Time-Dependent Resonant Inelastic X-ray Scattering of Pyrazine at the Nitrogen K-Edge: A Quantum Dynamics Approach

Abstract

We calculate resonant inelastic X-ray scattering spectra of pyrazine at the nitrogen K-edge in the time domain including wavepacket dynamics in both the valence and core-excited state manifolds. Upon resonant excitation, we observe ultrafast non-adiabatic population transfer between core-excited states within the core-hole lifetime, leading to molecular symmetry distortions. Importantly, our time-domain approach inherently contains the ability to manipulate the dynamics of this process by detuning the excitation energy, which effectively shortens the scattering duration. We also explore the impact of pulsed incident X-ray radiation, which provides a foundation for state-of-the-art time-resolved experiments with coherent pulsed light sources.

Introduction

The development of high-brilliance synchroton X-ray radiation sources¹⁻³ and X-ray free electron lasers⁴⁻⁷ has enabled techniques that require high photon flux such as resonant inelastic X-ray scattering (RIXS)⁸ and its extension into the ultrafast time domain.⁹⁻¹⁷ RIXS constitutes a Raman scattering process in which the system is resonantly excited into short-lived core-hole states and spontaneously decays back to the electronic ground and excited states.^8,18 This technique combines the element specificity of core-level spectroscopy with the ability to reach valence-excited states across a wide spectral range (>20 eV) even in the condensed phase¹⁹ and at a spectral resolution that is not limited by the core-hole lifetime broadening, making it a versatile and promising tool to study the local electronic structure in complex molecular systems. RIXS to ground and valence-excited states has already been applied in various areas such as the nature of hydrogen-bond interactions,²⁰⁻²² the photochemistry of transition-metal complexes,²³⁻²⁹ ultrafast dynamics of molecules on surfaces,^30,31 proton transfer dynamics,^13,32,33 or core-excited state dynamics.³⁴⁻³⁶

Within the dipole approximation, RIXS follows electronic symmetry selection rules that offer valuable insights into the symmetry of occupied and unoccupied orbitals in molecules.³⁷⁻⁴⁰ For instance, by strictly adhering to the dipole selection rules during both the absorption and emission steps, the parity of the system must remain unchanged throughout the RIXS process. However, molecules with equivalent atoms always possess multicenter core orbitals that are delocalized over these atoms, resulting in nearly degenerate core-excited states, commonly referred to as the core-hole localization problem.⁴¹⁻⁴⁶ These states couple vibronically through a non-totally symmetric normal mode, thus corresponding to a symmetry-allowed conical intersection⁴⁷ and resulting in a final localization of the core holes.^18,37,48,49 This dynamical distortion of symmetry can enable transitions that are otherwise electronically symmetry-forbidden^38,39,50 and significantly impact the intensity of their RIXS signal.⁴⁹

One effective method to control or even prevent symmetry breaking in RIXS is to adjust the excitation energy to below the X-ray absorption resonance.^51,52 More generally, this detuning of the excitation energy allows for the manipulation of the effective lifetime in the core-excited states and, consequently, the scattering duration.^51,53,54 Hence, by varying the detuning, it is possible to control and manipulate dynamical processes within the femtosecond scattering duration, such as dissociation,^53,55 vibrational collapse,^56,57 or symmetry distortion.^36,58−60

There are different theoretical approaches to simulating RIXS spectra, which can be broadly classified into time-independent and time-dependent methods. In general, the quantum mechanical description of X-ray Raman scattering is based on second-order time-dependent perturbation theory which gives rise to the well-known Kramers–Heisenberg–Dirac (KHD) expression for the polarizabilty tensor in the frequency domain.⁶¹⁻⁶³ Starting from this sum-over-states form, time-independent methods rely on an eigenstate representation of the system, ideally considering all molecular vibronic states that contribute to the resonance effect.^64,65 Although various efficient techniques have been developed,⁶⁶⁻⁶⁹ as the size and complexity of the molecular system increase, summing over all the eigenstates of the vibrational modes in the molecular excited state becomes rapidly computationally demanding. Therefore, time-dependent strategies based on wave packet propagation on the excited state manifold have been proposed as alternatives, where the scattering amplitude is determined by the half-Fourier transform of the time-dependent overlap between the final and initial vibrational states.^63,70 Various levels of theory have been employed, including real-time propagation,^71,72 Green’s function techniques,^73,74 and methods based on time-dependent density functional theory,^75,76 demonstrating the extensive potential and applicability of time-domain formalisms.

In the present work, we examine the RIXS process of pyrazine at the nitrogen K-edge by using a comprehensive time-domain approach. We employ a full quantum mechanical treatment of both, the nuclear and electronic degrees of freedom, under symmetry-selective excitations in a diabatic representation of the Hamiltonian within the multiconfiguration time-dependent Hartree (MCTDH)⁷⁷ framework. To accurately depict dynamic processes occurring within the ultrashort core-hole lifetime and their manipulation through changes in the excitation frequency, we explicitly incorporate nuclear motion on the core-excited state manifold. Additionally, we explicitly introduce an arbitrary coherent spectral distribution, e.g., an incoming ultrashort X-ray pulse, and investigate how it manifests in the resulting spectra, thereby enabling an optimal interplay between theory and experiment.

Methodology

Model Hamiltonian

The full Hamiltonian H employed in the nuclear quantum dynamics simulations consists of a molecular Hamiltonian H_mol supplemented by the interaction with an external electromagnetic field H_int acting as a time-dependent perturbation

1

Following the semi-classical ansatz of ref (78), the coupling to the external field is assumed to be accurately captured by the dipole approximation

2

where is the electric field of the photon beam and μ_αβ is the transition dipole moment between the electronic states |α⟩ and |β⟩, and where the Condon approximation and a single polarization direction are assumed. Within the rotating wave approximation, the electric field is represented by

3

with the temporal envelope function of the laser pulse centered at t₀ and with carrier frequency ω₀. In particular, we refer to a continuous wave (CW) experiment if .

As in ref (78), we decouple the valence and core-excited states,^79,80 yielding the following matrix representation of the molecular Hamiltonian

4

where H_v and H_c are sub-Hamiltonians acting on the manifolds of the valence and core-excited electronic states, respectively. In both subspaces, a vibronic coupling model up to second order^47,48,81 is employed, leading to coupled potential energy surfaces (PESs) in a diabatic representation.^82,83 In this framework, each submatrix in eq 4 is expressed as

5

where the zeroth-order Hamiltonian is constituted by the ground-state Hamiltonian in the harmonic approximation

6

with ω_i representing the frequency of mode Q_i. The diabatic potential matrix W_x covers all changes in the excited state surfaces compared to the ground state, whose matrix elements are in this work given by

7

8

with the vertical energy displacement E^(α) of the α-th electronic state, the linear intrastate coupling constant κ^(α)_i related to the gradients of the adiabatic potentials at the Franck–Condon point, the quadratic intrastate coupling constant γ^(α)_i enclosing changes in the frequencies, and the linear interstate coupling constants λ^(αβ)_i belonging to the non-adiabatic interaction between the α-th and β-th electronic states under displacements of the Q_i mode.

For strongly anharmonic modes, the harmonic expression of the diabatic potentials is replaced by Morse potentials

9

where D^(α)₀ denotes the state-specific dissociation energy, α^(α)_i defines the curvature of the potential, and Q₀ is the equilibrium position.

The parameters required to describe the diabatic molecular Hamiltonian were obtained from electronic structure calculations performed at the coupled cluster singles and doubles (CCSD) level of theory⁸⁴ and its extensions for excited states using Dunning’s correlation consistent basis set aug-cc-pVDZ.⁸⁵ Ground-state geometry optimization and normal mode evaluation were computed using the quantum chemistry software package Gaussian.⁸⁶ Excited-state calculations were performed using the quantum chemistry software package Q-Chem⁸⁷ where adiabatic energies, energy gradients, and non-adiabatic coupling terms for the valence-excited states were obtained employing the equation-of-motion (EOM-) CCSD approach⁸⁸ while core-excited state properties were computed using the frozen-core/core–valence-separated (fc-CVS-) EOM-CCSD method.⁸⁹ The latter two quantities were projected from Cartesian coordinates to the normal mode coordinate representation; for this purpose, the VCHam tools within the Heidelberg MCTDH package were used⁹⁰

10

11

where H_el denotes the electronic Hamiltonian. In the case of the most relevant electronic states, the corresponding excited state parameters were refitted through a least-squares procedure applied to a series of ab initio single-point energy calculations along each normal mode.

Derivation of Spectral Properties

The underlying mechanism of RIXS is a two-photon process that first involves interaction with an incident X-ray beam, creating an evolving wavepacket on the core-excited state manifold, followed by spontaneous emission of a photon. The time-dependent picture of RIXS can hence be retrieved by second-order time-dependent perturbation theory for the light–matter interaction.^70,91

Assuming a monochromatic and perturbative CW excitation, the RIXS cross-section I_RIXS can be obtained from the time evolution of the second-order correction to the system wave function⁹²

12

Following Lee and Heller,^70,92 the time-dependent second-order perturbation wave function is recursively defined by

13

with the first-order wave function

14

where |ϕ_i⟩ ≡ |ϕ_i(−∞)⟩ is the initial unperturbed wave function with eigenenergy E_i, μ_I and μ_S are the transition dipole moment operators along the direction of the incident and the scattering field, ε_I and ε_S are monochromatic with frequency and , respectively, and where is defined by . Furthermore, Γ_c denotes the intrinsic lifetime broadening of the core-excited states. Changing the variables τ = t′ – t″, the first-order wave function can be rewritten as

15

16

with the Raman wave function given by

17

Although the Raman wave function is not an eigenfunction of either Hamiltonian involved, it forms a pseudo-time-independent intermediate state, which contains all dynamical information prior to the scattering event. Finally, inserting eqs 16 and 13 into eq 12 and including the inverse final state lifetime Γ_f leads to a time-dependent expression for the total RIXS spectrum

18

where and is an evolving wavepacket on the ground and valence-excited state manifold. Furthermore, defines the energy loss of the system.

Performing the time-integral in eq 18 yields the frequency-dependent form of the RIXS cross section

19

where the scattering amplitude α_fi is governed by the well-known KHD formula

20

with |i⟩, |n⟩, and |f⟩ being the initial, intermediate, and final vibronic eigenstates involved in the RIXS process with energies E_i, E_n, and E_f, respectively. Furthermore, the Lorentzian line shape function

21

with full width at half-maximum (fwhm) Γ causes a phenomenological broadening. The KHD expression thus provides a static perspective of the RIXS event, where the scattering amplitudes are dependent on the Franck–Condon overlaps of all relevant states as well as the amount of detuning from each intermediate state.

Both, the time-dependent (eq 18) and time-independent (eq 19) expressions for the RIXS cross sections rely on CW conditions and thus neglect any contributions stemming from the spectral content of the incident radiation. In order to incorporate the spectral content required for a finite duration of the incident field, we go beyond the CW picture and assume an -shaped X-ray pulse as given in eq 3, describing a more realistic situation for time-resolved experiments. Without loss of generality, the pulse is centered at t₀ = 0 and has the carrier frequency . The first-order wave function now reads

22

while the second-order wave function is still recursively defined as given in eq 13. Using the Fourier representation of the envelope function

23

and again changing the variables τ = t′ – t″, the first-order wave function can be rewritten as

where * denotes the convolution and is given by

28

Analogously to eqs 13 and 16, we recursively define by

29

yielding the following expression for the second-order wave function

30

since the convolution is linear. Within the spontaneous emission process, a transition from every eigenstate component of a wavepacket occurs independently, populating a different final state for large enough t (see Appendix A) and thus

and use of eq 12 leads to

33

This dependence demonstrates that the impact of a limited pulse duration has two distinct aspects. First, when longitudinally coherent X-ray pulses with a wide spectral range are used, more resonances are encompassed within the coherent pulse excitation. This not only affects the finer vibronic substructure and peak intensity in the final spectra but also decreases the spectral sensitivity of the detuning effect. Additionally, the finite duration of the incident X-ray beam causes a general broadening of the RIXS spectra, given by the convolution with the squared absolute value of the Fourier transform of the incident field envelope function. Note that an incoherent spectral distribution as encountered at synchrotron radiation facilities is not described by a spectral convolution but rather a spectrally weighted average.

Quantum Dynamics

The nuclear time-dependent Schrödinger equation was solved employing the Heidelberg implementation of the multilayer (ML-) MCTDH method.⁹³⁻⁹⁶ In this approach, MCTDH is recursively applied through different layers of effective modes, within which the equations of motion are determined from the Dirac–Frenkel variational principle.^97,98 The layer structure as well as the discrete variable representation, number of grid points, and single-particle function (SPF) basis size were adapted from the ML-2 model in ref (93) without modifications except for the number of electronic states. In particular, this ML ansatz for a full dimensional wave function including all vibrational normal modes was proven to offer a reasonable balance of quality and computational effort. The details are reprinted in Figure 1 for completeness. Core-excited state propagations were run until the pseudo-time-independent intermediate state is reached while dynamics happening in the valence states were run for 100 fs. The data output was written every 0.1 fs, guaranteeing a sufficient time resolution and frequency span for subsequent Fourier transforms.

Figure 1.

Tree structure of the ML-MCTDH simulations, including all 24 normal modes. The maximum depth of the tree is five layers where the first one separates the 24 vibrational coordinates and the discrete electronic degree of freedom in the single-set formulation. The circles represent the node in the layer structure, and the number n of SPFs (blue) is given next to the link lines. The last layer comprises the vibrational normal modes, where the number N of primitive functions (red) is used to represent the grid. All values are taken from ref (93).

Results and Discussion

Parametrization of the Model Hamiltonian

Pyrazine (C₄H₄N₂) belongs to point group D_2h at neutral ground-state equilibrium geometry and possesses 24 vibrational normal modes. The vibrational frequencies computed at the CCSD/aug-cc-pVDZ level of theory are listed in Table 1 along with their symmetry and a comparison to the experimental data.

Table 1. Harmonic Ground-State Vibrational Frequencies (in cm^–1) Obtained at the CCSD/aug-cc-pVDZ Level along with the Experimental Data^99a.

mode	Wilson	symmetry	cm–1	Exp.
ν1	ν16a	au	350	341
ν2	ν16b	b3u	425	420
ν3	ν6a	ag	604	596
ν4	ν6b	b3g	709	704
ν5	ν4	b2g	737	756
ν6	ν11	b3u	805	785
ν7	ν10a	b1g	943	919
ν8	ν5	b2g	950	983
ν9	ν17a	au	980	960
ν10	ν12	b1u	1033	1021
ν11	ν1	ag	1038	1015
ν12	ν18b	b2u	1091	1063
ν13	ν14	b2u	1146	1149
ν14	ν18a	b1u	1162	1136
ν15	ν9a	ag	1253	1230
ν16	ν3	b3g	1367	1346
ν17	ν19b	b2u	1441	1416
ν18	ν19a	b1u	1518	1484
ν19	ν8b	b3g	1595	1525
ν20	ν8a	ag	1650	1582
ν21	ν7b	b3u	3196	3040
ν22	ν13	b1u	3197	3012
ν23	ν20b	b2u	3213	3063
ν24	ν2	ag	3218	3055

^aThe modes are labeled by ascending frequency and compared to Wilson’s notation.¹⁰⁰

The diabatic Hamiltonian H_mol comprises two sub-Hamiltonian H_v and H_c, accounting for the valence- and core-excited states dynamics, respectively. H_c involves two core-excited states that are nearly degenerate at the Franck–Condon point. In the vicinity of that point, the next energetically higher electronic states are well separated and can hence be disregarded. The four vibrational modes with symmetry B_1u are potential candidates to vibronically couple the two nearly degenerate core-excited states where the conical intersection formed along these modes takes place exactly at the Franck–Condon point. Cuts of the PESs along the two strongest coupling modes, ν₁₀(B_1u) and ν₁₈(B_1u), are shown in Figure 2. The lifetime of the core-excited states is assumed to be 8 fs¹⁰¹ captured by an imaginary energy term in the core Hamiltonian H_c.

Figure 2.

Cuts through the diabatic PESs along the ν₁₀(b_1u) and ν₁₈(b_1u) vibrational normal modes that mainly drive the symmetry breaking in the core-excited states. Label and symmetry for each state can be found in Table 2. The adiabatic energies (black points) are obtained from ab initio calculations using the (fc-CVS-) EOM-CCSD/aug-cc-pVDZ method.

Since RIXS has the ability to reach higher-lying valence-excited states, we include 20 electronic states in H_v. These states consist of the ground state and the 19 energetically lowest valence-excited singlet states, covering an approximate spectral range of 10 eV. The computed vertical excitation energies and symmetries for all electronic states are listed in Table 2. It is worth noting that the ordering of very closely lying states can differ between different levels of theory.^102,103 This concerns not only S₂(B_2u) and S₃(A_u) but also the region with a high density of states. To maintain a consistent model within this study, we treat every valence-excited state at the same level of theory without any adjustment of the vertical excitation energy.

Table 2. State Symmetries and Vertical Excitation Energies E^(α) (in eV) at the Franck–Condon Point of All Electronic States Considered in This Work^a.

state	symmetry	E(α)
S0	Ag	0.00
S1	B3u	4.32
S2	B2u	5.07
S3	Au	5.13
S4	B2g	6.01
S5	Ag	6.68
S6	B1u	6.91
S7	B1g	7.02
S8	B1g	7.11
S9	B2u	7.28
S10	B1u	7.47
S11	B3u	7.66
S12	B2u	7.94
S13	B3g	8.00
S14	Ag	8.04
S15	Au	8.12
S16	B1u	8.14
S17	B1g	8.35
S18	Au	8.36
S19	B2g	8.53
X1	B2g	402.30
X2	B3u	402.30

^aValence-excited state energies were evaluated using EOM-CCSD while core-excited state energies were obtained using (fc-CVS-) EOM-CCSD. In both cases, the aug-cc-pVDZ basis set was used.

Both sub-Hamiltonians, H_v and H_c, were then approximated by a vibronic coupling model including all vibrational degrees of freedom. The highest frequency mode ν₂₄ was fitted by Morse potentials to account for anharmonicity. Since the density of valence-excited states above 6 eV increases rapidly, we used the linear intra- and interstate coupling constants as obtained from EOM-CCSD/aug-cc-pvDZ calculations at the Franck–Condon point to parametrize these electronic states (S₅ – S₁₉). The linear coupling parameters, which mainly drive the core-excited state dynamics at short times, are listed in Table 3. A complete list of all parameters is further provided in the Supporting Information.

Table 3. Linear Intra- and Interstate Coupling Constants κ⁽ⁿ⁾_i and λ^(nm)_i, Respectively, for the Core-Excited States, X_n, Obtained in This Work.

	κ3	κ11	κ15	κ20
X1	0.02738	–0.04034	0.05568	0.10433
X2	0.02627	–0.04050	0.05619	0.10457

	λ10	λ14	λ18	κ22
(X1, X2)	0.08680	0.01326	0.09910	0.03014

The transition dipole moments and oscillator strengths for each pair of valence- and core-excited states were computed at the Franck–Condon point using the same level of theory. We neglect transitions where the transition dipole moment was below 0.01 and thus less than 10% of the strongest core–valence transition. The corresponding transition dipole moments are listed in Table 4.

Table 4. Transition Dipole Moments μ_fi between States i and f Obtained from fc-CVS-EOM-CCSD/aug-cc-pVDZ Calculations.

state transition	μfi
X2 ← S0	0.10
X1 ← S1	0.06
X2 ← S4	0.06
X1 ← S6	0.02
X1 ← S16	0.04
X1 ← S18	0.04

Symmetry Distortion Induced by Core-Excited State Dynamics

We consider the scattering event initiated by the resonant absorption of an X-ray photon, which promotes an electron from a nitrogen (N-)1s orbital to the lowest unoccupied molecular orbital (LUMO). Pyrazine contains two equivalent N atoms, so a linear combination of both N-1s orbitals builds symmetric and antisymmetric core-orbitals that are nearly degenerate. Excitation from these core-orbitals leads in turn to two nearly degenerate core-excited states, X₁(B_2g) and X₂(B_3u). Although only transitions from the antisymmetric core orbitals are allowed by symmetry, both core-excited states must be considered to account for nonadiabatic transitions induced by vibronic coupling. Figure 3 shows the static X-ray absorption spectrum, resulting from an electronic transition between the antisymmetric core orbital and the LUMO. The absorption spectrum was obtained from the Fourier transform of the autocorrelation function of the dipole-operated ground state, thus equivalent to a δ-pulse excitation.^92,104,105

Figure 3.

Computed static X-ray absorption spectrum of pyrazine at the nitrogen K-edge. The vertical lines indicate the excitation energies used to simulate the RIXS spectra. Here, the energy related to the dashed black line corresponds to the vertical excitation energy of X₂ used to calculate the RIXS spectrum shown in this section. The energies indicated by the colored lines relate to the detuning effect discussed in the next section.

The related diabatic population transfer is illustrated in Figure 4. After an instantaneous, vertical excitation at time 0, the bright X₂(B_3u) state rapidly depopulates into the dark X₁(B_2g) state, leading to an almost equal distribution of population between the two states within the short core-hole lifetime. This ultrafast, nonadiabatic population transfer can be traced back to the symmetry-allowed conical intersection between these states, which is formed in the immediate vicinity of the Franck–Condon point (see Figure 2) and primarily driven by the two asymmetric normal modes ν₁₀ and ν₁₈. These dynamics result in a final localization of the core-hole.

Figure 4.

Core-excited state populations of the diabatic X₁(B_2g) (green) and X₂(B_3u) (magenta) states starting from X₂(B_3u) at the Franck–Condon point. The solid lines present the total diabatic state population including the core-hole decay while the dashed lines correspond to the normalized diabatic state population.

The impact of symmetry breaking induced by core-excited state dynamics is clearly seen in the RIXS spectrum of pyrazine at the nitrogen K-edge presented in Figure 5. The spectrum was obtained using eq 18, assuming a monochromatic X-ray photon beam with carrier frequency equal to the vertical excitation energy from the ground state to the excited state X₂. It displays five prominent bands arising from six electronic transitions. The asymmetric shape of the elastic peak already suggests that the system undergoes nuclear displacements on the core-excited state manifold. As expected from the transition dipole moments (see Table 4), the elastic peak is the dominant feature of the spectrum but without obscuring any contributions from inelastic transition due to the significant energy gap between the ground and valence-excited states. Moreover, the spectrum reveals four bands of inelastic electronic transitions located at approximately 4.5, 6.0, 7.0, 8.1, and 8.6 eV where the latter two exhibit a large spectral overlap and are thus barely distinguishable in the overall spectrum. While the emission band located at 6.0 eV is attributable to the electronic X₂ → S₄ transition, the other four bands originate from population of the optically dark X₁(B_2g) state which can only be accessed through nonadiabatic population transfer from the bright X₂(B_3u) state. Consequently, the symmetry breaking caused by ultrafast core-excited state dynamics allows for four additional electronic transitions that would otherwise be forbidden for two consecutive dipole transitions (i.e., by quadrupole selection rules) within this spectral range.

Figure 5.

Simulated RIXS spectrum of pyrazine at the nitrogen K-edge. The contributions of the transitions to S₀, S₁, S₄, S₆, S₁₆, and S₁₈ to the total spectrum are highlighted in purple, blue, green, cyan, red, and orange, respectively. The elastic peak (purple) is downscaled to 25% for better visualization. Before performing the Fourier transformation in eq 18, the window function was applied to reduce the Gibbs phenomenon. Here, we assume a damping time of T = 30 fs in order to include broadening caused by dephasing mechanisms.

Dynamical Control by Detuning

Dynamical processes in the core-excited states can be controlled by detuning the frequency of the incident radiation that is used to prepare the intermediate state of the system. The effective scattering duration in RIXS is determined by the time the wavepacket spends in the intermediate core-excited states, i.e., the time interval [0, τ] that contributes to the Raman wave function (eq 17). Due to the uncertainty relation for the energy, the propagation time shortens the further the excitation energy is from resonance and is given by¹⁰⁶

34

where Ω is the amount of detuning and Γ_c inversely determines the core-hole lifetime.

The dependence of the RIXS signal for pyrazine on the incoming photon frequency, ω_I, is shown in Figure 6. The four inelastic emission channels stemming from the dark X₁(B_2g) state exhibit a strongly detuning-dependent intensity. In particular, when far from resonance, i.e., for incoming excitation energies below ∼402.0 eV, these bands almost disappear. Moreover, in this region, the elastic peak exhibits a symmetric Lorentzian line shape indicative of a single transition. Both observations show that the effective scattering duration becomes so short for large detuning that dynamical effects are negligible. In contrast, for excitation energies greater than 402.0 eV, the signature of vibrational progressions is apparent on the positive energy side of the elastic peak, and inelastic Raman transitions stemming from both core-excited states, X₁ and X₂, are prominent. Their shapes differ for different excitation energies, reflecting the nuclear motion in the core-excited states. In particular, for excitation energies greater than the vertical excitation energy, a rather broad and asymmetrical shape can be observed for each loss peak.

Figure 6.

Dependence of the RIXS profile on the incident photon energy. All spectra are independently normalized to their respective elastic peak height before rescaling the signals in the gray spectral region to 25%. The energies of the incident radiation used for these calculation are also highlighted in the photoabsorption band, as shown in Figure 3.

Spectral Distribution due to Finite Pulse Duration

While the previous sections are based on light–matter interaction with monochromatic plane X-ray waves, in the following section, we investigate the influence of the incident X-ray spectrum on the resolution of the RIXS spectra. The RIXS cross sections are calculated using eq 33 where the external electric field was given by a normalized Gaussian-shaped X-ray pulse, i.e., the pulse envelope function is

35

where t₀ denotes the pulse center and the standard deviation σ is linked to the temporal fwhm duration F_t by . According to eq 33, a spectral broadening of the RIXS spectrum is caused by the convolution with the square of the absolute value of the Fourier transform of the envelope function. For an envelope function of the form in eq 35, this is again a Gaussian function

36

with spectral fwhm . In particular, the fwhm ratio is here F_t·F_ω ≈ 3.921.

Due to the reciprocal connection between temporal and spectral width, shorter pulses create spectrally broader wavepackets and vice versa, leading to different signatures of the RIXS signal. This aspect is described by the term

37

in eq 33 and can be considered independently from the general broadening of the signal described by the convolution with although both aspects have the same origin. Figure 7 illustrates both effects on the RIXS spectra of pyrazine for different incoming X-ray pulses where the temporal duration was varied, but the carrier frequency of the pulse was kept fixed at 402.3 eV. The spectra shown in blue are evaluated in accordance with eq 37, while the orange shadowed spectra represent the total cross section as obtained from eq 33.

Figure 7.

Dependence of RIXS signals on the duration of the incident radiation field. Each spectrum was derived using eq 33 (orange shadow) and eq 37 (blue line), where a Gaussian X-ray pulse was used to trigger the core-excited state dynamics. The carrier frequency was kept fixed while the temporal standard deviation varies from 1 to 8 fs (equivalent to Γ_c/8 to Γ_c). The same window function as that in Figure 5 was applied before performing the Fourier transformation.

In general, the different RIXS spectra are in good agreement. However, the relative intensities vary depending on the length of the X-ray pulse. Specifically, emission channels starting from the dark X₁ at around 6.0, 7.0, and 8.4 eV appear to be more likely for shorter pulse duration. Furthermore, the shape of the spectral lines changes slightly depending on the distribution of the field. This is evident in the X₁ → S₁ emission band, where distinct progressions become more prominent with longer pulse duration.

While for pulses with FWHM of about a third of the dephasing time of the valence excitations or longer, the broadening resulting from the spectral distribution of the field is less significant compared to other dephasing effects, shorter pulses cause a more pronounced broadening of the signal, as shown by the orange shadow. This effect should hence be considered for optimal spectral and temporal resolution as well as the detuning capability. Especially for very short pulses, the finer details of the vibronic structure in the emission bands are almost completely suppressed, which can complicate the analysis of core-excited state dynamics when the effectiveness of detuning as a control mechanism is also reduced.

Conclusions

In this study, we benchmarked a fully time-dependent approach within the MCTDH framework to simulate RIXS spectra. We used pyrazine as a model system to examine this resonant scattering process at the nitrogen K-edge, considering wavepacket dynamics in all of the electronic states of the model. Our findings reveal ultrafast symmetry breaking of the molecule in the core-excited states, leading to a significant alteration of the symmetry selection rules. This behavior underscores the crucial role of non-adiabatic nuclear core-excited state dynamics for the correct description of RIXS signals. Furthermore, we explored the detuning dependence of the RIXS spectra in our time-dependent approach. Our results highlight the importance of detuning as a control mechanism for ultrafast dynamical processes, which allows for predicting not only the intensity of both inelastic and elastic emission channels but also the vibrational substructure of individual electronic transitions. Additionally, we investigated how the duration of a coherent X-ray pulse affects the RIXS spectrum.

In summary, we have described RIXS as a fully time-dependent process including electronic and nuclear dynamics in all core- and valence-excited states of the model and with a temporally arbitrary incident X-ray field. Our work demonstrates how the spectral distribution of the X-ray pulse impacts the individual RIXS pathways, and how it leads to a general broadening effect of the loss spectra. This research provides insights into the intricacies of inelastic X-ray scattering in the presence of core-hole symmetry breaking, under detuning, and for excitations with coherent X-ray pulses on the order of or shorter than the core-hole lifetime. Furthermore, our time-dependent simulation approach can be used for modeling ultrafast RIXS experiments at pulsed X-ray sources such as free-electron lasers.

Appendix A

Emission from a Wavepacket

In order to derive an expression for the probability that a photon with energy ℏω is emitted from a wavepacket, we consider a system containing two electronic states. The total Hamiltonian reads

38

39

where the Hamiltonian describes the unperturbed system and the interaction Hamiltonian is treated as perturbation. Here, the molecular Hamiltonian of the two-state system is given by

40

where E_n and |n⟩, n ∈ {i, f}, are the eigenenergies and eigenstates, respectively, of the initial and final electronic state. The light Hamiltonian is further defined through

41

with the vacuum state |vac⟩ where we ignore directions and polarizations for simplicity. For an interaction operator between the molecule and photon, the light–matter interaction term can be written as

42

43

Assuming a wavepacket has been initially created as a superposition of the initial electronic state, i.e.,

44

the transition rate Γ_ω of emitting photon with energy ℏω is defined by

45

where the transition probability P_ω(t)≔|⟨ω|ψ(t)⟩|² can be evaluated using first-order perturbation theory

46

where |ψ⁽⁰⁾(0)⟩ = |ψ(0)⟩ and |ψ⁽¹⁾(0)⟩ = 0. Considering only induced transitions between two different states, the probability then reads

47

where the first-order correction ψ⁽¹⁾ to the wavefunction of the system ψ is defined by

48

with

49

50

51

52

As ∫dω⟨ω′|ω⟩ = ∫dωδ(ω′ – ω) = 1, it follows

53

54

55

56

where Ω_if≔ω_i – ω_f. Hence, the emission probability per unit time in first-order correction is given by

57

58

59

For Ω_if ≠ ω and Ω_jf ≠ ω, it holds

60

and in the same way

61

Therefore, it follows for the product of a zero and bounded sequence

62

and the transition rate Γ_ω thus vanishes for Ω_if, Ω_jf ≠ ω.

Let now Ω_if ≠ ω but Ω_jf = ω. Applying the rule of l’Hopital

63

we can show that the fraction above is purely imaginary. Hence, it follows

64

65

66

67

68

and thus again Γ_ω → 0 for t → ∞. Analogously, Γ_ω is zero for Ω_if = ω and Ω_jf ≠ ω. Hence, only the case Ω_if = ω and Ω_jf = ω, so in particular i = j, contributes to the transition rate. Starting from eq 59, we get

69

70

71

72

73

74

where |i_γ(t)⟩ denotes the propagation of the projected eigenstate |i⟩ onto the final excited state manifold. So, every eigenstate from the wavepacket emits independently, where the contribution of a certain eigenstate is determined by the evolution of the projected wavepacket |i_γ(t)⟩ weighted by the initial population of state |i⟩. In particular, the spontaneous emission from a wavepacket is incoherent, in the sense that every transition from this wavepacket populates a different and hence orthogonal final state of the material system and the electromagnetic field.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c01259.

• On-diagonal linear intrastate coupling constants κ⁽ⁿ⁾_i, off-diagonal linear interstate coupling constants λ^(nm)_i, and on-diagonal bilinear (quadratic) coupling constants γ⁽ⁿ⁾_i and the parameters concerning the Morse potential (PDF)

The authors declare no competing financial interest.