Property derivatives dipole moments

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

The most important electric properties are dipole moment and polarizability derivatives. The theory of dipole moment derivatives has been worked out by Bratoz (1958) and by Gerratt and Mills (1968). Both papers use the obvious definition of the dipole moment derivative the change of the dipole moment with nuclear coordinates. Komornicki and Mclver (1979) have pointed out, in an influential paper, that the alternative definition, namely the derivative of the geometrical force with respect to the electric field, is more useful, as there are only three field components versus 3N nuclear coordinates. Similarly, the polarizability derivatives can be defined as the second derivatives of the forces with respect to the field components. Komornicki and Mclver (1979) suggest numerical differentiation with respect to the field... 

The symmetiy selection rales aie derived by studying the effect of synunetry operations on die matrix elements. The rales stem from the property of dipole moment matrix elements to be invariant with respect to a symmetiy operation (R)... 

The function/( C) may have a very simple form, as is the case for the calculation of the molecular weight from the relative atomic masses. In most cases, however,/( Cj will be very complicated when it comes to describe the structure by quantum mechanical means and the property may be derived directly from the wavefunction for example, the dipole moment may be obtained by applying the dipole operator. 

When you perform a single point semi-empirical or ab initio calculation, you obtain the energy and the first derivatives of the energy with respect to Cartesian displacement of the atoms. Since the wave function for the molecule is computed in the process, there are a number of other molecular properties that could be available to you. Molecular properties are basically an average over the wave function of certain operators describing the property. For example, the electronic dipole operator is basically just the operator for the position of an electron and the electronic contribution to the dipole moment is... 

This is not a unique way of classifying molecular properties. For example, Dykstra et al. (1990) concentrate on the response of a system to an apphed external field the electric dipole moment can be defined as the first derivative of the energy with respect to the field, and so on. I will stick with the Boys and Cook nomenclature as a broad basis for discussion. 

Besides these response properties of a molecule we will also devote one section in this chapter to the experimentally important infrared intensities, which are needed to complement the theoretically predicted frequencies for the complete computational simulation of an IR spectrum. This discussion belongs in the present chapter because the infrared intensities are related to the derivative of the permanent electric dipole moment p with respect to geometrical parameters. 

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