Zero-point energy factors

Equation 4.79a points out the Reduced Isotopic Partition Function Ratio (RPFR) may be considered as the product of three factors the product factor (PF), the excitation factor (EXC), and the zero-point energy factor (ZPE). Note that in terms of RPFR s, the isotope effects corresponding to Equations 4.65, 4.66, and 4.68 can be written... 

Thus the secondary and primary deuterium isotope effects determined in this study also indicate that entropy and zero-point energy factors associated with breaking of the C—D bond and migration in the activated complexes are important for the structural isomerization to yield butene-1 and butene-2, and reaction path B (equation 197) must be included in the mechanistic considerations concerning the cyclopropane isomerization. But the higher activation energy for isobutane formation (g = 64.3 kcal mol" ) than that for butene-2 or butene-1 (Q = 62.0 0.6 kcal mol" ) indicates also that the rupture of... 

Here the zero point energy is temporarily suppressed. Now the exponential is a product of independent factors. Thus one gets... 

The second, third, and fourth corrections to [MPd/b-Jl lG(d,p)] are analogous to A (- -). The zero point energy has been discussed in detail (scale factor 0.8929 see Scott and Radom, 1996), leaving only HLC, called the higher level correction, a purely empirical correction added to make up for the practical necessity of basis set and Cl truncation. In effect, thermodynamic variables are calculated by methods described immediately below and HLC is adjusted to give the best fit to a selected group of experimental results presumed to be reliable. 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

When comparing calculated results to thermodynamic quantities extrapolated to zero Kelvin, the zero point energy needs to be added to the total energy. As with the frequencies themselves, this predicted quantity is scaled to eliminate known systematic errors in frequency calculations. Accordingly, if you have not specified a scale factor via input to the Reodlsotopes option, you will need to multiply the values in the output by the appropriate scale factor (see page 64). 

The carbon dioxide zero-point energies in the table are scaled, using the scaling factors listed on page 64. ... 

RB3LYP/6-31 + G //RB3LYP/6-31 + G, cf. (99JOC 3113) A//r includes the zero-point energy correction scaling factor, 0.98. 

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. 

The first factor is responsible for normal isotope effects, which arise because the bonds being affected by deuteriation are weakened in the transition state, but the absolute effect is greater on the bonds to deuterium rather than protium because the former have higher vibrational frequencies (typically by a factor of ca 1.37). This factor essentially reflects zero-point energy effects, so it becomes progressively more important at lower internal energies. 

The MMI (mass moment of inertia), EXC (excitation factor), and ZPE (zero point energy) terms are defined on successive lines of Equation 4.145. For reactions involving heavier isotopes the effects are no longer concentrated in the ZPE term and it is convenient to apply the Teller-Redlich product rule (Section 3.5.1) and eliminate the moments of inertia by using Equations 4.79,4.79a, and 4.141, thus obtaining an equivalent relation... 

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... 

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... 

Fig. 1 Lewis and Funderburk found that the H/D primary kinetic isotope effects (25 °C in aqueous t-butyl alcohol) for proton abstraction from 2-nitropropane by pyridine derivatives all exceed the maximum isotope effect that could have been derived from the isotopic difference in reactant-state zero-point energies alone (a value around 7). The magnitude of the isotope effect increases with the degree of steric hindrance to reaction presented by the pyridine derivative, the identical results for 2,6-lutidine and 2,4,6-collidine ruling out any role for electronic effects of the substituents. The temperature dependence shown for 2,4,6-collidine is exceedingly anomalous the pre-exponential factor Ahis expected to be near unity but is instead about 1/7, while the value of AH — AH = 3030 cal/mol would have generated an isotope effect at 25 °C of 165 if the pre-exponential factor had indeed been unity.

In a crystalline antiferromagnet the moment on each ion is less than it would be on the free ion. There are two separate phenomena involved here. One is the zero-point energy of the spin waves, which reduces the moment on each ion by a factor (Anderson 1952, Ziman 1952)... 

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