Potential energy surfaces and MO calculations

We have made use of the PES in the discussion on TST without any elaboration on what a PES is. Here we define what is meant by PES, discuss how to compute the PES, and the pertinent results that may be extracted from it (e.g., the vibrational frequencies). 

Assume there is a function, Y, called a wavefunction, which is a complete 

Where T and Tei are the kinetic energy operators for the nuclei and electrons, and Vmi, Vne and Vee are the electrostatic potential energies arising from intemuclear, nucleus-electron, and interelectronic interactions. At this point, the Born-Oppenheimer approximation is invoked, i.e., the assumption that the movement of electrons is much faster than that of the nuclei and therefore the two can be decoupled from one another. This is mathematically represented by a separation of variables in the wavefunction, Y(R,r)= (r(R)) %(R), where 1 and % are the electronic and nuclear wavefunctions, respectively, and O is a function of r parameterized by R. Thus, with some rearrangement, Equation (21) becomes 

Equation (24) gives the electronic energy. The electronic wavefunctions O theoretically have an infinite basis but are in practice approximated by a wavefunction O with a finite basis. The size of the wavefunction and its ability to reproduce properties, such as the polarizability and dififuseness of the electron orbitals, have implications on the accuracy of the solutions. 

Vibrational frequencies may be extracted from the PES by performing a normal mode analysis. This analysis of the normal vibrations of the molecular configurations is a difficult topic and can be pursued efficiently only with the aid of group theory and advanced matrix algebra. In essence, the 3 translational, 3 rotational and 3N-6 vibrational modes (2 rotational and 3N-5 vibrational modes for linear molecules) may be determined by a coordinate transformation such that all the vibrations separate and become independent normal modes, each performing oscillatory motion at a well defined vibrational frequency. As a more concrete illustration, assume harmonic vibrations and separable rotations. The PES can thus be approximated by a quadratic form in the coordinates 

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