Density Functional Theory

In Density Functional Theory (DFT) the electronic density ρ(x) is the key quantity. The "Hohenberg-Kohn theorem" establishes that the density is in a one-to-one correspondence with the external potential, v_ext (r), e.g. the potential determined by the positive ions which at the end is the only quantity that changes in the Hamiltonian when passing from a condensed matter system to another. Thanks to this theorem, all the ground-state observables can be expressed as unique functionals of the density, O[ρ(x)], without needing to resort to the complicated many-body wave function. In particular, the total energy of the system can also be expressed as an unique functional of the density, E[ρ(r)].

The Hohenberg-Kohn theorem further provides a variational principle which states that the exact ground state density of the system is that which minimizes the total energy, E₀ = min_ρ E[ρ(r)] . If we know the total energy functional expression in terms of the density, the solution to the problem can be found by numerically minimizing this functional.

The problem can, however, be reformulated in other terms: in parallel to the real electron-electron interacting system, one can introduce a ficticious non-interacting system, called the Kohn-Sham system, which features an effective external potential such that the electronic density of this system exactly coincides by construction with the electronic density of the real system. The calculation of the electronic density is hence simpler within this system than within the real system. One needs to solve a one-particle Schroedinger equation, with a Hamiltonian containing a kinetic and an effective external potential term. Then the states of the systems are filled with a Fermi-Dirac distribution until all the electrons are accounted for up to the Fermi level. The density is hence calculated by . This constitutes the Kohn-Sham set of equations and the scheme is known as the Kohn-Sham scheme. The only problem is now that we have to provide appropriate forms to the effective Kohn-Sham potential, which contains the real external potential, the Hartree classical repulsion term, and an unknown term, the exchange-correlation potential. This term need to be approximated. The most used approximations are the Local-Density Approximation (LDA), or the Generalized Gradient Approximation (GGA).

DFT is, in principle, an exact theory for predicting ground state observables, such as the ground state energy, the electronic density, the atomic structure (lattice parameters, atomic positions), but also (thanks to perturbation theory), elastic constants, bulk moduli, phonon and vibrational frequencies.

To access excited-state properties, one needs to introduce a complication into the theory, which is the time-dependence, thus passing to Time-Dependent Density-Functional Theory (TDDFT). The "Runge-Gross" extends the Hohenberg-Kohn theorem to time-dependent external potentials and densities, O[ρ(r,t)]. TDDFT can in principle access excited-state properties, in particular the neutral excitations (excitations in which the system does not undergo a change in the charge, the number of electrons being kept constant). These include: optical spectroscopies such as optical absorption, reflectivity, real and imaginary indexes of refraction, etc.; Dielectric spectroscopies, such as Electron Energy-Loss Spectroscopy (EELS), Inelastic X-Ray Scattering Spectroscopy (IXSS), and so on.

TDDFT is versatile and computationally efficient, but the accuracy of the result may be affected by the approximation that we always need to make for the exchange-correlation functional.

MBPT

Green's function theory, also called (improperly) Many-Body Perturbation Theory (MBPT), is a Quantum Field Theory based on a formalism of second quantization of operators. The fundamental degree of freedom is the Green's function or propagator, $G(r_1,t_1,r_2,t_2)$, which represents the probability amplitude for the propagation of an electron from $r_1,t_1$ to $r_2,t_2$. The main advantages of this theory are that:

1. it avoids having indices running over many particles;
2. fermionic antisymmetrization is automatically imposed;
3. systems with varying number of particles can be treated;
4. and most importantly, all the physics of the system is condensed inside the Green's function.

As in any other quantum field theory (for example QED), the many-body system can be expanded in perturbation theory, with the coupling being the many-body interaction term. The Green's function (as well as any other quantity of the theory, such as the self-energy or the polarization) can be calculated at a given order of perturbation theory. A Feynman diagrammatic analysis is hence possible. The theory at the first order is equivalent to Hartree-Fock theory.

However the coupling is not small (compare to, for example, the electron-ion interaction) and the expansion does not converge. The second order is not necessarily smaller than the first. Hence one needs to resort to more complicated methods to solve the theory, such as partial resummations of diagrams at all orders, or better, iterative methods.

In iterative schemes one introduces new quantities into the theory but relating them to the old, in the hope that at the end one can succeed in closing the equations. Indeed, MBPT can be solved thanks to a set of five integro-differential equations, called the Hedin equations, that have to be solved iteratively until self-consistency is achieved.

So far, nobody has solved the Hedin equations for a real system. Approximations are required to simplify the problem. Among the most widely used approximate schemes are the GW approximation for the self-energy and the Bethe-Salpeter Equation approach and its related approximations.