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

fermion statistical mechanics to derive the kinetic energy for this system as (Thomas 1927; Fermi 1927)

Note that the various T and V terms defined in Eqs. (8.3)–(8.5) are functions of the density, while the density itself is a function of three-dimensional spatial coordinates. A function whose argument is also a function is called a ‘functional’, and thus the T and V terms are ‘density functionals’. The Thomas–Fermi equations, together with an assumed variational principle, represented the first effort to define a density functional theory (DFT); the energy is computed with no reference to a wave function. However, while these equations are of significant historical interest, the underlying assumptions are sufficiently inaccurate that they find no use in modern chemistry (in Thomas–Fermi DFT, all molecules are unstable relative to dissociation into their constituent atoms. . .)

One large approximation is the use of Eq. (8.4) for the interelectronic repulsion, since it ignores the energetic effects associated with correlation and exchange. It is useful to introduce the concept of a ‘hole function’, which is defined so that it corrects for the energetic errors introduced by assuming classical behavior. In particular, we write

The l.h.s. of Eq. (8.6) is the exact QM interelectronic repulsion. The second term on the r.h.s. corrects for the errors in the first term (the classical expression) using the hole function h associated with ρ (the notation h(r1; r2) emphasizes that the hole is centered on the position of electron 1, and is evaluated from there as a function of the remaining spatial coordinates defining r2; note, then, that not only does the value of h vary as a function of r2 for a given value of r1, but the precise form of h itself can vary as a function of r1).

The simplest way to gain a better appreciation for the hole function is to consider the case of a one-electron system. Obviously, the l.h.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since ρ must be greater than or equal to zero throughout space. In the one-electron case, it should be clear that h is simply the negative of the density, but in the many-electron case, the exact form of the hole function can rarely be established. Besides the self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well.

By construction, HF theory avoids any self-interaction error and exactly evaluates the exchange energy (it is only the correlation energy that it approximates); however, it is time-consuming to evaluate the four-index integrals from which these various energies are calculated. While Slater (1951) was examining how to speed up HF calculations he was aware that one consequence of the Pauli principle is that the Fermi exchange hole is larger than the correlation hole, i.e., exchange corrections to the classical interelectronic repulsion are significantly larger than correlation corrections (typically between one and two orders of magnitude). So, Slater proposed to ignore the latter, and adopted a simple approximation

for the former. In particular, he suggested that the exchange hole about any position could be approximated as a sphere of constant potential with a radius depending on the magnitude of the density at that position. Within this approximation, the exchange energy Ex is determined as

Within Slater’s derivation, the value for the constant α is 1, and Eq. (8.7) defines so-called ‘Slater exchange’.

Starting from the uniform electron gas, Bloch and Dirac had derived a similar expression several years previously, except that in that case α = 23 (Bloch, F. 1929 and Dirac, P. A. M. 1930). The combination of this expression with Eqs. (8.3)–(8.5) defines the Thomas–Fermi–Dirac model, although it too remains sufficiently inaccurate that it fails to see any modern use.

Given the differing values of α in Eq. (8.7) as a function of different derivations, many early workers saw fit to treat it as an empirical value, and computations employing Eq. (8.7) along these lines are termed Xα calculations (or sometimes Hartree–Fock–Slater calculations in the older literature). Empirical analysis in a variety of different systems suggests that α = 34 provides more accurate results than either α = 1 or α = 23 . This particular DFT methodology has largely fallen out of favor in the face of more modern functionals, but still sees occasional use, particularly within the inorganic community.

8.2 Rigorous Foundation

The work described in the previous section was provocative in its simplicity compared to wave-function-based approaches. As a result, early DFT models found widespread use in the solid-state physics community (where the enormous system size required to mimic the properties of a solid puts a premium on simplicity). However, fairly large errors in molecular calculations, and the failure of the theories to be rigorously founded (no variational principle had been established), led to their having little impact on chemistry. This state of affairs was set to change when Hohenberg and Kohn (1964) proved two theorems critical to establishing DFT as a legitimate quantum chemical methodology. Each of the two theorems will be presented here in somewhat abbreviated form.

8.2.1 The Hohenberg–Kohn Existence Theorem

In the language of DFT, electrons interact with one another and with an ‘external potential’. Thus, in the uniform electron gas, the external potential is the uniformly distributed positive charge, and in a molecule, the external potential is the attraction to the nuclei given by the usual expression. As noted previously, to establish a dependence of the energy on the density, and in the Hohenberg–Kohn theorem it is the ground-state density that is employed, it is sufficient to show that this density determines the Hamiltonian operator. Also as noted previously, integration of the density gives the number of electrons, so all that remains to

define the operator is determination of the external potential (i.e., the charges and positions of the nuclei). The proof that the ground-state density determines the external potential proceeds via reductio ad absurdum, that is, we show that an assumption to the contrary generates an impossible result.

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density ρ0. We will call these two potentials va and vb and the different Hamiltonian operators in which they appear Ha and Hb . With each Hamiltonian will be associated a ground-state wave function 0 and its associated eigenvalue E0. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.,

Since the potentials v are one-electron operators, the integral in the last line of Eq. (8.9) can be written in terms of the ground-state density

E0,a < [va (r) − vb (r)]ρ0(r)dr + E0,b (8.10)

As we have made no distinction between a and b, we can interchange the indices in Eq. (8.10) to arrive at the equally valid

<[vb(r) − va (r) + va (r) − vb(r)]ρ0(r)dr + E0,b + E0,a

where the assumption that the ground-state densities associated with wave functions a and b were the same permits us to eliminate the integrals as they must sum to zero. However, we are left with an impossible result (that the sum of the two energies is less than itself), which must indicate that our initial assumption was incorrect. So, the non-degenerate ground-state density must determine the external potential, and thus the Hamiltonian, and thus the wave

function. Note moreover that the Hamiltonian determines not just the ground-state wave function, but all excited-state wave functions as well, so there is a tremendous amount of information coded in the density.

The non-degenerate ground-state character of the density was used to ensure the validity of the variational inequalities. A question that naturally arises is: of what utility, if any, are the densities of excited states? Using group theory, Gunnarsson and Lundqvist (1976a,b) proved that the Hohenberg–Kohn existence theorem can be extended to the lowest energy (non-degenerate) state within each irreducible representation of the molecular point group. Thus, for instance, the densities of the lowest energy Ag and B1u states of p-benzyne each uniquely determine their respective wave functions, energies, etc. (these states are singlet and triplet, respectively, see Figure 7.5), but nothing can be said about the density of the triplet A state of N -protonated 2,5-didehydropyridine, since there is a lower energy singlet state belonging to the same A irrep (to which the Hohenberg–Kohn existence theorem does apply, see again Figure 7.5). The development of DFT formalisms to handle arbitrary excited states remains a subject of active research, as discussed in more detail in Chapter 14.

8.2.2 The Hohenberg–Kohn Variational Theorem

The first theorem of Hohenberg and Kohn is an existence theorem. As such, it is provocative with potential, but altogether unhelpful in providing any indication of how to predict the density of a system. Just as with MO theory, we need a means to optimize our fundamental quantity. Hohenberg and Kohn showed in a second theorem that, also just as with MO theory, the density obeys a variational principle.

To proceed, first, assume we have some well-behaved candidate density that integrates to the proper number of electrons, N . In that case, the first theorem indicates that this density determines a candidate wave function and Hamiltonian. That being the case, we can evaluate the energy expectation value

which, by the variational principle of MO theory, must be greater than or equal to the true ground-state energy.

So, in principle, we can keep choosing different densities and those that provide lower energies, as calculated by Eq. (8.13), are closer to correct. Such a procedure is, of course, rather unsatisfying on at least two levels. First, we have no prescription for how to go about choosing improved candidate densities rationally, and second, insofar as the motivation for DFT was to avoid solving the Schrodinger¨ equation, computing the energy as the expectation value of the Hamiltonian is no advance – we know how to do that already.

The difficulty lies in the nature of the functional itself. Up to this point, we have indicated that there are mappings from the density onto the Hamiltonian and the wave function, and hence the energy, but we have not suggested any mechanical means by which the density can be used as an argument in some general, characteristic variational equation, e.g., with terms along the lines of Eqs. (8.5) and (8.7), to determine the energy directly without recourse to the wave function. Such an approach first appeared in 1965.

8.3 Kohn–Sham Self-consistent Field Methodology

The discussion above has emphasized that the density determines the external potential, which determines the Hamiltonian, which determines the wave function. And, of course, with the Hamiltonian and wave function in hand, the energy can be computed. However, if one attempts to proceed in this direction, there is no simplification over MO theory, since the final step is still solution of the Schrodinger¨ equation, and this is prohibitively difficult in most instances. The difficulty derives from the electron–electron interaction term in the correct Hamiltonian. In a key breakthrough, Kohn and Sham (1965) realized that things would be considerably simpler if only the Hamiltonian operator were one for a non-interacting system of electrons (Kohn and Sham 1965). Such a Hamiltonian can be expressed as a sum of one-electron operators, has eigenfunctions that are Slater determinants of the individual one-electron eigenfunctions, and has eigenvalues that are simply the sum of the one-electron eigenvalues (see Eq. (7.43) and surrounding discussion).

The crucial bit of cleverness, then, is to take as a starting point a fictitious system of non-interacting electrons that have for their overall ground-state density the same density as some real system of interest where the electrons do interact (note that since the density determines the position and atomic numbers of the nuclei (see Eq. (8.2)), these quantities are necessarily identical in the non-interacting and in the real systems). Next, we divide the energy functional into specific components to facilitate further analysis, in particular

where the terms on the r.h.s. refer, respectively, to the kinetic energy of the non-interacting electrons, the nuclear–electron interaction (Eq. (8.3)), the classical electron–electron repulsion (Eq. (8.4)), the correction to the kinetic energy deriving from the interacting nature of the electrons, and all non-classical corrections to the electron–electron repulsion energy.

Note that, for a non-interacting system of electrons, the kinetic energy is just the sum of the individual electronic kinetic energies. Within an orbital expression for the density, Eq. (8.14) may then be rewritten as

where N is the number of electrons and we have used that the density for a Slaterdeterminantal wave function (which is an exact eigenfunction for the non-interacting system) is simply

i=1

Indeed, the similarities with HF theory extend well beyond the mathematical technology offered by a common variational principle. For instance, the kinetic energy and nuclear attraction components of matrix elements of K are identical to those for F. Furthermore, if the density appearing in the classical interelectronic repulsion operator is expressed in the same basis functions used for the Kohn–Sham orbitals, then the result is that the same four-index electron-repulsion integrals appear in K as are found in F (historically, this made it fairly simple to modify existing codes for carrying out HF calculations to also perform DFT computations). Finally, insofar as the density is required for computation of the secular matrix elements, but the density is determined using the orbitals derived from solution of the secular equation (according to Eq. (8.16)), the Kohn–Sham process must be carried out as an iterative SCF procedure.

Of course, there is a key difference between HF theory and DFT – as we have derived it so far, DFT contains no approximations: it is exact. All we need to know is Exc as a function of ρ . . . Alas, while Hohenberg and Kohn proved that a functional of the density must exist, their proofs provide no guidance whatsoever as to its form. As a result, considerable research effort has gone into finding functions of the density that may be expected to reasonably approximate Exc, and a discussion of these is the subject of the next section. We close here by emphasizing that the key contrast between HF and DFT (in the limit of an infinite basis set) is that HF is a deliberately approximate theory, whose development was in part motivated by an ability to solve the relevant equations exactly, while DFT is an exact theory, but the relevant equations must be solved approximately because a key operator has unknown form.

It should also be pointed out that although exact DFT is variational, this is not true once approximations for Exc are adopted. Thus, for instance, the BPW91 functional described in Section 8.4.2 predicts an energy for the H atom of −0.5042 Eh, but the exact result is −0.5. Note that the H atom is a one-electron system for which the Schrodinger¨ solution can be solved exactly – there is no correlation energy. However, because the BPW91 Exc for this system slightly exceeds the classical self-interaction energy (third term on the r.h.s. of Eq. (8.15)), which is 100 percent in error for this one-electron system, the energy is predicted to be slightly below the exact result. Both exact and approximate DFT are size-consistent.

The Kohn–Sham methodology has many similarities, and a few important differences, to the HF approach. We will, however, delay briefly a full discussion of how exactly to carry out a KS calculation, as it is instructive first to consider how to go about determining Exc.

8.4 Exchange-correlation Functionals

As already emphasized above, in principle Exc not only accounts for the difference between the classical and quantum mechanical electron–electron repulsion, but it also includes the difference in kinetic energy between the fictitious non-interacting system and the real system. In practice, however, most modern functionals do not attempt to compute this portion explicitly. Instead, they either ignore the term, or they attempt to construct a hole function that is analogous to that of Eq. (8.6) except that it also incorporates the kinetic energy difference between the interacting and non-interacting systems. Furthermore, in many functionals

empirical parameters appear, which necessarily introduce some kinetic energy correction if they are based on experiment.

In discussing the nature of various functionals, it is convenient to adopt some of the notation commonly used in the field. For instance, the functional dependence of Exc on the electron density is expressed as an interaction between the electron density and an ‘energy density’ εxc that is dependent on the electron density, viz.

Exc[ρ(r)] = ρ(r)εxc[ρ(r)]dr (8.22)

The energy density εxc is always treated as a sum of individual exchange and correlation contributions. Note that there is some potential for nomenclature confusion here because two different kinds of densities are involved: the electron density is a per unit volume density, while the energy density is a per particle density. In any case, within this formalism, it is clear from inspection of Eq. (8.7) that the Slater exchange energy density, for example, is

Another convention expresses the electron density in terms of an effective radius such that exactly one electron would be contained within the sphere defined by that radius were it to have the same density throughout as its center, i.e.,

Lastly, we have ignored the issue of spin up to this point. Spin can be dealt with easily enough in DFT – one simply needs to use individual functionals of the α and β densities – but there is again a notational convention that sees widespread use. The spin densities at any position are typically expressed in terms of ζ , the normalized spin polarization

8.4.1 Local Density Approximation

The term local density approximation (LDA) was originally used to indicate any density functional theory where the value of εxc at some position r could be computed exclusively from the value of ρ at that position, i.e., the ‘local’ value of ρ. In principle, then, the only requirement on ρ is that it be single-valued at every position, and it can otherwise be wildly ill-behaved (recall that there are cusps in the density at the nucleus, so some ill-behavior

in ρ has already been noted). In practice, the only functionals conforming to this definition that have seen much application are those that derive from analysis of the uniform electron gas (where the density has the same value at every position), and as a result LDA is often used more narrowly to imply that it is these exchange and correlation functionals that are being employed.

The distinction is probably best indicated by example. Following from Eq. (8.7) and the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be computed exactly, and is given by Eq. (8.23) with the constant α equal to 23 . However, the Slater approach takes a value for α of 1, and the Xα model most typically uses 34 . All of these models have the same ‘local’ dependence on the density, but only the first is typically referred to as LDA, while the other two are referred to by name as Slater (S) and Xα .

The LDA, Slater, and Xα methods can all be extended to the spin-polarized regime using

where the superscript-zero exchange energy density is given by Eq. (8.23) with the appropriate value of α (referring here to the empirical constant, not the electron spin), and the superscript-one energy is the analogous expression derived from consideration of a uniform electron gas composed only of electrons of like spin. Noting that ζ = 0 everywhere for an unpolarized system, it is immediately apparent that the second term in Eq. (8.26) is zero for that special case. Systems including spin polarization (e.g., open-shell systems) must use the spin-polarized formalism, and its greater generality is sometimes distinguished by the sobriquet ‘local spin density approximation’ (LSDA).

As for the correlation energy density, even for the ‘simple’ uniform electron gas no analytical derivation of this functional has proven possible (although some analytical details about the zeroand infinite-density limits can be established). However, using quantum Monte Carlo techniques, Ceperley and Alder (1980) computed the total energy for uniform electron gases of several different densities to very high numerical accuracy. By subtracting the analytical exchange energy for each case, they were able to determine the correlation energy in these systems. Vosko, Wilk, and Nusair (1980) later designed local functionals of the density fitting to these results (and the analytical low and high density limits). In particular, they proposed one spin-polarized functional completely analogous to Eq. (8.26) in terms of its dependence on ζ , but with the unpolarized and fully polarized energy densities expressed (now in terms of rS instead of ρ, see Eq. (8.24)) as

(8.27)

where different sets of empirical constants A, x0, b, and c are used for i = 0 and i = 1. Vosko, Wilk, and Nusair actually proposed several different fitting schemes, varying the functional forms of both Eq. (8.26) and (8.27). The two forms that have come to be most widely used tend to be referred to as VWN and VWN5, and in most cases give reasonably similar results. LSDA calculations that employ a combination of Slater exchange and the VWN correlation energy expression are sometimes referred to as using the SVWN method.

It is fairly obvious that Eq. (8.27) represents an utter violation of the promise with respect to intuitive equations found in the preface to this book; not only can every term not be assigned an intuitive meaning, it is rather difficult to assign any term such a meaning. However, the virtue of this momentary failure of authorial fidelity is that it allows the highlighting of several important details associated with DFT in general. First, it is apparent just how complex the correlation energy functional in a completely general system may be expected to be, and how difficult a task a first principles analysis may be. Secondly, it indicates the extent to which most modern DFT approaches can legitimately be described as semiempirical methods, in that they include empirically optimized constants and functional forms (albeit there are considerably fewer of these constants and they tend to be more globally used than in, say, semiempirical MO theory – in this respect they are rather like the parameterized electron correlation methods discussed in Section 7.7). Lastly, it should be fairly apparent that solution of the integral in Eq. (8.22) employing the VWN correlation functional is highly unlikely to be accomplished analytically.

In regard to this latter point, the evaluation of the integrals involving the exchange and correlation energy densities in DFT poses something of a mathematical challenge. Most modern electronic structure codes carry out this integration numerically on a grid (much along the lines already discussed in Section 3.4 in the context of Monte Carlo methods, except that the grid points are not sampled randomly, but exhaustively). Through the use of efficient quadratures, grid sizes can be kept manageable from the standpoint of computational resources. Usually, some default grid density is employed unless a user specifies otherwise – it must be noted that in certain situations the numerical noise associated with the default grid density can lead to problems, particularly when first and second derivatives are computed in order to optimize geometries (see Section 2.4), compute vibrational spectra (see Section 9.3.2), etc. For most calculations, however, the numerical noise falls very comfortably below the level of chemical interpretation, and no special care need be taken. So-called ‘grid-free’ integration schemes have been proposed, where in essence the exchange-correlation energy density is expressed in a basis set and advantage is taken of linear algebraic techniques to replace the numerical problem with a ‘smooth’ error associated with basis-set truncation. However, developmental work has not necessarily indicated this approach to be any more robust because very large basis sets are required to maintain reasonable accuracy (Zheng and Almlof¨ 1993).

Returning from this mathematical digression, let us make clear the steps involved in a LSDA calculation. These are summarized in Figure 8.1, and they are for the most part quite similar to those associated with a HF calculation. There are some important differences, however. For instance, step 1 is to choose a basis set. In DFT, there are sometimes several different basis sets involved in a calculation. First, there is the basis set from which the KS