Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Andrés
Moreno Carrascosa
and
Adam
Kirrander
*

EaStCHEM, School of Chemistry, University of Edinburgh, David Brewster Road, EH9 3FJ Edinburgh, UK. E-mail: Adam.Kirrander@ed.ac.uk

Received
30th March 2017
, Accepted 26th April 2017

First published on 26th April 2017

Nonresonant inelastic electron and X-ray scattering cross sections for bound-to-bound transitions in atoms and molecules are calculated directly from ab initio electronic wavefunctions. The approach exploits analytical integrals of Gaussian-type functions over the scattering operator, which leads to accurate and efficient calculations. The results are validated by comparison to analytical cross sections in H and He^{+}, and by comparison to experimental results and previous theory for closed-shell He and Ne atoms, open-shell C and Na atoms, and the N_{2} molecule, with both inner-shell and valence electronic transitions considered. The method is appropriate for use in conjunction with quantum molecular dynamics simulations and for the analysis of new ultrafast X-ray scattering experiments.

New X-ray Free-Electron Lasers (XFELs), in turn, generate high intensity and short duration pulses^{13–19} that enable time-resolved X-ray scattering,^{20–25} and thus ultrafast imaging of photochemical dynamics.^{26} An attractive feature of such experiments is that they provide direct access to the evolution of molecular geometry via the elastic scattering.^{27} However, questions remain regarding the degree to which inelastic contributions to the scattering signal can be ignored when analysing experiments, especially in regions where the separation of different electronic states is small.^{28–31} For instance, the inelastic contributions are known to be important for imaging of electronic wavepackets in atoms.^{32–35} A full theoretical analysis of ultrafast X-ray scattering, beyond the conventional elastic approximation, will require matrix elements corresponding to IXS,^{31} which provides an important incentive for the work presented in this article. In addition, it is conceivable that once the appropriate theoretical and computational tools for a more detailed analysis of ultrafast X-ray scattering experiments are in place, more detailed information can be extracted regarding the electron dynamics that accompanies the structural dynamics of a photochemical process.

In the following, we outline a method for the calculation of IXS from ab initio electronic structure calculations in atoms and molecules, based on our previously developed code for the prediction of elastic X-ray scattering.^{36–38} An important objective is to match the level of accuracy required for quantum molecular dynamics simulations of photochemical reactions,^{31} which generally implies a high-level multiconfigurational description of the electronic structure (e.g. CASSCF, CASPT2, or MRCI), while not quite reaching the level of sophistication possible when evaluating IXS from atomic targets or very small molecules (e.g. R-matrix theory) in order to maintain a necessary degree of computational efficiency. In the following, we will outline the theory, present our computational approach, and demonstrate that we can calculate IXS accurately.

(1) |

(2) |

(3) |

(4) |

The matrix elements L_{βα} = 〈Ψ_{β}||Ψ_{α}〉 in eqn (3) originate from the · terms in the interaction Hamiltonian,^{2} where is the vector potential of the electromagnetic field, at first order of perturbation theory. The competing contributions from the terms in the Hamiltonian, which for scattering appear in second order, are sufficiently small to be disregarded.^{40} Diagonal matrix elements, L_{αα} = 〈Ψ_{α}||Ψ_{α}〉, correspond to elastic scattering and are equivalent to the Fourier transform of the target electron density, a circumstance that underpins the role of elastic scattering in structure determination.^{1} Further details regarding the calculation of elastic scattering from ab initio wavefunctions can be found in ref. 36 and 41. The remaining off-diagonal, α ≠ β, matrix elements correspond to nonresonant IXS, also referred to as Compton scattering, and are the focus of this article. The elastic and inelastic matrix elements for X-ray scattering, L_{βα}, are a necessary ingredient in detailed treatments of ultrafast X-ray scattering from non-stationary quantum states by coherent X-ray sources such as XFELs, see e.g.ref. 31, and the requirement for these matrix elements is one of the motivations for the work presented in this article.

There is an immediate link between IXS and inelastic scattering of fast charged particles, such as electrons, which has been exploited extensively in Electron Energy-Loss Spectroscopy (EELS).^{42} The inelastic scattering of electrons is described by the same matrix elements as IXS,^{43,44} although the approximations involved are more severe for electrons than X-rays.^{45} Formally, eqn (1) pertains to electron scattering if the Thomson differential cross section, (dI/dΩ)_{Th}, is replaced by the corresponding Rutherford cross section, (dI/dΩ)_{Ru}. The scattering elements for elastic electron scattering are not quite identical to X-ray scattering, since they contain additional contributions from electron-nuclei scattering.

The similarity between electron and X-ray scattering in the first Born approximation can be emphasized by the use of generalised oscillator strengths (GOS).^{44} In brief, the GOS renormalizes the spectra using the Bethe f-sum rule,^{43}

(5) |

(6) |

(7) |

(8) |

Scattering matrix elements L_{βα} between electronic states β and α are thus given by,

(9) |

To evaluate the matrix elements, we note that is the sum of one-electron operators, leading to three standard cases for the evaluation of the brackets on the right-hand side of eqn (9).^{48} The first case occurs if the two Slater determinants are identical,

(10) |

(11) |

2.2.1 Evaluation of matrix elements.
The next step requires the evaluation of the integrals that contribute to the matrix elements in eqn (3), corresponding to the brackets listed in eqn (10) and (11). These one-electron integrals over spatial orbitals, expressed in a Gaussian basis, can be evaluated analytically as outlined in the following. The molecular orbitals ϕ_{j}(r_{j}) are obtained as linear combinations of the basis functions G_{k}(r),

where ^{j}_{k} are the molecular orbital expansion coefficients. The total number of basis functions G_{k}(r) is N_{BF}, with j ∈ N_{MO} = N_{BF}. Each basis function G_{k}(r), in turn, is a contraction of Gaussian-type orbitals (GTOs), g_{s}(r), such that,

where μ^{k}_{s} are the basis set contraction coefficients for the primitive GTOs. A Cartesian Gaussian-type orbital centered at coordinates r_{s} = (x_{s},y_{s},z_{s}) has the form,

with exponent γ_{s}, Cartesian orbital angular momentum L_{s} = l_{s} + m_{s} + n_{s}, and normalisation constant _{s},

where !! denotes the double factorial. The usage of Cartesian GTOs is convenient in the present context, but there is a direct mapping between Cartesian and spherical Gaussians.^{49} If spherical Gaussians are used the mathematics of the analytic Fourier transform takes a different form.^{50}

where we use the Gaussian product theorem^{51} to rewrite the product as,

where is the pre-factor and is the new Gaussian centered at with exponent . Since the Cartesian coordinates (x,y,z) are linearly independent and each Gaussian function can be written as a product of x, y and z components,

the problem is reduced to the solution of one-dimensional Fourier transforms . These can be determined analytically, as has been shown and tabulated in a previous publication.^{36}

(12) |

(13) |

g_{s}(r) = _{s}(x − x_{s})^{ls}(y − y_{s})^{ms}(z − z_{s})^{ns}e^{−γs(r−rs)2}, | (14) |

(15) |

The one-electron bracket in eqn (10) and (11) can then be evaluated as,

(16) |

(17) |

(18) |

(19) |

The IXS cross sections in this article are given in terms of the dynamic structure factor, S(q,ω_{β}), or the generalized oscillator strength, GOS(q,), with the choice between the two representations determined by the source of the reference data used for comparison. All calculated data is given at perfect energy resolution, i.e. with no averaging over energy (Δω = 0 in eqn (6)). The results for Ne and N_{2} are rotationally averaged to match published data. Furthermore, the astute reader will notice that some graphs show the cross sections as a function of q, while others as a function of q^{2}. The choice, again, reflects the source of the reference data, with EELS measurements (or IXS measurements that compare to EELS data) generally shown as a function of q^{2} in order to offset the small angle of scattering for EELS.

He |
^{1}S_{0}(1s2s) |
^{1}P_{1}(1s2p) |
||
---|---|---|---|---|

E (eV) | ΔE (%) | E (eV) | ΔE (%) | |

Exp.^{57} |
20.615 | — | 21.218 | — |

PVQZ | 20.793 | 0.8 | 23.943 | 12.8 |

PV5Z | 20.748 | 0.6 | 23.078 | 8.8 |

PV6Z | 20.684 | 0.3 | 22.667 | 6.8 |

d-PV5Z | 20.000 | 3.0 | 20.680 | 2.5 |

Fig. 2 Calculated dynamic structure factor, S(q,ω), in He for the ^{1}S_{0}(1s2s) ← ^{1}S_{0}(1s^{2}) transition compared to results from Cann and Thakkar.^{58} The numerical calculations are performed with CASSCF(2,10) and four Dunning basis sets (aug-cc-PVQZ, aug-cc-PV5Z, aug-cc-PV6Z, and d-aug-cc-PV5Z). |

In Fig. 3 we compare our calculated dynamic structure factors for the two transitions ^{1}S_{0}(1s2s) ← ^{1}S_{0}(1s^{2}) and ^{1}P_{1}(1s2p) ← ^{1}S_{0}(1s^{2}) in He to experimental results by Xie et al.^{7} and reference calculations by Cann and Thakkar.^{58} The agreement between present and previous calculations and the experimental data is good, with the reference calculations reproducing experiments slightly better for q < 2 a.u. The fact that the energy convergence for the ^{1}S_{0} state is better than for the ^{1}P_{1} state in our calculations (see Table 1) appears to have little effect on the agreement between the dynamic structure factor for the two transitions in our calculations and the experimental results, with the convergence of ^{1}S_{0}(1s2s) ← ^{1}S_{0}(1s^{2}) only marginally better than for ^{1}P_{1}(1s2p) ← ^{1}S_{0}(1s^{2}). Notably, for both transitions the best agreement with experimental data is achieved with the basis set that yields the best energy convergence for that state (see Table 1).

Fig. 3 Calculated dynamic structure factor, S(q,ω), in He for the ^{1}S_{0}(1s2s) ← ^{1}S_{0}(1s^{2}) and ^{1}P_{1}(1s2p) ← ^{1}S_{0}(1s^{2}) transitions compared to theory by Cann and Thakkar^{58} and experiments by Xie et al.^{7} The ab initio calculations are done at the CASSCF(2,10)/aug-cc-PV6Z and the CASSCF(2,10)/d-aug-cc-PV5Z levels, with the d-aug results identified by the label “(diff.)”. |

4.3.1 Ne.
The first multi-electron atom that we consider is the closed-shell rare-gas atom Ne, where we investigate inelastic excitations from the outer subshell np^{6} electrons. These have been studied before, theoretically^{46,59} and experimentally using both EELS^{60} and IXS.^{10} The IXS measurements by Zhu et al.^{10} demonstrate elegantly that the intensity of the EELS measurements at high q is increased by contamination from high-order Born terms corresponding to multiple scattering. For X-ray scattering, only first-order Born terms contribute due to the weaker interaction.

In the following, we focus the comparison on the previous benchmark random-phase with exchange (RPAE) calculations by Amusia et al.^{46} The excitations studied are characterized by the dependency on the total angular momentum, and following the lead of Amusia et al. we discuss the cross sections in terms of the monopole, dipole, and quadrupole transitions, respectively. We perform our ab initio calculations at the CASSCF(10,9)/aug-cc-PVTZ level of theory. Note that when using a general-use ab initio electronic structure package, one has to pay careful attention to symmetry and multiplicity in order to isolate different contributions correctly for an atom. The energies of the excited states of Ne involved in the monopole, dipole, and quadrupole transitions from the ground state are listed in Table 2.

Ne |
E
_{exp} (eV) |
E
_{calc} (eV) |
ΔE (%) |
---|---|---|---|

2s2p^{5}3s[1/2]_{1} |
16.715 | 16.554 | 1.0 |

2s2p^{5}3p[1/2]_{0} |
18.555 | 18.290 | 1.4 |

2s2p^{5}3p[3/2]_{2} |
18.704 | 18.720 | 0.1 |

In Fig. 4 we compare our results with those by Amusia et al.^{46} for the monopole and quadrupole 3p ← 2p and the dipole 3s ← 2p transitions. Note that the cross sections have been rotationally averaged (see Section 2). Overall, the agreement is very good, with the only notable discrepancy occurring for the dipole 3s ← 2p transition, where the low-q peak in our calculations is marginally shifted to lower values of q compared to Amusia et al., although the height and width of the peak agree almost perfectly. The Amusia et al. calculations have been compared to the recent IXS experiments by Zhu et al.,^{10} and the agreement for the monopole 2p^{5}3p[1/2]_{0}, the dipole 2p^{5}3s[1/2]_{1}, and the quadrupole 2p^{5}3p[5/2,3/2]_{2} were found to be quite good, which carries over to our present calculations.

Fig. 4 Dynamic structure factor, S(q,ω), in Ne for the 3s ← 2p dipolar and 3p ← 2p monopolar and quadrupolar transitions compared to results by Amusia et al.^{46} |

4.3.2 C and Na.
Next we consider two open-shell atoms, C and Na, which provides an opportunity to examine cross sections for inner shell excitations in higher multiplicity systems with unpaired electrons in the ground state and a significant degree of electron correlation. The energy convergence of the CASSCF/aug-cc-PVTZ calculations are shown in Table 3.

The IXS cross sections for the transitions from the ground state of the C atom to the first two inner-shell excited states, i.e.^{3}P([He]2s2p^{3}) ← ^{3}P([He]2s^{2}2p^{2}) and ^{3}D([He]2s2p^{3}) ← ^{3}P([He]2s^{2}2p^{2}), are shown in Fig. 5. Our ab initio calculations, done at the CASSCF(6,5)/aug-cc-PVTZ level, agree well with the RPAE calculations by Chen and Msezane,^{61} also included in Fig. 5. For q^{2} → 0 the GOSs should converge to the optical oscillator strength of the transitions. In our calculations these values are 0.0615 and 0.1130, respectively, which agrees reasonably well with the experimental values of 0.0634 and 0.0718,^{63} as well as previous theory.^{61} Further improvements in the oscillator strength would most likely require CASPT2 level corrections.

Fig. 5 Generalized oscillator strengths, GOS(q,ω), in C for the two transitions ^{3}P_{0}(2s2p^{3}) ← ^{3}P_{0}(2s^{2}2p^{2}) and ^{3}D_{0}(2s2p^{3}) ← ^{3}P_{0}(2s^{2}2p^{2}). The current ab initio calculations using CASSCF(6,5)/aug-cc-PVTZ are compared to RPAE calculations by Chen and Msezane.^{61} |

In Na, we have done the calculations at the CASSCF(11,9)/aug-cc-PVQZ level of theory. We consider an inner shell excitation from the doublet ground state, i.e. the ^{2}P([He]2s^{2}2p^{5}3s^{2}) ← ^{2}S([He]2s^{2}2p^{6}3s) transition, which has a very low oscillator strength compared to outer electron excitations. The cross sections, shown in Fig. 6 compare well to previous theory at the HF and RPAE level^{61} and EELS experiments by Bielschowsky et al.^{62}

Bradley et al.^{5} have identified deviations from first Born approximation scattering in the EELS signal at high q by comparison to IXS, along the lines of similar observations in Ne discussed earlier. A detailed analysis of TD-DFT theory and experiments in ref. 6, shows further that the a^{1}Π_{g} ← X^{1}Σ^{+}_{g} transition occurs in a region where there are additional contributions from the octupolar w^{1}Δ_{u} ← X^{1}Σ^{+}_{g} transition in the experimental signal, although in the following we focus on the transition to the a^{1}Π_{g} state.

The energy for the transition obtained using SA-CASSCF(14,12)/aug-cc-PVCTZ is within 0.3% of the experimental^{42} value. Table 4 shows the experimental and theoretical energies E for the a^{1}Π_{g} state in N_{2}, as well as the percentage error, ΔE, compared to experimental values from Leung.^{42} Also included are the results for a MRCI(14,10)/aug-cc-PVCTZ calculation, which in principle should perform better than CASSCF, but due to computational problems had to be run at lower symmetry which adversely affected the energy convergence.

N_{2} |
E (eV) | ΔE (%) |
---|---|---|

Exp.^{42} |
9.300 | — |

CASSCF(14,12) | 9.332 | 0.3 |

MRCI(14,10) | 9.700 | 4.3 |

The generalized oscillator strength, GOS(q,ω), that we have calculated is in good agreement with the experimental results from Leung et al.^{42} and Barbieri et al.,^{64} shown in Fig. 7, as well as recent theoretical calculations by Giannerini.^{65} The MRCI results provide a slightly lower scattering cross section, but the difference is small. The calculated cross sections are below the experiments at high values of q. As discussed above, in terms of comparison to EELS the reason for this difference is primarily the failure of the first Born approximation in EELS. For IXS the discrepancy is smaller, and is due to additional contributions from the w^{1}Δ_{u} state in the Lyman–Birge–Hopfield band.^{6}

Fig. 7 Generalized oscillator strength, GOS(q,ω), for the a^{1}Π_{g} ← X^{1}Σ^{+}_{g} transition in N_{2}. Our CASSCF and MRCI ab initio results are compared to experimental results from Leung et al.^{42} and Barbieri et al.,^{64} and to calculations by Giannerini et al.^{65} |

Finally, a brief remark regarding the energy convergence of the ab initio calculations, as summarized in Tables 1–4. In C and Na the number of Slater determinants is restricted to correctly isolate the inner-shell transitions considered, which impacts on the treatment of electron correlation. The valence transition considered in N_{2} allows for greater flexibility in the choice of active space, leading to a good account of static electron correlation, with the calculation close to full CI.

The approach presented in this paper covers the ground between these two extremes. It calculates inelastic scattering matrix elements at a level of ab initio theory congruent with state-of-the-art quantum molecular dynamics simulations, and is therefore well placed to evaluate inelastic contributions to the signals observed in ultrafast X-ray scattering experiments.^{31} As discussed in the Introduction, an important motivation for this work is the prospect of identifying electronic transitions in time-dependent ultrafast X-ray scattering experiments, which could enable complete characterization of reaction paths using X-ray scattering. Achieving the same insights today requires the combination of ultrafast X-ray scattering with a different experimental technique, e.g. time-resolved photoelectron spectroscopy.^{70} Finally, the accuracy of the matrix elements calculated by our method is only limited by the quality of the ab initio wavefunctions, and can be systematically improved by adjustments of the ab initio method and basis. Our approach involves a direct summation over all multipole matrix elements, aiding immediate comparison to experiments.

The link between IXS and EELS suggests that the codes developed here could be useful for detailed analysis of ultrafast electron diffraction (UED) data, as long as the nuclear-scattering contribution is included in the elastic terms.^{71} Future extensions of this work would be to include the effect of nuclear motion in the IXS signal, as we have recently done for elastic scattering,^{37,38} and to consider Compton ionization by the inclusion of continuum states either via multichannel quantum defect formalism^{72–74} or a Dyson orbital approach.^{75} We also aim to examine in greater detail the mapping of the wavefunction in momentum space made possible by inelastic measurements.

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