- 3.1.1 The Hohenberg-Kohn theorems
- 3.1.2 The constrained search formulation
- 3.1.3 Exchange and correlation
- 3.1.4 The Kohn-Sham equations
- 3.1.5 The local density approximation

3.1 Density-functional theory

In this section we will describe the remarkable theorems of density-functional theory (DFT) which allow us to find ground-state properties of a system without dealing directly with the many-electron state . We deal with a system of electrons moving in a static potential, and adopt a conventional normalisation in which .

As a result of the Born-Oppenheimer approximation, the Coulomb potential
arising from the nuclei is treated as a static external potential
:

(3.1) |

(3.2) |

(3.3) |

(3.4) |

- The external potential
is uniquely
determined by the corresponding ground-state electronic density, to within an additive constant.
Proof by

*reductio ad absurdum*: assume that a second different external potential with ground-state gives rise to the same density . The ground-state energies are and where and . Taking as a trial wave-function for the Hamiltonian , we obtain the strict inequality

(3.5)

whereas taking as a trial wave-function for gives

(3.6)

and adding these two equations together results in the contradiction

Thus, at least in principle, the ground-state density determines (to within a constant) the external potential of the Schrödinger equation of which it is a solution. The external potential and number of electrons determine all the ground-state properties of the system since the Hamiltonian and ground-state wave-function are determined by them.So for all densities which are ground-state densities for some external potential (-representable) the functional is unique and well-defined, since determines the external potential and (and therefore ) and thence . Now a functional for an arbitrary external potential unrelated to the determined by can be defined:

(3.7) - For all -representable densities ,
where is now the ground-state energy for electrons in the external
potential .
Proof of this energy variational principle: by the first theorem, a given determines its own external potential and ground-state . If this state is used as a trial state for the Hamiltonian with external potential , we have

(3.8)

Following Levy [20,21] we define a functional of the
density for the operator (defined above) as:

(3.10) |

with equality only if . This holds for all densities which can be obtained from an -electron wave-function (-representable). But from the definition of (3.9) we must also have

(3.12) |

(3.13) |

Thus the ground-state density minimises the functional
and the minimum value is the ground-state electronic energy. Note that the
requirement for non-degeneracy of the ground-state has disappeared, and
further that instead of considering only -representable densities, we
can now consider -representable densities. The requirements of
-representability are much weaker and satisfied by any
well-behaved density, indeed the only condition [22] is
proper differentiability i.e. that the quantity

is real and finite.

The remarkable results of density-functional theory are the existence of the universal functional , which is independent of the external potential, and that instead of dealing with a function of variables (the many-electron wave-function) we can instead deal with a function of only three variables (the density). The complexity of the problem has thus been much reduced, and we note here that this complexity now scales linearly with system-size , so that quantum-mechanical calculations based on density-functional theory can in principle be performed with an effort which scales linearly with system-size.

The exact form of the universal functional is unknown. The Thomas-Fermi
functional [23,24,25]

(3.14) |

The failure to find accurate expressions for the density-functional is a result of the complexity of the many-body problem which is at the heart of the definition of the universal functional. For the electron gas, a system of many interacting particles, the effects of exchange and correlation are crucial to an accurate description of its behaviour. In a non-interacting system, the antisymmetry of the wave-function requires that particles with the same spin occupy distinct orthogonal orbitals, and this results in the particles becoming spatially separated. In an interacting system such as the electron gas in which all the particles repel each other, exchange will thus lead to a lowering of the energy. Moreover, the interactions cause the motion of the particles to become correlated to further reduce the energy of interaction. Thus it is impossible to treat the electrons as independent particles. These effects are completely neglected by the Thomas-Fermi model, and must in part account for its failure, the other source of error being the local approximation for the kinetic energy.

In order to take advantage of the power of DFT without sacrificing accuracy
(i.e. including exchange and correlation effects) we follow the method
of Kohn and Sham [33] to map the problem of the system of
interacting electrons onto a fictitious system of non-interacting ``electrons''.
We write the variational problem for the Hohenberg-Kohn density-functional,
introducing a Lagrange multiplier to constrain the number of
electrons to be :

(3.16) |

Using this separation, equation 3.15 can be rewritten:

and the exchange-correlation potential is

(3.19) |

The crucial point to note here is that equation 3.17 is
precisely the same equation
which would be obtained for a non-interacting system of particles moving in
an external potential
. To find the ground-state
density
for this non-interacting system we simply solve the one-electron
Schrödinger equations;

(3.21) |

(3.22) |

Since the Kohn-Sham potential depends upon the density it is necessary to solve these equations self-consistently i.e. having made a guess for the form of the density, the Schrödinger equation is solved to obtain a set of orbitals from which a new density is constructed, and the process repeated until the input and output densities are the same. In practice there is no problem converging to the ground-state minimum because of the convex nature of the density-functional [34].

The energy of the non-interacting system, the sum of one-electron
eigenvalues, is

(3.23) |

which, compared to the interacting system, double-counts the Hartree energy and over-counts the exchange-correlation energy so that the interacting energy is

(3.24) |

Direct solution of the Schrödinger equation for the extended non-interacting orbitals requires a computational effort which scales as the cube of the system-size , due to the cost of diagonalising the Hamiltonian or orthogonalising the orbitals, whereas the original complexity of finding a minimum of the Hohenberg-Kohn functional only required an effort which scaled linearly with . Thus a linear-scaling method must modify this Kohn-Sham scheme.

The results so far are exact, provided that the functional form of
is
known. The problem of determining the functional form of the universal
Hohenberg-Kohn density functional has now been transferred to this one term,
and therefore this term is not known exactly.
Remarkably, it is possible to make simple approximations for the
exchange-correlation energy which work extremely well, and the simplest of
these, which is the approximation adopted in this work, is the * local
density approximation* (LDA).

In the LDA, the contribution to the exchange-correlation energy from each
infinitesimal volume in space,
, is taken to be the value
it would have if the whole of space were filled with a homogeneous electron
gas with the same density as is found in
i.e.

(3.25) |

(3.26) |

The exchange-correlation energy for the homogeneous electron gas has been
calculated by Ceperley and Alder [35] using Monte Carlo
methods
and in this work we use a parameterisation by Perdew and Zunger
[36]. The LDA is exact in the limit of slowly-varying
densities, however, the density in systems of interest is generally rapidly
varying, and the LDA would appear to be a crude approximation in these
cases. Its use is justified * a posteriori* by its surprising success
at predicting physical properties in real systems.
This success may be due in part to the fact that the sum rule for the
exchange-correlation hole, which must be obeyed by the real functional, is
reproduced by the LDA [37].
We can connect the interacting and non-interacting
systems using a variable coupling constant which varies between 0
and 1.
We replace the Coulomb interaction by

and vary in the presence of an external potential so that the ground-state density for all values of is the same [38]. The Hamiltonian is therefore

(3.27) |

(3.28) |

(3.29) |

(3.30) |