Multidirectional Angular Electronic Flux during Adiabatic Attosecond

Article pubs.acs.org/JPCA

Multidirectional Angular Electronic Flux during Adiabatic Attosecond Charge Migration in Excited Benzene Gunter Hermann,† ChunMei Liu,† Jörn Manz,*,†,‡,§ Beate Paulus,† Jhon Fredy Pérez-Torres,*,† Vincent Pohl,† and Jean Christophe Tremblay† †

Freie Universität Berlin, Institut für Chemie und Biochemie, Takustrasse 3, 14195 Berlin, Germany State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China § Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China ‡

ABSTRACT: Recently, adiabatic attosecond charge migration (AACM) has been monitored and simulated for the first time, with application to the oriented iodoacetylene cation where AACM starts from the initial superposition of the ground state (φ0) and an electronic excited state (φ1). Here, we develop the theory for electronic fluxes during AACM in ring-shaped molecules, with application to oriented benzene prepared in the superposition of the ground and first excited singlet states. The initial state and its time evolution are analogous to coherent tunneling where φ0 and φ1 have different meanings; however, they denote the wave functions of the lowest tunneling doublet. This analogy suggests to transfer the theory of electronic fluxes during coherent tunneling to AACM, with suitable modifications which account for (i) the different time scales and (ii) the different electronic states, and which make use of (iii) the preparation of the initial state for AACM by a linearly polarized laser pulse. Application to benzene yields the multidirectional angular electronic flux with a pincer-motion type pattern during AACM: this unequivocal result confirms a previous working hypothesis. Moreover, the theory of AACM allows quantification of the electronic flux; that is, the maximum number of electrons (out of 42) which flow concertedly during AACM in benzene is 6 × 0.08 = 0.48. electronic densities, and with the underlying electronic flux from one or several domains of the molecule to other domains. AACM, by preparation and subsequent time evolution of a superposition state such as eq 1, has been monitored experimentally and simulated theoretically, for the first time, in a recent benchmark publication with application to the oriented iodoacetylene cation.3 The publication3 may be considered as a culmination of previous work on attosecond charge migration, from the first experimental evidence of the phenomenon in an isolated peptide4 via quantum model simulations (see ref 5 and the references therein) to the first design of so-called π/2-laser pulses which generate the decisive initial superposition state, eq 1,1 and to the first quantum simulations of the laser preparations of two different initial states of this type, eq 1, combined with subsequent AACM in benzene;2 see below for the details. All publications1−5 did, however, not yet determine the electronic fluxes during AACM. In summary, the first goal for this paper is to develop the theory

1. INTRODUCTION The purpose of this article is twofold: first, to develop the theory for angular electronic fluxes during adiabatic attosecond charge migration (AACM) in an excited ring-shaped molecule. As a specific example, we consider the scenario where initially, at time t = 0, the molecule is prepared in a ”reactant” (R) superposition state ψR =

1 (φ + φe) 2 g

(1)

where φg and φe denote the wave functions of the electronic ground (g) state and an excited (e) eigenstate, respectively. The wave function eq 1 corresponds to 50% excitation of the molecule from the ground to the excited eigenstate, e.g. by a laser pulse. Since the superposition state eq 1 is not an eigenstate, it evolves in time, typically in the attosecond time domain, see e.g. refs 1−3. On this ultrashort time scale, the nuclei are essentially frozen. As a consequence, the effects of kinetic couplings, also called nonadiabatic couplings, are negligible; that is, the initial state, eq 1, evolves adiabatically. This scenario is referred to as ”adiabatic attosecond charge migration”. Irrespective of the robust 0.5:0.5 probabilities of occupying the electronic ground and excited states, the resulting time evolution is associated with nonstationary © XXXX American Chemical Society

Special Issue: Piergiorgio Casavecchia and Antonio Lagana Festschrift Received: February 25, 2016 Revised: April 5, 2016

A

DOI: 10.1021/acs.jpca.6b01948 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A

Figure 1. AACM in benzene, with period τ ̅ = 848 as, starting from the S0 + S1 superposition state, eq 2. The molecule is aligned in the x−y laboratory plane. The angular electronic flux at ϕ is defined as the flux through the half plane at ϕ, perpendicular to the molecular plane. (a) Series of Lewis-type structures, from reactant (R, t = 0) via charge equilibration in all bonds (t = τ ̅ /4 ) to product (P, t = τ ̅ /2 ), and back to R (t = τ ̅ ) (schematic). (b) Corresponding snapshots of the three-dimensional (3d) one-electron density, illustrated by 2d color-coded contour plots in the plane at z = 0.4 a0, parallel to the molecular plane ( in units of 1/a03). The red curved arrows in panel (a) symbolize the multidirectional, pincermotion type pattern of the angular electronic flux, confirming the working hypothesis of ref 2. The number 0.08 at one of the arrows is the maximum yield of electrons which flow from R to P through the half plane at ϕm = π/6, during the first half period, 0 ≤ t ≤ τ ̅ /2 . The same number applies to all six arrows. The corresponding maximum number of all electrons which flow concertedly is 6× 0.08 = 0.48. [The curved arrows in the second cartoon of the top row are adapted from the table-of-contents figure of ref 2. Copyright 2011 American Chemical Society. (Note that Figure 6 of ref 2 shows the same orientation with respect to a coordinate system which has been rotated by 90° around the z-axis.)]

of angular electronic fluxes during AACM in a ring-shaped molecule. For comparison, Takatsuka and co-workers6−11 have developed the theory for electronic fluxes which are associated with diabatic transitions, with many applications; see also refs 12 and 13. Those transitions depend on nuclear motions, typically on much longer time scales (at least ten femtoseconds, or more) compared to the present attosecond time domain. The second goal is to present the first application, exemplarily to charge migration in an excited superposition state of oriented benzene. Specifically, we assume that the benzene molecule is oriented with its molecular plane in the x− y plane (see section 2 for the details) and that it has been prepared by means of an x-polarized laser pulse in the ”reactant” state ψR =

1 (φ + φ1) 2 0

ψP =

1 ( −φ0 + φ1) 2

(3)

represent the ”reactant” and ”product” (P) states, which are localized in two opposite potential wells, labeled R and P. Now it just so happens that, recently, we have derived the theory of concerted electronic and nuclear fluxes during coherent tunneling; see the Perspective Article14 and the literature cited therein. The formal equivalence of the initial state, eq 2, thus offers an ideal opportunity for analogous solutions of the two tasks which have been defined above: first, to transfer the theory and methods for electronic fluxes from coherent tunneling14 (see also ref 17) to AACM. This transfer cannot be done, however, by a simple copy−paste approach, due to three fundamental differences: (i) the time scales of tunneling (from picoseconds to kiloseconds,14 and beyond15) are much longer compared to ultrafast charge migration (attoseconds1−5). (ii) coherent tunneling is in the electronic ground state, whereas charge migration involves the ground and excited states, cf. eqs 1 and 2. (iii) The preparation of the reactant state, eq 2, is irrelevant for tunneling,14−17 whereas for charge migration, it turns out to be helpful. For these reasons, we shall present the theory in a selfcontained manner which allows explanation of the important consequences of items (i)−(iii). The second opportunity is to apply the general theory to the first calculation of the angular electronic flux during AACM, specifically for the model system, oriented benzene, adapted from ref 2. This shall provide the answers to the questions about (a) the angular directions and (b) the maximum number of electrons which flow during AACM. In fact, the present application to benzene profits from experience with the previous examples of ring-shaped molecules14 (e.g., trapezoidal B4) or fragments of molecules (e.g., the six-membered ring of carbon atoms in semibullvalene14,16,17). As an appetizer, Figure 1a shows a sequence of Lewis-type structures of benzene which illustrate its time evolution, starting from the reactant configuration R, eq 2, adapted from ref 2. This is supported by corresponding snapshots of the time

(2)

where φ0 and φ1 represent the electronic ground (S0) and first excited singlet (S1) states, respectively, as suggested in ref 2. We shall address and answerfor the first timethe following questions: (a) In which angular direction(s)e.g. clockwise or anticlockwisedo the electrons flow during AACM? and (b) what is the maximum number of electrons to migrate? The present investigation is motivated by three recent developments. First, the benchmark publication, ref 3 calls for calculations of electronic fluxes during AACM. Second, ref 2 already suggests that after preparation of benzene in the superposition state, eq 2, charge migration should proceed with a pincer-motion type pattern of the angular electronic fluxes, but this conjecture was left as a stimulating working hypothesis, without any derivation and without specifying the number of flowing electrons. Third, we note that the initial state, eq 2, is formally identical to the ”reactant” state of coherent tunneling in a double well potential14 (see also refs 15−17), even though the meanings of the states φ0 and φ1 are entirely different: For charge migration starting from ψR, they represent the electronic ground and first excited states. In contrast, in coherent tunneling, φ0 and φ1 constitute the lowest tunneling doublet, and ψR as well as B


Article


These densities are normalized in (3Ne + 3Nn = 3 × 42 + 3 × 12)-dimensional space of all (Ne = 42) electrons and all (Nn = 12) nuclei. Spin coordinates are implicit in the notations in eqs 5 and 6, without writing them explicitly; for example, the wave functions are antisymmetric with respect to commutations of electrons. The reactant state, eq 2, evolves as

evolution of the one-electron density; cf. Figure 1b. The curved arrows in Figure 1a indicate the working hypothesis of ref 2, that is, pincer-motion type directions of the underlying angular electronic flux. But the same time evolution of the electronic density could result from entirely different, for example, clockwise or anticlockwise flux patterns; see, e.g., Figure 1 of ref 18. The goal of this paper is not only to determine the angular electronic flux directions unequivocally, but also to quantify the flux. We shall see that this task is nontrivial, and in fact, we shall come back to discuss alternative flux patterns, at the end of section 4.

ψ (q , Q , t ) =

(7)

where E0 and E1 denote the eigenenergies of the molecular ground and first excited singlet states, respectively. The resulting time evolution of the density is

2. MODEL, THEORY, AND METHODS The model for AACM in benzene is adapted from ref 2. Accordingly, we assume that the molecular center of mass is located at the origin of a laboratory-fixed system of Cartesian coordinates x, y, z, and the benzene molecule has been oriented in the x−y plane, using, e.g., the methods of refs 3 and 19−21. The specific orientation is illustrated in Figure 1. For convenience, the six carbon nuclei and the protons are labeled C1, C2, C3, C4, C5, C6 (i = 1, ..., 6) and H1, H2, H3, H4, H5, H6 (i = 7, ..., 12), with nuclear coordinates Qi = Q1, ..., Q6 and Q7, ..., Q12, respectively. We use the same short hand notation Qi either for Cartesian coordinates Qi = (Xi,Yi,Zi) or for cylindrical ones Qi = (Ri,Φi,Zi), i = 1, ..., 12. Subsequently, we shall also employ corresponding Cartesian coordinates qi = (xi,yi,zi) or cylindrical ones qi = (ri,ϕi,zi) of the electrons, i = 1, ..., 42. As in ref 2, we consider the scenario where the oriented benzene has been prepared in the initial (t = 0) ”reactant” state, eq 2, by means of an x-polarized laser pulse. This ”reactant” state (R) is illustrated schematically by the Lewis-type structure in Figure 1a, also adapted from ref 2. The model of ref 2 assumes that nuclear motions are negligible during laser excitation. Hence, the nuclei are centered at the equilibrium geometry of benzene in its electronic ground state S0. The corresponding equilibrium values Q̅ i of the nuclear coordinates (indicated by bars) are

ρ(q , Q , t ) = |ψ (q , Q , t )|2 = ρR (q , Q ) + [ρP (q , Q ) − ρR (q , Q )] sin 2(πt /τ ) ≡ ρR (q , Q ) + Δρ(q , Q ) sin 2(πt /τ ) (8)

where Δρ denotes the difference of the densities of P and R. Obviously, the density, eq 8, evolves periodically, from ρR at t = 0, τ, 2τ, ... to ρP at t = τ/2, 3τ/2, 5τ/2, .... The transitions R → P → R → ··· are illustrated symbolically in Figure 1; for the first period, 0 ≤ t ≤ τ. The period τ is related to the energy gap ΔE = E1 − E0 by Typical molecular energy gaps are in the domain from one to few electronvoltsthis puts the time scale τ of AACM into the domain from a few hundred attoseconds to a few femtoseconds. The first modification of the theory, from coherent tunneling14 to AACM, accounts for the different rather long versus ultrashort time scales, respectively; see item (i) in section 1. Accordingly, the nuclei are frozen on the attosecond time scale of AACMeffects of vibrations, rotations, or translations are negligible.22 This allows replacement, eqs 2, 3, 5−9, by the approximations eqs 2′, 3′, and 5′−9′,

R̅ 7 = R̅ 8 = ... = R12 ̅ = 4.67a0 Φ̅ 1 = Φ̅ 7 = π /6 Φ̅ 2 = Φ̅ 8 = π /6 + π /3 ... Φ̅6 = Φ̅ 12 = π /6 + 5π /3 (4)

(5)

ρP (q , Q ) = |ψP(q , Q )|2

(6)

ψR(q , Q̅ ) =

1 [φ (q , Q̅ ) + φ1(q , Q̅ )] 2 0

(2′)

ψP(q , Q̅ ) =

1 [−φ0(q , Q̅ ) + φ1(q , Q̅ )] 2

(3′)

ρR (q , Q̅ ) = |ψR(q , Q̅ )|2

(5′)

ρP (q , Q̅ ) = |ψP(q , Q̅ )|2

(6′)

ψ (q , Q̅ , t ) =

The theory for the resulting electronic fluxes during AACM is adapted from the theory of the fluxes during coherent tunneling,14 with three modifications which have been anticipated in items (i), (ii), and (iii) of the motivation in section 1. Accordingly, the ”product” state (P) is defined by eq 3, analogous to the ”reactant” state (R), cf. eq 2. Figure 1a shows the corresponding Lewis-type structure of P, analogous to R. The densities of R and P, depending on the coordinates of all nuclei Q = {Q1, ..., Q12} and all electrons q = {q1, ..., q42}, are ρR (q , Q ) = |ψR(q , Q )|2

(9)

ΔE ·τ = h

R1̅ = R̅ 2 = ... = R̅ 6 = 2.64a0

Z1̅ = Z̅2 = ... = Z12 ̅ =0

1 [φ (q , Q )e−iE0t / ℏ + φ1(q , Q )e−iE1t / ℏ] 2 0

1 [φ (q , Q̅ )e−iE0̅ t / ℏ + φ1(q , Q̅ )e−iE1̅ t / ℏ] 2 0 (7′)

ρ(q , Q̅ , t ) = |ψ (q , Q̅ , t )|2 = ρR (q , Q̅ ) + [ρP (q , Q̅ ) − ρR (q , Q̅ )] sin 2(πt /τ ̅ ) ≡ ρR (q , Q̅ ) + Δρ(q , Q̅ ) sin 2(πt /τ ̅ ) (8′)

ΔE ̅ ·τ ̅ = (E1̅ − E0̅ ) ·τ ̅ = h

(9′)

where Q̅ denotes the nuclear equilibrium coordinates, eq 4. The electronic densities in eqs 5′, 6′, and 8′ depend parametrically on the nuclear coordinates Q̅ . They are normalized in (3Ne = 3× 42)-dimensional space of all electrons. C


Article


jϕ (ϕ , Q̅ , t ) is the angular electronic flux. Its physical unit is 1/ time. The angular electronic flux eq 16 is the target quantity of this paper. It can be obtained by integrating eq 14 over the angle ϕ,

The wave functions φk (q , Q̅ ) and energies Ek̅ of the electronic eigenstates k = 0 and 1 are obtained as solutions of the electronic Schrödinger equation Helφk (q , Q̅ , t ) = Ek̅ φk (q , Q̅ , t )

(10)

for fixed nuclear coordinates Q̅ . The nonrelativistic electronic Hamiltonian Hel = Te + V consists of two terms, i.e., Te for the kinetic energy of all electrons, plus the Coulomb interactions V = Vee + Vnn + Ven between all electrons, between all the nuclei, and between the electrons and nuclei, respectively. The electronic density ρ(q , Q̅ , t ) is related to the flux density j ⃗ (q , Q̅ , t ) by the continuity equation for all electrons14,23 ρ̇(q , Q̅ , t ) + div j ⃗ (q , Q̅ , t ) = 0

jϕ (ϕ , Q̅ , t ) = jϕ (ϕ0 , Q̅ , t ) −

∂ j (ϕ , Q̅ , t ) = 0 ∂ϕ ϕ

= jϕ (ϕ0 , Q̅ , t ) −

(11)

∫0

∞

∫−∞ dzρ3d (r , ϕ , z , Q̅ , t )

∫0

∞

∫−∞ dzj3d (r , ϕ , z , Q̅ , t )

ϕ

dϕ′Δρϕ (ϕ′, Q̅ )·(π /τ ̅ )

where ρR , ϕ (ϕ , Q̅ ) =

∫0

∞

∞

r dr

∫−∞ dzρ3d ,R (r , ϕ , z , Q̅ )

r dr

∫−∞ dzρ3d ,P (r , ϕ , z , Q̅ )

(19)

and ρP , ϕ (ϕ , Q̅ ) = (12)

∫0

∞

∞

(20)

are the angular densities of R and P, and Δρϕ (ϕ , Q̅ ) =

(13)

∫0

∞

∞

r dr

∫−∞ dzΔρ3d (r , ϕ , z , Q̅ )

= ρP , ϕ (ϕ , Q̅ ) − ρR , ϕ (ϕ , Q̅ )

(21)

is the difference between the angular densities of P and R. Equation 18 expresses the angular electronic flux as sum of two terms. The first term jϕ (ϕ0 , Q̅ , t ) is called the ”boundary value” of the angular electronic flux at a specific angle ϕ0. Its determination calls for two modifications of the theory of electronic fluxes during coherent tunneling, which are necessary for transfer to AACM; see items (ii) and (iii) of the Introduction. Item (ii) points to an advantage of coherent tunneling which cannot be transferred: it proceeds in the electronic ground state. This allows employment of a special symmetry selection rule for the directions of the electronic fluxes which has been derived in ref 24. Accordingly, if the molecular system has a symmetry plane that coincides with the half plane at ϕ0, then the angular electronic flux at ϕ0 is equal to zero, jϕ (ϕ0 , Q̅ , t ) = 0. For coherent tunneling, the first term in eq 18 thus vanishes at the system’s mirror plane. Unfortunately, the derivation of this simple boundary value cannot be transferred from coherent tunneling to AACM, because the representative wave function eq 7′ is not in the electronic ground state. Now, let us employ item (iii) for an alternative determination of the boundary value jϕ (ϕ0 , Q̅ , t ). Accordingly, it makes use of the preparation of the initial state eq 2′here this is the linearly x-polarized laser pulse with electric field F⃗(t) = (F(t),0,0), adapted from ref 2. Specifically, the laser pulse consists of three subpulses for the series of transitions S0 → S0 + S3 → S0 + S5 → S0 + S1. All subpulses are x-polarized, and all transitions are symmetry-allowed. The “detour” from S0 via two excited states to S0 + S1 allows the preparation of the target state, eq 2′, whereas direct access (S0 → S0 + S1) is symmetry

(14)

(15)

∞

dr

∫ϕ

(18)

to the angular electronic flux jϕ (ϕ , Q̅ , t ) =

dϕ′[ρP , ϕ (ϕ′, Q̅ )

× sin(2πt /τ ̅ )

∞

r dr

ϕ

0

Equation 14 relates the angular density ρϕ (ϕ , Q̅ , t ) =

∫ϕ

− ρR , ϕ (ϕ′, Q̅ )]·(π /τ ̅ ) sin(2πt /τ ̅ )

⃗ is the 3d where ρ3d is the one-electron density in 3d space, j3d one-electron flux, and r ⃗ denotes the electron position. In principle, ρ3d is normalized in 3d space, but here we adapt the ubiquitious renormalization of ρ3d such that ∫ ρ3d ( r ⃗ , Q̅ , t )d r ⃗ = Ne. The flux j3d⃗ is renormalized accordingly, such that the 3d continuity, eq 13, guarantees conservation of the number of electrons. The time evolution of ρ3d ( r ⃗ , Q̅ , t ) is illustrated by snapshots in Figure 1b. Note that knowledge of ρ3d ( r ⃗ , Q̅ , t ) does not suffice to determine j3d⃗ ( r ⃗ , Q̅ , t ). The continuity eq 13 may be reduced further from 3d to 1d, by integration over two coordinates. Here, we integrate over half planes (0 ≤ r< ∞, −∞ < z< +∞) at fixed angles ϕ, in order to obtain the 1d angular continuity equation ρϕ̇ (ϕ , Q̅ , t ) +

(17)

0

Integration of eq 11 over the coordinates of all electrons but one yields the familiar reduced continuity equation ρ̇3d ( r ⃗ , Q̅ , t ) + ∇r ⃗ j3d⃗ ( r ⃗ , Q̅ , t ) = 0

dϕ′ρϕ̇ (ϕ′, Q̅ , t )

Inserting eq 8′ into eq 17, we obtain

iℏ [ψ *(q , Q̅ , t )∇ri ψ (q , Q̅ , t ) 2me

− ψ (q , Q̅ , t )∇ri ψ *(q , Q̅ , t )]

ϕ

0

where ρ̇(q , Q̅ , t ) is the time derivative of the density of all electrons, for the present scenario of fixed nuclei (Q̅ ). Moreover, div = (∇1, ∇2, ..., ∇42) is the divergence of all ⃗ ) comprises the flux density of all electrons, and j ⃗ = (j1⃗ , j2⃗ , ..., j42 electrons, ji ⃗ (q , Q̅ , t ) = −

∫ϕ

jϕ (ϕ , Q̅ , t ) = jϕ (ϕ0 , Q̅ , t ) −

(16)

The angular density, eq 15, is normalized to the number Ne of electrons, and the angular electronic flux, eq 16, is normalized accordingly, such that the angular continuity, eq 14, guarantees conservation of the number Ne of electrons. Since eq 16 integrates the 3d flux density over the half plane at angle ϕ, D


Article

The Journal of Physical Chemistry A forbidden. The corresponding electronic Hamiltonian is, in the semiclassical dipole approximation H(t ) = Hel + F(t )e(x1 + x 2 + ... + x42)

ρR , ϕ (ϕ , Q̅ ) = ρR , ϕ (ϕ + 2π /3, Q̅ ) = ρR , ϕ (ϕ + 4π /3, Q̅ ) = ρR , ϕ (− ϕ , Q̅ ) = ρR , ϕ (− ϕ + 2π /3, Q̅ )

(22)

= ρR , ϕ (− ϕ + 4π /3, Q̅ )

The laser-driven wave function is obtained as solution of the time-dependent Schrödinger equation d iℏ ψ (q , Q̅ , t ) = H(t )ψ (q , Q̅ , t ) dt

ρP , ϕ (ϕ , Q̅ ) = ρR , ϕ ( −ϕ + π /3, Q̅ ) = ρR , ϕ (ϕ + π , Q̅ ) (31)

(23)

for the angular electron densities of R and P, and

starting from the ground state, ψ (q , Q̅ , t = 0) = φ0(q , Q̅ ). During the laser pulse, the symmetry D6h of the isolated benzene molecule and its electronic ground state φ0(q , Q̅ ) are reduced to C2v. All symmetry operations {E,C2x,σxy,σxz} of C2v commute with the time-dependent Hamiltonian, eq 22. As a consequence, the wave function eq 7′, prepared with an xpolarized laser, is an eigenfunction of all those symmetry operations, in particular σxzψ (q , Q̅ , t ) = ±ψ (q , Q̅ , t )

jϕ (ϕ , Q̅ , t ) = jϕ (ϕ + 2π /3, Q̅ , t ) = jϕ (ϕ + 4π /3, Q̅ , t ) = −jϕ ( −ϕ , Q̅ , t ) = −jϕ ( −ϕ + 2π /3, Q̅ , t ) = −jϕ ( −ϕ + 4π /3, Q̅ , t )

(24)

σxzjy (q , Q̅ , t ) = −jy (q , Q̅ , t )

jϕ (ϕ0 = π /2, Q̅ , t ) = 0

Hence, the y-component of the electronic flux density vanishes on the x−z mirror plane. This property is robust with respect to the subsequent reductions to the one-electron flux density, and subsequently to the angular electronic flux. The x−z plane coincides with the half plane at ϕ0 = 0. Hence, we obtain the boundary value of the angular electronic flux density

ϕ

jϕ (ϕ , Q̅ , t ) = −

implying symmetry relations, eq 28′ and 30′−32′, analogous to eqs 28 and 30−32. The relation eq 27 allows determination of the angular electronic flux without explicit calculations of the angular densities of R and P. The angular electronic flux is thus obtained by steps Sa−Sd: Sa: Evaluate the density difference Δρϕ (ϕ , Q̅ ), eq 21.

(26)

Sb: Evaluate the yield yϕ (ϕ , Q̅ ), eq 29.

ϕ

Sc: Evaluate the energy gap ΔE ̅ = E1̅ − E0̅ and the period τ ̅ of charge migration, eq 9′. Sd: Combine the results from steps Sb and Sc to evaluate the angular electronic flux jϕ (ϕ , Q̅ , t ), eq 27. For the first step (Sa), the electronic Schrödinger equation, eq 10, is solved for the ground and first excited singlet states of benzene, at the state-averaged CASSCF(6,12) level of quantum chemistry, as implemented in MOLPRO.25 The wave functions are computed using a cc-pVTZ basis,26 and the excitation energy is extrapolated at the complete basis set limit.27 The calculation starts with solving the Roothaan equations (a representation of the restricted Hartree−Fock RHF equations) for the canonical molecular spin orbitals (MOs) χk (r , Q̅ ) or χk ̅ (r , Q̅ ), where the quantum numbers k and k ̅ refer to α-spins (spin-up) or β-spins (spin-down), respectively. The Slater determinant for the ground state in the RHF is written as Ψ0(q , Q̅ ); Slater determinants representing single excitations from spin orbitals χa to χr, or double excitations from χa,χb to rs χr,χs, are written as Ψ ra(q , Q̅ ) and Ψ ab (q , Q̅ ), with analogous notations for spin orbitals with α- and β-spins, respectively. The wave functions φ0(q , Q̅ ) and φ1(q , Q̅ ) in eqs 2′ and 3′ are then approximated as

dϕ′Δρϕ (ϕ′, Q̅ ) ·(π /τ ̅ ) sin(2πt /τ ̅ ) (27)

Apparently, this is the product of angular times temporal factors. The temporal factor is sinusoidal, with period τ ̅ . This implies the temporal symmetries ⎛ ⎞ ⎛ ⎞ τ τ jϕ (ϕ , Q̅ , t ) = jϕ ⎜ϕ , Q̅ , ̅ − t ⎟ = −jϕ ⎜ϕ , Q̅ , ̅ + t ⎟ ⎝ ⎠ ⎝ ⎠ 2 2 = −jϕ (ϕ , Q̅ , τ ̅ − t )

(28)

It is illuminating to calculate the number of electrons which flow through the half plane at ϕ during the first half of the period, i.e., during charge migration from R to P, also called the electron yield (y) at angle ϕ, yϕ (ϕ , Q̅ ) =

∫0

τ ̅ /2

dtjϕ (ϕ , Q̅ , t ) = −

∫0

∫π /2 dϕ′Δρϕ (ϕ′, Q̅ )·(π /τ ̅) sin(2πt /τ ̅) (27′)

for preparation by an x-polarized laser pulse. By chance, this is the same boundary value as for coherent tunneling, but the derivations are entirely different. The angular electronic flux, eq 18, thus reduces to the second term

∫0

(26′)

for preparation by a y-polarized laser pulse. The angular electronic flux, eq 18, thus reduces to

(25)

jϕ (ϕ0 = 0, Q̅ , t ) = 0

(32)

for the angular electronic flux. We note in passing that analogous results can be derived for preparations of the initial state by a linearly y-polarized laser pulse. As a consequence, the electronic flux vanishes on the y−z mirror plane, such that eqs 26 and 27 are replaced by

As the next consequence, the y-component of the flux density, eq 12, is antisymmetric with respect to reflection at the x−z mirror plane,

jϕ (ϕ , Q̅ , t ) = −

(30)

ϕ

dϕ′Δρϕ (ϕ′, Q̅ ) (29)

The angular factor of the angular electronic flux eq 27 is thus the electron yield. After the laser pulse (i.e., for t ≥ 0), the benzene molecule has D3d symmetry.2 This implies the symmetries E


Article


Figure 2. Selected canonical molecular orbitals (MOs) χ20 (a), χ21 (b) of the dominant Slater determinant representing the electronic ground state (S0) of benzene, and the corresponding MOs χ22 (c), χ23 (d), which replace χ20, χ21 in the dominant Slater determinants representing the first electronic excited singlet state (S1). The MOs are illustrated by color-coded contour plots in the plane at z = 1.5a0 parallel to the x−y plane (in units of 1/a03/2).

φi(q , Q̅ ) = D0(i)Ψ0(q , Q̅ ) +

has been prepared in the ”reactant” state (eq 2′), by means of a linearly x-polarized laser pulse. Step (Sa): For calculating the difference of the one-electron densities, Δρ(q , Q̅ ) = ρP (q , Q̅ ) − ρR (q , Q̅ ), we use the state-

∑ Dar(i)Ψra(q , Q̅ ) ar

+

∑ Dabrs(i)Ψrsab(q , Q̅ ) + ... abrs

(33)

averaged CASSCF(6,12) method with a cc-pVTZ basis set.26 Only the A1g ground state and the first B1u excited state are included in the self-consistent averaging procedure aimed at improving the orbitals. The active space is adapted from ref 29 and is chosen such that two sets of π molecular orbitals are included; that is, it consists of the MOs (1a2u, 1e1g, 1e2u, 1b1g) and (2a2u, 2e1g, 2e2u, 2b1g). Since MOLPRO can only handle Abelian point groups, the calculations are carried out in the subgroup D2h of D6h, yielding the following assignment for these 12 MOs: (1b1u, 1b3g, 1b2g, 1au, 2b1u, 2b3g) and (3b1u, 2b3g, 3b2g, 2au, 4b1u, 4b3g). Accordingly, 36 inner-valence electrons are kept frozen in 18 MOs, labeled χ1, ..., χ18, and the active space has 12 MOs, which we dub as χ19, χ20, ..., χ30, for the remaining six electrons. Most important for the subsequent results are two pairs of degenerate MOs 1e1g, i.e., χ20 and χ21, as well as 1e2u, i.e., χ22 and χ23, illustrated in Figures 2a and 2b as well as 2c and 2d, respectively. The resulting wave functions φ0(q , Q̅ ) and φ1(q , Q̅ ) are dominated by a single determinant

where the sums include spin orbitals with α- as well as β-spins. The MOs included in the active space of the CASSCF(6,12) calculation are reoptimized within this procedure in order to improve the description of the desired electronic states. All post-processing, e.g., the grid representation of the electronic density and the integration thereof, is performed with our open-source python toolbox ORBKIT.28 The density difference, eq 8′, is then rewritten as Δρ(q , Q̅ ) = ρP (q , Q̅ ) − ρR (q , Q̅ ) 1 {[− φ0(q , Q̅ ) + φ1(q , Q̅ )]2 2 − [φ0(q , Q̅ ) + φ1(q , Q̅ )]2 } =

= − 2φ0(q , Q̅ )· φ1(q , Q̅ ) (34)

with φ0(q , Q̅ ) and φ1(q , Q̅ ) expressed in terms of the expansions in eq 33. Integration of eq 34 over the coordinates of all electrons but one (see the steps from eq 11 to eq 13) yields the corresponding 3d one-electron density difference Δρ3d (r , Q̅ ) in terms of those MOs of φ0(q , Q̅ ) which are replaced in φ1(q , Q̅ ); see the example in section 3. This compact reduction of the (3Ne = 3 × 42)-dimensional expression, eq 34, to 3d profits from the orthonormality of the canonical MOs χk (q , Q̅ ), somewhat analogous to the derivations in ref 1. The resulting 3d density difference Δρ3d (r , Q̅ ) is then reduced to the 1d angular density difference Δρϕ (ϕ , Q̅ ), using eq 21. The integration, eq 21, is carried out on fine numerical grids, ri = i·Δr, i = 0, ..., 600, Δr = 0.01a0 and zk = k·Δz, k = 0, ±1, ±2, ..., ±600, Δz = 0.01a0, exploiting the symmetry of the densities with respect to the x−y mirror plane. The angular grid for the integration, eq 29, is ϕl = l·Δϕ,l = 1, ..., 360, Δϕ = 2π/360, again exploiting symmetry. After this demanding step (Sa), it is straightforward to carry out steps Sb−Sd.

for the ground state, φ0(q , Q̅ ) = D0(0)Ψ0(q , Q̅ ) + ....

(35)

with coefficient D(0) 0 ≈ 0.94, and four Slater determinants (or two configuration state functions) for the target excited state 22(1) 22 23(1) 23 φ1(q , Q̅ ) = D20 Ψ20(q , Q̅ ) + D21 Ψ21(q , Q̅ ) 23(1) 23 22(1) 22 + D20 Ψ 20(q , Q̅ ) + D21 Ψ 21(q , Q̅ ) + ...

(36)

The coefficients of the four configurations have the same absolute values, i.e., 22(1) 23(1) 22(1) 23(1) D20 = −D20 =D21 = −D21 ≈ 0.45. Subsequently,

we keep these one and four Slater determinants with the dominant coefficients in expansions eq 35 and eq 36, respectively, while neglecting all others, each of which contributes to less than a percent of the total wave function. Accordingly, the density difference, eq 8′, is rewritten as

3. RESULTS AND DISCUSSION The angular electronic flux during AACM in excited benzene is calculated, using the model, theory, and methods of section 2. Below we present the successive steps Sa−Sd. The calculations are for the scenario where initially (t = 0) the benzene molecule F


Article


wave functions. It is important to recognize that any correction to the difference density would appear as an additive term of the same form as the last line in eq 37. The next largest corrections to the difference density come from the products of r,s(0) r(1) coefficients (Da,b Da ). These corrections are at least 1 order of magnitude smaller than the dominant ones. In the unlikely event that all these terms are in phase, the corrections could add up significantly. Nonetheless, this would lead to only a marginal change of the difference density and, therefore, of the one-electron fluxes, in the present case. The simplifications proposed by retaining the dominant determinants have the further important advantage of offering a transparent formalism that allows an illuminating interpretation of the one-electron fluxes in terms of molecular orbitals. Step (Sb): Integration of the angular density difference Δρϕ (ϕ , Q̅ ) yields the angular electron yield yϕ (ϕ , Q̅ ); cf. eq

Δρ(q , Q̅ ) = ρP (q , Q̅ ) − ρR (q , Q̅ ) ≈ − 2D0(0)Ψ0(q , Q̅ )

22(1) 22 Ψ20(q , Q̅ ) [D20

23(1) 23 23(1) 23 22(1) 22 + D21 Ψ21(q , Q̅ ) + D20 Ψ 20(q , Q̅ ) + D21 Ψ 21(q , Q̅ )] 22(1) ≈ − 2D0(0)D20 Ψ0(q , Q̅ )

[Ψ22 20(q , Q̅ )

22 23 + Ψ23 21(q , Q̅ ) − Ψ 20(q , Q̅ ) − Ψ 21(q , Q̅ )]

(37)

Integration over all electrons but one yields the difference of the 3d one-electron densities of P minus R, Δρ3d ( r ⃗ , Q̅ ) = ρ3d, P ( r ⃗ , Q̅ ) − ρ3d, r ( r ⃗ , Q̅ ) 22 [χ20 ( r ⃗ , Q̅ )χ22 ( r ⃗ , Q̅ ) + χ21 ( r ⃗ , Q̅ ) ≈ − 4D0(0)D20 χ23 ( r ⃗ , Q̅ )]

(38)

29. The result is shown in Figure 3b. Apparently, the yields have maxima and minima at the angles Φ1, Φ3, Φ5 and Φ2, Φ4, Φ6 of the carbon nuclei C1, C3, C5 and C2, C4, C6, respectively. These maxima and minima correspond to extreme values of the angular electronic fluxes at the time t = τ ̅ /4 , halfway from R to P. They are labeled by arrows in forward and backward directions in Figure 3b. These arrows correspond to the curved arrows in anticlockwise and clockwise directions, respectively, shown in Figure 1a for the cartoon at t = τ ̅ /4 . The present result (Figure 3b) thus confirms the working hypothesis of ref 2 unequivocally, that is, a multidirectional, pincer-motion type pattern of the angular electronic flux. The absolute values of the maximum and minimum yields, i.e.,

As anticipated in section 2, Δρ3d ( r ⃗ , Q̅ ) is thus expressed in terms of the MOs of φ0(q , Q̅ ), which are replaced in φ1(q,Q), focusing on the dominant Slater determinants in the expansions in eqs 35 and 36. Finally, the corresponding difference of the angular densities of P and R, Δρϕ (ϕ , Q̅ ), is obtained by integrating Δρ3d ( r ⃗ , Q̅ ) over the half planes at ϕ (∫ r dr and ∫ dz; cf. eq 21). The result is illustrated in Figure 3a. Note that the selection of only the small number of dominant determinants is a good approximation to using the full CAS

max|yϕ (ϕ , Q̅ )| = |yϕ (ϕm , Q̅ )| = yϕ (Φ1, Q̅ ) = 0.08 ± 0.02 (39)

correspond to the maximum number of electrons which flow from one bond to the neighboring bond (illustrated as double and single bonds in Figure 1a), at t = τ ̅ /4 , i.e., halfway between R and P. The value 0.08 of the maximum number of flowing electrons is also written in Figure 1a. The uncertainty ±0.02 (i.e., ca. 20%) in eq 39 is a conservative estimate which takes into account that the dominant Slater determinants in eqs 35 and 36 represent more than 80% of the electronic ground and first excited states, respectively. This uncertainty does not affect any of the subsequent conclusions. The maximum number of electrons which flow concertedly through six equivalent half planes at the angles Φ1, ..., Φ6 of the carbon nuclei C1, ..., C6, in alternating angular directions, is 6 × 0.08 = 0.48; cf. Figure 1a. Step (Sc): Solution of the electronic Schrödinger equation, eq 10, at the present level of quantum chemistry yields the S0 → S1 excitation energy E1̅ − E0̅ = 4.88 eV, in satisfactory agreement with the experimental result (4.8−5.1) eV.30 For the calculation of the energy gap, the results are extrapolated to the complete basis set limit. The corresponding period of AACM is τ ̅ = 848 as; cf. eq 9′. Step (Sd): The angular electronic flux jϕ (ϕ , Q̅ , t ) is

Figure 3. (a) Difference Δρϕ = ρP,ϕ − ρR,ϕ of the angular densities of the product (P) and the reactant (R). The notation is as in Figure 1. (b) Angular yield yϕ of electrons which flow from R to P through half planes at angle ϕ during the time 0 ≤ t ≤ τ ̅ /2 . The angular yield yϕ is proportional to the angular electronic flux from R to P. The six vertical dashed black lines mark the angular positions Φ1, Φ2, ..., Φ6 of carbon nuclei C1, C2, ..., C6. Maximum positive and minimum negative fluxes at Φ1, Φ3, Φ5 and at Φ2, Φ4, Φ6 are indicated by horizontal arrows → and ←, respectively. These arrows correspond to the anticlockwise and clockwise curved arrows of the pincer-motion type pattern of fluxes shown in Figure 1a (t = τ ̅ /4 ), respectively. Angles of zero flux are highlighted by solid black lines.

obtained by combining the results of steps Sb and Sc, by means of eq 27. The result is shown in Figure 4. The temporal and angular symmetries of the angular electronic flux, eqs 28 and 32, are obvious from Figure 4. G


Article


induce significant variations of the flux patterns.36 The present theory may also be tailored to applications to nonring-shaped, e.g. linear, molecules such as the example of ref 3, the iodoacetylene cation. The present theory is fundamental, allowing various extensions. For example, the present determination of the boundary value of zero angular electronic flux at the x−z plane of the aligned benzene profits from information about the preparation of the initial state, eq 2, by means of a linearly xpolarized laser pulse.3 In principle, the same initial state, eq 2, may also be prepared by different methods, e.g. by optimal control, which results in a non-x-polarized laser pulse, as documented in ref 2. In this scenario, one may determine the symmetry of the initial state (here: D3d; see ref 2), and employ not only one specific boundary value, such as in eq 26, but also alternative ones. For example, for D3d symmetry, the angular electronic flux vanishes on all vertical or dihedral mirror planes of the one-electron density of the initial state. This is documented in the set of lines of zero angular electronic flux, shown in Figure 4. In practice, this allows some rewarding flexibility for the preparation of the initial state with different (not only linear x-) polarizations of the laser pulse in those mirror planes, or equivalently, with corresponding molecular alignments. The present theory may also be extended to determine not only the angular electronic flux, but even the underlying electronic 3d flux densities. The key to this demanding property is inherent in eq 38, which relates the difference Δρ3d (r , Q̅ ) = ρ3d, P (r , Q̅ ) − ρ3d, R (r , Q̅ ) of the one-electron densities of P and R to a selective, rather small set of molecular orbitals χk (r , Q̅ ) which are involved in the excitation from the ground state φ0 to the superposition state, eq 2. As a working hypothesis, these molecular orbitals should also play a pivotal role in direct evaluations of the flux densities, starting from expressions, eq 12, with subsequent integrations over all electrons but one, similar to the step from eq 11 to eq 13. This approach to the flux density during AACM may reveal a powerful alternative to previous approaches to electronic flux densities during adiabatic molecular processes, e.g. during vibrations or dissociations; see, e.g., refs 37−41. Work along these lines is in progress. The ultimate challenge is, of course, to monitor electronic fluxes during AACM experimentally. This possibility is supported by two recent developments. First, the example of ref 42 shows that it is now possible to measure intramolecular nuclear fluxes during adiabatic processes, e.g. coherent vibrations, with spatiotemporal resolutions down to 5 pm and 200 as. The method of ref 42 consists of two steps: Step 1 derives the time dependent nuclear density from experimental pump−probe spectra: this step had already been achieved and was adapted from refs 43 and 44 with applications to vibrating Na2 and D2+, respectively. Step 2 employs the continuity equation in order to convert the time dependent nuclear densities to nuclear fluxes. This step invokes the proper boundary value for the nuclear fluxes, which is obvious for linear molecules; that is, nuclear fluxes vanish at very large (“infinite”) bond length.42 Analogous steps should enable the experimental measurement of electronic fluxes during AACM. Consider first charge migration in a linear molecule. Here, the recent breakthrough in ref 3 provides the second encouragement; that is, for step 1, we anticipate that the rapid development of strategies and techniques based on high

Figure 4. Time evolution of the angular electronic flux jϕ (in units of 10−4 as−1) illustrated by a color-coded contour plot for the angular and temporal domains 0 ≤ ϕ ≤ 2π, 0 ≤ t ≤ τ ̅ = 848 as. The six vertical dashed black lines mark the angular positions Φ1, Φ2, ..., Φ6 of carbon nuclei C1, C2, ..., C6. Angles of zero flux are highlighted by solid black lines.

4. CONCLUSION The present development of the theory for electronic fluxes during AACM and the first application, exemplarily to oriented benzene, starting from the initial superposition state eq 2, has several general and specific consequences. Two specific results include the first unequivocal determination of the angular electronic flux patterns, confirming the previous working hypothesis of ref 2 and the first quantification of the angular electronic flux, as summarized in Figure 1a. Accordingly, the maximum number of electrons which flow concertedly is equal to 6 × 0.08 = 0.48, cf. Figure 1a. This number is somewhat surprising, calling for an explanation. In fact, it is smaller than the number of six electrons which should flow according to the scheme of double bond shifting which is suggested by the Lewis-type structures of R to P, as illustrated symbolically in Figure 1a. The reason for the apparent ”decrease” of the number of flowing electrons is that Lewis-type structures are idealized, limiting cartoons of the electron densities which suggest that electrons are localized in double or single bonds such that the number of electrons in double bonds exceeds those in single bonds by two. In reality, however, the electrons are delocalized such that part of the electrons which are assigned to double bonds in Lewis-type structures actually penetrate into the neighboring domains of single bonds. As a consequence, a significant fraction of electrons do not need to flow from double to single bondsthey are already there in the target domain(s), without flowing. The number of electrons which flow concertedly from R to P is, therefore, significantly smaller than the number suggested by Lewis structures. The effect of delocalization of electrons, resulting in rather small numbers of flowing electrons, has already been noted for electronic fluxes during molecular reactions or vibrations in the electronic ground state;18,31−34 see also ref 35. A general consequence of the present development of the theory for electronic fluxes during AACM in ring-shaped molecules is that it opens myriads of applications. These include not only oriented benzene prepared in the initial state, eq 2, but also in an arbitrarily large variety of different superpositions of two electronic states φa and φb, and also in an arbitrarily large set of other ring-shaped molecules prepared in analogous initial states (see, e.g., ref 1). It is easy to predict that different initial superpositions of eigenstates induce different flux patterns, just as for coherent tunneling, where, e.g., preparations in more and more excited tunneling doublets H


Article


(9) Okuyama, M.; Takatsuka, K. Dynamical Electron Mechanism of Double Proton Transfer in Formic Acid Dimer. Bull. Chem. Soc. Jpn. 2012, 85, 217−227. (10) Yamamoto, K.; Takatsuka, K. An Electron Dynamics Mechanism of Charge Separation in the Initial-Stage Dynamics of Photoinduced Water Splitting in X-Mn-Water (X = OH, OCaH) and Electron-Proton Acceptors. ChemPhysChem 2015, 16, 2534−2537. (11) Takatsuka, K.; Yonehara, T.; Hanasaki, K.; Arasaki, Y. Chemical Theory beyond the Born-Oppenheimer Paradigm: Nonadiabatic Electronic and Nuclear Dynamics in Chemical Reactions; World Scientific: 2015. (12) Patchkovskii, S. Electronic Currents and Born-Oppenheimer Molecular Dynamics. J. Chem. Phys. 2012, 137, 084109. (13) Gomez, T.; Hermann, G.; Zarate, X.; Pérez-Torres, J. F.; Tremblay, J. C. Imaging the Ultrafast Photoelectron Transfer Process in Alizarin-TiO2. Molecules 2015, 20, 13830−13853. (14) Bredtmann, T.; Diestler, D. J.; Li, S.-D.; Manz, J.; Pérez-Torres, J. F.; Tian, W.-J.; Wu, Y.-B.; Yang, Y.; Zhai, H.-J. Quantum Theory of Concerted Electronic and Nuclear Fluxes Associated with Adiabatic Intramolecular Processes. Phys. Chem. Chem. Phys. 2015, 17, 29421− 29464. (15) Hund, F. Zur Deutung der Molekelspektren. III. Bemerkungen über das Schwingungs−und Rotationspektrum bei Molekeln mit mehr als zwei Kernen. (Remarks on the Vibrational and Rotational Spectra of Molecules with More Than Two Nuclei). Z. Phys. 1927, 43, 805− 826. (16) Bredtmann, T.; Manz, J. Electronic Bond-to-Bond Fluxes in Pericyclic Reactions: Synchronous or Asynchronous? Angew. Chem., Int. Ed. 2011, 50, 12652−12654. (17) Bredtmann, T.; Zhao, J.-M.; Manz, J. Concerted Electronic and Nuclear Fluxes During Coherent Tunnelling in Asymmetric Double Well Potentials. J. Phys. Chem. A 2016, DOI: 10.1021/acs.jpca.5b11295. (18) Barth, I.; Hege, H.-C.; Ikeda, H.; Kenfack, A.; Koppitz, M.; Manz, J.; Marquardt, F.; Paramonov, G. K. Concerted Quantum Effects of Electronic and Nuclear Fluxes in Molecules. Chem. Phys. Lett. 2009, 481, 118−123. (19) Filsinger, F.; Küpper, J.; Meijer, G.; Holmegaard, L.; Nielsen, J. H.; Nevo, I.; Hansen, J. L.; Stapelfeldt, H. Quantum-State Selection, Alignment, and Orientation of Large Molecules Using Static Electric and Laser Fields. J. Chem. Phys. 2009, 131, 064309. (20) Fleischer, S.; Zhou, Y.; Field, R. W.; Nelson, K. A. Molecular Orientation and Alignment by Intense Single-Cycle THz Pulses. Phys. Rev. Lett. 2011, 107, 163603. (21) Spanner, M.; Patchkovskii, S.; Frumker, E.; Corkum, P. Mechanisms of Two-Color Laser-Induced Field-Free Molecular Orientation. Phys. Rev. Lett. 2012, 109, 113001. (22) Ulusoy, I. S.; Nest, M. Remarks on the Validity of the Fixed Nuclei Approximation in Quantum Electron Dynamics. J. Phys. Chem. A 2012, 116, 11107−11110. (23) Schrödinger, E. Quantisierung als Eigenwertproblem. Ann. Phys. (Leipzig, Ger.) 1926, 81, 109−139. (24) Manz, J.; Yamamoto, K. A Selection Rule for the Directions of Electronic Fluxes During Unimolecular Pericyclic Reactions in the Electronic Ground State. Mol. Phys. 2012, 110, 517−530. (25) 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; see http://www.molpro.net (accessed Feb 18, 2016). (26) 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. (27) Jensen, F. Introduction to Computational Chemistry; John Wiley & Sons: 2007. (28) Hermann, G.; Pohl, V.; Tremblay, J. C.; Paulus, B.; Hege, H.-C.; Schild, A. ORBKIT: A Modular Python Toolbox for Cross-Platform Postprocessing of Quantum Chemical Wavefunction Data. J. Comput. Chem. 2016, DOI: 10.1002/jcc.24358.

harmonic generation (HHG) will provide the experimental time evolution of the 3d one-electron density, again with sub10-pm and attosecond spatiotemporal resolution. The 3d result may then be reduced to 1d components, e.g. for the electronic density that travels from one end of the molecule to the other. For step 2, the 1d electron density should then be converted into the corresponding 1d electronic flux, again by means of the continuity equation. As in ref 42, this second step calls for the proper boundary value, but for linear molecules, this is again obvious; that is, the electronic flux vanishes at large distances, far away from the molecule. For ring-shaped molecules, we suggest to apply the analogous two-step strategy from HHG spectra via angular electronic densities to angular electronic fluxes, with a proper boundary value. As we have shown in section 2, for ring-shaped molecules, the boundary value may be imposed experimentally by specific preparation of an initial state of the oriented molecule, by means of a laser pulse with proper (e.g., linear2 or circular1) polarization. In conclusion, the present theory with the strategies and methods should pave the way to the first experimental measurements of electronic fluxes during AACM.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Dr. Venkatraman Mohan (Honeywell, Morris Plains, NJ), Dr. Axel Schild (Halle), and Prof. Yonggang Yang (Taiyuan) for valuable hints during the preparation of the manuscript. Financial support by the Deutsche Forschungsgemeinschaft (projects Ma 515/27-1, Pe 2297/1-1, Tr 1109/2-1) and the Elsa-Neumann foundation of the Land Berlin is also gratefully acknowledged.

■

REFERENCES

(1) 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. (2) Ulusoy, I. S.; Nest, M. Correlated Electron Dynamics: How Aromaticity Can Be Controlled. J. Am. Chem. Soc. 2011, 133, 20230− 20236. (3) Kraus, P.; Mignolet, B.; Baykusheva, D.; Rupenyan, A.; Hornỳ, L.; Penka, E.; 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. (4) Weinkauf, R.; Schanen, P.; Metsala, A.; Schlag, E.; Bürgle, M.; Kessler, H. Highly Efficient Charge Transfer in Peptide Cations in the Gas Phase: Threshold Effects and Mechanism. J. Phys. Chem. 1996, 100, 18567−18585. (5) Remacle, F.; Levine, R. An Electronic Time Scale in Chemistry. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 6793−6798. (6) Okuyama, M.; Takatsuka, K. Electron Flux in Molecules Induced by Nuclear Motion. Chem. Phys. Lett. 2009, 476, 109−115. (7) Nagashima, K.; Takatsuka, K. Electron-Wavepacket Reaction Dynamics in Proton Transfer of Formamide. J. Phys. Chem. A 2009, 113, 15240−15249. (8) Takatsuka, K.; Yonehara, T. Exploring Dynamical Electron Theory Beyond the Born-Oppenheimer Framework: From Chemical Reactivity to Non-Adiabatically Coupled Electronic and Nuclear Wavepackets On-the-fly under Laser Field. Phys. Chem. Chem. Phys. 2011, 13, 4987−5016. I


Article

The Journal of Physical Chemistry A (29) Matos, J. M. O.; Roos, B. O.; Malmqvist, P.-Å. A CASSCF-CCI Study of the Valence and Lower Excited States of the Benzene Molecule. J. Chem. Phys. 1987, 86, 1458−1466. (30) Doering, J. Electronic Energy Levels of Benzene Below 7 eV. J. Chem. Phys. 1977, 67, 4065−4070. (31) Kenfack, A.; Barth, I.; Marquardt, F.; Paulus, B. Molecular Isotopic Effects on Coupled Electronic and Nuclear Fluxes. Phys. Rev. A: At., Mol., Opt. Phys. 2010, 82, 062502. (32) Hege, H.-C.; Manz, J.; Marquardt, F.; Paulus, B.; Schild, A. Electron Flux during Pericyclic Reactions in the Tunneling Limit: Quantum Simulation for Cyclooctatetraene. Chem. Phys. 2010, 376, 46−55. (33) Andrae, D.; Barth, I.; Bredtmann, T.; Hege, H.-C.; Manz, J.; Marquardt, F.; Paulus, B. Electronic Quantum Fluxes during Pericyclic Reactions Exemplified for the Cope Rearrangement of Semibullvalene. J. Phys. Chem. B 2011, 115, 5476−5483. (34) Bredtmann, T.; Hupf, E.; Paulus, B. Electronic Fluxes during Large Amplitude Vibrations of Single, Double and Triple Bonds. Phys. Chem. Chem. Phys. 2012, 14, 15494−15501. (35) Hermann, G.; Paulus, B.; Pérez-Torres, J.; Pohl, V. Electronic and Nuclear Flux Densities in the H2 Molecule. Phys. Rev. A: At., Mol., Opt. Phys. 2014, 89, 052504. (36) Liu, C.; Manz, J.; Yang, Y. Staircase Patterns of Nuclear Fluxes during Coherent Tunneling in Excited Doublets of Symmetric Double Well Potentials. Phys. Chem. Chem. Phys. 2016, 18, 5048−5055. (37) Diestler, D. J.; Kenfack, A.; Manz, J.; Paulus, B. CoupledChannels Quantum Theory of Electronic Flux Density in Electronically Adiabatic Processes: Application to the Hydrogen Molecule Ion. J. Phys. Chem. A 2012, 116, 2736−2742. (38) Diestler, D. J.; Kenfack, A.; Manz, J.; Paulus, B.; Pérez-Torres, J. F.; Pohl, V. Computation of the Electronic Flux Density in the BornOppenheimer Approximation. J. Phys. Chem. A 2013, 117, 8519−8527. (39) Manz, J.; Pérez-Torres, J. F.; Yang, Y. Vibrating H+2 (2Σ+g , JM = 00) Ion as a Pulsating Quantum Bubble in the Laboratory Frame. J. Phys. Chem. A 2014, 118, 8411−8425. (40) Pérez-Torres, J. F. Dissociating H+2 (2Σ+g , JM = 00) Ion as an Exploding Quantum Bubble. J. Phys. Chem. A 2015, 119, 2895−2901. (41) Pohl, V.; Tremblay, J. C. Adiabatic Electronic Flux Density: A Born-Oppenheimer Broken-Symmetry Ansatz. Phys. Rev. A: At., Mol., Opt. Phys. 2016, 93, 012504. (42) Manz, J.; Pérez-Torres, J. F.; Yang, Y. Nuclear Fluxes in Diatomic Molecules Deduced from Pump-Probe Spectra with Spatiotemporal Resolutions Down to 5 pm and 200 asec. Phys. Rev. Lett. 2013, 111, 153004. (43) Frohnmeyer, T.; Baumert, T. Femtosecond Pump-Probe Photoelectron Spectroscopy on Na2: A Tool to Study Basic Coherent Control Schemes. Appl. Phys. B: Lasers Opt. 2000, 71, 259−266. (44) Ergler, T.; Rudenko, A.; Feuerstein, B.; Zrost, K.; Schröter, C.; Moshammer, R.; Ullrich, J. Spatiotemporal Imaging of Ultrafast Molecular Motion: Collapse and Revival of the D2+ Nuclear Wave Packet. Phys. Rev. Lett. 2006, 97, 193001.

J


Multidirectional Angular Electronic Flux during Adiabatic Attosecond

Recommend Documents