Construction of All Representations

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

These density matrices are themselves quadratic functions of the Cl coefficients and they reflect all of the permutational symmetry of the determinental functions used in constructing T they are a compact representation of all of the Slater-Condon rules as applied to the particular CSFs which appear in Tk They contain all information about the spin-orbital occupancy of the CSFs in Tk The one- and two- electron integrals < (f>i I f I (f>j > and < (f>i(f>j I g I ( >k4>i > contain all of the information about the magnitudes of the kinetic and Coulombic interaction energies. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

Any one-parameter set of transformations obeying this rule under successive applications can be said to be a representation of the dilatation group. Thus relation (8.5) identifies the RG mapping constructed in Sect. 8.1 as a representation of dilatations in the space of models parameterized by /, n, / . We now work out some properties of the dilatation group represented in some general space with coordinates Y = Yj = Yi. Y , Y3,.... This space for instance may be the space of all Hamiltonians, the coordinates Y> being the coupling constants. 

At this point we are sufficiently equipped to consider briefly the methods used to approximate the wave functions constructed in the restricted subspace of orbitals. So far the only approximation was to restrict the orbital basis set. It is convenient to establish something that might be considered to be the exact solution of the electronic structure problem in this setting. This is the full configuration interaction (FCI) solution. In order to find one it is necessary to construct all possible Slater determinants for N electrons allowed in the basis of 2M spin-orbitals. In this context each Slater determinant bears the name of a basis configuration and constructing them all means that we have their full set. Then the matrix representation of the Hamiltonian in the basis of the configurations ( >K is constructed ... 

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