Polyatomic system extended formulation

Thus, with these proofs two important additions to the previous traditional understanding of the JTE emerged (1) Any polyatomic system may be subject to the JTE, and (2) if there are instabilities and distortions of high-symmetry configurations, they are due to, and only to the JTE. Together with the previously achieved understanding of the role of quadratic terms of the vibronic coupling, the extended formulation of the JTE that includes the latest achievements in this field is as follows [1,8] ... 

The extended formulation of the JTE above - the law of instability - states that the necessary and sufficient condition for instability of the high-symmetry configuration of any polyatomic system is the presence of two or more electronic states... 

Furthermore their incidence are very dependent upon the nature of the properties [52,53], Due to eomputer limitations, basis sets cannot be extended indefinitely and direct numerical evaluations seem the ultimate solution for molecules [54], In position space this is a viable alternative for diatomic molecules [55,56], but it cannot be extended easily to polyatomic systems. Formulated in momentum space, the HF equations have not explicit solutions and the difficulties to express them in terms of basis functions are analogous to those encountered in the r-space. However the momentum space HF equations give way to numerical approaches in which Coulombic interactions become tractable even for polyatomic molecules [7] among other advantages, these equations, Eqs. 13 and 20, do not require coordinate systems adapted to the geometry of the molecules to remove Coulombic singularities. In both equations the only singular contribution comes from the q 2 factor. 

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