Zero-point energy correction

That is, the sum of the electronic energy and nuclear repulsion energy of the molecule at the specified nuclear configuration. This quantity is commonly referred to as the total energy. However, more complete and accurate energy predictions require a thermal or zero-point energy correction (see Chapter 4, p. 68). 

We will provide you with the difference between the HF/6-31G(d) zero-point energy corrections for the two isomers, so you will not need to run frequency calculations ... 

How well do your results agree with the experimental value of about 3.4 kcal-mol i Since this observation is very approximate, we will not worry about zero-point energy corrections in this exercise. 

You ll need to run five calculations at each model chemistry oxygen atom, chlorine atom, O2, CIO and ozone (but don t forget that you can obtain lower-level energies from a higher-level calculation). Use the experimental geometries for the various molecules and the following scaled zero-point energy corrections ... 

The entries in this first section d the table perform geonMlry opiirniialions and compute zero-point energy corrections and final total enerm values th the same model chemistry. 

Compute the frequencies at each optimized geometry using the same method to obtain the zero point energy corrections. 

Evaluation of frequencies for zero-point energy corrections. 

The influence of zero-point energies is shown in Table 11.28. As the HF frequencies are systematically too high, the HF/6-31G(d,p) values are scaled by 0.92, and the MP2/ 6-31G(d,p) values are similarly scaled by 0.97. The change in stabilities by zero-point energy corrections is less than 2 kcal/mol, and the difference between the HF and MP2 values is less than 0.7 kcal/mol. 

Table 11.28 Zero-point energy corrections (kcal/mol)...

By assuming additivity in the style of the G2 procedure (Section 5.5), the CCSD(T)/ 6-31G(d,p) results may be combined with the changes due to basis set enlargement to 6-31 lG(2df,2pd) at the MP2 level and the zero-point energy corrections calculated at the MP2/6-31G(d,p) level. The results are shown in Table 11.31. From the observed accuracy of 2 kcal/mol for structures 2-8, the energetics of the species 9-11 may be assumed to be reliable to the same level of accuracy. 

There is an evolution with time the older calculations correspond to isolated molecules in the gas phase without any corrections, the more recent ones include solvent effects, with different approximations, and also some corrections, like ZPE (zero-point energy correction). The contributions of some authors to the understanding of tautomerism have been significant. Some of their contributions are collected in Table II. 

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

Table 3 shows that the small activation enthalpies of the reactions (3) and (4) are clearly affected by the zero point energy corrections. But the relative order of the activation enthalpies remains the same with or without the corrections. The activation entropies have great negative values, which is of mechanistic interest (see part 4.3.1). However, because of their similarity, when comparing the three reactions to one another they have only small importance, e.g. for estimation of copolymerization parameters (see part 4.3.2). 

In this paper we present the relative energies of the isomers of the phenylenediamines, dihydroxybenzenes and difluorobenzenes from ab initio calculations using large basis sets and including correlation corrections at the MP-2 level. These calculations were done at the geometry optimized structures. We also include zero-point energy corrections based on our calculated force fields. 

Note 0 The 6-31 G(d) basis set was used. Including zero-point energy corrections. 

Figure 5.54 displays the IRC reaction profile for the fluoride-exchange reaction (5.87). Because the reaction coordinate is the IRC rather than R. c, the barrier profile differs somewhat from that shown in Fig. 3.86. The activation energy is calculated to be 9.54 kcal mol-1 (without zero-point-energy correction), corresponding to an appreciable rate of chemical exchange under ambient thermal conditions.

The DFT results of Table II (which include the zero point energy correction) have been computed by considering the lowest values of the two sets of Table I. The results are clearly good for n=l and n=2, but wrong for higher n a clear indication that the minima we have reached are far from being close to the absolute ones. Therefore, the question remains whether for n=5, one water molecule is in a second hydration shell. 

The presentation of the theoretical BDEs is organized as follows. We list the calculated De values and their zero-point energy corrected counterparts (denoted by Do) for TM compounds that belong to different classes of molecules. This leads in some cases to double presentation, as some species belong to more than one class. However, we believe that the ordering chosen here facilitates comparison between compounds as well as between methods in a balanced way. 

For chemical purposes, substitution of total energy hypersurfaces by those based on the heat of formation seems more reasonable, with the difference given by the zero point energy corrections. However, their calculations from first principles can be rather cumbersome (12) and, moreover, for a given variation of some nuclear coordinates it usually can be assumed that the change in zero point energy is small compared to that of the total energy. On the other hand, se eral semiempirical quantum chemical procedures which are appropriately parametrized often yield satisfactory approximations for molecular heats of formation (10) and, therefore, AH hypersurfaces have become rather common. 

Fortunately, it is relatively simple to estimate from harmonic transition-state theory whether quantum tunneling is important or not. Applying multidimensional transition-state theory, Eq. (6.15), requires finding the vibrational frequencies of the system of interest at energy minimum A (v, V2,. . . , vN) and transition state (vj,. v, , ). Using these frequencies, we can define the zero-point energy corrected activation energy ... 

A final complication with the version of transition-state theory we have used is that it is based on a classical description of the system s energy. But as we discussed in Section 5.4, the minimum energy of a configuration of atoms should more correctly be defined using the classical minimum energy plus a zero-point energy correction. It is not too difficult to incorporate this idea into transition-state theory. The net result is that Eq. (6.15) should be modified... 

Here, A zp is the zero-point energy corrected activation energy defined in Eq. (6.22). At intermediate temperatures, Eq. (6.24) smoothly connects these two limiting cases. In many examples that do not involve H atoms, the difference between the classical and zero-point corrected results is small enough to be unimportant. 

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