14. Density Functional Theory (DFT)
The quantum mechanics is based on the wavefunction: the wavefunction contains all the
information there is about the system. But the wave function itself is essentially uninterpretable –
it is an “inscrutable oracle that returns valuably accurate answers when questioned by quantum
mechanical operators, but it offers little by way of sparking intuition”.
The density functional theory tries to circumvent the wavefunction and use a physical
observable, the electron density  (r ) which is a real function of only three spatial variables
(vector r).
14.1 The Hohenberg-Kohn Theorems
The H-K theorems are the foundation of the modern DFT. There are two:
1. The existence theorem states that if N interacting electrons move in an external
potential Vext(r), the ground-state energy is a unique functional of the density (r).
Therefore, the ground state electron density is sufficient to construct the full Hamilton
operator and hence to calculate - in principle - any ground state property of the system
without the knowledge of the many electron wavefunction.
Alternatively formulated, this means that any ground state property can be expressed in
terms of the ground state electron density (r).
2. The variational theorem states that the ground state energy can be obtained
variationally: the density that minimizes the total energy is the exact ground state density
(0):
E[  (r)]  E0  E[  0 (r)]
(99)
Although the Hohenberg-Kohn theorems are extremely powerful, they do not offer any
prescription or method for computing the ground-state density of a system in practice.
14.2 The Kohn-Sham methodology
According to the first K-H theorem, density determines the external potential, which determines
the Hamiltonian, which in turn determines the energy and the wave function. Unfortunately, if
one attempts to proceed in this direction, there is no simplification over MO theory, since the
final step is still the solution of the Schrodinger equation.
As we know, the main (actually, only) difficulty in solving the Schrodinger equation for a
molecule derives from the electron–electron interaction. For a non-interacting system of
electrons it would be a trivial problem. The Hamiltonian would be simply a sum of one-electron
operators, its eigenfunctions would be Slater determinants of the individual one-electron
eigenfunctions, and eigenvalues would simply be the sum of the one-electron eigenvalues (see
the section on MP2, eqn. 87).
Because non-interacting electrons are a trivial problem, Kohn & Sham proposed the following
clever idea to get around the difficulty:
Take as a starting point a fictitious system of non-interacting electrons, but such that their overall
ground-state density is exactly the same as for the real system of interest.
The energy functional can then be divided into components as follows:
E[  (r)]  Tni [  (r)]  Vne[  (r)]  Vee [  (r)]  T [  (r)]  V [  (r)]
where:
Tni [  (r)]
Vne [  (r)]
is kinetic energy of the non-interacting electrons
is nuclear-electron interaction, which is easily written in term of the
electron density:
Vne [  (r )] 
nuclei
ZI
 |r r
I
Vee [  (r)]
(100)
I
|
 (r)dr
(101)
is classical electron-electron repulsion:
Vee [  (r)] 
1  (r1 )  (r2 )
dr1dr2
2  | r1  r2 |
(102)
T [  (r)]
is correction to the kinetic energy deriving from the interacting nature of
the electrons
V [  (r)]
contains all non-classical corrections to the electron–electron repulsion
energy (which pretty much means the quantum mechanical exchange and
electron correlation)
What this means is essentially an equation for the energy of non-interacting system of electrons
(first three terms), but with corrections added (last two terms) so that the density and therefore
the energy ends up being the correct one.
14.3 The Kohn-Sham SCF equations
For a non-interacting system of electrons, the kinetic energy is just the sum of the individual
electronic kinetic energies. Unfortunately we do not know how to write the kinetic energy in
terms of the electronic density (recall that the kinetic energy operator has a Laplacian, i.e. the
second derivatives of the wavefunction).
Therefore, we have to take a step back and revive the wavefunction. The wavefunction, as
before, is formed as a Slater determinant from the fictitious spinorbitals i for non-interacting
electrons, which are called Kohn-Sham (KS) orbitals:
 1,2,..., N    1  2  3 ... N
  1 (1)  2 (1)  3 (1)  4 (1)

  (2)  2 (2)  3 (2)  4 (2)
1  1

 (3)  2 (3)  3 (3)  4 (3)
N!  ... 1
...
...
...

  (N )  (N )  (N )  (N )
3
2
4
 1
...  N (1) 

...  N (2) 

...  N (3) 
... ...

...  N ( N ) 
(103)
This determinant would therefore be an exact wavefunction if the electrons were non-interacting.
Using the KS spinorbitals, the electron density is:
 (r)     i  i dri
N
(104)
i 1
where N is the number of electrons. Then:
nuclei
  1 2 

ZI
E[  (r )]      i      i dri     i
 i dri 
| r  rI |
 2 
i 1 
I

N
 1  (r ' )

    i  
dr '  i dri
i 1
 2 | ri  r ' | 
N
(105)
 E XC [  (r )]
The individual terms are, again, the kinetic energy (non-interacting electrons) and electronnuclear attraction (the first line of eqn. 105), followed by the Coulombic energy of electronelectron repulsion (the second line).
The correction terms T [  (r)] and V [  (r)] are all lumped in to the last term E XC [  (r)] (the
third line) typically referred to as the exchange-correlation energy. This is something of a
misnomer, because the term includes not only the effects of quantum mechanical exchange and
correlation, but also the correction for the difference in kinetic energy between the fictitious noninteracting system and the real one. This E XC [  (r)] term is the important one because it defined
the density functional: the first three are always the same.
Now we are back on familiar ground. The next step is to apply the variational principle to find
the orbitals  that minimize the energy, just as we did when deriving HF equations. It is perhaps
not surprising that we end up with a set of equations that look identical to HF:
hiKS  i   i  i
(106)
The are called the Kohn-Sham (KS) equations and just like HF are single-electron equations for
the KS spinorbitals The KS one-electron operator is defined as:
1 2 nuclei Z I
 (r' )
hiKS    i  

dr'  VˆXC
2
| ri  r' |
I | r  rI |
(107)
where:
E
VˆXC  XC
(108)

is the so-called functional derivative, which behaves the same way as regular derivatives. VXC is
perhaps best described as the one-electron operator for which the expectation value of the KS
Slater determinant is EXC.
As for determination of the KS orbitals, the same approach as before is used, where the MOs are
expressed as linear combinations of AOs {} (LCAO), whose coefficients are varied to minimize
the energy. This leads to equations analogous to Roothan-Hall:
M
M
 1
 1
 K  Ci   i  S  Ci
i = 1, 2, … M
(109)
except that the elements of the Fock operator F are replaced by the elements of Kohn-Sham
operators K defined by:
 1 2
ZA


H core
 
   h1        
A r RA
 2
J     P      P




 dr


  r1  r1  r2   r2 
r1  r2
dr1 dr2
(110)
 ˆ XC
VXC
 dr
    V
Note that there are very close similarities to HF Roothan-Hall equations (42):

the kinetic energy and nuclear attraction components of matrix elements of K are
identical to those for F.

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 same four-index
integrals appear in K as are found in F This makes it fairly simple to modify codes
for carrying out HF calculations to also perform DFT computations.

the solution to Kohn–Sham equations is obtained by an iterative SCF procedure,
essentially identical to that for solving HF equations.

There are also important differences 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.
Unfortunately, there is no rigorous way to derive EXC and considerable research effort has gone
and is still going into finding functions of the density that may be expected to reasonably
approximate EXC.
The key differences between HF and DFT can therefore be summarized as follows:


HF is a deliberately approximate theory, whose development was motivated by an ability
to solve the relevant equations exactly
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. Approximate DFT therefore looses the advantage of ab
initio variational theory: that the exact energy is approached from above. In other words DFT can
overestimate the correlation energy and give the total energy which is lower than the true one.
On the other hand, both exact and approximate DFT are size-consistent.