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Quantum Design of π‑Electron Ring Currents in Polycyclic Aromatic Hydrocarbons: Parallel and Antiparallel Ring Currents in Naphthalene Hirobumi Mineo*,†,‡ and Yuichi Fujimura*,§,∥ †

Atomic Molecular and Optical Physics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City, Vietnam § Department of Applied Chemistry, Institute of Molecular Science, and Center for Interdisciplinary Molecular Science, National Chiao-Tung University, Hsinchu 30010 Taiwan ∥ Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan ‡

S Supporting Information *

ABSTRACT: Control of π-electrons in polycyclic aromatic hydrocarbons (PAHs) is one of the fundamental issues in optoelectronics for ultrafast optical switching devices. We have proposed an effective scenario for design of the generation of coherent ring currents in naphthalene (D2h), which is the smallest unit of planar PAHs. It has been demonstrated by using quantum chemical calculations and quantum optimal control (QOC) simulations that two types of ring currents, parallel and antiparallel, can be generated by resonance excitations by two linearly polarized lasers. A parallel (antiparallel) ring current means that the currents of two benzene rings run in the same (opposite) directions. The two types of ring currents may be experimentally identified by magnetic force microscopy. The QOC simulations indicate that a parallel ring current can be generated by using continuous wave and Gaussian pulse lasers with their time delay without relying on a sophisticated experimental apparatus. The present results provide a guiding principle of coherent πelectronics in PAHs for next-generation organic optical switching devices.

P

currents obtained by solving the time-dependent Schrödinger equation with the analytical electric fields exhibit almost the same magnitudes as those obtained by the QOC procedure. This indicates an effective experimental scenario for design of ring currents in PAHs without relying on a sophisticated QOC apparatus. Let us consider naphthalene belonging to the D2h point group, which is the simplest PAH with two fused aromatic (benzene) rings. The Z axis in the laboratory fixed space is placed along the principal rotational axis (see Figure 1). The X axis is parallel to the long axis of naphthalene. Because a coherent ring current can be generated on each aromatic ring, three types of coherent ring currents can in principle be classified, as shown in Figure 1. These are called (a) parallel ring currents because two ring currents on the two benzene rings run in the same direction, (b) antiparallel ring current because they run in opposite directions, and (c) localized ring current. In the localized ring current, the ring current localizes in one of the two benzene rings. The patterns of ring currents of naphthalene shown in Scheme 1 can be expressed as JL = JR for (a) parallel ring

olycyclic aromatic hydrocarbons (PAHs), which are twodimensional (2D), are typical organic materials. Attention has recently been paid to PAHs as organic molecular electronic materials, particularly for field-effect transistor devices.1−7 In addition, from the viewpoint of the ultrafast coherent electromagnetic response of π electrons in aromatic ring molecules,8−25 it is important to explore the possibility of such an optical response in PAHs. Control of electrons in atoms and molecules has been possible because of rapid advances in laser science and technology as well as quantum dynamical theory of electrons in molecules.26−35 The control of π-electrons in PAHs by lasers is one of the attractive research targets to create ring currents and current-induced magnetic fields. This is expected to serve as a fundamental research for next-generation organic optical switching devices. In this Letter, we propose a quantum design of generation of coherent ring currents in PAHs consisting of linearly fused benzene rings, which are called acenes or polyacenes. It has long been understood that PAHs of D2h symmetry in the point group cannot generate any ring current by optical excitations because there are no degenerate electronic excited states. We demonstrate that ring current can be generated by using a quantum optimal control (QOC) procedure.36−38 By analyzing the QOC results, we derived electric fields of analytical expressions for two linearly polarized lasers with a time delay. We show that the ring © 2017 American Chemical Society

Received: March 24, 2017 Accepted: April 20, 2017 Published: April 20, 2017 2019

DOI: 10.1021/acs.jpclett.7b00704 J. Phys. Chem. Lett. 2017, 8, 2019−2025

Letter

The Journal of Physical Chemistry Letters

belongs to the B1g representation of point group D2h, and an antiparallel current belongs to B2u. A parallel ring current (B1g) can be generated by a coherent excitation of two electronic excited states with B2u and B3u, while an antiparallel ring current (B2u), is generated by a coherent excitation of two electronic excited states with opposite inversion symmetries, u and g. Therefore, three excited states with B2u, B3u, and B1g are the minimum number of the π-electronic excited states for the generation of both parallel and antiparallel ring currents. On the basis of the above symmetry consideration, we adopt a three-electronic-excited-state model. Here, we mention that translations along the X and Y axes belong to B3u and B2u, respectively, while a rotation around the principal symmetry axis Z belongs to B1g. Let us now briefly describe an expression for a timedependent ring current induced by linearly polarized laser pulses.18,19 A ring current passing through surface S at time t is defined in terms of the expectation value of current density operator j(r) as19 J (t ) ≡ Figure 1. Energy levels of three lower π−π* excited states and directions of electronic transition dipole moments for generation of coherent ring currents of naphthalene. The black (red) arrow represents the transition dipole moments parallel to the X (Y) axis.

∫S d2r n·⟨Ψ(t )|j(r)|Ψ(t )⟩

where j(r) is defined as j(r) =

(1)

eℏ (∇⃗ 2me

− ∇⃖). Here, ∇⃗ (∇⃖ )

denotes the nabla operating the atomic orbital on the righthand (left-hand) side. In eq 1, Ψ(t), the wave function for the time-dependent Schrödinger equation, is expressed by a linear combination of the ground-state Φ0 and excited-state Φα configurations as Ψ(t) = C0(t )Φ0 + ∑α Cα(t )Φα . In our treatment, surface S was set at the center cutting the Ci−Cj bond. As an example, the ring current along benzene ring χ at time t, Jχ(t), is given by Jχ(t) = ∑α , β Jχ , αβ Im(Cα(t )Cβ*(t )), in which Jχ,αβ is the time-independent expectation value of the current operator over two excited-state configurations in benzene ring χ.19 The time-dependent coefficients of Ψ(t) can be determined by a quantum optimal control (QOC) procedure for π-electron dynamics induced by linearly polarized electric fields of lasers F(t).20 Naphthalene is assumed to be fixed in a space by an orientation-laser control39−41 or attached on a surface by linker bonds. For simplicity, we omit effects of nuclear motions in this Letter and adopt the frozen-nuclear approximation. The Hamiltonian of the π-electrons H(t) is given in the dipole approximation as

Scheme 1. Schematic Representation of Three Types of Coherent Ring Currents in Naphthalene (D2h)a

a

(a) Parallel ring current, which runs through the two aromatic rings in the same (clockwise or anticlockwise) direction; (b) antiparallel ring currents, which run through the two aromatic rings in opposite directions; and (c) localized ring current, which runs on the left- or right-hand side aromatic ring. Red arrows indicate the direction of ring current. JL (JR) denotes the ring current along the benzene ring fused in the left- (right-) hand side.

H (t ) = H 0 − μ · F (t )

(2)

where H0 is the π-electron Hamiltonian and μ is the electric dipole moment operator. Let us define a target operator Ô T at final time (control time) T. The optimal field F(t) is given as20 F(t ) = −

currents; JL = −JR for (b) antiparallel ring currents; and JL = 0, JR ≠ 0 or JL ≠ 0, JR = 0 for localized ring currents. In this Letter, we focus on parallel and antiparallel ring currents because these two ring currents are the fundamental units of ring currents, and the localized ones can be constructed by a coherent superposition of the two fundamental units of the ring currents.22 Localized ring currents in PAHs will be described in detail elsewhere. As a result of symmetry consideration, we can identify the πelectronic excited states that make a contribution to parallel and antiparallel ring currents. We note that a parallel ring current

1 Im⟨ξ(t )|μ|Ψ(t )⟩ α0

(3)

which is derived by maximizing the expectation value of target operator Ô T at t = T under the condition of minimum laser field fluence. α0 is a penalty factor chosen to weight the intensity of the laser field. The wave function Ψ(t) satisfies the timedependent Schrödinger equation iℏ

∂ Ψ(t ) = (H0 − μ·F(t )) Ψ(t ) ∂t

(4a)

with initial condition Ψ(0) = Φ0. 2020


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Figure 2. Quantum optimal control (QOC) simulations for a parallel coherent ring current in the clockwise direction JCC. (a) Temporal behaviors of the π-electron ring current Jχ(t) on aromatic ring χ = L, R. (b) Temporal behaviors in populations of the ground state (S0) and three excited states (S3, S4, and Sg). (c) X-component of optimized electric field F(t). (d) Y-component of time-dependent optimized electric field F(t). (e) Mechanism of the parallel ring current generation: simultaneous coherent resonant excitation by two lasers with delay time δ. The red (blue) arrow denotes the laser with Y- (X-) linearly polarized electric field of the laser.

In eq 3, ξ(t), which is a time-dependent Lagrange multiplier for the constraint eq 4a, is introduced to satisfy the timedependent Schrödinger equation with ξ(T) = Ô TΨ(T) at the final time as

S3, S4, and Sg (see the Supporting Information). Two excited states S3 (B3u) with excitation energies E3 = 5.81 and S4 (B2u) E4 = 6.04 correspond to the third excited state with its absorption peak at 5.86 eV and the fourth excited state with a peak at 6.09 eV in absorption spectra, respectively,43−46 and for the other state with g symmetry, we adopted Sg (B1g) with Eg = 6.13 [eV] from the TDDFT calculation results. Sg (B1g) is unidentified in the one-photon absorption spectrum. The electronic excited-state energies and the nonzero transition dipole moments between the ground and excited states and those between these excited states are shown in Figure 1. The calculated nonzero transition dipole moment vectors between the ground and excited states are μS0,S3 = (−3.45, 0, 0) and μS0,S4 = (0, −1.23, 0), and those between two excited states are μS3,Sg = (0, −0.66, 0) and μS4,Sg = (0.49, 0, 0) [a.u.]. A penalty factor α0 = 5.0[a.u.] is used in this QOC procedure. Generation of Parallel Ring Currents. The target operators for parallel ring current in the anticlockwise direction, i.e., JL > 0, JR > 0, and that in the clockwise direction, i.e., JL < 0, JR < 0, can be expressed in terms of ΦS3(B3u) and ΦS4(B2u) as Ô T = 1 |Φ + iΦS4⟩⟨ΦS3 + iΦS4 |(= 1 (|Φ S3 ⟩ + i|Φ S4 ⟩)(⟨Φ S3 | − 2 S3

∂ ξ(t ) = (H0 − μ·F(t ))ξ(t ) (4b) ∂t Here, target operator Ô T is given as Ô T = |ξ(T )⟩⟨ξ(T )| = |∑a Ca(T )Φa⟩⟨∑b Cb(T )Φb|. The target operator for each type of ring current is set to maximize the coherence between the relevant two excited states. To apply the QOC procedure to naphthalene, we determined three electronic excited states (B2u, B3u, and B1g) by using the ab initio molecular orbital (MO) and timedependent density functional theory (TDDFT) methods. The molecular geometry of naphthalene was optimized within the MP2/6-311+g(d,p) level theory by using Gaussian 09.42 The electronic states and transition dipole moments between the ground and excited states and those between two excited states were calculated within the TDDFT/6-311+g(d,p) level of theory under the Tamm−Dancoff approximation (TDA). To generate parallel and antiparallel ring currents, we chose three lower π−π* excited states from the TDDFT calculation results, iℏ

2

i⟨ΦS4|)) and 2021

1 |Φ 2 S3

− iΦS4 ⟩⟨ΦS3 − iΦS4 |, respectively. Here, DOI: 10.1021/acs.jpclett.7b00704 J. Phys. Chem. Lett. 2017, 8, 2019−2025

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Figure 3. QOC simulations of antiparallel ring current JAC. Control time T was set to T = 60 fs. (a) Time-dependent ring currents, JL, JR, and JB. (b) Time evolution of four electronic states. Panels c and d are the optimized electric fields of the UV laser and IR laser, respectively. Panels e and f are the Fourier transformed spectra FX(ω) and FY(ω), respectively. (g) Mechanism of the JAC antiparallel ring current formation: sequential UV and IR resonant excitations.

the π-electron ring current rotating anticlockwise is defined as positive. The ring current for the two relevant excited state configurations α and β were calculated using eqs 3 and 4a,b. The time-independent expectation values of the current operator for parallel ring currents Jχ,S3S4 were given as JL,S3S4 = JR,S3S4 = −113.0 [μA]. We remark that real π-electronic excited-state wave function |ΦS3⟩ belonging to B3u representation behaves as a real function X and |ΦS4⟩ belonging to B2u behaves as a real function Y, and complex superpositions of |ΦS3⟩ and |ΦS4⟩ provide the approximate eigenstates of the angular momentum ±ℏ, i.e., |±⟩ = 2−1/2(|ΦS3⟩ ∓ i|ΦS4⟩). Figure 2 shows the QOC results for generation of a parallel ring current in the clockwise direction, JL > 0, JR > 0. Here, T =

30 fs was set for the control time. Figure 2a shows that JR(t) = JL(t), and the bond current of bridging between two benzene rings vanishes. It should be noted that the ring currents have a sharp-rising time dependence with a peak at T, without any oscillatory behaviors. In general, the time dependence on the ring currents calculated by using the QOC procedure depends on T. For example, the QOC procedure for longer time T = 60 fs resulted in oscillatory behaviors in time-dependent ring currents (see the Supporting Information). Figure 2b shows the temporal behaviors in populations for four electronic states. The S3 population starts to rapidly increase at around 20 fs after the S4 population starts to increase, and both S3 and S4 states result in equal population at the final time T = 30 fs. Time2022


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c and d of Figure 3 show the time-dependent optimal electric fields of the ultraviolet (UV) and infrared (IR) lasers, respectively. Panels e and f of Figure 3 show the Fourier transformations of the time-dependent optimal electric fields in panels c and d. These figures indicate that the UV laser induces resonance transition from the ground state to S3, while the IR laser induces resonance transition from S3 to Sg. The excited state S3 plays two kinds of key roles for generation of antiparallel ring current: one is that the excited state is one of the two components in the coherent state, and the other is that it is the intermediate state to supply energy from S0 to Sg, which is the other component of the coherent state for generation of an antiparallel ring current. The resonant transition from S3 to Sg is induced by an IR laser. It should be noted from the selection rule in the dipole approximation that the excited state Sg (B1g) cannot be directly excited by any one-photon optical excitation from the ground state. It should also be noted that naphthalene (S0) is not excited by the IR laser because there are no C−C stretching vibrational modes with fundamental frequency of 2680 cm−1,47 which is the difference in frequency between the two excited states, S3 and Sg. The generation mechanism of the antiparallel ring current is a formation of electronic coherence between the two excited states having different parity by sequential resonant excitations of UV and IR lasers, as shown in Figure 3f. For the counterpart of the antiparallel ring current, JCA, the same treatment as that adopted in JAC can be used only by taking the complex conjugate of the target operator for JCA. We now briefly mention how we can identify the two ring currents, i.e., parallel and antiparallel ones. Figure 4 shows the calculated current-induced magnetic fields for a parallel ring current and an antiparallel ring current on the (X, 0, Z) surface. Here, |JL| = |JR| = 60 μA was adopted. The expressions for the current-induced magnetic fields are shown in the Supporting

dependent degree of coherence between S3 and S4, which is a measure of electronic coherence and defined as Im(CS3(t) CS4(t)*)/(|CS3(t)∥CS4(t)|), is shown in the Supporting Information. Panels c and d of Figure 2 show the electric fields, FX(t) and FY(t), of the optimal laser pulses, respectively. The Fourier transformed spectra of FX(t) and FY(t) indicate that the dominant optical processes are resonant transition from the ground state S0 to excited state S3 and that from the ground state S0 to excited state S4, respectively. The essential point for the generation of the parallel ring current without oscillating behaviors is that there is a time gap between two excitation processes induced by FX(t) and FY(t) lasers. Figure 2c,d clearly shows that S4 is first resonantly excited by the FY(t) laser with slowly increasing amplitude, and after 20 fs, S3 is excited by the second laser pulse with a duration of ∼10 fs. Figure 2c,d shows simple, sinusoidal electric fields. The generation mechanism of the parallel ring current is a simultaneous coherent resonant excitation with time delay δ, as shown in Figure 2e. This suggests that the time-dependent parallel ring current can be obtained by using analytical forms of the electric fields of two lasers without any optimal control. Let us now calculate the time evolution of a parallel ring current by solving the time-dependent Schrödinger equation with two electric fields with analytical forms FY(t) = FY sin(ωS4t) and FX(t) = FX sin(ωS3t − δ)f(t) with f(t) = exp(−(t (E − E )

− T)2/(cT)2) and with ωSα ≡ α ℏ 0 for α = 3 and 4. Here, δ, the relative phase between the two electric fields, is δ = 0.2π. The field amplitudes are FX = 2.05 GV/m and FY = 0.575 GV/ m with c = 0.2 [a.u.]. Determination of parameters in the analytical forms for the two electric fields of lasers is briefly described in the Supporting Information. In Figure 2a, the dotted line denotes the time-dependent ring current calculated in the analytical electric fields of the two lasers. It can be seen that the ring current exhibits almost the same time-dependent behaviors as that in the QOC method. This indicates the possibility of the generation of a parallel ring current in naphthalene by using analytical electric fields of two linearly polarized lasers. So far we have considered parallel ring current, JCC. For the counterpart of the parallel ring current, JAA, the same treatment as that adopted in JCC can be used only by changing relative phase of the two lasers by π. Generation of Antiparallel Ring Currents. From symmetry considerations, antiparallel ring currents can be generated by creation of a coherence between two electronic excited states, S3 (B3u) and Sg (B1g). The target operator for antiparallel ring current JL < 0, JR > 0 and that for JL > 0, JR < 0, can be 1 e x p r e s s e d a s Ô T = 2 |ΦS3 + iΦSg ⟩⟨ΦS3 + iΦSg | a n d 1 |Φ 2 S3

− iΦSg ⟩⟨ΦS3 − iΦSg |, respectively. The resultant expectation value of the current operator is given as JL,S3Sg = −JR,S3Sg = 67.1 [μA], and that for the bond current between two dividing benzene rings is given as JB,S3Sg = 107.0 [μA]. The bond current rotating in the Y direction is defined as positive. Figure 3 shows the QOC results of generation of antiparallel ring currents for naphthalene. T = 60 fs was set for the control time. Figure 3a shows that JR(t) = −JL(t) and shows the nonzero bond current of bridging between two benzene rings, JB(t). The maximum value of the antiparallel current is created at the target time after several peak values of the ring current were obtained. For shorter control times, for example, for T = 30 fs, no converged results were obtained. Figure 3b shows the time evolutions of the four electronic-state populations. Panels

Figure 4. Magnetic fields induced by the parallel ring current of JL = JR = 60 μA (upper figure) and antiparallel ring current of JL = −JR = 60 μA (lower figure). Blue stick bar indicates the frame of naphthalene located on the (X, Y, 0) surface. The unit of the magnetic field is tesla (T). An arrow representing 0.3 T is indicated in each panel for reference of the magnetic field strength. 2023


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The Journal of Physical Chemistry Letters Information. The two figures can be clearly identified: the directions of the induced magnetic field in χ = L and R are the same for the parallel ring current, while they are the opposite directions for the antiparallel ring current. This indicates the possibility of identifying two types of ring currents by measuring induced magnetic fields. Advances in time-resolved scanning microscopy (STM) and magnetic force microscopy (MFM) may make it possible to observe induced magnetic fields in an ultrashort time.48−51 We have proposed a quantum design of coherent π-electron ring currents in naphthalene (D2h), one of the simplest PAHs. It has been demonstrated by using quantum chemical considerations and QOC simulations that two types of ring currents, parallel and antiparallel, in naphthalene can be generated by two linearly polarized lasers. The mechanisms by which the two types of ring currents are generated have been clarified by analyzing both the time-dependent population changes and the electric fields of the two lasers. The QOC results suggest an experimental scenario for generation of a parallel ring current in PAHs by solving the time-dependent Schrödinger equation with both continuous wave and pulse lasers having an appropriate time delay. The key is to determine the time delay (relative phase) between the two lasers. The approach to generation of coherent π-electron ring currents, which was applied to naphthalene, can be immediately applied to other planar polycyclic aromatic hydrocarbon systems with more than three aromatic rings, such as anthracene and tetracene. The ring current patterns for larger PAHs increase in number compared with those in naphthalene. The selection of a target current pattern of interest is essential for quantum design of coherent ring currents in large aromatic ring molecules. The next stage of the present research is to develop an effective method for maintaining the ring currents as long as possible. Quantum design of a ring current was performed in nuclear frozen approximation. As found in previous studies,52,53 incorporation of nuclear motion with π-electron motions is also very important in order to design ring currents for a more realistic molecular system.

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ment, and the National Science Council of Taiwan for partial support.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00704. Data of quantum chemical calculations; the QOC results of JCC(t) at T = 60 fs; degree of electronic coherence for JCC(t) and JAC(t); analytical treatment for JCC(t); expressions for current-induced magnetic fields (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

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

Yuichi Fujimura: 0000-0001-7961-8169 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y.F. thanks Professor M. Hayashi for his critical comments on the QOC simulations, Professor S. H. Lin for his encourage2024


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