ABSTRACT
Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value hHi of the Hamiltonian (atomic units are used)
H ¼ X i
1 2 r2i þ vext(~ri)
þ 1 2
1 j~ri ~rjj (7:1)
of a system of N electrons. In the expression above, vext(~r) is the potential with which the electrons are moving in. For example, in an atom vext(~r)¼(Z=r), in a molecule vext(~r) ¼
P i Zi=~r ~Ri
, where Zi indicates the nuclear charge on the ith atom of the molecule and ~Ri its position, and vext(~r) ¼ 12 kr2 if electrons are moving in a harmonic potential. While in conventional theory the expectation value hHi is a functional E[] of the wave function , in DFT it is [3] a functional E[r] of the ground-state density r. The density is given in terms of the wave function as
r(~r) ¼ N ð C(~r,~r2,~r3 . . .~rN)j j2d~r2d~r3 . . . d~rN (7:2)
The equation satisfied by the wave function , the Schrödinger equation, is obtained by minimizing the functional E[] with respect to , with the energy of the system appearing as a Lagrange multiplier to ensure the normalization of the wave function. Similarly in DFT, the equation for the density is obtained by minimizing the functional E[r] with respect to the density r and leads to the Euler equation
dE[r]
dr(~r) ¼ m (7:3)
where m appears as a Lagrange multiplier to ensure that the density integrates to the correct number of electrons N. The physical interpretation of m as the chemical potential and its derivatives has been discussed in other chapters of this book. For the purposes of this chapter, we note that the chemical potential of an electronic system equals the negative of its ionization energy when an electron is removed from it and negative of its electron affinity when an electron is added to it [4].
