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1 Introduction

Although the Hamiltonian governing the electron dynamics in a molecule or solid is readily written down, a direct numerical solution of the corresponding Schrödinger equation is not feasible except for very small systems. As a consequence, various approximation schemes have been developed for practical ab initio calculations. In quantum chemistry, post Hartree–Fock methods, such as configuration interaction or coupled cluster (see the Chap. Tensor Product Approximation (DMRG) and Coupled Cluster Method in Quantum Chemistry), have traditionally played a leading role. However, the computational cost rises so rapidly with the number of electrons \(N\) that solids and even large molecules cannot be treated in this way. In order to overcome this severe limitation, modern theoretical approaches abandon the many-electron wave function \(\varPsi (\mathbf {r}_1,\ldots ,\mathbf {r}_N)\) as the central quantity and replace it by other mathematical objects whose complexity is less dependent on the system size. The most famous example in this category is density functional theory [17, 20], which is based on the electron density \(n(\mathbf {r})\), a real-valued function of just three spatial coordinates (see the Chap. Levy-Lieb Principle Meets Quantum Monte Carlo). Due to its relatively simple implementation and its high computational efficiency (see the Chap. Computational Techniques for Density Functional Based Molecular Dynamics Calculations in Plane-Wave and Localized Basis Sets), density functional theory quickly became the dominant ab initio method in solid-state physics and is now also widely used in quantum chemistry.

As the Hohenberg–Kohn theorem [17] that underlies density functional theory establishes a one-to-one correspondence between the ground-state electron density and the external potential, and hence the entire Hamiltonian, all observable system properties can be deduced, in principle, just from the knowledge of \(n(\mathbf {r})\). However, the theorem does not yield explicit formulas for the relevant functionals. As accurate quantitative approximations are only available for the total energy in the electronic ground state, density functional theory itself is widely regarded as a ground-state method, although this restriction is in fact not fundamental. In actual calculations, the eigenvalues of the one-particle Kohn–Sham equations are routinely taken as excitation energies, but this interpretation has no formal justification [29] and leads to systematic deviations from experiments (see the Chap. Application of (Kohn–Sham) Density Functional Theory to Real Materials). For example, the Kohn–Sham eigenvalue gap underestimates the fundamental band gap of semiconductors as measured in photoemission experiments by as much as 50 %, an observation that applies not only to the popular local-density approximation or to generalized gradient approximations, but also to more elaborate orbital-dependent functionals like the random-phase approximation [12, 21]. Therefore, alternative approaches outside density functional theory that offer a more reliable quantitative description of electronic excitation spectra fill an important niche.

Many-body perturbation theory, which was developed by theoretical physicists in the 1950s and 1960s, is precisely such a method [24]. It is based on Green functions and, like density functional theory, constitutes a formally exact reformulation of the quantum-mechanical many-particle problem without wave functions. While Green functions are more difficult to compute than the much simpler electron density, the distinct advantage of many-body perturbation theory lies in the fact that the explicit functional forms of the total energy as well as other ground-state and excited-state properties are known exactly. Therefore, no systematic errors are incurred if excitation spectra are obtained in this way, although approximations for the treatment of exchange and correlation are still necessary in actual implementations. In solid-state physics, virtually all practical calculations rely on the so-called \(GW\) approximation [13], which has become so ubiquitous in this field that it is frequently even used as a synonym for many-body perturbation theory as a whole. Despite an ongoing debate about the merits of different possible flavors, which vary principally in the degree of self-consistency, the \(GW\) approximation has been immensely successful, yielding band structures and related spectroscopic data in very good quantitative agreement with experimental measurements for a wide range of materials [3]. However, there are also notable failures, in particular for strongly correlated systems.

In the following I briefly sketch the foundations of many-body perturbation theory, focusing on Green functions and their relation to physical observables. Then I discuss the \(GW\) approximation, including its practical implementation as well as the performance in actual ab initio calculations, before offering an outlook. Hartree atomic units are used throughout unless indicated otherwise.

2 Green Functions

Many-body perturbation theory is specifically designed to describe such excitation processes as direct or inverse photoelectron spectroscopy, where the particle number in the sample changes due to the emission or the injection of electrons under external irradiation. Consequently, its formalism is cast in the language of second quantization, which is particularly suited for this purpose. In this notation, the Hamiltonian of an interacting many-electron system is written as

$$\begin{aligned} \hat{H} = \int \hat{\psi }^\dagger (\mathbf {x}) h_0(\mathbf {x}) \hat{\psi }(\mathbf {x}) \,\mathrm{{d}}\mathbf {x} + \frac{1}{2} \int \hat{\psi }^\dagger (\mathbf {x}) \hat{\psi }^\dagger (\mathbf {x}^{\prime }) v(\mathbf {r}-\mathbf {r}^{\prime }) \hat{\psi }(\mathbf {x}^{\prime }) \hat{\psi }(\mathbf {x}) \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\mathbf {x}^{\prime }. \end{aligned}$$
(1)

Here \(\mathbf {x} = \{\mathbf {r},\sigma \}\) is the shorthand notation for a combination of the spatial and spin coordinates, the integrals are understood as spatial integrations and a sum over the two possible spin orientations. The one-body Hamiltonian \(h_0(\mathbf {x}) = -\nabla ^2 / 2 + V_\mathrm {ext}(\mathbf {x})\) comprises the kinetic energy as well as the external potential, and \(v(\mathbf {r}) = 1 / r\) denotes the two-body Coulomb interaction, which does not depend on the spin orientations. Furthermore, \(\hat{\psi }^\dagger (\mathbf {x})\) and \(\hat{\psi }(\mathbf {x})\) denote the creation and annihilation field operators for an electron at the position \(\mathbf {x}\), respectively. The eigenstates and energy eigenvalues of the Hamiltonian are given by

$$\begin{aligned} \hat{H} | \varPsi ^N_\nu \rangle = E^N_\nu | \varPsi ^N_\nu \rangle , \end{aligned}$$
(2)

where \(N\) indicates the number of electrons. The corresponding Green function in the frequency domain is then defined as [24]

$$\begin{aligned} G(\mathbf {x},\mathbf {x}^{\prime };\omega )&= \frac{\mathrm {i}}{2 \pi } \int \limits _{-\infty }^0 \langle \varPsi ^N_0 | \hat{\psi }^\dagger (\mathbf {x}^{\prime },0) \hat{\psi }(\mathbf {x},\tau ) | \varPsi ^N_0 \rangle \mathrm{{e}}^{\mathrm {i}( \omega - \mathrm {i}\eta ) \tau } \,\mathrm{{d}}\tau \\&- \frac{\mathrm {i}}{2 \pi } \int \limits _0^\infty \langle \varPsi ^N_0 | \hat{\psi }(\mathbf {x},\tau ) \hat{\psi }^\dagger (\mathbf {x}^{\prime },0) | \varPsi ^N_0 \rangle \mathrm{{e}}^{\mathrm {i}( \omega + \mathrm {i}\eta ) \tau } \,\mathrm{{d}}\tau \nonumber \end{aligned}$$
(3)

in terms of the stationary ground-state wave function \(| \varPsi ^N_0 \rangle \) and the time-dependent field operators in the Heisenberg picture

$$\begin{aligned} \hat{\psi }(\mathbf {x},\tau ) = \mathrm{{e}}^{\mathrm {i}\hat{H} \tau } \hat{\psi }(\mathbf {x}) \mathrm{{e}}^{-\mathrm {i}\hat{H} \tau }. \end{aligned}$$
(4)

The matrix element in the first term of (3) describes a process, analogous to direct photoemission in experimental spectroscopy, in which one electron is removed from the sample, creating an intermediate state with \(N-1\) electrons that is propagated for a time \(| \tau |\) and eventually projected back onto the initial state. The intermediate state is, in general, not an eigenstate of the Hamiltonian, but it can be expanded as a linear combination of eigenstates, each of which rotates with its own characteristic phase that equals the corresponding eigenvalue. The Fourier transformation hence yields the eigenvalue spectrum of the ionized system. Likewise, the second term describes the addition of an electron in analogy to inverse photoemission. An infinitesimal \(\eta > 0\) is added to the exponents to ensure the convergence of the integrals.

The significance of the Green function and its relation to the excitation spectrum become even clearer if a complete set of eigenstates, which satisfy

$$\begin{aligned} 1 = \sum _\nu | \varPsi ^{N \pm 1}_\nu \rangle \langle \varPsi ^{N \pm 1}_\nu |, \end{aligned}$$
(5)

is inserted between the creation and annihilation operators in (3). In this case the integrations can be performed analytically and lead to the Lehmann representation

$$\begin{aligned} G(\mathbf {x},\mathbf {x}^{\prime };\omega ) = \sum _\nu \frac{\psi ^{N-1}_\nu (\mathbf {x}) {\psi ^{N-1}_\nu }^*(\mathbf {x}^{\prime })}{\omega - \varepsilon ^{N-1}_\nu - \mathrm {i}\eta } + \sum _\nu \frac{\psi ^{N+1}_\nu (\mathbf {x}) {\psi ^{N+1}_\nu }^*(\mathbf {x}^{\prime })}{\omega - \varepsilon ^{N+1}_\nu + \mathrm {i}\eta } . \end{aligned}$$
(6)

The amplitudes

$$\begin{aligned} \psi ^{N-1}_\nu (\mathbf {x}) = \langle \varPsi ^{N-1}_\nu | \hat{\psi }(\mathbf {x}) | \varPsi ^N_0 \rangle \quad \text{ and }\quad \psi ^{N+1}_\nu (\mathbf {x}) = \langle \varPsi ^N_0 | \hat{\psi }(\mathbf {x}) | \varPsi ^{N+1}_\nu \rangle \end{aligned}$$
(7)

are generalized overlap functions between wave functions with \(N\) and with \(N \pm 1\) electrons. They are neither normalized nor orthogonal but fulfill the completeness relation

$$\begin{aligned} \sum _\nu \psi ^{N-1}_\nu (\mathbf {x}) {\psi ^{N-1}_\nu }^*(\mathbf {x}^{\prime }) + \sum _\nu \psi ^{N+1}_\nu (\mathbf {x}) {\psi ^{N+1}_\nu }^*(\mathbf {x}^{\prime }) = \delta (\mathbf {x} - \mathbf {x}^{\prime }) . \end{aligned}$$
(8)

Furthermore, the values

$$\begin{aligned} \varepsilon ^{N-1}_\nu = E^N_0 - E^{N-1}_\nu \quad \text{ and }\quad \varepsilon ^{N+1}_\nu = E^{N+1}_\nu - E^N_0 \end{aligned}$$
(9)

correspond to the total-energy differences between the ground state of the original \(N\)-electron system and all eigenstates with \(N \pm 1\) electrons. In atoms or molecules, \(I = -\varepsilon ^{N-1}_0\) equals the first ionization energy, while \(A = -\varepsilon ^{N+1}_0\) equals the electron affinity. In solids, \(\varepsilon ^{N-1}_\nu \) and \(\varepsilon ^{N+1}_\nu \) correspond to the energies of the valence and the conduction bands, respectively, and the exact fundamental band gap of an insulator as measured in photoemission spectroscopy is given by \(\varepsilon _\mathrm {gap} = \varepsilon ^{N+1}_0 - \varepsilon ^{N-1}_0\).

According to (6), the excitation energies can be obtained from the positions of the poles of the Green function, but for practical purposes it is much more convenient to examine the spectral function

$$\begin{aligned} A(\mathbf {x},\mathbf {x}^{\prime };\omega ) = \frac{\mathop {\mathrm {sgn}}(\mu - \omega )}{\pi } \mathop {\mathrm {Im}} G(\mathbf {x},\mathbf {x}^{\prime };\omega ), \end{aligned}$$
(10)

where the sign function ensures that the diagonal elements \(A(\mathbf {x},\mathbf {x};\omega )\) are positive. The chemical potential \(\mu \), which separates the occupied valence states \(\varepsilon ^{N-1}_\nu \) from the higher-lying unoccupied states \(\varepsilon ^{N+1}_\nu \), is determined by the sum rule

$$\begin{aligned} \int \limits _{-\infty }^\mu \int A(\mathbf {x},\mathbf {x};\omega ) \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\omega = N . \end{aligned}$$
(11)

For atoms and molecules, this condition is consistent with the Mulliken definition \(\mu = -(A + I)/2\). In solids, \(\mu \) corresponds to the Fermi energy. From the Lehmann representation one obtains the expression

$$\begin{aligned} A(\mathbf {x},\mathbf {x}^{\prime };\omega )&= \sum _\nu \psi ^{N-1}_\nu (\mathbf {x}) {\psi ^{N-1}_\nu }^*(\mathbf {x}^{\prime }) \delta (\omega - \varepsilon ^{N-1}_\nu )\\&+ \sum _\nu \psi ^{N+1}_\nu (\mathbf {x}) {\psi ^{N+1}_\nu }^*(\mathbf {x}^{\prime }) \delta (\omega - \varepsilon ^{N+1}_\nu ) ,\nonumber \end{aligned}$$
(12)

which shows that the excitation energies can be identified in a simple manner from the positions of the peaks in the frequency-dependent spectral function. In fact, the projection onto plane waves

$$\begin{aligned} A(\mathbf {k};\omega ) = \frac{1}{V} \int \mathrm{{e}}^{-\mathrm {i}\mathbf {k} \cdot \mathbf {r}} A(\mathbf {x},\mathbf {x}^{\prime };\omega ) \mathrm{{e}}^{\mathrm {i}\mathbf {k} \cdot \mathbf {r}^{\prime }} \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\mathbf {x}^{\prime }, \end{aligned}$$
(13)

where \(V\) denotes the volume of the system, can be directly related to experimentally measured angle-resolved photoemission spectra, where \(\mathbf {k}\) corresponds to the wave vector of the emitted or injected electron, and may be analyzed in the same way.

In addition to the excited states of ionized configurations, the Green function also yields information about ground-state properties of the original non-ionized system. The expectation value of an arbitrary one-body quantum-mechanical operator \(\hat{O}(\mathbf {x})\) can be computed according to

$$\begin{aligned} \langle \varPsi ^N_0 | \hat{O}(\mathbf {x}) | \varPsi ^N_0 \rangle = \int \limits _{-\infty }^\mu \int \lim _{\mathbf {x}^{\prime } \rightarrow \mathbf {x}} \left[ O(\mathbf {x}) A(\mathbf {x},\mathbf {x}^{\prime };\omega ) \right] \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\omega . \end{aligned}$$
(14)

As an example, the spin-resolved ground-state electron density is given by

$$\begin{aligned} n(\mathbf {x}) = \int \limits _{-\infty }^\mu A(\mathbf {x},\mathbf {x};\omega ) \,\mathrm{{d}}\omega , \end{aligned}$$
(15)

which implies \(n(\mathbf {r}) = 2 n(\mathbf {x})\) for non-magnetic systems. This result is also important in a wider context of ab initio methods, because it provides an independent way of calculating the electron density without the typical approximations used in practical implementations of density functional theory. Indeed, early applications of many-body perturbation theory to semiconductors exploited this approach to numerically derive an accurate exchange-correlation potential from the density and compare the resulting eigenvalue spectrum with that of the local-density approximation [11, 12], finding little difference in the size of the eigenvalue gap. This result is generally interpreted as evidence that the underestimation of the fundamental band gap is in fact inherent in the Kohn–Sham scheme and does not stem from the widely used local or semi-local approximations for the exchange-correlation functional.

In general, explicit expressions for the expectation values of two-body operators require a higher-order Green function, which may be constructed in analogy to (3) but with an additional pair of creation and annihilation operators. The ground-state total energy, however, is given by the Galitskii-Migdal formula [10]

$$\begin{aligned} E^N_0 = \frac{1}{2} \int \limits _{-\infty }^\mu \int \lim _{\mathbf {x}^{\prime } \rightarrow \mathbf {x}} \left[ \left( \omega + h_0{(\mathbf {x})} \right) A(\mathbf {x},\mathbf {x}^{\prime };\omega ) \right] \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\omega . \end{aligned}$$
(16)

Although results have been reported for atoms and small molecules [31] as well as the homogeneous electron gas [16] and other model systems, ab initio total-energy calculations within many-body perturbation theory are rare in practice because of the high numerical effort that is required to compute the Green function, including its low-frequency tail, with the necessary accuracy.

3 The Self-Energy

As outlined in the previous section, the extraction of both ground-state and excited-state properties from the Green function is, in principle, straightforward: While the excitation energies correspond to the peaks of the frequency-dependent spectral function and can be obtained immediately from a visual inspection, ground-state expectation values are given by relatively simple integral expressions, such as (16) for the total energy or (14) for other observables. The respective functionals, i.e., the mathematical relations between these quantities and the Green function, are hence known exactly, and, in contrast to density functional theory, no approximations are required at this stage. On the other hand, it is not obvious how to construct the Green function in the first place, because the original definition (3) involves the unknown many-body wave function \(| \varPsi ^N_0 \rangle \) and thus cannot be used for practical purposes. However, perturbation theory provides an expansion of this wave function in terms of the wave functions of the corresponding non-interacting system, i.e., the Slater determinants made from the eigenstates \(\varphi _j(\mathbf {x})\) of the one-body Hamiltonian \(h_0(\mathbf {x})\), and the Coulomb potential \(v(\mathbf {r})\). The resulting expansion of \(G\) can be written solely in terms of the Green function of the non-interacting system

$$\begin{aligned} G_0(\mathbf {x},\mathbf {x}^{\prime };\omega ) = \sum _{j = 1}^N \frac{\varphi _j(\mathbf {x}) \varphi _j^*(\mathbf {x}^{\prime })}{\omega - \varepsilon _j - \mathrm {i}\eta } + \sum _{j = N+1}^\infty \frac{\varphi _j(\mathbf {x}) \varphi _j^*(\mathbf {x}^{\prime })}{\omega - \varepsilon _j + \mathrm {i}\eta }, \end{aligned}$$
(17)

where \(\varepsilon _j\) denotes the eigenvalues corresponding to \(\varphi _j(\mathbf {x})\) in ascending order, and the Coulomb potential. Incidentally, it is this important property that gave many-body perturbation theory its name.

Formal expressions for individual terms of this expansion series can be derived, most conveniently with the help of Feynman diagrams, but the number of terms as well as their complexity grow rapidly with the order of the perturbation. In practice, it is hence useful to employ Dyson’s equation [6]

$$\begin{aligned} G(\mathbf {x},\mathbf {x}^{\prime };\omega ) = G_0(\mathbf {x},\mathbf {x}^{\prime };\omega ) + \int G_0(\mathbf {x},\mathbf {x}^{\prime \prime };\omega ) M(\mathbf {x}^{\prime \prime },\mathbf {x}^{\prime \prime \prime };\omega ) G(\mathbf {x}^{\prime \prime \prime },\mathbf {x}^{\prime };\omega ) \,\mathrm{{d}}\mathbf {x}^{\prime \prime } \,\mathrm{{d}}\mathbf {x}^{\prime \prime \prime }, \end{aligned}$$
(18)

which is formally equivalent to a geometric series. As a consequence, only the set of “irreducible” diagrams, i.e., Feynman diagrams that do not fall apart into two disconnected parts if a single instance of \(G_0\) is removed, must be incorporated into the operator \(M\), resulting in a significant simplification. An even smaller subset of so-called “skeleton” diagrams suffices if the expansion is rewritten in terms of the full Green function \(G\) rather than \(G_0\), but at the expense that \(M\) must then be calculated self-consistently together with Dyson’s equation instead of a once-only evaluation with a known function. By convention, \(M\) is further decomposed according to

$$\begin{aligned} M(\mathbf {x},\mathbf {x}^{\prime };\omega ) = V_\mathrm {H}(\mathbf {r}) \delta (\mathbf {x} - \mathbf {x}^{\prime }) + \varSigma (\mathbf {x},\mathbf {x}^{\prime };\omega ) \end{aligned}$$
(19)

into the local, frequency-independent Hartree potential \(V_\mathrm {H}(\mathbf {r}) = \int v(\mathbf {r} - \mathbf {r}^{\prime }) n(\mathbf {x}^{\prime }) \,\mathrm{{d}}\mathbf {x}^{\prime }\) and a non-local, frequency-dependent part \(\varSigma \). The latter incorporates all quantum-mechanical exchange and correlation effects that are not contained in the classical electrostatic Hartree potential. Historically, \(M\) was known either as the self-energy or as the mass operator, owing to the fact that its inclusion changes the dispersion of the energy bands and hence the associated effective masses in solids. However, the term self-energy is nowadays usually applied to the exchange-correlation part \(\varSigma \). The name relates to its physical meaning in the description of photoemission spectra: If an electron is injected into a sample, then its energy is not only governed by the existing effective potential, but the appearance of the electron itself changes the local charge arrangement due to the Coulomb interaction with all the other electrons and the creation of an exchange-correlation hole. The self-energy describes precisely this self-induced potential and its effect on the electronic energy level.

Another useful consequence of introducing the self-energy becomes apparent if one exploits the relation \([ \omega - h_0(\mathbf {x}) ] G_0(\mathbf {x},\mathbf {x}^{\prime };\omega ) = \delta (\mathbf {x} - \mathbf {x}^{\prime })\) and rewrites Dyson’s equation as

$$\begin{aligned} \quad [ \omega - h_0(\mathbf {x}) - V_\mathrm {H}(\mathbf {r})] G(\mathbf {x},\mathbf {x}^{\prime };\omega ) - \int \varSigma (\mathbf {x},\mathbf {x}^{\prime \prime };\omega ) G(\mathbf {x}^{\prime \prime },\mathbf {x}^{\prime };\omega ) \,\mathrm{{d}}\mathbf {x}^{\prime \prime } = \delta (\mathbf {x} - \mathbf {x}^{\prime }).\quad \end{aligned}$$
(20)

If the Lehmann representation (6) is inserted and the limit \(\omega \rightarrow \varepsilon ^{N \pm 1}_\nu \) is taken, then the divergent terms on the left-hand side must cancel in order to match the right-hand side. This requirement leads to the so-called quasiparticle equation

$$\begin{aligned}{}[ h_0(\mathbf {x}) + V_\mathrm {H}(\mathbf {r})] \psi ^{N \pm 1}_\nu (\mathbf {x}) + \int \varSigma (\mathbf {x},\mathbf {x}^{\prime };\varepsilon ^{N \pm 1}_\nu ) \psi ^{N \pm 1}_\nu (\mathbf {x}^{\prime }) \,\mathrm{{d}}\mathbf {x}^{\prime } = \varepsilon ^{N \pm 1}_\nu \psi ^{N \pm 1}_\nu (\mathbf {x}) . \end{aligned}$$
(21)

A quasiparticle designates the combination of an electron or hole and its surrounding polarization cloud. Quasiparticles represent a very fruitful concept and constitute the elementary excitations created in photoemission experiments. They possess a well-defined energy and momentum, which are probed in band-structure measurements, but, unlike electrons, have a finite spatial extent. Surprisingly, (21) demonstrates that the excitation energies \(\varepsilon ^{N \pm 1}_\nu \) can in fact be obtained directly from a formally exact single-particle equation with the self-energy as a non-local, frequency-dependent potential. Unlike the Schrödinger equation, however, this is a non-linear eigenvalue equation, as the energy eigenvalue also appears in the argument of the self-energy.

The suggestive form of the quasiparticle equation also makes it possible to place other schemes within the context of many-body perturbation theory. In particular, the Hartree–Fock approximation corresponds to the exchange-only self-energy

$$\begin{aligned} \varSigma _\mathrm {x}(\mathbf {x},\mathbf {x}^{\prime }) = -\sum _{j=1}^{N} \frac{\varphi ^\mathrm {HF}_j(\mathbf {x}) {\varphi ^\mathrm {HF}_j}^*(\mathbf {x}^{\prime })}{|\mathbf {r} - \mathbf {r}^{\prime }|} = \frac{\mathrm {i}}{2 \pi } \int G_\mathrm {HF}(\mathbf {x},\mathbf {x}^{\prime };\omega \,+\, \omega ^{\prime }) v(\mathbf {r}-\mathbf {r}^{\prime }) \mathrm{{e}}^{\mathrm {i}\delta \omega ^{\prime }} \,\mathrm{{d}}\omega ^{\prime }, \end{aligned}$$
(22)

where the Hartree–Fock Green function \(G_\mathrm {HF}\) is defined just as in (17) but with the Hartree–Fock orbitals \(\varphi ^\mathrm {HF}_j(\mathbf {x})\) and eigenvalues instead of those of the non-interacting system, and \(\delta \) is a positive infinitesimal that serves to filter out the occupied states by forcing the closure of the frequency contour across the upper complex half-plane. The Hartree–Fock approximation reflects the non-locality of the full self-energy but is frequency-independent due to the neglect of correlation. Formally, it corresponds to the “skeleton” expansion of \(\varSigma \) to first order in the Coulomb potential, evaluated self-consistently in conjunction with Dyson’s equation.

4 The GW Approximation

The formal expression of the self-energy as an infinite series of Feynman diagrams naturally suggests improving upon the Hartree–Fock approximation by continuing the expansion to higher orders in the Coulomb potential. Such an approach is viable for finite systems and has indeed been demonstrated for atoms and small molecules [4], where it is essentially equivalent to Møller-Plesset perturbation theory [26]. For extended solids, however, the expansion cannot simply be terminated at some finite order, because divergent terms that are only canceled by higher contributions appear in every order beyond the first due to the long-range nature of the Coulomb potential. Quinn and Ferrell [30] calculated the first non-divergent correction beyond Hartree–Fock for the homogeneous electron gas in the high-density limit, where the kinetic energy dominates and the expansion in terms of the Coulomb potential is equivalent to an expansion by powers of the Wigner-Seitz radius \(r_\mathrm {S} = (4 \pi n / 3)^{-1/3}\). Their expression, which involves the sum over an infinite number of Feynman diagrams as illustrated in Fig. 1, already anticipated what would later become known as the \(GW\) approximation, but its usefulness beyond the high-density limit of the homogeneous electron gas was not yet recognized. This credit belongs to Hedin [13], who in 1965 derived a set of coupled integral equations for the so-called vertex function \(\Gamma \), the polarizability \(P\), the dynamically screened Coulomb interaction \(W\) and the self-energy \(\varSigma \). Together with Dyson’s equation (18) these form a closed system, whose solution, in principle, yields the exact Green function. Unfortunately, however, the equation for the vertex function contains a functional derivative of the self-energy with respect to the Green function, which cannot be evaluated by standard numerical techniques, so that a straightforward automated iterative solution is not possible.

Fig. 1
figure 1

Diagrammatic form of the self-energy \(\varSigma \) in the \(GW\) approximation. Arrows, wavy lines and dashed lines represent the Green function \(G_\mathrm {KS}\), the dynamically screened Coulomb interaction \(W\) and the bare Coulomb potential \(v\), respectively. According to the rules of Feynman diagrams, the first identity is a concise representation of Eq. (23), while the second translates into Eqs. (25) and (26). A physical interpretation of these diagrams is that the \(GW\) approximation describes dynamic correlation in an electron gas by the emission and subsequent reabsorption of virtual plasmons

Full size image

To circumvent this problem, Hedin proposed to choose a suitable explicit starting point and then evaluate the equations iteratively in an algebraic way. The functional derivative can be carried out analytically in this case, but the resulting terms modify the formal expressions for the self-energy and the other quantities in each iteration. After one complete cycle starting from the Hartree approximation, he arrived at

$$\begin{aligned} \varSigma (\mathbf {x},\mathbf {x}^{\prime };\omega ) = \frac{\mathrm {i}}{2 \pi } \int G_\mathrm {KS}(\mathbf {x},\mathbf {x}^{\prime };\omega +\omega ^{\prime }) W(\mathbf {r},\mathbf {r}^{\prime };\omega ^{\prime }) \mathrm{{e}}^{\mathrm {i}\delta \omega ^{\prime }} \,\mathrm{{d}}\omega ^{\prime } , \end{aligned}$$
(23)

for which he coined the name \(GW\) approximation that eventually stuck. Its formal similarity to the Hartree–Fock approximation (22) is evident, but the bare Coulomb potential \(v\) is now screened. According to the derivation, the \(GW\) self-energy should be evaluated with the Hartree Green function, but as all practical implementations employ the more accurate and readily available Kohn–Sham Green function \(G_\mathrm {KS}\), the latter is inserted here. It is defined in analogy to (17), but with the eigenfunctions and eigenvalues of the Kohn–Sham equation

$$\begin{aligned}{}[ h_0(\mathbf {x}) + V_\mathrm {H}(\mathbf {r}) + V_\mathrm {xc}(\mathbf {x})] \varphi ^\mathrm {KS}_j(\mathbf {x}) = \varepsilon ^\mathrm {KS}_j \varphi ^\mathrm {KS}_j(\mathbf {x}) , \end{aligned}$$
(24)

which involves the Hartree potential \(V_\mathrm {H}(\mathbf {r})\) and the local (multiplicative) exchange-correlation potential. Within density functional theory the latter is defined as the functional derivative \(V_\mathrm {xc}(\mathbf {x}) = \delta E_\mathrm {xc}[n] / \delta n(\mathbf {x})\) of the exchange-correlation energy with respect to the density. The dynamically screened interaction

$$\begin{aligned} W(\mathbf {r},\mathbf {r}^{\prime };\omega ) = v(\mathbf {r}-\mathbf {r}^{\prime }) + \int v(\mathbf {r}-\mathbf {r}^{\prime \prime }) P(\mathbf {x}^{\prime \prime },\mathbf {x}^{\prime \prime \prime };\omega ) W(\mathbf {r}^{\prime \prime \prime },\mathbf {r}^{\prime };\omega ) \,\mathrm{{d}}\mathbf {x}^{\prime \prime } \,\mathrm{{d}}\mathbf {x}^{\prime \prime \prime } \end{aligned}$$
(25)

is evaluated with the polarizability in the random-phase approximation

$$\begin{aligned} P(\mathbf {x},\mathbf {x}^{\prime };\omega ) = -\frac{\mathrm {i}}{2 \pi } \int G_\mathrm {KS}(\mathbf {x},\mathbf {x}^{\prime };\omega +\omega ^{\prime }) G_\mathrm {KS}(\mathbf {x}^{\prime },\mathbf {x};\omega ^{\prime }) \,\mathrm{{d}}\omega ^{\prime }. \end{aligned}$$
(26)

The objection that the insertion of \(G_\mathrm {KS}\) is inconsistent with the derivation of the \(GW\) expression for the self-energy can be removed if density functional theory rather than the Hartree approximation is chosen as the starting point for the iteration. As it turns out, the self-energy can still be written in the same form as (23) in this case, albeit with a slightly modified screened interaction [19] that includes the so-called exchange-correlation kernel defined within the context of time-dependent density functional theory (see the Chap. Time-dependent Density Functional Theory). Numerical results suggest that these modifications only have a very small influence on the quasiparticle band structure [5, 25], however, and it is hence considered safe to ignore them.

The good performance of the \(GW\) approximation (23) for quasiparticle properties as discussed below seemingly implies that the self-energy is already converged after one iteration of Hedin’s equations, but this is in fact not true, because other features are still poorly described at this stage. In particular, the quasiparticles constitute only a part of the electronic excitation spectrum. By definition, the term quasiparticle refers to an excitation that can be described in terms of transitions between single-particle states. Their energies can be measured in photoemission experiments if the entire energy of an incident photon is transfered to the photoelectron. Occasionally, however, additional secondary excitations, such as one or more plasmons, which reduce the energy of the emitted electron, are created in this process. The spectral function hence contains a variety of satellite resonances that are offset from the main quasiparticle peaks and do not feature in the band structure. In metals, for example, satellites are observed at intervals equal to integer multiples of the plasmon frequency. As seen most clearly from the interpretation of the Feynman diagrams in Fig. 1, the \(GW\) approximation only describes the coupling of electrons or holes to a single plasmon and is thus unable to predict these multiple satellites; instead, this requires the more elaborate and hence rarely invoked cumulant expansion [2]. As another example, nickel features a prominent satellite resonance at 6 eV below the valence band, which originates from an Auger excitation within the 3\(d\) shell. Again, this feature is absent in the \(GW\) approximation but can be reproduced if appropriate additional diagrammatic terms are added to the self-energy [35].

The deviations between the \(GW\) approximation and the exact self-energy stem from two sources: the lack of self-consistency, i.e., use of \(G_\mathrm {KS}\) rather than \(G\), and the neglect of vertex corrections in the form of a non-trivial vertex function \(\Gamma \) [13]. These two effects are interrelated and partially cancel each other. In particular, a self-consistent evaluation of the self-energy without simultaneous vertex corrections leads to a deterioration of the spectrum: The valence band width of metals becomes larger, not smaller, than the dispersion of free electrons [16, 33], while the band gaps of semiconductors overshoot the experimental values by a similar amount as the local-density approximation underestimates them [33]. In addition, the satellites in the spectral function are not improved [16]. As a consequence, the self-consistent \(GW\) approximation, which is also computationally much more demanding, is not used in actual ab initio simulations, although, at a fundamental level, it has some favorable features. For example, it exactly fulfills the sum rule (11), which is violated by the standard non-self-consistent variant, albeit only by a small margin [32].

Even without self-consistency, the numerical cost of \(GW\) calculations is much higher than for density functional theory due to the frequency-dependence and non-locality of the Green function and related quantities. Additional simplifications are hence routinely employed. In particular, the formal similarity between the quasiparticle equation (21) and the Kohn–Sham equation (24) can be exploited to calculate the quasiparticle energies in a perturbative manner according to

$$\begin{aligned} \varepsilon _j = \varepsilon ^\mathrm {KS}_j + \int {\varphi ^\mathrm {KS}_j}^*(\mathbf {x}) \left[ \varSigma (\mathbf {x},\mathbf {x}^{\prime };\varepsilon _j) - V_\mathrm {xc}(\mathbf {x}) \delta (\mathbf {x} - \mathbf {x}^{\prime }) \right] \varphi ^\mathrm {KS}_j(\mathbf {x}^{\prime }) \,\mathrm{{d}}\mathbf {x} \,\mathrm{{d}}\mathbf {x}^{\prime }. \end{aligned}$$
(27)

This approach is further justified by the large overlap between the quasiparticle and Kohn–Sham orbitals [19] and has the advantage that only diagonal matrix elements of the self-energy in the Kohn–Sham basis are required. From a theoretical point of view it is interesting to note that this expression can be interpreted as an implicit density functional for the excitation energies, because the right-hand side involves only the Kohn–Sham orbitals and eigenvalues, which in turn depend on the density.

5 Applications

After the \(GW\) approximation was proposed in 1965, it was quickly realized to be a useful tool, but applications were initially limited to the homogeneous electron gas [14]. The result that attracted most attention was the fact that the \(GW\) approximation, in its standard non-self-consistent form, predicts a smaller valence band width and consequently a larger effective mass than for free electrons [25] in agreement with angle-resolved photoemission experiments on sodium or other simple metals. This contrasts with density functional theory, which always yields the free-electron band width due to the constant exchange-correlation potential in homogeneous systems, or the Hartree–Fock approximation, which yields an even larger band width. It was thus accepted that the \(GW\) approximation cures some of the well-known shortcomings of simpler mean-field schemes. Other information that can be directly obtained from the self-energy include the momentum distribution and the Compton profile [28], which are also in good quantitative agreement with measured data for metals.

Fig. 2
figure 2

Calculated electronic band structure of silicon. The data sets refer to the Kohn–Sham eigenvalues within the local-density approximation (LDA) and the quasiparticle energies obtained from the \(GW\) approximation according to Eq. (27). The principal effect of the self-energy correction is the widening of the band gap, but the dispersion of the bands also changes slightly

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The first \(GW\) calculations for real materials were only reported in the mid 1980s, when computers became widely available for academic research. For alkali metals like sodium these confirmed the previous results for the homogeneous electron gas [27], but the majority of early applications focused on semiconductors, where the systematic underestimation of the experimentally measured band gaps obtained with the local-density approximation (LDA) seemed the most pressing problem in need of a solution. Indeed, the results reported independently by Hybertsen and Louie [18, 19] and by Godby, Schlüter and Sham [11, 12] immediately established that the \(GW\) approximation yields band gaps in excellent agreement with the experimental measurements for typical semiconductors like silicon or gallium arsenide. Although these early calculations employed a number of additional simplifications, they were confirmed by more elaborate later studies. As an illustration, Fig. 2 shows the band structure of silicon calculated within the LDA and the \(GW\) approximation. The LDA bands are qualitatively correct, but the eigenvalue gap of 0.47 eV lies far below the experimental value of 1.17 eV. If self-energy corrections are added as in (27), then the gap increases to 1.21 eV. Besides the widening of the band gap, the figure shows only a minor modification of the dispersion in the case of silicon. However, this is not true for all materials, and the \(GW\) approximation is hence more than a “scissors operator” that merely adjusts the distance between valence and conduction bands. As a consequence, the effective masses are also modified in general [7].

Fig. 3
figure 3

Theoretical versus experimental band gaps for a wide variety of semiconductors and insulators. Unlike the local-density approximation (LDA), the \(GW\) approximation is in very good agreement with the experimental values, except for certain strongly correlated systems like NiO. Due to the limited space, not all materials for which results are shown in the figure are individually labeled

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Figure 3 displays results, in part calculated by the author and in part taken from [3], for a wide range of semiconductors and insulators. The arrangement of theoretical versus experimental band gaps means that corresponding LDA and \(GW\) results for the same material can be connected by vertical lines. The systematic underestimation within the LDA is clearly visible, as all data points lie far below the diagonal. In contrast, the \(GW\) approximation is consistently close to the experimental values. A notable exception is NiO [1], a Mott–Hubbard insulator where the energy gap is not a band-structure effect but due to strong electronic correlation. The LDA yields almost no gap at all in this case, and the perturbative \(GW\) correction consequently also fails. In fact, the picture of individual, weakly interacting quasiparticles, which the \(GW\) approximation assumes, itself becomes questionable for materials like NiO. Novel functionals that improve upon the LDA for strongly correlated systems have already been developed (see the Chap. Density Functional Theory for Strongly-Interacting Electrons) and may offer a better starting point for many-body perturbation theory in the future. At present, however, non-perturbative methods like dynamical mean-field theory (see the Chap. Electronic Structure Calculations with LDA + DMFT) provide the most reliable description.

For a long period \(GW\) calculations were almost exclusively performed for bulk semiconductors, because practical implementations relied on pseudopotentials and plane-wave basis sets, which effectively restrict the range of possible applications. With a new generation of all-electron codes [9, 22, 34] it is equally possible to treat materials with transition-metal and rare-earth elements, however. The quasiparticle energies obtained from all-electron and pseudopotential calculations within the \(GW\) approximation differ, in principle, due to the inexact core-valence partitioning and the use of pseudo wave functions in the latter approach, but the large quantitative deviation reported in early all-electron calculations [23] is now believed to stem from incomplete convergence with respect to the number of unoccupied states included in (17). In fact, carefully converged all-electron calculations yield band gaps that are close both to the experimental values and to established pseudopotential results.

In a separate development, the availability of increasingly powerful computer resources now enables applications to more complex systems that were previously impossible. The maximum system size is still significantly smaller than in density functional theory, but calculations for supercells with up to a few hundred atoms are within reach. This suffices to investigate, for example, the electronic structure of point defects at semiconductor surfaces [15]. However, surface calculations within the \(GW\) approximation require a lot of care, because they are easily influenced by details of the artificial slab geometry usually employed in actual simulations due to the inherent non-locality of the Green function and the self-energy [8].

Finally, although most applications of the \(GW\) approximation are concerned with solids, it has also been successfully applied to atoms and molecules [31], where it improves significantly upon the Hartree–Fock approximation. Although electronic screening is often considered weak in molecular systems, its inclusion at the level of the \(GW\) self-energy yields ionization potentials that are in much better agreement with experimental measurements than the unscreened Hartree–Fock values.

6 Outlook

The \(GW\) approximation for the electronic self-energy is a highly successful scheme that allows accurate quantitative calculations of electronic excitation energies. It is most famous for yielding accurate quasiparticle band gaps of semiconductors, but other quantities like valence band widths or effective masses are also improved with respect to density functional theory. While the \(GW\) approximation is not appropriate for strongly correlated systems, it can be applied to bulk solids, surfaces and defects as well as atoms and molecules. Recent all-electron implementations now also allow the treatment of transition-metal and rare-earth compounds. The future will thus see a growing number of applications in different areas where an accurate knowledge of the electronic and optical properties of materials is essential. It is expected that this will then spur more powerful implementations, not least via efficient parallelization, a resource largely untapped until now, that overcome the present size limitation of just over one hundred atoms per supercell. Finally, the results reported so far suggest that the \(GW\) approximation, in combination with density functional theory, is also a promising approach to determine molecular energy levels in systems that are too large for conventional quantum-chemical techniques.