436 Computational Chemistry

Also, the more rigorous method of locating the two transition states and comparing their energies was, in the author’s hands, straightforward and not excessively demanding of time. Nevertheless, the large amount of work which has been done using these ideas suggests that they offer a useful approach to interpreting and predicting chemical reactivity. Even an apparently unrelated property, or rather complex of properties, namely aromaticity, has been subjected to analysis in terms of hardness [121]. As Parr and Yang say, “This is perhaps an oversimplified view of chemical reactivity, but it is useful” [122].

7.3.6 Visualization

The only cases for which one might anticipate differences between DFT and wavefunction theory as regards visualization (sections 5.5.6, 6.3.6) are those involving orbitals: as explained in section 7.2.3.2, The Kohn-Sham equations, the orbitals of currently popular DFT methods were introduced to make the calculation of the electron density tractable, and in a pure DFT theory orbitals do not exist. Thus, electron density, spin density, and electrostatic potential can be visualized in DFT calculations just as in ab initio or semiempirical work. However, visualization of orbitals, so important in wavefunction work (especially the HOMO and LUMO, which in frontier orbital theory [114] strongly influence reactivity) is not possible in a pure DFT approach. However, in currently popular DFT calculations one can visualize the KS orbitals, which are qualitatively much like wavefunction orbitals [93] (section 7.3.5, Ionization energies and electron affinities).

7.4STRENGTHS AND WEAKNESSES OF DFT

Strengths

DFT includes electron correlation in its theoretical basis, in contrast to wavefunction methods, which must take correlation into account by add-ons (Møller-Plesset perturbation, configuration interaction, coupled-cluster) to ab initio HF theory, or by parameterization in semiempirical methods. Because it has correlation fundamentally built in, DFT can calculate geometries and relative energies with an accuracy comparable to MP2 calculations, in roughly the same time as needed for HF calculations. Aiding this, DFT calculations are basis-set-saturated more easily than are ab initio: limiting results are approached with smaller basis sets than for ab initio calculations. Calculations of post-HF accuracy can thus be done on bigger molecules than ab initio methods make possible. DFT appears to be the method of choice for geometry and energy calculations on transition metal compounds, for which conventional ab initio calculations often give poor results [58, 123].

DFT works with electron density, which can be measured and is easily intuitively grasped [4], rather than a wavefunction, a mathematical entity whose physical meaning is still controversial.

sion for the energy, is unknown, and no one knows how to systematically improve our approximations to it. In contrast, ab initio energies can be systematically lowered by

Density Functional Calculations 437

using bigger basis sets and by expanding the correlation method: MP2, MP3,..., or more determinants in the CI approach. It is true that for a particular purpose 6-311G* may not be better than and MP3 is certainly not necessarily better than MP2, but bigger basis sets and higher correlation levels will eventually approach an exact solution of the Schrödinger equation. The accuracy of DFT is being gradually improved by modifying functionals, not according to some grand theoretical prescription, but rather with the aid of experience and intuition, and checking the calculations against experiment. This makes DFT somewhat semiempirical. Some functionals contain parameters which must be fitted to experiment; these methods are even more heavily empirical. Since the functionals are not based purely on fundamental theory, one should be cautious about applying DFT to very novel molecules. Of course the semiempirical character of current DFT is not a fundamental feature of the basic method, but arises only from our ignorance of the exact exchange-correlation functional. Because our functionals are only approximate, DFT as used today is not variational (the calculated energy could be lower than the actual energy).

DFT is not as accurate as the highest-level ab initio methods, like QCISD(T) and CCSD(T) (but it can handle much bigger molecules than can these methods). Even gradient-corrected functionals apparently are unable to handle van der Waals interactions [124], although they do give good energies and structures for hydrogen-bonded species [125].

DFT today is mainly a ground-state theory, although ways of applying it to excited states are being developed.

7.5 SUMMARY OF CHAPTER 7

Density functional theory is based on the two Hohenberg–Kohn theorems, which state that the ground-state properties of an atom or molecule are determined by its electron density function, and that a trial electron density must give an energy greater than or equal to the true energy. Actually, the latter theorem is true only if the exact functional (see below) is used; with the approximate functionals in use today, DFT is not variational – it can give an energy below the true energy. In the Kohn–Sham approach the energy of a system is formulated as a deviation from the energy of an idealized system with noninteracting electrons. The energy of the idealized system can be calculated exactly since its wavefunction (in the Kohn-Sham approach wavefunctions and orbitals were introduced as a mathematical convenience to get at the electron density) can be represented exactly by a Slater determinant. The relatively small difference between the real energy and the energy of the idealized system contains the exchange-correlation functional, the only unknown term in the expression for the DFT energy; the approximation of this functional is the main problem in DFT. From the energy equation, by minimizing the energy with respect to the Kohn–Sham orbitals the Kohn–Sham equations can be derived, analogously to the HF equations. The molecular orbitals of the KS equations are expanded with basis functions and matrix methods are used to iteratively find the energy, and to get a set of molecular orbitals, the KS orbitals, which are qualitatively similar to the orbitals of wavefunction theory.

438 Computational Chemistry

The simplest version of DFT, the LDA, which treats the electron density as constant or only very slightly varying from point to point in an atom or molecule, and also pairs two electrons of opposite spin in each KS orbital, is little used nowadays. It has been largely replaced by methods which use gradient-corrected (“nonlocal”) functionals and which assign one set of spatial orbitals to electrons, and another set of orbitals to this latter “unrestricted” assignment of electrons constitutes the LSDA. The best results appear to come from so-called hybrid functionals, which include some contribution from HF type exchange, using KS orbitals. The most popular current DFT method is the LSDA gradient-corrected hybrid method which uses the B3LYP (Becke three-parameter Lee–Yang–Parr) functional.

Gradient-corrected and, especially, hybrid functionals, give good to excellent geometries. Gradient-corrected and hybrid functionals usually give fairly good reaction energies, but, especially for isodesmic-type reactions, the improvement over orcalculations does not seem to be dramatic (as far as the relative energies of normal, ground-state organic molecules goes; for energies and geometries of transition metal compounds, DFT is the method of choice). For homolytic disso– ciation, correlated methods (e.g. B3LYP, and MP2) are vastly better than HF-level calculations; these methods also tend to give fairly good activation barriers.

DFT gives reasonable IR frequencies and intensities, comparable to those from MP2 calculations. Dipole moments from DFT appear to be more accurate than those from MP2, and B3LYP/6-31G* moments on AM1 geometries are good. Time-dependent DFT (TDDFT) is the best method (with the possible exception of semiempirical methods parameterized for the type of molecule of interest) for calculating UV spectra reasonably quickly. DFT is said to be better than HF (but not as good as MP2) for calculating NMR spectra. Good first ionization energies are obtained from B3LYP/6-31 + G*//B3LYP/3-21G(*) energy differences (using AM1 geometries makes little difference, at least with normal molecules). These values are somewhat better than the ab initio MP2 energy difference values, and are considerably better than MP2 Koopmans’ theorem IEs. Rough estimates of electron affinities can be obtained from the negative LUMOs from LSDA functionals (gradient-corrected functionals give much worse estimates). For conjugated molecules, HOMO–LUMO gaps from hybrid functionals agreed well with theUV transitions. The mutually related concepts of electronic chemical potential, electronegativity, hardness, softness, and the Fukui function are usually discussed within the context of DFT. They are readily calculated from ionization energy, electron affinity, and atom charges.

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440 Computational Chemistry

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