^{1}, S. H. Lin

^{1,2}and Y. Fujimura

^{1,3,a)}

### Abstract

The results of a theoretical investigation of coherent π-electron dynamics for nonplanar (*P*)-2,2′-biphenol induced by ultrashort linearly polarized UV pulses are presented. Expressions for the time-dependent coherent angular momentum and ring current are derived by using the density matrix method. The time dependence of these coherences is determined by the off-diagonal density matrix element, which can be obtained by solving the coupled equations of motion of the electronic-state density matrix. Dephasing effects on coherent angular momentum and ring current are taken into account within the Markov approximation. The magnitudes of the electronic angular momentum and current are expressed as the sum of expectation values of the corresponding operators in the two phenol rings (*L* and *R* rings). Here, *L* (*R*) denotes the phenol ring in the left (right)-hand side of (*P*)-2,2′-biphenol. We define the bond current between the nearest neighbor carbon atoms C_{ i } and C_{ j } as an electric current through a half plane perpendicular to the C_{ i }–C_{ j } bond. The bond current can be expressed in terms of the inter-atomic bond current. The inter-atomic bond current (bond current) depends on the position of the half plane on the bond and has the maximum value at the center. The coherent ring current in each ring is defined by averaging over the bond currents. Since (*P*)-2,2′-biphenol is nonplanar, the resultant angular momentum is not one-dimensional. Simulations of the time-dependent coherent angular momentum and ring current of (*P*)-2,2′-biphenol excited by ultrashort linearly polarized UV pulses are carried out using the molecular parameters obtained by the time-dependent density functional theory (TD-DFT) method. Oscillatory behaviors in the time-dependent angular momentum (ring current), which can be called angular momentum (ring current) quantum beats, are classified by the symmetry of the coherent state, symmetric or antisymmetric. The bond current of the bridge bond linking the *L* and *R* rings is zero for the symmetric coherent state, while it is nonzero for the antisymmetric coherent state. The magnitudes of ring current and ring current-induced magnetic field are also evaluated, and their possibility as a control parameter in ultrafast switching devices is discussed. The present results give a detailed description of the theoretical treatment reported in our previous paper [H. Mineo, M. Yamaki, Y. Teranish, M. Hayashi, S. H. Lin, and Y. Fujimura, J. Am. Chem. Soc.134, 14279 (Year: 2012)10.1021/ja3047848].

We are grateful to Professor M. Hayashi for his useful comments on the electronic structure theory of molecules. This work was supported by a JSPS Research Grant (No. 23550003) and the National Science Council of Taiwan. H.M. would like to thank Professor J.-L. Kuo for his critical comments and supports.

I. INTRODUCTION

II. COHERENT π-ELECTRON ANGULAR MOMENTUM AND CURRENT

A. Equations of motion for π electrons in a pulsed laser field

B. Coherent electric angular momentum

C. Coherent ring current

III. RESULTS AND DISCUSSION

A. Generation of two-state electronic coherence by linearly polarized UV pulses

B. Angular momentum quantum beats

C. Bondcurrents and time evolution of coherent ring current

D. Coherent ring current-induced magnetic field

E. Bridge bondcurrent density

IV. SUMMARY AND CONCLUSION

### Key Topics

- Ring currents
- 54.0
- Angular momentum
- 43.0
- Coherence
- 32.0
- Chemical bonds
- 29.0
- Excited states
- 26.0

## Figures

(a) Geometrical structure of (*P*)-2,2′-biphenol and transition moment vectors. (b) Three electronic excited states and optical transitions to create the coherent electronic states.

(a) Geometrical structure of (*P*)-2,2′-biphenol and transition moment vectors. (b) Three electronic excited states and optical transitions to create the coherent electronic states.

Occupied and unoccupied MOs relevant to the coherent electronic states. *a*, *b* _{1}, and *b* _{2} electronic states have electronic configurations of 46 → 52, 46 → 53, and 46 →51, respectively. A (B) denotes the totally symmetric (antisymmetric) irreducible representation of the C_{2} point group.

Occupied and unoccupied MOs relevant to the coherent electronic states. *a*, *b* _{1}, and *b* _{2} electronic states have electronic configurations of 46 → 52, 46 → 53, and 46 →51, respectively. A (B) denotes the totally symmetric (antisymmetric) irreducible representation of the C_{2} point group.

Time evolution of coherent angular momentum (left-hand side) created by π pulse excitation (right-hand side): (a) for (*b* _{1} + *b* _{2}) electronic coherence created by pulse with the amplitude of 0.19 TW/cm^{2}, (b) for (*a* + *b* _{1}) electronic coherence created by pulse with the amplitude of 0.83 TW/cm^{2}, and (c) for (*a* + *b* _{2}) electronic coherence created by pulse with the amplitude of 3.32 TW/cm^{2}. Dephasing constants were set as γ_{b1b2} = γ_{ab1} = γ_{ab2} = 0.01 eV (1/50 fs^{−1}).

Time evolution of coherent angular momentum (left-hand side) created by π pulse excitation (right-hand side): (a) for (*b* _{1} + *b* _{2}) electronic coherence created by pulse with the amplitude of 0.19 TW/cm^{2}, (b) for (*a* + *b* _{1}) electronic coherence created by pulse with the amplitude of 0.83 TW/cm^{2}, and (c) for (*a* + *b* _{2}) electronic coherence created by pulse with the amplitude of 3.32 TW/cm^{2}. Dephasing constants were set as γ_{b1b2} = γ_{ab1} = γ_{ab2} = 0.01 eV (1/50 fs^{−1}).

Bond currents *J* _{ ij } for the three types of electronic coherence at *t* = *t** when the maximum coherence is created, i.e., , , or . denotes the averaged ring current given by Eq. (14) . The blue arrows above C–C bonds denote the initial direction of the currents. Dephasing effects were omitted. Note that the bridge current *J* _{1,7} = 0 for (*b* _{1} + *b* _{2}) electronic coherence, while *J* _{1,7} ≠ 0 for the other two (*a* + *b* _{1}) and (*a* + *b* _{2}) electronic coherences. The same magnitudes of pulses as those in Fig. 3 were used.

Bond currents *J* _{ ij } for the three types of electronic coherence at *t* = *t** when the maximum coherence is created, i.e., , , or . denotes the averaged ring current given by Eq. (14) . The blue arrows above C–C bonds denote the initial direction of the currents. Dephasing effects were omitted. Note that the bridge current *J* _{1,7} = 0 for (*b* _{1} + *b* _{2}) electronic coherence, while *J* _{1,7} ≠ 0 for the other two (*a* + *b* _{1}) and (*a* + *b* _{2}) electronic coherences. The same magnitudes of pulses as those in Fig. 3 were used.

Inter-atomic bond current as a function of θ. The values are normalized by . θ, (0 ⩽ θ ⩽ π/3) is defined in the inserted figure. Here, *r* _{ i } and *r* _{ j } are the positions of two atomic sites of the bond. *s* is the position of the half plane *S*. . The red broken line refers to the first term of and the blue dotted line refers to the second term. The black line indicates the total value of .

Inter-atomic bond current as a function of θ. The values are normalized by . θ, (0 ⩽ θ ⩽ π/3) is defined in the inserted figure. Here, *r* _{ i } and *r* _{ j } are the positions of two atomic sites of the bond. *s* is the position of the half plane *S*. . The red broken line refers to the first term of and the blue dotted line refers to the second term. The black line indicates the total value of .

Time evolution of coherent ring currents created by a π pulse excitation. *J* _{R} and *J* _{L} denote the ring currents created on *R* and *L* rings, respectively. The same pulse excitation conditions as those shown in Fig. 3 are used. *J* _{B} denotes the ring current created on the bridging bond: (a) for (*b* _{1} + *b* _{2}) electronic coherence, (b) for (*a* + *b* _{1}) electronic coherence, and (c) for (*a* + *b* _{2}) electronic coherence. Dephasing constants were set as γ_{b1b2} = 0.01 eV (1/50 fs^{−1}) and 0.05 eV (1/10 fs^{−1}) for (a); γ_{ab1} = γ_{ab2} = 0.01 eV for (b) and (c). The same magnitudes of pulses as those in Fig. 3 were used.

Time evolution of coherent ring currents created by a π pulse excitation. *J* _{R} and *J* _{L} denote the ring currents created on *R* and *L* rings, respectively. The same pulse excitation conditions as those shown in Fig. 3 are used. *J* _{B} denotes the ring current created on the bridging bond: (a) for (*b* _{1} + *b* _{2}) electronic coherence, (b) for (*a* + *b* _{1}) electronic coherence, and (c) for (*a* + *b* _{2}) electronic coherence. Dephasing constants were set as γ_{b1b2} = 0.01 eV (1/50 fs^{−1}) and 0.05 eV (1/10 fs^{−1}) for (a); γ_{ab1} = γ_{ab2} = 0.01 eV for (b) and (c). The same magnitudes of pulses as those in Fig. 3 were used.

Induced magnetic fields for the (*b* _{1} + *b* _{2}) electronic coherence as a function of height *h* from the center of aromatic ring *K* at the time of maximum coherence *t* = *t**. and *B* _{ K }(*t**, *h*) are the induced magnetic field calculated by a SRL model and that calculated by a distribution of the current density, respectively.

Induced magnetic fields for the (*b* _{1} + *b* _{2}) electronic coherence as a function of height *h* from the center of aromatic ring *K* at the time of maximum coherence *t* = *t**. and *B* _{ K }(*t**, *h*) are the induced magnetic field calculated by a SRL model and that calculated by a distribution of the current density, respectively.

Contour plot of the bridge bond current density [μA/ Å^{2}] on the perpendicular plane at the bond center, which is calculated at the maximum value for the (*a* + *b* _{1}) electronic coherence at *t* = *t**. The region in which the current density flows from the back (*L* ring) to the front (*R* ring) is denoted by positive values, while the region in which the current density flows from the front to the back is denoted by negative values.

Contour plot of the bridge bond current density [μA/ Å^{2}] on the perpendicular plane at the bond center, which is calculated at the maximum value for the (*a* + *b* _{1}) electronic coherence at *t* = *t**. The region in which the current density flows from the back (*L* ring) to the front (*R* ring) is denoted by positive values, while the region in which the current density flows from the front to the back is denoted by negative values.

(a) Relation between the molecular frame denoted by *xy* and that of the new frame *XY* for evaluation of angular momentum at *O*, the direction of which is perpendicular to the molecular plane, *xOy*. is the angular momentum operator. (b) Coordinates on the new frame.

(a) Relation between the molecular frame denoted by *xy* and that of the new frame *XY* for evaluation of angular momentum at *O*, the direction of which is perpendicular to the molecular plane, *xOy*. is the angular momentum operator. (b) Coordinates on the new frame.

Coordinates for the inter-atomic bond current from atom *j* to the nearest neighbor atom *i*, in a hexagonal ring, which determines the current passing through a half-plane *S* perpendicular to the bond at the center *R* _{ c }.

Coordinates for the inter-atomic bond current from atom *j* to the nearest neighbor atom *i*, in a hexagonal ring, which determines the current passing through a half-plane *S* perpendicular to the bond at the center *R* _{ c }.

Coordinates for the inter-atomic bond current, , which is perpendicular to a half-plane *S* (x = 0) at carbon nuclei C_{ j } of bond C_{ i }–C_{ j }.

Coordinates for the inter-atomic bond current, , which is perpendicular to a half-plane *S* (x = 0) at carbon nuclei C_{ j } of bond C_{ i }–C_{ j }.

Coordinates for the inter-atomic bond current in a general case. The half plane *S* is set at a position between carbon atomic sites, C_{ i } and C_{ j }. *N* is the number of carbon atoms. Angle θ is defined in the range 0 ⩽ θ ⩽ θ_{ N } in which θ_{ N } is the angle between C_{ i } and C_{ j } from the center of the ring.

Coordinates for the inter-atomic bond current in a general case. The half plane *S* is set at a position between carbon atomic sites, C_{ i } and C_{ j }. *N* is the number of carbon atoms. Angle θ is defined in the range 0 ⩽ θ ⩽ θ_{ N } in which θ_{ N } is the angle between C_{ i } and C_{ j } from the center of the ring.

Coordinates for the induced magnetic field created at the center of ring *K* at height *h* by a ring current density ⟨*J*(*t*)⟩_{ K } *f*(*x*, *z*)*dxdz*. The bond between *i* and *j* lies on the *y*-axis.

Coordinates for the induced magnetic field created at the center of ring *K* at height *h* by a ring current density ⟨*J*(*t*)⟩_{ K } *f*(*x*, *z*)*dxdz*. The bond between *i* and *j* lies on the *y*-axis.

## Tables

Angular momenta of the two phenol rings, *l* _{ L } and *l* _{ R }, and the resultant angular momenta *l* _{ X } an *l* _{ Z } at the maximum coherence. ^{ a }

Angular momenta of the two phenol rings, *l* _{ L } and *l* _{ R }, and the resultant angular momenta *l* _{ X } an *l* _{ Z } at the maximum coherence. ^{ a }

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