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(which are of course of the covalenttype) is explained in the book by Foresman and Frisch [1e]. Energy calculations are discussed further in section 5.5.2.

5.5APPLICATIONS OF THE AB INITIO METHOD

An extremely useful book by Hehre [32] discusses critically the merits of various computational levels (ab initio and others) for calculating molecular properties, and contains a wealth of information, admonitory and tabular, on this general subject.

5.5.1 Geometries

It is probably the case that the two parameters most frequently sought from ab initio calculations (and most semiempirical and DFT calculations too) are geometries and (section 5.5.2) energies, although this is not to say that other quantities, like vibrational frequencies (section 5.5.3) and parameters arising from electron distribution (section 5.5.4) are unimportant. Molecular geometries are important: they can reveal subtle effects oftheoretical importance, and in designing new drugs or materials [82] the shapes of the candidates for particular roles must be known with reasonable accuracy – e.g. docking a putative drug into the active site of an enzyme requires that we know the shape of the drug and the active site. While the creation of new pharmaceuticals or materials can be realized with the aid of molecular mechanics (chapter 3) or semiempirical methods (chapter 6), the increasingly facile application of ab initio techniques to large molecules makes it likely that this method will play a more important role in such utilitarian pursuits. Novel molecules of theoretical interest can be studied reliably only by ab initio methods, or possibly by density functional theory (chapter 7), which is closer in theoretical tenor to the ab initio, rather than semiempirical, approach. The theory behind geometry optimizations was outlined in section 2.4, and some results of optimizations with different basis sets and electron correlation methods have been given (sections 5.3.3 and 5.4). Extensive discussions of the virtues and shortcomings of various ab initio levels for calculating geometries can be found in [1e,g,38].

Molecular geometries or structures refer to the bond lengths, bond angles, and dihedral angles that are defined by two, three and four, respectively, atomic nuclei. In speaking of the distance, say, between two “atoms” we really mean the internuclear distance, unless we are considering nonbonded interactions, when we might also wish to examine the separation of the van der Waals surfaces. In comparing calculated and experimental structures we must remember that calculated geometries correspond to a fictional frozen-nuclei molecule, one with no ZPE (section 5.2.3.6d), while experimental geometries are averaged over the amplitudes of the various vibrations [83].

Furthermore, different methods measure somewhat different things. The most widelyused experimental methods for finding geometric parameters are X-ray diffraction, electron diffraction and microwave spectroscopy. X-ray diffraction determines geometries in a crystal lattice, where they may be somewhat different than in the gas phase to which ab initio reactions usually apply (although structures and energies can be calculated taking solvent effects into account, as explained in section 5.5.2). X-ray diffraction depends on the scattering of photons by the electrons around nuclei, while

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electron diffraction depends on the scattering of electrons by the nuclei, and microwave spectroscopy measures rotational energy levels, which depend on nuclear positions. Neutron diffraction, which is less used than these three methods, depends on scattering by atomic nuclei.

The main differences are between X-ray diffraction (which probes nuclear positions via electron location) on the one hand and electron diffraction, microwave spectroscopy and neutron diffraction (which probe nuclear positions more directly), on the other hand. The differences result from (1) the fact that X-ray diffraction measures distances between mean nuclear positions, while the other methods measure essentially average distances, and (2) from errors in internuclear distances caused by the nonisotropic (uneven) electron distribution around atoms. The mean vs. average distinction is illustrated here:

Suppose that nucleus A is fixed and nucleus B is vibrating in an arc as indicated. The distance between the mean positions is r (shown), but on the average B is further away than r.

Differences resulting from nonisotropic electron distribution are significant only for H–X bond lengths: X-rays see electrons rather than nuclei, and the simplest inference of a nuclear position is to place it at the center of a sphere whose surface is defined by the electron density around it. However, since a hydrogen atom has only one electron, for a bonded hydrogen there is relatively little electron density left over from covalent sharing to blanket the nucleus, and so the proton, unlike other nuclei, is not essentially at the center of an approximate sphere defined by its surrounding electron density:

Clearly, the X-ray-inferred H–X distance will be less than the internuclear distance measured by electron diffraction, neutron diffraction, or microwave spectroscopy, methods which see nuclei rather than electrons. These and other sources of error that can arise in experimental bond length measurements (like bond length, bond angles and dihedral angles will obviously also depend on nuclear positions) are detailed by Burkert and Allinger [84], who mention nine (!) kinds of internuclear distance r, and a comprehensive reference to the techniques of structure determination may be found in the book edited by Domenicano and Hargittai [85a]. Despite all these problems with defining and measuring molecular geometry (see e.g. [85b], we will adopt the position that it is meaningful to speak of experimental geometries to within 0.01 Å for bond lengths, and to within 0.5° for bond angles and dihedrals [86].

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Let us briefly compare HF/3-21G, HF/6-31G* and MP2/6-31G* geometries. Figure 5.23 gives bond lengths and angles calculated at these three levels and experimental bond lengths and angles, for 20 molecules. The geometries shown in Fig. 5.23 are analyzed in Table 5.7, and Table 5.8 provides information on dihedral angles in eight molecules. There should be little difference between MP2(full) geometries and the MP2(FC) geometries used here. This (admittedly limited) survey suggests that:

geometries are almost as good as HF/6-31G* geometries.

MP2/6-31G* geometries are on the whole slightly but significantly better than HF/6-31G* geometries, although individual MP2 parameters are sometimes a bit worse.

and HF/6-31G* C–H bond lengths are consistently slightly (ca. 0.01–0.03 and ca. 0.01 Å, respectively) shorter than experimental, while MP2/6-31G* C–H bond lengths are not systematically overor underestimated.

HF/6-31G* O–H bonds are consistently slightly (ca. 0.01 Å) shorter than experimental, while MP2/6-31G* O–H bond lengths are consistently slightly (ca. 0.01 Å) longer. O–H bond lengths are not consistently overor underestimated.

None of the three levels consistently overor underestimates C–C bond lengths. HF/6-31G* C–X (X = O, N, Cl, S) bond lengths tend to be underestimated

slightly (ca. 0.015 Å) while MP2/6-31G* C–X bond lengths may tend to be slightly (ca. 0.01 Å) overestimated. C–X bond lengths are not consistently overor underestimated.

HF/6-31G* bond angles may tend to be slightly larger (ca. 1°) than experimental, while MP2/6-31G* angles may tend to be slightly (0.7°) smaller.

bond angles are not consistently overor underestimated. Dihedrals do not seem to be consistently over-or underestimated by any of the three levels. The level breaks down completely for HOOH, where a dihedral angle of 180°, far from the experimental 119.1°, is calculated; omitting this error of 61° and the

suspect because of its anomalously large deviation from all three calculated results, and because it is among those dihedrals which are said to be suspect or having a large or unknown error (designated X in Harmony et al. – see reference in Table 5.8). The error for the HOOH dihedral represents a clear failure of the level and is an example of a case which provides an argument for using the 6-31G* rather than the 3-21G basis, although the latter is much faster and often of comparable accuracy (of course with correlated methods like MP2 a smaller basis than 6-31G* should not be used, as pointed out in section 5.4). The errors in calculated dihedral angles are ca. 2–3° for HF/6-31G*, and ca. 2° for MP2/6-31G*: omitting the dihedral errors of 8.6° and 5.9° from the sample lowers the HF error from 2.9° to 2.3° and the MP2 error from 2.3 to 1.9.

The accuracy of ab initio geometries is astonishing, in view of the approximations present: the basis set is small and the 6-31G* is only moderately large, and so these probably cannot approximate closely the true wavefunction; the HF method does not account properly for electron correlation, and the MP2 method is only the simplest

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Ab initio calculations 257

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approach to handling electron correlation; the Hamiltonian in both the HF and the MP2 methods used here neglects relativity and spin-orbit coupling. Yet with all these approximations the largest error (Table 5.7) in bond lengths is only 0.033 A level for HCHO) and the largest error in bond angles is only 3.2° level for

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the MP2/6-31G* level. The mean error in 18 bond angles is only 1.3° and 1.0° at the

dral) is 2.9° (HF/6-31G*) and 2.3° (MP2/6-31G*). If we agree that errors in calculated bond lengths, angles and dihedrals of up to 0.02 Å, 3° and 4° respectively correspond

fied basis used in CBS calculations – section 5.5.2.2b) are 0.012, 0.016 and 0.015 Å, respectively [86].

The calculations summarized in Tables 7 and 8 are in reasonable accord with conclusions based on information available ca. 1985 and given by Hehre, Radom, Schleyer and Pople [87]: HF/6-31G* parameters for A–H, A/B single and A/B multiple bonds are usually accurate to 0.01, 0.03 and 0.02 A, respectively, bond angles to ca. 2° and dihedral angles to ca. 3°, with HF/3-21G(*) values being not quite as good. MP2 bond lengths appear to be somewhat better, and bond angles are usually accurate to ca. 1 °, and dihedral angles to ca. 2°. These conclusions from Hehre et al., hold for molecules composed of first-row elements (Li to F) and hydrogen; for elements beyond the first

ones is not that the geometries are much better, but rather that for a stationary point, MP2 optimizations followed by frequency calculations are more likely to give the correct curvature of the potential energy surface (chapter 2) for the species than are HF optimizations/frequencies. In other words, the correlated calculation tells us more reliably whether the species is a relative minimum or merely a transition state (or even a higher-order saddle point; see chapter 2). Thus fluorodiazomethane [71] and several oxirenes [44] are (apparently correctly) predicted by MP2 optimizations to be not minima, while HF optimizations indicate them to be minima. The interesting hexaazabenzene is predicted to be a minimum at the HF/6-31G* level, but a hilltop with two imaginary frequencies at the MP2/6-31G* level [88]. For transition states, in contrast to ground states, we don’t have experimental geometries, but correlation effects can certainly be important for their energies (section 5.5.2.2b), and MP2/6-31G* geometries for transition states are probably significantly better in general than HF/6-31G* ones.

Suppose we want something better than “fairly good” structures? Experienced workers in computational chemistry have said [89]

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When we speak of “accurate” geometries, we generally refer to bond lengths that are within about 0.01–0.02 Å of experiment and bond and dihedral angles that are within about 1–2° of the experimentally-measured value (with the lower end of both ranges being more desirable).

Even by these somewhat exacting criteria, MP2/6-31G* and even HF/6-31G* calculations are not, in the cases studied here, far wanting; the worst deviations from experimental values seem to be for dihedral angles, and these may be the least reliable experimentally. However, since some larger deviations from experiment are seen in our sample, it must be conceded that HF/6-31G* and even MP2/6-31G* calculations cannot be relied on to provide “accurate” (sometimes called high-quality) geometries. Furthermore, there are some molecules that are particularly recalcitrant to accurate calculation of geometry (and sometimes other characteristics); two notorious examples are FOOF (dioxygen difluoride) and ozone (these have been described as “pathological” [90]). Here are the HF/6-31G*, MP2(FC)/6-31G* and experimental [91] geometries:

The errors in the calculated geometries are (HF/6-31G*/MP2/6-31G*):

FOOF: FO length, –0.208/–0.080 A; OO length, 0.094/0.076 Å

FOO angle, –3.7°/–2.6°

FOOF dihedral, –3.4°/–1.7°

OO length, –0.068/0.028 Å

OOO angle, 2.2/–0.5

These calculated geometries do not satisfy even our “fairly good” criterion and are well short of being “accurate”; the bond lengths are particularly bad. Using the 6-311++G** basis (for FOOF, 88 vs. 60 basis functions; for 66 vs. 45 basis functions) we get for calculated geometries (errors) using HF/6-311 + +G**:

FOOF: FO length, 1.353 Å (–0.222); OO length, 1.300 (0.083) Å

FOO angle, 106.5° (–3.0)

FOOF dihedral, 85.3° (–2.2)

OO length, 1.194 Å (–0.078)

OOO angle, 119.4° (2.6)

Thus with a much larger basis, but still using the Hartree-Fock method, the FOOF geometry is about the same and the geometry has become even worse than at the HF/6-31G* level! Better geometries were obtained by going beyond the MP2 correlational level; we get for calculated geometries (errors) using CCSD(T)/6-31G*:

FOOF: FO length, 1.539 Å (–0,036); OO length, 1.276 (0.059) Å