which is hermitian and includes the Coulomb and exchange interactions explicitly. For this approximation of r we took care to include a coefficient of normalizatíon in order to maintain the proper trace for r. Even so, we found the surprising result that the total energy, after minimizing the energy was below the experimental energy.