Density Functional Theory: A primer

Sergio AragonSan Francisco State University

AbstractA condensed description of Hartree-Fock methods and Density Functional theory is presented. We show that the mathematical form of both methods is the same and that DFT achieves accuracy better than or equal to HF//MP2 methods more economically and for a much larger class of molecules, including transition metals.

I. Wavefunctions & Schroedinger’s Equation

To study and predict the properties of atomic and molecular systems (including solid state matter), one must use Quantum Mechanics. For molecular systems, the energy shifts caused by the presence of magnetic dipoles of the electrons and nuclei are small compared to chemical binding energies, and can usually be treated successfully by perturbation theory. In addition, because the chemistry focus is on the valence electrons, the core electrons can be treated approximately and thus relativistic effects, which affect mostly the core, can be neglected. For chemistry and the solid state, then, the basic characterization of structure and reactivity can be obtained from the non-relativistic Schroedinger equation without magnetic effects. For the characterization of structure, the time independent Schroedinger Equation suffices, while to study the interaction of molecular systems with electromagnetic radiation, the full time dependent Schroedinger Equation is necessary. The following discussion is restricted to the time independent Schroedinger Equation in the Born-Oppenheimer approximation in which the electronic structure is computed at fixed nuclear positions. For a molecular system, it is:

H (x1,x2,…,xN; R1,…,RNn) = E (x1,x2,…,xN; R1,…,RNn) (1)

Where x1,x2,…,xN represent the spin and cartesian coordinates of the N electrons in the molecule, and R1,…,RNn are the nuclear coordinates of the Nn nuclei in the molecule. The Hamiltonian operator is given by,

H = Te + Tn +Ven +Vee +Vnn (2)

where the kinetic and potential energies of N electrons (e) and Nn nuclei (n) are given by,

Te = (3)

Tn =

Ven =

Vee =

Vnn =

The solutions to this equation must be obtained with several important restrictions. First, the wavefunction must be normalizable, and, second, it must be antisymmetric with respect to the exchange of any pair of electron spin-coordinates xi, xj because electrons are Fermions. This requirement is equivalent to the usually stated Pauli Exclusion Principle: no two electrons of the same spin can occupy the same space point at the same time. Equivalently, (x1,x1,…) =0. This implies that two electrons of the same spin are automatically kept away from each other by the antisymmetry requirement and they do not move independently of each other. Their motion is correlated, regardless of the fact that they are electrically charged objects.

Example I: The Hydrogen Molecule

A simple example of the above requirements is the approximate MO-LCAO minimal basis set wavefunction of the 2-electron hydrogen molecule. Let R = |R1-R2|, then the unnormalized wavefunction may be written as,

(x1,x2; R) = [1s1(r1;R)+1s2(r1;R)][ 1s1(r2;R)+1s2(r2;R)][(1)(2)-(1)(2)].

The spatial part of the wavefunction containing the 1s orbitals (each centered on nucleus 1 and 2) is symmetric with respect to the exchange of electron coordinates r1 and r2, but the spin part of the wavefunction is antisymmetric with respect to the exchange of electron 1 for 2. This wavefunction satisfies the Pauli exclusion principle (i.e., it is overall antisymmetric). Note that the meaning of the coordinate x1

implies the simultaneous specification of the electron Cartesian coordinates r1 and the spin state of the electron (or). The normalization can be done in confocal elliptical coordinates and is omitted for simplicity. This wavefunction is too crude to yield an accurate value for the coordinate R (it is too large) and for the ground state energy E0 (it is not negative enough), but it does give us an idea to start from.

Exercise: Show that the above wavefunction can be written as a single determinant.

The molecular structure problem entails finding the values of the R1,…,RNn nuclear coordinates at which the energy E is minimum. This value of the energy, E0, is called the ground state energy, and the resulting structure, the ground state molecular geometry. We would also like to know a variety of other properties of the ground state such as the electric moments (dipole, quadrupole, ….), the electrostatic potential on the “boundary of the molecule”, the effective electric charges on the nuclei, and other such properties.

If we solve eq. (1) completely, we obtain much more than that. In fact, quantum mechanics tells us that all possible measurable properties of the isolated molecular system are contained in the wavefunctions (x1,x2,…,xN; R1,…,RNn) for all the states of the molecule. In order for all this information to be contained there, the wavefunction is very complex. It is a high dimensional object, dependent on 3 N electron Cartesian coordinates, N electron spin coordinates, and 3 Nn nuclear coordinates. For a molecule such as benzene, with 12 nuclei and 42 electrons, this object exists in 162-6=156 dimensional Cartesian space! The determinant has 42! = 1.4 1051 terms! We can’t even make a picture of it. If we were to store it as a numerical object, with a resolution of 0.1 au out to 10 au, we would need 10312 numbers to store, at single precision (4 bytes/number), at 2 Gbytes/cm2 (counting electrical connections access), we would need more than 10293 Km2 of surface to store that information. The Earth has a surface area of less than 109 Km2. The promised knowledge hidden in the Schroedinger Equation is not quite easily accessible. We must make do with much much less. How much less can we do with?

The situation is not as hopeless as this trivial computation makes it sound. First of all, the majority of the storage space can be avoided by using a basis set of functions, usually centered around the nuclei, to expand the unknown wavefunction. Then only a smaller set of coefficients and parameters are required. The remaining problem, however, is that the mathematical representation requires the use of an infinite set of basis functions and in practice we must use only a finite set. The choice of basis set becomes a crucial point of departure.

Furthermore, the fundamental behavior of the electrons is to exquisitively avoid each other in a multi-electron system. The motion of the electrons is highly correlated. Electrons of the same spin are automatically kept away from each other by the antisymmetry requirement (exchange correlation or Fermi correlation), but electrons of different spin must be kept away from each other by the spatial form of the wavefunction (Coulomb correlation). The latter can only be done approximately in practice.

In wavefunction based methods, the simplest treatment with useable accuracy is the Hartree-Fock method. In this method, one assumes that the wavefunction can be written as the antisymmetric combination of one-electron orbitals (a Slater determinant) and thereby replaces the exact Hamiltonian of eq. (2) by that of a set of non-interacting “electrons” where each moves in the average field of the rest of them (a Mean Field Theory). The orbitals obey non-linear integro-differential Fock equations, and the solution must be obtained by iterative methods until self-consistence is achieved (an SCF method).

Example II: The Hartree-Fock Method

For an even number of electrons, one Slater determinant suffices to setup the Hartree-Fock equations. Let the wavefunction be composed of the spin-orbitals i = i or i. Then,

HF = (4)

The fundamental principle that enables us to obtain an approximate solution to the Schroedinger Equation is the Variational Principle: E = <|H|> E0. We vary the spin-orbitals themselves in order to minimize the value of E, obtaining a quality approximation to the ground state energy. Substituting the Hartree-Fock wavefunction into the expectation value, we obtain,

EHF = <HF |H|HF > =

(5)

where the is a one electron operator containing the electron

kinetic energy and the electron-nucleus attraction. The double integrals in the double summation are the Coulomb and Exchange integrals, respectively. The antisymmetry requirement makes the Exchange integral appear. The variational principle, applied to eq. (5) yields the integro-differential equation that the spin-orbitals must satisfy:

(6)

where the effective Hartree-Fock potential VHF is a operator containing both a Coulomb (J) and a non-local Exchange operator (K) added over all electrons (the mean field). These equations represent a fictitious system of non-interacting

particles that approximates the electron behavior. Furthermore, if we assume that upon removal of a given electron (ionization) from the system, that the distribution of the rest of the electrons does not change (which is not true), then the energy required to ionize the system from the orbital is precisely the eigenvalue . This is a statement of Koopman’s theorem and provides a physical interpretation for the orbital eigenvalues. Note that it is approximate, with unknown error.

In actual practice, the HF spin-orbitals are expanded in a basis set and the Variational problem becomes a simpler one of finding the optimum values of the expansion coefficients. When this is done, the practical equations are called the Roothaan-Hartree-Fock equations. This has been discussed in detail previously.

The Hartree-Fock method satisfies the exchange requirements perfectly, but misses badly in the rest of the electron correlation effects. To obtain chemical accuracy, post-Hartree-Fock methods must be used in addition, such as Moller-Plesset perturbation theory, Configuration Interaction, or Coupled Cluster methods. These are computationally expensive, limiting the size of the system to which they can be applied, and of very limited application to transition metals. What else can we do?

II. Electron density instead of the wavefunction

Can we obtain the ground state properties of the molecular system without explicitly constructing the very complex wavefunction? In 1964, Hohenberg & Kohn demonstrated that, surprisingly, the answer is YES. The reason this is of fundamental importance is that the electron density, for any molecular system, is a function of only three spatial coordinates (for a given set of nuclear positions)! Let’s define it:

(r1) = N (7)

This is the integral over all electron cartesian and spin coordinates, except the Cartesian ones of electron 1, of the probability distribution defined by the wavefunction, multiplied by the number of electrons in the system. The integral of the density over all space equals N. The Hohenberg-Kohn theorems are deceptively simple:

Theorem 1: The external potential, Ven is determined, up to an additive constant by the electron density. Since (r1) determines the number of electrons N, then the density also determines the ground state wavefunction and all other electronic properties of the system.

Proof: Prove by contradiction. Assume that there exist two different external potentials V=Ven +Vnn and V’=Ven’ +Vnn’ which both give the same electron density (r1). Then we have two Hamiltonians, H and H’ with the same ground state density and different and ’. Now we use the variational principle, taking ’ as a trial function for the H Hamiltonian, to obtain:

E0 < <’’<’'’<’'’

= E0' + <’V-V'’In addition, we can take the as a trial function for the H’ Hamiltonian, to obtain:

E0’ < <'<<’

= E0 + <V’-V

Now we recognize that the expectation value of the difference in the external potentials differ only in sign because we assumed that the electron density is the same. When we add the two equations, we then obtain the contradiction:

E0 + E0’ < E0’ + E0 (10)

Thus we conclude there there exists a unique map between the external potential V and the ground state density. This implies that all the energies, including the total energy, is a functional of the density. We write this as E = E[]. The density determines the Hamiltonian, and thereby, the wavefunction.

Detour: What is a functional?

We give an intuitive definition by comparing the situation to the more familiar notion of function. A function f: R R is a map (or assignment) between one number of the set R (say the real numbers) to another number of the set R. We usually denote this assignment by y = f(x), and we agree that to each value of x there corresponds only one value of y. On the other hand, a functional is a map between a set of functions and a number; F: {f(x)} R, and we denote this by F[f(x)] = y. A simple example is the definite integration operator:

. The specification of the integrable function g(x) produces a

number defined in terms of the constants a and b.

Theorem 2: Variational Principle

Suppose we have a trial density ’. Then this density defines its own wavefunction ’, and the expectation value of the true Hamiltonian satisfies the variational principle:

<’’Vee[ ’] E0[0] = <00

Thus, the correct density is the one that produces the minimum energy. Is there any system for which we can explicitly write out the energy functional? The box below gives such an example.

Thomas-Fermi-Dirac Atomic Model

The uniform electron gas consists of N=Z electrons in a volume which is uniformily positive so as to keep the entire system electrically neutral. For this system, Thomas and Fermi (1927) computed the kinetic energy, and Dirac (1930) computed the Exchange correlation energy:ETFD[] =

(12)

The values of the constants C are of order unity. This model reproduces the Madelung Rule for the orbital energy filling of electrons in atoms, but is not otherwise accurate, and when applied to molecules, it produces no binding. However, for the electron gas, the Coulomb correlation can also be computed, albeit, numerically (Ceperly & Alder,1980) and this is included in many modern functionals.

The Kohn-Sham Equations

One of the main problems of the Thomas-Fermi-Dirac theory is that the kinetic energy functional is not representative of the very inhomogenous electron density in atoms and molecules. The KS method relies on the introduction of a fictitious reference systems of non-interacting electrons that is constructed to have the same electron density as the system of interest. These electrons must move in a complex potential that takes into account the actual forms of electron correlation and the difference between the kinetic energy functional of the reference system and the real system.

The reference system of ficticious non-interacting particles has a rigorous solution in terms of single electron wavefunctions, or molecular orbitals. These are called the Kohn-Sham orbitals. Thus, we can express the kinetic energy exactly for the reference system as the sum of the expectation value of the Laplacian for each “electron”.

(13)

provided that the density of the real system, = S = . Then, the

energy functional of the real system is written as:

E[] = TS[] + J[] + Vnn + Exc[] (14)

Where all the unknown pieces are collected (swept under the rug!) into the Exchange Correlation energy,

Exc[SEee[J[

The above expression for the Exchange-Correlation energy includes the error in the kinetic energy, the correlation effects, and also the correction for the self –interaction of the electron that has been included in the classical Coulomb integral J.

Now we must apply the variational principle and the variables to be adjusted are the KS orbitals themselves. In practice, this is done just like in the HF method, a basis set expansion is used to represent the KS orbitals. Thus, similar computational engines are required for both DFT and HF theory and it is not then a surprise that computational packages have them often both available as options. We write the entire energy functionalin terms of the KS orbitals:

(16)

The setting the functional derivative of this expression to zero yields the equations that the KS orbitals must satisfy to provide the same density for both systems:

(17)

where the effective KS potential in which the non-interacting particles move contains the classical electrostatic interaction, the Exchange-Correlation potential and the electron nucleus attraction.

(18)

Note that eq. 17 is identical in mathematical form to the HF equations satisfied by the Hartree-Fock orbitals. Thus, the same methods are used to solve them. Since the unknown density is required to write out the KS potential, the equations are also pseudo-eigenvalue equations and must be solved by an SCF method. The fundamental difference lies in the form of the VKS potential versus the VHF potential. Each of them has an ordinary Coulomb repulsion term and an Exchange term. In the HF case, the Exchange terms is known and exact for the model, while the Coulomb correlation term is missing.

In the DFT case, the Vxc term includes UNKNOWN functional forms for the Coulomb and Exchange correlation and the correction to the kinetic energy.

Why is this advantegous? It might not seem we have gained anything because we knew ahead of time that we needed to include all these correlation effects, and pasting an unknown term in the equations as a place holder may not seem like a great advance. The reason that it does represent a fundamental advance is that we don’t have to completely guess what to put in for Vxc because we do have the exactly soluble uniform electron gas.

III. Approximations for the Exchange-Correlation Functional

The simplest approximation that one can make is to imagine that at every point in space we can use the value of the density that the uniform electron gas would have at that point, and allow it to vary from point to point. This is called the Local Density Approximation (LDA). In more detail, we write:

, (19)

where the is the Dirac/Slater Exchange energy of the gas.

The Coulomb correlation piece is obtained from interpolations of the Monte Carlo data of Ceperly and Alder (1980). The typical LDA approximation is the SWVN implementation that implies Slater Exchange plus Coulomb correlation obtained by Vosko, Wilk & Nussair (1980). Perdew & Wang (1992) made a more accurate implementation.

Schematic of the LDA (Koch & Holthausen).

The LDA by its self does not contain sufficient accuracy for chemical applications. It is necessary to include terms that explicitly take into account the spatial variation of the density. This the formulation of functionals within the Generalized Gradient Approximation (GGA) is what made good accuracy density functional theory possible. As above, approximations are made separately for the Exchange and Coulomb portions.

The Coulomb expressions are quite complex and we will skip writing them down. The popular choices are:

LYP : Lee, Parr &Yang (1988)PW91: Perdew & Wang (1991)P86: Perdew (1986)

These can be combined with Exchange Correlation functionals. These have the form,

, (20)

Two commonly used choices for the Exchange functional are:

Becke (1988): (21)

Perdew (1986) : (22)

The b parameter in the Becke form is fit to accurate densities of a large number of atoms, while that of Perdew does not have any empirical parameters. Typical combinations in common use give rise to the functionals:

Becke & Perdew/Wang: BPW91Becke & Lee/Parr/Yang: BLYP

These functionals are all LOCAL in the mathematical sense, but go beyond the Local Density Approximation.

Hybrid Functionals

It should be clear that the HF method has a systematic error. The lack of Coulomb correlation implies that the energies are higher than they should be as charges are allowed to get closer to each other than they should be, thus the bond lengths tend to be smaller and the binding energies larger than experiment. In DFT, on the other hand, the systematic errors occur in precisely the opposite direction. Thus a combination of the HF Exchange and the functional exchange should improve things considerably. The popular B3LYP functional is the prime example of this kind. The performance is excellent, with

an RMS error of about 2 kcal/mol on the large G2 molecule reference set. This functional has the form:

(23)

The parameters a,b,c control the relative amounts of HF Exchange and Coulomb correlation from the various sources and they are determined empirically (by fitting to data). The performance is very good, as we will discuss further in the next section.

IV. Performance

V. References

1. W. Koch, M. C. Holthausen, “A Chemist’s Guide to Density Functional Theory”, Wiley-VCH: New York, 2000.

2.