FURTHER DEVELOPMENTS IN DENSITY FUNCTIONAL THEORY (DFT)

Density-Functional Theory

3. FURTHER DEVELOPMENTS IN DENSITY FUNCTIONAL THEORY (DFT)

vðrÞ; ð31Þ

allowing us to identify l as the chemical potential for the electrons in this system [4].

The introduction of the chemical potential in the density-functional variational principle, Eq. (30), is analogous to the transformation to the grand canonical ensemble in statistical mechanics [5].

Simplifying Eq. (29) yields a functional diﬀerential equation for the ground-state density,

yEv½q

yqðrÞ ¼ l ¼ constant; ð32Þ

that is directly analogous to Schro¨dinger’s partial diﬀerential equation for the wave function. This is most apparent when the Schro¨dinger equation is written in the form

ˆHC

C ¼ E ¼ constant: ð33Þ

Equations (32) and (33) possess comparable importance and similar utility in the density-functional and wave-functional approaches to quantum mechanical systems, respectively.

3. FURTHER DEVELOPMENTS IN DENSITY FUNCTIONAL THEORY (DFT)

3.1. The Kohn–Sham Equations

In principle, solving Eq. (32) provides a straightforward approach to the ground-state electronic energy, ground-state electron density, and from the density, certain other properties of the ground-state system. Unlike the analogous Eq. (33), however, where the energy functional is known from the Hamiltonian operator, the exact expression for Ev[q] is not known, or, more accurately, it is not known in an explicit and computational tractable form. What we have done is to trade an extremely diﬃcult computational problem (solving the Schro¨dinger equation) for an extremely challeng-ing theoretical problem (ﬁndchalleng-ing accurate approximations for Ev[q]).

The core problem is ﬁnding adequate approximations for the Hohenberg–Kohn functional [cf. Eq. (21)]

F½q ¼ E ½q

R

qðrÞv½q; rdr: ð34Þ

Indeed, in the formative years of quantum mechanics, there was substantial interest in determining the properties of systems directly from the electron density, with key

developments being the Thomas–Fermi theory (1927–1928) [6,7], the Dirac [8]

exchange correction (1930), the Wigner [9] correction for the correlation energy (1934), and the Weizsacker [10] functional for the kinetic energy (1935), In 1964, Hohenberg and Kohn proved that there exists a functional such that the solution to Eq. (32) gives the exact ground-state electron density and the exact ground-state energy of any system of electrons. Decomposing the Hohenberg–Kohn functional into its kinetic energy and electron–electron repulsion energy components,

F½quT½q þ Vee½q; ð35Þ

one observes that the primary problem is not the inadequate approximations for Vee[q]

(even the most primitive of models, combining Coulomb repulsion, Dirac exchange, and Wigner correlation, gives a rather reasonable result), but the inadequate approx-imation of the kinetic energy. Most notably, the kinetic energy functionals of Thomas–

Fermi and Weizsacker fail to adequately account for the inﬂuence of the Pauli exclusion principle on electrons [11,12], Still, as the simplest explicit density-functional theory methods, Thomas–Fermi type models possess substantial theoretical impor-tance and hence have been studied extensively [13,14].

In the pursuit of quantitative accuracy from density-functional theory, one typically abandons the idea of expressing the kinetic energy directly in terms of the electron density. Instead, one introduces a set of auxiliary functions, the Kohn–Sham orbitals, {wi[q;r]}^li= 1, themselves functionals of the ground-state electron density, and computes an accurate approximation to the kinetic energy from these orbitals according to the formula [15],

The limits on the summation indicate that the sum only runs over those orbitals which are occupied (according to the aufbau rule); moreover, in this section, we restrict ourselves to closed shell systems and so each orbital is doubly occupied. Fortunately, the correction to the Kohn–Sham kinetic energy, T_s[q], is quite small. For notational simplicity and compactness, we shall henceforth employ atomic units.

To motivate the Kohn–Sham method, we return to molecular Hamiltonian [Eq.

(2)] and note that, were it not for the electron–electron repulsion terms coupling the electrons, we could write the Hamiltonian operator as a sum of one-electron operators and solve Schro¨dinger equation by separation of variables. This motivates the idea of replacing the electron–electron repulsion operator by an average local representation thereof, w(r), which we may term the ‘‘internal potential.’’ The Hamiltonian operator becomes

and, upon separation of variables, solving the Schro¨dinger equation is equivalent to solving the one-electron eigenproblems

j²

2 þ v ðrÞ þ wðrÞ

w_iðrÞ ¼ eiw_iðrÞ: ð38Þ

The associated approximation to the system’s ground-state wave function is obtained as the antisymmetric product of lowest-energy orbitals, {wi(r)}iN/2= 1

, with appropriate spin factors; we denote this as Slater determinate A. Excepting certain limiting cases, A is an inadequate approximation to the true wave function, as obtained by solving the Schro¨dinger equation.

Before proceeding further, it is necessary that we choose a method for con-structing the internal potential, w(r). The insight of Kohn and Sham was to choose the internal potential so that the systems deﬁned by the true Hamiltonian [Eq. (11)] and the model Hamiltonian [Eq. (37)] have the same ground-state electron density [15], This can occur only if the ground-state energy density functionals, Ev[q] [cf. Eq. (21)], and

are minimized by the same electron density. Referring to Eq. (32), we may express this condition as:

where we have used deﬁnitions (35) and (36). Insofar as the zero of energy is arbitrary, the Kohn–Sham chemical potential, lKS, is usually (but not always [16]) taken to be the same as the true chemical potential.For historical reasons and computational facility, the classical electrostatic repulsion energy functional,

J½qu1 2

R R

_q_{ðrÞqðr VÞ}

r r V

j j drdr V ð42Þ

is usually separated from the‘‘non-electrostatic’’ terms in Vee[q], which are combined with Tc[q] to form the exchange-correlation energy,

Exc½quVee½q J½q þ Tc½q: ð43Þ

The equation for the internal potential then becomes wðrÞ ¼ yJ½q

For notational simplicity, this is generally simpliﬁed further by deﬁning the electro-static potential, vJ½q; r u _yqðrÞ^yJ½q, and the exchange-correlation potential, vxc½q; r u

yExc½q

yqðrÞ. Substitution of these results into Eq. (37) yields the celebrated Kohn–Sham equations [15]

Because Eq. (45) depends on the electron density and Eq. (46) depends on the Kohn–

Sham orbitals, these two equations must be solved self-consistently. The procedure for solving the Kohn–Sham system, then, is to guess an electron density, construct vJ[q;r]+vxc[q;r], and solve Eq. (45), subsequently obtaining a new electron density from Eq. (46). Unless the electron density from Eq. (46) equals the‘‘guess density,’’

one proceeds to construct a (suitably improved) guess for the electron density and repeats the process until the input density and the output density are the same.

The Kohn–Sham wave function, AKS, is not expected to be a good approxima-tion to the exact wave funcapproxima-tion; indeed, it is a worse approximaapproxima-tion to the exact wave function than the Hartree–Fock wave function. However, unlike the electron density obtained from the Hartree–Fock equations, the Kohn–Sham method yields, in principle, the exact electron density. Thus we do not need to use the Kohn–Sham wave function to compute the properties of chemical systems. Rather, motivated by the ﬁrst Hohenberg–Kohn theorem, we compute properties directly from the Kohn–

Sham electron density. How one does this, for any given system and for any property of interest, is an active topic of research.

Because of its critical role in constructing the potential energy surface [cf. Eq. (3)]

for a molecule, thence in the prediction of molecular structure and chemical reactivity, we mention how one may compute the electronic energy of a system using the Kohn–

Sham method. In particular, one has E½v; NuEv½q_g:s:

This general form, in which the value of a property is computed expressed in terms of its value for the Kohn–Sham system plus a correction dependent on the exchange-correlation energy, recurs throughout Kohn–Sham density-functional theory.

3.2. Spin Density-Functional Theory

The Kohn–Sham equations as presented in the previous section are most useful for systems in which all electrons are paired. For systems with nonvanishing total spin, it is more convenient computationally (but by no means essential theoretically) to, taking a

cue from unrestricted Hartree–Fock theory, construct spin-dependent Kohn–Sham equations. Brieﬂy, then, the key elements of spin density-functional theory are

(a) The spin density for the a and b spin electrons, qa(r)+qb(r) = q(r).

(b) The exchange-correlation spin density functional, Exc[qa,qb], and its func-tional derivatives, v_xc;r½q_a_;q_b; r u ^yE_yq^xc^½q_r_ðrÞ^a^;q^b ðr ¼ a; bÞ:

j²

Similar to the spin-compensated case, the solution of the unrestricted Kohn–Sham equations starts with the external potential and the number of spin a and spin b electrons in the state of interest (denoted Naand Nb, respectively); then Eqs. (48) and (49) are solved until consistency is achieved. Using the Kohn–Sham orbitals and orbital energies, one then computes the total energy of the system using the spin-dependent generalization of Eq. (47),

For simplicity, we shall, throughout the remainder of this document, treat only the original Kohn–Sham equations, Eqs. (45) and (46).

3.3. Exchange-Correlation Energy Functionals

After solving the Kohn–Sham system, one may evaluate the total electronic energy of the system using Eq. (47) or Eq. (50), as appropriate. From these expressions for the energy and the dependence of the Kohn–Sham potential upon the functional derivative of the exchange-correlation energy functional, it is clear that the accuracy of a density-functional method is entirely dependent upon choosing an appropriate exchange-correlation energy functional, Exc[q]. Indeed, if a practical and exact form for Exc[q] was known, then Kohn–Sham calculations employing this functional would give the exact energy and the exact ground-state electron density.

No useful explicit form for Exc[q] is known, but approximate exchange-corre-lation energy functionals often provide an excellent approximation to the energetic properties of the molecule. To explain how approximate exchange-correlation func-tionals work (and when they fail), recall the essence of the Kohn–Sham method: the Kohn–Sham method constructs the model system with the energy functional

E^KS_v_KS½qu A ˆTD AE þ

R

qðrÞvKSðrÞdr; ð51Þ

that has the exact same ground-state electron density as the real system of interest, which is associated with the energy expression,

E_v½q ¼ C ˆT þ ˆVD eeCE þ

R

qðrÞvðrÞdr: ð52Þ

Starting from these disparate systems, we construct a whole range of intermedi-ate systems, all with identical electron density but with incrementally increasing strengths for the electron–electron repulsion term:

E^k_v½qu CD ^k ˆTþ k ˆVeeC^kE þ

R

qðrÞvkðrÞdr; ð53Þ

clearly k = 0 corresponds to the Kohn–Sham model and k = 1 corresponds to the system of interest. From the fundamental theorem of calculus,

Exc½q ¼ T ½q þ Vee½q Ts½q J½q

Application of the Hellmann–Feynman theorem yields a simple and useful expression for the exchange-correlation energy [19,20]

This approach to the exchange-correlation energy is known as the adiabatic con-nection formalism [21–24].

Deﬁning q2k

(x,xV) to be the probability of observing a pair of electrons, one at x and one at xV,

we obtain from Eq. (55) the working expression

Exc½q ¼1

(x,xV) averaged over the adiabatic connec-tion path. If, in analogy to the classical theory of liquids, we deﬁne the exchange-correlation hole as

hðx; xVÞuq2ðx; xVÞ qðxÞqðxVÞ ð58Þ

then we may write the exchange-correlation energy in the compact form Exc½qu1

R

qðxÞ

R

_qðxVÞhðx; xVÞ x xV

j j dxVdx: ð59Þ

Approximate exchange-correlation density functionals diﬀer in the approxima-tions they make to the innermost integral in Eq. (59). A common assumption is that the exchange-correlation charge, qxc(x,xV) u q(xV)h(x,xV), when spherically averaged about xV = x: is strongly localized near x. This is often a good assumption: it is often true that q_xc(x,jxxVj) takes its minimum value at jxxVj = 0 [where qxc(x,0) is slightly larger thanq(x)] and decays rapidly and monotonically to its limiting asymptotic value

[

jxx Vj!llim q_xcðx; x xVj jÞ ¼ 0 ð61Þ

asjxxVj increases [25]. This suggests that information about the electron density at and near the point x may be used to form an eﬀective approximation to qxc(x,jxxVj).

The simplest choice, of course, is to express qxc(x,jxxVj) as a function of the electron density at x; this yields the class of models for the exchange-correlation energy known as local-density approximations (LDAs) [15],

E^LDA_xc ½qu

R

qðxÞfðqðxÞÞdx: ð62Þ

Recognizing that information about how the electron density changes in the vicinity of the point x is also relevant; numerous other approximations express qxc(x,jxxVj) as a function of not only the density at the point x, but also the derivatives thereof. These generalized gradient approximations (GGAs) to the exchange-correlation energy take the general form [26–28]

E_xc^CCA½qu

R

qðxÞf qðxÞ; jqðxÞ; j ²qðxÞ; . . .

dx: ð63Þ

Most recently developed functionals use either the GGA for or a generalization thereof. Of particular importance are hybrid functionals, which express the total exchange-correlation as a sum of an ‘‘exact’’ exchange (Hartree–Fock) term and a GGA term [29,30].

From this argument, one expects that neither LDAs nor GGAs are reliable when the exchange correlation charge is delocalized over several atomic regions. Such behavior is in fact observed: modern generalized gradient approximations are reliable and accurate when qxc(x,jxxVj) is localized and centered on x; errors are typically no more than a few kilocalories per mole. In some cases, however, qxc(x,jxxVj) does not increase more or less monotonically as one moves away from x, instead having additional minima in other nearby portions of the molecule. In such cases, errors are often an order of magnitude larger, frequently tens of kilocalories per mole. Even when qxc(x,jxxVj) is rather delocalized, however, generalized gradient approxima-tions sometimes give reasonable results, owing mostly to the cancellation of errors [and helped by the fact that the factor of inverse distance in Eq. (59) helps to reduce the energetic importance of the ‘‘far away portion’’ of the exchange-correlation charge].

Given the importance of localized exchange-correlation charges to the accuracy of approximate density functionals, a rule of thumb for deciding when qxc(x,jxxVj) is

likely to be localized is of great utility. One such rule, due to Zhang and Yang [31], is that a dissociating molecule will tend to have a nonlocalized hole when the ionization potential of one of the dissociating fragments resembles the electron aﬃnity of the other. Another rule was proposed by Gritsenko, Ensing, Schipper, and Baerends (GESB) [32,33], who present a semiempirical argument that when the number of bonding electrons divided by the number of atomic orbitals composing a bond is an integer, then qxc(x,jxxVj) is usually localized, increasing more or less monotonically as one moves away from the reference point x. By contrast, systems with 1-electron 2-center bonds (as H+2

) [31], 2-electron 3-center bonds (as the bridging bonds in diborane), 4-electron 3-center bonds (as the transition state in an SN2 reaction) [33], and 3-electron, 2-center bonds (as F2) tend to have exchange correlation charges with two or more signiﬁcant minima and are problematic for every known density-func-tional theoretic technique [32], including many much more elaborate than the simple generalized gradient approximations. Indeed, systems of these types usually have substantial multireference character; such systems, then, are often problematic for conventional wave function-based approaches also.

Fortunately, for a wide range of molecular structure, including ‘‘normal’’

covalent bonds, ionic bonds, and closed-shell/closed-shell interactions, and for the most common reaction pathways (simple bond formation and cleavage), the GESB rule indicates that qxc(x,jxxVj) is relatively well localized. Modern GGAs are accurate and reliable predictors of molecular structure and reactivity for such systems. On the other hand, when the GESB rule suggests that qxc(x,jxxVj) is not localized, then it is essential that one carefully check the results of one’s calculations, preferably by recourse to experiment or to a more conventional wave function-based technique.

Density-functional theory can still be useful in these cases, especially since the energy obtained from density-functional calculations on these systems tends to be system-atically too low: in assuming the exchange-correlation charge to be more highly localized than it actually is, GGAs typically overestimate the attraction between the electron and its hole in Eq. (59), thereby overestimating the magnitude of the ex-change-correlation energy.

3.4. Linear-Scaling Methods for Solving the Kohn–Sham System

The Kohn–Sham construction is a pragmatic one, justiﬁed by computational utility.

Of special computational utility is the fact that each Kohn–Sham orbital experiences the same potential and that this potential, in turn, is a functional of the electron density alone. This allows us to rewrite the Kohn–Sham energy in terms of the ﬁrst-order density matrix,

where the electron density is given by

qðrÞucðr; rÞ ð66Þ

and we have deﬁned the Kohn–Sham potential according to

v_KSðrÞuvðrÞ þ vJ½q; r þ vxc½q; r: ð67Þ

To see what may be gained by such a construction, consider that the normal Kohn–Sham procedure requires finding the eigenvalues and eigenvectors of the Kohn–Sham equations; computational cost thus increases as the cube of the size of the system. However, due to the particular structure of the Kohn–Sham equations, we do not need to find the Kohn–Sham eigenvalues and eigenvectors: computing the internal potential only requires the electron density, while computing the kinetic energy only requires the first-order density matrix. Because c(r,rV) depends on two spatial coordinates instead of just one (like the electron density), the computational cost inherent in finding c(r,rV) formally increases only as the square of the system’s size.

One may be tempted then to ﬁnd the density matrix by minimizing the Kohn–

Sham energy, Eq. (65), with respect to the density matrix subject to the constraint that N¼

R

qðrÞdr ¼

R

cðr; rÞdr: ð68Þ

Unfortunately, this does not give the correct answer, giving instead the state where all the electrons are in the lowest energy Kohn–Sham orbital; this violates the Pauli exclusion principle. Satisfying the Pauli exclusion principle requires that every state of the system be occupied by no fewer than zero and no more than two electrons (one with spin a and one with spin b). This indicates that the eigenvalues of the ﬁrst-order density matrix [it follows from the deﬁning Eq. (64) that the eigenvectors of c(r,rV) are the Kohn–Sham orbitals]

R

cðr; rVÞwiðrVÞdrV ¼ niw_iðrÞ ð69Þ

must be between zero and two,

0 V ni V 2: ð70Þ

The eigenvalues of the ﬁrst-order density matrix are identiﬁed with the occupation numbers for their associated Kohn–Sham orbitals.

For the minimum energy state of a closed shell system, all the Kohn–Sham orbital occupation numbers will be either 0 or 2, which allows us to replace the constraint (70) with the more compact idempotency condition

cðr; rVÞu1 2

R

cðr; xÞcðx; rVÞdx: ð71Þ

The‘‘divide-and-conquer’’ method [34,35] was the ﬁrst Kohn–Sham algorithm which delivered linear scaling: computational costs that grow linearly with the size of the system. In this technique, one ﬁrst projects the density matrix onto a basis set, typically

a set of Gaussian functions,fe^fⁱ^j^rR^a^j²g, centered at each atomic center, a. We may

onto this basis set; here S is the overlap matrix, Sai;bju

R

v_aið Þvr _bjð Þdr:r ð74Þ

Because the basis functions decay strongly as one moves away from the atom on which they are centered, hai;bjc 0 when jR^aRbj is large. In particular, this means that the hai;bjcan be neglected wheneverjRaRbj is greater than some threshold, Rh. Typically Rhf7 A˚ = 13.5 Bohr.

Computationally, one proceeds as follows. Starting at subsystem a (which could be an atom, but is more generally a molecular fragment), one constructs the subsystem Hamiltonian matrix hci;dj(a) In addition to the aforementioned cutoff on the off-diagonal elements (Rh), there is an additional system-dependent cutoff on hci;dj(a)

: we do not calculate matrix elements in which either of the two centers, c or d, are further than some‘‘buﬀer distance,’’ Rb, from an atom in the subsystem a. Typically, Rbf6 A˚ = 11.5 Bohr is suﬃcient to ensure that the interactions between the subsystem of interest and the neighboring systems are accurately modeled [36,37].

Given the subsystem Hamiltonian, hci;dj(a)

, one may solve the Kohn–Sham equations directly. Projecting the Kohn–Sham equations onto the nonorthogonal basis set yields a generalized eigenvalue problem for the molecular fragment:

(r) is the kth Kohn–Sham orbital for subsystem a and Sci;dj(a)

is using the cutoﬀ scheme from Eq. (75). We then deﬁne the density matrix of the subsystem as

c^{ð Þ}_ci;^a_yj¼X

where fT(le^(a)k) is the Fermi distribution function at temperature T and

The Mulliken-like partitioning function [38] deﬁned by Eq. (79) ensures that we include in the density matrix associated with fragment a only those elements which have at least one basis function centered in the subsystem of interest, and we fully include only those elements that have both indices in the subsystem of interest. (The weighting factor of one-half is motivated by symmetry, pab(a)

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