Symmetry orbital integrals

If symmetry orbitals are used in place of atomic orbitals, then and 8ik will become integrals over these orbitals and they will have to be broken down to integrals over the atomic orbitals before the Hiickel approximations are made. 

Integrals and Matrix Elements over Symmetry Orbitals 4.1 Selection Rules... 

Integrals over Atomic Orbitals and Symmetry Orbitals... 

We now consider one-electron integrals over symmetry orbitals. 

A two-electron integral over symmetry orbitals for D2h is given by... 

The fivefold symmetry has been reduced to symmetry through a plane. These symmetry orbitals can be found easily, merely by redoing the calculations using slightly modified values for C, (e.g. 1.01/7 for its resonance integral). 

Note The MOs of an ML complex can be found by interacting the metal orbitals with the symmetry orbitals of the ligands.21 The foregoing method, where we calculate the Hiickel MOs and then zero the resonance integral, permits symmetry orbitals to be obtained without recourse to group theory. 

In order to proceed we first need to know the atomic basis functions from which we can construct the symmetry orbitals and how to evaluate the one-centre and multi-centre integrals of H and S. Finally one has to find an efficient and accurate way to describe the molecular potential in the self-consistent-field calculations. 

All two-center, two-electron integrals were set to the integral over s symmetry orbitals, a requirement for rotational invariance discussed above ... 

By incorporating these symmetries in the 4-spinor basis functions, as we have done in our BERTHA code [50-54], we can make substantial computational economies in computing interaction integrals. The angular stracture of Dirac 4-spinors described here is also exploited by the major computer package TSYM, which utilizes projection operators to construct relativistic molecular symmetry orbitals for double valued representations of point groups [77-79]. 

As we shall see in a later chapter, if we make the further constraint on the SCF method that the molecular orbitals are symmetry-adapted, then the above technique can be extended in such a way that only the symmetry-distinct integrals need be computed and stored What is more, the restriction to permutational symmetries can be removed so that the full molecular point group symmetry can be used, which may involve operations which induce linear combinations among the basis functions. 

Independently of the use of symmetry to facilitate the calculation of energy integrals in the LCAO approximation, it is always possible to reduce the dimension of the matrix equations involved in an effective-Hamiltonian model of molecular electronic structure by the use of any molecular symmetry. The prototype of these approximations is the LCAOSCF model and in this chapter we look at the effects of using symmetry orbitals on the implementation of this model. 

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