Off-diagonal force constant

Inclusion of electron correlation normally lowers the force constants, since the correlation energy increases as a function of bond length. This usually means that vibrational frequencies decrease, although there are exceptions (vibrational frequencies also depend on off-diagonal force constants). The values calculated at the MP2 and CCSD(T) levels are shown in Tables 11.14 and 11.15. 

For a non-linear molecule of N atoms, there are 3N — 6 (3N — 5 if the molecule is linear) internal vibrational coordinates. If we wish to include the off-diagonal force constants then there are (3N — 6) diagonal and (3N — 6)2 — (3N — 6) off-diagonal terms. Only half of the off-diagonal force constants are unique, since (for example) kco.cs must equal kcs.co- In other words, the force constant matrix has to be symmetric. This gives 1/2(3JV — 6)(3N - 5) independent force constants, a number that usually far exceeds the available experimental vibration frequencies. A complete determination of all force constants requires analysis of the spectra of many isotopically substituted molecules. Many of the off-diagonal terms turn out to be very small, and spectroscopists have developed systematic simplifications to the force field in order to make as many of the off-diagonal terms as possible, vanish. 

The diagonal and off-diagonal force constants of the internal harmonic force field... 

Because of their small magnitudes, all off-diagonal force constants are ignored. The Urey-Bradley force field (10) takes into account 1-3 nonbonded interactions, Eq. (5). If/,/, and/" are harmonic force constants, then the Urey-Bradley potential is given by... 

Occupation number 103, 119, 203 Off-diagonal force constant 39 One-electron operator 86 Onsager model 259 Open shell 118... 

By putting all but one off-diagonal force constant equal to zero and that one equal to unity, this test is not difficult to apply. 

Ab initio calculations produce a potential energy surface, from whose second derivatives at the minima one can obtain all of the quadratic (diagonal and off-diagonal) force constants. This comprises a complete force field, but problems exist ... 

From the previous equation, we can conclude that the spectral width increases with the displacements (related to the gradient g), the difference between the diagonal force constants in the final state F r and the squared frequencies in the initial state co, and finally the Duschinsky mixings (related to the off-diagonal force constants F )... 

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