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# The Insignificance of Long-Distance Interactions for Molecular Frequencies. Analyses of Sparse Forceconstant Matrices

The experimentally observed group additivity of temperature dependent molecular
properties of organic molecules i.e. the heat capacity, suggests that molecular
vibrational frequencies primarily are determined by short-distance atomic interactions.
Based on the molecular graph a consistent hierarchy of molecular force constant (FC)
matrices has be constructed. These sparse and low-density matrices systematically
include the force constants for the uncoupled atomic displacement, neighbour (1-2),
next-neighbour (1-3) and longer-range atomic interactions. Such FC matrices were
constructed for a number of hydrocarbons and hetero atomic molecules, and
the error on the vibrational frequencies decrease rapidly with the inclusion of more
interactions in the matrices. With inclusion of 1-4 interactions average relative errors
between 1.3 % and 3.8 % are obtained based on B3LYP/6-311(d,p) calculations. The
largest relative errors are found for low frequency vibrations below 50 cm-1
. The
results were largely independent of the calculational level. The insignificance of the
long-distance interactions was confirmed by the improvement in the calculated
frequencies of hetero atomic molecules when FC matrices with only short-distance
interactions is complemented with matric elements from a FC matrix of a hydrocarbon
with similar geometry. Further confirmation is provided by calculations of molecular
frequencies from the FC matrices constructed from fragments and supplemented with
matrix elements for the interactions between atoms belonging to different fragments
taken from appropriate compounds. The difference between the vibrational frequencies
obtained in this way without the molecular FC matrix an those obtained from the full
FC matrix are within a few cm-1
.