International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2022 
International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007521 Green's functions applied to the theory of spectroscopy^{a}Department of Physics, University of Washington, Seattle, WA 981951560, USA Green's functions are a powerful analytical and computational tool for ab initio calculations of Xray spectra. For example, Green's functions provide an efficient means for calculations over broad energy ranges since manybody effects can be incorporated naturally in terms of the electron selfenergy. Here, their role in the theory of Xray absorption and related spectroscopies is discussed, with particular focus on manybody effects such as quasiparticle energy shifts and lifetimes, corehole interactions and electron–phonon interactions. In addition, the cumulant expansion for the Green's function is reviewed and compared with the usual GW Dyson equation approach for including multielectron excitations that produce satellite features in Xray spectra. Keywords: Xray absorption; XANES; manybody effects. 
Green's functions have a long history in physics, and there are numerous texts that discuss their behaviour as applied in condensed matter physics (Doniach & Sondheimer, 1974; Mahan, 1990). Here, we focus on the use of Green's functions to describe Xray interactions with matter. Green's functions have been used in multiple ways to describe these interactions. They include, for example, the use of multiplescattering Green's functions to describe XAS and related spectra (Gonis & Butler, 2000; Rehr & Albers, 2000), Hedin's approach for calculating the electron selfenergy in condensed matter (Hedin, 1965; Aryasetiawan & Gunnarsson, 1998) and the use of the Bethe–Salpeter equation to describe excitonic effects in optical and Xray spectra (Leng et al., 2016; Vinson et al., 2011; Olovsson et al., 2009). For example, Green's functions G(E) for the oneelectron Schrodinger equation with Hamiltonian h are solutions of the inhomogeneous equation (E − h)G(E) = 1, but are not unique. Particular solutions depend on boundary conditions. Thus, G(E) can be defined via the resolvent of the effective oneelectron Hamiltonian h aswhere δ produces an infinitesimal shift of the pole below the real axis and the plus sign is chosen to enforce causality.
In order to discuss the relation between spectroscopy and Green's functions, with particular focus on Xray absorption, we begin by writing down the absorption cross section σ(ω) for an Xray of energy ℏω in terms of the oneelectron Fermi's golden rule, i.e. Here, i〉 and f〉 denote the core level in question and unoccupied states of the system calculated in the presence of a core hole, respectively, while E_{i} and E_{f} denote their energies and d is the transition operator. This formula can be related to the spectral representation of the singleparticle Green's function G(E) by rewriting it asand noting that the spectral sum is proportional to the imaginary part of the Green's function G(E),The imaginary part of G(E) is also referred to as the spectral function A(E) = −(1/π)ImG(E), which characterizes the poles of the Green's function. Consequently, the XAS from core level i is given by a single matrix element of the Green's function,While equivalent to Fermi's golden rule, this representation can be much more efficient, since it avoids the need to calculate finalstate eigenfunctions and eigenvalues and the summation over final states. For XAS, the dipole approximation d = ɛ · r is usually very good for most systems. Due to dipole selection rules (l → l ± 1), which are dominated by the l + 1 term, and the limited variation of the transitionmatrix elements as a function of energy, the XAS signal is roughly proportional to a particular angular momentumprojected density of states ρ_{l}(E), for example the projected density of pstates for the K edge. This can be derived by choosing the final unoccupied states on an angular momentum L = (l, m) and scatteringstate basis, so thatwhere M_{l}(E) is the radial dipole matrix element of the absorbing atom and E = (1/2)k^{2} = E_{i} + ℏω is the photoelectron energy. Thus, in the absence of strong manybody effects, since the transitionmatrix elements usually vary smoothly with energy, Xray absorption nearedge structure (XANES) is a good measure of the unoccupied projected electronic density of final states of the system.
While the above derivation relies on a system of effective noninteracting electrons, equation (1) still holds for interacting electrons, as long as the Green's function is defined via an effective Hamiltonian h that includes the electron selfenergy operator Σ(E), which accounts for the effects of electron–electron interactions. Of course, Σ(E) is an energydependent, nonlocal and complexvalued operator, and hence must be approximated. The effects of these manybody electron–electron or electron–phonon interactions via various approximations to the Green's function and selfenergy are the topic of the next section.
When a corelevel electron is ejected from the system, the hole left behind gives an attractive interaction with the photoelectron. This interaction between the effective photoelectron and hole is exceedingly complex due to the dynamic screening of the valence electrons. In many cases, however, simple approximations are very useful. In particular, using the system of fully relaxed electrons in the presence of a frozen core hole, an approximation known as the finalstate rule, can be quite good for a variety of materials. This is especially true at the K edge (Rehr et al., 2005), where mixing between hole states is not possible. However, in cases such as the L edges of transition metals this approach must be improved.
A more advanced method is that based on the Bethe–Salpeter equation (BSE), where the excitation caused by the Xray is treated via a twoparticle (particle–hole) Green's function. Formally, the twoparticle Green's function can be described through the Green's functions of the individual particles and the effective interaction between them (Salpeter & Bethe, 1951). To treat optical and Xray interactions with matter, this concept been extended to condensed matter systems and has been approximated in various ways (Strinati, 1984; Leng et al., 2016). Probably the most common approach is to assume a static effective interaction, which then leads to an equation based on a twoparticle Hamiltonian, for example where Δ is the transition operator, which creates a particle–hole excitation, and H_{BSE} = h_{e} − h_{h} + V_{I} is an effective particle–hole Hamiltonian consisting of the independent electron and hole parts h_{e} and h_{h} and an effective interaction V_{I} between them. This interaction is the sum of a screened Coulomb, or direct, term V_{d} as well as an exchange term V_{x} (Shirley, 1998; Vinson et al., 2011),While the screened Coulomb interaction W = ɛ^{−1}v above is in principle frequencydependent, most applications use the adiabatic approximation, V_{d} = W(ω = 0), which is quite reasonable for optical or Xray absorption. Fig. 1 (Vinson et al., 2011) shows a comparison of the Kedge XANES spectrum of lithium fluoride using the finalstate rule compared with that using the BSE approach, along with experimental results (Olovsson et al., 2009). It can be seen that for this case the finalstate rule approximation is quite good, although there are differences very close to the edge. The differences between the finalstate rule and the BSE are larger for, for example, the L edges of transition metals (Vinson & Rehr, 2012).
As the photoelectron travels through the material it interacts with the valence electrons, producing various excitations, for example particle–hole and plasmon excitations. The main effect of these excitations is to produce an effective quasiparticle, which has a shifted energy and finite lifetime, both of which are characterized by the quasiparticle selfenergy. The selfenergy in the context of condensed matter is analogous to that of Feynman, with the definition of the Fermi sea corresponding to electronic states below the chemical potential. In this sense, one speaks of `holes' rather than positrons. Similar manybody perturbation and diagrammatic techniques can be used to describe the creation and annihilation of particle–hole states, as are used to describe electron–positron creation and annihilation. The selfenergy can be estimated via Hedin's GW approximation (GWA) Σ = iGW (Hedin, 1965), where G is the singleparticle Green's function and W = ɛ^{−1}v is the screened Coulomb interaction. While there are a variety of codes and methods for calculating or approximating the GW selfenergy (Gonze et al., 2005; Shishkin et al., 2007; Soininen et al., 2003; Chantler & Bourke, 2014), in the interest of computational efficiency we represent the dielectric screening function ɛ^{−1}(q, ω) as a sum of poles, i.e. where ω_{i} and g_{i} are the excitation frequencies and strengths. We use a simple approximate dispersion = + /3 + q^{4}/4. This preserves the well known Bethe ridge [ω(q) ∝ q^{2}/2] at high momentum transfer, and is reasonably consistent with calculations based on the random phase approximation (Lundqvist, 1967). In any case, the effects of the dispersion are rather weak since the selfenergy involves an integral over both energy and momentum transfer. The strengths and positions of the poles are found by matching to an ab initio calculation of the loss function at zero momentum transfer,Within these approximations, the total selfenergy is given by a sum of plasmonpole selfenergies (Lundqvist, 1967), each evaluated with a different plasmon frequency, and weighted according the weights in the model above, i.e. In practice, the calculation of the zero momentum loss function can be performed using an elemental atomic database for efficient calculations, or using more advanced techniques such as the Bethe–Salpeter equation method, as implemented, for example, in the code AI2NBSE (Lawler et al., 2008), or taken from experimental data. Fig. 2 shows a calculation of the loss function of copper metal using the atomic database compared with experimental data (Palik, 1985), along with the pole model representation using 20 poles. Fig. 3 shows the final calculated Kedge XANES of copper compared with the results based on the plasmonpole model as well as experimental results (Newville, 2007). Clear improvement is seen in the agreement of peak amplitudes and positions between theoretical and experimental data when using the manypole model. In the extended Xray absorption finestructure (EXAFS) region, this selfenergy broadening results in an inelastic mean free path (IMFP) λ = k/Im[Σ(E)].

Calculated loss function of copper (blue dashed line) compared with the experimental result (black crosses; Palik, 1985). Vertical green lines show the manypole model representation with 20 poles. 
Another manybody effect that can be extremely important corresponds to multiple electron excitations. While quasiparticle theory can explain many of the features of XANES spectra, some systems have strong manyelectron excitations which produce satellite peaks in the spectrum that correspond to the simultaneous production of a photoelectron as well some neutral excitation of the valence system, such as a plasmon, particle–hole or chargetransfer excitation. While these multielectron excitations affect all spectra, they are especially relevant to corelevel Xray photoelectron spectroscopy, where they lead to a reduction in the main (quasiparticle) peak as well as to satellite peaks.
One way to interpret these effects is that the sudden appearance of the core hole causes intrinsic excitations of the valence system, while the photoelectron causes extrinsic excitations as it travels through the material. Due to the quantummechanical nature of these excitations the amplitudes, rather than the intensities, are additive, thus there is interference between the intrinsic and extrinsic effects (Hedin, 1999; Campbell et al., 2002). These excitations again can be described by the electron selfenergy Σ(E), and are strongly related to the spectral function A(ω) = −(1/π)Im[G(ω)]. Consequently, the manybody XAS can be given by a convolution of the quasiparticle XAS μ^{(1)}(ω) described in Section 1 and the spectral function, i.e.
There are a variety of theoretical methods for approximating the Green's function, which produce results of varying accuracy. The standard method is to define the Green's function in terms of the Dyson equation, i.e. where G^{0} is the Green's function of the noninteracting system. While this definition is formally exact, it depends on the knowledge of the exact selfenergy, which is not known. Thus, the selfenergy is approximated, most commonly via the GW approximation. This method produces good quasiparticle properties for a wide variety of materials, but its predicted satellite (manyelectron) features cannot fully explain experimental measurements, particularly those involving multiple plasmon excitations. An alternative approach which has shown great promise is the cumulant expansion (Langreth, 1970; Guzzo et al., 2011; Aryasetiawan et al., 1996; Kas et al., 2014; Lischner et al., 2013; Zhou et al., 2015), in which the Green's function is represented as an exponential in time, , where the cumulant is given byThis form characterizes an isolated core electron interacting with a system of bosons with an excitation spectrum characterized by β(ω), which explains its success in treating multipleplasmon excitations. The cumulant function C(t) can be found in a variety of ways. For corelevel spectra, C(t) is related to the response function and can be found in frequency space (Langreth, 1970) or via realtime timedependent densityfunctional theory (Kas et al., 2015). For valence electrons, the cumulant can be related to the GW selfenergy (Hedin, 1980; Gunnarsson et al., 1994; Guzzo et al., 2011; Almbladh & Hedin, 1983). Fig. 4 shows a comparison of the Xray photoelectron spectrum (XPS) of silicon (closely related to the spectral function) based on the Dyson equation (labelled GW in the plot) and that of the cumulant expansion with (black) or without (green dotted/dashed line) extrinsic and interference effects, along with experimental Xray photoemission results (blue crosses) for silicon (Guzzo et al., 2011). The main (quasiparticle) peaks between −15 and 0 eV are well represented in all theoretical results, while the GW spectral function fails to reproduce the oneplasmon and twoplasmon satellite peaks at about −25 and −42 eV, respectively. Instead, the GW theory produces only one peak at an energy difference ω ≃ 1.5ω_{p} from the main peaks, where ω_{p} is the plasmon frequency. The cumulantbased theories, on the other hand, are able to reproduce the multipleplasmon peaks, and the inclusion of extrinsic and interference effects produces nearquantitative results.
This theory can also be extended for use in XAS by representing the particle–hole Green's function in terms of an exponential form (Zhou et al., 2015; Kas et al., 2016). In this case interference terms are very important, although they are quite difficult to include beyond simple models (Kas et al., 2016; Campbell et al., 2002). While the cumulant approach has mainly been used to describe multiple satellites associated with quasibosonic excitations such as plasmons and phonons, it has also been used to model other neutral excitations such as chargetransfer excitations. In order to show the utility of such a theory, we present the calculated Kedge XANES of NiO (red) in Fig. 5 compared with experimental results (black crosses; Calandra et al., 2012) as well as the singleparticle calculation (without multielectron excitations; blue). In NiO, as in many of the transitionmetal oxides, a strong chargetransfer satellite causes a doubling of the peaks seen in the singleparticle spectrum (as indicated by the arrow in the figure), with a shift in this case of approximately 7 eV.
Green's functions can also be used to calculate phonon properties, which are especially important for calculations and analysis of EXAFS (Vila et al., 2007; Vila, Hayashi et al., 2018; Vila, Spencer et al., 2018). Phonon properties such as the meansquare relative displacement (MSRD) σ^{2}, commonly used in EXAFS analysis, can be calculated from the vibrational density of states (VDOS), which is related to the lattice dynamical Green's function,where D is the dynamical matrix of force constants. The MSRD for a given EXAFS path Γ is related to the VDOS projected onto that path,where Γ〉 is a seed vector corresponding to a normalized displacement along the path Γ. This formula can be implemented efficiently via Lanczos inversion techniques. Finally, the MSRD is calculated via the Debye integral,where β = 1/k_{B}T and μ_{Γ} is the reduced mass associated with the path. Similar Debye integrals can be used to calculate other quantities such as the crystallographic Debye–Waller factors or the vibrational free energy.
Fig. 6 shows the theoretical EXAFS MSRD in germanium calculated using either LDA or hGGA exchange–correlation functionals (Vila et al., 2007) compared with calculations based on the commonly used correlated Debye model, as well as with experimental results (Dalba et al., 1999). Clearly, the ab initio results compare very well with the experimental results and are a significant improvement over the correlated Debye model for this system. It should be noted that germanium has acoustic modes that are described well by a Debye model, but also has optical modes that are more appropriately described by an Einstein model. While a combination of the models has been used in some cases, the use of one or the other to model EXAFS Debye–Waller factors is prolific. Fig. 7 presents the fitted EXAFS χ(k) of the myoglobin–carbon monoxide complex using ab initio MSRDs. As this is a complex molecule, the use of calculated MSRDs is essential to reduce the number of free parameters in the fit. Other methods based on molecular dynamics (MD) can also be used to incorporate vibrational effects (Cauët et al., 2010; Vila et al., 2008). In some cases, such as near the edge, configurational averaging is necessary in order to take vibrational disorder into account, since this disorder breaks the local symmetry, an effect that cannot be accounted for via standard EXAFS damping factors. In other cases DFT/MD simulations have been used to calculate the MSRD, which can then be applied to EXAFS simulations (Vila et al., 2012).
In addition to XAS, Green's functions can be used in calculations of many other spectroscopic quantities. Here, we give a brief summary of several of these other spectroscopies and some of the Green's function methods that have been used to calculate them. Firstly, there are several quantities that are nearly identical to XAS in terms of the formalism. In particular, Xray emission (XES), electron energyloss spectroscopy (EELS) and nonresonant inelastic Xray scattering (NRIXS) are formally similar to XAS in that they are also given by Fermi's golden rule, but with different transition matrices. Thus, Green's function methods can be used in much the same way as with XAS (Jorissen et al., 2010; Soininen et al., 2005; Rehr & Albers, 2000). In XES, the transition operator is the same as in XAS, but the transitions are from occupied valence states to the unoccupied core level. In EELS and NRIXS the transition operator is , which replaces the dipole operator ɛ · r in XAS. Thus, NRIXS and EELS are nearly identical to XAS in the limit of low momentum transfer q → 0, with q taking the place of the polarization vector . However, at high momentum transfer the spectrum is appreciably different since transitions well beyond the dipole approximation are not negligible. Resonant inelastic Xray scattering can also be written with various levels of approximation in terms of either the oneparticle Green's function (Kas et al., 2011) or the particle–hole Green's function as in the Bethe–Salpeter equation (Shirley, 1998). Finally, Compton scattering can also be described within a Green's function formalism (Mattern et al., 2012) within the impulse approximation.
Funding information
This work was supported primarily through the US DOE Office of Science BES Grant DEFG0297ER45623, with computational support from NERSC, a DOE Office of Science User Facility, under Contract No. DEAC0205CH11231. Recent extensions are supported by the Theory Institute for Materials and Energy Spectroscopies (TIMES) at SLAC which is funded by the US DOE Office of Science BES, Division of Materials Sciences and Engineering under Contract No. DEAC0276SF00515.
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