Density Functional Theory - Derivation and Formalism

Derivation and Formalism

As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born–Oppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction satisfying the many-electron time-independent Schrödinger equation

where, for the -electron system, is the Hamiltonian, is the total energy, is the kinetic energy, is the potential energy from the external field due to positively charged nuclei, and is the electron-electron interaction energy. The operators and are called universal operators as they are the same for any -electron system, while is system dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction term .

There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the Hartree–Fock method, more sophisticated approaches are usually categorized as post-Hartree–Fock methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.

Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with, onto a single-body problem without . In DFT the key variable is the particle density which for a normalized is given by

This relation can be reversed, i.e. for a given ground-state density it is possible, in principle, to calculate the corresponding ground-state wavefunction . In other words, is a unique functional of ,

and consequently the ground-state expectation value of an observable is also a functional of

In particular, the ground-state energy is a functional of

where the contribution of the external potential can be written explicitly in terms of the ground-state density

More generally, the contribution of the external potential can be written explicitly in terms of the density ,

The functionals and are called universal functionals, while is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified, one then has to minimize the functional

with respect to, assuming one has got reliable expressions for and . A successful minimization of the energy functional will yield the ground-state density and thus all other ground-state observables.

The variational problems of minimizing the energy functional can be solved by applying the Lagrangian method of undetermined multipliers. First, one considers an energy functional that doesn't explicitly have an electron-electron interaction energy term,

where denotes the kinetic energy operator and is an external effective potential in which the particles are moving, so that .

Thus, one can solve the so-called Kohn–Sham equations of this auxiliary non-interacting system,

which yields the orbitals that reproduce the density of the original many-body system

The effective single-particle potential can be written in more detail as

where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term is called the exchange-correlation potential. Here, includes all the many-particle interactions. Since the Hartree term and depend on, which depends on the, which in turn depend on, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e., iterative) way. Usually one starts with an initial guess for, then calculates the corresponding and solves the Kohn-Sham equations for the . From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation called Harris functional DFT is an alternative approach to this.