International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

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

J. J. Kas,a* J. J. Rehra and F. D. Vilaa

aDepartment of Physics, University of Washington, Seattle, WA 98195-1560, USA
Correspondence e-mail:  joshua.j.kas@gmail.com

Green's functions are a powerful analytical and computational tool for ab initio calculations of X-ray spectra. For example, Green's functions provide an efficient means for calculations over broad energy ranges since many-body effects can be incorporated naturally in terms of the electron self-energy. Here, their role in the theory of X-ray absorption and related spectroscopies is discussed, with particular focus on many-body effects such as quasi-particle energy shifts and lifetimes, core-hole 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 multi-electron excitations that produce satellite features in X-ray spectra.

Keywords: X-ray absorption; XANES; many-body effects.

1. Green's functions: relation to X-ray absorption spectroscopy (XAS)

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[link]; Mahan, 1990[link]). Here, we focus on the use of Green's functions to describe X-ray interactions with matter. Green's functions have been used in multiple ways to describe these interactions. They include, for example, the use of multiple-scattering Green's functions to describe XAS and related spectra (Gonis & Butler, 2000[link]; Rehr & Albers, 2000[link]), Hedin's approach for calculating the electron self-energy in condensed matter (Hedin, 1965[link]; Aryasetiawan & Gunnarsson, 1998[link]) and the use of the Bethe–Salpeter equation to describe excitonic effects in optical and X-ray spectra (Leng et al., 2016[link]; Vinson et al., 2011[link]; Olovsson et al., 2009[link]). For example, Green's functions G(E) for the one-electron Schrodinger equation with Hamiltonian h are solutions of the inhomogeneous equation (Eh)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 one-electron Hamiltonian h as[G(E) = {{1} \over {E-h+i\delta}},\eqno (1)]where 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 X-ray absorption, we begin by writing down the absorption cross section σ(ω) for an X-ray of energy ℏω in terms of the one-electron Fermi's golden rule, i.e. [\sigma(\omega)\propto\textstyle\sum \limits_{\rm f}|\langle {\rm i}| {d} |{\rm f}\rangle|^{2}\delta(\omega-E_{\rm f}+E_{\rm i}). \eqno (2)]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 Ei and Ef denote their energies and d is the transition operator. This formula can be related to the spectral representation of the single-particle Green's function G(E) by rewriting it as[\sigma(\omega)\propto\langle {\rm i}|{d} \textstyle \sum \limits_{\rm f}|{\rm f}\rangle\langle {\rm f}|\delta(\omega-E_{\rm f}+E_{\rm i}){d}^{\dagger}| {\rm i}\rangle \eqno (3)]and noting that the spectral sum is proportional to the imaginary part of the Green's function G(E),[{\textstyle \sum \limits_{\rm f}}|{\rm f}\rangle\langle {\rm f}|\delta(\omega-E_{\rm f}+E_{\rm i}) = -{{1} \over {\pi}}{\rm Im}[G(\omega+E_{\rm i})]. \eqno (4)]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,[\eqalignno {\sigma(\omega) & \propto -{1 \over \pi} {\rm Im} [\langle {\rm i}|{d} G(\omega+E_{\rm i}){d}^{\dagger}|{\rm i}\rangle] \cr & = \langle {\rm i}|{d} A(\omega+E_{\rm i}){d}^{\dagger}|{\rm i}\rangle. & (5)}]While equivalent to Fermi's golden rule, this representation can be much more efficient, since it avoids the need to calculate final-state 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 (ll ± 1), which are dominated by the l + 1 term, and the limited variation of the transition-matrix elements as a function of energy, the XAS signal is roughly proportional to a particular angular momentum-projected density of states ρl(E), for example the projected density of p-states for the K edge. This can be derived by choosing the final unoccupied states on an angular momentum L = (l, m) and scattering-state basis, so that[\sigma(\omega) = |M_{l}(E)|^{2}\rho_{l}(E), \eqno (6)]where Ml(E) is the radial dipole matrix element of the absorbing atom and E = (1/2)k2 = Ei + ℏω is the photoelectron energy. Thus, in the absence of strong many-body effects, since the transition-matrix elements usually vary smoothly with energy, X-ray absorption near-edge 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 non-interacting electrons, equation (1)[link] still holds for interacting electrons, as long as the Green's function is defined via an effective Hamiltonian h that includes the electron self-energy operator Σ(E), which accounts for the effects of electron–electron interactions. Of course, Σ(E) is an energy-dependent, nonlocal and complex-valued operator, and hence must be approximated. The effects of these many-body electron–electron or electron–phonon interactions via various approximations to the Green's function and self-energy are the topic of the next section.

2. Many-body effects

2.1. Core-hole interaction effects

When a core-level 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 final-state rule, can be quite good for a variety of materials. This is especially true at the K edge (Rehr et al., 2005[link]), 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 X-ray is treated via a two-particle (particle–hole) Green's function. Formally, the two-particle Green's function can be described through the Green's functions of the individual particles and the effective interaction between them (Salpeter & Bethe, 1951[link]). To treat optical and X-ray interactions with matter, this concept been extended to condensed matter systems and has been approximated in various ways (Strinati, 1984[link]; Leng et al., 2016[link]). Probably the most common approach is to assume a static effective interaction, which then leads to an equation based on a two-particle Hamiltonian, for example [\mu(\omega)\propto-{1 \over {\pi}}\left\langle 0\left |\Delta^{\dagger}{1 \over {\omega-H_{\rm BSE}}} \Delta\right|0\right\rangle, \eqno (7)]where Δ is the transition operator, which creates a particle–hole excitation, and HBSE = hehh + VI is an effective particle–hole Hamiltonian consisting of the independent electron and hole parts he and hh and an effective interaction VI between them. This interaction is the sum of a screened Coulomb, or direct, term Vd as well as an exchange term Vx (Shirley, 1998[link]; Vinson et al., 2011[link]),[\eqalignno {V_{\rm d} & = W({\bf r},{\bf r}^{\prime},\omega),\cr V_{\rm x} & = {1 \over {|{\bf r}-{\bf r}^{\prime}|}}.& (8)}]While the screened Coulomb interaction W = ɛ−1v above is in principle frequency-dependent, most applications use the adiabatic approximation, Vd = W(ω = 0), which is quite reasonable for optical or X-ray absorption. Fig. 1[link] (Vinson et al., 2011[link]) shows a comparison of the K-edge XANES spectrum of lithium fluoride using the final-state rule compared with that using the BSE approach, along with experimental results (Olovsson et al., 2009[link]). It can be seen that for this case the final-state rule approximation is quite good, although there are differences very close to the edge. The differences between the final-state rule and the BSE are larger for, for example, the L edges of transition metals (Vinson & Rehr, 2012[link]).

[Figure 1]

Figure 1

Calculated F K-edge XAS of lithium fluoride using the final-state rule approximation (light blue dotted/dashed line) or BSE (green dashed line) compared with experimental results (dotted line; Olovsson et al., 2009[link]). Reprinted with permission from Vinson et al. (2011[link]). Copyright 2011 by the American Physical Society.

2.2. Quasi-particle effects: the GW approximation and the many-pole model

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 quasi-particle, which has a shifted energy and finite lifetime, both of which are characterized by the quasi-particle self-energy. The self-energy 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 many-body 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 self-energy can be estimated via Hedin's GW approximation (GWA) Σ = iGW (Hedin, 1965[link]), where G is the single-particle Green's function and W = ɛ−1v is the screened Coulomb interaction. While there are a variety of codes and methods for calculating or approximating the GW self-energy (Gonze et al., 2005[link]; Shishkin et al., 2007[link]; Soininen et al., 2003[link]; Chantler & Bourke, 2014[link]), in the interest of computational efficiency we represent the dielectric screening function ɛ−1(qω) as a sum of poles, i.e. [-{\rm Im}[\varepsilon^{-1}(q,\omega)] = {{\pi} \over {2}}\textstyle \sum\limits_{i}g_{i}\omega _{i}\delta[\omega-\omega_{i}(q)], \eqno (9)]where ωi and gi are the excitation frequencies and strengths. We use a simple approximate dispersion [\omega_{i}(q)^{2}] = [\omega_{i}^{2}] + [v_{F}^{2}q^{2}]/3 + q4/4. This preserves the well known Bethe ridge [ω(q) ∝ q2/2] at high momentum transfer, and is reasonably consistent with calculations based on the random phase approximation (Lundqvist, 1967[link]). In any case, the effects of the dispersion are rather weak since the self-energy 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,[L(\omega) = -{\rm Im}[\varepsilon^{-1}(q=0,\omega)].]Within these approximations, the total self-energy is given by a sum of plasmon-pole self-energies (Lundqvist, 1967[link]), each evaluated with a different plasmon frequency, and weighted according the weights in the model above, i.e. [\Sigma(E) = \textstyle \sum \limits_{i}g_{i}\Sigma_{\rm pp}(E,\omega_{i}).\eqno(10)]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[link]), or taken from experimental data. Fig. 2[link] shows a calculation of the loss function of copper metal using the atomic database compared with experimental data (Palik, 1985[link]), along with the pole model representation using 20 poles. Fig. 3[link] shows the final calculated K-edge XANES of copper compared with the results based on the plasmon-pole model as well as experimental results (Newville, 2007[link]). Clear improvement is seen in the agreement of peak amplitudes and positions between theoretical and experimental data when using the many-pole model. In the extended X-ray absorption fine-structure (EXAFS) region, this self-energy broadening results in an inelastic mean free path (IMFP) λ = k/|Im[Σ(E)]|.

[Figure 2]

Figure 2

Calculated loss function of copper (blue dashed line) compared with the experimental result (black crosses; Palik, 1985[link]). Vertical green lines show the many-pole model representation with 20 poles.

[Figure 3]

Figure 3

Cu K-edge XANES calculated using the many-pole self-energy model (blue dashed line) and the plasmon pole model (red solid line), compared with experimental results (black asterisks; Newville, 2007[link]).

2.3. Multi-electron excitations

Another many-body effect that can be extremely important corresponds to multiple electron excitations. While quasi-particle theory can explain many of the features of XANES spectra, some systems have strong many-electron 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 charge-transfer excitation. While these multi-electron excitations affect all spectra, they are especially relevant to core-level X-ray 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 quantum-mechanical nature of these excitations the amplitudes, rather than the intensities, are additive, thus there is interference between the intrinsic and extrinsic effects (Hedin, 1999[link]; Campbell et al., 2002[link]). These excitations again can be described by the electron self-energy Σ(E), and are strongly related to the spectral function A(ω) = −(1/π)Im[G(ω)]. Consequently, the many-body XAS can be given by a convolution of the quasi-particle XAS μ(1)(ω) described in Section 1[link] and the spectral function, i.e. [\mu(\omega) = \textstyle \int A(\omega,\omega^{\prime})\mu^{(1)}(\omega^{\prime}). \eqno (11)]

2.3.1. Cumulant and GW approximations

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. [G(E) = G^{0}(E)+G^{0}(E)\Sigma(E)G(E), \eqno (12)]where G0 is the Green's function of the non-interacting system. While this definition is formally exact, it depends on the knowledge of the exact self-energy, which is not known. Thus, the self-energy is approximated, most commonly via the GW approximation. This method produces good quasi-particle properties for a wide variety of materials, but its predicted satellite (many-electron) 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[link]; Guzzo et al., 2011[link]; Aryasetiawan et al., 1996[link]; Kas et al., 2014[link]; Lischner et al., 2013[link]; Zhou et al., 2015[link]), in which the Green's function is represented as an exponential in time, [G(t) = G^{0}(t)\exp[C(t)]], where the cumulant is given by[C(t) = {\int} {\rm d}\omega{{\beta(\omega)} \over {\omega^{2}}}[\exp(-i\omega t) +i\omega t-1]. \eqno (13)]This 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 multiple-plasmon excitations. The cumulant function C(t) can be found in a variety of ways. For core-level spectra, C(t) is related to the response function and can be found in frequency space (Langreth, 1970[link]) or via real-time time-dependent density-functional theory (Kas et al., 2015[link]). For valence electrons, the cumulant can be related to the GW self-energy (Hedin, 1980[link]; Gunnarsson et al., 1994[link]; Guzzo et al., 2011[link]; Almbladh & Hedin, 1983[link]). Fig. 4[link] shows a comparison of the X-ray 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 X-ray photoemission results (blue crosses) for silicon (Guzzo et al., 2011[link]). The main (quasi-particle) peaks between −15 and 0 eV are well represented in all theoretical results, while the GW spectral function fails to reproduce the one-plasmon and two-plasmon satellite peaks at about −25 and −42 eV, respectively. Instead, the GW theory produces only one peak at an energy difference ω ≃ 1.5ωp from the main peaks, where ωp is the plasmon frequency. The cumulant-based theories, on the other hand, are able to reproduce the multiple-plasmon peaks, and the inclusion of extrinsic and interference effects produces near-quantitative results.

[Figure 4]

Figure 4

Valence XPS of silicon calculated using the cumulant approach including intrinsic and extrinsic excitations as well as the interference between them (black solid line), only intrinsic effects (green dashed/dotted line) or the GW (Dyson equation) approach (red dashed line), compared with experimental results (blue crosses). Reprinted with permission from Guzzo et al. (2011[link]). Copyright 2011 American Physical Society.

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[link]; Kas et al., 2016[link]). In this case interference terms are very important, although they are quite difficult to include beyond simple models (Kas et al., 2016[link]; Campbell et al., 2002[link]). While the cumulant approach has mainly been used to describe multiple satellites associated with quasi-bosonic excitations such as plasmons and phonons, it has also been used to model other neutral excitations such as charge-transfer excitations. In order to show the utility of such a theory, we present the calculated K-edge XANES of NiO (red) in Fig. 5[link] compared with experimental results (black crosses; Calandra et al., 2012[link]) as well as the single-particle calculation (without multi-electron excitations; blue). In NiO, as in many of the transition-metal oxides, a strong charge-transfer satellite causes a doubling of the peaks seen in the single-particle spectrum (as indicated by the arrow in the figure), with a shift in this case of approximately 7 eV.

[Figure 5]

Figure 5

Ni K-edge XANES of nickel(II) oxide calculated using the cumulant method to include many-body effects (red solid line) and using only the single-particle theory (blue solid line), compared with experimental results (black line with plus symbols; Calandra et al., 2012[link]). The arrow shows the effects of the many-body charge-transfer satellite, which is reproduced by the cumulant approach.

2.4. Vibrational effects: ab initio Debye–Waller factors

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[link]; Vila, Hayashi et al., 2018[link]; Vila, Spencer et al., 2018[link]). Phonon properties such as the mean-square relative displacement (MSRD) σ2, 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,[G(\omega) = 1/(\omega-{\bf D}+i\delta), \eqno (14)]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,[\rho_{\Gamma}(\omega) = {{2\omega} \over {\pi}}{\rm Im}[\langle\Gamma|G(\omega)|\Gamma\rangle], \eqno (15)]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,[\sigma_{\Gamma}^{2}(T) = {{\hbar} \over {2\mu_{\Gamma}}} {\textstyle \int \limits_{0}^{\infty}} {{\rho_{\Gamma}(\omega)} \over {\omega}}\coth{{\beta\hbar\omega} \over {2}}\,{\rm d}\omega,\eqno (16)]where β = 1/kBT and μΓ 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[link] shows the theoretical EXAFS MSRD in germanium calculated using either LDA or hGGA exchange–correlation functionals (Vila et al., 2007[link]) compared with calculations based on the commonly used correlated Debye model, as well as with experimental results (Dalba et al., 1999[link]). 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[link] 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[link]; Vila et al., 2008[link]). 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[link]).

[Figure 6]

Figure 6

Results of ab initio calculated first-shell MSRD of germanium using different exchange–correlation potentials. The results are compared with experiment (Dalba et al., 1999[link]), as well as calculations using the correlated Debye model. Reprinted with permission from Vila et al. (2007[link]). Copyright 2007 American Physical Society,

[Figure 7]

Figure 7

EXAFS best fit of the myoglobin–carbon monoxide complex using ab initio Debye–Waller factors compared with experimental data. The fit residual is also shown. Reprinted with permission from Veronesi et al. (2010[link]). Copyright (2010) Americal Physical Society.

3. Green's functions and other spectroscopic quantities

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, X-ray emission (XES), electron energy-loss spectroscopy (EELS) and nonresonant inelastic X-ray 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[link]; Soininen et al., 2005[link]; Rehr & Albers, 2000[link]). 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 [\exp(i{\bf q}\cdot{\bf r})], 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 [\hat{\boldvarepsilon}]. However, at high momentum transfer the spectrum is appreciably different since transitions well beyond the dipole approximation are not negligible. Resonant inelastic X-ray scattering can also be written with various levels of approximation in terms of either the one-particle Green's function (Kas et al., 2011[link]) or the particle–hole Green's function as in the Bethe–Salpeter equation (Shirley, 1998[link]). Finally, Compton scattering can also be described within a Green's function formalism (Mattern et al., 2012[link]) within the impulse approximation.

Funding information

This work was supported primarily through the US DOE Office of Science BES Grant DE-FG02-97ER45623, with computational support from NERSC, a DOE Office of Science User Facility, under Contract No. DE-AC02-05CH11231. 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. DE-AC02-76SF00515.

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