Visualizations of Photoinduced Charge Transfer and Electron-Hole

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C: Energy Conversion and Storage; Energy and Charge Transport

Visualizations of Photoinduced Charge Transfer and Electron-Hole Coherence in Two-Photon Absorptions Xijiao Mu, Jingang Wang, and Mengtao Sun J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b00700 • Publication Date (Web): 10 May 2019 Downloaded from http://pubs.acs.org on May 10, 2019

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The Journal of Physical Chemistry

Visualizations of Photoinduced Charge Transfer and Electron-Hole Coherence in Two-Photon Absorptions Xijiao Mu,1,† Jingang Wang,2,† Mengtao Sun1, * 1. School of Mathematics and Physics, Center for Green Innovation, Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, Beijing, 100083, P.R. China. 2. College of Science, Liaoning Shihua University, Fushun, 113001, P.R. China * Corresponding author. Email: [email protected] (M. Sun). † Contributed Equally. ABSTRACT: We provide a visualization method to observe the photoinduced charge transfer and electron-hole coherence in the two-photon absorption (TPA), especially includes the optical properties in the “hidden” intermediate states in TPA. A new method for analyzing the characteristics of two-photon excitation is proposed by using a two-step process, which is closest to the physical reality process. Theses visualization methods include Transition density matrix, Charge density difference and Transition dipole moment in TPA involving both ground and excited-states wavefunctions and the transition dipole moments among electric transitions from ground to intermediate and to final excited states. Our visualization method of photoinduced CT and electron-hole coherence in TPA can promote deeper understanding and design molecules with large cross sections in TPA.

1．Introduction Simultaneous two-photon absorption (TPA) is a nonlinear optical process. This phenomenon was first predicted by Gppert-Mayer in 19311 who calculated the transition probability for a two-quantum absorption process. Two-photon absorption can create excited states with photons of half the nominal excitation energy, which can provide improved penetration in absorbing or scattering media. TPA is a third-order nonlinear optical process. Two-photon absorption is the I2 dependence of the process, which allows for excitation of chromophores with a high degree of spatial selectivity in three dimensions through the use of a tightly

focused laser beam.2 Two-photon absorption are in great demand for variety of applications, such as two-photon– excited fluorescence microscopy, 3-5 3D optical memory,6 nanofabrication,7 and up conversion lasing,8 solar cell,9,10 nondestructive imaging of biological tissues,11,12 photodynamic therapy.13 Theoretical analysis of TPA have been implemented to interpret experimental measurements of TPA, which is very useful to understand structure-property relations. Coupled with finite field techniques, the ab-initio methods are widely used to calculate off-resonant NLO responses,14 more general method is the

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time dependent perturbation theory. The sum-over-states (SOS) model is an essentially practical methods, involving calculating both ground and excited-states wavefunctions and the transition dipole moments between them.15,16 The transition dipole moment in TPA includes two parts as three-state and two-state models, 16 respectively. Richter, M., and S. Mukamel 𝛿𝑡𝑝 = 8

∑

|⟨𝑓|𝜇|𝑗⟩⟨𝑗|𝜇|𝑔⟩|2

(

𝑗 ≠𝑔 𝑗 ≠𝑓

𝜔𝑗 ―

𝜔𝑓 2

)

2

et al. demonstrated that the two-photon absorption process is a two-step process in which an intermediate state must be experienced during two-photon 17,18. excitation The complete two-photon absorption probability is the sum of the processes that traverse all intermediate states,16

(1 + 2𝑐𝑜𝑠

+ 𝛤2𝑓

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2

|𝛥𝜇𝑓𝑔|2|⟨𝑓|𝜇|𝑔⟩|2

𝜃 𝑗) + 8

𝜔𝑓

( ) 2

(1 + 2𝑐𝑜𝑠2 𝜙)

2

+ 𝛤2𝑓 (1)

where

g , f

and

j

are the ground

state wave function, final state wave function, and the intermediate state of the two-photon transition, respectively. The two Dirac brackets in the first term of the formula are the transition dipole moments from the ground state to the intermediate state and the intermediate state to the final state. The corresponding angle

 j and is the angle between the

Figure 1. The schematic diagrams of (a) onestep process in one photon absorption, and (b) two-step processes in two photon absorption

two transition dipole moment vectors. The  fg in the second term of the formula is

the difference between the final state and the ground state permanent dipole moment. The the vector of f  g .

 is the angle between of  fg

and the vector of

 j and  f are the energy of

intermediate and final state, respectively. The  f is the lifetime of the excited state.

Eq.(1) describes the two-photon absorption cross section of a two-step process. In Figure 1(a), there are two onestep process in one photon absorption. As shown in Figure 1(b), a and b represent two different intermediate states in twophoton absorption. This diagram illustrates that the same final two-photon absorption state may experience different intermediate states, which may be quite large for a real system. When studying the two-photon process, we not only care about the shape of the absorption spectrum (two-photon absorption cross section), but also the electronic motion behavior (such as charge transfer characteristics) when two-photon


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excitation. Therefore, it is advantageous to use Eq. (1) to calculate two-photon absorption. This is because it reflects the two-step process and can focus on the nature of the intermediate state. It is also a factor that plays a crucial role in twophoton excitation. For the three-state model in Eq. (1), the first part of the formula, which is the transition from the ground state to the intermediate state; and the second part of the formula, which is the transition from the intermediate state to the final state. For the two-state model, it includes the transition from the ground state to the final state; and the difference of permanent dipoles between final excited state and ground state. For the symmetric system, the two-state model can be ignored, since the contribution in Eq. (1) is vanished small, compared with the three-state model.19

chain in which a six-membered conjugated ring and an alkyne group alternate, and a ferrocene is attached to both ends. In the calculation and analysis, the atomic number increases from left to right. We firstly build up the theoretical analysis method, and then, perform the visualization of charge transfer and electron-hole coherence in TPA, including the hidden “intermediate” states. Lastly, conclusions are summarized to point out the importance of our developed visualization method of CT in TPA.

2．Methods In this work, we expect to give a twophoton absorption analysis method close to the actual physical process. So, in this section we will give a detailed wave function analysis method. This way, all the analysis methods in the text can be fully realized. Firstly, we present the existing mature methods for obtaining one-photon excited states and their wave functions. Secondly, a method of transition density matrix and charge differential density is given. Finally, we give the formula for quickly calculating the dipole moment of the excited state.

Figure 2. Schematic diagram of molecular structure of three similar conjugated molecules.

2.1 One photon excited state and its wave function calculation

In this paper, based on the few-state model in Eq. (1), we theoretical study the photoinduced charge transfer and electron-hole coherence in TPA, especially includes optical physical properties at “hidden” intermediate state. The chosen symmetric systems of three molecules can be seen from figure. 2, based on the recent experimental report in Ref. [19]. The molecule is composed of a hydrocarbon

From the foregoing, the key to calculating the two-photon absorption cross section by the two-step method is to calculate the transition dipole moment from the ground state to each excited state and the transition dipole moment between the excited state and the excited state. According to the Frank Condon principle, the transition dipole moment and the wave function of the ground state and the excited state are determined. Therefore, it



is necessary to first generate a wave function using the existing quantum chemical code. Optical properties of symmetric molecular models in Fig. 1 are calculated with quantum chemical method. All the quantum chemical calculations were done with Gaussian 16 A03 software.20 The geometry of graphene is optimized with density functional theory (DFT),21 B3LYP functional,22 and 6-31G(d) basis set. The electronic transitions of three molecules were calculated with time dependent DFT (TD-DFT),23 CAM-B3LYP functional24 and 6-31G(d) basis set. Since the selected molecular system has no heavy atoms, the properties of the directional wave function, especially the transition dipole moment, can be completely described using the 2-zeta basis set group. Note that the long-rangecorrected functional (CAM-B3LYP) was employed for the non-Coulomb part of exchange functional. Two-photon absorption spectroscopy, two-photon absorption cross section, and two-photon transition dipole moment integration are performed using a self-program. The charge transfer on the electronic transition is visualized with charge different density. The transition density matrix map is done by the Multiwfn 3.6 program.25 The density isosurface is visualized by the VMD program.26 Note that the benzene ring that connected to a long-saturated hydrocarbon chain is ignored. Since we are studying the excitation characteristics of the molecules excited by photons (mainly charge transfer characteristics), the influence of saturated hydrocarbon chains on charge transfer is very small. This approximation is reliable.10 The benzene ring and the conjugated triple-bond in the main skeleton of the molecule are very

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strong in charge transferability, and it can be considered that the main skeleton of the molecule communicates the charge transfer of ferrocene at both ends. The molecular skeleton will act as a "bridge."

2.2 Transition density matrix (TDM) In order to analyze the electronic excitation characteristics of the intermediate state during two-photon excitation. The transition density matrix is an effective tool for analyzing charge transfer between atoms in a molecule. The core part of the transition density matrix is to obtain the transition density. The core part of the transition density matrix is to obtain the transition density. In basis function representation, the TDM can be written as:

Ptran  i v

occ



vir j

wij Ci Cvj

(2)

where 𝐶𝜇𝑖 denotes the expansion coefficient of basis function 𝜇 in molecular orbital (MO) i. It is worth to note in passing that the TDM in real space, which can be constructed easily via TDM in basis function representations: 𝑇(𝒓;𝒓′) = ∑𝜇∑𝜈𝑃𝑡𝑟𝑎𝑛 𝜇𝜈 𝜒𝜇(𝒓)𝜒𝜈(𝒓′) (3) The off-diagonal elements of TDM essentially represent the coupling between various basis functions during electronic excitation. Assume there are only two basis set functions and the excitation can be perfectly represented as i → j orbital transition, then the TDM could be explicitly written as： 𝐶𝜈𝑖𝐶𝜇𝑗 𝐶𝜈𝑖𝐶𝜈𝑗 𝑃𝑡𝑟𝑎𝑛 (4) 𝜇𝜈 = 𝐶 𝐶 𝜇𝑖 𝜇𝑗 𝐶𝜇𝑖𝐶𝜈𝑗

[

]

If magnitude of off-diagonal element ≠ 𝜈) is large, it implies that basis 𝜇 𝜈 functions and significantly participate in hole (occupied orbital i) and electron (virtual orbital j), respectively, 𝑃𝑡𝑟𝑎𝑛 𝜇𝜈 (𝜇


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and in this case we can say that the two basis set functions are strongly coupled during the excitation. Since the transition density is contributed by different basis functions (base functions belonging to different atomic centers), the transition represented by this transition density can be considered as charge transfer excitation. The diagonal terms are also meaningful, if element 𝑃𝑡𝑟𝑎𝑛 has large magnitude, that 𝜇𝜇 𝜇 means basis function must simultaneously have large contribution to both hole and electron. Contrary to the previous situation, since the transition density is contributed by the same basis function (the same atomic center basis function), this excitation is a local excitation.

2.3 Charge difference density and integral curve Charge density differential is a good way to visualize the characteristics of electron motion during electronic excitation using 3D visualization. Since the two-photon process involves a series of intermediate states, the CDD algorithm is no longer consistent with traditional methods. A new method is needed to generate the density matrix of the intermediate state. Firstly, generating density matrix of excited state 2

𝑷𝐸𝑥𝑐𝑖𝑡𝑒𝑑 = 𝑷𝐺𝑟𝑜𝑢𝑛𝑑 + ∑𝑖→𝑗(𝑤𝑗𝑖) (𝑷𝑗 ― 𝑷𝑖) + ∑𝑖←𝑗(𝑤𝑗𝑖)2(𝑷𝑗 ― 𝑷𝑖)

(5)

where 𝑷𝐺𝑟𝑜𝑢𝑛𝑑 is density matrix of ground state wavefunction, the matrix like 𝑷𝑖 is density matrix constructed solely by orbital i, it can be evaluated as below, 𝑷𝑖 = 𝑪𝑖𝑪𝑇𝑖 (6) where 𝑪𝑖 is column vector of expansion coefficients of orbital i. Secondly, diagonalizing the 𝑷𝐸𝑥𝑐𝑖𝑡𝑒𝑑 to yield nature orbitals (NOs). Each NO is an eigenvector

of 𝑷𝐸𝑥𝑐𝑖𝑡𝑒𝑑, the accompanying eigenvalue is occupation number of the NO. Finally Exporting information of basis function and NOs coefficients to a file. Then use the files of different excited states to make a difference and obtain the charge differential density of the transition between the excited states. This density allows efficient analysis of charge transfer characteristics during excitation in real space. After obtaining the difference in charge density, these lattice points can be integrated in a certain direction to obtain an integral curve. 𝑧

𝑧

+∞

+∞

𝐼(𝑧′) = ∫𝑧′ 𝐼𝐿𝑜𝑐𝑎𝑙(𝑧)𝑑𝑧 = ∫𝑧′ ∫ ―∞∫ ―∞𝑝(𝑥,𝑦,𝑧) 0

0

(7) Drawing these integral curves can be more quantitatively discussed in relation to the charge density transfer distribution. A method for analyzing charge transfer distribution also has a natural transition orbit (NTO) analysis.27,28 This orbit is obtained by a unitary orbital transformation of the molecular orbital. The eigenvalue of the orbital wave function represents the extent to which the orbit can describe the excited state (always less than 1). This method of analysis is physically complete, but for complex transition behavior, the eigenvalue of the orbital wave function may be less than one. In other words, n NTOs are needed to jointly describe the excitation process. Since the NTO investigation of the transition process requires simultaneous observation of occupied orbits and virtual orbits, it may take 2n NTOs to approximate the transition behavior. However, the method of charge differential density can show where electron density is reduced (holes) and where electron density is increased (electrons) in the same real space. So, for each two-photon excitation, we 𝑑𝑥𝑑𝑦𝑑𝑧



always need only two graphs to analyze. A further advantage of this analytical method over NTO is the ability to separately analyze the charge transfer in two steps in a two-photon transition. This allows a good analysis of the overall local excitation, but internally the two charge transfer excitations with opposite charge transfer directions.

2.4 Transition dipole moment The key to calculating the two-photon absorption cross section using Eq.(1) is to calculate the transition dipole moment between states. Note that there are many kinds of transition dipole moment, including transition electric dipole moment, transition magnetic dipole moment, transition velocity dipole moment and so on. The word "transition dipole moment" commonly refers to transition electric dipole moment. X, Y and Z components of transition electric dipole moment density can be written as negative of product of X, Y and Z coordinate variables and transition density, respectively: 𝑇𝑖(𝒓) = ―𝑖𝑇(𝒓),𝑖 = 𝑋,𝑌,𝑍 (8) Integrating transition electric dipole moment density over the whole space yields transition dipole moment 𝜇， 𝜇𝑖 = ∫𝑇𝑖(𝒓)𝑑𝒓,𝑖 = 𝑋,𝑌,𝑍 (9) Obviously, one can conveniently study contribution to transition electric dipole moment of various molecular regions by plotting transition electric dipole moment density. In fact, there is an external electric field fitting calculation method for the permanent dipole moment used in Eq 1. This method is based on the classic definition of the dipole moment. The energy under the disturbance of different external electric fields is used to fit the

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dipole moment. The fit formula by external electric field is defined as:

1 Ei  F   Ei (0)   F   F 2 2

(10)

where Ei is the excitation energy of the ith excited state.



is the difference

between permanent dipole moments. F is the external electric field strength.  is a change in polarizability.

3. Results and discussion The left half of figure 3 is the onephoton absorption curve (left axis) and the oscillator strength of the single photon transition (right axis). As can be seen from the figure, the one-photon absorption spectrum of the three molecules is generally composed of three parts. The rising and falling edges of the highest energy absorption peaks of molecules 8 and 9 have shoulder peaks. For the lowest energy peak, the molecules 9 and 10 are S9, but the molecular 8 is the S14. This may be caused by the six-center two-electron orbital of the benzene group introduced into the molecule. For the excited state in which the energy is in the middle, it is a weak one-photon excited state. Although this has little effect on the one-photon absorption spectrum, it will likely become the middle bridge between two-photon transitions. As shown in Fig. 3(g), the single photon absorption spectra of three molecules were compared. It can be seen from the figure that the high energy peaks of the three molecules are almost identical, but the energy of the low energy peaks gradually decreases from the molecule 8 to the molecule 10. The three molecules grow from the numerator to the conjugated chain of the molecule 10, so this low energy peak is undesired as the conjugated chain


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of the molecule grows. According to the above analysis, it is indicated that the high energy peak of the molecule is determined by ferrocene, and the low energy peak is determined by the conjugate chain (probably a charge transfer transition). The right half of Figure 3 is the two-photon absorption spectrum (left axis) and absorption cross section (right axis). The two-photon absorption spectra of the three molecules are composed of one main peak and several very weak accessory peaks. Since the peak intensity of the subsidiary peak is very weak, we only analyze the main peak. Figure 3 (h) is a comparison of the two-photon absorption spectra of three

molecules. It can be seen from the figure that as the molecular conjugate chain grows, the two-photon absorption peak gradually shifts red. According to the previous discussion of one-photon absorption, it is stated that the two-photon absorption peaks are all contributed by the conjugated chains in the molecule. The red shift of the molecules 8 to 9 is much stronger than the red shift of the molecules 9 to 10. This is due to the introduction of a six-center two-electron orbit (benzene ring) from the molecular conjugated chain starting from the molecule 9. It is shown that the influence of the 6c-2e orbit on the absorption transition is very large.



Figure 3. OPA and TPA spectrum of 8((a) and (b)), 9((c) and (d)) and 10 ((e) and (f)) molecule and their joint comparison ((g) and (h)).


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Table 1. The main two-photon absorption process and its transition dipole moment value Molecule

TPA states 34

8

41 43 22

9

10

24 19

Process

Integral Value

⟨𝜙𝑆0|𝜇|𝜙𝑆14⟩→⟨𝜙S14|𝜇|𝜙S34⟩ ⟨𝜙𝑆0|𝜇|𝜙S33⟩→⟨𝜙S33|𝜇|𝜙S34⟩ ⟨𝜙𝑆0|𝜇|𝜙𝑆14⟩→⟨𝜙S14|𝜇|𝜙S41⟩ ⟨𝜙𝑆0|𝜇|𝜙S33⟩→⟨𝜙S33|𝜇|𝜙S41⟩ ⟨𝜙𝑆0|𝜇|𝜙𝑆14⟩→⟨𝜙S14|𝜇|𝜙S41⟩ ⟨𝜙𝑆0|𝜇|𝜙S33⟩→⟨𝜙S33|𝜇|𝜙S43⟩ ⟨𝜙𝑆0|𝜇|𝜙S9⟩→⟨𝜙S9|𝜇|𝜙S22⟩ ⟨𝜙𝑆0|𝜇|𝜙S9⟩→⟨𝜙S9|𝜇|𝜙S24⟩ ⟨𝜙𝑆0|𝜇|𝜙𝑆17⟩→⟨𝜙S17|𝜇|𝜙S24⟩ ⟨𝜙𝑆0|𝜇|𝜙S9⟩→⟨𝜙S9|𝜇|𝜙S19⟩ ⟨𝜙𝑆0|𝜇|𝜙𝑆16⟩→⟨𝜙S16|𝜇|𝜙S19⟩

Two-photon absorption is the process of two transitions (Formula 1), which is a single photon excitation and a transition from an excited state to an excited state. If the transition probability (transition dipole moment) of these two processes is relatively high, a strong two-photon absorption peak can be observed. We calculated the transition dipole moment matrix elements in two-photon absorption separately and presented in Table 1. As shown about Table 1, in general, the first process of the two-photon transition (the transition from the ground state to the intermediate state) has a higher probability of transition. In contrast, the probability of transition between the excited state and the excited state is relatively small. The horizontal comparison between the three molecules reveals that the longer the molecular chain, the larger the transition dipole moment of the first process in the two-photon transition. The first step in the two-photon transition is determined by the conjugate chain in the molecule. In order to thoroughly analyze the charge transfer characteristics during the twophoton transition. We plot the charge differential density and transition density

5.395→0.671 3.368→0.711 5.395→0.780 3.368→0.771 5.395→0.373 3.368→0.876 30.024→1.655 30.024→0.542 3.846→1.183 57.056→0.506 7.210→2.068

matrix for each step in Figure 4. Since the two-photon absorption main peak of molecule 8 is composed of three twophoton transitions. It is therefore necessary to analyze the characteristics of these three excitations. First, analyze the S34 twophoton excited state. The first process of the S34 two-photon excited state of molecule 8 can be analyzed in conjunction with Figures 3(a) and 4(c), which is the excitation from S0 to S14. Figure 4(a) shows the charge differential density, where the green isosurface represents where electrons are reduced (holes) and the pink isosurface is where electrons increase (electrons). The charge transfer characteristics are not apparent from the figure, and may be local excitation. However, it can be confirmed from the transition density matrix of Fig. 4(c) that it is a local excitation—the transition density is concentrated on the conjugate chain in the middle of the molecule. The second process is the transition from S14 to S34. The presence of charge transfer characteristics can be seen in conjunction with Figures 4(b) and 4(d). The pink isosurface is concentrated on the conjugated chain in the middle of the molecule, and the



transition density matrix can also prove this. The charge transfers integral curve of the molecule 8 is shown in Figure S1(a) in supporting information (SI). The region that the charge transfer integral curve is less than zero represents the decrease of electron, i.e., that is the region of hole. Conversely, the region with a curve being larger than zero represents an increase in electron, i.e., that is the region of electron. Its absolute value represents the amount of charge transfer. Thus, the curve can quantitatively describe the degree of charge transfer. The two processes of this path are represented by a green curve and a black curve. It was found that the green curve squats at 0, which proves that the process is local excitation (or the charge transfer length is very small). However, the black curve has a positive and negative value on both sides of the molecular axis of 0 angstroms, indicating that there is a weak charge transfer characteristic. Another path for the S34 two-photon excited state is the S33 state as the intermediate state, and then the S33 to the S34. The transition dipole moment of this path is slightly smaller than the first path, indicating that the contribution in the entire excited state is also slightly smaller than the first path, but

its contribution is still not to be underestimated. Another path for the S34 two-photon excited state is the S33 state as the intermediate state, and then the S33 to the S34. The transition dipole moment of this path is slightly smaller than the first path, indicating that the contribution in the entire excited state is also slightly smaller than the first path, but its contribution is still not to be underestimated. By analyzing its charge transition characteristics (See, Figure 4 (e), (f), (g) and (h)), it is found that both processes of this path have strong charge transfer characteristics. The electrons of the two processes (pink isosurface) are all concentrated on the conjugate chain in the middle of the molecule. This can also be proved from the transition density matrix. The charge transfers integral curve of the molecule 8 is shown in Figure S1(a) in SI. The charge transfer integral curve for this path is the green and black dashed lines. The positive and negative values of these two dotted lines are very different, indicating that there is a strong charge transfer.


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Figure. 4 The two-photon absorption charge differential density and transition density matrix of molecule 8. The charge differential density (a) of S0 → S14 and the transition density matrix (c). S14→ S34 charge differential density (b) and transition density matrix (d). The charge differential density (e) and the transition density matrix (g) of S0 → S33. The charge differential density (f) of S33 → S34 and the transition density matrix (h).



Figure 5. The two-photon absorption differential charge density and transition density matrix of molecule 8. The differential charge density (a) of S14 → S41 and the transition density matrix (c). S33 →S41 differential charge density (b) and transition density matrix (d).

Figure 6. The two-photon absorption differential charge density and transition density matrix of molecule 8. The charge differential density (a) of S14 → S43 and the transition density matrix (c). S33 →S43 charge differential density (b) and transition density matrix (d).

The second excited state constituting the main two-photon absorption peak of the molecule 8 is the S41. This excited state also has two paths, that is, S14 is the

intermediate state and S33 is the intermediate state. The charge differential density and transition density matrix from the ground state to the S14 and S33 states are


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shown in Figure 4. As shown in Figure 5, it is a matrix of differential charge density and transition density from the S14 and S33 states to the S41 excited state. Firstly, let's analyze the first path. The transition from S14 to S41 shows the characteristics of local excitation, but there is a weak charge transfer. The integral value of the charge transfer curve (red curve) in Figure S1(a) in SI oscillates up and down around 0, but the overall trend has a positive and negative difference around 0 on the x-axis, so the overall local excitation is mixed with a weak charge transfer. And because according to the previous discussion, the transition from the ground state to S14 is a local excitation. Therefore, the S41 twophoton excited state is a local charge excited with a weak charge transfer. The second path of the S41 is the intermediate state of S33. The charge transfer behavior on this path is completely different from the first path. According to the foregoing, the ground state to S33 is a significant charge transfer excitation. In turn, the transition behavior from S33 to S41 is analyzed and it is found that this transition is also a charge transfer excitation. However, unlike the previous case, the portion where the charge density is reduced (hole, green isosurface) is concentrated on the conjugated chain at the center of the molecule, and the portion where the charge density is increased (electron, pink isosurface) is near the iron atom. The case of the first process (S0→S33)

and the charge transition behavior of the second process are completely opposite. In other words, the charge transfer direction of the second process is the reverse of the first process. This can be seen from Figure S1(a) in SI. The integral curve of the first step transition (green dashed line) and the second step transition (red dashed line) is exactly the opposite. Therefore, the transition path of the S41 two-photon excited state with S33 as the intermediate state is local excitation. Combined with the conclusion that the first path is a local excitation, it can be concluded that the two-photon excited state is a local excitation as a whole. Next, we use the same discussion paradigm to discuss the third two-photon excited state of the main peak of the molecule 8, namely the S43 twophoton excited state. Some conclusions can be drawn by combining the charge differential density and transition density matrix (See, Figure 6) and the charge transfer integral curve (Figure S1(b) in SI). Firstly, the first process of the first path is local excitation, and the second process is a weak local charge transfer feature for the overall local excitation. Second, the two processes of the second path are charge transfer excitation, but the charge transfer direction is completely opposite (similar to the S41 two-photon excited state), indicating that the second path is generally local excitation. Finally, the S43 two-photon excited state is a local excitation.



Figure 7. The two-photon absorption charge differential density and transition density matrix of molecule 9. The charge differential density (a) of S0 → S9 and the transition density matrix (b). S9→ S22 charge differential density (c) and transition density matrix (d). The charge differential density (e) of S9 → S24 and the transition density matrix (f).

The conjugate chain of the molecule 9 is longer than the molecule 8, and a sixcenter orbit (benzene group) is introduced. The two-photon absorption main peak of molecule 9 is mainly composed of two excited states, which are the S22 and S24 two-photon excited states, respectively. Combined with the differential charge density and transition density matrix (See Figure 7), the transition from the ground state to the ninth excited state is a local excitation and is a localized local excitation on the conjugate chain of the molecule. The transition from the S9 to the S22 and the S24 is a charge transfer excitation. The holes are concentrated on the iron atoms (Figure 7(c) and (e)), indicating that the iron atoms provide electrons. The electron transfer on the iron atom is first received by the conjugated orbit of the linked the ring of ferrocene and then transferred to the middle portion of the molecule.

Finally, the transition characteristics of the molecule 10 are analyzed. The conjugated chain of the molecule 10 is longer than the molecule 9. The main peak of molecule 10 is determined by only one S19 two-photon excited state. This excited state has two main transition processes, with the S9 and S16 being intermediate. The transition from the ground state to the ninth excited state is a charge transfer excitation (this is evidenced by the black curve in the charge transfer integral curve). The charge is transferred from both sides of the molecular conjugated chain to the central region (see Figure 8). The transition of the iron atom from the S9 to the S19 excited state is also a charge transfer excitation, in which the charge transfer direction is opposite to the first step (Figure S1(c) in SI). On the other hand, the first step of the second transition path is the charge transfer excitation of the iron


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atom to the conjugate chain in the middle of the molecule, and the second step is the excitation of the molecule to the iron atom. The charge transfer excitation directions of these two steps are also completely

opposite (See, Figure S1(c) in SI). In summary, the S19 of the molecule 10 is internally charge-excited and is macroscopically localized.

Figure 8. The molecule 10’s two-photon absorption charge differential density of S0 → S9 (a), S9 → S19 (b), S0 → S16 (c), S16 → S19 (d); and the transition density matrix map of S0 → S9 (e), S0 → S16 (f), S9 → S19 (g), S16 → S19 (h).



Figure 9. The asymmetric molecule 9’s molecular structure (a), one-photon absorption spectrum(b), two-photon absorption spectrum compared with symmetric 9(c) and the comparison of first and second item spectrum of Eq 1 (d). The two-photon absorption charge differential density of S0 → S5 (e), S5 → S15 (f); and the transition density matrix map of S0 → S5 (g), S5 → S15 (h).


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The structures of these three molecules are bilaterally symmetric, so the permanent dipole moments of different excited states will cancel each other out. This is disadvantageous for the second term of Eq 1. Therefore, we use the intermediate product in the molecular synthesis process, that is, the asymmetric molecule 9 to carry out the influence of the second term in Eq 1. As mentioned earlier, the second term in Equation 1 represents the difference in the permanent dipole moment between the initial state and the final state. If this permanent dipole moment is canceled out, it is not observable in the result of the twophoton absorption cross section29. And if the system under study is an asymmetric molecular structure, then the contribution of the second item is very prominent. The molecular structure of the synthetic intermediate of molecule 9, that is, the asymmetric molecule 9, is shown in Figure 9(a). A ferrocene group in the molecule is not present and is replaced by a benzoic acid group. This situation leads to a difference in the number of electrons and local energy levels at both ends of the molecular structure. In other words, the electron chemistry potentials on both sides of the molecule are different. Figure 9(b) shows the one photon absorption spectrum of the asymmetric molecule 9 and its oscillator strength. The most absorptive intensity is the S5 at 382.67 nm. The other two absorption peaks are the 11st excited state at 279.67 nm and the 48th excited state at 197.47 nm, respectively. Figure 9(c) shows a comparison of the twophoton absorption between the asymmetric molecule 9 and the symmetric molecule 9. It can be seen intuitively that the two-photon absorption intensity of the asymmetric molecule 9 is enhanced, and a new two-photon absorption peak appears

at 410 nm in the high energy region. Here we can see the effect of the second term of Eq 1. This part of the enhancement is contributed by the difference in the permanent dipole moment. In order to more clearly reflect the contribution of the second term of Eq 1, Figure 9(d) plots the comparison of the two-photon absorption spectra produced by the first and second terms of Eq 1. in two-photon absorption. Firstly, an absorption peak (red curve) contributed by the first excited state appeared at 1176 nm, and an absorption peak appeared at 764 nm. This absorption peak is contributed by the S5 (single photon absorption at 382.67). Secondly, there are two strong absorption peaks at the position where the two-photon absorption is strong. The relative peak-to-strong relationship between the two strong absorption peaks is different. The S15 is the strongest in the two-photon absorption spectrum of the asymmetric molecule 9 (Fig. 9(c)). However, analyzing the contribution of the second permanent dipole moment, respectively. It is found that the S48 is that the peak at 394 nm is stronger than the S15. This indicates that the newly appearing peak in the twophoton absorption spectrum is mainly the contribution of the difference of the permanent dipole moment, which is the second term of Eq 1. Since the S15 is the main contributor of the strongest two-photon absorption peak of the asymmetric molecule 9, (86.85% contribution). Therefore, we analyzed the excitation process of the S15 excited state. In the two-photon absorption process, the main path of the S15 is the transition from the S5 to the intermediate state. It is apparent that the vibrator strength of the S5 is high in single photon absorption, which provides a basis for the S5 to become



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a good intermediate state. From figure 9 (e ~ h) it can be concluded that this excitation process is first internal charge transfer on the long chain of molecules. The process from the S5 to the S15 is a significant charge transfer transition of the molecule to the benzoic acid group. This method can also perform accurate two-photon excitation characteristics for asymmetric molecules.

(91436102, 11374353 and 11874084), the

4. Conclusion

1. Goppert-Mayer, M., Über Elementarakte mit

National Basic Research Program of China (Grant number 2016YFA0200802), and the fundamental Research Funds for the Central Universities in China.

References zwei Quantensprüngen. Ann. Phys. Lpz. 1931,

Based on the analysis and discussion of three molecules, some laws about charge transfer in long-chain molecules were discovered. Firstly, the molecular conjugated chain does enhance the degree of charge transfer. For the molecules discussed in this paper, as the molecular conjugated chain becomes longer, the degree of charge transfer becomes stronger. Secondly, as the molecular conjugated chain grows longer, the contribution of ferrocene on both sides of the molecule to charge transfer becomes weaker. Finally, the multi-center orbital in the conjugated chain can significantly promote the red shift of the charge transfer absorption peak. In order to analyze the two-photon excitation process, this paper proposes an analysis method that best fits the physical picture in the two-photon transition process, that is, the method that can analyze the excitation behavior of the intermediate state. And this method is universal for a variety of molecular structures, including symmetric and asymmetric molecules.

9, 273-294.

2. Albota, M.; Beljonne, D.; Brédas, J.; Ehrlich, J. E.; Fu, J.; Heikal, A. A.; Hess, S. E.; Kogej T.; Levin, M. D.; Marder, S. R.; McCord-Maughon, D. J.; Perry, W.; Röckel, H.; Rumi, M.; Subramaniam, G.; Webb, W. W.; Wu, X.; Xu, C. Design of Organic Molecules with Large Two-Photon Absorption Cross Section, Science 1998, 281, 1653-1656.

3. Denk, W.; Strickler, J. H.; Webb, W. W.; Twophoton

laser

scanning

fluorescence

microscopy. Science 1990, 248, 73-76.

4. Köhler, R. H.; Cao, J.; Zipfel, W. R.; Webb, W. W.; Hanson, M. R. Exchange of protein molecules

through

connections

between

higher plant plastids. Science 1997, 276, 20392042.

5. Li, R.; Zhang, Y. J.; Xu, X. F.; Zhou, Y.; Chen, M. D.; Sun, M. T. Optical characterizations of two-dimensional materials using nonlinear optical microscopies of CARS, TPEF, and SHG, Nanophotonics, 2018, 7, 873–881

6. Parthenopoulos, D. A.; Rentzepis, P. M. Threedimensional optical storage, Science 1989, 245, 843-845.

7. Cumpston, B. H.; Ananthavel, S. P.; Barlow, S.; Dyer, D. L.; Ehrlich, J. E.; Erskine, L. L.; Heikal, A. A.; Kuebler, S. M.; Lee, I.-Y. S.; McCord-

Acknowledgments:

Maughon, D. J. Two-Photon Polymerization

This work was supported by the National

Storage and Microfabrication. Nature 1999, 398,

Initiators for Three-Dimensional Optical Data

Natural Science Foundation of China

51-53.

8. Bauer, C.; Schnabel, B.; Kley, E. B.; Scherf, U.;


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


Giessen, H.; Mahrt, R. F. Two ‐ Photon

particle

Pumped Lasing from a Two ‐ Dimensional

entanglement. Phys. Rev. A, 2011, 83, 063805(1-

Photonic Bandgap Structure with Polymeric

9).

Gain Material. Adv Mater 2002, 14, 673-676.

resonances

induced

by

photon

18. Dorfman K. E.; Schlawin F.; Mukamel S.

9. Li, Y. Z.; Xu, B. B.; Song, P.; Ma, F. C.; Sun, M.

Nonlinear optical signals and spectroscopy

T., D–a− Π–a System: Light Harvesting,

with quantum light. Rev. Mod. Phys, 2016, 88,

Charge Transfer, and Molecular Designing. J

045008.

Phys Chem C 2017, 121, 12546-12561.

19. Mikhaylov, A.; Uudsemaa, M.; Trummal, A.;

10. Li, Y. Z.; Pullerits, T.; Zhao, M. Y.; Sun, M. T.,

Arias, E.; Moggio, I.; Ziolo, R.; Cooper, T. M.;

Theoretical characterization of the PC60BM:

Rebane, A., Spontaneous Symmetry Breaking

PDDTT model for an organic solar cell. J. Phys.

Facilitates Metal-to-Ligand Charge Transfer:

Chem. C 2011, 115, 21865-21873.

A Quantitative Two-Photon Absorption Study

11. Miller, M. J.; Wei, S. H.; Parker, I.; Cahalan, M. D., Two-photon imaging of lymphocyte motility and antigen response in intact lymph node. Science 2002, 296, 1869-1873.

12. Li, R.; Wang, X. X.; Zhou, Y.; Zong, H.; Chen, M. D.; Sun, M. T., Advances in nonlinear optical

of Ferrocenephenyleneethynylene Oligomers, J. Phys. Chem. Lett. 2018, 9, 1893−1899.

20. Frisch M. J.; Trucks G. W.; Schlegel H. B., et al. Gaussian 16, Revision A.3; Gaussian, Inc.: Wallingford CT, 2016.

21. Kohn, W.; Sham, L. J., Self-Consistent

microscopy for biophotonics, Nanophotonics,

Equations

2018, 12, 033007.

Correlation Effects. Phys. Rev. 1965, 140,

13. Kim, S.; Ohulchanskyy, T. Y.; Pudavar, H. E.; Pandey, R. K.; Prasad, P. N., Organically

Including

Exchange

and

A1133-A1138.

22. Woon D. E.; Dunning Jr T. H., Gaussian basis

modified silica nanoparticles co-encapsulating

sets

photosensitizing

calculations. III. The atoms aluminum through

drug

and

aggregation-

enhanced two-photon absorbing fluorescent

for

use

in

correlated

molecular

argon. J. Chem. Phys. 1993, 98, 1358-1371.

dye aggregates for two-photon photodynamic

23. Furche F.; Ahlrichs R. Adiabatic time-

therapy. J. Am. Chem, Soc. 2007, 129, 2669-2675.

dependent density functional methods for

14. Kanis, D. R.; Ratner, M. A.; Marks, T. J., Design

excited state properties. J. Chem. Phys, 2002,

and construction of molecular assemblies with

117, 7433-7447.

large second-order optical nonlinearities.

24. Yanai T.; Tew D. P.; Handy N. C.; A new

Quantum chemical aspects. Chem. Rev. 1994,

hybrid exchange–correlation functional using

94, 195-242.

the Coulomb-attenuating method (CAM-

15. Bredas, J. L.; Adant, C.; Tackx, P.; Persoons, A.;

B3LYP). Chem. Phys. Lett. 2004, 393, 51-57.

Pierce, B., Third-order nonlinear optical

25. Lu T.; Chen F. W. Multiwfn: a multifunctional

response in organic materials: theoretical and

wavefunction analyzer. J. Comput. Chem., 2012,

experimental aspects. Chem. Rev. 1994, 94, 243-

33, 580-592.

26. Humphrey W.; Dalke A.; Schulten K. VMD:

278.

16. Sun, M. T.; Chen, J. N.; Xu, H. X. Visualizations of transition dipoles, charge transfer, and electron-hole coherence on electronic state transitions between excited states for twophoton absorption. J. Chem. Phys, 2008, 128,

visual molecular dynamics. J. Mol. Graph. Model., 1996, 14, 33-38.

27. Martin R. L. Natural transition orbitals. J. Chem. Phys. 2003, 118, 4775-4777.

28. Mu, X. J.; Cai, K. S.; Wei, W. J.; Li, Y. E.; Wang, Z.; Wang, J. G. Dependence of UV–Visible

064106.

17. Richter, M.; Mukamel, S. Collective two-

Absorption Characteristics on the Migration



Distance and the Hyperconjugation Effect of a

centrosymmetric and asymmetric fluorene

Methine Chain. J. Phys. Chem. C, 2018, 122,

derivatives in two-photon absorption. Chem

7831-7837.

Phys, 2009, 359, 166-172.

29. Sun, M. T.; Ding, Y.; Zhao, L. L.; Ma, F. C. Photoexcitation

mechanisms

of


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489x275mm (96 x 96 DPI)


Visualizations of Photoinduced Charge Transfer and Electron-Hole

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