Density Functional Theory: A primer
Sergio Aragon
San Francisco State University
Abstract
A 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 = 
Tn = 
2
2 me
2
2
Ne



2
1
 2k

k 1 m k
Nn
(3)
Zk e2
Ven =  


 1 k 1 40 | r  Rk |
Ne N n
Ne
Vee =
Ne
e2
 


 1    4 0 | r  r |
Zk Z je2


Vnn = 
j 1 k  j 4 0 | R j  Rk |
Nn Nn
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,
1 ( x1 )  2 ( x1 ) . .  N ( x1 )
1 ( x2 )
.
. .  N ( x2 )
1
HF =
(4)
.
.
. .
.
N!
.
.
. .
.
1 ( x N )
.
. .  N (xN )
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 > =
1 N N
1
1


|
h
|



[    | |          | |    ]  Vnn





2  1  1
r12
r12
 1
N
(5)
1 2 Nn Z k
where the h      
is a one electron operator containing the electron
2
k 1 rk
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 spinorbitals must satisfy:
{h  VHF ( x  )}      
(6)
N
VHF ( x  )   [ J ( x  ) K ( x  )]
 1
J ( x1 )   |  ( x2 ) | 2
dx2
r12
K ( x1 )   ( x1 )  [   ( x2 ) *   ( x2 )
dx2
] ( x1 )
r12
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-HartreeFock 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-HartreeFock 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  |  x1  x 2 ... x N R1 ... R Nn  | 2 d 1 dx 2 ...dx 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:
b
F [ g ]   g ( x)dx . The specification of the integrable function g(x) produces a
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[   (r )V (r )dr ’]  E0[0] = <00


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:




 
 (r )  1  (r1 )  (r2 )  
5/3 
ETFD[] = C F   (r )dr  Z 
dr  
dr1dr2  C X   4 / 3 (r )dr
r
2
r12
(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 KohnSham 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”.
TS [  S ]  
1 N
  j |  2j |  j 

2 j 1
provided that the density of the real system,  = S =
(13)
N


j 1
energy functional of the real system is written as:

|  j (r ,  ) | 2 . Then, the
E[] = TS[] + J[] + Vnn + Exc[]
(14)
Where all the unknown pieces are collected (swept under the rug!) into the Exchange
Correlation energy,
Exc[SEee[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 functional
in terms of the KS orbitals:
1 N
1 N N
1
2
E[  ]      j |  j |  j       j  i |
|  j  i   Exc[  ] 
2 j 1
2 j 1 i 1
r12
Nn N
Nn Nn
 a b
1
(16)



|
|





a
j
j
ra1
a 1 j 1
a 1 b 1 rab
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:
Nn
Z
1
{  2j   a  VKS ( x j )} j   j  j
2
a 1 raj
(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.

 (r2 ) 


VKS (r1 )  
dr2  Vxc(r1 )
r12
(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 pseudoeigenvalue 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:

 
E xcLDA [  ]    (r ) xc (  (r )) dr
,



 xc (  (r ))   x (  (r ))   c (  (r ))
(19)
 1/ 3

3  3 (( r )) 
where the  x (  (r ))   
 is the Dirac/Slater Exchange energy of the gas.
4 

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,
 
E xCGA  E xLDA   F (s ) 4 / 3 (r )dr

, s 
|  |
 4 / 3
(20)
Two commonly used choices for the Exchange functional are:
Becke (1988): F B 
s
(21)
1  6s Sinh 1 ( s )


1 / 15
(22)
F 86  1  1.296 x 2  14 x 4  0.2 x 6
s
x
(24 2 )1 / 3
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:
Perdew (1986) :
Becke & Perdew/Wang: BPW91
Becke & 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:
E xcB 3 LYP  (1  a) E xLSD  aE xc 0  bE cB 88  cE cLYP  (1  c) E cLSD
(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.