Heat of formation errors

Table 3.1 Average heat of formation error in kcaL/mol (number of compounds)...

AfHg = Author s suggested value for the heat of formation. Error = — AfH ... 

The apparent accuracy of 20-40 kJ/mol for calculating heats of formation with semi-empirical methods is slightly misleading. Normally the interest is in relative energies of different species, and since the heat of formation errors are essentially random, relative energies may not be predicted as well (two random errors of 40kJ/mol may... 

A more useful quantity for comparison with experiment is the heat of formation, which is defined as the enthalpy change when one mole of a compound is formed from its constituent elements in their standard states. The heat of formation can thus be calculated by subtracting the heats of atomisation of the elements and the atomic ionisation energies from the total energy. Unfortunately, ab initio calculations that do not include electron correlation (which we will discuss in Chapter 3) provide uniformly poor estimates of heats of formation w ith errors in bond dissociation energies of 25-40 kcal/mol, even at the Hartree-Fock limit for diatomic molecules. 

Many problems with MNDO involve cases where the NDO approximation electron-electron repulsion is most important. AMI is an improvement over MNDO, even though it uses the same basic approximation. It is generally the most accurate semi-empirical method in HyperChem and is the method of choice for most problems. Altering part of the theoretical framework (the function describing repulsion between atomic cores) and assigning new parameters improves the performance of AMI. It deals with hydrogen bonds properly, produces accurate predictions of activation barriers for many reactions, and predicts heats of formation of molecules with an error that is about 40 percent smaller than with MNDO. 

Heats of formation, molecular geometries, ionization potentials and dipole moments are calculated by the MNDO method for a large number of molecules. The MNDO results are compared with the corresponding MINDO/3 results on a statistical basis. For the properties investigated, the mean absolute errors in MNDO are uniformly smaller than those in MINDO/3 by a factor of about 2. Major improvements of MNDO over MINDO/3 are found for the heats of formation of unsaturated systems and molecules with NN bonds, for bond angles, for higher ionization potentials, and for dipole moments of compounds with heteroatoms. 

Some typical errors in heat of formation for the MNDO, AMI and PM3 methods are given in Table 3.1. The exact numbers of course depend on which, and how many, compounds have been selected for comparison, thus the numbers should only be taken as a guideline for the accuracy expected. Some typical errors in bond distances are given in Table 3.2. 

It should be noted that the G2-1 data set, with two exceptions (SO2 and CO2), only includes data for molecules containing one or two heavy (non-hydrogen) atoms. It is likely that the typical error for a given model to a certain extent depends on the size of the system, i.e. the G2 method is presumably not able to predict the heat of formation of... 

For the gas hydrates it is not possible to make an entirely unambiguous comparison of the observed heat of hydrate formation from ice (or water) and the gaseous solute with the calculated energy of binding of the solute in the ft lattice, because AH = Hfi—Ha is not known. If one assumes AH = 0, it is found that the hydrates of krypton, xenon, methane, and ethane have heats of formation which agree within the experimental error with the energies calculated from Eq. 39 for details the reader is referred to ref. 30. 

Extension of the method to nonisostructural metal halides, some of which yield erroneous AHf values via Bom-Haber cycles, is shown in Fig. 1. All curves are nonlinear with the bow increasing in the expected order T1(I) < Pb(II) < Bi(III) < Ag(I). For the first transition metal dihalides, however, straight lines can be drawn within the limits of enthalpy errors except for Zn(II) or Mn(II) salts. Thus heats of formation of the fluorides can be extrapolated linearly from the other three halides to a first approximation. 

In an overall assessment, the established semiempirical methods perform reasonably for the molecules in the G2 neutral test set. With an almost negligible computational effort, they provide heats of formation with typical errors around 7 kcal/mol. The semiempirical OM1 and OM2 approaches that go beyond the MNDO model and are still under development promise an improved accuracy (see Table 8.1). 

In our own validation sets, experimental heats of formation are preferentially taken from recognized standard compilations [38-40]. If there are enough experimental data for a given element, we normally only use reference values that are accurate to 2 kcal/mol. If there is a lack of reliable data, we may accept experimental heats of formation with a quoted experimental error of up to 5 kcal/mol. This choice is motivated by the target accuracy of the established semiempirical methods. If experimental data are missing for a small molecule of interest, we consider it legitimate [18] to employ computed heats of formation from high-level ab initio methods as substitutes. 

In the course of the MNDO/d development [15-18] we have generated new validation sets for second-row and heavier elements. Those for Na, Mg, Al, Si, P, S, Cl, Br, I, Zn, Cd, and Hg have been published [16-18], The corresponding statistical evaluations for heats of formation [18] are summarized in Table 8.3. It is obvious that MNDO/d shows by far the smallest errors followed by PM3 and AMI. All four semiempirical methods perform reasonably well for normalvalent compounds, especially when considering that more effort has traditionally been spent on the parameterization of the first-row elements. For hy-pervalent compounds, however, the errors are huge in MNDO and AMI, and still substantial in PM3, in spite of the determined attempt to reduce these errors in the PM3 parameterization [20], Therefore it seems likely that the improvements in MNDO/d are due to the use of an spd basis set [16-18]. 

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