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

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 271

orbitals. Such dependence re-introduces a self-consistency requirement into the minimization of the energy from Eq. (8.39), and this approach is called self-consistent charge densityfunctional tight-binding (SCC-DFTB) theory. Thus, for a realistic representation of charge redistribution, one must sacrifice the higher efficiency of DFTB for an SCF approach. Nevertheless, SCC-DFTB is about as fast as a semiempirical NDDO model and many promising applications have begun to appear. For example, Elstner et al. (2003) found SCC-DFTB to compare favorably to B3LYP and MP2 calculations with the 6-311+G(d,p) basis set for structural and energetic properties associated with biological model systems coordinating zinc. One feature requiring further attention, however, is that in very large molecules like biopolymers there are likely to be non-bonded interactions, e.g., dispersion, between different sections of the molecule. Dispersion is not well treated by SCC-DFTB; in a QM MD study of the protein crambin, Liu et al. found that inclusion of an ad hoc scaled r−6 potential between non-bonded atoms (i.e., the attractive portion of a Lennard-Jones potential, cf. Eq. (2.14)) was required to maintain a structure in acceptable agreement with experiment.

8.5Advantages and Disadvantages of DFT Compared to MO Theory

Since 1990 there has been an enormous amount of comparison between DFT and alternative methods based on the molecular wave function. The bottom line from all of this work is that, as a rule, DFT is the most cost-effective method to achieve a given level of accuracy, sometimes by a very wide margin. There are, however, significant exceptions to this rule, deriving either from inadequacies in modern functionals or intrinsic limitations in the KS approach for determining the density. This section describes some of these cases.

8.5.1Densities vs. Wave Functions

The most fundamental difference between DFT and MO theory must never be forgotten: DFT optimizes an electron density while MO theory optimizes a wave function. So, to determine a particular molecular property using DFT, we need to know how that property depends on the density, while to determine the same property using a wave function, we need to know the correct quantum mechanical operator. As there are more well-characterized operators then there are generic property functionals of the density, wave functions clearly have broader utility. As a simple example, consider the total energy of interelectronic repulsion. Even if we had the exact density for some system, we do not know the exact exchange-correlation energy functional, and thus we cannot compute the exact interelectronic repulsion. However, with the exact wave function it is a simple matter of evaluating the expectation value for the interelectronic repulsion operator to determine this energy,

	Eee	=			1			(8.43)
					rij
				i<j
	where i and j run over all electrons.

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Another key example is in the area of dynamics, where transition probabilities depend on matrix elements between different wave functions. Because densities do not have phases as wave functions do, multistate resonance effects, interference effects, etc., are not readily evaluated within a DFT formalism.

Because of the mechanical details of the KS formalism, it is easy to become confused about whether there is a KS ‘wave function’. Early work in the field tended to resist any attempts to interpret the KS orbitals, viewing them as pure mathematical constructs useful only in construction of the density. In practice, however, the shapes of KS orbitals tend to be remarkably similar to canonical HF MOs, and they can be quite useful in qualitative analysis of chemical properties. If we think of the procedure by which they are generated, there are indeed a number of reasons to prefer KS orbitals to HF orbitals. For instance, all KS orbitals, occupied and virtual, are subject to the same external potential. HF orbitals, on the other hand, experience varying potentials, and, in particular, HF virtual orbitals experience the potential that would be felt by an extra electron being added to the molecule. As a result, HF virtual orbitals tend to be too high in energy and anomalously diffuse compared to KS virtual orbitals. (In exact DFT, it can also be shown that the eigenvalue of the highest KS MO is the exact first ionization potential, i.e., there is a direct analogy to Koopmans’ theorem for this orbital – in practice, however, approximate functionals are quite bad at predicting IPs in this fashion without applying some sort of correction scheme, e.g., an empirical linear scaling of the eigenvalues).

In point of fact, there is a DFT wave function; it is just not clear how useful it should be considered to be. Recall that the Slater determinant formed from the KS orbitals is the exact wave function for the fictional non-interacting system having the same density as the real system. This KS Slater determinant has certain interesting properties by comparison to its HF analogs. In open-shell systems, KS determinants usually show extremely low levels of spin contamination, even for cases where HF determinants are pathologically bad (Baker, Scheiner, and Andzelm 1993). For instance, the spin contamination in planar triplet phenylnitrenium cation (PhNH+) is very high at the UHF/cc-pVDZ level (see Section 6.3.3) as judged by an expectation value for S2 of 2.50. At the BLYP/cc-pVDZ level, on the other hand the expectation value for S2 over the KS determinant is 2.01, very close to the proper eigenvalue of 2.0. The high spin contamination at the UHF level leads to the planar structure being erroneously determined to be a minimum, while at the BLYP level it is correctly identified as a TS structure for rotation about the C–N bond (Cramer, Dulles, and Falvey 1994).

While it is by no means guaranteed that the expectation value for S2 over the KS determinant has any bearing at all on its expectation value over the exact wave function corresponding to the KS density (see Grafenstein¨ and Cremer 2001), it is an empirical fact that DFT is generally much more robust in dealing with open-shell systems where HF methods show high spin contamination (recall that high HF spin contamination makes post-HF methods of questionable utility, so DFT can be a happy last resort). Note, incidentally, that expectation values of S2 are sensitive to the amount of HF exchange in the functional. A ‘pure’ functional nearly always shows very small spin contamination, and each added percent of HF exchange tends to titrate in a corresponding percentage of the spin

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 273

contamination exhibited by the HF wave function. This behavior can mitigate the utility of hybrid functionals in some open-shell systems.

Finally, one clear utility of a wave function is that excited states can be generated as linear combinations of determinants derived from exciting one or more electrons from occupied to virtual orbitals (see Section 14.1). Although the Hohenberg–Kohn theorem makes it clear that the density alone carries sufficient information to determine the excited-state wave functions, it is only very recently that progress has been made on applying DFT to excited states (the exception being in symmetric molecules, where the lowest energy state in each spatial irreducible representation is amenable to a simple SCF treatment as already noted in Section 8.2.1). Additional discussion on this subject is deferred to Section 14.2.1.

8.5.2Computational Efficiency

The formal scaling behavior of DFT has already been noted to be in principle no worse than N 3, where N is the number of basis functions used to represent the KS orbitals. This is better than HF by a factor of N , and very substantially better than other methods that, like DFT, also include electron correlation (see Table 7.4). Of course, scaling refers to how time increases with size, but says nothing about the absolute amount of time for a given molecule. As a rule, for programs that use approximately the same routines and algorithms to carry out HF and DFT calculations, the cost of a DFT calculation on a moderately sized molecule, say 15 heavy atoms, is double that of the HF calculation with the same basis set.

However, it is possible to do very much better than that in programs optimized for DFT. One area where DFT enjoys a clear advantage over HF is in its ability to use basis functions that are not necessarily contracted Gaussians. Recall that the motivation for using contracted GTOs is that arbitrary four-center two-electron integrals can be solved analytically. In most electronic structure programs where DFT was added as a new feature to an existing HF code, the representation of the density in the classical electron-repulsion operator is carried out using the KS orbital basis functions. Thus, the net effect is to create a four-index integral, and these codes inevitably continue to use contracted GTOs as basis functions. However, if the density is represented using an auxiliary basis set, or even represented numerically, other options are readily available for the KS orbital basis set, including Slater-type functions. STOs enjoy the advantage that fewer of them are required (since, inter alia, they have correct cusp behavior at the nuclei) and certain advantages associated with symmetry can more readily be taken, so they speed up calculations considerably. The widely used Amsterdam Density Functional code (ADF) makes use of STO basis functions covering atomic numbers 1 to 118 (Snijders, Baerends, and Vernooijs 1982; van Lenthe and Baerends 2003; Chong et al. 2004).

Another interesting possibility is the use of plane waves as basis sets in periodic infinite systems (e.g., metals, crystalline solids, or liquids represented using periodic boundary conditions). While it takes an enormous number of plane waves to properly represent the decidedly aperiodic densities that are possible within the unit cells of interesting chemical systems, the necessary integrals are particularly simple to solve, and thus this approach sees considerable use in dynamics and solid-state physics (Dovesi et al. 2000).

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Even in cases where contracted GTOs are chosen as basis sets, DFT offers the advantage that convergence with respect to basis-set size tends to be more rapid than for MO techniques (particularly correlated MO theories). Thus, polarized valence double-ζ basis sets are quite adequate for a wide variety of calculations, and very good convergence in many properties can be seen at the level of employing polarized triple-ζ basis sets. Extensive studies of basis set effects on functional performance and parameterization have been carried out by Jensen (2002a, 2002b, 2003) and Boese, Martin, and Handy (2003). They found, inter alia, that for most functionals Pople-type basis sets provide much better accuracy than cc-pVnZ basis sets of similar size, that adding diffuse functions offers substantial improvement over using the non-augmented analog basis (a point also made by Lynch, Zhao, and Truhlar (2003), particularly for the computation of barrier heights or conformational energies in molecules containing multiple lone pairs of electrons), that satisfactory convergence is generally arrived at for most properties of interest by the time triple-ζ basis sets are used, and finally that the optimal values for parameters that are included in various functionals are sensitive to choice of basis set size. Thus, the optimal percent HF exchange for HCTH/407 was about 28% with double-ζ basis sets, but about 18% with triple-ζ basis sets. Jensen (2002b, 2003) found that, with reoptimization of the polarization exponents for DFT, the pc-n basis sets were always able to provide the best accuracy for a given basis set size.

Besides issues associated with basis sets, considerable progress has been made in developing linear-scaling algorithms for DFT. In this regard, DFT is somewhat simpler than MO theoretical techniques because all potentials are local (this refers to ‘pure’ DFT – incorporation of HF exchange introduces the non-local exchange operator). Thus, one promising technique is the ‘divide-and-conquer’ formalism of Yang and co-workers, where a large system is divided up into a number of smaller regions, within each of which a KS SCF is carried out representing the other regions in a simplified fashion (Yang and Lee 1995). The total cost of matrix diagonalization is thereby reduced from N 3 scaling to M(N/M)3 scaling where M is the number of sub-regions. Since the number of basis functions in each sub-region (N/M) tends to be close to some fixed value irrespective of N , the overall scaling goes as order M, i.e., linear. Of course, all the algorithms developed to facilitate linear scaling in computing Coulomb interactions in HF and MD calculations (e.g., fast multipole methods) can be used in DFT calculations as well.

As a final point with regard to efficiency, note that SCF convergence in DFT is sometimes more problematic than in HF. Because of the similarities between the KS and HF orbitals, this problem can often be very effectively alleviated by using the HF orbitals as an initial guess for the KS orbitals. Because the HF orbitals can usually be generated quite quickly, the extra step can ultimately be time-saving if it sufficiently improves the KS SCF convergence.

8.5.3 Limitations of the KS Formalism

It is important to emphasize that nearly all applications of DFT to molecular systems are undertaken within the context of the Kohn–Sham SCF approach. The motivation for this choice is that it permits the kinetic energy to be computed as the expectation value of the kinetic-energy operator over the KS single determinant, avoiding the tricky issue of

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 275

determining the kinetic energy as a functional of the density. However, as has already been discussed in the context of MO theory, some chemical systems are not well described by a single Slater determinant. The application of DFT to such systems is both technically and conceptually problematic.

To illustrate this point, let us return to the cases of p-benzyne and N -protonated 2,5- pyridyne already discussed at length in Section 7.6.2. When restricted DFT is applied to the closed-shell singlet states of these molecules, the predicted splittings between the singlet and triplet states at the BPW91/cc-pVDZ level are 3.1 and 3.7 kcal mol−1, respectively. Comparing to the last line of Table 7.2, we see that these predictions are in error by about 8 kcal mol−1 and are qualitatively incorrect about which state is the ground state. A careful analysis indicates that there is no problem with the triplet state, but that the singlet state is predicted to be insufficiently stable as a consequence of enforcing a single-determinantal description as part of the KS formalism (this also results in rather poor predicted geometries for the singlets).

In cases like this, showing high degrees of non-dynamical correlation, there are two primary approaches to correcting for inadequacies in the KS treatment. In the first approach, the remedy is fairly simple: an unrestricted KS formalism is applied and the wave function for the singlet is allowed to break spin symmetry. That is, even though the singlet is closedshell, the α and β orbitals are permitted to be spatially different. When this unrestricted formalism is applied to p-benzyne and N -protonated 2,5-pyridyne, the S–T splittings are predicted to be −3.6 and −3.9 kcal mol−1, respectively, in dramatically improved agreement with experiment/best estimates (singlet geometries are also improved).

Similar results have been obtained in transition-metal compounds containing two metal atoms that are antiferromagnetically coupled. An adequate description of the singlet state sometimes requires a broken-symmetry SCF, and inspection of the KS orbitals afterwards typically indicates the highest energy α electron(s) to be well localized on one metal atom while the corresponding highest energy β electron(s) can be found on the other metal atom. The transition from a stable restricted DFT solution to a broken-symmetry one takes place as the distance between the metal atoms increases and covalent-like bonding gives way to more distant antiferromagnetic interactions (Lovell et al. 1996; Cramer, Smith, and Tolman 1996; Adamo et al. 1999).

What is to be made of these broken-symmetry singlet KS wave functions? One interpretation is to invoke the variational principle and assert that, insofar as they lower the energy, they must provide better densities and one is fully justified in using them. While pragmatic, this view is somewhat unsatisfying in a number of respects. One troubling issue is that the expectation value of the total spin operator for the KS determinant is often significantly in excess of the expected exact value. Thus, in the case of the singlet arynes discussed above, S2 values of zero are expected, but computed values are on the order of 0.2 for broken-symmetry solutions. If these were HF wave functions, we would take such a value as being indicative of a fair degree of spin contamination. However, it is by no means obvious that S2 for the non-interacting KS wave function is in any way indicative of what S2 may be for the interacting wave function corresponding to the final KS density. It may not be spin contaminated at all. On the other hand, DFT energies where broken-symmetry KS wave functions have S2

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values of 1.0 (i.e., they are equal mixtures of singlet and triplet) can be usefully interpreted as being the average of the singlet and triplet state energies (see below and Section 14.4), so it does not seem that one can ignore spin contamination as an issue entirely.

Indeed, if we have a situation where a higher-spin state is lower in energy than a corresponding lower-spin state generated by a single spin flip (e.g., triplet and singlet), it is almost always the case that a wave function corresponding to the higher-energy lower-spin state will be unstable to symmetry breaking that can mix in character of the lower-energy higherspin state. A good example is phenylnitrene (PhN) whose triplet ground state and first three singlet excited states are depicted in Figure 8.3. To properly model the S2 state, which is closed-shell in character, we must use a restricted DFT formalism, and when this is done the resulting state energy splitting agrees well with experiment (Johnson and Cramer 2001). Note that the S3 state cannot be handled by standard DFT formalisms because, since it has the same spatial symmetry as the S2 state, restricted DFT variationally collapses to the latter. As for the broken-symmetry approach, when that is applied to a system having an equal number of α and β electrons, a KS wave function is obtained having an S2 value of 1.0 which, as mentioned above, is best interpreted as a 50:50 mixture of T0 and S1.

An important point with respect to phenylnitrene is that the S1 state is not multideterminantal because of the mixing of different electronic configurations having similar weights. Instead, it is intrinsically two-determinantal because it is an open-shell singlet. In order to better accommodate systems like this, some efforts have been undertaken to develop multideterminantal DFT formalisms. Grafenstein,¨ Kraka, and Cremer (1998) have proposed a restricted open-shell singlet (ROSS) methodology specifically for such states, while Filatov and Shaik (1998) have advanced a more general MCSCF-like formalism named restricted open-shell Kohn–Sham (ROKS) theory. Khait and Hoffman (2004) have also described a general multireference spin-adapted DFT. The details of these methods are sufficiently

N N

3A2 (T0)

1A2 (S1)

N N

11A1 (S2)

21A1 (S3)

Figure 8.3 Configuration cartoons for the ground-state triplet and first three singlet states of phenylnitrene. Note that the cartoon for S1 glosses over its two-determinantal character

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 277

						•
						•
						•
			Orbital					b2
			energies					2e−
								a1
						•
	R					•
						•
c1	C	↑
		↑							b2
	R
									a1
		R	↑					R ↑
	+ c2		↑						↑
			C					C
		R						R
	1Ψ = c1 ...a12 + c2 ...b22							3Ψ = ...a11b21

Figure 8.4 Illustration of the two-configurational character of singlet carbenes compared to their triplet congeners. Single-determinantal methodologies are entirely appropriate for the triplet, but begin to fail for the singlets as the weights c1 and c2 grow closer to one another. Empirically, KS-DFT methods are less sensitive to this instability than HF, but they ultimately fail as well, at least within a restricted formalism

technical and the range of their applications still sufficiently narrow that we will not discuss them at any more length.

As a practical point, however, returning to non-dynamical correlation as it gives rise to wave functions best described in MO theory as a combination of configurations, DFT seems to be less sensitive to this issue than HF theory. Thus, for example, as has already been mentioned, HF theory usually does very poorly with singlet–triplet splittings in carbenes and isoelectronic analogs because the singlet states are best described as mixtures of two (or more) configurations (Figure 8.4). Speaking very roughly, the weights of the two dominant configurations are in the range of 85:15 for some of the more typical cases. Interestingly, DFT usually does very well indeed for these same systems, and it is not until the weights of the individual configurations become much nearer one another (e.g., the 60:40 balance for p-benzyne discussed above) that DFT too begins to badly underestimate the stability of the singlet. In one case, the ring opening of methylenecyclopropane to closed-shell singlet trimethylenemethane (see Section 7.1) was examined and the BP86/cc-pVDZ level of theory was found to be accurate up to about a 75:25 mixing of the dominant configurations for TMM, at which point it began to diverge from a multireference description of the reaction coordinate (Cramer and Smith 1996).

Note that the generally lower sensitivity of DFT to multireference character is again dependent on the amount of HF exchange included in the functional. Pure DFT functionals seem to be most robust, and inclusion of HF exchange introduces a proportional degree of instability for systems with multiconfigurational character. Thus, the generally higher

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accuracy of the ACM methods can fail to extend to systems where they are methodologically less stable owing to their Hartree–Fock component.

8.5.4 Systematic Improvability

In molecular orbital theory, there is a clear and well defined path to the exact solution of the Schrodinger¨ equation. All we need do is express our wave function as a linear combination of all possible configurations (full CI) and choose a basis set that is infinite in size, and we have arrived. While such a goal is essentially never practicable, at least the path to it can be followed unambiguously until computational resources fail.

With density functional theory, the situation is much less clear when it comes to evaluating how to do a ‘better’ calculation. One thing that seems fairly clear is that, as a general rule, results from MGGA functionals tend to improve on those from GGA functionals, which in turn drastically improve on those from LSDA. Somewhat less clear is the status of hybrid functionals. The best ones are competitive in quality with the best MGGA functionals (and B3LYP seems to continue to be the ‘magic’ functional in project after project) subject to the caveat that in certain situations the presence of HF exchange may cause problems that are associated with the single-determinant KS formalism to become manifest more quickly. As for basis-set effects, just as with MO theory one can examine convergence with respect to basis-set size, but there is no guarantee that this may not lead to increased errors since errors associated with basis-set incompleteness may offset errors associated with approximate functionals.

All that being said, experience dictates that, across a surprisingly wide variety of systems, DFT tends to be remarkably robust. Thus, unless a problem falls into one of a few classes of well characterized problems for DFT, there is good reason to be optimistic about any particular calculation.

Finally, it seems clear that routes to further improve DFT must be associated with better defining hole functions in arbitrary systems. In particular, the current generation of functionals has reached a point where finding efficient algorithms for correction of the classical self-interaction error are likely to have the largest qualitative (and quantitative) impact.

8.5.5 Worst-case Scenarios

Certain failures of modern DFT should be anticipated, and others are readily explained after some thought about the forms of current functionals. One clear problem with modern functionals is that they make the energy a function entirely of the local density and possibly the density gradient. As such, they are incapable of properly describing London dispersion forces, which, as noted in Section 2.2.4, derive entirely from electron correlation at ‘long range’. Adding HF exchange to the DFT functional cannot entirely alleviate this problem, since the HF level of theory, while non-local, does not account in any way for opposite-spin electron correlation.

So, even though noble-gas dimers like He2, Ne2, etc., exhibit potential energy minima at van der Waals contact, DFT predicts the potential energy curve for these diatomics to

8.5 ADVANTAGES AND DISADVANTAGES OF DFT COMPARED TO MO THEORY 279

be purely repulsive, at least as long as saturated basis sets are used. If an incomplete basis set is used, it is possible for BSSE to introduce a spurious minimum in the association curve at about the right position, but this is purely fortuitous – the physics of dispersion is simply not included in the functional(s). This is an area of active developmental research (see, for instance, Lein, Dobson, and Gross 1999) and indeed Adamo and Barone (2002) have reported that mPBE does reasonably well for noble-gas-dimer geometries and energies with saturated basis sets, and Xu and Goddard (2004a) have shown that XLYP and X3LYP give reasonable results for the dimers of He and Ne.

Other problems with non-bonded complexes have also been documented with DFT. In hydrogen bonded systems, heavy-atom–heavy-atom distances tend to be rather variable as a function of functional. Hobza and Sponer (1999) have examined a large number of nucleic acid base pairs and found, by comparison to X-ray crystal structures and high levels of correlated MO theory, that heavy-atom–heavy-atom distances predicted by GGA DFT functionals are typically too short by about 0.1 A˚ . Nevertheless, interaction energies are often reasonably well predicted at these levels. Critical to accurate prediction, however, is that a basis set including diffuse functions be employed, as large errors can otherwise be observed, particularly for intramolecular hydrogen bonds (an interesting comparison is provided by Ma, Schaefer, and Allinger 1998 and Lii, Ma, and Allinger 1999; see also Lynch, Zhao, and Truhlar 2003). Staroverov et al. (2003) have provided an analysis of 16 different functionals for the energetics and geometrics of 11 hydrogen-bonded systems with the 6-311++G(3df,3pd) basis set; their results suggest that B3LYP and TPSS are both fairly robust, with mean unsigned errors for dissociation of about 0.5 kcal mol−1 and for H-bond lengths of about 0.02 A˚ . Xu and Goddard (2004a, 2004b) observe similarly good performance for X3LYP applied to the water dimer.

Somewhat more problematic are intermolecular complexes bound together by charge transfer interactions. Modern DFT functionals have a tendency to predict such interactions to be stronger than they should be. Thus, Ruiz, Salahub, and Vela (1995) showed that some pure DFT functionals overestimated the binding of ethylene and molecular fluorine by as much as 20 kcal mol−1. Including HF exchange in the functional alleviates the problem to some extent, but only by cancellation of errors, since HF theory incorrectly predicts the interaction between ethylene and molecular fluorine to be purely repulsive. Dative bonds have also been found to be problematic for many functionals. Gilbert (2004) found that heterolytic B−N bond dissociation energies in amine-boranes were underestimated by standard functionals, and that inclusion of a substantial fraction of HF exchange (e.g., as in MPW1K) was required to improve agreement with experiment for this quantity.

A problem with DFT that is not restricted to intermolecular complexes is what might be called ‘overdelocalization’. In part because of problems in correcting for the classical self-interaction energy, many functionals overstabilize systems having more highly delocalized densities over more localized alternatives. Such an imbalance can lead to erroneous predictions of higher symmetry structures being preferred over lower symmetry ones, as has been observed, for instance, for phosphoranyl radical structures (Lim et al. 1996), transitionstate structures for cationic [4+3] cycloadditions (Cramer and Barrows 1998), and in the comparison of cumulenes to poly-ynes (Woodcock, Schaefer, and Schreiner 2002). It can

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also lead to very poor predictions along coordinates for bond dissociation (Bally and Sastry 1997; Zhang and Wang 1998; Grafenstein,¨ Kraka, and Cremer 2004), nucleophilic substitution (Adamo and Barone 1998; Gritsenko et al. 2000), competing cycloaddition pathways (Jones et al. 2002), and rotation about single bonds in conjugated systems, like the benzylic bond in styrene (Choi, Kertesz, and Karpfen 1997).

Note that, because electron correlation often stabilizes delocalized electronic structures over localized ones, HF theory tends to be inaccurate for such systems in the opposite direction from DFT, and thus, again, hybrid ACM functionals tend to show improved performance by an offsetting of errors.

A number of different methods have been proposed to introduce a self-interaction correction into the Kohn–Sham formalism (Perdew and Zunger 1981; Kummel¨ and Perdew 2003; Grafenstein,¨ Kraka, and Cremer 2004). This correction is particularly useful in situations with odd numbers of electrons distributed over more than one atom, e.g., in transition-state structures (Patchkovskii and Ziegler 2002). Unfortunately, the correction introduces an additional level of self-consistency into the KS SCF process because it depends on the KS orbitals, and it tends to be difficult and time-consuming to converge the relevant equations. However, future developments in non-local correlation functionals may be able to correct for self-interaction error in a more efficient manner.

8.6 General Performance Overview of DFT

While the cases noted in the immediately preceding section illustrate certain pathological failures of current DFT functionals, the general picture for DFT is really quite bright. For the ‘average’ problem, DFT is the method of choice to achieve a particular level of accuracy at lowest cost. With the appearance of each new functional, there has tended to be at least one paper benchmarking the performance of that functional on a variety of standard test sets (for energies, structures, etc.) and there is now a rather large body of data that is somewhat scattered and disjoint with respect to individual functional performance. The comparisons made below are designed to provide as broad a coverage as possible without becoming unwieldy, and as such are not necessarily exhaustive.

8.6.1 Energetics

Exact DFT is an ab initio theory (even if most modern implementations may be regarded as having a semiempirical flavor) and like other such theories its quality with respect to energetic predictions is usually judged based on its performance for atomization energies. Table 8.1 collects average unsigned and maximum absolute errors in atomization energies as computed for various functionals, and for some other computational methodologies, over several different test sets of increasing complexity. The G2/97 and G3/99 test sets (columns D and E) include substituted hydrocarbons, radicals, inorganic hydrides, unsaturated ring hydrocarbons, and polyhalogenated organics and inorganics. While effort is made to describe all levels of theory accurately, it should be noted that in many cases geometries are optimized (and zero-point vibrational energies computed) using a basis set smaller than that used to compute the atomization energies. In addition, some results for the larger test sets include