Harmonic zero-point energy

A first consideration is the treatment of the bound vibrational modes, which in the TST/W results shown in Fig. 27.1, use the harmonic approximation. The total harmonic zero-point energy at the saddle point (for stretch and bend vibrations) is much higher for the Mu reaction, 10.2 kcal mobf than for the H reaction,... 

Within the harmonic approximation, one can easily obtain the total vibrational zero point energy, e t(s), and the vibrational partition function factor, Q (T,s), contained in Q (T,s) from the generalized normal mode frequencies discussed above. Specifically, can be written as the sum of the harmonic zero point energies in each generalized bound mode m, e9jin(s), while Q (T,s) can be written as the product of the harmonic vibrational partition functions for each generalized bound mode m, Qvib.m(T,s), where e j (s) and Qvib.m given by standard formulas found in almost any physical chemistry text. However, the... 

The first correction we can apply is the harmonic zero-point energy (ZPE) correction, which is the energy difference between the vibrational ground state and the equilibrium structure. This is given by... 

The difference between the harmonic zero-point energy (i.e., one-half the sum of the harmonic frequencies) and its anharmonic counterpart becomes quite non-negligible for larger molecules for example, in C2H4, the best zero-point energy amounts to 31.60 kcal mol compared with S/o>//2 = 31.98 kcal mol and L/v,72 = 30.87 kcal mol"7 While the differences may seem paltry in the context of chemical accuracy , in a calibration calculation they may obscure most or all of the remaining discrepancy between theory and experiment. 

The classical potential energy barrier. Including harmonic zero-point energies lowers the barrier by 0.3-0.5 kcal/mol. 

Tlris is the Schrodinger equation for a simple harmonic oscillator. The energies of the system are given by E = (i + ) x liw and the zero-point energy is Hlj. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The MP2/TZDP optimized structures were then used to calculate the stationary state geometry force constants and harmonic vibrational frequencies, also at the MP2 level. These results serve several purposes. Firstly, they test that the calculated geometry is really an energy minimum by showing all real frequencies in the normal coordinate analysis. Secondly, they provide values of the zero-point energy (ZPE) that can be used... 

The vibrational levels corresponding to n = 0,1,2... are evenly spaced. Like the particle confined to a line segment, the harmonic oscillator also has zero-point energy Eq = hu. 

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. 

An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48], These frequencies are used to evaluate the zero-point energy Ezpe and thermal effects. 

It has already been noted that the new quantum theory and the Schrodinger equation were introduced in 1926. This theory led to a solution for the hydrogen atom energy levels which agrees with Bohr theory. It also led to harmonic oscillator energy levels which differ from those of the older quantum mechanics by including a zero-point energy term. The developments of M. Born and J. R. Oppenheimer followed soon thereafter referred to as the Born-Oppenheimer approximation, these developments are the cornerstone of most modern considerations of isotope effects. 

Fig. 4.1 The zero point energy or low temperature approximation As temperature drops and u increases above u 4 the harmonic oscillator partition function Q (Harm. Osc.) is better and better approximated by the zero point energy term, exp(—u/2). For a typical CH stretching frequency, v = 3000 cm-1, u 4 at 1050 K and it is reasonable to use the ZPE approximation for that frequency at temperatures below 1000 k...

Pople JA, Scott AP, Wong MW, Radom L (1993) Scaling factors for obtaining fundamental vibrational frequencies and zero-point energies from HF/6-31G and MP2/6-31G harmonic frequencies. Israel J Chem 33 345-350... 

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