Simulating Xâray Spectroscopies and Calculating Core-Excited States

Review Cite This: Chem. Rev. 2018, 118, 7208−7248

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Simulating X‑ray Spectroscopies and Calculating Core-Excited States of Molecules Patrick Norman*,† and Andreas Dreuw*,‡ †

Chem. Rev. 2018.118:7208-7248. Downloaded from pubs.acs.org by UNIV OF SUSSEX on 08/09/18. For personal use only.

Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden ‡ Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany ABSTRACT: During the past decade, the research field of computational X-ray spectroscopy has witnessed an advancement triggered by the development of advanced synchrotron light sources and X-ray free electron lasers that in turn has enabled new sophisticated experiments with needs for supporting theoretical investigations. Following a discussion about fundamental conceptual aspects of the physical nature of core excitations and the concomitant requirements on theoretical methods, an overview is given of the major developments made in electronic-structure theory for the purpose of simulating advanced X-ray spectroscopies, covering methods based on density-functional theory as well as wave function theory. The capabilities of these theoretical approaches are illustrated by an overview of simulations of selected linear and nonlinear X-ray spectroscopies, including X-ray absorption spectroscopy (XAS), X-ray natural circular dichroism (XNCD), X-ray emission spectroscopy (XES), resonant inelastic X-ray scattering (RIXS), and X-ray two-photon absorption (XTPA).

CONTENTS 1. Introduction 2. Fundamental Aspects about Core Excitation Processes 2.1. Character of Electronically Excited States 2.2. Electronic Relaxation and Polarization Effects 2.3. Short Wavelengths and Electric-Dipole Approximation 3. Theoretical Approaches 3.1. Time-Domain versus Frequency-Domain Approaches 3.2. Complex Polarization Propagator (CPP) Approach 3.3. Core−Valence Separation (CVS) Approximation 3.4. Density Functional Theory (DFT) Based Approaches 3.4.1. ΔSCF and Extensions 3.4.2. Transition-Potential DFT 3.4.3. Real-Time TDDFT 3.4.4. Linear Response TDDFT 3.5. Wave Function Theory Based Approaches 3.5.1. Single-Reference Methods 3.5.2. Multireference Methods 3.6. Visualizing and Characterizing Core-Excited States 4. Applications in X-ray Spectroscopies 4.1. X-ray Absorption Spectroscopy (XAS) 4.2. X-ray Natural Circular Dichroism (XNCD) 4.3. X-ray Emission Spectroscopy (XES) © 2018 American Chemical Society

4.4. Resonant Inelastic X-ray Scattering (RIXS) 4.5. X-ray Two-Photon Absorption (XTPA) 5. Summary and Outlook Author Information Corresponding Authors ORCID Notes Biographies Acknowledgments References

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1. INTRODUCTION The history of X-ray spectroscopy is over 100 years long. A beautiful overview of the pioneering development in X-ray absorption spectroscopy (XAS) that followed Rö ntgen’s discovery of X-rays in 1895 is provided in the first chapter of the book titled X-ray Absorption and X-ray Emission Spectroscopy: Theory and Applications written by the editors Lamberti and van Bokhoven,1 in which the birth of X-ray spectroscopy is attributed to the experiments in 1913 by de Broglie when the K-edge absorption spectra of silver and bromide atoms in photographic plates were observed. But it was with the introduction of the second generation synchrotron radiation facilities dedicated to X-ray spectroscopy that the field went from being a curiosity to becoming a scientific discipline. Thereafter followed the development of undulators that

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Special Issue: Theoretical Modeling of Excited State Processes Received: March 10, 2018 Published: June 12, 2018 7208

DOI: 10.1021/acs.chemrev.8b00156 Chem. Rev. 2018, 118, 7208−7248

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need to include a treatment of spin−orbit effects and state multiplets as well as a combination of the external X-ray radiation and the static magnetic field, and as of today, we find that there is little to report along these lines within the framework of theoretical chemistry and it has therefore been left out in the present work. We note, however, that in the case of K-edge XMCD on systems that are single reference in nature things are different and there does exist an available computational strategy based on the complex polarization propagator approach described in section 3.2. This computational strategy has been demonstrated in the UV/vis region6,7 but applies equally well to the X-ray region of the spectrum. As a final note on our selection of spectroscopies to include in this review, it may appear odd to leave out X-ray photoelectron spectroscopy (XPS) given its historical and contemporary importance. But to address XPS in theoretical calculations boils down in practice to either a direct determination of the electronic density of the core-ionized state or an indirect response/propagator theory calculation of the energy separation of the ground and core-ionized states. The former approach is covered in this review since this technique forms an integral component in some of the discussed methods, and the latter approach can be seen as merely a special case of a polarization propagator-based calculation of an electronic transition to a very diffuse state at the Rydberg limit. With this in mind, we felt it motivated not to dedicate a separate section to XPS. Our review consists of three main sections. In section 2, we discuss some of the most fundamental aspects of core excitation processes that are central to all method development that takes place in the field. In section 3, we first provide an overview of different and general ways to approach the simulation of spectroscopies, namely (i) real-time propagation of electronic wave functions, (ii) secular equations defined in the frequency domain, and (iii) complex response functions. Thereafter follows a subsection devoted to a key approximation in theoretical X-ray spectroscopy, namely the core−valence separation, which has truly enabled method development in the field. Finally, section 3 is completed by providing an overview of the molecular electronic structure theory methods that are employed in contemporary applied work together with a presentation of a means to, based on such calculations, visualize, analyze, and characterize electronic transitions in the X-ray region. Since the most widely used theoretical methods in chemistry are Kohn−Sham density functional theory (DFT) for the electronic ground state and time-dependent DFT for excited states, we cover certain aspects of DFT in some detail in section 3.4. In particular the issue of the electronic selfinteraction error (SIE) in DFT is of concern in X-ray spectroscopies, and it will therefore be discussed at some length. In section 4, we review how these methods have been used to simulate the discussed selection of X-ray spectroscopies. In doing so, we do focus on original work in the sense that it demonstrates a methodological novelty rather than necessarily providing valuable physicochemical or biochemical insights about individual or classes of systems, or, in other words, it is the method rather than the molecule that takes center stage in this review.

resulted in a paradigm shift in synchrotron science with orders of magnitude higher brightness and much increased beam qualitysynchrotrons built in this era belong to the third generation. The state of the art in terms of synchrotron facilities is today represented by the MAX IV Laboratory in Sweden2 that by some is referred to as belonging to a next and thereby fourth generation of sources and by others as representing an ongoing development of the third generation of sources.3 The present limit in time resolution, or pulse length, provided by synchrotron sources is on the order of 100 fs, and in order to reach the true molecular time scales, radiation sources known as X-ray free electron lasers (XFELs) have been developed and are often considered as representing the fourth generation of X-ray radiation sourcesthe Linac Coherent Light Source at Stanford being the first one as it was taken into operation in 2009.4 Today, it is indisputable to claim that X-ray spectroscopy has become an indispensable tool in materials and life sciences. Experiments conducted at synchrotron and XFEL facilities are not only monetarily expensive and labor intensive but also complex in several respects such as sample preparation, detector construction and operation, data acquisition, and, not least, data analysis. Theoretical simulations play a critical role in the analysis of experimental spectra, and alongside the development of sophisticated X-ray spectroscopies (see ref 1 for an account) there has been a concomitant development of theoretical methods. An early work dedicated to the calculation of core-hole states in atomic systems based on first-principles methods is the one by Bagus from 1965,5 i.e., some three decades after the starting point of experimental X-ray spectroscopy. These calculations of core-hole states came as a byproduct of the development of a self-consistent field scheme for optimizing general hole states, somewhat analogous to the early experiments which were conducted by means of radiation that came as a byproduct from high-energy physics (first generation synchrotron facilities). Since then several generations of methods have been developed and employed in connection with theoretical X-ray science, but at the same time it will be noticed in this review how many of the fundamental issues of core excitation processes remain at the heart of method development to date. To continue the analogy, one finds a parallel to the experimental striving to gain perfect control over the light-beam properties in the theoretical striving to gain perfect control over the electronic relaxation and polarization in the excited state. This review will present the methods in molecular electronic structure theory that are employed in contemporary theoretical X-ray spectroscopy but exclude those based on periodic boundary conditions and targeting crystal solids. The characteristic features of these methods will be pointed out, including strengths as well as weaknesses, in discussions regarding a selected set of linear and nonlinear X-ray spectroscopies. This selection is knowingly incomplete but, at the same time, is arguably representative for the theoretical tools that have been developed in theoretical chemistry. We have chosen to include near-edge (but not extended) X-ray absorption spectroscopy (XAS), X-ray natural circular dichroism (XNCD), X-ray emission spectroscopy (XES), resonant inelastic X-ray scattering (RIXS), and X-ray two-photon absorption (XTPA). We note that X-ray magnetic circular dichroism (XMCD) spectroscopy is widely adopted in the characterization of transition metal complexes and then usually measured at the L-edge. This is challenging for theory since an appropriate treatment would

2. FUNDAMENTAL ASPECTS ABOUT CORE EXCITATION PROCESSES From a principle theoretical perspective, spectroscopic core and valence excitation processes share many aspects. After invoking 7209


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excitation occurs to a continuum resonance, then the final electron state is short-lived and has the character of a plane wave and τ is governed by τe, which in this case has a characteristic natural lifetime broadening on the order of 10 eV. In comparison to low-lying states in the pre-edge region, such as π*-resonances, higher-lying states in the post-edge region, such as σ*-resonances, can effectively couple to the continuum resonances which then results in a natural lifetime broadening that is intermediate to the two mentioned extremes. The vast majority of implementations of standard electronic structure theory methods in chemistry are based on the use of Gaussian basis functions that in turn are ill-suited to describe the final-state wave functions of continuum resonances, i.e., photoionization processes. However, it has been demonstrated at the algebraic diagrammatic construction (ADC) and coupled cluster (CC) levels of theory that accurate valence photoionization cross sections can be determined from a finite Gaussian basis set pseudospectrum of bound states via the Stieltjes imaging technique,13,14 as well as via so-called complex absorbing potentials (CAPs).15 A CAP is added to the electronic Hamiltonian that absorbs the outgoing electron making its wave function square integrable, and standard quantum chemical methods applicable.16,17 Although this approach is in principle also applicable to the core region, it is hampered by being numerically rather elaborate and far from straightforward to employ in practice and the resulting energy resolution is quite low. A more attractive way to address the post-edge region is to include a better suited set of one-particle functions in the basis set. Montuoro and Moccia have shown that this is possible to achieve by using a combination of Slatertype orbitals and B-splines to compute valence photoionization spectra of N2 and C2H4 in the random phase approximation (RPA).18 Marante et al.19 have successfully used a combination of Gaussian functions and B-splines and solved the timedependent Schrödinger equation for the hydrogen atom. Decleva and co-workers have used B-splines together with the ADC method to produce accurate photoionization cross sections for noble gas atoms and notably are able to correctly reproduce the position and shape of the argon Cooper minimum.20 A further development by this group to consider the multichannel scattering problem with a B-spline basis has also been recently presented but then in conjunction with the configuration interaction singles (CIS) method.21 Therefore, while substantial progress has been made to describe systems with a combination of bound and unbound electrons, there is still quite some way to go before it can, in general, be considered a standard tool for polyatomic molecules and, in particular, the performance of the discussed mixed-basis-set approaches in the X-ray region of the spectrum is yet to be benchmarked.

the Born−Oppenheimer approximation, one arrives at a nonrelativistic electronic Hamiltonian that takes the form N

Ĥ =

i=1

+

⎛ 1 [p̂ + eA(ri , t )]2 − eϕ(ri , t ) ⎝ 2me i

∑⎜

⎞ e B(ri , t ) ·sî ⎟ + V (r1 , ..., rN ) me ⎠

(1)

where the interaction of the molecular system and the external electromagnetic radiation field has been introduced by the principle of minimal coupling,8 introducing the vector potential A and the scalar potential ϕ as well as the spin-Zeeman term involving the external magnetic field B itself. In view of eq 1, it appears as if core and valence spectroscopies should be possible to treat by means of identical theoretical approaches. In practice, however, it turns out that there are several physical aspects in X-ray spectroscopies that enter and which have led to the development of targeted and specific methods. For one, it is clear that core electrons are more strongly bound and thereby more prone to display relativistic effects. In K-edge spectroscopies, relativistic effects are scalar in nature and corrections made to calculations based on the Hamiltonian in eq 1 typically amount to mere spectral shifts. In L-edge spectroscopies, on the other hand, it becomes necessary to consider spin−orbit interactions to even obtain a qualitatively correct X-ray spectrum. But let us set this aspects aside for a moment and discuss some other fundamental differences between core and valence excitation processes. 2.1. Character of Electronically Excited States

Molecular excitation processes due to the absorption of photons in the soft X-ray region ( 0 ⎣ ωn0 − ω − iγn0

ω ⟨0|V̂β |n⟩⟨n|Ω̂|0⟩ ⎤ ⎥ + ωn0 + ω + iγn0 ⎥⎦

(29)

1 ω ω ⟨⟨Ω̂; V̂β 1 , Vγ̂ 2⟩⟩ = 2 ∑ 71,2 ℏ ω ω ⎡ ̂ ⟨0|Ω|m⟩⟨m|V̂β 1 |n⟩⟨n|Vγ̂ 2|0⟩ ⎢ ∑ ⎢ m , n > 0 ⎣ (ωm0 − ωσ − iγm0)(ωn0 − ω2 − iγn0) ω

+

+

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ω

⟨0|V̂β 1|m⟩⟨m|Ω̂|n⟩⟨n|Vγ̂ 2|0⟩ (ωm0 + ω1 + iγm0)(ωn0 − ω2 − iγn0) ⎤ ⎥ + ω1 + iγm0)(ωn0 + ωσ + iγn0) ⎥⎦

ω ω ⟨0|V̂β 1|m⟩⟨m|Vγ̂ 2 |n⟩⟨n|Ω̂|0⟩

(ωm0

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ω ω ω ⟨⟨Ω̂; V̂β 1 , Vγ̂ 2 , Vδ̂ 3⟩⟩

1 =− 3 ℏ +

+

+

−

ω ω ω ⎧ ⎡ ⟨0|Ω̂|n⟩⟨n|V̂β 1 |m⟩⟨m|Vγ̂ 2 |p⟩⟨p|Vδ̂ 3|0⟩ ⎪ ⎢ ∑ 71,2,3⎨ ∑ ⎢ ⎪ n , m , p > 0 ⎣ (ωn0 − ωσ − iγn0)(ωm0 − (ω2 + ω3) − iγm0)(ωp0 − ω3 − iγp0) ⎩ ω ω ω ⟨0|V̂β 1|n⟩⟨n|Ω̂|m⟩⟨m|Vγ̂ 2 |p⟩⟨p|Vδ̂ 3|0⟩

(ωn0 + ω1 + iγn0)(ωm0 − (ω2 + ω3) − iγm0)(ωp0 − ω3 − iγp0) ω ω ω ⟨0|Vδ̂ 3|n⟩⟨n|Vγ̂ 2 |m⟩⟨m|Ω̂|p⟩⟨p|V̂β 1|0⟩

(ωn0 + ω3 + iγn0)(ωm0 + (ω2 + ω3) + iγm0)(ωp0 − ω1 − iγp0) ⎤ ⎥ + (ω2 + ω3) + iγm0)(ωp0 + ωσ + iγp0) ⎥⎦

ω ω ω ⟨0|Vδ̂ 3|n⟩⟨n|Vγ̂ 2 |m⟩⟨m|V̂β 1 |p⟩⟨p|Ω̂|0⟩

(ωn0 + ω3 + iγn0)(ωm0

ω ω ω ⎡ ⟨0|Ω̂|n⟩⟨n|V̂β 1|0⟩⟨0|Vγ̂ 2|m⟩⟨m|Vδ̂ 3|0⟩ ⎢ ⎢ n , m > 0 ⎣ (ωn0 − ωσ − iγn0)(ωm0 + ω2 + iγm0)(ωm0 − ω3 − iγm0)

∑

ω

+

+

ω

(ωn0 + ωσ + iγn0)(ωm0 + ω3 + iγm0)(ωm0 − ω2 − iγm0) ω ω ω ⟨0|Vγ̂ 2|n⟩⟨n|Vδ̂ 3|0⟩⟨0|Ω̂|m⟩⟨m|V̂β 1|0⟩

(ωn0 + ω2 + iγn0)(ωm0 − ωσ − iγm0)(ωm0 − ω1 − iγm0) ω

+

ω

⟨0|V̂β 1|n⟩⟨n|Ω̂|0⟩⟨0|Vδ̂ 3|m⟩⟨m|Vγ̂ 2|0⟩

ω

ω

⎤⎫ ⎪ ⎥⎬ + ωσ + iγm0) ⎥⎦⎪ ⎭

⟨0|Vδ̂ 3|n⟩⟨n|Vγ̂ 2|0⟩⟨0|V̂β 1|m⟩⟨m|Ω̅ |0⟩ (ωn0 − ω2 − iγn0)(ωm0 + ω1 + iγm0)(ωm0

where the symbols ∑ 71,2 and ∑ 71,2,3 denote the sum of terms obtained by performing, respectively, the two and six permutations given by interchanging the pairs of perturbation operators and optical frequencies: (V̂ ωβ 1,ω1), (V̂ ωγ 2,ω2), (V̂ ωδ 3,ω3). The overline labels fluctuation operators, for instance, Ω̂ = Ω̂ − ⟨0|Ω̂|0⟩. The linear response equation in eq 29 is seen to contain one-photon resonances, which in the X-ray region can occur for core-excited intermediate states |n⟩. In the vicinity of such a resonance (ω ≈ ωn0), the second term in the summation can be safely neglected, and this is known as the rotating wave approximation. The cubic response function in eq 31 contains in addition to one-photon resonances also two- and three-photon resonances, which with the ever higher light intensities produced in X-ray radiation facilities will become increasingly relevant to consider. An X-ray two-photon resonance will occur in any one of the four terms in the first sum in eq 31 for core-excited states |m⟩, whereas for intermediate states |n⟩ and |p⟩ there is no conclusion of this kind to be drawn, and they can consequently refer to valenceand core-excited states in the dominant terms in the summation. The frequency ωσ is positive and refers to the frequency component at hand in the time-dependent expectation value in eq 16, which leads us to conclude that, for a monochromatic external field, either two or three (but not one) of the frequencies ω1, ω2, and ω3 are positive and two-photon resonances in eq 31 are therefore bound to be found in terms one and two of the first sum. Under conditions of two-photon resonance, it is consequently to be expected that all other terms in the somewhat daunting expression in eq 31 are far less important and reduced expressions may be adopted in the calculation without significant loss of accuracy (corresponding

(31)

to the rotating wave approximation adopted to the linear response equation). From a physical point of view, the real and imaginary parts of complex response functions of the above form will have quite different interpretations and correspond to dispersive and absorptive spectroscopies interrelated by the Kramers−Kronig relations.79 Prominent examples of such interrelated spectroscopies are provided in Table 2. If we consider the calculation of Table 2. Real and Imaginary Parts of a Selection of Response Functions Involving Electric and Magnetic Dipole Operators and Their Respective Connections to Spectroscopies response function

real part

imaginary part

⟨⟨μ̂ ;μ̂⟩⟩ω

refractive index, polarizability

⟨⟨m̂ ;μ̂⟩⟩ω

electronic circular dichroism

⟨⟨μ̂ ;μ̂,m̂ ⟩⟩ω,0 ⟨⟨μ̂ ;μ̂,μ̂,μ̂ ⟩⟩ω,−ω,ω

magnetic circular dichroism intensity-dependent refractive index

vis/UV/X-ray absorption optical rotatory dispersion Faraday rotation two-photon absorption

linear absorption spectra, then the cross section takes the form79 ω σ (1)(ω) = Im[α̅ ( −ω; ω)] ε0c (32) which is the correspondence of eq 11 in the electric-dipole approximation. The overline on the polarizability indicates the isotropic tensor average as representative for a randomly oriented sample, whereas individual tensor components may be considered in regard with oriented samples. In the complex polarization propagator (CPP) approach, the simulation of a 7218


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Figure 5. Characteristic CPP linear response equation solver convergence (left) and resulting linear absorption spectrum (right). Bars depict the number of unresolved equations, and black squares show the number of trial vectors added in a given iteration. After 14 iterations all 100 response equations are converged and the resulting response function values are shown as triangles in the right panel (interpolation of data points is shown as green solid line). This example calculation refers to C60 at the B3LYP/Sadlej-pVTZ level of theory and is taken from ref 110.

of the intensity-dependent refractive index (IDRI) optical process.

linear absorption spectrum thus amounts to the calculation of the imaginary part of the electric-dipole polarizability, or generalized polarizability if eq 11 is adopted, over the spectral range, or energy window, of interest. This puts the treatment of UV/vis and X-ray spectroscopies on equal footing from a computational perspective and amounts merely to changing the user-defined energy window. Thus, spectral features inside this energy window are resolved on a grid of frequencies, each spawning a linear response equation to be solved. With a window width of some 10 eV, one needs to solve around 100 equations to obtain a spectral resolution that is sufficiently high to enable a comparison against high-resolution X-ray absorption spectra, as illustrated in Figure 5. The response equations are solved with an efficient iterative algorithm using a common reduced-space representation of the whole set of response equations.110,111 The wall time of the calculation of the spectrum depends almost exclusively on the number of iterations in the linear response equation solver, which in prototypical applications remain at the same level as in the case of performing a conventional Davidson scheme for finding the lowest few eigenstates and eigenvalues of the secular matrix. The key feature of the CPP approach to be stressed once more is that it provides a uniform computational treatment of spectroscopies regardless of the frequency of the external radiation. Once the complex response functions in eqs 29, 30, and 31 have been implemented for a given electronic structure theory method, then this method can immediately be employed for simulation of molecular responses in the X-ray spectral region without needing to consider the fact that the electronic transitions of interest are represented by high-lying roots of the generalized eigenvalue equation, embedded in a pseudorepresentation of the continuum. A striking example is the direct access to X-ray two-photon absorption cross sections from the relation112 σ (2)(ω) =

ℏω 2 Im[γ ̅( −ω; ω , −ω , ω)] 4ε0 2c0 2

3.3. Core−Valence Separation (CVS) Approximation

The diagonalization of an approximate Hamiltonian matrix is typically performed using iterative diagonalization schemes, for example, the Davidson algorithm, and thereby providing a description of the energetically lowest eigenvalues in the spectrum.113 This technique has proven remarkably successful in the calculation of valence-excited electronic states and the corresponding UV/vis spectra. However, it makes the calculation of core-excited states tedious, because core-excited electronic states are located in the high energy X-ray region of the electronic spectrum (see Figure 6) and one has to compute

Figure 6. Schematic representation of the structure of the shifted electronic Hamiltonian matrix, MIJ = ⟨ΨI|(Ĥ − E0)|ΨJ⟩. The couplings (black triangles) between core-excited (CE, red) and valence-excited (VE, blue) states are generally very small but prevent the direct diagonalization of the core-excited space.

all excited states energetically below the core excitations. This implies a large computational effort that makes the treatment of systems with more than 10 electrons or so unrealistic. A solution would be a direct diagonalization of the coreexcited space only, which, however, is in principle rigorously not possible due to couplings between valence- and core-

(33)

which will be discussed in further detail in section 4.5. Also here the overline refers to the isotropic tensor average and in this case the relevant tensor is the second-order hyperpolarizability 7219


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Figure 7. In the CVS approximation, the coupling between the singly core-excited states and the singly valence-excited state as well as the doubly core-excited states is exactly set to zero (light blue), and thus the full matrix shrinks to a CVS matrix of reduced dimension containing exclusively core-excited states.

excited115 electronic states. The same approximation has been introduced in STEX calculations,116,117 but in this context, it has been referred to as the restricted-channel approximation.117,118 Stener et al. and Ray et al. used an index restriction corresponding to the CVS approximation within TDDFT.119,120 Lopata et al. introduced the restricted-excitation window (REW) approximation to TDDFT by restricting the single-excitation space to core-excited states only, which however is again essentially identical to the CVS approximation.57 In the context of wave-function-based or propagatorbased ab initio methods, the CVS approximation has been applied to the algebraic diagrammatic construction scheme for the polarization propagator up to third order listing ADC(2), ADC(2)-x,121−124 and ADC(3)25 and to coupled-cluster approaches including CC2, CCSD, CC3, and CCSDR(3).126,127

excited states (Figure 6). However, these couplings are usually very small due to the strong energetic separation of valenceand core-excited states and the core orbitals are strongly localized in space. Therefore, these couplings can typically be neglected, which corresponds to the so-called core−valence separation approximation (CVS) to the electronic Hamiltonian matrix. As a consequence, the following types of Coulomb integrals practically vanish and are set to exactly zero within the CVS approximation: ⟨Ip|qr ⟩ = ⟨pI |qr ⟩ = ⟨pq|Ir ⟩ = ⟨pq|rI ⟩ = 0 ⟨IJ |pq⟩ = ⟨pq|IJ ⟩ = 0

(34)

⟨IJ |Kp⟩ = ⟨IJ |pK ⟩ = ⟨Ip|JK ⟩ = ⟨pI |JK ⟩ = 0

where capital letters I, J, and K refer to core orbitals and lowercase letters p, q, and r refer to valence orbitals. Neglect of these coupling integrals separates the singly core-excited states exactly from the valence-excited and doubly core-excited ones. Therefore, only elements found in matrix blocks MIa,Kc (particle (p)−hole (h), p−h), MIjab,Kc (2p−2h, p−h), MIa,Klcd (p−h, 2p− 2h), and MIjab,Klcd (2p−2h, 2p−2h) need to be considered, where j and l represent occupied valence orbitals and a, b, c, and d represent virtual orbitals. As a consequence, the matrix M, representing the shifted electronic Hamiltonian (Ĥ − E0) of some approximate-state method such as CIS, RPA, TDDFT, ADC, or CC is effectively reduced to take into account singly core-excited states only, and a diagonalization of this reduced matrix leads to significant computational savings compared to conventional approaches; see Figure 7 for an illustration. The CVS approximation is generally applicable, and can in principle be applied to any excited state method. For methods with large excitation spaces containing doubly or even triply excited states, the dimension of the matrix to be diagonalized shrinks substantially and the computational effort decreases drastically. Since only core-excited states are included in the matrix M, the core-excited states are now obtained as the lowest eigenvalues of the matrix. Hence the conventional iterative Davidson diagonalization scheme can be adopted without need for any additional changes to the existing computer programs besides the adaptation of the matrix elements in accordance with Figure 7. The CVS approximation was introduced by Cederbaum et al. as early as 1980 in the context of core-ionized114 and core-

3.4. Density Functional Theory (DFT) Based Approaches

The most widely used quantum chemical methods in all areas of chemistry and physics are Kohn−Sham density functional theory (KS-DFT) for the electronic ground state128 and linearresponse time-dependent DFT for excited electronic states.92,101,129 Although KS-DFT is in principle a formally exact theory had the exact exchange−correlation (xc) functional been known, it functions in practice like a semiempirical method since an appropriate approximate xc-functional has to be chosen for the application at hand. Not surprising, the quality of DFT and TDDFT calculations of core-excited states therefore varies largely depending on the chosen flavor of xcfunctional. But we are not going to enter into a discussion of finding the most appropriate xc-functional for the calculation of X-ray spectroscopies, as this issue is largely application dependent and has already been discussed extensively in the literature; see, for example, refs 130 and 131. Instead, we are going to limit the discussion to more fundamental issues of the prominent DFT-based methods that are used for the calculation of core-excited states. 3.4.1. ΔSCF and Extensions. A viable approach to the computation of core-excited states is to tweak a ground-state method to converge onto the higher-energy solution and to take the difference between the total energies of the ground state and the excited state to obtain excitation energies. n 0 ℏωex = ΔEtot = Etot − Etot

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(35) DOI: 10.1021/acs.chemrev.8b00156 Chem. Rev. 2018, 118, 7208−7248

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Hartree−Fock exchange operator, it will lead to the SIE given in eq 36. Virtual orbitals will consequently not see an SIE in accordance with that for occupied orbitals. Instead, Imamura and Nakai defined an SIE for virtual orbitals in the context of electronic transitions and which depends on the occupied core orbital from which an excitation takes place.146

By setting constraints on the occupation numbers of orbitals of, for example, different irreducible representations of the molecular point group symmetry, it is generally possible to converge an SCF procedure, Hartree−Fock or DFT, onto the lowest solution of each irreducible representation. These solutions correspond to local minima in the SCF parameter space and have been shown to indeed represent physical excited states.132 Already in 1992, Schmitt and Schirmer have tested the suitability of relaxed core-hole Hartree−Fock references for the calculation of core-excited states.133 If desired, suitable postHartree−Fock methods can be used afterward, which however have scarcely been applied in the context of core-excited states.134,135 In the following, we will thus focus on ΔDFT and variants thereof, since DFT includes dynamic electron correlation required for core-excited states by virtue of the xc-functional and ΔDFT provides an excellent compromise between accuracy and computational effort. To force convergence of DFT to higher-energy electronic states becomes more involved when the molecule under consideration has no symmetry or one is interested in higher excited states of a certain irreducible representation, which is generally the case when core-excited states are addressed. But also here, it is possible to obtain the lowest state of a particular class of core-excited states, e.g., from C(1s) of one particular carbon atom by starting the SCF procedure with an appropriate initial guess with the corresponding core hole. Using some sort of root-homing algorithm,5,136,137 for instance, the maximumoverlap-method (MOM),138 or replacement of all equivalent cores by an effective core potential except for the targeted atom,131 enables the convergence of the SCF procedure to the desired energetically lowest C(1s) core-excited state as the overlap between core and valence orbitals is very small and prevents a variational collapse to the electronic ground state. Alternatively, undesired core states can be localized and frozen so that they cannot mix with the targeted core-hole state. Thereby also a sequence of variationally determined mutually orthogonal core-excited states of the same symmetry can be generated.139 Although this is rather straightforward for smaller molecules, it becomes cumbersome and tedious for large molecular systems. The same applies to core-ionized states, which can be computed by the ΔDFT approach and serve as input for other methods used for calculating core-excited states; see sections 3.4.2 and 3.5.1. DFT-based methods generally suffer from self-interaction error (SIE), which corresponds to the sum of Coulomb and exchange self-interactions that remain because of the use of approximate exchange functionals. In more detail, the Coulomb self-repulsion of each electron included in the Coulomb operator is exactly canceled by the nonlocal exchange selfattraction in Hartree−Fock theory, but this is no longer the case when the exchange operator is replaced by an approximate exchange functional.128 For occupied orbitals, the SIE is defined as140 ΔϵSIE = ⟨ii|ii⟩ − ⟨i|vxc(r)|i⟩ i

ΔϵSIE a (i → a) = −c HF⟨ia|ia⟩ + ⟨ia|ai⟩ + (1 − c HF)⟨ia|fxc |ia⟩ (37)

where the first term in the right-hand side of eq 37 describes the electron−hole attraction, the second describes the electron−hole exchange, and the remainder represents the contribution of the xc-kernel given by f xc = δvxc/δρ. In chargetransfer excited states within linear-response TDDFT, this SIE had previously been coined as the electron-transfer selfinteraction error as it describes the spurious interaction of the excited electron in virtual orbital a with itself still being in the previously occupied core orbital i.147,148 It is, however, not present in ΔDFT as, in this case, the orbital a is treated like an occupied orbital. Instead, as will be seen in section 3.4.4, the SIE in eq 37 dominates the results of TDDFT calculations of core-excited states as it prevents the excited electron from “feeling” the positive ion core and relax accordingly. Imamura and Nakai evaluated eq 37 for the π*-orbital in CO and found values of −14.89, −13.28, and −7.91 eV at BLYP, B3LYP, and BHHLYP level of theory,146 respectively; i.e., these SIEs are 1 order of magnitude larger than those of the core orbitals and they are strongly dependent on the amount of exact Hartree− Fock exchange in the xc-functional. Using the BLYP, B3LYP, and BHHLYP xc-functional in ΔDFT as well as ΔHartree−Fock, the excitation energy of the lowest C(1s) → π* core-excited state of CO is calculated to be 286.36, 286.47, 287.02, and 287.14 eV,146 respectively, which agrees surprisingly well with the experimental value of 287.4 eV in view of the associated low computational effort. However, using single determinant KS-DFT, the obtained core-excited singlet state is given as a 50:50 mixture of the true singlet and triplet states. Since the exchange interaction between the remaining core electron and the excited electron is usually negligible, it is not problematic in most cases, but for the C(1s) → π* of CO, in particular, it is quite large and should be accounted for. In general, the triplet can be described as a single determinant and by computing both singlet and triplet states one can extract the correct singlet excitation energy, which is discussed by Takahashi and Pettersson.131 SIE is of course not the only source of error in calculations of core-electron binding and excitation energies at the ΔDFT level, and the influence of DFT xc-functionals has been extensively investigated by Takahashi and Pettersson.131 However, since the SIEs of core orbitals are small in comparison with core excitation energies, on the other hand, ΔDFT results are more accurate and less sensitive to the employed xc-functional than those of TDDFT. Besley et al.134 have evaluated the performance of ΔDFT/ B3LYP for some 40 core-excited states of first- and second-row elements by performing a comparison against TDDFT/B3LYP, ΔMP2, and experimental results. It turned out that ΔDFT/ B3LYP and ΔMP2 give rise to similar accuracies with meanabsolute errors of 0.5 and 0.7 eV for first-row elements and 1.5 and 1.8 eV for second-row elements, respectively, and both approaches clearly outperformed TDDFT/B3LYP when it

(36)

and usually SIE is larger in locally compact core orbitals than in spatially extended valence orbitals. Imamura and Nakai have determined the ground-state SIE of the C(1s) orbital of CO and shown that it amounts to 0.76, 1.30, and 0.40 eV at the BLYP,141,142 B3LYP,143,144 and BHHLYP145 levels of theory, respectively.146 The source origin of this SIE in the SCF approximation is the Coulomb operator with its sum over occupied orbitals, and, without a cancellation from the full exact 7221


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principle, it is based on the idea that the excited electron can be separately treated from the core hole, which is the basic assumption underlying eq 2 and which is also physically reasonable. TP-DFT is an approximation of Slater’s transitionstate method163,164 by neglecting the half excited electron and using the half-occupied core as a potential for the excited states. It can be employed using different xc-functionals, and it has the desirable feature for the user to obtain all core-excited states in one calculation, similar to the static-exchange (STEX) approach; see section 3.5.1. Slater’s ground-state Xα method corresponds to a KS-DFT calculation with a statistical exchange−correlation potential that includes an undetermined factor α, which, in turn, is determined by the criterion that the resulting total energy corresponds to the Hartree−Fock energy.165 In analogy to ΔDFT, two individual SCFs would have to be performed to obtain a ΔXα value for an excitation energy, but, as a circumvention, the so-called transition-state method has been devised. The key idea is here to define a transition state for the excitation process as an artificial state, in which only a half electron is excited into a virtual orbital, while the other half remains in the originally occupied one. It is assumed that the obtained orbitals represent the initial as well as final state appropriately, and as a consequence, the difference of the oneparticle energies of the half-occupied orbitals corresponds to the excitation energy correctly up to second order.162,164 The transition state theory applies to any DFT functional and also to HF and is not restricted to the initially applied Xα statistical exchange−correlation potential. In TP-DFT, the idea of transition state theory is exploited to create a suitable reference state for core-excited states by performing a DFT calculation for a partially core-ionized state or, in other words, for a fictitious model state in which the core orbital is only partially occupied. Full removal of the core electron to create a fully core-ionized state has generally proved to lead to too strongly contracted reference orbitals.162 Within TP-DFT the degree of partial occupation of the core orbital is not rigorously defined and gives some freedom of choice for tuning the amount of orbital relaxation to properly simulate the finally desired core-excited state. The fractional occupation number can thus be seen as a fitting parameter of the fictitious molecular model system. For water, it has been shown that the half-occupied core (HCH) approach strikes the best balance, in accordance with Slater’s transition state Xα method.162,166 The resulting molecular orbitals of the partially ionized molecular model system are used as a basis in which to represent the Hamiltonian of the N-electron system, thereby providing a potential for the excited electron. A diagonalization of this Hamiltonian matrix yields a representation of all excited states. It can provide a good approximation for core-excited states as long as a suitable fractional occupation number has been chosen in the initial DFT calculation. TP-DFT includes in principle all interactions between the excited electron and the frozen molecular ion core, the density of which is, however, kept frozen and not allowed to relax when interacting with the excited electron. Since TP-DFT is a single determinant theory, the approach does not distinguish between singlet and triplet states. To correct for this missing relaxation of the ion core and to obtain a corrected absolute energy scale, additional energy corrections need to be introduced. Errors originating from the chosen xc-functional, from relativistic effects, or from basis set incompleteness can also be accounted for.131 Relaxation effects

comes to the determination of core excitation energies in absolute terms.134 Nevertheless, in simulating core excitation spectra, an individual DFT calculation has to be performed for each relevant core-excited state, followed by the calculation of the associated transition moments. It turns out that there are problematic issues involved with both these steps. First, it becomes technically more and more involved, sometimes impossible, to converge the SCF procedure to the desired coreexcited state the higher they lie above the lowest core-excited state of its symmetry class, and, in addition, the core-excited states need to be known prior to their computation. Second, wave functions (or Kohn−Sham determinants) obtained by individual and separate SCF procedures are generally not orthogonal, making the calculation of transition moments and other transition properties questionable, or at least not straightforward. Moreover, unrestricted SCF solutions of core-excited states, which have an open-shell electronic structure, are usually also spin-contaminated; i.e., they represent mixtures of singlet, triplet, and higher spin states. However, it has been shown for valence-excited states that the resulting errors are small,138 and the same is likely to be true for coreexcited states. For smaller molecules, up to 5−10 variationally determined orthogonal core-excited states can be routinely obtained,149 and the core-excited states need not be known prior to their computation but can be successively constructed.139 For larger molecules with several identical atoms, these approaches become very tedious and cumbersome. To circumvent the issue of nonorthogonality, Evangelista et al. developed an orthogonality-constrained DFT (OCDFT)150,151 approach which is a particular form of constrained DFT,152−154 allowing for the determination of electronic excitation energies within the framework of a timeindependent variational formulation of DFT. Within OCDFT, the minimization of the excited-state Kohn−Sham functional is equivalent to a ground state DFT calculation augmented with orthogonality constraints with respect to the ground-state Kohn−Sham determinant. Derricotte and Evangelista have further extended OCDFT to a multistate formulation,155 which provides excited states in subsequent DFT calculations orthogonal to all previous states. They have tested this approach for the calculation of core-excited states and, not surprisingly, obtained a practically identical accuracy as conventional ΔDFT with a mean absolute error of 0.4 eV for first-row elements and 1.6 eV for second-row elements, when the B3LYP xc-functional is used. OCDFT is clearly superior to TDDFT in absolute accuracy of vertical core excitation energies.155 OCDFT can also be seen as a methodological link between ΔDFT and the constricted variational DFT (CV-DFT) approach provided by Ziegler et al.156−159 and in which excited-state Kohn−Sham determinants are variationally optimized under the constraint that a full electron is moved from the occupied to the virtual orbital space. Several singly excited states are obtained in one CV-DFT calculation and their orthogonality is ensured, as it has a very similar numerical structure as TDA/TDDFT; see section 3.4.4. Due to its close relation to ΔDFT and OCDFT, one can expect CV-DFT to exhibit a similar accuracy for core-excited states, but the method has yet to be benchmarked in this context. 3.4.2. Transition-Potential DFT. Transition-potential DFT (TP-DFT) is a methodological approach specifically designed to simulate X-ray absorption spectra.160−162 In 7222


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are easiest taken care of by performing a corresponding ΔDFT calculation for the lowest core-excited state, as described in section 3.4.1, and shifting the calculated spectrum accordingly. The remaining differences between the calculated and experimental core excitation spectra are mainly associated with the core level. Hence, it is possible to estimate them as the difference between a calculated and experimental core-electron binding energy (CEBE), which can also be added as a constant shift to the previously ΔDFT-shifted TP-DFT spectrum,162 to achieve an even better agreement between experimental and simulated X-ray absorption spectra. 3.4.3. Real-Time TDDFT. Time-dependent Kohn−Sham DFT or, equivalently, real-time, time-dependent densityfunctional theory (RT-TDDFT) is the time-dependent analogue of ground-state Kohn−Sham DFT. It relies on a one-to-one mapping theorem of the time-dependent oneelectron density and the many-body time-dependent wave function, which was first proven by Runge and Gross in 1984.129 The resulting time-dependent Kohn−Sham equation takes the form iℏ

KS ∂ ϕ(r , t ) = fi ̂ (r , t ) ϕi(r , t ) ∂t i

due to the faster oscillations in this region of the spectrum. However, if one is exclusively interested in core excitation spectra the total propagation times can be significantly shortened. The same authors employed the CVB3LYP functional within RT-TDDFT and linear-response TDDFT. This functional has been specifically designed for the determination of both core- and valence-excitation energies with high accuracy, and it has been shown to provide accurate core excitation energies of 1s orbitals of several second-row elements.42 Lopata et al. employed RT-TDDFT to simulate core-level near-edge X-ray absorption spectra of a variety of different chemical systems ranging from the oxygen K-edge of water up to the ruthenium L3-edge of the hexaamineruthenium ion. The simulated spectra were shifted by −4 to 19 eV compared to the experimental values.57 3.4.4. Linear Response TDDFT. Linear-response timedependent density functional theory (TDDFT) is one of the most widely used excited-state methods, mostly for valenceexcited states of medium-sized and large molecular systems.101 Successful applications are found in many scientific areas such as photochemistry, photobiology, and materials science, and not surprisingly, TDDFT has also been widely used to study core excitation processes.175 In TDDFT, the linear response of the electron density to an external oscillating time-dependent electromagnetic field is found by means of the time-dependent evolution of a noninteracting Kohn−Sham system, which is otherwise described by the time-dependent Kohn−Sham equation; see section 3.4.3. To derive TDDFT expressions, the linear response function given in eq 21 is evaluated for a Kohn− Sham reference state and a manifold of single-electron excitation and de-excitation operators. In the limit of an infinitesimal perturbation, Casida’s equation is obtained,92 which is identical in structure and form to the secular equation in RPA, or equivalently TDHF,92,101 following the fact that their reference state parametrizations can be chosen identical:79

(38)

with occ

ρ (t ) =

∑ |ϕi(r , t )|2

(39)

i

̂ KS

The time-dependent Kohn−Sham operator f (r,t) contains the time-dependent xc-potential functional vxc[ρ(r,t)](r,t), which depends explicitly on time as well implicitly via the time-dependent electron density.129,167 Propagation of eq 38 from an appropriate initial state using numerical integration schemes, for example, the fourth-order Runge−Kutta method,168 yields the electron density trajectory, whose Fourier transform gives access to all excitation energies and the complete excitation spectrum as described in section 3.1. The initial state is usually represented by the static unperturbed electron density obtained from a ground-state DFT calculation. In all practical applications, the adiabatic local density approximation is applied so that the originally nonlocal (in time) time-dependent xc-kernel is replaced with a timeindependent local one based on the assumption that the density varies only slowly with time. This approximation allows for the use of a standard local ground-state xc-potential in the TDDFT framework. Nowadays several implementations employing different classes of basis functions exist: plane waves,169−171 numerical grids,172,173 and Gaussian basis functions.174 Repisky et al. have also recently provided a relativistic four-component implementation of RT-TDDFT.46 An explicit time propagation of the Dirac−Kohn−Sham density matrix is performed, allowing for the simulation of molecular spectroscopies involving strong electromagnetic fields and simultaneously treating relativistic scalar and spin−orbit corrections variationally. This method is ideally suited to treat core excitations in elements, for which relativistic effects cannot be covered by scalar relativistic correction schemes.46 Akama and Nakai proposed to use a short-time Fourier transform (STFT) technique to analyze density trajectories obtained from RT-TDDFT simulations using Gaussian-type basis functions to obtain not just the valence but also the core excitation energies.43 For the latter, the time steps of the numerical propagation scheme of eq 38 need to be substantially reduced

⎛ A B ⎞⎛ X ⎞ ⎛ 1 0 ⎞⎛ X ⎞ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ = ω ⎜ ⎝ 0 −1⎠⎝ Y ⎠ ⎝ B* A*⎠⎝ Y ⎠

(40)

Here, the elements of the submatrices A and B are given for hybrid exchange−correlation functionals as Aia , jb = δijδab(ϵa − ϵi) + (ia|jb) − c HF(ij|ab) + (1 − c HF)(ia|fxc |jb) Bia , jb = (ia|bj) − c HF(ib|aj) + (1 − c HF)(ia|fxc |bj)

(41)

where the two-electron integrals are given in the Mulliken notation and f xc refers to the xc-kernel that in the local adiabatic density approximation is given as the second functional derivative of the chosen ground-state xc-functional:

fxc =

δ 2Exc(r ) δρ(r )2

(42)

As a consequence, the quality of the results obtained in TDDFT calculations strongly depends on the chosen xcfunctional. For cHF = 0, i.e., xc-functionals not containing nonlocal orbital exchange, eq 40 describes the equation for pure xc-functionals whereas, for cHF = 1, it is the standard linearresponse TDHF equation. Calculations employing hybrid functionals are linear combinations of both limiting cases. When matrix blocks B in eq 40 are neglected, one obtains the 7223


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so-called Tamm−Dancoff approximation, resulting in TDA/ TDDFT or configuration interaction singles in the case of TDHF.176 For the calculation of core-excited states, linear-response TDDFT calculations are plagued by large errors in excitation energies when standard xc-functionals are used, and these errors are usually strongly xc-functional dependent and, in particular, change strongly with the amount of nonlocal orbital exchange in the xc-functional. The errors are generally attributed to the self-interaction error (SIE) in the occupied ground-state Kohn−Sham core orbitals (see eq 36) and in the virtual orbitals of excited Kohn−Sham determinants (eq 37). The latter SIE is closely related to the problem of TDDFT in combination with charge-transfer (CT) states when xcfunctionals with no or small fractions of nonlocal orbital exchange are employed. In this case, the SIE in virtual orbitals is not canceled out by the response of the local xc-kernel, as a consequence of the fact that the initially occupied orbital i does not spatially overlap with the virtual orbital a in long-range CT transitions.147,148,177 In this context, this error has previously been termed electron-transfer SIE (ET-SIE),148 and it can be related to a missing discontinuity with particle number in the approximate xc-potential.177 In core-excited states, the spatially localized core orbitals have only small spatial overlaps with the virtual orbitals, leading to the same effect as in CT states, namely that the excited electron still interacts with itself in the core orbital. From a physical perspective, the presence of this SIE prevents the excited electron from “feeling” the attraction of the positive ion core; i.e., the electron−hole interaction is simply missing. Inspecting eq 41, it becomes apparent that the response term originating from the exchange operator cHF(ij|ab) describes the electron−hole attraction. To counter this error in TDDFT calculation of core-excited states, specific xc-functionals have been developed. A first attempt was made by Imamura and Nakai by combining the modified Leeuwen−Baerends xc-functional for short- and longrange interactions with the Becke88 exchange for the valence intermediate region.178 The resulting BmLBLYP functional yielded typical errors for core-excited states of about 1.5 eV compared to over 13 eV when the standard B3LYP hybrid functional is employed.178 A further step forward was made by the development of the CV-B3LYP and CVR-B3LYP xcfunctionals, specifically designed to be accurate for all kinds of excited states including core excitations by using a statedependent amount of HF exchange. The error in core excitation energies has thereby been reduced to below 1 eV.179,180 Song et al. optimized their LCgau-DFT approach181 for the calculation of core-excited states and termed the resulting specialized xc-functional LCgau-core-BOP,182 which exhibits a mean absolute error of 0.7 eV for a selected test set of core-excited states. Verma and Bartlett demonstrated recently ionization-energy-corrected exchange and correlation potentials to yield core excitation energies of C, N, and O K-edge spectra with a mean absolute error (MAE) of 0.77 and a maximum error of 2.6 eV.183 A major step forward was taken by Besley et al. in their developing short- and long-range corrected xc-functionals for core-excited states in order to cancel the ultrashort-range SIE present in the core orbital as well as the long-range ET-SIE present in the virtual orbital.184 In these functionals, the Coulomb operator is written in the following way:

csr(1 − erf(μsr r12)) csr(1 − erf(μsr r12)) 1 = − r12 r12 r12 +

c lr erf(μ lr r12) r12

−

c lr erf(μ lr r12) r12

+

1 r12

(43)

and the first and third terms are evaluated using nonlocal orbital exchange while the remaining terms are treated with DFT. Following this procedure and using the B88 exchange and LYP correlation functionals for the DFT part and fitting the corresponding parameters, csr, μsr, clr, and μlr, the so-called short-range corrected functionals SRC1-BLYP and SRC2-BLYP have been developed.184 The mean absolute error in the excitation energies of a selected set of core-excited states of first-row atoms is substantially reduced from 12.7 and 12.9 eV for core−valence and core−Rydberg excited states at the B3LYP level to 0.3 and 0.4 eV at the SRC2-BLYP level, respectively. For second-row atoms, the improvement is even more pronounced, from a mean average error of 51.6 eV at B3LYP level to 0.4 eV at SRC2-BLYP level. For the second-row atoms a precomputed relativistic correction for the core-orbital energies was taken into account.184 In the vast majority of linear-response TDDFT calculations making direct and explicit reference to the individual coreexcited states, eq 40 is solved by iterative diagonalization using, for example, the Davidson procedure.113 For this objective, the CVS approximation as described in section 3.3 is applied, which means that an index restriction is applied to the occupied orbitals i and j in eq 41 to correspond to core orbitals. In this way, valence-excited states are excluded from the calculation and the core-excited states are obtained as the eigenvectors with lowest eigenvalues.57,119,120,185 A slightly different route to target the core-excited states is followed by Liang et al. and Lestrange et al., who employed an energy-specific algorithm to converge eigenvalues of the TDDFT linear response matrix in a selected energy range.70,186 Therefore, they could employ the full TDDFT matrix without index restriction, and hence, all singly excited Kohn−Sham determinants can contribute to the core-excited states without making any a priori selection. 3.5. Wave Function Theory Based Approaches

In recent years, substantial progress has been made in the development of ab initio methods for the calculation of coreexcited states using wave-function-based approaches. This progress comprises theoretical model chemistries based on configuration interaction (CI), Møller−Plesset (MP), and coupled cluster (CC) methods, often in combination with a response theory or polarization propagator scheme, e.g., the algebraic diagrammatic construction (ADC) approach. In addition, multireference approaches have also been employed and found particularly important in the context of core excitations in transition-metal complexes due to the multireference character of the ground electronic state. Organic molecules containing elements from the main groups of the periodic table usually possess, on the other hand, well-defined single-reference electronic ground states, and single-reference methods therefore suffice for comprehensive studies of their Xray spectra. 3.5.1. Single-Reference Methods. The conceptually most simple excited state method is configuration interaction singles (CIS).187 It builds upon the ground-state Hartree−Fock (HF) determinant Φ0 by constructing all possible singly excited determinants Φai replacing one occupied orbital ϕi with a virtual 7224


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electron in each α and β core spin−orbital. The main conclusion of this work is that the inclusion of orbital relaxation effects is crucial to observing a major gain in accuracy, but the details of how the core-excited reference is generated have only a minor influence. For larger molecules, the CVS approximation is (as expected) proven to have only a negligible influence on the results.135 Asmuruf and Besley have tested the CIS(D) and the semiempirical spin-opposite-scaled (SOS) CIS(D) approaches for the description of core-excited states using the CVS approximation for the underlying CIS calculation.190 The CIS(D) method incorporates a perturbative doubles correction to the CIS excitation energies, which has been demonstrated to lead to an improved agreement with experiment.191,192 SOSCIS(D) is an extension of the SOS-MP2 method, in which the opposite-spin components of the energy are scaled with semiempirical factors yielding generally an improved performance compared to CIS(D).193 However, core excitation spectra computed with CIS(D) are often qualitatively incorrect with Rydberg states positioned too low in energy. A specialized SOS-CIS(D) version, denoted cSOS-CIS(D), taking only the direct perturbative correction to the excitation energy into account, leads to improved theoretical NEXAFS spectra, and can provide a reliable basis for the interpretation of experimental data.190 When it is combined with the CVS approximation, the algebraic diagrammatic construction (ADC) scheme for the polarization propagator provides an elegant route to obtain core excitation energies.121,122,125,194 The ADC scheme for the polarization propagator was originally introduced by Schirmer in 198284 based on the Green’s function formalism, and it is today available up to third order in perturbation theory.107,195−199 The ADC secular equation corresponds to a Hermitian CI-type eigenvalue equation of the form

orbital ϕa. Hence, the excitation space comprises nocc × nvirt singly excited determinants. Since the HF ground state is orthogonal to the single-excitation space by virtue of the Brillouin theorem, the HF ground state energy can be subtracted from the electronic Hamiltonian, and the corresponding shifted Hamiltonian is represented in the basis of singly excited determinants. This yields the CIS equation Ax = ωx

(44)

with matrix elements Aia , jb = δijδab(ϵa − ϵi) + (ia|jb) − (ij|ab)

(45)

The two-electron integrals are given in Mulliken’s notation. This is also identical with eq 41 after invoking the Tamm− Dancoff approximation and setting cHF = 1. In general, the same accuracy can be expected with CIS for the excited states as with Hartree−Fock for the ground state.101 For core-excited states, however, a standard CIS approach starting from the electronic ground state as reference is not well-suited for the problem at hand and it is clear that what is missing are the double-excited determinants needed to describe the large orbital relaxation effects that are generally present in core-excited states. A viable pathway to circumvent the problem with the limited orbital relaxation effects is to use a reference state that already takes into account the orbital relaxation due to the core hole. This idea was put into practice in 1994 by Ågren and coworkers in their development of the static exchange (STEX) method.116−118,188 It is recognized, however, that the STEX approach is closely related to the approach of Decleva et al.189 that had been presented a few years earlier. Within STEX, an initial Hartree−Fock calculation is performed for the coreionized state with (N − 1) electrons to capture the relaxation of the molecular orbitals due to the core hole. The resulting molecular orbitals are then used to set up the CIS Hamiltonian of the N-electron system, and this so-called STEX matrix is to be diagonalized within the so-called restricted channel approximation or, equivalently, the CVS approximation, in which excitations are restricted to originate from the selected core orbital(s) belonging to the atomic center of interest. This gives access to core-excited states associated with excitations from an atomic shell (for instance a K- or L2,3-shell) of a single atomic center. Hence, one STEX calculation needs to be performed for each core-excited atom. Also, the relaxation (or polarization) of the valence molecular orbitals with respect to the excited electron is neglected, leading to term values that are too small. Finally, the excitation energies of the core-excited states are eventually obtained by setting the lowest eigenvalue of the STEX matrix to the value of the separately computed ΔSCF value for the core-ionization potential. The suitability of core-relaxed Hartree−Fock orbitals for the calculation of core-excited states was tested already in 1992 by Schmitt and Schirmer in their exploiting of the so-called relaxed-core Hartree−Fock approximation.133 Recently, Ehlert and Klamroth performed a comprehensive study of the influence of different orbital references for use in CIS calculations of core-excited states.135 These authors focused on the exploitation of different core-excited reference states, all designed to capture orbital relaxation effects at the SCF level. In their core-hole reference (CHR) CIS schemes, they used, for example, restricted Hartree−Fock and Kohn−Sham orbitals, core-excited restricted open-shell Hartree−Fock or Kohn− Sham orbitals, and core-excited unrestricted Hartree−Fock orbitals with the hole in the α-core spin−orbital or with half an

MY = ΩY

(46)

in which the ADC matrix, M, is the representation of the electronic Hamiltonian shifted by the corresponding Møller− Plesset ground state energy to order n, EMPn 0 : MIJ = ⟨ΨĨ |Ĥ − E0MPn|ΨĨ ⟩

(47)

and |Ψ̃n⟩ are the so-called intermediate states85 that form an orthogonal basis constructed by formal excitation of the corresponding MPn ground-state wave function and subsequent orthogonalization. Alternatively, employing the MPn reference state in the ISR formalism results in the same ADC(n) equations.107 The hierarchy of ADC schemes offers not only the advantages of leading to Hermitian eigenvalue equations and, at the same time, being size-consistent and size-intensive,84,107 but also they readily provide access to excited-state properties via the ISR formalism.85 The operator corresponding to the desired excited-state property, e.g., the electric-dipole operator, needs to be represented in the known ISR basis, DIJ, and it can thereafter be contracted with eigenvectors of the ADC matrix according to ⟨μ⟩̂ IJ = YI†DYJ

(48)

For the calculation of core-excited states, the CVS approximation needs to be applied to avoid the full diagonalization of the otherwise huge ADC matrix, giving rise to the so-called CVS-ADC(2), CVS-ADC(2)-x, and CVS7225


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ques, to efficiently obtain high-energy solutions without scanning through low-energy states.215 3.5.2. Multireference Methods. Multiconfiguration selfconsistent field (MCSCF) or multireference configuration interaction (MRCI) methods216,217 are generally required for the simulation of X-ray spectroscopies when the underlying electronic ground state exhibits multireference character. This is typically the case when, for example, L-edge absorption spectra of transition-metal complexes are studied, or when the electronic ground state of an open-shell organic molecule is near-degenerate. In such situations, a balanced description of dynamic and static electron correlation becomes imperative to find. While the first is already captured by advanced singlereference methods, the latter requires a multireference treatment. Multireference (MR) configuration interaction (CI) methods constitute one of the conceptually oldest approaches to multireference problems.218 Herein, the main configurations of a CI reference ground state are expanded individually in truncated CI expansions and collected in one huge MRCI matrix making these methods computationally demanding. Although MRD-CI calculations for the core−valence excited states of N2 have been reported already in 1977,219 the method has scarcely been used to investigate X-ray spectra. In these early calculations, Butscher et al. used a root-homing algorithm to converge the iterative diagonalization of the MRD-CI calculation to the desired core−valence excited states. Recently, Coe and Paterson adapted the method of Monte Carlo configuration interaction to calculate core-excited states and used this to simulate X-ray absorption and emission spectra.220 Their calculations for a set of small molecules up to the size of methanol with strong multireference character should provide useful data for improving approximations in methods for larger systems such as time-dependent density functional theory and single-reference wave function theories. A different multireference scheme based on a CI expansion and including a second-order energy correction has been proposed by Lisini and Decleva and has been termed as the QDPT-CI scheme.221 Later, it was employed to investigate the XAS spectrum of Cl2.222 For the QDPT-CI calculations of the core-excited states, optimized orbitals obtained for the appropriately core-ionized electronic ground state have been used. To obtain correct absolute XAS energies, the corresponding core-ionization potential needs to be added to the calculated excitation energies at the QDPT-CI level, similar to what is done in the previously described STEX method in section 3.5.1. An alternative multireference approach is provided by the multiconfiguration SCF family of methods216,223 comprising the complete and restricted active space self-consistent field, CASSCF224,225 and RASSCF,226 methods as well as their second-order perturbation-theory corrected variants CASPT2227,228 and RASPT2.229,230 The common denominator of all these methods is the a priori choice of the molecular orbitals relevant to the desired excitation process and which therefore need to be included into the CAS/RAS space, and the choice of state(s) for which the orbitals are to be optimized during the calculation (more than one state at a time is considered in state-averaged optimizations). In CAS calculations, active and inactive orbital spaces are defined, and in the former a full CI calculation is performed whereas the inactive orbitals are treated only at the SCF level. In RAS calculations, the active space is divided further into RAS1, RAS2, and RAS3

ADC(3) models. Out of these three methods, CVS-ADC(2)-x provides the most accurate results for core-excited states of second-row elements due to a favorable and stable error compensation of a combination of missing orbital relaxation, electron correlation, and relativistic effects.125,200 In addition, the ADC scheme has recently been extended to the complex polarization propagator formalism201,202 allowing for the calculation of static polarizabilities and C6 dispersion coefficients and RIXS scattering amplitudes; see further section 4.4. In recent years, equation-of-motion and linear-response coupled-cluster approaches have also been developed to investigate core-excited states. In brief, coupled-cluster methods for the calculation of excited states rely on a formulation either via the equation-of-motion formalism resulting in the EOM-CC schemes,203−205 or via linear-response theory giving LR-CC schemes.206−208 The excited-state wave functions are in both schemes parametrized according to ̂ Ψn = Rê T Φ0

(49)

T̂

in which e Φ0 corresponds to the coupled-cluster ground state while the linear excitation operator R̂ creates excited states. With this ansatz, both the EOM and LR formalism lead to a matrix-eigenvalue equation H̅ ΨkR = ℏωk ΨkR

(50)

in which H̅ is the similarity-transformed Hamiltonian and ΨRk and ℏωk are the corresponding kth right-handed eigenvector and eigenvalue, respectively. Note that also a left-handed eigenvector ΨLk exists due to the non-Hermiticity of H̅ , which is generally needed for the calculation of ground-to-excited-state transition moments and excited-state properties. Equation 50 therefore needs to be solved twice, once for the right-handed and once for the left-handed eigenvectors. The operators T̂ and R̂ determine the excitation level of the corresponding coupledcluster method, e.g., with both including single- and doubleexcitation operators, the EOM-CCSD or LR-CCSD schemes are obtained. Although the energy expressions are identical for EOM- and LR-CCSD, the transition moments differ, and while they are not size-intensive in the EOM-CC formulation, in LRCC they are.209 In the context of X-ray spectroscopies, Nooijen and Bartlett applied the EA-EOM-CCSD approach in connection with a core-ionized reference state to capture most orbital relaxation effects already at the SCF level to calculated core-excited states of a selected set of small molecules.210 Therefore, they used an open-shell variant of EA-EOM-CCSD and had to prevent the reoccupation of the core orbital in the EOM-CCSD treatment by index restriction.210 Coriani et al. adapted the hierarchy of CC methods to adopt the complex polarization propagator framework and thereby allowing for the calculation of coreexcited states and X-ray absorption spectra; see section 4.1.211,212 Most recently, Coriani and Koch implemented the CVS approximation within several different CC models comprising CCS, CC2, CCSD, CC3, and also CCSDR(3) demonstrating its accuracy and robustness.126,127 Using the CVS-CC methods, a multilevel CC approach for core excitation energies has also been provided recently.213,214 A different approach to core-excited states using CC theory has been taken by Peng et al., who have developed an energy-specific nonHermitian Davidson eigensolver, facilitated by the energyscreening, eigenvector-bracketing, and growing window techni7226


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core excitation spectra of molecules with a suitably chosen model space containing core-excited configurations.239 Unlike standard single-reference CC methods, FSMRCC does not require any special technique to reach convergence in singles and doubles approximation and it gives a performance similar to that of EOM-CCSD.239 However, due to the enormous computational effort involved with MR-CC calculations, this method is unlikely to become relevant for real-life physicochemical or biophysical applications.

subspaces and the number of maximum holes in RAS1 and maximum electrons in RAS3 is fixed. Orbital optimization is in CAS/RAS approaches necessary, since the limited excitation spaces of CAS and RAS calculations do otherwise not provide sufficient flexibility to allow for proper orbital relaxation. Although these methods are widely used for the investigation of valence-excited states and UV/vis photochemistry, they are more scarcely applied in X-ray spectroscopies. Alagia et al. used CASSCF to investigate the soft X-ray absorption spectrum of the allyl free radical, which possesses some multireference ground-state character due to its electronic open-shell structure.231 The key to successfully converging the CASSCF calculations to the desired C(1s) valence excited states is to constrain the occupation number of the 1s orbital to one, i.e., by fixing the core hole. Recently, RASSCF has been applied to study metal K- and L-edges,232,233 and it was demonstrated by Pinjari et al. that the RAS approach can be used to accurately simulate experimental L-edge XAS spectra of first-row transition-metal complexes without the use of any fitting parameters.232 They managed to converge the RASSCF wave function to states at the iron L-edge by including the 2p orbitals in the RAS1 space and by setting the maximum hole number in RAS1 to one. Considering the molecular point group symmetry, the Fe(2p) valence excited states are obtained as the lowest states in a particular irreducible representation. Overall a deviation of 1 eV in peak positions, 30% for the relative intensity of major peaks and 50% for minor peaks, was obtained. Using a similar computational setup, Guo et al. studied the RIXS spectra of ferrous and ferric hexacyano iron.233 Slightly later, Kunnus et al. extended the investigation to also include second-order corrections to account for dynamic electron-correlation via the RASPT2 scheme.234 Thereby, the donation and back-donation interactions in ferric and ferrous hexacyanide aqueous solutions could be quantified with unprecedented detail, which demonstrates the potential of soft X-ray RIXS for detailed studies of local chemical bonding of functional transition-metal compounds. A third class of multireference approaches are the multireference coupled cluster (MR-CC) methods,235,236 in which the exponential single-reference ansatz of CC theory is generalized in a multireference formulation. Unfortunately, the formulation of CC theory in a multireference setup is much less straightforward than in the standard single-reference case. In principle, two families of MR-CC approaches exist: multistate and state-specific formulations.235 In the context of core-excited states, state-specific formulations are clearly advantageous as the desired core-excited state can be addressed specifically. In this context, Brabec et al. have employed iterative statespecific MRCC methods (SS-MRCC) to study core excitations of water and N2, and they have shown that SS-MRCC with single and double excitations are comparable in accuracy to high-level single reference equation-of-motion coupled cluster (EOMCC) results. The SS-MRCC calculations are converged to the desired core-excited state by choosing a corresponding core-excited reference space.237 Sen et al. used different variants of MR-CC for the calculation of core excitation spectra and obtained promising results for a set of very small test molecules. However, these state-specific MRCC approaches always require two separate calculations for the ground and the core-excited states.238 In contrast, Fock space multireference coupled cluster (FSMRCC) yields excitation energies directly, and Dutta et al. have demonstrated that FSMRCC is able to accurately describe

3.6. Visualizing and Characterizing Core-Excited States

It has been emphasized already several times that orbital relaxation effects play an important role in the electronic structure of core-excited states, and an accurate treatment of these by means of our adopted quantum chemical models is necessary for finding a physically correct description of the underlying electronic transition processes in X-ray spectroscopies. While their importance is easily recognized in benchmark studies comparing results, e.g., excitation energies, and including quantum chemical methods that properly capture orbital relaxation, the electronic relaxation process itself is more elusive and difficult to grasp in detail. An excellent means to visualize and thus to understand orbital relaxation effects occurring in core-excited states in particular, is to investigate core excitations by virtue of the one-electron transition density matrix (1TDM), which reveals the vertical nature of the electronic transition.240−242 A given element of the 1TDM between the ground state, |0⟩, and excited state, |n⟩, is defined as241 0n Dpq = ⟨0|ap†̂ aq̂ |n⟩

(51)

â†p

where and âq are the creation and annihilation operators defined in the second quantization formalism and p and q are general orbital indices. Associated with the 1TDM, one identifies the so-called electron−hole amplitude of the polarization propagator as241 χ (rh, re) =

∑ Dpq0nϕp*(rh) ϕq(re) (52)

p,q

The interpretation of eqs 51 and 52 is in this context that the operator â†p is considered to act to the left to create a hole in orbital ϕp, whereas the operator âq acts to the right to annihilate an electron in orbital ϕq. Based on the electron−hole amplitude, one defines the hole and particle densities, respectively, as241 ρh (rh) = ρe (re) =

∫ |χ(rh, re)|2 d3re

(53)

∫ |χ(rh, re)|2 d3rh

(54)

In parallel, the one-electron difference density matrix (1DDM) should be considered and which involves the relaxed electronic structures of the ground and excited states.241 A given element of the 1DDM is defined as241 nn 00 Δpq = Dpq − Dpq = ⟨n|ap†̂ aq̂ |n⟩ − ⟨0|ap†̂ aq̂ |0⟩

(55)

i.e., the 1DDM corresponds to the difference of the two statedensity matrices and the corresponding difference-density that integrates to zero is given by ρΔ (r) =

∑ Δpqϕp*(r) ϕq(r) p,q

7227

(56) DOI: 10.1021/acs.chemrev.8b00156 Chem. Rev. 2018, 118, 7208−7248

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The detachment density ρd(r) corresponds to that part of the ground state electron density that is removed upon excitation and rearranged and added as attachment density ρa(r). Based on the matrix Δ, they are defined as follows.241 First, diagonalize Δ according to WTΔW = diag(d1, d 2 , ...)

(57)

Second, restore only the negative eigenvalues

min(di , 0) → di

(58)

Third, back-transform to find the detachment density Δd = Wdiag(d1, d 2 , ...)WT

(59)

or, alternatively ρd (r) =

∑ Δdpqϕp*(r) ϕq(r) p,q

(60)

Likewise, this recipe provides access to the attachment density ρa(r) with the only difference that the positive instead of the negative eigenvalues are restored in eq 58. Since the 1TDM describes the nature of the vertical transition and the 1DDM corresponds to the difference of the electron densities of the ground and core-excited states, a comparison of the results from these two conceptually different approaches yields insight into the nature of the orbital relaxation processes occurring in excited electronic states in general and in core-excited states in particular. In Figure 8, as an example, the natures of a selection of oxygen K-edge core-excited singlet states of cytosine are illustrated based on calculations performed at the CVS/ ADC(2)-x level of theory.194 The analysis of the hole, ρh(r), and electron, ρe(r), densities based on the exciton analysis of the 1TDM reveals the vertical character of the core−valence excitation;243−245 see the upper three panels of Figure 8. While the core−S1 state is clearly due to an O(1s) → π* transition, the core−S3 and core−S4 states correspond to core-electron excitations into Rydberg orbitals. In these hole−electron plots, the strongly localized character of the O(1s) orbital is clearly conveyed. Let us now instead turn to the plots of the detachment, ρd(r), and attachment, ρa(r), densities for the same states that are found in the lower three panels of Figure 8.241,246 These represent a decomposition of the 1DDM in parts that are detached and attached upon excitation; the final electronic structure of the core-excited states is revealed to take a strong orbital relaxation into account. Inspecting these plots, the overall nature of the electronic structure remains the same; most strikingly, however, the core hole has seemingly expanded. This expansion is due to the relaxation of the total electron density due to the strong attraction of the positive charge initially localized strictly to the oxygen nucleus. Also, the attachment densities of the O(1s)−Rydberg transitions to the final states core−S3 and core−S4 are more compact in comparison to the corresponding electron densities, which is a relaxation that is due to the attraction to the core hole. Summarizing, hole/electron plots characterize the vertical nature of the electronic transition while detachment/attachment density plots show the nature of the final core-excited state. Their comparison nicely visualizes the drastic orbital relaxation processes occurring in core-excited states. In contrast, in typical valence-electron transitions, the differences

Figure 8. Comparison of electron and hole densities with the detachment and attachment densities for the first, third, and fourth O(1s) core-excited states in cytosine. Results are obtained at the CVS/ ADC(2)-x level of theory and plotted for isodensity values in atomic units of 0.0064 (opaque), 0.0016 (colored transparent), and 0.0004 (transparent).194

when comparing results from the 1TDM and 1DDM analysis tools are hardly visible by eye inspection.

4. APPLICATIONS IN X-RAY SPECTROSCOPIES 4.1. X-ray Absorption Spectroscopy (XAS)

The spectral regions that are most rich in details are found close to the ionization edges, and they are studied in near-edge X-ray absorption fine structure (NEXAFS) spectroscopy. The name “X-ray absorption near-edge structure” (XANES) is synonymous, but the terminology “NEXAFS” is more commonly adopted in surface and molecular physics. Our discussion in this section will be focused on the development of methods in electronic structure theory that are designed to address this near-edge spectroscopy, but we note that X-ray absorption spectroscopy (XAS) also includes the study of the extended X-ray absorption fine structure (EXAFS) regions found at higher energies, above the ionization edges, and for which we refer to the review by Rehr and Albers for a treatment.247 The standard reference work for NEXAFS spectroscopy is the book by Stöhr,10 but, at the same time, we would also like to mention the tutorial review by Hähner that provides an accessible reading for novices to the field.248 The most commonly adopted starting point for calculations of XAS spectra is the expression for the transition moment in the electric-dipole approximation e 0→f MED,velocity = ⟨0|ϵ ·p̂ |f ⟩ me (61) 7228


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in line came the derivation and development of the third-order ADC model, or ADC(3). In combination with the CVS approximation, ADC(3) was applied to the calculation of coreexcited states, and it was with some disappointment that it could be concluded that the performance did not reach the same high level that had been demonstrated earlier for ADC(2)-x.125 It could thereby be confirmed that ADC(2)-x represents a sweet spot in terms of error cancellation when it comes to the calculation of XAS spectra. The intermediate-state representation (ISR) represents an alternative way to formulate and derive propagators in the ADC framework, and it was introduced for core-excited states to provide a means to get direct access to core-excited-state properties such as electron densities and dipole moments.194 In 1994, an alternative approach to the calculation of XAS spectra was developed by Ågren and co-workers.116 Rather than improving on the RPA by the introduction of dynamical electron correlation and two-electron excitations in the propagator, they instead in a sense made a further approximation to the RPA propagator by adopting the Tamm−Dancoff approximation to reach the CIS model but explicitly introduced electronic relaxation by taking the reference state orbitals of the core-ionized state, as explained in detail in section 3.5.1. In hindsight, a key reason as to why this so-called STEX method became recognized as an important theoretical advancement in XAS despite the fact that it represented a methodological step backward compared to the higher-order propagator method, ADC(2), is that it benefited from the earlier development of atom-orbital direct SCF methods to become applicable to medium- and large-sized systemsone must also remember that standard Kohn−Sham TDDFT had not yet entered in the field of X-ray sciences as a competitor to treat large systems. In a review from 1997, Ågren et al. provided an overview of STEX studies to series of carbonbased materials including poly(ethyleneterephthalate) and polyacenes as well as a copper cluster surface adsorbate.117 Later, the STEX method was developed in combination with a relativistic four-component Hartree−Fock reference state and could be applied to calculations of L-edge spectra, where a treatment of spin−orbit effects becomes imperative to produce accurate simulations.118,254−256 The main limitations of the STEX method are (i) the lack of electronic polarization effects due to the core-excited electron, leading to XAS spectral compressions that in the worst case are atomic-site dependent; (ii) the need to perform a single calculation for every inequivalent atomic center, hampering the application to large systems with low symmetry; and (iii) the lack of a route to systematically improve on the method, making it difficult to see that STEX will remain all that relevant in future work on simulations of XAS. An improvement on STEX in the sense of addressing the lack of a treatment of electron correlation was provided by the introduction of DFT into calculations of core excitation spectra. In 1995, Stener et al. combined DFT with the transition-state method of Slater to form the so-called transition-potential DFT approach (as described in section 3.4.2) and applied it to the calculation of K-edge absorption spectra for a selection of carbonyl compounds.257 TP-DFT is essentially a single-particle approach in its description of core-electron excitations, and as such there is no reference made to spin conservation in the molecular system. Hu and Chong showed, however, how the singlet−triplet spin splitting can be treated in TP-DFT by the formation of a minimal spin-adapted configuration to represent

which is the result of introducing the approximation of exp(−ik·r) ≈ 1 in eq 13 and with p̂ being a short-hand notation for the N-electron linear momentum operator. The corresponding equation in the length gauge is 0→f MED,length = ⟨0|ϵ ·μ̂ |f ⟩

(62)

where μ̂ is the electric-dipole operator. Written in this form, eqs 61 and 62 differ by a factor of iωf 0 that needs to be taken into account in the determination of the corresponding oscillator strength that is proportional to the experimental intensity of the corresponding transition band. In terms of the length gauge expression, the dimensionless oscillator strength for a randomly oriented system takes the form f 0→f =

2meωf 0 3ℏe 2

⟨0|μ̂ |f ⟩·⟨f |μ̂ |0⟩

(63)

Early calculations of XAS spectra were performed with the semiempirical multiple-scattering Xα approach as reviewed in ref 10 and ab initio CI wave functions as in the work by Butscher and co-workers on a series of small molecules including nitrogen, acetylene, and ethylene.219,249 But a cornerstone for the development of contemporary firstprinciples electronic structure theory methods to calculate XAS spectra was laid by Cederbaum et al.114 when they designed a Hamiltonian that is computationally enabling yet still quantitatively accurate for the description of core-hole processes. The adopted approximation is known as the core− valence separation, and it is described in detail in section 3.3 and was combined with the ADC(2) method by Barth and Schirmer in 1985 to study core excitation spectra of nitrogen and carbon monoxide.250 With the efficient inclusion of dynamic electron correlation in the ADC approach, this work represented a major advancement in the field of theoretical XAS, and an impressive agreement between theory and experiment was demonstrated. In a following publication using the same approach, it is reported that a capturing of a double-electron excited state is made at the carbon K-edge in formaldehyde.251 This state was found only 2.3 eV above the lowest single-electron excited π*-resonance, which is interesting and somewhat remarkable considering that the relaxation of a double-electron excited state does in principle require triple excitations in the propagator. The CVS/ADC(2) approach was used to study the vibrational structure in XAS involved with symmetric stretch and bending modes in water, ammonia, and methane,252 and it was used to determine the parameters required for a vibronic coupling model in a study of ethylene and some of its isotopomers.253 In 2014, Dreuw and co-workers presented a new program implementation of the strict ADC(2) and its extended variant ADC(2)-x methods, and with use of the CVS approximation they demonstrated XAS calculations for a series of medium-sized molecules including porphyrin and perylenetetracarboxylic dianhydride (PTCDA).121 Their benchmark study showed that excitation energies are generally overestimated with use of the ADC(2) method, whereas an excellent agreement between theory and experiment was found using the ADC(2)-x method.121 Based on this modern program platform, the group further derived and implemented the unrestricted ADC scheme and determined the XAS spectra for a series of radicalsthe largest being the anthracene cation and their conclusion regarding method performance were largely the same for radicals as for closed-shell species, namely that the ADC(2)-x method performed excellently well.122 Next 7229


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the excited state.258 Also the implementation of TP-DFT made by Triguero et al.160 has been recognized and formed the basis for a large number of applications over the years as exemplified by the early study of the XAS spectra of fullerenes C60 and C70,259 investigations of cyclic organic compounds,139,260,261 and the long series of papers devoted to the study of water (see ref 262 for a review). There are several things to balance in TPDFT calculations including the choice of functional that is a general question in DFT to the specific issue of choosing the charge of the core holeranging in typical applications from a half core hole (HCH) to a full core hole (FCH); see, for example, the discussion in refs 263 and 264 concerned with calculations on water. In addition, there is the possibility of performing a more state-specific treatment with use of the exact Slater transition states for a selection of transitions, and performing separate SCF optimizations in each case with onehalf electron excited from the core orbital.265 It is clear that TPDFT has led to successful assignments of a large number of experimental spectra over the years, but given it is a singleparticle approach with a tunable parameter in terms of the corehole occupation, it does not offer a way to systematically develop methods of improved accuracy with the goal of reaching high-precision predictive calculations. In 2003, Stener and co-workers adopted the Tamm−Dancoff and CVS approximations in combination with time-dependent DFT for the calculation of K- and L-edge XAS spectra of TiCl4.119 This approach represents an advancement in comparison with TP-DFT in its providing a many-particle representation of core-electron transitions with a proper treatment of spin symmetry, and it is noted that the authors performed a bridging investigation by using the HCH transition-state orbitals in an otherwise regular TDDFT calculation.119 In this original implementation of the TDDFT approach to XAS, a mix of functionals is used, employing the simple adiabatic local density approximation (ALDA) for the exchange−correlation kernel in the evaluation of the coupling matrix of the TDDFT eigenvalue equation. Later applications followed from this group demonstrating XAS spectra for alkaline earth oxides, oxomolybdenum complexes, and titanium oxide,266−268 but with a focus on the L-edges of inorganic compounds it was difficult to assess the performance of the method per se as there was no treatment made of spin−orbit effects. The same TDDFT methodology was later implemented by Besley and co-workers and applied to organic systems.184,269 In this work increased attention was paid to the dependence of spectral features on the choice of functional, noting the strong dependence on the amount of exact Hartree−Fock exchange.269 The self-interaction error at short interelectronic distances was shown to be the reason for the systematic TDDFT underestimation of core excitation energies, and shortrange-corrected hybrid functionals were designed as a remedy.184 By the time Besley and Asmuruf wrote their review paper in 2010,175 it was made clear that TDDFT had entered into the arena of near-edge X-ray absorption and emission spectroscopies to stay, and today, these calculations can be performed with a computational efficiency that enables applications to systems as large as chlorophyll a and fullerenes C60 and C70.270 As an alternative to invoking the CVS and Tamm−Dancoff approximations, time-dependent DFT can also be applied to XAS by means of the energy-specific (ES) technique that provides solutions to the eigenvalue equation in agreement with full-space TDDFT results.70

In 2006, an alternative route to enabling the use of standard electronic structure theory methods in X-ray spectroscopies was presented by Norman and co-workers.271−273 In their use of the complex polarization propagator approach35,109,274 (see section 3.2 for a description), a calculation is made of the imaginary part of the electric-dipole polarizability to obtain in a direct manner the linear absorption cross section given in eq 32. In conjunction with DFT, the CPP approach relieves the needs to adopt the Tamm−Dancoff and the CVS approximations since there is no explicit reference made to the eigenvectors of the generalized eigenvalue equation. In fact, this represents the key difference between CPP-DFT and TDDFT in applications regarding linear X-ray absorption spectroscopy, and it makes CPP-DFT perfectly suited as a platform for the development of codes directed toward a treatment of large-scale systems where the near-edge spectrum may involve hundreds or maybe even thousands of underlying electronic transitions. A demonstration along this line is provided by the study of the effects of πstacking interactions on the XAS spectrum of phthalocyanine.275 With regard to applications involving large complex physicochemical and biochemical systems, it is noted that the CPP-DFT approach has been combined with various polarizable embedding schemes to describe liquid276 and heterogeneous277 environments. A methodological comparison of TPDFT and CPP-DFT was recently performed in a study on water clusters, showing that the latter but not the former approach could correctly reproduce the sensitive balance of intensity distribution in between the pre-edge and the postedge and that some 2000 MM water molecules were needed to be considered in the embedding region in order to reach XAS spectral convergence.265 With regard to relativistic effects, inclusion of spin−orbit effects has been achieved at the fourcomponent CPP-DFT level,278 demonstrating that the sensitive balance of intensities between the L2,3-edges is accurately reproduced at this level of theory and also showing that nonrelativistic or scalar-relativistic calculations are not trustworthy at the L-edges of the spectrum.279 In contrast to what has been expressed in the literature, the characterization of the underlying core-excited states in the XAS spectrum is made just as easily in calculations based on CPP-DFT as those based on TDDFT. But instead of analyzing the eigenvector, one identifies large elements in the imaginary part of the complex response vector at a frequency corresponding to the transition energy of the peak of interest in the XAS spectrum and backtraces these elements to the associated pairs of occupied− virtual orbitals. Summarizing the use of DFT response theory for calculations of XAS spectra, we note that there exist several techniques (CVS, CPP, ES) to target the X-ray region when solving the secular equation. There are large errors due to self-interactions, but it is known how to deal with them in applied work; e.g., since absolute transition energies are not often of main concern in characterization work, one can simply apply overall spectral shifts. The resulting DFT spectra often compare well with those obtained with more sophisticated wave function correlated approaches, and DFT response theory approaches typically outperform methods such as STEX and TP-DFT even though these explicitly address the issue of electronic relaxation; see Figure 9 for one illustrative comparison of method performance. As we see it, the single most important concern in the application of DFT response theory in XAS is that its good performance is not founded in its basic formulation, which, at the end of the day, boils down to a first-order propagator 7230


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oxygen, and neon, respectively.211 Apart from the fact the long chain lengths are required to converge spectrum calculations in the X-ray region, a troublesome characteristic of the Lanczos chain driven equation solver is the lack of precise error control of spectral convergence. A reduced-space algorithm was for this reason developed to solve the linear response equations in damped coupled cluster theory, and the XAS spectrum of ethylene was determined as an example calculation.281 The CPP-CC method has also been combined with a polarizable embedding model to enable simulations in the condensed phase, and results were presented for liquid water and aqueous acrolein showing atomic site-specific solvation shifts in XAS spectra.282 As made clear in this review, the CPP approach enables computational access to the UV/vis and X-ray regions of the spectrum on equal footing, but an alternative route to address the X-ray region is to introduce the CVS approximation. This was done by Coriani and Koch for the hierarchy of CC models (including CC3), and CCSD XAS spectra of uracil at the carbon and oxygen K-edges were presented and showed an excellent agreement with experiment after overall shifts of 3−4 eV had been applied.126,127 In this context, we also note the energy-specific eigenvalue solver technique for equation-of-motion CCSD that has been presented by Peng et al.215 and which provides an alternative means to the CVS approximation to target the X-ray region of the spectrum. The referenced works clearly demonstrate the high accuracy that can be reached at the level of CCSD (and higher) for XAS spectrum calculations (transition energies and intensities), but it is also clear that the associated high computational cost is prohibitive in applications involving medium-sized (and larger) molecules. The development of multilevel coupled cluster by Koch and co-workers is for this reason of high interest.213 In this approach the local region involving a specific core excitation is treated with a high-accuracy CC method whereas the rest of the system is treated more approximately. Using propenal and butanal as example systems, it was demonstrated that errors involved with using multilevel CCSD in comparison with regular CCSD could be made negligibly small given that appropriate active spaces were selected.213 This need for a selection of active spaces makes the approach somewhat cumbersome to apply in the general case, but in cases when one has a focus on a specific moiety, it can become a powerful tool to obtain an accurate description of the associated core excitations. A common feature for all methods discussed so far in connection with XAS applications is that they are based on single-reference states. There is an important class of systems for which the use of single-reference methods becomes questionable, and that is the first-row transition metal complexes that are frequently studied in metal L-edge XAS. Response theory approaches have been developed for multiconfiguration SCF wave functions,35 and these are in principle well-suited to be used in this context, but in practice, the lack of dynamic electron correlation will inflict a poor quality on the resulting XAS spectra. A methodology for the determination of transition metal XAS spectra has been developed by Lundberg and Odelius and co-workers232,283−285 based on the restricted active-space SCF approach with account made for scalar relativistic as well as spin−orbit effectsthe latter by means of the atomic mean-field approximation. The use of the stateaveraged formalism means that a common set of molecular orbitals is optimized for all final core-hole states and transition moments between the excited states and the separate-state

Figure 9. Chemical shifts (in eV) between the two π*-resonances in carbon K-edge XAS spectra of fluoro-substituted ethylenes. Data are taken from ref 280 with experimental reference data listed below each respective illustration of the molecular structures. The DFT results are obtained with use of the CAM-B3LYP(100%) functional.280

approach constrained to single-electron excitations and deexcitations in the operator manifold. We note that we are here referring to performance in terms of spectrum quality and setting aside the issue of computational efficiency and speed. In 2012, Coriani and co-workers combined the CPP approach with coupled cluster wave function theory to enable the calculation of XAS spectra of small molecules.211,212 The complex linear response equations were solved by means of an asymmetric Lanczos chain algorithm that provides a solution for the entire frequency region, which in practice means that relatively long chain lengths were needed in order to converge absorption spectra in the X-ray region based on eq 32. XAS spectra were presented for water, ammonia, neon, and carbon monoxide using a hierarchy of coupled cluster methods including CCS (being the same as CIS), CC2, CCSD, and CCSD with noniterative triple-corrected excitation energies CCSDR(3). Results at the CCSD level showed relative peak intensities in good agreement with experiment with discrepancies in absolute unshifted transition energies on the order of 1−2 eV due to incomplete treatment of electronic relaxation and correlation. With inclusion of triple excitations, errors in energetics became less than 0.9 eV, thereby capturing 90, 95, and 98% of the relaxation-correlation energies for carbon, 7231


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optimized ground state are determined by the restricted activespace state-interaction (RASSI) approach. Based on studies of iron complexes, it is shown that the RAS approach provides Ledge spectra of high accuracy if dynamic correlation is taken into account at the RASPT2 level of theory and it thereby enables reliable spectral assignments to be made of transition metal complexes.285 The obvious limitation of any multireference wave function approach is the steep computational scaling with system size. As a cost-efficient alternative to calculate XAS spectra of large transition metal complexes, e.g., deoxymyoglobin with 703 atoms, Neese and co-workers have developed the pair natural orbital restricted open-shell configuration interaction (PNO-ROCIS) approach,286 which represents a computationally more efficient version of their earlier developed DFT-ROCIS method.287,288 The ground state is represented by a high-spin (HS) single-reference state, which excludes application to more complex systems with antiferromagnetically coupled ground states. But in cases when a HS representation is adequate, the DFT-ROCIS approach has been shown to be sufficiently accurate to provide XAS spectra in quantitative agreement with experiment.286

[η]

R n = Im⟨0|μα̂ |n⟩⟨n|m̂ α |0⟩

n←0 M md = ⟨FCI|m̂ α |HF⟩

(64)

(65)

where ν̃ (cm−1) is the wavenumber of the incident beam, M (g mol−1) is the molar mass, and β (au) is given as β=−

1 (G′xx + G′ yy + G′zz ) 3ω

(69)

In line with the core−valence separation discussed in section 3.3, the significant FCI expansion coefficients of the coreexcited state are all associated with configurations that have a vacancy in the core orbital. Furthermore, since m̂ α is a oneelectron operator, only single-excited determinants in |FCI⟩ will contribute to transition matrix elements, which thus reduces down to a one-electron integral in between the initial core and final virtual orbitals. The spherical 1s-core orbital, however, is an eigenfunction of the magnetic dipole operator with eigenvalue 0, so the transition moment integral, in the absence of core polarization, will vanish. Therfore, the two contributions to the magnetic transition matrix elements are identified as polarization of the core state due to the molecular field and valence-excited determinant in the FCI expansion, i.e., a breakdown of the CVS approximation. A starting point for calculations of X-ray CD in theoretical chemistry is provided by the work of Alagna et al.296 in which RPA calculations using the CVS approximation were presented for the rotatory strengths of propylene oxide at the K-edges of carbon and oxygen. The next step in the line of method development was taken by Ågren and co-workers when they used the STEX approach (see section 3.5.1) to obtain near carbon K-edge spectra for twisted ethylene, propylene oxide, trans-1,2-dimethylcyclopropane, and a selection of amino acids.297−299 The STEX approach was later benchmarked against CIS (referred to as STEX with unrelaxed orbitals in the original work), RPA, and separate-state Hartree−Fock by Kimberg and Kosugi,300 and, although not explicitly stated in the publication, the CIS and RPA methods applied to core

where N (m−3) is the number density of molecules, μ0 is the permeability of vacuum, and G′(ω) (C2 m3 J−1 s−1) is given by the real part of the mixed electric-dipole−magnetic-dipole linear response function of the form given in eq 29.292,293 The associated molar ellipticity (deg cm3 g−1 dm−1) is defined as294 βν 2̃ M

(68)

where μ̂ α and m̂ α are components of the electric and magnetic dipole moment operators, respectively, and an implicit Einstein summation is assumed of the repeated operator index. To calculate rotatory strengths and apply Lorentzian broadenings will result in a CD spectrum that is in perfect agreement with what one gets from a calculation of the real part of a complex linear response function,292 and the choice of approach should be largely based on computational convenience. CD at the K-edge puts some particular demands on the accuracy in the theoretical calculation as compared to other Xray spectroscopies. The following line of arguments was put forward by Villaume and Norman.295 The electronic ground state of a closed-shell system with large HOMO−LUMO gap is reasonably well described by the Hartree−Fock wave function, |0⟩ = |HF⟩, and let us assume that the excited state is obtained at the level of full CI (FCI), |n⟩ = |FCI⟩, based on these canonical Hartree−Fock orbitals. The magnetic transition dipole moment in eq 68 then takes the form

The standard reference for a treatment of the theory of optical activities in molecular systems is the book by Barron,289 and the amount of experimental and theoretical work in the field is vast as demonstrated in the two comprehensive volumes edited by Berova et al.290 While the underlying theory of optical activity in the X-ray region is the same as for the UV/vis region, there are a comparatively minuscule number of publications devoted to simulations of molecular optical activities in the X-ray region of the spectrum owing to the fact that intense circularly polarized X-ray beams are only recently available and the experimental activity has been limited, as reviewed in the chapter by Goulon et al.291 The calculations of molecular optical activities in the X-ray region have been focused on the determination of the circular dichroism (CD) for randomly oriented systems such as gases and liquids where the electric dipole−electric quadrupole vanishes, and we will therefore only consider the electricdipole−magnetic-dipole contribution in the following. The CD for a monochromatic light beam of frequency ω (rad s−1) propagating over a path length l (m) is described by the ellipticity (rad)289

[η] = 0.1343 × 10−3

(67)

where the molar ellipticity has been transformed into units of deg cm2 dmol−1. Just as in absorption spectroscopy, it also possible in calculations of CD to determine the signal strength from the residues of the pertinent linear response function. The correspondence of the oscillator strength in absorption spectroscopy is known as the rotatory strength, Rn, in CD spectroscopy, describing the transition from the ground state, |0⟩, to excited state, |n⟩,

4.2. X-ray Natural Circular Dichroism (XNCD)

1 η = − ωμ0 ln G′αα (ω) 3

M = 3298.8Δϵ 100

(66)

Alternatively, the CD is expressed in terms of the differential extinction coefficient Δϵ in units of L mol−1 cm−1: 7232


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Figure 10. Comparison of XAS and XNCD spectra of noradrenaline (neutral form), showing a spectral richness in the latter case. Peak assignments in the XAS spectrum are provided, referring to the atomic labeling in the inset illustration of the molecular structure. Results refer to CPP-DFT calculations with data taken from ref 295.

see the six lowest spectral features in the XNCD spectrum for noradrenaline in Figure 10. Using the transition-potential DFT method, Pettersson and co-workers addressed the effect of intermolecular interactions in molecular crystals on the XNCD spectra of alanine and serine.302 In their approach, a half-occupied core orbital is introduced at the site of the core hole and other sites are described by means of effective-core potentials. For a given element (carbon in this case), spectra are obtained for all inequivalent atomic sites in this manner and these are thereafter added together to form the final spectrum. With respect to the focus of their work, namely to compare calculated X-ray CD spectra of isolated molecules with those of the crystal model, it is found that molar intensities are reduced in the latter case and that caution is called for when comparing theoretical calculation on molecules and experiments performed in the solid phase. Based on the CVS and Tamm−Dancoff approximations, Zhang et al. presented time-dependent DFT calculations of Xray CD for a series of systems including chlorophenylethanol (CPEO), n-chlorohexahelicene (nCHHC), and 1-bromo-nchloronona-2,4,6,8-tetraene-1-amine (BnCTA).303 In accordance with the systematic alteration of the spatial separation of the chiral center and the benzenediol discussed above, the authors of this work study the dependence of the XNCD response of a probe with respect to its distance to the chiral centers in CPEO and BnCTA. In the case of nCHHC, where there is no chiral center but the chirality stems from the helical molecular structure, the probe is placed at the tip of the helicene. The probe is very local and is represented by a single chlorine atom and the XNCD is studied at the chlorine L2,3edges. The study concludes that (i) the CD response strongly decreases with an increased spatial separation between probe and chiral center and (ii) in comparison to optical CD that is routinely used to determine the absolute configuration of large molecules, XNCD can better reveal the relative configuration of different chemical groups in a molecule as a consequence of the

excitations normally refer to the adoption of the CVS approximation. Among the listed approaches, the authors conclude that the RPA method performs best, which is surprising given its lack of a treatment of electronic relaxation effects in the excited state (see section 2.2) and which is explicitly considered in the separate-state Hartree−Fock and STEX approaches. On the other hand, it is also clear that the use of different and nonorthogonal orbital sets for the ground and excited states introduces an issue of gauge-origin dependence as discussed in ref 255 and which is likely to be far more critical in calculations of XNCD as compared to XAS. From the perspective of gauge-origin dependence and introducing electron correlation in the reference state, a more rigorous approach was taken by Norman and co-workers in their adopting the CPP-DFT technique with use of gaugeincluding London atomic orbitals to in a direct manner calculate the tensor G′(ω) in eq 64.292,295,301 The XNCD response could thereby be determined for L-alanine,292 providing an assignment of one of the few existing experimental noncrystalline molecular spectra, and the C84 fullerene,301 amply demonstrating the superior resolution of electronic states in carbon-rich systems by means of XNCD as compared to XAS.292 They also showed the intramolecular induction of chirality in the X-ray region. With a molecular plane of symmetry it is well-known that the optical activity vanishes, because the electric- and magnetic-dipole operators (μ̂ α and m̂ α) for one and the same molecular axis α span different irreducible representations, which leads to a vanishing rotatory strength in eq 68. From this viewpoint, it is expected that strong π*-resonances in the XAS spectrum become invisible in the XNCD spectrum due to the local molecular planarity. Villaume and Norman295 studied noradrenaline and L-DOPA with a systematic increase in the spatial separation of the chiral center and the achiral benzenediol and demonstrated that effective transfer of chirality takes place at the carbon K-edge; 7233


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locality of core excitations.303 On the technical side of things, the authors conclude that the calculations are sensitive to the approximations made in the theoretical model with respect to the electronic structure theory method and also the lack of treatment of a realistic dynamical environment. In view of the adopted level of electronic structure theory, it is noted that spin−orbit interactions are not included in the Hamiltonian despite the focus being on the L2,3-edges. The obtained chlorine XNCD spectra for this reason alone cannot be expected to reach quantitative accuracy. For instance, it was recently demonstrated in a XAS study of the silicon L2,3-edges in silane derivatives that spin−orbit interactions are essential to include in the simulations in order to reproduce the correct ratios of intensities between spin−orbit-split features in the spectrum,279 because the observed ratios of peak intensities deviate strongly from the 1:2 statistical ratio that, in an ad hoc manner, is typically used to split peaks as determined by means of nonrelativistic or scalar relativistic calculations.

Ågren and co-workers continued the line of development by considering not only transition energies but also XES intensities, and they tested the validity of the earlier introduced one-center approximation by conducting a Hartree−Fock study on N2, CO, H2O, and NH3. This work represents the first ab initio study of XES intensities, and as a main conclusion, the authors report that the one-center approximation is well justified when applied to second-row elements in the periodic table.306 With progress in experimental XES spectroscopy, weak multielectron satellites could be observed and in response multiconfiguration CI approaches were developed.307 This phenomenon was referred to as radiative electron rearrangement (RER), or semi-Auger transitions due to their pictorial similarity with the Auger process, and the radiative transitions thus involve the filling of the core hole while also exciting weaker bound electrons. Various types of correlation satellites in the XES of CO and N2 were defined and explored by Ågren et al. with theuse of separate-state optimized CI wave functions with mutually nonorthogonal orbitals.307,308 The effects of the short lifetimes of the core-excited states enters into the theory and calculations not only in terms of lifetime broadenings but also as an interference in the various vibronic decay channels. For CO, these effects were investigated at the next natural level of theory following CI, namely multiconfiguration SCF.309 To evaluate transition moments for wave functions expressed in terms of mutually nonorthogonal orbitals comes at high computational cost as compared to the situation of using a common set of orthogonal orbitals for the initial and final states. In the Hartree−Fock scheme, this was handled by means of cofactor calculations, but for CI wave functions the use of nonorthogonal orbitals had put a severe limit on the length of the CI vectors that could be employed in the calculations. A remedy was put forward by Ågren et al. in their adopting a biorthogonalization scheme to avoid the cofactor calculation, allowing for efficient calculation of XES intensities using restricted and complete active space wave functions that in turn were optimized with a second-order Hessian-based algorithm to enable stable wave function optimization with full relaxation of the core hole.310 In comparison to MCSCF, the SCF optimization of a corehole state in the Hartree−Fock or Kohn−Sham approximation is easier, and a simple maximum-overlap method (MOM) has been presented by Besley and co-workers138 which bases the orbital occupation from one SCF iteration to the next on the principle of maximizing orbital overlaps. This technique has been shown to effectively eliminate a variational collapse134 and can also be applied to the optimization of double core-hole states.311 Based on the optimized single core-hole reference state, it was demonstrated that XES spectra were available by means of otherwise conventional response theory calculations in which the transitions of interest are found as those with negative eigenvalues in the spectral representation of the linear response functions.175 This is a very elegant idea and it makes the calculation of XES spectra closely analogous to the calculation of XAS spectra, and the approach is also applicable to any given electronic structure theory method at hand. For a series of small molecules including methane derivatives, acetylene, ammonia, and water, X-ray emission energies were presented at the levels of CIS, CIS with a perturbative inclusion of doubles, equation-of-motion CCSD, and TDDFT with a selection of functionals.175 The conclusions to be drawn from this set of data are (i) CCSD with its inclusion of double excitations and thereby a treatment of electronic relaxation in

4.3. X-ray Emission Spectroscopy (XES)

Following core-electron ionization, the system is found in a high-lying electronically excited state that decays by the deexcitation of a valence electron to refill the core hole in combination with the release of energy by either the emittance of another valence electron to result in a final state with two valence holes or the emittance of a fluorescence photon to result in a final state with a single valence hole. The study of the electron emission process is known as Auger electron spectroscopy, and the study of the photon emission process is known as X-ray emission spectroscopy (XES). From the point of view of photophysical characterization, XES is complementary to XAS in the sense that it probes the occupied molecular valence orbitals whereas XAS probes the unoccupied ones. But from the point of view of theoretical calculations, the two spectroscopies are in a sense the opposites of one another, because in XES the initial state is the one with a core hole and the final state has filled core orbitals whereas in XAS the opposite is true. The electronic-structure optimization of the wave function representing the core-ionized reference state therefore becomes a central issue in the calculation of XES, which is not necessarily true in simulations of XAS as we have seen in section 4.1. There are important considerations to take into account in this process such as (i) the fact that the coreionized reference state is embedded in a continuum so that the optimization may be subject to a variational collapse and (ii) the large relaxation effects discussed in section 2.2 that result in nonorthogonal orbital representations of the initial and final state wave functions in the emission process. It is necessary to find a balanced treatment of the dynamic core-hole effects and valence-hole relaxation, and if this is not reached in the calculation, it is argued that it is better to rely on initial and final states determined in the ground state potential.304 Pioneering work in the optimization of core-hole states of molecules was carried out by Bagus and co-workers,5,25,305 providing evidence that both the large electronic relaxation and the spin-configuration splitting are well described at the level of Hartree−Fock theory with the use of separate-state optimization of the ground and excited states. For instance, the contemporary experimental reference data for the spinconfiguration splitting of the oxygen and nitrogen 1s−1 states in NO+ (that is, 3Π and 1Π) were reported to be 0.53 ± 0.02 and 1.41 ± 0.02 eV, respectively, which compared favorably with the calculated values of 0.48 and 1.35 eV.305 7234


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Figure 11. X-ray emission energies for electronic transitions at the near carbon K-edge in methane, acetylene, carbon monoxide, and methanol. Data are taken from ref 175. For convenience, a solid line has been drawn on the diagonal, representing a perfect agreement between theory and experiment.

a spectroscopy that has seen rapid development and increased popularity in recent years owing to the installation of fourth generation synchrotron radiation sources.316 There are several fundamental differences between the underlying physics in XES and RIXS spectroscopies: (i) The XES process is initiated by core-electron ionization, whereas, in RIXS, the molecular charge is preserved. (ii) XES is fundamentally a photon-out fluorescence process, whereas RIXS is a photon-in−photon-out Raman scattering process. (iii) The core-excited state in XES is a cationic state represented by a single wave function, whereas the coherent excitation of the intermediate states in RIXS will cause interference effects.317 Early RIXS experiments date back to the 1970s (see the review by Ament et al.316) and the basic theoretical description was given by Kramers and Heisenberg and also Dirac in the 1920s introducing the so-called Kramers−Heisenberg−Dirac (KHD) expression for the scattering amplitude,318,319 but we refer to the review by Gel’mukhanov and Ågren for a modern account of the theory of X-ray Raman scattering.320 For a molecular system residing in the ground state |0⟩, the inelastic scattering process involving the final state |f ⟩ is described in second-order time-dependent perturbation theory by the KHD f 0 321 scattering amplitude, - αβ :

core−valence (and valence−core in XES) transitions is about as accurate for XES as for XAS (the reported mean absolute deviation with respect to experiment is equal to 0.5 eV175) and (ii) the determination of absolute emission energies at the TDDFT levels is severely hampered by the self-interaction error in the standard functionals (also analogous to the case of XAS). In Figure 11, we present the carbon K-edge X-ray emission energies from ref 175 as obtained at the levels of CCSD and DFT with use of the B3LYP functional and which nicely illustrate the drawn conclusions. Figure 11 shows the excellent agreement between CCSD and experimental data as well as the large overestimation of X-ray emission energies at the TDDFT level with the use of standard functionals. Later work has been devoted to tuning the functional with respect to the amount of exact Hartree−Fock exchange, resulting in TDDFT values of X-ray emission energies in closer agreement with experiment.312−314 But it is noted already in Figure 11 that the chemical shifts in the X-ray emission energies are well captured at the DFT level, which is an observation that also applies to XAS spectra as discussed in section 4.1. Based on the same computational strategy, successful TDDFT/B3LYP simulations are also reported by Zhang et al.315 for the XES spectra associated with valence to core transitions in transition metal complexes involving Fe, Mn, and Cr. Overall large red shifts are applied in these calculations to align theoretical spectra with the experimental ones, but it is convincingly demonstrated that both relative peak positions and relative intensities are in good agreement with experiment. Considering the sensitive nature of the weak valence to core transitions in transition metal complexes that inflicts strong requirements on the experimental setup, this study provides strong evidence for the merits of the computational approach of using response or propagator theory techniques in XES.

f0 (ω) = - αβ

⎡ ⟨f |μα̂ |n⟩⟨n|μβ̂ |0⟩

1 ℏ

∑⎢

+

⟨f |μβ̂ |n⟩⟨n|μα̂ |0⟩ ⎤ ⎥ ωn0 + (ω′ + iγn0) ⎥⎦

n

⎢⎣ ωn0 − (ω + iγn0)

(70)

where μ̂α is the electric dipole operator, ℏωn0 = En − E0 is the transition energy, ℏω and ℏω′ = ℏ(ω − ωf 0) are the energies of the incident and scattered photons, respectively, and ℏγn0 is the half-width half-maximum (HWHM) lifetime broadening of intermediate state |n⟩. It is noted that, with f = 0, we get an expression pertinent to elastic Rayleigh scattering with an amplitude - 00 αβ(ω) that is equal to the electric-dipole

4.4. Resonant Inelastic X-ray Scattering (RIXS)

X-ray emission following resonant or near-resonant coreelectron excitations in the NEXAFS regions of the spectrum is known as resonant inelastic X-ray scattering (RIXS), which is 7235


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polarizability ααβ(−ω;ω) introduced in eq 32 and given by the linear response function in eq 29 with Ω̂ = μ̂ α and V̂ ωβ = −μ̂ β. But also note that in elastic X-ray scattering there is also to be considered a term describing the amplitude of Thomson scattering.320 Just as in calculations of the polarizability under resonance or near-resonance conditions, it is well motivated in calculation of the resonant inelastic scattering amplitude to introduce the rotating wave approximation and ignore the second term in the sum in eq 70 to arrive at f0 (ω) - αβ

1 = ℏ

∑ n

Similar in spirit but more approximate in the description of potential energy surfaces and the time evolution, one finds the recently developed semiclassical Kramers−Heisenberg (SCKH) approach by Ljungberg that is designed to be able to describe more complex systems such as liquids and surface adsorbates.327 In these approaches, electronic structure theory calculations take a backstage role and become the supplier of the potential energy surfaces and electronic transition moments that are required as input data for the evaluation of eq 72. This in turn means that computational requirements and limitations on this part of the work are identical to what has already been discussed in connection with XAS and XES. We will therefore not further review the underlying electronic structure theory calculations in RIXS that are merely providing electronic transition energies and moments and which are not representing a methodological advancement per se. For randomly oriented systems, the scattering cross section is proportional to the transition strength, σ0f, averaged over all molecular orientations as well as over the polarization of the emitted radiation and depending on the angle θ between the polarization vector of the incident photon and the propagation vector of the scattered photon:202

⟨vf |μαFN |vn⟩⟨vn|μβN 0 |0⟩ ωn0 − (ω + iγn0)

(71)

where the Born−Oppenheimer approximation has been adopted as to represent vibronic states as products of electronic and vibrational states. The sum includes all intermediate vibronic states |n⟩ = |N⟩|vn⟩ and integration over electronic coordinates has been carried out in the numerator, resulting in the introduction of the nuclear-configuration-dependent electronic transition moments μFN and μN0. For small molecules, it sometime happens that the lowest electronic core-excited states are well separated and the summation in eq 71 can be restricted to the single electronic state in resonance with the external X-ray field, leading to a replacement of the summation over n by a summation over the vibrational levels vn of the resonant electronic state. Further, assuming a weak dependence of electronic transition moments on the nuclear configurations, these can be brought outside the integrations in eq 71 and replaced by their values at the ground state equilibrium geometry so that the scattering amplitude can alternatively be written as f0 - αβ (ω) = μαFN μβN 0

e

i ℏ

∑∫

∞

e

∫0

∞

+

1 ω′ 15 ω

⎡

∑ ⎢⎛⎝2 − α ,β

⎣

⎜

⎞ 1 sin 2 θ ⎟-αβ- *αβ ⎠ 2

⎤ ⎛3 1⎞ sin 2 θ − ⎟(-αβ- *βα + -αα- *ββ)⎥ ⎝4 ⎠ ⎦ 2

⎜

(73)

Therefore, with a focus on the observable, it is seen that the absolute square of the scattering amplitude is the pertinent quantity in the calculation. Two limiting case are discussed in ref 262: First, there is the situation when the energy spacings of the intermediate states in eq 71 are large in comparison to the lifetime broadening, in which case the excitation channels act independently and one arrives at an approximate expression of the form

̂N

⟨vf |e−iHn t / ℏ|vn⟩⟨vn|0⟩

0 vn −i([ΔEad − E0]/ ℏ− ω)t −γt

i = μαFN μβN 0 ℏ

σθ0f =

dt

⟨vf |0(t )⟩e−i([ΔEad − E0]/ ℏ− ω)t e−γt dt (72)

f0 2 |- αβ | =

where Ĥ Nn is the Hamiltonian of nuclear motion in the excited electronic state N and for which |vn⟩ is an eigenstate and γ denotes an adopted common broadening of the intermediate vibronic states. The transition energy ℏωn0 has been written as the sum of the adiabatic excitation energy ΔEad and the difference in vibrational energiesE0 here denotes the lowest vibrational energy of the electronic ground state. After the resolution of identity has been identified as the sum over intermediate vibrational states, the time propagation operator is left to act on the ground vibrational state |0⟩. As a result, the time-dependent state |0(t)⟩ is obtained and thus refers to the time propagation of the initial wave packet on the excited state potential energy surface. Equation 72 represents the principle for the time-domain representations of RIXS that provide a full vibronic spectral resolution. For small molecules with a correspondingly small number of vibrational degrees of freedom, Gel’mukhanov and co-workers have shown in a series of papers that such timedomain calculations of the nuclear dynamics in the core-excited state can be performed to accurately describe the vibrational structure in resonant X-ray absorption and scattering spectroscopies,322 which in turn displays strong dependencies on the character of the excited state (bound or dissociative) and nuclear isotope substitutions (hydrogen or deuterium).323−326

1 ℏ2

∑ n

|⟨vf |μαFN |vn⟩|2 |⟨vn|μβN 0 |0⟩|2 (ωn0 − ω)2 + γn0 2

(74)

For polyatomic molecules it is clearly not reasonable to expect energy separations of vibronic levels to be large in comparison to the lifetime broadenings of the short-lived coreexcited states. The opposite may be a more relevant case, i.e., an assumption that the lifetime broadening is large and the coherent excitation spans over a large number of intermediate vibrational levels. Neglecting the dependence of the vibrational energies in the denominator, one can then identify a resolution of the identity as the sum over vibrational states in the numerator which results in the following expression: f0 2 |- αβ | =

1 ℏ2

∑ N

|⟨vf |μαFN μβN 0 |0⟩|2 (ωN 0 − ω)2 + γ 2

(75)

A very similar analysis with numerical illustrations and comparisons has been presented for the optical Raman spectrum of ethylene,328 which concludes that this approximation may be quite reasonable. It also points the way in a new direction of method development in response and propagator theory, because it becomes relevant to define and implement response functions in electronic structure theory that relates to inelastic scattering whereas in all response functions listed in eqs 7236


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Figure 12. RIXS map of water at the ADC(3/2) level of theory, showing sideways to the right the corresponding XAS spectrum. The transition strengths in eq 73 are determined for θ = 45°. The dominating excitation (in blue) and de-excitation (in red) channels in the evaluation of eq 76 are illustrated to the left, but the inclusion of all many-particle intermediate electronic states is made in the calculation. Data are taken from ref 202. The sizes and colors of the circles encode the magnitude of the transition strengths.

29−31 the initial and final states are equal to the electronic ground state, |0⟩. This line of development was recently pursued by Rehn et al.,202 and in essence, they derived and implemented the ADC propagator matrix equation corresponding to the full electronic KHD scattering amplitude. F0 (ω) = - αβ

effects may be strong and consequently hampering the use of few-state models in the calculation. 4.5. X-ray Two-Photon Absorption (XTPA)

The theory of two-photon absorption (TPA) dates back to the work of Göppert-Mayer in 1931,329 but it was not until the advent of the laser that it became an experimental tool for photophysical characterization: the first experimental demonstration of TPA was given by Kaiser and Garrett in 1961 using a ruby laser.330 In this perspective, the review by Mahr provides a relatively early experimental and theoretical account of the field and offers a clear presentation of the underlying theory in one of its sections.31 In addition to the direct focus on the connection between the two-photon cross section and the third-order susceptibility, which is the macroscopic counterpart of the molecular hyperpolarizability in eq 33, this presentation includes a derivation of the two-photon transition matrix element, which is the two-photon counterpart of the onephoton transition matrix element in eq 13, with explicit formulas provided also for the lowest-order multipole moment contributions.31 A word of caution to be remembered here is that expansions based on transition moments rather than cross sections lead to gauge-origin-dependent results as shown by Jacob and co-workers36 for one-photon absorption (OPA); see discussion in section 2.3. However, this is not a concern in the electric-dipole approximation in which the length-gauge expression for the two-photon transition dipole moment takes the form

⎡ ⟨F |μα̂ |N ⟩⟨N |μβ̂ |0⟩

1 ℏ

∑⎢

+

⟨F |μβ̂ |N ⟩⟨N |μα̂ |0⟩ ⎤ ⎥ ωN 0 + (ω′ + iγ ) ⎥⎦

N

⎢⎣ ωN 0 − (ω + iγ )

(76)

In their approach, the low-lying valence-excited states that are the final states |F⟩ in eq 76 are represented by eigenvectors of the secular matrix which can be found by employment of a regular Davidson algorithm. The treatment of the complete set of electronic intermediate states, on the other hand, is performed with use of the CPP technique described in section 3.2 and will cover a “window” of frequencies of the incident radiation. An illustration of technique is provided in Figure 12, where the photon excitation energies cover the oxygen near Kedge region of 538−541 eV and the resulting valence state excitation energies are given by the differences of excitation and emission energies. One very interesting aspect of RIXS spectroscopy is illustrated in Figure 12, and that is the fact that dipole-forbidden states become accessible (just as in twophoton absorption spectroscopy) and we note that the 11A2 state in fact has the largest RIXS cross section (shown by the large red circle in Figure 12). It is noted in the original work that the ratio of cross sections observed in the experiment for states 21A1 and 11B1 as well as 11B2 and 11A2 are well reproduced in the simulations, which brings merit to the approach of calculating electronic RIXS cross sections. This complex propagator technique is likely to become an important tool in applications involving larger molecules with a higher density of intermediate electronic states and where interference

0→f (ω1 , ω2) = Sαβ

⎡ ⟨0|μα̂ |k⟩⟨k|μβ̂ |f ⟩

1 ℏ

∑⎢

+

⟨0|μβ̂ |k⟩⟨k|μα̂ |f ⟩ ⎤ ⎥ ωk 0 − ω2 ⎥⎦

k

⎢⎣

ωk 0 − ω1

(77)

where conservation of energy is enforced by ω2 = ωf − ω1 and, with use of a monochromatic light source, we consequently have ω1 = ω2 = ωf 0/2. Whereas OPA as described in the electric-dipole approximation does not depend on the light 7237


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Figure 13. Characteristics of X-ray two-photon absorption. The one-photon energy in XTPA falls in the region in between the valence ionization potential and the semibound core-excited states (left). The 1s → 3s two-photon transition in neon has contributing channels in eq 77 from (i) core excitations followed by valence de-excitations and (ii) valence excitations followed by core excitations (right).

Figure 14. Illustration of the TPA spectral equivalence between calculations based on eqs 33 and 77, respectively, with data taken from ref 333 (left). XAS and XTPA spectra of water with data taken from ref 112 (right).

ization propagator is that it provides a very direct connection between a sum-over-states expression and its corresponding matrix representation that is used in the actual calculation. In contrast, the more indirect connection made by means of the residue of a first-order residue of the quadratic response function is less concrete and more elusive at first sight. The main differences from a computational perspective between UV/vis and X-ray TPA are illustrated in Figure 13. First, the photon energy in monochromatic XTPA is such that it falls in between the valence ionization potential and the transition energies of the semibound core-excited states in the near-edge region. This means that there will be a large number of states in the calculations and infinitely many in the real system with excitation energies smaller than the photon energy. Spurious resonances from the discrete representation of the continuum in the electronic Hessian may cause a divergence in the calculation. Second, in the calculation of the two-photon transition moment in eq 77, a representation of the final state in terms of an eigenvector of the generalized eigenvalue equation is required. In the X-ray region this can be accomplished by

polarization, TPA, on the other hand, displays a polarization dependence as shown by McClain with an expression for the two-photon transition strength that reads as331 δ TP =

* + GSαβSαβ * + HSαβSβα *] ∑ [FSααSββ α ,β

(78)

where F, G, and H are coefficients that depend on the polarization of the light and the notation for the two-photon transition matrix element in eq 77 has been reduced to Sαβ. A computational approach to determine the two-photon transition moments of molecular systems based on standard electronic structure theory was established by Olsen and Jørgensen34 when they showed that the sum-over-states expression in eq 77 can be identified as a first-order residue of the first-order nonlinear response function in eq 30. Dreuw and co-workers332 have presented a way to calculate the twophoton transition matrix element in eq 77 with the use of the ADC scheme for the polarization propagator and employing the intermediate state representation (ISR). An appealing feature of this ADC/ISR technique to represent the polar7238


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finding theoretical descriptions and computational approaches to address nonlinear X-ray spectroscopies that demands also the inclusion of the valence-excited states. This aspect is, by the way, also relevant in the linear regime, for example, in resonant inelastic X-ray scattering spectroscopy. Since the final states in the RIXS process are molecular valence-excited states (section 4.4), the accurate simulation of RIXS spectra requires an accurate descriptions of both core- and valence-excited states. With the recent development of X-ray free electron lasers, one can expect nonlinear X-ray spectroscopies to become significantly more important in the future, and the development of accurate and computationally affordable methods for the simulation of such spectroscopies is urgently needed. With the advent of fourth-generation synchrotrons, the control of the polarization of the X-ray radiation is taken to a new level, allowing for polarization-dependent experiments to be carried out. A prominent example is circular dichroism spectroscopy, which in the UV/vis region of the spectrum has become an indispensable research tool, relying on the difference in absorption of left and right circularly polarized light by a chiral system. On can expect that circular dichroism as well as other polarization anisotropy dependent X-ray spectroscopies will be developed and utilized in the future. In this context, wave function correlated methods are urgently needed to allow for the simulation of XNCD and related spectroscopies to assist the interpretation and development of emerging polarization-dependent X-ray spectroscopies. Important aspects of X-ray phenomena and spectroscopies in general that need to be addressed in future efforts to provide theoretical methods for their description comprise the consideration of environments, electronic decay processes, and nuclear dynamics. In chemistry, biochemistry, and materials science alike, the investigated molecules are normally embedded in an environment; i.e., they are rather found in the condensed phase than in the gas phase. Hence, it is important to include these effects in the theoretical modeling of X-ray spectroscopies, via polarizable continuum models for solution or embedding schemes for more static environments. Since core-excited states exhibit excitation energies high above the ionization potential, they lie within and interact with the electronic continuum and thus they are prone to electron emission. In the gas phase, one can often observe resonant Auger decay of core-excited states, but in environments the situation is usually more involved, since the emission of an electron can also occur at a neighboring site, initially not excited, via so-called resonant intermolecular Coulombic decay processes. Since standard quantum chemical methods are generally not suited to treat unbound electronic states, method development is required to address X-ray induced electronic decay theoretically. A third aspect that is largely neglected in the simulation of X-ray spectroscopies so far is nuclear motion. This is partially justified, since core-excited states generally exhibit very short lifetimes. However, a larger influence of nuclear dynamics on nonlinear spectroscopies and electronic decay processes can be expected. Hence, the development of theoretical methods based on a classical or quantum description of nuclear motion in the context of X-ray spectroscopies represents another promising avenue of future research.

means of the core−valence separation (CVS) approximation discussed in section 3.3. But valence-to-virtual excitations cannot be eliminated in the calculation of eq 77 since there will be contributing terms (or channels) in the summation that involve first a valence-to-virtual excitation to be followed by a core-to-valence excitation. A way to circumvent the issue of resolving two-photon states in the X-ray region in calculations is to instead turn to eq 33. Based on a pragmatic introduction of damping terms, damped cubic response functions have been developed in the quasienergy formalism334,335 and a density-matrix-based timedependent DFT formalism.336 A physically more motivated approach is to turn to eq 27, which leads to expressions for the approximate-state response functions that differ in their details. This CPP approach was taken by Norman and co-workers333 and applied to the X-ray region to calculate the XTPA spectra of neon and water.112 Whether one performs calculations of TPA cross sections based on the imaginary part of the secondorder hyperpolarizability or the two-photon transition moment is largely a matter of taste and leads to spectra that are in close agreement as is demonstrated in the left panel of Figure 14. It is clearly seen, however, that the two approaches are in agreement only as long as one can afford to resolve the eigenstates in the calculation of the two-photon moments, which in the provided illustration for neon limits the covered energy range to a few electronvolts in the UV/vis region. For larger systems with a denser density of states, this computational issue becomes increasingly problematic. In the CPP approach, on the other hand, there are no restrictions imposed on the selection of the energy window in focus, which is illustrated in the right panel of Figure 14, which presents the XTPA spectrum of water together with the linear XAS counterpart. In water the lowest core excitation gives rise to the 4a1 band in the spectrum that is both one- and two-photon-allowed. The comparison of the XAS and XTPA spectra is therefore interesting in their probing the p- and s-characters of the final state, respectively, providing complementary information about the excited state.

5. SUMMARY AND OUTLOOK During the past 10 years, the research field of X-ray spectroscopy has witnessed an enormous advancement in experimental techniques and in theoretical methods and methodologies for simulations. This progress is mostly driven by the development of advanced light sources on the experimental side and the emerging possibilities to perform, for example, also nonlinear X-ray spectroscopies that require an extremely high photon flux. Concomitantly, theoretical developments have followed as reviewed here, ranging from density functional theory to wave function theory as well as response and polarization propagator theories built upon these and together allowing for the simulation of X-ray absorption spectroscopy (XAS), X-ray natural circular dichroism (XNCD), X-ray emission spectroscopy (XES), resonant inelastic X-ray scattering (RIXS), and X-ray two-photon absorption (XTPA). While before methods were especially designed for the simulation of X-ray spectroscopiese.g., the static exchange (STEX) method (section 3.5.1), the TP-DFT approach (section 3.4.2), and the CVS approximation (section 3.3) most modern developments aim at a uniform and balanced treatment of core-excited and valence-excited states by means of response or propagator theory and thereby automatically benefiting from the formal and practical virtues of these approaches. This is important because it opens up the door for

AUTHOR INFORMATION Corresponding Authors

*E-mail: [email protected]. 7239


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*E-mail: [email protected].

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ORCID

Patrick Norman: 0000-0002-1191-4954 Andreas Dreuw: 0000-0002-5862-5113 Notes

The authors declare no competing financial interest. Biographies Patrick Norman received his Ph.D. in 1998 from Linköping University, Sweden. He did a postdoc at the University of Odense, Denmark, and a series of postdoctoral secondments at the University of Ottawa, Canada. In 2008, he became a senior researcher of the Swedish Research Council in the subject matter of theoretical chemistry. This appointment came with a six-year funding for research and led to the promotion to professor in 2010. In 2012, Norman became the Director of National Supercomputer Centre (NSC), alongside his faculty position. In 2016, he left Linköping University and became Head of Theoretical Chemistry and Biology at KTH Royal Institute of Technology in Stockholm. Andreas Dreuw received his Ph.D. in theoretical chemistry from Heidelberg University in 2001. After a two-year postdoc at the University of California Berkeley, he joined the Goethe University of Frankfurt first as an Emmy-Noether fellow and then as a Heisenberg Professor for Theoretical Chemistry. Since 2011, Andreas Dreuw has held the chair for Theoretical and Computational Chemistry at the Interdisciplinary Center for Scientific Computing, Heidelberg University. His research interests comprise the development of electronic structure methods and their application in photochemistry, mechanochemistry, biophysics, and materials science.

ACKNOWLEDGMENTS P.N. acknowledges financial support from the Knut and Alice Wallenberg Foundation (Grant KAW-2013.0020) and the Swedish Research Council (Grant 621-2014-4646) as well as a fellowship from Heidelberg University to become a visiting professor at the Interdisciplinary Center for Scientific Computing. REFERENCES (1) X-Ray Absorption and X-Ray Emission Spectroscopy; Theory and Applications; van Bokhoven, J. A., Lamberti, C., Eds.; Wiley: 2016. (2) Tavares, P. F.; Leemann, S. C.; Sjöström, M.; Andersson, Å. The Max IV Storage Ring Project. J. Synchrotron Radiat. 2014, 21, 862− 877. (3) Synchrotron Radiation: Basics, Methods and Applications; Mobilio, S., Boscherini, F., Meneghini, C., Eds.; Springer: 2014. (4) X-Ray Free Electron Lasers: Applications in Materials, Chemistry and Biology; Bergmann, U., Yachandra, V. K., Yano, J., Eds.; Energy and Environment Series 18; Royal Society of Chemistry: 2017. (5) Bagus, P. S. Self-Consistent-Field Wave Functions for Hole States of Some Ne-Like and Ar-Like Ions. Phys. Rev. 1965, 139, A619−A634. (6) Solheim, H.; Ruud, K.; Coriani, S.; Norman, P. The a and B Terms of Magnetic Circular Dichroism Revisited. J. Phys. Chem. A 2008, 112, 9615. (7) Solheim, H.; Ruud, K.; Coriani, S.; Norman, P. Complex Polarization Propagator Calculations of Magnetic Circular Dichroism Spectra. J. Chem. Phys. 2008, 128, 094103. (8) Helgaker, T.; Coriani, S.; Jørgensen, P.; Kristensen, K.; Olsen, J.; Ruud, K. Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations. Chem. Rev. 2012, 112, 543−631. (9) Hercules, D. M. Electron Spectroscopy: Applications for Chemical Analysis. J. Chem. Educ. 2004, 81, 1751−1766. 7240


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