Truncated moment expansion, electrical

Electrical moments are useful because at long distances from a molecule the total electronic distribution can be increasingly well represented as a truncated multipole expansion, and thus molecular interactions can be approximated as multipole-multipole interactions (charge-charge, charge-dipole, dipole-dipole, etc.), which are computationally particularly... 

However, Eq. (2.21) is not very convenient in the context of intramolecular electrostatic interactions. In a protein, for instance, how can one derive the electrostatic interactions between spatially adjacent amide groups (which have large local electrical moments) In principle, one could attempt to define moment expansions for functional groups that recur with high frequency in molecules, but such an approach poses several difficulties. First, there is no good experimental way in which to measure (or even define) such local moments, making parameterization difficult at best. Furthermore, such an approach would be computationally quite intensive, as evaluation of the moment potentials is tedious. Finally, the convergence of Eq. (2.20) at short distances can be quite slow with respect to the point of truncation in the electrical moments. 

The three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

Using harmonic oscillator wavefiinctions as basis functions, truncating the Taylor expansion of the geometry dependence of the electric and magnetic dipole moments at linear order as done in Eq. 2.28 (in some contexts referred to as the Placzek approximation [249]) and following the discussion in Section 2.1.3 for the two vibrational transition moments, the rotational strength can be shown to be... 

As mentioned in the introduction, most of the QM calculations of vibrational frequencies are performed within the double-harmonic approximation, that is, the truncation of the expansion of the potential energy as a function of the nuclear coordinates to the quadratic term (mechanical harmonic approximation) and the consideration of the hnear term only in the expansion of the dipole moment as a function of the nuclear coordinates (electric harmonic approximation). In such a framework, the QM calculation of vibrational frequencies can be reduced to the evaluation of the components of the Hessian matrix followed by diagonalization of the corresponding mass-weighted matrix [1]. Let us start by writing out the energy of a... 

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