In this lecture we present many-body perturbation theory as a method to determine quasiparticle excitations in solids, especially electronic band structures, accurately from first principles.The main ingredient is the electronic self-energy that, in principle, contains all many-body exchange and correlation effects beyond the Hartree potential.As its exact mathematical expression is unknown, approximations must be used in practical calculations.The approximation is obtained using a systematic algebraic approach on the basis of Green function techniques.It constitutes an expansion of the self-energy up to linear order in the screened Coulomb potential, which describes the interaction between the quasiparticles and includes dynamic screening through the creation of exchange-correlation holes around the bare particles.The implementation of the approximation relies on a perturbative treatment starting from density functional theory.Besides a detailed mathematical discussion we focus on the underlying physical concepts and show some illustrative applications.
A perturbation theory is developed for the correlation energy
Ec[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+
tsumi${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(
ri). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]=
ec,2[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$
ec,3[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$
ec,4[n]+..., where each
ec,j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy
ec,2[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}
Ec[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as
Ec[n] in finite insulators and semiconductors.Approximations to
Ec[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
Background There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development , carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development , where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory. Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways. Classical derivation of Rayleigh–Schrödinger perturbation theory The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).