A Polarization Propagator for Nonlinear X-ray Spectroscopies - The

Subscriber access provided by UNIV OF PITTSBURGH

Letter

A Polarization Propagator for Nonlinear X-Ray Spectroscopies Tobias Fahleson, Hans Ågren, and Patrick Norman J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b00750 • Publication Date (Web): 09 May 2016 Downloaded from http://pubs.acs.org on May 14, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry Letters

A Polarization Propagator for Nonlinear X-ray Spectroscopies Tobias Fahleson,1 Hans ˚ Agren,2 and Patrick Norman1,2∗ 1

Department of Physics, Chemistry and Biology, Link¨ oping University, SE-581 83 Link¨ oping, Sweden

2

Division of Theoretical Chemistry and Biology, School of Biotechnology, KTH Royal Institute of Technology, SE-106 91, Stockholm, Sweden E-mail: [email protected]

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract A complex polarization propagator approach has been developed to third order and implemented in density functional theory (DFT), allowing for the direct calculation of nonlinear molecular properties in the X-ray wavelength regime without explicitly addressing the excited state manifold. We demonstrate the utility of this propagator method for the modeling of coherent near-edge X-ray two-photon absorption (XTPA) using, as an example, DFT as the underlying electronic structure model. Results are compared with the corresponding near-edge X-ray absorption fine structure (NEXAFS) spectra, illuminating the differences in the role of symmetry, localization, and correlation between the two spectroscopies. The ramifications of this new technique for nonlinear X-ray research are briefly discussed.

Graphical TOC Entry TOC ENTRY REQUIRED

2


Page 2 of 17

Page 3 of 17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

the implementation of two- or multi-photon X-ray absorption can offer confocal imaging in unprecedented detail at the atomic or sub-atomic scale. New effects and processes will likely be revealed, like manipulation at ultra-small, even atomic-scale, confinements (”Xray scissors” or ”X-ray tweezers”) as well as ultra-sharp 3D imaging using confocal XTPA. The change of penetration depth and avoidance of self-absorption may open unexplored possibilities for two-photon induced X-ray fluorescence marking and imaging, and there are many new aspects to be considered such as field splitting and broadening of spectral lines. 10–12 As for modeling of X-ray absorption and emission processes, the strong valence relaxation and also the embedding of semi-bound core excited states in the valence-ionized continuum are complicating factors to deal with. Treatments of relaxation effects have traditionally favored state-specific methods, where the core-hole induced relaxation is accounted for by self-consistent field procedures, 15,16 but which leads to non-orthogonal and interacting states. Propagator, or response, technologies pose a viable and universal alternative to such state-bystate evaluations of spectra—without explicitly addressing the excited states, requiring only a ground state optimization, and maintaining favorable features such as state orthogonality, size consistency, sum rules, and operator gauge invariance. 17 Furthermore being applicable and implemented to all standard electronic structure methods (wave function and density based), they nowadays form the standard in spectroscopic calculations. 18 One way to calculate propagators in the X-ray regime is to impose restrictions on the excitation channels based on a core-valence separation. 19,20 As shown by the present authors, 21,22 it is also possible to reach full channel solutions (complete random phase approximation) by extending the propagator theory into the complex domain. 23,24 By going from a real resonance-divergent propagator theory to its complex counterpart, absorption spectra are obtained directly from the imaginary part of polarization tensor, without having to determine eigenvalues of the electronic Hessian. This technique has by now shown its versatility as a precise computational tool for linear X-ray absorption spectroscopy. In the present work, we generalize the complex polarization propagator (CPP) approach

4


Page 4 of 17

Page 5 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


to third order and demonstrate its applicability to coherent near-edge XTPA by providing benchmark results for the cross sections of a series of small systems including neon, hydrogen fluoride, water, and silane. We provide a brief presentation of the basics of the relevant theory as expressed in the exact-state basis. All details regarding the derivation and implementation of the nonlinear CPP in the approximate time-dependent DFT framework is, however, left out and will appear in a subsequent publication.

Theory The induced macroscopic polarization can be written as P = PL + PNL ,

(1)

where a division is made between the linear and nonlinear parts that also constitute a separation between single- and multi-photon absorption processes. Introducing the electric susceptibilities, the components of the polarization vector oscillating with frequency ω are expressed in terms of a Taylor series in the electric field strengths 1 (3) (1) Pi (ω) = χij (−ω; ω)Ejω + χijkl (−ω; ω, −ω, ω)Ejω Ek−ω Elω + . . . . 6

(2)

The nonlinear polarization also contains other frequencies but none of these contribute to the absorption of radiation, which is related to the induced current density according to dI = dz

d dt

absorbedenergy volume

= −α(1) I − α(2) I 2 − . . .

= hj · EiT = T

∂P(t) ·E ∂t

(3)

T

where linear and nonlinear absorption coefficients have been introduced. By carrying out the time-averaging integration h. . .iT over one period of oscillation, expressions for the absorption

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 17

cross sections for a randomly oriented molecular sample with number density N are identified as 1 (1) α (ω) N i h ω ω = Im χ(1) (−ω; ω) = Im α(−ω; ω) , ε0 c 0 N ε0 c 0 ~ω (2) σ (2) (ω) = α (ω) N i h (3) ~ω 2 ~ω 2 Im χ (−ω; ω, −ω, ω) = = γ(−ω; ω, −ω, ω) , Im 4ε20 c20 N 4ε20 c20

σ (1) (ω) =

(4)

(5)

where, in the last step, we have introduced the isotropic tensor averages of the molecular polarizability α and second-order hyperpolarizability γ. Conventional time-dependent DFT is based on the Schrödinger equation under the assumption of an infinite lifetime of all excited states, leading to response functions that are real-valued and divergent under conditions of resonance. The introduction of spontaneous and collision-induced relaxation in quantum mechanics is achieved in the Liouville density matrix formulation, leading to response functions that are complex-valued and provide a sound description also when the external electromagnetic fields are in resonance with transition frequencies in the system. A wave function based complex polarization propagator based on the Ehrenfest theorem has been presented by Norman and co-workers. 18,24 The CPP approach is the wave function correspondence of response theory in the density matrix formalism for pure states with the advantage of serving as the foundation for the development of approximate state methods in quantum chemistry. Apart from the CPP approach, damped linear and nonlinear response functions have also been developed in the quasi-energy framework 25,26 and a density matrix based time-dependent DFT formalism. 27 In these approaches, a pragmatic damping of resonances is introduced by the introduction of complex excitation energies that results in similar expressions for response functions, although one can note that an argumentative damping of higher-order Hessian matrices is required 25 and which leads to detailed matrix equations that differ in their details in comparison to those arising from the 6


Page 7 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


CPP approach. We report here the derivation and implementation of second-order nonlinear response function in the CPP Kohn–Sham DFT framework, allowing for the calculation of the complex hyperpolarizability with a sum-over-states expression that is equal to

X

(6)

P1,2,3 γαβγδ (−ωσ ; ω1 , ω2 , ω3 ) = ~ " X′ h0|ˆ µα |kihk|ˆ µβ |lihl|ˆ µγ |mihm|ˆ µδ |0i −3

k,l,m

(ωk0 − ωσ − iγk )(ωl0 − ω2 − ω3 − iγl )(ωm0 − ω3 − iγm )

µα |lihl|ˆ µγ |mihm|ˆ µδ |0i h0|ˆ µβ |kihk|ˆ (ωk0 + ω1 + iγk )(ωl0 − ω2 − ω3 − iγl )(ωm0 − ω3 − iγm ) µγ |lihl|ˆ µα |mihm|ˆ µδ |0i h0|ˆ µβ |kihk|ˆ + (ωk0 + ω1 + iγk )(ωl0 + ω1 + ω2 + iγl )(ωm0 − ω3 − iγm ) +

h0|ˆ µβ |kihk|ˆ µγ |lihl|ˆ µδ |mihm|ˆ µα |0i + (ωk0 + ω1 + iγk )(ωl0 + ω1 + ω2 + iγl )(ωm0 + ωσ + iγm ) X′ h0|ˆ µα |kihk|ˆ µβ |0ih0|ˆ µγ |lihl|ˆ µδ |0i − (ωk0 − ωσ − iγk )(ωk0 − ω1 − iγk )(ωl0 − ω3 − iγl ) k,l

#

h0|ˆ µα |kihk|ˆ µβ |0ih0|ˆ µγ |lihl|ˆ µδ |0i (ωk0 − ω1 − iγk )(ωl0 + ω2 + iγl )(ωl0 − ω3 − iγl ) h0|ˆ µβ |kihk|ˆ µα |0ih0|ˆ µδ |lihl|ˆ µγ |0i + (ωk0 + ωσ + iγk )(ωk0 + ω1 + iγk )(ωl0 + ω3 + iγl ) h0|ˆ µβ |kihk|ˆ µα |0ih0|ˆ µδ |lihl|ˆ µγ |0i + , (ωk0 + ω1 + iγk )(ωl0 − ω2 − iγl )(ωl0 + ω3 + iγl ) +

where µ ˆα denotes the electric-dipole operator along molecular axis α (with an overbar to denote a corresponding fluctuation operator), and ~ωn0 and γn are the excitation energy and inverse lifetime of excited state |ni, respectively. The possibility to determine the ground state hyperpolarizability by Eq. (6), or rather its counterpart in the DFT approximation, offers a direct way to calculate the TPA cross section in Eq. (5) in the visible, ultra-violet, and X-ray regions of the spectrum without optimizing anything more than the ground state electron density as well as solving sets of standard linear response equations (see Ref. 28 for details how this can be done efficiently). This approach stands in stark contrast to the conventional technique to determine TPA spectra

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

that involves the calculation of two-photon matrix elements, involving an explicit reference to the final state as well as its coupling to all intermediate excited states. With targeted final states that are core excited and embedded in a continuum of valence-ionized states this approach is severely hampered—a core-valence separation, which has proven accurate in the linear regime, 20 is difficult to adopt here since the coupling to all intermediate states needs to be accounted for. We believe that our proposed CPP approach will be a boon to theoretical work in nonlinear X-ray spectroscopy, illustrated in the present work by calculations of the X-ray two-photon absorption spectra of a series of small molecules.

Calculations Structure optimizations of water (H2 O), hydrogen fluoride (HF), and silane (SiH4 ) were done with Dunning’s correlation-consistent basis sets 29 taug-cc-pVTZ, taug-cc-pVTZ, and cc-pVTZ, respectively, at the DFT/B3LYP 30 level of theory. Property calculations were performed with use of a Coulomb-attenuated B3LYP 31 functional parametrized with 100% asymptotic exact exchange (CAM-B3LYP100% ) as described in Ref. 22 to account for the hole–electron Coulomb interaction—this level of theory has proven to provide NEXAFS spectra of coupled cluster singles and doubles (CCSD) quality as long as overall spectral shifts are applied to remedy effects of DFT self-interaction. 32 Benchmark work is required to assess the quality of DFT results in regard with nonlinear X-ray spectroscopies, but this is beyond the scope of the present work and future studies need to be conducted. In the present work, all applied spectral shifts are given in the figure legends. The taug-cc-pVTZ basis set was used for the calculation of molecular spectra and, for neon (Ne), this basis set was further augmented with core polarizing orbitals, taug-cc-pCVTZ, as well as Rydberg sand p-functions as suggested by Kaufmann et al. 33 All calculation were performed with a local version of the Dalton program. 34 The experimental NEXAFS spectrum of Ne 35 shows three distinct Rydberg transitions

8


Page 8 of 17

Page 9 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1s → np (n = 3, 4, 5) that are, energy- and intensity-wise, all well reproduced in the theoretical spectrum (see Fig. 1a). The quality of the CAM-B3LYP100% spectrum reaches that of the CCSD spectrum presented previously by us. 36 A comparison with the corresponding XTPA spectrum in Fig. 1a shows the stark differences in atomic selection rules between one- and two-photon absorption spectroscopies. The relevant expression for the coherent two-photon transition matrix element is

Sαβ

1 X h0|ˆ µβ |kihk|ˆ µα |f i µα |kihk|ˆ µβ |f i h0|ˆ = + , ~ k ωk0 − ωf 0 /2 ωk0 − ωf 0 /2

(7)

which shows that the lowest 1s → 3s peak at 865.3 eV in the theoretical XTPA spectrum has two main contributing excitation channels namely (i) core-excitation 1s → 3p followed by valence de-excitation 3p → 3s and (ii) valence excitation 2p → 3s followed by core-excitation 1s → 2p. The next peak in this Rydberg series, i.e., 1s → 4s, is found at 868.1 eV and is significantly less intense. The near-edge XTPA spectrum of Ne is dominated by the strong band at 869.5 eV associated with the 1s → 3d transition for which the two main contributing excitation channels are (i) core-excitation 1s → 3p followed by valence excitation 3p → 3d and (ii) valence excitation 2p → 3d followed by core-excitation 1s → 2p. The experimental NEXAFS spectrum of HF shows three distinct peaks in terms of the 4σ ∗ transition to the nonbonding molecular orbital and the atomic Rydberg 3p and 4p transitions. 37 Our theoretical spectrum stands in good agreement in terms of relative intensities but shows a too small separation (by some 0.8 eV) between the first and second peak. Unlike Ne, the first transition is not atomic-like and it is therefore both one- and two-photon allowed. In fact, it is seen in Fig. 1 to be the dominant spectral feature in both NEXAFS and XTPA spectra. The np Rydberg transitions are two-photon forbidden and instead we note a spectrally rich region between 345–347 eV associated with Rydberg transitions with ∆l = 0, 2. The experimental NEXAFS spectrum of gaseous H2 O 38 shows three distinct peaks namely

9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

4a1 , 2b2 , and Rydberg state 2b1 . A comprehensive theoretical X-ray absorption study of water in gas, liquid, and solid phases has been presented by Fransson et al., 39 revealing that the CAM-B3LYP100% spectrum does not quite match the accuracy of the CCSD spectrum. While intensities are well described, the transition energy of the Rydberg transition is underestimated by some 0.5 eV at the DFT level (see Fig. 1c). The lowest 4a1 band associated with a core electron excitation to the bond polarized σ ∗ -orbital is both one- and two-photon active, in fact it is the dominating feature in the XTPA spectrum. The 2b2 transition, on the other hand, is two-photon inactive and the first Rydberg transition 2b1 is weak in two-photon mode. It is clear that the XTPA spectrum is affected by a ”background” signal causing the spectrum to slope. The reason for this is the appearance of an unphysical one-photon resonance in relative proximity of the two-photon region. Since we adopt a localized basis set, the continuum will become discretized and can coincidentally overlap with the XTPA spectrum. The identification of the XTPA spectrum is unproblematic in this case by disregarding the slope, but this issue can also be technically avoided by the exclusion of electron excitation operators in the propagator that are associated with high-energy valence excitations (& 100 eV). The experimental near gas phase silicon K-edge spectrum for SiH4 shows three peaks attributed as t2 and Rydberg 4p and 5p, 40 which are all well captured in the theoretical spectrum in Fig. 1d. Silane spans the Td point group and due to this high degree of symmetry the NEXAFS and XTPA are complementary and non-overlapping, like in the atom. For XTPA, it is the a1 transition that is seen in the spectrum at an energy of about 1843 eV. Comparing σ (2) results for the different systems, we note that, out of the assigned peaks discussed above, the 3d transition in Ne is the most intense (100%), followed by the 4a1 transition in H2 O (19%), the 4σ ∗ transition in HF (16%), and the a1 transition in SiH4 (2.5%).

10


Page 10 of 17

Page 11 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Conclusions By the derivation and implementation of the second-order nonlinear complex polarization propagator, we have demonstrated that first-principles calculations of X-ray two-photon absorption cross sections becomes a straightforward matter. Our first results obtained for neon and a series of small and highly symmetric molecules reveal the complementarity of oneand two-photon X-ray absorption spectroscopies in their probing of different electronically excited states. We expect that our work provides a first breakthrough in theoretical nonlinear X-ray sciences to be followed by that of many others as to reach the same level of advanced theoretical support platform as we see today in linear X-ray spectroscopies. Such a development will be crucial for optimizing the scientific output of the fourth generation synchrotron sources and X-ray free electron lasers that have arrived or are being built.

Acknowledgement Financial support from the Knut and Alice Wallenberg Foundation (Grant No. KAW2013.0020) and the Swedish Research Council (Grant No. 621-2014-4646) is acknowledged. The calculations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputer Centre (NSC), Sweden.

References (1) Doumy, G.; Roedig, C.; Son, S.-K.; Blaga, C. I.; DiChiara, A. D.; Santra, R.; Berrah, N.; Bostedt, C.; Bozek, J. D.; Bucksbaum, P. H. et al. Nonlinear Atomic Response to Intense Ultrashort X Rays. Phys. Rev. Lett. 2011, 106, 083002–083005. (2) Glover, T. E.; Fritz, D. M.; Cammarata, M.; Allison, T. K.; Coh, S.; Feldkamp, J. M.; Lemke, H.; Zhu, D.; Feng, Y.; Coffee, R. N. et al. X-ray and Optical Wave Mixing. Nature 2012, 488, 603–608. 11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(3) Shwartz, S.; Fuchs, M.; Hastings, J. B.; Inubushi, Y.; Ishikawa, T.; Katayama, T.; Reis, D. A.; Sato, T.; Tono, K.; Yabashi, M. et al. X-Ray Second Harmonic Generation. Phys. Rev. Lett. 2014, 112, 163901–163904. (4) Tamasaku, K.; Shigemasa, E.; Inubushi, Y.; Katayama, T.; Sawada, K.; Yumoto, H.; Ohashi, H.; Mimura, H.; Yabashi, M.; Yamauchi, K. et al. X-ray Two-Photon Absorption Competing Against Single and Sequential Multiphoton Processes. Nature Photon. 2014, 8, 313–316. (5) Fuchs, M.; Trigo, M.; Chen, J.; Ghimire, S.; Shwartz, S.; Kozina, M.; Jiang, M.; Henighan, T.; Bray, C.; Ndabashimiye, G. et al. Anomalous Nonlinear X-ray Compton Scattering. Nature Phys. 2015, 11, 964–970. (6) Palffy, A. X-Ray Physics: Straight Outta Compton. Nature Phys. 2015, 11, 893–894. (7) Biggs, J. D.; Zhang, Y.; Healion, D.; Mukamel, S. Two-Dimensional Stimulated Resonance Raman Spectroscopy of Molecules with Broadband X-ray Pulses. J. Chem. Phys. 2012, 136, 174117–174132. (8) Mukamel, S.; Healion, D.; Zhang, Y.; Biggs, J. D. Multidimensional Attosecond Resonant X-Ray Spectroscopy of Molecules: Lessons from the Optical Regime. Annu. Rev. Phys. Chem. 2013, 64, 101–127. (9) Zhang, Y.; Hua, W.; Bennett, K.; Mukamel, S. Nonlinear Spectroscopy of Core and Valence Excitations Using Short X-Ray Pulses: Simulation Challenges. Top. Curr. Chem. 2016, 368, 273–345. (10) Liu, J.-C.; Sun, Y.-P.; Wang, C.-K.; ˚ Agren, H.; Gelmukhanov, F. The Propagation of a Strong X-ray Pulse Followed by Pulse Slowdown and Compression, Amplified Spontaneous Emission and Lasing without Inversion. J. Phys. B: At. Mol. Opt. Phys. 2009, 42, 201001–201005.

12


Page 12 of 17

Page 13 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(11) Sun, Y.-P.; Liu, J.-C.; Gelmukhanov, F. Slowdown and Compression of Strong X-ray Free-Electron Pulses Propagating through Mg Vapor. Europhys. Lett. 2009, 87, 64002– 64006. (12) Liu, J.-C.; Sun, Y.-P.; Wang, C.-K.; ˚ Agren, H.; Gelmukhanov, F. Auger Effect in the Presence of Strong X-ray Pulses. Phys. Rev. A 2010, 81, 043412–043418. (13) Bhawalkar, J. D.; He, G. S.; Prasad, P. N. Nonlinear Multiphoton Processes in Organic and Polymeric Materials. Rep. Prog. Phys. 1996, 59, 1041–1070, Erratum Rep. Prog. Phys., 60, 689, (1997). (14) He, G. S.; Tan, L. S.; Zheng, Q.; Prasad, P. N. Multiphoton Absorbing Materials:Molecular Designs, Characterizations, and Applications. Chem. Rev. 2008, 108, 1245–1330. (15) 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. (16) Jensen, H. J. Aa.; Jørgensen, P.; ˚ Agren, H. Efficient Optimization of Large Scale MCSCF Wave Functions with a Restricted Step Algorithm. J. Chem. Phys. 1987, 87, 451–466. (17) Olsen, J.; Jørgensen, P. Linear and Nonlinear Response Functions for Exact State and for an MCSCF State. J. Chem. Phys. 1985, 82, 3235–3264. (18) Norman, P. A Perspective on Nonresonant and Resonant Electronic Response Theory for Time-Dependent Molecular Properties. Phys. Chem. Chem. Phys. 2011, 13, 20519– 20535. (19) Barth, A.; Cederbaum, L. S. Many-Body Theory of Core-Valence Excitations. Phys. Rev. A 1981, 23, 1038–1061.

13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(20) Wenzel, J.; Wormit, M.; Dreuw, A. Calculating Core-Level Excitations and Xray Absorption Spectra of Medium-Sized Closed-Shell Molecules with the AlgebraicDiagrammatic Construction Scheme for the Polarization Propagator. J. Comput. Chem. 2014, 35, 1900–1915. (21) Ekström, U.; Norman, P.; Carravetta, V.; ˚ Agren, H. Polarization Propagator for X-ray Spectra. Phys. Rev. Lett. 2006, 97, 143001–143004. (22) Ekström, U.; Norman, P. X-ray Absorption Spectra from the Resonant-Convergent First-Order Polarization Propagator Approach. Phys. Rev. A 2006, 74, 042722–042728. (23) Norman, P.; Bishop, D. M.; Jensen, H. J. Aa.; Oddershede, J. Near-Resonant Absorption in the Time-Dependent Self-Consistent Field and Multi-Configurational SelfConsistent Field Approximations. J. Chem. Phys. 2001, 115, 10323–10334. (24) Norman, P.; Bishop, D. M.; Jensen, H. J. Aa.; Oddershede, J. Nonlinear Response Theory with Relaxation: the First Hyperpolarizability. J. Chem. Phys. 2005, 123, 194103–194120. (25) Kristensen, K.; Kauczor, J.; Kjærgaard, T.; Jørgensen, P. Quasienergy Formulation of Damped Response Theory. J. Chem. Phys. 2009, 131, 044112–044133. (26) Kristensen, K.; Kauczor, J.; Thorvaldsen, A. J.; Jørgensen, P.; Kjærgaard, T.; Rizzo, A. Damped Response Theory Description of Two-Photon Absorption. J. Chem. Phys. 2011, 134, 214104–214117. (27) Hu, Z.; Autschbach, J.; Jensen, L. Simulating Third-Order Nonlinear Optical Properties using Damped Cubic Response Theory within Time-Dependent Density Functional Theory. J. Chem. Theory Comput. 2016, 12, 1294–1304. (28) Kauczor, J.; Norman, P. Efficient Calculations of Molecular Linear Response Properties for Spectral Regions. J. Chem. Theory Comput. 2014, 10, 2449–2455. 14


Page 14 of 17

Page 15 of 17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(29) Dunning Jr., T. H. Gaussian Basis Sets for use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (30) Becke, A. D. Density-Functional Thermochemistry. III. The role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648–5652. (31) Yanai, T.; Tew, D. P.; Handy, N. C. A new Hybrid Exchange-Correlation Functional using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393, 51–57. (32) Fransson, T.; Coriani, S.; Christiansen, O.; Norman, P. Carbon X-ray Absorption Spectra of Fluoroethenes and Acetone: a Study at the Coupled Cluster, Density Functional, and Static-Exchange Levels of Theory. J. Chem. Phys. 2013, 138, 124311–124322. (33) Kaufmann, K.; Baumeister, W.; Jungen, M. Universal Gaussian Basis Sets for an Optimum Representation of Rydberg and Continuum Wavefunctions. J. Phys. B: At. Mol. Opt. Phys. 1989, 22, 2223–2240. (34) Aidas, K.; Angeli, C.; Bak, K. L.; Bakken, V.; Bast, R.; Boman, L.; Christiansen, O.; Cimiraglia, R.; Coriani, S.; Dahle, P. et al. The Dalton Quantum Chemistry Program System. WIREs Computational Molecular Science 2014, 4, 269. (35) Coreno, M.; Avaldi, L.; Camilloni, R.; Prince, K. C.; de Simone, M.; Karvonen, J.; Colle, R.; Simonucci, S. Measurement and Ab Initio Calculation of the Ne Photoabsorption Spectrum in the Region of the K Edge. Phys. Rev. A 1999, 59, 2494–2497. (36) Coriani, S.; Christiansen, O.; Fransson, T.; Norman, P. Coupled-Cluster Response Theory for Near-Edge X-ray Absorption Fine Structure of Atoms and Molecules. Phys. Rev. A 2012, 85, 022507–022514. (37) Hitchcock, A. P.; Brion, C. E. K-Shell Excitation of HF and F2 Studied by Electron Energy-Loss Spectroscopy. J. Phys. B: At. Mol. Phys. 1981, 14, 4399–4413. 15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(38) Schirmer, J.; Trofimov, A. B.; Randall, K. J.; Feldhaus, J.; Bradshaw, A. M.; Ma, Y.; Chen, C. T.; Sette, F. K-shell Excitation of the Water, Ammonia, and Methane Molecules using High-Resolution Photoabsorption Spectroscopy. Phys. Rev. A 1993, 47, 1136–1147. (39) Fransson, T.; Zhovtobriukh, I.; Coriani, S.; ; Wikfeldt, K. T.; Norman, P.; Pettersson, L. G. M. Requirements of First-Principles Calculations of X-ray Absorption Spectra of Liquid Water. Phys. Chem. Chem. Phys. 2016, 18, 566–583. (40) Bodeur, S.; Millié, P.; Nenner, I. Single- and Multiple-Electron Effects in the Si 1s Photoabsorption Spectra of SiX4 (X = H, D, F, Cl, Br, CH3 , C2 H5 , OCH3 , OC2 H5 ) Molecules: Experiment and Theory. Phys. Rev. A 1990, 41, 252–263.

16


Page 16 of 17

Page 17 of 17


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

A Polarization Propagator for Nonlinear X-ray Spectroscopies - The

Recommend Documents