Molecular geometry, dependence Hamiltonian integrals

The Hamiltonian integrals depend on the molecular geometry in two ways. The first is trivial and arises because the Coulomb interactions between the electrons and the nuclei depend on the geometry. The second is more complicated and arises because the orbitals are themselves functions of the geometry. The reason for this is that the MOs are expanded in a finite set of AOs fixed on the nuclear centers. A consequence of using a finite set of AOs is that we are presented with a different basis set at each geometry. 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

The one-electron Hamiltonian only involves differences of geometrical parameters the nuclear attraction terms involve the electron-nucleus distance not the absolute position of either particle. Likewise the molecular integrals dependence on molecular geometry is only via inter-centre distance the fact that the basis functions are atom-centred does not induce any dependence of the integrals on absolute position of the integrals. 

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