Cramer C.J. Essentials of Computational Chemistry Theories and Models

6.2	BASIS SETS	171
Table 6.1 STO-3G 2sp basis set for oxygen
α2sp	c2s	c2p
5.0331527	−0.099967	0.155916
1.1695944	0.399513	0.607684
0.3803892	0.700115	0.391957

defined for each type of orbital core through valence. Thus for H and He, there is only a 1s function. For Li to Ne, there are five functions, 1s, 2s, 2px , 2py , and 2pz. For Na to Ar, 3s, 3px , 3py , and 3pz are added to the second-row set, making a total of nine functions, etc. This number is the absolute minimum required, and it is certainly nowhere near the infinite basis set limit. Other minimal basis sets include the MINI sets of Huzinaga and co-workers, which are named MINI-1, MINI-2, etc., and vary in the number of primitives used for different kinds of functions.

One way to increase the flexibility of a basis set is to ‘decontract’ it. That is, we might imagine taking the STO-3G basis set, and instead of constructing each basis function as a sum of three Gaussians, we could construct two basis functions for each AO, the first being a contraction of the first two primitive Gaussians, while the second would simply be the normalized third primitive. This prescription would not double the size of our basis set, since we would have all the same individual integrals to evaluate as previously, but the size of our secular equation would be increased. A basis set with two functions for each AO is called a ‘double-ζ ’ basis. Of course, we could decontract further, and treat each primitive as a fullfledged basis function, in which case we would have a ‘triple-ζ ’ basis, and we could then decide to add more functions indefinitely creating higher and higher multiple-ζ basis sets. Modern examples of such basis sets are the cc-pCVDZ, cc-pCVTZ, etc. sets of Dunning and co-workers, where the acronym stands for ‘correlation-consistent polarized Core and Valence (Double/Triple/etc.) Zeta’ (Woon and Dunning 1995); correlation consistency and polarization are described in more detail below.

The advantage of such a scheme is, naturally, that these increasingly large basis sets must come closer and closer to the HF limit. Let us step back for a moment, however, and consider the chemical consequences of providing extra basis functions for a given AO. Recall that a final MO from an HF calculation is a linear combination of all of the basis functions. Indeed, if we were to examine the 1s core orbital resulting from an HF calculation on atomic oxygen using the fully uncontracted set of STO-3G Gaussian primitives as a basis (i.e., a triple-ζ basis), we might well find it to be a linear combination of the 1s functions very similar to that defining the STO-3G contracted oxygen 1s function. And, if we were to look at the MOs resulting from an equivalent calculation on, say, formaldehyde (H2C=O), we would probably find another orbital which we would assign as the oxygen 1s orbital having very similar AO coefficients. Indeed, we would find this same orbital little changed in almost any molecule incorporating oxygen we might choose to examine. The reason for this is that core orbitals are only weakly affected by chemical bonding.

Valence orbitals, on the other hand, can vary widely as a function of chemical bonding. Atoms bonded to significantly more electronegative elements take on partial positive charge

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from loss of valence electrons, and thus their remaining density is distributed more compactly. The reverse is true when the bonding is to a more electropositive element. From a chemical standpoint, then, there is more to be gained by having flexibility in the valence basis functions than in the core, and recognition of this phenomenon led to the development of so-called ‘split-valence’ or ‘valence-multiple-ζ ’ basis sets. In such basis sets, core orbitals continue to be represented by a single (contracted) basis function, while valence orbitals are split into arbitrarily many functions.

Amongst the most widely used split-valence basis sets are those of Pople et al. These basis sets include 3-21G, 6-21G, 4-31G, 6-31G, and 6-311G. The nomenclature is a guide to the contraction scheme. The first number indicates the number of primitives used in the contracted core functions. The numbers after the hyphen indicate the numbers of primitives used in the valence functions – if there are two such numbers, it is a valence-double-ζ basis, if there are three, valence-triple-ζ . This notation is somewhat more informative than the older style noted in the previous section. Thus, for a calculation on water, for instance, the 6-311G basis would be represented (11s5p/5s)/[4s3p/3s]. The latter notation does not specify how many primitives are devoted to which contracted basis functions, while 6-311G makes this point clear. Like the STO-MG basis sets, the split-valence sets use sp basis functions having common exponents.

An interesting question arises for split-valence and multiple-ζ basis sets: how should one go about choosing exponents and coefficients for the contracted functions? As the basis is no longer minimal, there is no particular virtue in fitting to STOs (which were originally used because they were thought to represent the optimal single-function approximation to an AO). Pople and co-workers, like most other researchers in the field, relied on the variational principle. That is, some test set of atoms and/or molecules was established, and exponents and coefficients were optimized so as to give the minimum energy over the test set. In the end, just as the name of a force field refers to its functional form and a list of all its parameters, so too the name of a basis set refers to its contraction scheme and a list of all of its exponents and coefficients for each atom.

One feature of the Pople basis sets is that they use a so-called ‘segmented’ contraction. This implies that the primitives used for one basis function are not used for another of the same angular momentum (e.g., no common primitives between the 2s and 3s basis functions for phosphorus). Such a contraction scheme is typical of older basis sets. Other segmented split-valence basis sets include the MIDI and MAXI basis sets of Huzinaga and co-workers, which are named MIDI-1, MIDI-2, etc., MAXI-1, MAXI-2, etc. and vary in the number of primitives used for different kinds of functions.

The Pople basis sets have seen sufficient use in the literature that certain trends have clearly emerged. While a more complete discussion of the utility of HF theory and its basisset dependence appears at the end of this chapter, we note here that, in general, the 4-31G basis set is inferior to the less expensive 3-21G, so there is little point in ever using it. The 6-21G basis set is obsolete.

An alternative method to carrying out a segmented contraction is to use a so-called ‘general’ contraction (Raffenetti 1973). In a general contraction, there is a single set of primitives that are used in all contracted basis functions, but they appear with different coefficients

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in each. The general contraction scheme has some technical advantages over the segmented one. One advantage in terms of efficiency is that integrals involving the same primitives, i.e., those occurring in the final line of Eq. (6.4), need in principle be calculated only once, and the value can be stored for later reuse as needed. Examples of split-valence basis sets using general contractions are the cc-pVDZ, cc-pVTZ, etc. sets of Dunning and co-workers, where the acronym stands for ‘correlation-consistent polarized Valence (Double/Triple/etc.) Zeta’ (Dunning 1989; Woon and Dunning 1993). The ‘correlation-consistent’ part of the name implies that the exponents and contraction coefficients were variationally optimized not only for HF calculations, but also for calculations including electron correlation, methods for which are described in Chapter 7. The subject of polarization is what we turn to next.

6.2.4Polarization Functions

The distinction between atomic orbitals and basis functions in molecular calculations has been emphasized several times now. An illustrative example of why the two should not necessarily be thought of as equivalent is offered by ammonia, NH3. The inversion barrier for interconversion between equivalent pyramidal minima in ammonia has been measured to be 5.8 kcal mol−1. However, a HF calculation with the equivalent of an infinite, atomcentered basis set of s and p functions predicts the planar geometry of ammonia to be a minimum-energy structure!

The problem with the calculation is that s and p functions centered on the atoms do not provide sufficient mathematical flexibility to adequately describe the wave function for the pyramidal geometry. This is true even though the atoms nitrogen and hydrogen can individually be reasonably well described entirely by s and p functions. The molecular orbitals, which are eigenfunctions of a Schrodinger¨ equation involving multiple nuclei at various positions in space, require more mathematical flexibility than do the atoms.

Because of the utility of AO-like GTOs, this flexibility is almost always added in the form of basis functions corresponding to one quantum number of higher angular momentum than the valence orbitals. Thus, for a first-row atom, the most useful polarization functions are d GTOs, and for hydrogen, p GTOs. Figure 6.3 illustrates how a d function on oxygen can polarize a p function to improve the description of the O–H bonds in the water molecule. The use of p functions to polarize hydrogen s functions has already been mentioned in Section 4.3.1. [An alternative way to introduce polarization is to allow basis functions not to be centered on atoms. Such floating Gaussian orbitals (FLOGOs) are illustrated on the left-hand side of Figure 4.1. While the use of FLOGOs reduces the need to work with integrals involving high-angular-momentum functions, the process of geometry optimization is rendered considerably more complicated, so they are rarely employed in modern calculations.] Adding d functions to the nitrogen basis set causes HF theory to predict correctly a pyramidal minimum for ammonia, although some error in prediction of the inversion barrier still exists even at the HF limit because of the failure to account for electron correlation.

A variety of other molecular properties prove to be sensitive to the presence of polarization functions. While a more complete discussion occurs in Section 6.4, we note here that d functions on second-row atoms are absolutely required in order to make reasonable

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O

HH

O	+	O	=	O

HH

Figure 6.3 The MO formed by interaction between the antisymmetric combination of H 1s orbitals and the oxygen px orbital (see also Figure 6.7). Bonding interactions are enhanced by mixing a small amount of O dxz character into the MO

predictions for the geometries of molecules including such atoms in formally hypervalent bonding situations, e.g., phosphates, sulfoxides, siliconates, etc.

A variety of empirical rules exist for choosing the exponent(s) for a set of polarization functions. If only a single set is desired, one possible choice is to make the maximum in the radial density function, r2 , equal to that for the existing valence set (e.g., the 3d functions that best ‘overlap’ the 2p functions for a first-row atom – note that the radial density is used instead of the actual overlap integral because the latter, by symmetry, must be zero).

Because of the expense associated with adding polarization functions – the total number of functions begins to grow rather quickly with their inclusion – early calculations typically made use of only a single set. Pople and co-workers introduced a simple nomenclature scheme to indicate the presence of these functions, the ‘*’ (pronounced ‘star’). Thus, 6- 31G implies a set of d functions added to polarize the p functions in 6-31G. A second star implies p functions on H and He, e.g., 6-311G (Krishnan, Frisch, and Pople 1980).

Subsequent work has shown that there is a rough correspondence between the value of adding polarization functions and the value of decontracting the valence basis function(s). In particular, there is a rough equality between each decontraction step and adding one new set of polarization functions, including a new set of higher angular momentum. Put more succinctly, ‘balanced’ double-ζ basis sets should include d functions on heavy atoms and p functions on H, triple-ζ basis sets should include 1 set of f and 2 sets of d functions on heavy atoms, and 1 set of d and 2 sets of p functions on H, etc. This is the polarization prescription adopted by the cc-pVnZ basis sets of Dunning and co-workers already mentioned above, where n ranges over D (double), T (triple), Q (quadruple), five, and six (Wilson, van Mourik, and Dunning 1996). Thus, for cc-pV6Z, for example, each heavy atom has one i function, two h functions, three g functions, four f functions, five d functions, and six valence s and p functions, in addition to core functions (using the canonical numbers of these functions, we have 140 basis functions for a single second-row atom, so this basis set presently finds use only for the smallest of systems). Note that while it would be an unpleasant exercise to try to draw an i function, it is straightforwardly defined by taking the sum of i, j , and k equal to 6 in Eq. (6.2).

A somewhat more detailed analysis of the correct ratio of number of polarization functions to number of valence functions has been carried out by Jensen (2001) in the context of the

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polarization consistent basis sets pc-n, where n indicates the largest increment in angular momentum above the valence maximum used in the basis set. The ratio is not fixed, per se, but varies depending on an analysis of energetic convergence over a test set of first-row molecules. The pc-n basis sets are presently defined for H−F for n = 0–4.

Recognizing the tendency to use more than one set of polarization functions in modern calculations, the standard nomenclature for the Pople basis sets now typically includes an explicit enumeration of those functions instead of the star nomenclature. Thus, 6-31G(d) is to be preferred over 6-31G because the former obviously generalizes to allow names like 6-31G(3d2fg,2pd), which implies heavy atoms polarized by three sets of d functions, two sets of f functions, and a set of g functions, and hydrogen atoms by two sets of p functions and one of d (note that since this latter basis set is only valence double-ζ , it is somewhat unbalanced by having so many polarization functions).

A partially polarized basis set, MIDI! (where the ‘!’ is pronounced ‘bang’; in some electronic structure programs, the abbreviation MIDIX is employed to avoid complications associated with interpretation of the exclamation point), has been introduced by Cramer and Truhlar and co-workers, who adopted a different philosophy in its development (Easton et al. 1996; Li, Cramer, and Truhlar 1998). Rather than optimizing the basis set with respect to molecular energies, they sought to design an economical basis set for geometry optimizations and partial charge calculations on medium-sized molecules, including neutrals, cations, and anions, with special emphasis on functional groups that are important for biomolecules. The MIDI! basis set has d functions on all atoms heavier than H for which it is defined with the exception of carbon (i.e., on heteroatoms). Although much smaller than the 6-31G(d) basis set, for instance, in direct comparisons it yields more accurate geometries and charges as judged by comparison to much higher level calculations. For cases where p polarization functions on H might be expected to be important, the MIDIY basis set includes these as an extension to MIDI! (Lynch and Truhlar 2004).

Finally, an important nomenclature point is that most basis sets are defined to use the five spherical d functions, but an important exception is 6-31G (or 6-31G(d)), which is defined to use the six Cartesian d functions. Some electronic structure programs are not flexible about permitting the user to choose how many d functions are used, so it is important to check that a consistent scheme has been employed when comparing to existing literature data. To avoid ambiguity, it is helpful to modify the basis set name when the number of d functions employed is not the same as that assumed as the default, e.g., MIDI!(6d) to denote use of the six Cartesian d functions instead of the normal spherical five with the MIDI! basis. Another nomenclature issue of importance involves the basis set ‘3-21G ’. While this notation pervades the literature, it is ambiguous and should be avoided. Pople and co-workers suggested taking from 6-31G the polarization functions for second-row atoms and beyond and using them directly (i.e., without any reoptimization of exponents) with the smaller basis 3-21G. The motivation for this was to address the serious geometry problems that arise for hypervalent second-row atoms without d functions whilst maintaining a very cheap description of first-row atoms. To distinguish this situation from the normal ‘*’, they named this basis set 3-21G( ), and that is the notation that should always be used to emphasize that no d functions are present on first-row atoms.

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6.2.5 Diffuse Functions

The highest energy MOs of anions, highly excited electronic states, and loose supermolecular complexes, tend to be much more spatially diffuse than garden-variety MOs. When a basis set does not have the flexibility necessary to allow a weakly bound electron to localize far from the remaining density, significant errors in energies and other molecular properties can occur. To address this limitation, standard basis sets are often ‘augmented’ with diffuse basis functions when their use is warranted.

In the Pople family of basis sets, the presence of diffuse functions is indicated by a ‘+’ in the basis set name. Thus, 6-31+G(d) indicates that heavy atoms have been augmented with an additional one s and one set of p functions having small exponents. A second plus indicates the presence of diffuse s functions on H, e.g., 6-311++G(3df,2pd). For the Pople basis sets, the exponents for the diffuse functions were variationally optimized on the anionic one-heavy-atom hydrides, e.g., BH2−, and are the same for 3-21G, 6-31G, and 6-311G. In the general case, a rough rule of thumb is that diffuse functions should have an exponent about a factor of four smaller than the smallest valence exponent. Diffuse sp sets have also been defined for use in conjunction with the MIDI! and MIDIY basis sets, generating MIDIX+ and MIDIY+, respectively (Lynch and Truhlar 2004); the former basis set appears particularly efficient for the computation of accurate electron affinities.

In the Dunning family of cc-pVnZ basis sets, diffuse functions on all atoms are indicated by prefixing with ‘aug’. Moreover, one set of diffuse functions is added for each angular momentum already present. Thus, aug-cc-pVTZ has diffuse f, d, p, and s functions on heavy atoms and diffuse d, p, and s functions on H and He. An identical prescription for diffuse functions has been used by Jensen (2002) in connection with the pc-n basis sets.

Particularly for the calculation of acidities and electron affinities, diffuse functions are absolutely required. For instance, the acidity of HF (not Hartree-Fock in this case, but hydrogen fluoride) increases by 44 kcal/mol when the 6-31+G(d) basis set is used instead of unaugmented 6-31G(d).

6.2.6 The HF Limit

Solution of the HF equations with an infinite basis set is defined as the HF limit. Actually carrying out such a calculation is almost never a practical possibility. However, it is sometimes the case that one may extrapolate to the HF limit with a fair degree of confidence.

Of the basis sets discussed thus far, the cc-pVnZ and cc-pCVnZ examples were designed expressly for this purpose. As they increase in size in a consistent fashion with each increment of n, one can imagine plotting some particular computed property as a function of n−1 and extrapolating a curve fit through those points back to the intercept; the intercept corresponds to n = ∞, i.e., the infinite basis limit (Figure 6.4).

Note that certain issues do arise in how one should carry out this extrapolation. If the property is sensitive to geometry, should the geometry be optimized at each level, or should a single geometry be chosen, thereby permitting the extrapolating equation to account for basisset effects only? Are there any fundamental principles dictating what form the extrapolating

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Property
6	5	Q	T	D
		n−1

Figure 6.4 Use of an extrapolation procedure to estimate the expectation value for some property at the HF limit. The abscissa is marked off as n−1 in cc-pVnZ notation (see page 162). Note the sensitivity of the limiting value, which is to say the ordinate intercept, that might be expected based on the use of different curve-fitting procedures

equation should take, or can any arbitrary curve fitting approach be applied? In general, the answers to these questions are case-dependent, and the chemist cannot be completely removed from the calculation.

Note that the cost of the extrapolation procedure outlined above becomes increasingly large as points for n = 4, 5, and 6 are added. For systems having more than five or six atoms, these calculations can be staggeringly demanding in terms of computational resources.

A somewhat more common approach is one that does not try explicitly to extrapolate to the HF limit but uses similar concepts to try to correct for some basis-set incompleteness. The assumption is made that the effects of ‘orthogonal’ increases in basis set size can be considered to be additive (a substantial amount of work suggests that this assumption is typically not too bad, at least for molecular energies), and thus the individual effects can be summed together to estimate the full-basis-set result. This is best illustrated by example. Consider HF calculations carried out for the chemical warfare agent VX (C11H26NO2PS, Figure 6.5) with the following basis sets: 6-31G, 6-31++G, 6-31G(d,p), 6-311G, and 6-311++G(d,p). With these basis sets, the total number of basis functions for VX are 204, 378, 294, 294, and 542, respectively.

The additivity assumption can be expressed as

E[HF/6-311++G(d,p)] ≈ E[HF/6-31G]

+{E[HF/6-31G(d,p)] − E[HF/6-31G]}

+{E[HF/6-311G] − E[HF/6-31G]}

+ {E[HF/6-31++G] − E[HF/6-31G]}

(6.5)

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HF/6-311+ +G(d,p)

HF/6-31+ +G		O
HF/6-	H3C	P	OCH2CH3
311G
			SCH2CH2N(i −Pr)2
			VX
HF/6-31G	HF/6-31G(d,p)

Figure 6.5 The chemical warfare agent VX and a conceptual illustration of the additivity concept embodied in Eq. (6.5). Each boldface line in the additivity cube represents one line on the r.h.s. of the equation

where the notation ‘x/y’ implies ‘level of theory x using basis set y’. Each successive line on the r.h.s. of Eq. (6.5) reflects the incremental contribution from a particular basis set improvement – first polarization functions, then valence decontraction, then diffuse functions. As already noted, the calculation on the l.h.s. requires 542 basis functions. Although there are four different calculations on the r.h.s., if we recall that the amount of time for an HF calculation scales formally as the fourth power of the number of basis functions, the amount of time to carry out those four calculations, expressed as a fraction of the amount of time to carry out the full calculation, is (2044 + 3784 + 2944 + 2944)/5424 = 0.43. That is, evaluation of the r.h.s. of Eq. (6.5) takes less than half the time of evaluation of the l.h.s.

While the above schemes are interesting from a technical standpoint, it must be recalled that chemically there are potentially large errors associated with the HF approximation, so the HF limit is of more interest from a formal standpoint than from a chemical one. Thus, we will defer additional discussion of extrapolation and additivity concepts until Section 7.7, where it is re-examined in the additional context of accounting for electron correlation effects.

6.2.7 Effective Core Potentials

The periodic table is rich and complex, and very heavy elements pose rather distinct challenges to MO theory. First, there is the purely technical hurdle that such elements have large numbers of electrons, and there is thus a concomitant requirement to use a large number of basis functions to describe them. Of course, these extra electrons are mostly core electrons, and thus a minimal representation will probably be adequate. Nevertheless, if one wants to model a small cluster of uranium atoms, for instance, the basis set size quickly becomes intractable. Not surprisingly, more electrons means more energy associated with electron correlation, too.

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It was Hellmann (1935) who first proposed a rather radical solution to this problem – replace the electrons with analytical functions that would reasonably accurately, and much more efficiently, represent the combined nuclear–electronic core to the remaining electrons. Such functions are referred to as effective core potentials (ECPs). In a sense, we have already seen ECPs in a very crude form in semiempirical MO theory, where, since only valence electrons are treated, the ECP is a nuclear point charge reduced in magnitude by the number of core electrons.

In ab initio theory, ECPs are considerably more complex. They properly represent not only Coulomb repulsion effects, but also adherence to the Pauli principle (i.e., outlying atomic orbitals must be orthogonal to core orbitals having the same angular momentum). This being said, we will not dwell on the technical aspects of their construction. Interested readers are referred to the bibliography at the end of the chapter.

Note that were ECPs to do nothing more than reduce the scope of the electronic structure problem for heavy elements, they would still have great value. However, they have another virtue as well. The core electrons in very heavy elements reach velocities sufficiently near the speed of light that they manifest relativistic effects. A non-relativistic Hamiltonian operator is incapable of accounting for such effects, which can be significant for many chemical properties (see, for example, Kaltsoyannis 2003). A full discussion of modeling relativistic effects, while a fascinating topic, is well beyond the scope of this book, although some details are discussed in Section 7.4.4. We note here simply that, to the extent an ECP represents the behavior of an atomic core, relativistic effects can be folded in, and thereby removed from the problem of finding suitable wave functions for the remaining electrons.

A key issue in the construction of ECPs is just how many electrons to include in the core. So-called ‘large-core’ ECPs include everything but the outermost (valence) shell, while ‘small-core’ ECPs scale back to the next lower shell. Because polarization of the sub-valence shell can be chemically important in heavier metals, it is usually worth the extra cost to explicitly include that shell in the calculations. Thus, the most robust ECPs for the elements Sc –Zn, Y–Cd, and La –Hg, employ [Ne], [Ar], and [Kr] cores, respectively. There is less consensus on the small-core vs. large-core question for the non-metals.

Popular pseudopotentials in modern use include those of Hay and Wadt (sometimes also called the Los Alamos National Laboratory (or LANL) ECPs; Hay and Wadt 1985), those of Stevens et al. (1992), and the Stuttgart–Dresden pseudopotentials developed by Dolg and co-workers (2002). The Hay–Wadt ECPs are non-relativistic for the first row of transition metals while most others are not; as relativistic effects are usually quite small for this region of the periodic table, the distinction is not particularly important. Lovallo and Klobukowski (2003) have recently provided additional sets of both relativistic and non-relativistic ECPs for these metals. For the p block elements, Check et al. (2001) have optimized polarization and diffuse functions to be used in conjunction with the LANL double-ζ basis set.

Another recent set of pseudopotentials for the 4p, 5p, and 6p elements has been developed by Dyall (1998, 2002). These ECPs are designed to be the ECP-equivalent to the correlation-consistent basis sets of Dunning insofar as (i) prescriptions for double-ζ and

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triple-ζ contractions are provided, (ii) polarization functions of increasingly higher angular momentum are included for the triple-ζ case, and (iii) diffuse functions were optimized for negative ions at a correlated level of electronic structure theory.

6.2.8 Sources

Most electronic structure programs come with a library of built-in basis sets, to include many if not all of those mentioned above. A tremendously useful electronic resource is the Environmental Molecular Sciences Laboratory Gaussian Basis Set Order Form, a website that permits the download of a very large number of different basis sets formatted for a variety of different software packages. Moreover, the site has reference information that typically includes values for test calculations as published by the original authors. Since different software packages may have different conventions for how to deal with certain aspects of the basis set (e.g., five spherical vs. six Cartesian d functions), it is always a good idea to carry out such test calculations to ensure that the basis set is being used in a manner consistent with its definition and, hopefully, with previously reported calculations in the literature.

So, how to choose the ‘best’ basis set for the problem at hand? Obviously a fair rule of thumb is that bigger is better, keeping in mind issues of balance between valence decontraction and presence of polarization functions. As noted above, diffuse functions are warranted in certain specific situations, but in the absence of those situations, there tends to be no strong reason to include them.

Additionally, access to particular software packages may play some role in motivating the choice of basis set. Some packages are equipped to take advantage of efficiencies possible for such features as combined s and p exponents, or general contractions, while others are not, and there may thus be significant timing issues differentiating basis sets.

Finally, and perhaps most important for the vast majority of chemical problems where saturation of the basis set is not a practical possibility, the choice should consider the degree to which other results from that particular basis set at that particular level of theory are available for comparison. For instance, to the extent that there are an enormous number of HF/6-31G(d) results published, and thus a reasonably firm understanding of the specific successes and failures of the model, this can assist in the interpretation of new results – Pople has referred to the collection of all data from a given theoretical prescription as comprising a ‘model chemistry’ and emphasized the utility of analyzing theoretical performance (and future model development efforts) within such a framework.

6.3 Key Technical and Practical Points of Hartree–Fock Theory

A deep understanding of the underlying theory is, alas, of only limited value in successfully carrying out a HF calculation with any given software package. This section is not designed to supplant program users’ manuals, the utility of reading which cannot be overemphasized, but discusses aspects of practical HF calculations that are often glossed over in formal presentations of the theory.