Functional Theories

Three density functional theories (DFT), namely LDA, BLYP, and B3LYP, are included in this section. The simplest is the local spin density functional LDA (in the SVWN implementation), which uses the Slater exchange functional [59] and the Vosko, Wilk and Nusair [60] correlation functional. The BLYP functional uses the Becke 1988 exchange 

The three DFT methods under assessment have a wide range of mean absolute deviations (4.27 to 85.27 kcal/mol) for the energies in the G3/99 test set. Table 3.6 compares the performance of the DFT methods with the different variations of G3 theory for the subset of the 222 heats of formation. 

The ordering of the reliability of the methods is similar to the results for the G2/97 test set seen previously. As expected from its known tendency for substantial overbinding, the local density method (LDA) performs poorest with a mean absolute deviation of 134 kcal/mol. The BLYP functional has a mean absolute deviation of 9.3 kcal/mol, while the B3LYP functional performs the best with a mean absolute deviation of 4.8 kcal/mol. In our previous study on the G2/97 test set that included seven functionals, the B3LYP function also had the lowest mean absolute deviation. 

There is a significant increase in deviations of the data obtained with DFT methods for the heats of formation in the new G3-3 subset. The B3LYP and BLYP mean absolute deviations for the G3-3 subset are about two times larger than that in the G2/97 test set (8.21 kcal/mol vs. 3.08 kcal/mol and 13.32 kcal/mol vs. 7.25 kcal/mol, respectively). 

Average number of electron pairs per molecule in each subset (enthalpies only) of the G3/99 test set 

The interplay between theory and experiment is of paramount importance in understanding materials properties that in turn is important for technological applications when selecting materials with specific properties. The research in this area accordingly spans the complete range from basic research where the fundamental principles behind materials properties on the one side and their structure and composition on the other side are explored to applied research where one seeks the optimal combination of materials with pre-defined properties for specific applications. 

Depending on the properties and systems of interest one can choose different theoretical approaches for such studies. When focusing on the mechanical properties of macroscopic samples, the precise arrangement of the atoms and electrons is often of only secondary interest (although they ultimately dictate the mechanical properties) and might therefore not be considered in the models. On the other hand, when, e.g., studying the electronic properties of semiconductors or the reactivity of specific molecules, one needs to include explicitly both electronic and atomic degrees of freedom in the models. 

Methods based on the density-functional theory of Hohenberg, Kohn, and Sham1,2 represent one class of methods for theoretical studies of materials properties. They are so-called parameter-free methods, indicating that in principle such methods require only the types and positions of the nuclei as input. However, it also means that everything has to be calculated, making such calculations computationally heavy. Therefore, only for the absolutely simplest systems can the statement above be considered justified, whereas for more complex systems one has to apply one or more carefully chosen approximations. Furthermore, such methods are currently not able to study processes that take more than, say, some few ns, or to describe systems with more than a couple of 1000 atoms (an exception is that of infinite, periodic solids, as well as isolated impurities in such crystals and their surfaces). 

During the last 1-2 decades density-functional methods have become increasingly important in chemistry (see, e.g., ref. 3 and references therein), which 

Chemical Modelling Applications and Theory, Volume 1 The Royal Society of Chemistry, 2000 

Density functional theory (DFT) has become very popular in recent years. This is justified based on the pragmatic observation that it is less computationally intensive than other methods with similar accuracy. This theory has been developed more recently than other ah initio methods. Because of this, there are classes of problems not yet explored with this theory, making it all the more crucial to test the accuracy of the method before applying it to unknown systems. 

Density functional theory-based methods ultimately derive from quantum mechanics research from the 1920 s, especially the Thomas-Fermi-Dirac model, and from Slater s fundamental work in quantum chemistry in the 1950 s. The DFT approach is based upon a strategy of modeling electron correlation via general functionals of the electron density. 

Such methods owe their modern origins to the Fiohenberg-Kohn theorem, published in 1964, which demonstrated the existence of a unique functional which determines the ground state energy and density exactly. The theorem does not provide the form of this functional, however. 

Following on the work of Kohn and Sham, the approximate functionals employed by current DFT methods partition the electronic energy into several terms  

All terms except the nuclear-nuclear repulsion are functions of p, the electron density E is given by the following expression  

E +e +E corresponds to the classical energy of the charge distribution p. The E term in Equation 54 accounts for the remaining terms in the energy  

Density functional theory (DFT) is a form of quantum mechanics which uses the one-electron density function, p, instead of the more usual wave function, V , to describe a chemical system. Such a system is any collection of nuclei and electrons. It may be an atom, a molecule, an ion, a radical or several molecules in a state of interaction. 

Hohenberg and Kohn proved in 1964 that the ground-state energy of a chemical system is a functional of p only A functional is a recipe for turning a function into a number, just as a function is a recipe for turning a variable into a number. For example, the energy is also a functional of the wave function. The variational method is one recipe for turning V into a number, . 

H is the many-electron Hamiltonian operator, just as V is the many-electron wave function. The angle-brackets mean integration over the electronic coordinates. 

The density, p, can be obtained by squaring and integrating over the coordinates of all the electrons but one. This is then multiplied by N, the total number of electrons, to get the number of electrons per unit volume, p, which is a function of the three space coordinates only. It is a quantity easily visualized and experimentally measurable by X-ray diffraction, though the accuracy is not adequate for chemical purposes. 

This approximation arose because of the need to simplify the quantum 

In some of the subsequent sections we shall discuss different recent developments of density-functional theory. Some of the fundamentals of electronic-structure calculations that we presented in our first report will be very useM for that discussion, and we shall therefore repeat parts from that. We shall stress that these fundamental considerations can be found in many modem textbooks on methods of electronic-stracture calculations (see, e.g., ref 2). 

We consider a molecule with M nuclei and N electrons. The positions of the nuclei are denoted. i, Ri,, Rm and those of the electrons n, 2,. .., fjv. Moreover, we use Hartree atomic xmits and set accordingly nie= e = 4jt o = h=. The mass and charge of the Ath nucleus are then and Z, respectively. The combined coordinate jq denotes the position and spin coordinate of the rth electron. In the absence of external interactions and relativistic effects, the Hamilton operator for this system can then be written as a sum of five terms. 

The time-independent Schrodinger equation becomes then 

Except for some few special cases (we shall consider some of those in Section 8) one resorts to the Bom-Oppenheimer approximation in order to solve Eq. (3). 

Correlation effects, which are not included in the Hartree-Fock approximation, are also built into the approximate energy functionals that are used in modem DFT apphcations. DFT methods are thus able, in principle, to treat the entire periodic table. 

As with other semiempirical methods, much of the progress has to proceed with trial-and-error approaches. The B3LYP functional is quite commonly used now and often referred to in this book. There are both advantages and disadvantages to the use of B3LYP calculations versus MP2, which are currently the usual choices when dealing with relatively large systems. 

How does one determine a molecular structure using molecular mechanics One first identifies the molecule in question. What are the numbers and kinds of atoms present, and what are their connectivities Using this information, one could build the 

Only one actual problem in chemistry was solved using molecular mechanics as we use it now, by hand calculations. In a series of studies, Frank Westheimer described the following situation. Ortho-substituted biphenyl molecules are generally not co-planar because of the steric interference of the ortho substituents. A typical example is 2,2 -dibromo-4,4 -dicarboxybiphenyl. Consider the two structures of this molecule shown in Reaction (1)  

Since these structures are nonplanar, they are nonsuperimposable mirror images, that is, enantiomers. One can be converted to the other by a partial rotation about the central bond, and the transition state separating them is the planar form. (The carboxyl groups 

Most of the ideas of molecular orbital theory are familiar to organic chemists, even if the details of modem computational methods remain somewhat obscure. A very different and often less familiar approach to calculating structure and properties known as density functional theory (DFT) has become prominent in recent years. DFT calculates an observable property, electron density, instead of a nonobservable entity, a molecular orbital. It is important to note that afunctional is not the same as a function. A function acts on a set of variables to produce a number, but a functional acts on another function to produce a number. For example, a wave function is a function, but the dependence of energy on a wave function is a functional. A function is denoted while a functional is denoted F[/].  

Kohn and Sham proposed that the kinetic energy of the electrons could be calculated from a set of orbitals, Xr which are expressed with a basis set of functions for which the individual orbital coefficients are determined in a manner somewhat similar to that used to determine the coefficients of the orbitals in HF theory. DFT thus becomes a self-consistent procedure in which one starts with a hypothetical system of noninteracting electrons that have the same electron density as the system of interest, determines the corresponding wave functions, and uses the variational principle to minimize the energy of the system and produce a new electron density. That density serves as a basis for a new iteration of the procedure, and the process is repeated until convergence is achieved. 

The energy functional in DFT has several components, some of which can be determined in straightforward fashion. The exchange-correlation  

Notice that the first two terms correspond to the Hartree-Fock energy, equation (A. 66). The last term is the sum of all doubly excited configurations. 

In the Kohn Sham equations (A.116) [324, 325], the core Hamiltonian operator h( 1) has the same definition as in HF theory (equation A.6), as does the Coulomb operator, 7(1), although the latter is usually expressed as 

The Kohn-Sham equations are distinquished from the HF equations by the treatment of the exchange term, which in principle incorporates electron-electron correlation, 

Of course, because the exchange term will be different from HF theory, the DFT orbitals will also be different. 

An early approximation to Uxc(l) was to assume that it arises from a uniform (homogeneous) electron gas  

For transition metal systems, D FT methods generally lead to more accurate structures and vibrational energies than single-determinant HF methods [29,43], and they are often similar in quality to high-level post-HF methods. Since, in addition, DFT calculations are less computationally expensive (approximate scaling factor N ), they have become the method of choice for routine applications in the area of transition metal compounds [29, 43 6]. 

The basis of DFT is that the ground-state energy of a molecular system is a function of the electron density [47]. The Kohn-Sham equations provide a rigorous theoretical model for the all-electron correlation effects within a one-electron, orbital-based scheme [48]. Therefore, DFT is similar to the one-electron HF approach, but the exchange-correlation term. Vex, is different in DFT it is created by the functional Exc(C) and in real applications we need approximations for this functional. The quality of DFT calculations depends heavily on the functional. The simplest approximate 

D FT approach is the Xa method, which uses only the exchange part in a local density approximation (LDA, local value of the electron density rather than integration over space) [49, 50]. The currently available functionals for approximate D FT calculations can, in most cases, provide excellent accuracy for problems involving transition metal compounds. Therefore, DFThas replaced semi-empirical MO calculations in most areas of inorganic chemistry. 

The constraints on the domain of electron densities included in the variational search ensure that the electron density is Af-representable. -  

The primary difficulty in DFT is that the energy functional is unknown. This can be contrasted with the situation in wave function theory, where the formula for evaluating the energy is explicit and computationally feasible but the form of the exact wave function is unknown and—because it is very complicated— probably unknowable. In DFT, the formerly intractable wave function is replaced by the mathematically simple electron density, but the energy functional is unknown and— because it is very complicated—probably unknowable. 

Some progress can be made by decomposing the energy into contributions from kinetic energy, electron-nuclear attraction potential energy, and electron-electron repulsion potential energy 

The electron-nuclear attraction energy has a simple explicit form  

The electron-electron repulsion contribution is often decomposed into classical electrostatic repulsion energy, plus corrections for the Pauli principle (exchange) and electron correlation 

A pedagogical introduction to KS DFT can be found in a previous chapter in this series. While DFT methods are highly appealing in terms of their low cost, the description of anions by DFT, even strongly-bound ones like F , has 

DFT tends to yield positive HOMO eigenvalues, even for species with sizable EAs, suggesting that anions are unbound in DFT (in the KT sense). 

SIE causes DFT to overstabilize half-filled orbitals, and in the context of electron attachment to a closed-shell molecule (forming a doublet radical anion) this means that the anion is overstabilized with respect to the neutral molecule, possibly drastically. 

However, given the steady progress in functional development it is nowadays broadly recognized that DFT has an important role to play in anion electronic structure theory, even (with appropriate caveats, to be discussed) in the case of weakly-bound anions. At the same time, the literature is rife with egregious missteps and dubious conclusions because of ill-conceived DFT calculations for anions. In what follows, we attempt to sort this out and to address the two criticisms enumerated above. 

The effective one-electron potential in KS DFT is traditionally expressed as follows  

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The statistical mechanical approach, density functional theory, allows description of the solid-liquid interface based on knowledge of the liquid properties [60, 61], This approach has been applied to the solid-liquid interface for hard spheres where experimental data on colloidal suspensions and theory [62] both indicate 0.6 this... 

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Molecular dynamics and density functional theory studies (see Section IX-2) of the Lennard-Jones 6-12 system determine the interfacial tension for the solid-liquid and solid-vapor interfaces [47-49]. The dimensionless interfacial tension ya /kT, where a is the Lennard-Jones molecular size, increases from about 0.83 for the solid-liquid interface to 2.38 for the solid-vapor at the triple point [49], reflecting the large energy associated with a solid-vapor interface. 

Density functional theory from statistical mechanics is a means to describe the thermodynamics of the solid phase with information about the fluid [17-19]. In density functional theory, one makes an ansatz about the structure of the solid, usually describing the particle positions by Gaussian distributions around their lattice sites. The free... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

Ira N. Levine, Quantum Chemistry, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1991. (Source for density functional theory.)... 

Parr R G and Yang W 1994 Density-Functional Theory of Atoms and Molecules (New York Oxford)... 

A comprehensive treatment of density functional theory, an idea that is currently very popular in quantum chemistry. 

N -Fle [, ], Fle-F and Ne-F [Ml- Density-functional theory [ ] is currently unsuitable for the calculation of van der Waals interactions [90], but the situation could change. 

Parr B 2000 webpage http //net.chem.unc.edu/facultv/rap/cfrap01. html Professor Parr was among the first to push the density functional theory of Hohenberg and Kohn to bring it into the mainstream of electronic structure theory. For a good overview, see the book ... 

The Flohenberg-Kohn theorem and the basis of much of density functional theory are treated ... 

Becke A D 1995 Exchange-correlation approximations in density-functional theory Modern Eiectronic Structure Theory vol 2, ed D R Yarkony (Singapore World Scientific) pp 1022-46... 

Dunlap B I 1987 Symmetry and degeneracy in Xa and density functional theory Advances in Chemicai Physics vol LXIX, ed K P Lawley (New York Wiley-Interscience) pp 287-318... 

Parr R G 1983 Density functional theory Ann. Rev. Phys. Chem. 34 631 -56 Salahub D R, Lampson S FI and Messmer R P 1982 Is there correlation in Xa analysis of Flartree-Fock and LCAO Xa calculations for O3 Chem. Phys. Lett. 85 430-3... 

Janak J F 1978 Proof that dEldn- = r . in density-functional theory Phys. Rev. B 18 7165-8... 

Perdew J P, Parr R G, Levy M and Balduz J L Jr 1982 Density-functional theory for fractional particle number derivative discontinuities of the energy Phys. Rev. Lett. 49 1691-4... 

Fattebert J-L and Bernholc J 2000 Towards grid-based 0(N) density-functional theory methods optimized nonorthogonal orbitals and multigrid acceleration Phys. Rev. B 62 1713-22... 

Wang Y A and Carter E A 2000 Orbital-free kinetic-energy density functional theory Theoretical Methods in Condensed Phase Chemistry (Progress in Theoretical Chemistry and Physics Series) ed S D Schwartz (Boston Kluwer) pp 117-84... 

Wimmer E, Fu C L and Freeman A J 1985 Catalytic promotion and poisoning all-electron local-density-functional theory of CO on Ni(001) surfaces coadsorbed with K or S Phys. Rev. Lett. 55 2618-21... 

Hammer B, Hansen L B and Norskov J K 1999 Improved adsorption energetics within density functional theory using revised Perdew-Burke-Enerhof functionals Phys. Rev. B 59 7413-21... 

Dreizier R M and Gross E K U 1990 Density Functional Theory an Approach to the Quantum Many-body Problem (Berlin Springer)... 

Car R and Parrinello M 1985 Unified approach for molecular dynamics and density functional theory Phys. Rev. Lett. 55 2471... 

Yethira] A and Woodward C E 1995 Monte Carlo density functional theory of nonuniform polymer melts J Chem. Phys. 102 5499... 

Kierlik E and Rosinberg M L 1993 Perturbation density functional theory for polyatomic fluids III application to hard chain molecules in slitlike pores J Chem. Phys. 100 1716... 

Fraai]e J G E M 1993 Dynamic density functional theory for micro-phase separation kinetics of block copolymer melts J. Chem. Phys. 99 9202... 

Figure Cl. 1.6. Minimum energy stmctures for neutral Si clusters ( = 12-20) calculated using density functional theory witli tire local density approximation. Cohesive energies per atom are indicated. Note tire two nearly degenerate stmctures of Si g. Ho K M, Shvartsburg A A, Pan B, Lu Z Y, Wang C Z, Wacher J G, Fye J L and Jarrold M F 1998 Nature 392 582, figure 2.

Massobrio C, Pasquarello A and Corso A D 1998 Structural and electronic properties of small Cu clusters using generalized-gradient approximations within density functional theory J. Chem. Phys. 109 6626... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

Handy, N.C. Density functional theory. In Quantum mechanical simulation methods for studying biological systems, D. Bicout and M. Field, eds. Springer, Berlin (1996) 1-35. 

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