Zero point anharmonicity

Anharmonicity Internal Modes, Effect of Zero Point Anharmonicity... 

After transforming to Cartesian coordinates, the position and velocities must be corrected for anharmonicities in the potential surface so that the desired energy is obtained. This procedure can be used, for example, to include the effects of zero-point energy into a classical calculation. 

The corresponding level broadening equals half. In fact is the diagonal kinetic coefficient characterizing the rate of phonon-assisted escape from the ground state [Ambegaokar 1987]. In harmonic approximation for the well the only nonzero matrix element is that with /= 1,K0 Q /> = <5o, where is the zero-point spread of the harmonic oscillator. For an anharmonic potential, other matrix elements contribute to (2.52). 

The zero-point vibrational energy (Ezpv) is obtained from harmonic B3LYP/VTZ+1 frequencies scaled by 0.985 in the case of Wl theory. For W2 theory, anharmonic values of Ezpv from quar-tic force fields at the CCSD(T)/VQZ+1 (or comparable) level are preferred where this is not feasible, the same procedure as for Wl theory is followed as a fallback solution . 

W1/W2 theory and their variants would appear to represent a valuable addition to the computational chemist s toolbox, both for applications that require high-accuracy energetics for small molecules and as a potential source of parameterization data for more approximate methods. The extra cost of W2 theory (compared to W1 theory) does appear to translate into better results for heats of formation and electron affinities, but does not appear to be justified for ionization potentials and proton affinities, for which the W1 approach yields basically converged results. Explicit calculation of anharmonic zero-point energies (as opposed to scaling of harmonic ones) does lead to a further improvement in the quality of W2 heats of formation at the W1 level, the improvement is not sufficiently noticeable to justify the extra expense and difficulty. 

It is instructive to calculate the anharmonic correction to the zero point energy contribution to fractionation factors for isotope exchange equilibria involving hydrogen and deuterium. For example consider the exchange... 

Finally, the zero point vibration corrections (SET V) use to be much larger than the pseudopotential corrections. In the present case, these zero point corrections seems to give rise to unrrealistic values, probably because of the harmonic approximation used in the calculations. The torsion mode as well as its interactions with the remaining modes are indeed very anharmonic. 

Though the anharmonic components of the thermal motion decrease rapidly with temperature, as described in chapter 2, they will be present to some extent even if the motion is reduced to zero-point vibrations. 

The final term in (4.67) is a constant, and affects all vibrational levels equally. We see from (4.72) that oo is a result of anharmonicity in U. The molecular zero-point vibrational energy is... 

It should be noted that anharmonic ft corrections, in particular to the zero-point energy, are ignored in this section. 

Calculations of vibrational frequencies are never accurate enough to verify that the secondary IE arises entirely from zero-point energies. Therefore although they do confirm a role for zero-point energies, which was never at issue, they cannot exclude the possibility of an additional inductive effect arising from changes of the average electron distribution in an anharmonic potential. The question then is whether it is necessary to invoke anharmonicity to account for a part of these secondary IEs. 

In summary, Equation (9) can account for secondary IEs on acidity, in terms of changes of vibrational frequencies and zero-point energies. There is no need to invoke anharmonicity or inductive effects. 

---