Display of the Electric Potential

Suppose the molecule has N nuclei and n electrons then according to Coulomb s law the electric potential V u, (r) at point r arising from the nuclei is 

The motions of the nuclei (molecular vibrations) are small relative to the motions of the electrons here we consider the nuclei fixed in space, which defines an isolated vibrationless molecule. To obtain the electric potential V,.,(r) arising from the electrons we must integrate over the electronic motions. Let the gth electron be in the (p, state, and then the electric charge in the volume element dx given by the gth electron will be -. Therefore the electric 

In Eqs. [7] and [8], Z is the charge of the ath nucleus and e is the electronic charge (taken to be 1 in atomic units). The integration dr goes over all space. In terms of the density matrix defined by Eq. [2], the electric potential is 

Unlike net atomic charges, the molecular electric potential is a rigorously defined quantum mechanical property. If we can accurately calculate the density matrix by solving Schrodinger s equation, the electric potential V(r) can be obtained easily evaluation of Eq. [10] is much easier than the HF calculation. Furthermore, since the electric potential is the expectation value of a one-electron operator r - r the calculation of the electric potential is correct to one order higher than the wavefunction employed.  

Computer molecular display programs are available that draw contours or stipple patterns of the electric potential around molecules in color. For fast display the electric potential may be calculated from a Del Re model, or some other model based on electronegativity. Optionally, if sufficient computer power is available (or for small molecules) the exact operator electric potential from the molecular wavefunction may be displayed. Whereas these displays are quite valuable in showing qualitatively the location of positive and negative electric potential regions, they are normally not satisfactory for quantitative study of electrostatic interaction. 

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