Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics
Abstract
:1. Introduction
2. Models, Methods and Techniques
2.1. Linearly Polarized Laser Pulses in the Laboratory Frame
2.2. Orientations of the Nuclear Scaffolds
2.3. Nomenclature of Selected Symmetry Elements, Symmetry Operations and Subgroups of the Point Groups D_{6h} and D_{4h} of the Oriented Benzene and MgPorphyrin
2.4. Quantum Chemical Methods for the Electronic States with Their Energies, Wavefunctions, IRREPs, and Transition Dipole Matrix Elements
2.5. Symmetry Breaking of the Electronic Hamiltonians for the Oriented Model Benzene or MgPorphyrin by Linearly Polarized Laser Pulses
≠ 0 else
≠ 0 for laser polarizations e_{p} different from
e’_{1}, e’_{2}, e’_{3} and e”_{1}, e”_{2}, e”_{3} for oriented benzene
or for two sequential laser pulses with different polarizations
2.6. Quantum Dynamical Methods for the Propagation of the LaserDriven Electronic Superposition States
2.7. Some Symmetry Relations of the Wavefunctions Driven by Laser Pulses with Different Linear Polarizations
= [H_{e}({r_{i}}) + e C_{3} e_{1} * Σ_{i} r_{i} * $\mathcal{E}$(t)] Ψ_{e}_{2}({r_{i}},t)
= [H_{e}({r_{i}(t)})] Ψ_{e}_{2}({r_{i}},t)
= C_{3} Ψ_{A1g}({r_{i}}) = Ψ_{e}_{1}({C_{3}^{−1} r_{i}},t = 0).
= Σ_{m} c_{e}_{“1m}(t) C_{3} Ψ_{m}({r_{i}}).
e’_{1} → e’_{3} = C_{3}^{2} e’_{1} yields the rotation Ψ_{e}_{’1}({r_{i}},t) → Ψ_{e}_{’3}({r_{i}},t) = C_{3}^{2} Ψ_{e}_{’1}({r_{i}},t),
e”_{1} → e”_{3} = C_{3}^{2} e”_{1} yields the rotation Ψ_{e}_{“1}({r_{i}},t) → Ψ_{e}_{“3}({r_{i}},t) = C_{3}^{2} Ψ_{e}_{“1}({r_{i}},t),
2.8. Calculation of the OneElectron Density of the TimeDependent Electronic Superposition State
ρ_{p,dia}(r,t) = Σ_{m} c_{e}_{p,m}(t)^{2} ∫Ψ_{m}({r_{i}})^{2} dr_{2} dr_{3} .... _{r = r1},
ρ_{p,odi}(r,t) = ΣΣ_{m<n} [c_{e}_{p,m}(t)^{*} * c_{e}_{p,n}(t) ∫Ψ_{m}({r_{i}})^{*} * Ψ_{n}({r_{i}}) dr_{2} dr_{3} .... _{r = r1}
+ c_{e}_{p,m}(t) * c_{e}_{p,n}(t)^{*} ∫Ψ_{m}({r_{i}}) * Ψ_{n}({r_{i}})^{*} dr_{2} dr_{3} .... _{r = r1}.
<ρ_{p}(r)> = ρ_{p,dia}(r,t_{f}) = Σ_{m} P_{e}_{p,m}(t_{f}) ∫Ψ_{m}({r_{i}})^{2} dr_{2} dr_{3} .... _{r = r1},
≡ Σ_{m} P_{e}_{p,m}(t_{f}) ρ_{m}(r),
Δρ_{p}(r,t) = ρ_{p,odi}(r,t) = ΣΣ_{m<n} c_{e}_{p,m}(t_{f}) * c_{e}_{p,n}(t_{f})
* {exp[ i (E_{m} − E_{n}) t / ℏ] ∫Ψ_{m}({r_{i}})^{*} * Ψ_{n}({r_{i}}) dr_{2} dr_{3} .... _{r = r1}
+ exp[ i (E_{n} − E_{m}) t / ℏ] ∫Ψ_{m}({r_{i}}) * Ψ_{n}({r_{i}})^{*} dr_{2} dr_{3} .... _{r = r1}}
= ΣΣ_{m<n} c_{e}_{p,m}(t_{f}) * c_{e}_{p,n}(t_{f})
* 2 cos(ω_{mn} t) * ∫Ψ_{m}({r_{i}}) * Ψ_{n}({r_{i}}) dr_{2} dr_{3} .... _{r = r1}
2.9. ReOptimized π/2 and πLaser Pulses for Control of Electron Symmetry Breaking and Charge Migration
3. Results
3.1. Electron Symmetry Breaking and Charge Migration Induced by Linearly Polarized Laser Pulses in Oriented Benzene with D_{6h} Nuclear Scaffold
+ c_{e}_{’1&e’1,2A1g} (t_{f}) 2A_{1g}> + c_{e}_{‘1&e‘1,1E2g^x2y2} (t_{f}) 1E_{2g}^{x2−y2}>
+ c_{e}_{’1&e”1,2A1g} (t_{f}) 2A_{1g}> + c_{e}_{‘1&e“1,1E2g^xy} (t_{f}) 1E_{2g}^{xy}>
3.2. Electron Symmetry Breaking and Charge Migration Induced by Linearly Polarized Laser Pulses in Oriented MgPorphyrin with D_{4h} Nuclear Scaffold
3.3. Symmetry of OneElectron Densities of Superposition States in Oriented Benzene
E_{2g}^{xy}> = c_{e}_{‘1&e“1,1E2g^xy} (t) 1E_{2g}^{xy}>.
3.4. Symmetry of OneElectron Densities of Superposition States in Oriented MgPorphyrin
= {g=E,C_{2}”_{2,} σ_{h}, σ_{d2}} = C_{2v}”_{2}
= {g=E, C_{2}‘_{2}, σ_{h}, σ_{v2}}
3.5. Symmetry of OneElectron Densities of Oriented Molecules Driven by One or Several Linearly Polarized Laser Pulses
= U_{e}(t) g Ψ_{e}(t = 0)>
= U_{e}(t)Ψ_{e}(t = 0)>
= Ψ_{e}(t)> for g ϵ S(H_{e}(t)).
4. Conclusions
 (a)
 Specific designs of laser pulses with selective polarization vectors and electric fields yield a large variety of electron symmetry breakings, from the original symmetry point group G of the electron density for the electronic ground state to various subgroups of G for excited superposition states. This is demonstrated here by eight plus four examples for different laser pulses applied to oriented benzene and Mgporphyrin, cf. Table 3 and Table 4, respectively. The target subgroups of the electron symmetry breaking are determined by two different approaches. First, the laser pulse(s) induce(s) intramolecular charge migration which is represented by the timedependent oneelectron density. Its symmetry can be determined by inspection of its symmetry elements, cf. Section 3.1 and Section 3.2 and Figure 6, Figure 7 and Figure 8, Figure 10 and Figure 12. Alternatively, the symmetry subgroup can be determined by means of a systematic grouptheoretical approach which is developed in Appendix A and exemplified in Section 3.3 and Section 3.4. The results of the two approaches agree perfectly with each other.
 (b)
 One can make use of this variety (a) for laser control of electron symmetry.
 (c)
 Laser control of electron symmetry is, however, not ad libitum: For any chosen polarization(s) of the laser pulse, or series of laser pulses, the target electron symmetry must obey the theorem (70), (71), cf. Section 3.5, which means it must be a subgroup of G, and it must contain the symmetry elements of the symmetry group of the timedependent electronic Hamiltonian for the oriented molecule interacting with the laser pulse. All examples in Table 3 and Table 4 in Section 3.3 and Section 3.4 satisfy this rule. It may also be used to check previous assignments of laserdriven symmetry breakings, see the discussion at the end of Section 3.5.
 (d)
 The present examples are for linearly polarized laser pulses which prepare superpositions of electronic basis functions with two different IRREPs. One of these IRREPs is the totally symmetric one, cf. Table 3 and Table 4 in Section 3.3 and Section 3.4. This scenario makes the symmetry of the superposition state equal to the symmetry of the oneelectron density, and this facilitates the applications. The general grouptheoretical derivation in Appendix A is, however, ready for applications to more general cases. It is thus a challenge to design laser pulse(s) which break electron symmetry by preparation of superpositions of basis functions that do not belong to the totally symmetry IRREP, and/or with more than two IRREPs.
 (e)
 (f)
 At longer times, typically for t > 10 fs, the laserdriven electron density representing attosecond charge migration with broken electron symmetry will induce nuclear motions away from their initial equilibrium positions. As a working hypothesis, the nuclear and electron symmetries should adapt to each other, somewhat analogous to dynamical Jahn–Teller distortion on much longer time scales [14]. The underlying fundamental point group analyses for coupled electron and nuclear quantum dynamics driven by short laser pulses in the time domain from a few hundred to a few fs is largely terra incognita.
 (g)
 The literature has fascinating concepts for monitoring electron symmetry breaking in oriented molecules by short laser pulses, e.g., by attosecond photoionization of the superposition state [41], by timeresolved measurements of the asymmetries in photoelectron angular distributions [36,42], by high harmonic spectroscopy which is sensitive to electron symmetry [40,43,44] or by electron diffraction induced by ultrashort Xray pulses [6,45,46]. The present predictions call for experimental applications or extensions of the concepts. The pioneering Ref. [47] points even to practical applications, i.e., photodissociation of molecules with broken electron symmetry may cause asymmetric distributions of the photoproducts in the laboratory—ultimately this could pave the way to photoseparation of the products.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Symmetry Classification of Electronic Superposition States and Their Densities
= ∫ f (g^{−1} r_{1}, g^{−1} r_{2}, …, g^{−1} r_{N}) dτ = ρ(g^{−1} r_{1}) = g ρ(r_{1}).
$${D}_{6h}\phantom{\rule{0ex}{0ex}}\mathrm{State}$$

$${D}_{6h}$$

$${D}_{3h}^{\u2033}$$

$${D}_{3h}^{\prime}$$

$${D}_{2h,1}$$

$${C}_{2v,1}^{\u2033}$$

$${C}_{2v,1}^{\prime}$$

$${C}_{2h}$$

$${C}_{s}$$


$$\phantom{\rule{0ex}{0ex}}{E}_{2g}^{xy}\rangle $$
 E_{2g}  E‘  E‘  B_{1g}  B_{1}  B_{2}  A_{g}  A‘ 
$${E}_{2g}^{{x}^{2}{y}^{2}}\rangle $$
 A_{g}  A_{1}  A_{1}  A_{g}  A‘  
$${E}_{1u}^{y}\rangle $$
 E_{1u}  E‘  E‘  B_{2u}  B_{1}  A_{1}  B_{u}  A‘ 
$${E}_{1u}^{x}\rangle $$
 B_{3u}  A_{1}  B_{2}  B_{u}  A‘  
$${B}_{1u}\rangle $$
 B_{1u}  A_{2}‘  A_{1}‘  B_{2u}  B_{1}  A_{1}  B_{u}  A‘ 
$${B}_{2u}\rangle $$
 B_{2u}  A_{1}‘  A_{2}‘  B_{3u}  A_{1}  B_{2}  B_{u}  A‘ 
$${A}_{1g}\rangle $$
 A_{1g}  A_{1}‘  A_{1}‘  A_{g}  A_{1}  A_{1}  A_{g}  A‘ 
Appendix B
 Symmetry Properties of Transition Dipole Moments of Benzene and MgPorphyrin Interacting with Linearly Polarized Laser Pulses
Appendix B.1. Preliminary Remarks
Appendix B.2. Reduced Products and Their Bases
Appendix B.3. Some Transition Dipole Moments
Transition Ψ〉 → Φ〉 ^{(a)}  Integrals ^{(b,c)} 
$$\left{c}_{mn}^{\left(i\right)}\right$$
[ea_{0}]  cf. Fig. 

$${D}_{6h}$$
 
$${A}_{1g}\rangle \to {B}_{2u}\rangle $$
 (f)  
$$\hspace{1em}\hspace{1em}\to {B}_{1u}\rangle $$
 (f)  
$$\hspace{1em}\hspace{1em}\to {E}_{1u}\rangle $$

$$\hspace{1em}\hspace{1em}\left(\mathrm{a}\right)\langle m{A}_{1g}{d}_{x}n{E}_{1u}^{x}\rangle =\langle m{A}_{1g}{d}_{y}n{E}_{1u}^{y}\rangle ={c}_{mn}^{\left(1\right)}{}^{\left(\mathrm{d}\right)}\phantom{\rule{0ex}{0ex}}\left(\mathrm{f}\right)\langle m{A}_{1g}{d}_{x}n{E}_{1u}^{y}\rangle =\langle m{A}_{1g}{d}_{y}n{E}_{1u}^{x}\rangle =0$$

$$\left{c}_{11}^{\left(1\right)}\right=2.36\phantom{\rule{0ex}{0ex}}\left{c}_{12}^{\left(1\right)}\right=0.16\phantom{\rule{0ex}{0ex}}\left{c}_{13}^{\left(1\right)}\right=0.03\phantom{\rule{0ex}{0ex}}\left{c}_{21}^{\left(1\right)}\right=0.93\phantom{\rule{0ex}{0ex}}\left{c}_{22}^{\left(1\right)}\right=0.51$$
 5,9 
$$\hspace{1em}\hspace{1em}\to {E}_{2g}\rangle $$
 (f)  
$${E}_{1u}\rangle \to {B}_{2u}\rangle $$
 (f)  
$$\hspace{1em}\hspace{1em}\to {B}_{1u}\rangle $$
 (f)  
$$\hspace{1em}\hspace{1em}\to {E}_{2g}\rangle $$

$$\hspace{1em}\hspace{1em}\hspace{1em}\left(\mathrm{a}\right)\langle m{E}_{1u}^{x}{d}_{x}n{E}_{2g}^{{x}^{2}{y}^{2}}\rangle =\langle m{E}_{1u}^{x}{d}_{y}n{E}_{2g}^{xy}\rangle ={c}_{mn}^{\left(2\right)}\phantom{\rule{0ex}{0ex}}=\langle m{E}_{1u}^{y}{d}_{y}n{E}_{2g}^{{x}^{2}{y}^{2}}\rangle =\langle m{E}_{1u}^{y}{d}_{x}n{E}_{2g}^{xy}\rangle \phantom{\rule{0ex}{0ex}}\left(\mathrm{f}\right)\langle m{E}_{1u}^{x}{d}_{y}n{E}_{2g}^{{x}^{2}{y}^{2}}\rangle =\langle m{E}_{1u}^{x}{d}_{x}n{E}_{2g}^{xy}\rangle \phantom{\rule{0ex}{0ex}}=\langle m{E}_{1u}^{y}{d}_{x}n{E}_{2g}^{{x}^{2}{y}^{2}}\rangle =\langle m{E}_{1u}^{y}{d}_{y}n{E}_{2g}^{xy}\rangle =0$$

$$\left{c}_{11}^{\left(2\right)}\right=0.38\phantom{\rule{0ex}{0ex}}\left{c}_{21}^{\left(2\right)}\right=0.67$$
 9 
$${D}_{4\mathrm{h}}$$
 
$${A}_{1g}\rangle \to {E}_{1u}\rangle $$

$$\hspace{1em}\hspace{1em}\left(\mathrm{a}\right)\langle m{A}_{1g}{d}_{x}n{E}_{u}^{x}\rangle =\langle m{A}_{1g}{d}_{y}n{E}_{u}^{y}\rangle ={c}_{mn}^{\left(3\right)}{}^{\left(\mathrm{d}\right)}\phantom{\rule{0ex}{0ex}}\left(\mathrm{f}\right)\langle m{A}_{1g}{d}_{x}n{E}_{u}^{y}\rangle =\langle m{A}_{1g}{d}_{y}n{E}_{u}^{x}\rangle =0$$

$$\left{c}_{11}^{\left(3\right)}\right=5.41$$
 11 
Appendix B.4. The Invariance of 〈${A}_{1g}{d}_{x}{E}_{1u}^{x}$〉 with Respect to Rotation of ${d}_{x}{E}_{1u}^{x}$
References
 Ulusoy, I.S.; Nest, M. Correlated Electron Dynamics: How Aromaticity Can Be Controlled. J. Am. Chem. Soc. 2011, 133, 20230–20236. [Google Scholar] [CrossRef]
 Liu, C.; Manz, J.; Ohmori, K.; Sommer, C.; Takei, N.; Tremblay, J.C.; Zhang, Y. Attosecond Control of Restoration of Electronic Structure Symmetry. Phys. Rev. Lett. 2018, 121, 173201. [Google Scholar] [CrossRef] [PubMed]
 Liu, C.; Manz, J.; Tremblay, J.C. Comment on not only breaking but also restoring the symmetry of the electronic structure of a small molecule by means of laser pulses. Faraday Discuss. 2018, 212, 598–600. [Google Scholar]
 Liu, C.; Manz, J.; Tremblay, J.C. From Symmetry Breaking via Charge Migration to Symmetry Restoration in Electronic Ground and Excited States: Quantum Control on the Attosecond Time Scale. Appl. Sci. 2019, 9, 953. [Google Scholar] [CrossRef] [Green Version]
 Haase, D.; Manz, J.; Tremblay, J.C. Attosecond Charge Migration Can Break Electron Symmetry While Conserving Nuclear Symmetry. J. Phys. Chem. A 2020, 124, 3329–3334. [Google Scholar] [CrossRef] [PubMed]
 Bouakline, F.; Tremblay, J.C. Is it really possible to control aromaticity of benzene with light? Phys. Chem. Chem. Phys. 2020, 22, 15401–15412. [Google Scholar] [CrossRef]
 Mineo, H.; Lin, S.H.; Fujimura, Y. Vibrational effects on UV/Vis laserdriven πelectron ring currents in aromatic ring molecules. Chem. Phys. 2014, 442, 103–110. [Google Scholar] [CrossRef]
 Despré, V.; Marciniak, A.; Loriot, V.; Galbraith, M.C.E.; Rouzée, A.; Vrakking, M.J.J.; Lépine, F.; Kuleff, A.I. Attosecond Hole Migration in Benzene Molecules Surviving Nuclear Motion. J. Phys. Chem. Lett. 2015, 6, 426–431. [Google Scholar] [CrossRef]
 Hermann, G.; Liu, C.; Manz, J.; Paulus, B.; PérezTorres, J.F.; Pohl, V.; Tremblay, J.C. Multidirectional Angular Electronic Flux during Adiabatic Attosecond Charge Migration in Excited Benzene. J. Phys. Chem. A 2016, 120, 5360–5369. [Google Scholar] [CrossRef]
 Jia, D.; Manz, J.; Paulus, B.; Pohl, V.; Tremblay, J.C.; Yang, Y. Quantum control of electronic fluxes during adiabatic attosecond charge migration in degenerate superposition states of benzene. Chem. Phys. 2017, 482, 146–159. [Google Scholar] [CrossRef]
 Hermann, G.; Liu, C.; Manz, J.; Paulus, B.; Pohl, V.; Tremblay, J.C. Attosecond angular flux of partial charges on the carbon atoms of benzene in nonaromatic excited state. Chem. Phys. Lett. 2017, 683, 553–558. [Google Scholar] [CrossRef]
 Hermann, G.; Pohl, V.; Dixit, G.; Tremblay, J.C. Probing Electronic Fluxes via timeResolved XRay Scattering. Phys. Rev. Lett. 2020, 124, 013002. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Altmann, S.L.; Herzig, P. PointGroup Theory Tables; Clarendon: Oxford, UK, 1994; ISBN 0198552262. [Google Scholar]
 Bunker, P.R.; Jensen, P. Molecular Symmetry and Spectroscopy; National Research Council: Ottawa, ON, Canada, 1998; ISBN 0660175193. [Google Scholar]
 Barth, I.; Manz, J. Periodic Electron Circulation Induced by Circularly Polarized Laser Pulses: Quantum Model Simulations for Mg Porphyrin. Angew. Chem. Int. Ed. 2006, 45, 2962–2965. [Google Scholar] [CrossRef] [PubMed]
 Barth, I.; Manz, J.; Shigeta, Y.; Yagi, K. Unidirectional electronic ring current driven by a few cycle circularly polarized laser pulse: Quantum model simulations for Mgporphyrin. J. Am. Chem. Soc. 2006, 128, 7043–7049. [Google Scholar] [CrossRef] [PubMed]
 Dunning, T.H., Jr. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. [Google Scholar] [CrossRef]
 Werner, H.J.; Knowles, P.J.; Knizia, G.; Manby, F.R.; Schütz, M.; Celani, P.; Korona, T.; Lindh, R.; Mitrushenkov, A.; Rauhut, G.; et al. Molpro, Version 2012.1, A Package of Ab Initio Programs. 2012. Available online: http://www.molpro.net (accessed on 20 November 2020).
 Hashimoto, T.; Nakano, H.; Hirao, K. Theoretical study of the valence π→π* excited states of polyacenes: Benzene and naphthalene. J. Chem. Phys. 1996, 104, 6244–6258. [Google Scholar] [CrossRef]
 Yanai, T.; Tew, D.P.; Handy, N.C. A new hybrid exchange–correlation functional using the Coulombattenuating method (CAMB3LYP). Chem. Phys. Lett. 2004, 393, 51–57. [Google Scholar] [CrossRef] [Green Version]
 Woon, D.E.; Dunning, T.H., Jr. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358–1371. [Google Scholar] [CrossRef] [Green Version]
 Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Petersson, G.A.; Nakatsuji, H.; et al. Gaussian 16, Revision A.03; Gaussian Inc.: Wallingford, CT, USA, 2016. [Google Scholar]
 Sundholm, D. Density functional theory study of the electronic absorption spectrum of Mgporphyrin and MgetioporphyrinI. Chem. Phys. Lett. 2000, 317, 392–399. [Google Scholar] [CrossRef]
 Tremblay, J.C.; Klamroth, T.; Saalfrank, P. TimeDependent ConfigurationInteraction Calculations of LaserDriven Dynamics in Presence of Dissipation. J. Chem. Phys. 2008, 129, 084302. [Google Scholar] [CrossRef]
 Tremblay, J.C.; Krause, P.; Klamroth, T.; Saalfrank, P. The Effect of Energy and Phase Relaxation on Dynamic Polarizability Calculations. Phys. Rev. A 2010, 81, 063420. [Google Scholar] [CrossRef]
 Tremblay, J.C.; Klinkusch, S.; Klamroth, T.; Saalfrank, P. Dissipative ManyElectron Dynamics of Ionizing Systems. J. Chem. Phys. 2011, 134, 044311. [Google Scholar] [CrossRef] [PubMed]
 Klinkusch, S.; Tremblay, J.C. ResolutionofIdentity Stochastic TimeDependent Configuration Interaction for Dissipative Electron Dynamics in Strong Fields. J. Chem. Phys. 2016, 144, 184108. [Google Scholar] [CrossRef] [PubMed]
 Hermann, G.; Pohl, V.; Tremblay, J.C.; Paulus, B.; Hege, H.C.; Schild, A. ORBKIT—A modular Python Toolbox for CrossPlatform PostProcessing of Quantum Chemical Wavefunction Data. J. Comput. Chem. 2016, 37, 1511–1520. [Google Scholar] [CrossRef] [Green Version]
 Pohl, V.; Hermann, G.; Tremblay, J.C. An OpenSource Framework for Analyzing NElectron Dynamics: I. Multideterminantal Wave Functions. J. Comput. Chem. 2017, 38, 1515–1527. [Google Scholar] [CrossRef] [Green Version]
 Hermann, G.; Pohl, V.; Tremblay, J.C. An OpenSource Framework for Analyzing NElectron Dynamics: II. Hybrid Density Functional Theory/Configuration Interaction Methodology. J. Comput. Chem. 2017, 38, 2378–2387. [Google Scholar] [CrossRef]
 Tremblay, J.C.; Carrington, T., Jr. Using Preconditioned Adaptive Step Size RungeKutta Methods for Solving the TimeDependent Schrödinger Equation. J. Chem. Phys. 2004, 121, 11535. [Google Scholar] [CrossRef]
 Hunter, J.D. Matplotlib: A 2D graphics environment. Comput. Sci. Eng. 2007, 9, 90–95. [Google Scholar] [CrossRef]
 Tannor, D.J. Introduction to Quantum Mechanics: A TimeDependent Perspective; University Science Books: Sausalito, CA, USA, 2012. [Google Scholar]
 Jia, D.; Manz, J.; Schild, A.; Svoboda, V.; Yang, Y. From Nuclear Fluxes during Tunnelling to Electronic Fluxes During Charge Migration, in Tunnelling in Molecules: Nuclear Quantum Effects from Bio to Physical Chemistry; Kästner, J., Kozuch, S., Eds.; Theoretical and Computational Chemistry Series No. 18; Royal Society of Chemistry: Cambridge, UK, 2021; Chapter 5; pp. 167–191. [Google Scholar]
 Woodward, R.B.; Hoffmann, R. The Conservation of Orbital Symmetry. Angew. Chem. Int. Ed. 1969, 8, 781–853. [Google Scholar] [CrossRef]
 Chelkowski, S.; Yudin, G.L.; Bandrauk, A.D. Observing electron motion in molecules. J. Phys. B At. Mol. Opt. Phys. 2006, 39, S409–S417. [Google Scholar] [CrossRef]
 Kraus, P.M.; Mignolet, B.; Baykusheva, D.; Rupenyan, A.; Horný, L.; Penka, E.F.; Grassi, G.; Tolstikhin, O.I.; Schneider, J.; Jensen, F.; et al. Measurement and Laser Control of Attosecond Charge Migration in Ionized Iodoacetylene. Science 2015, 350, 790–795. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 Jia, D.; Manz, J.; Yang, Y. Generation of electronic flux during femtosecond π/2 laser pulse tailored to induce adiabatic attosecond charge migration in HCCI^{+}. J. Mod. Opt. 2017, 64, 960–970. [Google Scholar] [CrossRef]
 Remacle, F.; Kienberger, R.; Krausz, F.; Levine, R.D. On the feasibility of an ultrafast purely electronic reorganization in lithium hydride. Chem. Phys. 2007, 338, 342–347. [Google Scholar] [CrossRef]
 Yuan, K.J.; Bandrauk, A.D. Symmetry in Circularly Polarized Molecular HighOrder Harmonic Generation with Intense Circular Laser Pulses. Phys. Rev. A 2018, 97, 023408. [Google Scholar] [CrossRef]
 Yudin, G.L.; Chelkowski, S.; Itatani, J.; Bandrauk, A.D.; Corkum, P.B. Attosecond Photoionization of Coherently Coupled Electronic States. Phys. Rev. A 2005, 72, 051401. [Google Scholar] [CrossRef] [Green Version]
 Ivanov, M.Y.; Corkum, P.B.; Dietrich, P. Coherent Control and Collapse of Symmetry in a TwoLevel System in an Intense Laser Field. Laser Phys. 1993, 3, 375–380. [Google Scholar]
 Baykusheva, D.; Ahsan, M.S.; Lin, N.; Wörner, H.J. Bicircular HighHarmonic Spectroscopy Reveals Dynamical Symmetries of Atoms and Molecules. Phys. Rev. Lett. 2016, 116, 123001. [Google Scholar] [CrossRef] [Green Version]
 Liu, X.; Zhu, X.; Li, L.; Li, Y.; Zhang, Q.; Lan, P.; Lu, P. Selection Rules of HighOrderHarmonic Generation: Symmetries of Molecules and Laser Fields. Phys. Rev. A 2016, 94, 033410. [Google Scholar] [CrossRef] [Green Version]
 Yuan, K.J.; Bandrauk, A.D. Exploring Coherent Electron Excitation and Migration Dynamics by Electron Diffraction with Ultrashort Xray Pulses. Chem. Phys. 2017, 19, 25846–25852. [Google Scholar] [CrossRef]
 PopovaGorelova, D. Imaging Electron Dynamics with Ultrashort Light Pulses: A Theory Perspective. Appl. Sci. 2018, 8, 318. [Google Scholar] [CrossRef] [Green Version]
 Martín, F.; Fernández, J.; Havermeier, T.; Foucar, L.; Weber, T.; Kreidi, K.; Schöffler, M.; Schmidt, L.; Jahnke, T.; Jagutzki, O.; et al. Single PhotonInduced Symmetry Breaking of H_{2} Dissociation. Science 2007, 315, 629–633. [Google Scholar] [CrossRef] [PubMed] [Green Version]
 McWeeny, R. Symmetry, an Introduction to Group Theory and Its Applications; Pergamon: Oxford, UK, 1963; pp. 223–233. [Google Scholar]
$$\begin{array}{l}{D}_{6\mathrm{h}}=\{E,2{C}_{6},2{C}_{3},{C}_{2},3{C}_{2}^{\prime},3{C}_{2}^{\u2033},i,2{S}_{3},2{S}_{6},{\sigma}_{h},3{\sigma}_{d},3{\sigma}_{v}\}\\ {D}_{3\mathrm{h}}^{\prime}=\{E,2{C}_{3},3{C}_{2}^{\prime},{\sigma}_{h},2{S}_{3},3{\sigma}_{v}\}\\ {D}_{3\mathrm{h}}^{\u2033}=\{E,2{C}_{3},3{C}_{2}^{\u2033},{\sigma}_{h},2{S}_{3},3{\sigma}_{d}\}\\ {D}_{3\mathrm{d}}^{\prime}=\{E,2{C}_{3},3{C}_{2}^{\prime},i,2{S}_{6},3{\sigma}_{d}\}\\ {D}_{3\mathrm{d}}^{\u2033}=\{E,2{C}_{3},3{C}_{2}^{\u2033},i,2{S}_{6},3{\sigma}_{v}\}\\ {D}_{2\mathrm{h},\mathrm{k}}=\{E,{C}_{2},{C}_{2k}^{\prime},{C}_{2k}^{\u2033},i,{\sigma}_{h},{\sigma}_{dk},{\sigma}_{vk}\},\hspace{1em}k=1,2,3\\ {C}_{2\mathrm{v},\mathrm{k}}^{\prime}=\{E,{C}_{2k}^{\prime},{\sigma}_{h},{\sigma}_{vk}\},\hspace{1em}k=1,2,3,\hspace{1em}\subset \hspace{1em}{D}_{2\mathrm{h},\mathrm{k}},{D}_{3\mathrm{h}}^{\prime}\\ {C}_{2\mathrm{v},\mathrm{k}}^{\u2033}=\{E,{C}_{2k}^{\u2033},{\sigma}_{h},{\sigma}_{vk}\},\hspace{1em}k=1,2,3,\hspace{1em}\subset \hspace{1em}{D}_{2\mathrm{h},\mathrm{k}},{D}_{3\mathrm{h}}^{\u2033}\\ {C}_{2h}=\{E,{C}_{2},i,{\sigma}_{h}\}\hspace{1em}\subset \hspace{1em}{D}_{2\mathrm{h},\mathrm{k}},k=1,2,3\\ {C}_{s}\hspace{1em}=\{E,{\sigma}_{h}\}\hspace{1em}\subset \hspace{1em}{C}_{2\mathrm{h}}\end{array}$$

Benzene (G = ${D}_{6h})$ ^{(b)}  MgPorphyrin (G = ${D}_{4h})$  

Polarization vectors  ${e}_{1}^{\prime}$  ${e}_{2}^{\prime}$  ${e}_{3}^{\prime}$  ${e}_{1}^{\u2033}$  ${e}_{2}^{\u2033}$  ${e}_{3}^{\u2033}$  ${e}_{1}^{\prime}$ & ${e}_{1}^{\u2033}$  ${e}_{1}^{\prime}$  ${e}_{2}^{\prime}$  ${e}_{1}^{\u2033}$  ${e}_{2}^{\u2033}$  
=  ${e}_{y}$  ${C}_{3}{e}_{y}$  ${C}_{3}^{2}{e}_{y}$  ${e}_{x}$  ${C}_{3}{e}_{x}$  ${C}_{3}^{2}{e}_{x}$  ${e}_{y}$ & ${e}_{x}$  =  ${e}_{x}$  ${e}_{y}$  ${C}_{8}{e}_{x}$  ${C}_{8}{e}_{y}$  
parallel to binary rotation axes  ${C}_{21}^{\prime}$  ${C}_{22}^{\prime}$  ${C}_{23}^{\prime}$  ${C}_{21}^{\u2033}$  ${C}_{22}^{\u2033}$  ${C}_{23}^{\u2033}$  ${C}_{21}^{\prime},{C}_{21}^{\u2033}$  ${C}_{21}^{\prime}$  ${C}_{22}^{\prime}$  ${C}_{21}^{\u2033}$  ${C}_{22}^{\u2033}$  
associated symmetry planes  ${\sigma}_{v1}$  ${\sigma}_{v2}$  ${\sigma}_{v3}$  ${\sigma}_{d1}$  ${\sigma}_{d2}$  ${\sigma}_{d3}$  ${\sigma}_{h}$  ${\sigma}_{v1}$  ${\sigma}_{v2}$  ${\sigma}_{d1}$  ${\sigma}_{d2}$  
resulting subgroup S (${\mathrm{H}}_{\mathrm{e}}\left(\mathrm{t}\right))$  ${C}_{2v,1}^{\prime}$  ${C}_{2v,2}^{\prime}$  ${C}_{2v,3}^{\prime}$  ${C}_{2v,1}^{\u2033}$  ${C}_{2v,2}^{\u2033}$  ${C}_{2v,3}^{\u2033}$  ${C}_{s}$  ${C}_{2v,1}^{\prime}$  ${C}_{2v,2}^{\prime}$  ${C}_{2v,1}^{\u2033}$  ${C}_{2v,2}^{\u2033}$  
with symmetry operations  ${C}_{2v,k}^{\prime}$ = {E, ${C}_{2k}^{\prime}$, ${\sigma}_{h},$ ${\sigma}_{vk}$}, $k$= 1, 2, 3 {E,${\sigma}_{h}\}$ ${C}_{2v,k}^{\u2033}$ = {E, ${C}_{2k}^{\u2033},{\sigma}_{h}$, ${\sigma}_{dk}$}, $k$ = 1, 2, 3  ${C}_{2v,k}^{\prime}$ = {E, ${C}_{2k}^{\prime},{\sigma}_{h}$, ${\sigma}_{vk}$}, $k$= 1, 2 ${C}_{2v,k}^{\u2033}$ = {E, ${C}_{2k}^{\u2033},{\sigma}_{h}$, ${\sigma}_{dk}$}, $k$ = 1, 2 
Ψ〉 = 0〉 + m〉 ^{(a)}  A^{1g}〉 + $\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{x}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + ${\mathit{C}}_{\mathbf{3}}\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{x}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + ${\mathit{C}}_{\mathbf{3}}^{\mathbf{2}}$$\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{x}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + $\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{y}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + ${\mathit{C}}_{\mathbf{3}}$$\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{y}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + ${\mathit{C}}_{\mathbf{3}}^{\mathbf{2}}$$\mathbf{}{\mathit{E}}_{\mathbf{1}\mathit{u}}^{\mathit{y}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + $\mathbf{}{\mathit{E}}_{\mathbf{2}\mathit{g}}^{{\mathit{x}}^{\mathbf{2}}\mathbf{}{\mathit{y}}^{\mathbf{2}}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + $\mathbf{}{\mathit{E}}_{\mathbf{2}\mathit{g}}^{\mathit{x}\mathit{y}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + $\mathbf{}{\mathit{B}}_{\mathbf{2}\mathit{u}}\mathbf{\rangle}$  $\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}$ + $\mathbf{}{\mathit{B}}_{\mathbf{1}\mathit{u}}\mathbf{\rangle}$  

g ∈ D_{6h} ^{(b)}  
E  1  1  1  1  1  1  1  1  1  1  
2 C_{6}  
2 C_{3}  1  1  
C_{2}^{z}  1  1  
C_{21}‘  1  1  1  
C_{22}‘  1  1  
C_{23}‘  1  1  
C_{21}‘‘  1  1  1  
C_{22}‘‘  1  1  
C_{23}‘‘  1  1  
i  1  1  
2 S_{3}  1  1  
2 S_{6}  
σ_{h}  1  1  1  1  1  1  1  1  1  1  
σ_{v1}  1  1  1  
σ_{v2}  1  1  
σ_{v3}  1  1  
σ_{d1}  1  1  1  
σ_{d2}  1  1  
σ_{d3}  1  1  
${\mathrm{S}}_{\mathsf{\Psi}}\subseteq {\mathbf{D}}_{6\mathrm{h}}$ $\mathrm{IRREP}\mathrm{of}{\mathrm{S}}_{\mathsf{\Psi}}{}^{\left(\mathrm{c}\right)}$  ${C}_{2v,1}^{\u2033}$ ${A}_{1}$  ${C}_{2v,2}^{\u2033}$ ${A}_{1}$  ${C}_{2v,3}^{\u2033}$ ${A}_{1}$  ${C}_{2v,1}^{\prime}$ ${A}_{1}$  ${C}_{2v,2}^{\prime}$ ${A}_{1}$  ${C}_{2v,3}^{\prime}$ ${A}_{1}$  ${D}_{2h,1}$ ${A}_{g}$  ${C}_{2h}$ ${A}_{g}$  ${D}_{3h}^{\u2033}$ ${A}_{1}^{\prime}$  ${D}_{3h}^{\prime}$ ${A}_{1}^{\prime}$  
${\mathrm{S}}_{\mathsf{\rho}}\subseteq {\mathrm{D}}_{6\mathrm{h}}$  ${C}_{2v,1}^{\u2033}$  ${C}_{2v,2}^{\u2033}$  ${C}_{2v,3}^{\u2033}$  ${C}_{2v,1}^{\prime}$  ${C}_{2v,2}^{\prime}$  ${C}_{2v,3}^{\prime}$  ${D}_{2h,1}$  ${C}_{2h}$  ${D}_{3h}^{\u2033}$  ${D}_{3h}^{\prime}$  
Laser pulses p ^{(d)}  1  3  4  2  5  6  7&8  7&9  Ref. [1]  Ref. [1] 
$$\mathbf{}\mathsf{\Psi}\mathbf{\rangle}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{\rangle}\phantom{\rule{0ex}{0ex}}\mathbf{\hspace{1em}\hspace{1em}\hspace{1em}+}\phantom{\rule{0ex}{0ex}}$$

$$\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}\phantom{\rule{0ex}{0ex}}\mathbf{+}\phantom{\rule{0ex}{0ex}}\mathbf{}{\mathit{E}}_{\mathit{u}\mathbf{1}}^{\mathbf{\prime}}\mathbf{\rangle}$$

$$\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}\phantom{\rule{0ex}{0ex}}\mathbf{+}\phantom{\rule{0ex}{0ex}}\mathbf{}{\mathit{E}}_{\mathit{u}\mathbf{2}}^{\mathbf{\prime}}\mathbf{\rangle}$$

$$\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}\phantom{\rule{0ex}{0ex}}\mathbf{+}\phantom{\rule{0ex}{0ex}}\mathbf{}{\mathit{E}}_{\mathit{u}\mathbf{1}}^{\mathbf{\u2033}}\mathbf{\rangle}$$

$$\mathbf{}{\mathit{A}}_{\mathbf{1}\mathit{g}}\mathbf{\rangle}\phantom{\rule{0ex}{0ex}}\mathbf{+}\phantom{\rule{0ex}{0ex}}\mathbf{}{\mathit{E}}_{\mathit{u}\mathbf{2}}^{\mathbf{\u2033}}\mathbf{\rangle}$$
 

g ∈ D_{4h} ^{(b)}  
E  1  1  1  1  
C_{4}^{+} C_{4}^{−}  
C_{2}^{z}  
C_{21}‘  1  
C_{22}‘  1  
C_{21}‘‘  1  
C_{22}‘‘  1  
i  
S_{4}^{−} S_{4}^{+}  
σ_{h}  1  1  1  1  
σ_{v1}  1  
σ_{v2}  1  
σ_{d1}  1  
σ_{d2}  1  
$${\mathrm{S}}_{\mathsf{\Psi}}\subseteq {\mathbf{D}}_{4\mathrm{h}}$$

$${C}_{2v,1}^{\prime}$$

$${C}_{2v,2}^{\prime}$$

$${C}_{2v,1}^{\prime \prime}$$

$${C}_{2v,2}^{\prime \prime}$$
 
$$\mathrm{IRREP}\mathrm{of}{\mathrm{S}}_{\mathsf{\Psi}}{}^{\left(\mathrm{c}\right)}$$
 A_{1}  A_{1}  A_{1}  A_{1}  
S_{ρ} ⊆ D_{4h} 
$${C}_{2v,1}^{\prime}$$

$${C}_{2v,2}^{\prime}$$

$${C}_{2v,1}^{\prime \prime}$$

$${C}_{2v,2}^{\prime \prime}$$
 
Laser pulses p ^{(d)}  1  2  3  4 
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Haase, D.; Hermann, G.; Manz, J.; Pohl, V.; Tremblay, J.C. Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics. Symmetry 2021, 13, 205. https://doi.org/10.3390/sym13020205
Haase D, Hermann G, Manz J, Pohl V, Tremblay JC. Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics. Symmetry. 2021; 13(2):205. https://doi.org/10.3390/sym13020205
Chicago/Turabian StyleHaase, Dietrich, Gunter Hermann, Jörn Manz, Vincent Pohl, and Jean Christophe Tremblay. 2021. "Electron Symmetry Breaking during Attosecond Charge Migration Induced by Laser Pulses: Point Group Analyses for Quantum Dynamics" Symmetry 13, no. 2: 205. https://doi.org/10.3390/sym13020205