Matrices molecular orbital application

McWeeny, R., Proc. Roy. Soc. [London) A237, 355, (ii) "The density matrix in self-consistent field theory. II. Applications in the molecular orbital theory of conjugated systems."... 

A preliminary analysis of the absorption spectrum was given in Example 5.4-1 as an illustration of the application of the direct product (DP) rule for evaluating matrix elements, but the analysis was incomplete because at that stage we were not in a position to deduce the symmetry of the electronic states from electron configurations, so these were merely stated. A more complete analysis may now be given. The molecular orbitals (MOs)... 

Methods based on simulated Ab initio Molecular Orbital technique (SAMO) or cm the application of Linear Combination of Localized Orbitals have been proposed. The a priori advantages are a negligible cost (typically of the order of magnitude of an extended Huckel calculation) and the ab initio character of the approach. They suffer however from a rather tedious generation of a high number of matrix elements and it is still impossible to... 

For further details of the graph-theoretical aspects of HMO theory, see 1. Gutman and N. Trinajstic. Graph Theory and Molecular Orbitals , Fortschritte der Chemischen Forschung (Topics in Current Chemistry), 42, 49 (1973), and D. H. Rouvray, The Topological Matrix in Quantum Chemistry", Chapter 7 of Chemical Applications of Graph Theory (Editor A. T. Balaban), Academic Press, London, 1976. 

These third-order equations have been used in many applications in which molecular EAs have been computed for a wide variety of species as illustrated in Ref. [16]. Clearly, all the quantities needed to form the second- or third-order EOM matrix elements Hj. are ultimately expressed in terms of the orbital energies sj and two-electron integrals j, k l, h) evaluated in the basis of the neutral molecule s Hartree-Eock orbitals that form the starting point of the Mpller-Plesset theory. However, as with most electronic stmcture theories, much effort has been devoted to recasting the working EOM equations in a manner that involves the atomic orbital (AO) two-electron integrals rather than the molecular orbital based integrals. Because such technical matters of direct AO-driven calculations are outside the scope of this work, we will not delve into them further. 

The concept of the molecular orbital and their occupation is, however, not restricted to the HF model. It has much wider relevance and is applicable also for more accurate wave functions. For each wave function we can form the first-order reduced density matrix. This matrix is Hermitian and can be diagonalized. The basis for this diagonal form of the density matrix are the Natural Orbitals first introduced in quantum chemistry by Per-Olof Lowdin [4]. 

The antisymmetry of d (X) is a consequence of the orthonormality of the molecular orbitals, Eq. (43a). Here the adjective square has been emphasized in reference to the one-particle transition density matrix. The one particle transition density matrix is in general not symmetric, that is, the full or square matrix must be retained. However, in most electronic structure applications the associated one electron integrals, for example are symmetric, permitting the off-diagonal density matrix element to be stored in folded or triangular form. Since d is not symmetric, it is necessary to construct and store the transition density matrix in its unfolded or square form. 

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