Understanding of the OffâOn Response Mechanism in Caged

Article pubs.acs.org/JPCB

Understanding of the Off−On Response Mechanism in Caged Fluorophores Based on Quantum and Statistical Mechanics Kosuke Usui,† Stephan Irle,†,‡ and Daisuke Yokogawa*,†,‡ †

Department of Chemistry, Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan Institute of Transformative Bio-Molecules (WPI-ITbM), Nagoya University, Nagoya 464-8602, Japan

‡

S Supporting Information *

ABSTRACT: For many years, numerous fluorescent probes have been synthesized and applied to visualize molecules and cells. The development of such probes has accelerated biological and medical investigations. As our interests have been focused on more complicated systems in recent years, the search for probes with sensitive environment off−on response becomes increasingly important. For the design of such sophisticated probes, theoretical analyses of the electronically excited state are inevitable. Especially, understanding of the nonradiative decay process is highly desirable, although this is a challenging task. In this study, we propose an approach to treat the solvent fluctuation based on the reference interaction site model. It was applied to selected bioimaging probes to understand the importance of solvent fluctuation for their off−on response. We revealed that the this switching process involves the nonradiative decay through the charge transfer state, where the solvent relaxation supported the transition between excited and charge transfer states. In addition, energetically favorable solvent relaxation paths were found due to the consideration of multiple solvent configurations. Our approach makes it possible to understand the nonradiative decay facilitated by a detailed analysis and enables the design of novel fluorescent switching probes considering the effect of solvent fluctuation.

1. INTRODUCTION In recent years, fluorescent probes for bioimaging have been intensively developed, advancing practical applications for both in vivo and in vitro measurements.1,2 Development of probes and measuring techniques3−5 has enabled us to visualize targeted ions6−8,37 and molecules,9 and has contributed to investigate various biological phenomena. Because our interests have gradually shifted toward more complicated systems, the search for environment-sensitive fluorescent probes becomes a much more exciting topic than ever before. Such novel probes have been synthesized owing to the recent progress in synthetic chemistry. Thanks to the richness of experimental investigations, fascinating chemical frameworks have been proposed, and highly useful probes were designed.2 The framework utilizing the electron transfer (ET) between donor (or fluorophore) and acceptor, called donor−acceptor (D−A) framework, is one of them, because one can systematically tune the photophysical properties as well as the scission and generation of chemical bonds by changing the combination of donor and acceptor units. In this framework, especially, the controllable off−on response based on the ET process is essential for the quantitative detection of ions, radicals, molecules, and chemical reactions in solution. For example, the D−A type probes which were designed based on the idea of photoinduced electron transfer (PeT) enabled the imaging of the activity of an enzyme in living cells.10 Typically, a spacer (S) separates D and A units © XXXX American Chemical Society

in these molecular systems. Selective measurement of zinc ions has also been reported by utilizing the phosphorescent sensor.11 Probes which were turned on by the UV irradiation have also been applied to image living cells, and their emission intensity can be tuned by changing the acceptor.12 In most probes, the off−on response of photophysical properties against environmental and chemical changes is the most important mechanism for its function as a probe (Scheme 1). For the design of these probes, quantum chemical approaches have been employed. Time-dependent density functional theory (TD-DFT) method13−17 is one of the methods to simulate electronic and molecular structures of experimentally relevant molecules in their excited states and are frequently employed for the study of electronic and molecular structures and the assignment and understanding of spectroscopic data obtained in experiments. Since most bioimaging probes work in aqueous solution, consideration of solvent effect is a fundamental necessity. So far, several solvent models, such as dielectric continuum,18 explicit solvent, and reference interaction site models (RISM), have been combined with quantum chemical methods. Due to these theoretical developments, the radiative decay process including the solvent effect Received: March 4, 2016 Revised: April 28, 2016

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DOI: 10.1021/acs.jpcb.6b02298 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

solvation free energies, ΔFST1(x) and ΔFST2(x), were defined as follows:

Scheme 1. Bioimaging Probe Based on a Caged BODIPY D−S−A Framework Which Shows Off−On Response upon UV Irradiationa

ΔF ST1(x) = ΔF(x) − ΔF0ST1 + Vx′·xe

(1)

ΔF ST2(x) = ΔF(x) − ΔF0ST2 + Vx′·(x − 1)e

(2)

where ΔF(x) and Vx′ are the solvation free energy and the electrostatic potential at the solvation structure defined by x, respectively. The solvation free energy, ΔFSTn 0 (n = 1,2), is independent from x, and calculated with the equilibrium solvent structure at the n-th state (STn). The derivation and more detailed explanation for these two equations can be found elsewhere.21,22 Although such definitions were only designed for the ET process, they can be applied to general cases where the electron density distribution changes between ST1 and ST2. In the RISM-SCF-SEDD method, which is the solvent model used in the present study, the electron density distribution of a solute ρSTn is fitted with the linear combination of auxiliary basis sets (ABS), f i.

a

See ref 12. The bond between spacer and acceptor cleaves by the irradiation, and strong emission is observed.

becomes feasible, and thus, such quantum calculations have become a powerful tool to design bioimaging probes. Although the quantum chemical approach based on TDDFT becomes popular, the theoretical design of off−on switching probes is still challenging. This is because the energy transfer process must be investigated in detail, and understanding of the nonradiative decay process is highly important. There are few theoretical investigations toward the nonradiative decay of bioimaging molecules, although only relatively small molecules have been studied with accurate quantum chemical methods.19 Particularly, solvent fluctuation plays a vitally important role in the electronic state transition during the decay process of the excited state. Since the introduction of Marcus theory, the solvent fluctuation has been examined by many researchers. In most cases, only one solvent coordinate is adopted, while other reaction coordinates like nuclear motion are implicitly included in this one coordinate.20 To overcome the difficulties in the theoretical treatment of the nonradiative decay process, we here propose a new approach to study the effect of solvent fluctuation derived from multiple solvent coordinates in the RISM framework. Our approach was applied to existing bioimaging molecules to explain the nonradiative decay process including solvent configurations. Focusing on the treatment of solvent fluctuation, Hirata and co-workers proposed a method to handle nonequilibrium solvent structures in RISM, and they obtained the free energy profile for the ET process in a system containing two model solute species A and B.21,22 Following their approach, a detailed investigation of solvation and electronic structure in water was carried out for the exchange ET reaction between ruthenium complexes by Sato et al.23 In this work, we expand the previous ideas to consider more than two electronic states and solvent configurations. Using our approach, the nonradiative decay induced by solvent fluctuation can be understood.

ρSTn =

∑ dSTn,ifi

(3)

i

In terms of the set of expansion coefficients, dSTn, free energy equations of the two-state model have more general forms by replacing e with the electron density distribution change, dST2 − dST1. ΔF ST1(x) = ΔF(x) − ΔF0ST1 + Vx ·x(dST 2 − dST1)

(4)

ΔF ST 2(x) = ΔF(x) − ΔF0ST 2 + Vx ·(x − 1)(dST 2 − dST1) (5)

where Vx,i is the electrostatic potential on the i-th ABS. These two equations are more conveniently expressed by eq 6. ΔF STn(x) = ΔF(x) − ΔF0STn + Vx{(1 − x)dST1 + x dST 2 − dSTn} (n = 1, 2) (6)

Both electron density distributions, which correspond to dST1 and dST2, are mixed by x, and the nonequilibrium solvation structure is generated on the basis of this “hypothetical” electron density distribution, (1 − x)dST1 + xdST2. In other words, we assume a process where the electron density distribution changes continuously from ST1 to ST2 with increasing x; x = 0 (x = 1) gives the solvent structure at ST1 (ST2). In the present study, we expanded these formulas at first to a three-state model. In this three-state model, the hypothetical electron density distribution is defined as (1 − x − y)dST1 + xdST2 + ydST3. In addition to x, one more parameter, y, is introduced to include the solvent fluctuation in direction of the third state (ST3). We consider the cycle shown in Figure 1 to calculate the free energy, ΔFSTn (n = 1,2,3). After some mathematical derivation (see Appendix), ΔFSTn is given as

2. METHODS Before the description of our model, we briefly explain the existing two-state model.21,22 In previous studies, the electron transfer (ET) process between two particles A and B was discussed. In going from the first state (ST1) to the second state (ST2), one electron, e, transfers from one particle to the other. To calculate the solvation free energies, a parameter determining the solvation structure, x, was introduced, and

ΔF STn(x , y) = ΔF(x , y) − ΔF0STn + Vx , y{(1 − x − y)dST1 + x dST 2 + ydST 3 − dSTn} (n = 1, 2, 3)

(7)

The solvation free energy, ΔF, and electronic potential, Vx,y, are functions of x, and y, while ΔFSTn is the solvation free energy 0 B


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The Journal of Physical Chemistry B y=

simulated with the equilibrium solvent structure at the n-th state. When y = 0, this three-state model smoothly reduces to the familiar two-state model: (8)

Vx ,0 = Vx

(9)

(13)

In the application of our three-state model, we separate the fluorescent off−on switching probe into donor and acceptor molecules, using hydrogen termination. The advantage of this separation lies in avoiding the expensive quantum chemical calculation for the whole system. One of the frequently used frameworks to design bioimaging probes is the donor and acceptor pair connected by a spacer.12 For quantum chemical calculations, however, such system tends to be too large to perform calculations for the accurate prediction of molecular properties, and calculation methods and a level of theory may be limited due to its high computational cost. Besides, either the donor or the acceptor unit is often changed to tune and obtain desired functions in experimental studies. In this work, the electronic and solvent structures are determined for each isolated unit, and the distance between donor and acceptor is reduced. Two complexes containing donor (BODIPY fluorophore) and acceptor parts, 1-H and 2-H, were chosen as models as shown in Figure 3.12 Although they have the same

Figure 1. Schematic figure of the cycle to calculate solvation free energies at the nonequilibrium solvent structure, which is given as ΔFSTn. The solute with the hypothetical charge distribution is illustrated by dotted circles.

ΔF STn(x , 0) = ΔF STn(x)

t1 3 2 t1 + t 2 + t3

A general multistate (M-dimensional) model can be derived in the same manner. ΔF STn(x 2 , x3 , ..., xM ) = ΔF(x 2 , x3 , ..., xM ) − ΔF0STn M

+ Vx2 , x3 ,..., xM {(1 −

M

∑ xi)dST1 + ∑ xjdSTj − dSTn} i=2

(n = 1, 2, ... M )

Figure 3. Calculation models.

j=2

(10)

donor part [H], their acceptor parts are different from each other. By changing the acceptor from 1 to 2, the complex becomes an off−on response probe, and the turn-on mechanism and fluorescence difference between 1-H and 2-H are explained in Section 4. Before we close this section, it is useful to discuss the difference between our multistate model and the conventional two-state model. By employing eq 10, we can consider the multistate model. Herein, three surfaces, ground state (GS), excited state (ES), and charge transfer state (CT), are shown in Figure 4. The dashed line represents the solvent relaxation process upon excitation from GS via ES to the CT state along the t3 and t2 axes, which can be obtained by a two-state model

In the present study, we only apply the three-state model, and, for the purpose of visualization, a triangle with three independent coordinates is used as shown in Figure 2. The t1,

Figure 2. ”Triangular” coordinates defined for the three-state model; schematic figures of solvent orientation at points corresponding to the triangle vertices and midpoints of the connecting edges.

t2, and t3 denote the edges of the triangle graph and represent weighting factors to define solvent configurations. For instance, at point α in Figure 2, t1 and t2 take a value of 0.5, and the solvent configuration at this point is defined as a 50:50 mix of those of points A and B. The following relation holds true for all internal points of the triangle: t1 + t 2 + t3 = 1 (11)

Figure 4. Schematic figures of (a) solvation free energy surfaces and (b) contour plot for the plausible decay path. Paths shown by solid and dashed lines are allowed in three- and two-state model, respectively. The energy range represented in (b) is from E1 to E2 shown in (a).

They can be transformed to the Cartesian coordinates, x and y. 1 t1 + 2t3 x= 2 t1 + t 2 + t3 (12) C


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The Journal of Physical Chemistry B approach. However, we can see that it is possible to reach CT from ES directly without barrier or passing solvation state B via pathways inside the triangle, indicating the necessity of our three-state approach. Indeed, one of them is shown with curved solid arrows in Figure 4b, and a relaxation along it does not require any activation energy. As indicated above, the number of states to be considered corresponds to the degrees of freedom for solvent fluctuation, so that the reasonable selection from numberless solvent structures is possible in our approach.

3. COMPUTATIONAL DETAILS The geometries of the separated and hydrogen-terminated D and A units as well as of those of the combined D−S−A systems were optimized on the ground state surface using B3LYP24 with the 6-31+G(d) basis set25 in vacuo (see Figures S1 and S2). Frequency calculations proved that D and A units were minima on their respective ground state potential energy surfaces. Geometry optimizations and frequency calculations were performed with the Gaussian09 package.26 Gibbs free energies were calculated evaluated by multireference MP2 (MRMP2) using the GAMESS-US package.2728−31 Basis sets for MRMP2 calculations were selected as follows: cc-pVDZ for donor and methyl groups, and aug-cc-pVDZ for the remaining atoms.32 Active orbitals are shown in Figure S3. To include the solvent effect, the RISM-SCF-SEDD method33,34 as implemented in a modified version of the GAMESS-US package was applied with the closure approximation of Kovalenko and Hirata.35 Since bioimaging probes are mostly used in water, we chose water as solvent. It should be noted that the spacer (S) is arranged perpendicular to the donor, and the electron delocalization from donor to acceptor is therefore not possible. We further note that the geometry of the fluorophore unit does not significantly change between GS, ES, and CT states, as discussed below, which justifies the use of optimized ground state geometries. To compare the charge distribution between GS, ES, and CT states, natural population analysis was performed.36

Figure 5. (a) Superposition of GS, ES, and CT geometries of BODIPY. (b) Natural charges for GS. (c) Difference of natural charges between ES-GS and CT-GS. (d) Subtracted RDFs defined by eq 14.

functions (RDF) were analyzed. RMSD values of RDFs between different states can be regarded as an index for similarity of their solvent structures. The maximum RMSD value calculated from RDFs of GS and CT is 0.040 Å, while that of GS and ES is less than half (0.017 Å). As expected from comparison of geometrical and electronic structures in the gas phase, the solvation structure does not significantly change after excitation. To see the solvent configurational difference between GS and CT, a difference RDF, ΔgCT−GS(r), is defined. ΔgCT − GS(r ) = gCT(r ) − gGS(r )

(14)

gCT(r) and gGS(r) are RISM-SCF-SEDD RDFs at CT and GS, respectively. In Figure 5d, two sets of ΔgCT−GS(r) for the CA site are shown. The population of surrounding Hw around 2 Å decreases from GS to CT. On the contrary, that of Ow around 3 Å increases. By removing an electron from BODIPY, electrostatic repulsion between positively charged Hw and CA is enhanced, but this contribution seems to be compensated by the approach of the Ow site. This Hw leaving and Ow approaching behavior at CT can be observed for all other solute sites, and RDFs and subtracted RDFs are summarized in the Figure S4. In experiments, strong emission was observed from compound 1-H and [H], but 2-H showed weak emission. Compound 2-H is turned on by UV irradiation. We now turn our attention to the simulated diabatic free energy surfaces of the compound 2-H (Figure 6a, center). Two seams of crossing between ES and CT surfaces and between CT and GS surfaces are observed, and this result suggests the existence of nonradiative decay through the sequential state transition (ES → CT → GS) promoted by solvent fluctuation. If an acceptor is removed by UV irradiation (Figure 6a, right), ET from BODIPY to an acceptor is prohibited. This means that the CT surface disappears, and nonradiative decay through the CT state becomes impossible. In other words, the radiative decay or emission is enhanced. Therefore, we conclude that compound [H] shows emission, and existence of an acceptor controls the turn-on response of this probe. It should be noted that the

4. RESULTS AND DISCUSSION To understand the nature of the BODIPY framework, [H], we first discuss the results in the gas phase. The superposition of optimized geometries of this fluorophore at each state is shown in Figure 5a, and geometrical parameters are summarized in Figure S2. All geometries overlap greatly and are similar to a deviation of only up to 0.03 Å and 3.0°. When the GS geometry is considered as a reference, the root-mean-square deviation (RMSD) values of ES and CT are 0.0245 and 0.0234 Å, respectively. These small differences indicate that BODIPY has a rigid geometry against ionization and excitation. We conclude that GS geometry can be applied to subsequent calculations. Natural population analyses were performed to compare electronic structures of GS, ES, and CT (Figure 5b,c). In GS, charges are efficiently delocalized on the π framework, and only the boron atom is significantly charged by +1.32. Interestingly, this charge distribution does not change so much by excitation and even by ionization. The maximum change of charges is 0.14, and the electron delocalization suppresses the drastic change of electronic structure. These characteristic properties can be considered to reduce the change in solvent structure. Subsequently, the solution state at equilibrium was investigated to understand solute−solvent interaction and solvent reorganization. For this purpose, the radial distribution D


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Figure 6. (a) Computed PESs for compound 1-H (left), 2-H (center) and [H] (right), and (b) contour maps of the ES surfaces of 1-H and 2-H. Bold solid lines (red) represent the intersections between ES and CT surfaces.

The fluorescence intensity was regulated by acceptors in experiments, and the order of reduction potentials of them agreed with that of quantum yields. For prediction of the electron accepting capacity, we define the theoretical molecular electron affinities, EA, as

similarity of GS and ES surfaces is consistent with the small difference between their electronic structures. Now, let us discuss the fluorescent difference between 1-H and 2-H. In the case of compound 1-H, the CT surface is energetically higher than the ES surface, and the decay through the CT surface seems to be unfavorable. Activation free energies are estimated from contour maps of ES shown in Figure 6b. The activation barrier of 1-H is about 23 kcal/mol, but that of 2-H is less than 1 kcal/mol. Judging from these activation energies, the nonradiative decay process is likely to be feasible only for compound 2-H, which is consistent with the experimental results. As the quantum yield of compound 2-H has a nonzero value in experiments, the geometrical fluctuation at the excited state may play the important role for the nonradiative decay process. The importance of such effect has been already pointed out in our previous report.37 Although the geometrical fluctuation is ignored in the present study, we are considering the new methodology to treat both solvent and geometrical fluctuations at the same time. We also discuss the importance of the three-state model. If the two-state model is applied to compound 2-H, the solvent configurations are limited to only the sides of the triangle. For instance, the path through the contour line of zero is not permitted in Figure 6b, while such energetically favorable paths are allowed in the three-state model. Therefore, the multistate model is crucially important to understand the mechanism of the nonradiative decay process and free energy profiles in detail.

EA = [Eneutral + ZPVEneutral] − [Eanion + ZPVEanion] (15)

where Eneutral/anion and ZPVEneutral/anion are electronic energies and harmonic zero-point vibrational energies for the corresponding electronic structure, respectively. Simulated electron affinities and energies of the lowest unoccupied molecular orbitals (LUMO) are shown in Table 1. Compound 2 has the higher electron affinity than 1, and the ELUMO of 2 is lower than Table 1. Experimental Values of Quantum Yield, Φfl, and Reduction Potential, Ered, and Simulated Electron Affinity, EA (in eV), and Energy Level of LUMO, ELUMO (in eV) exptl

calcd

compound

Φfla

Ered

1 2

0.78 0.24

< −2.00 −1.28

b

c

EA

ELUMOc

−1.25 1.10

−0.39 (2.31) −2.60 (1.09)

a

Measured values for complexes 1-H and 2-H in acetonitrile. Determined in acetonitrile by cyclic voltammetry. The corresponding phenol ether was used as a reference. cSimulated at B3LYP/631+G(d), and the results at HF/6-31+G(d) are shown in parentheses. b

E


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approach, the design of off−on fluorescent probes considering the solvent fluctuation becomes possible, and we pointed out that the solvent fluctuation between multiple states is important to understand the solvent-mediated nonradiative decay process. In this study, we did not consider the geometry change in the nonradiative decay process because we want to focus on the nonradiative decay induced by solvent fluctuation. However, there are many examples where the geometry changes play a key role in the nonradiative decay. In our previous study, we pointed out the importance of geometry change in the nonradiative decay.5 On the basis of the present and the previous studies, we have studied the nonradiative decay induced by geometry change and solvent fluctuation. In the study, we have determined the important modes in the geometry change and solvent fluctuation, and we have constructed the energy surface along them.

that of 1. These results indicate that the electron accepting capacity of 2 is stronger than that of 1, and this trend shows good agreement with the order of experimental reductive potentials. The high electron accepting capacity lowers the CT surface. Therefore, compound 2-H has an energetically lower CT surface than 1-H, and thus the nonradiative decay process of 2-H (the “off” state) is enhanced. The fluorescence difference between 1-H and 2-H is thus explained. Upon dissociation of the acceptor unit after UV irradiation, compound 2-H becomes fluorescent as shown in Figure 6c, corresponding to the “on” state. Because BODIPY is a good acceptor, the charge transfer from the BODIPY to acceptor seems to be counterintuitive. When we computed absorption energy of 2-H in water using TD-CAM-B3LYP, the characters of the first and the second excited states are π(BODIPY) − π*(BODIPY) and π(BODIPY) − π*(acceptor 2), as shown in Table S1. Because of the low lying π* orbital of BODIPY, π−π* excitation in BODIPY is favorable. However, the CT state (the second excited state in Table S1) is greatly stabilized by solvation after solvent relaxation, which is expressed with ΔF(x,y) − ΔFSTn 0 in eq 7. The stability computed with CAM-B3LYP with RISMSCF-SEDD is −55.2 kcal/mol. This large stabilization by solvent induces the charge transfer from the BODIPY to the acceptor. Although the energy barrier from ES to CT is significantly small in the case of 2-H, as shown in Figure 6a, the quantum yield of 2-H is not zero. To discuss the quantum yield, we also have to consider the internal conversion rate between the two states, which are greatly affected by the orbital overlap between the donor and acceptor. In Figure S2, we showed the geometries of 2-H employed in this study. Because there is a spacer, the donor and the acceptor cannot maximize the overlap. We think that the insufficient overlap decreases the internal conversion rate, and there is a small probability to decay through radiative pathway, which gives nonzero quantum yield even in 2-H.

■

APPENDIX ΔFSTn is given by eq A.1. ΔF STn(x , y) = −ΔF1 − ΔF2 + ΔF3 + ΔF4 + ΔF5 + ΔF6 (A.1)

Each term in the right side is the free energy change of the corresponding process in Figure 1. ΔF1 = (⟨ΨSDΨSA|Ĥ |ΨSDΨSA⟩ + ΔF0STn) − (⟨ΨSDΨSA|Ĥ |ΨSDΨSA⟩ + Δμ2 )

(A.2)

ΔF2 = (⟨ΨSDΨSA|Ĥ |ΨSDΨSA⟩ + Δμ2 ) − (⟨ΨGDΨGA|Ĥ |ΨGDΨGA⟩) (A.3)

ΔF3 =

(⟨ΦGDΦGA |Ĥ |ΦGDΦGA ⟩)

−

(⟨ΨGDΨGA|Ĥ |ΨGDΨGA⟩)

(A.4)

ΔF4 = (⟨ΦSDΦSA |Ĥ |ΦSDΦSA ⟩ + Δμ4 ) − (⟨ΦGDΦGA |Ĥ |ΦGDΦGA ⟩) (A.5)

ΔF5 = (⟨ΦSDΦSA |Ĥ |ΦSDΦSA ⟩ + ΔF(x , y))

5. CONCLUSIONS We proposed a new approach which can treat multiple solvent configurations generated by interpolation between different solvent structures. This new approach gives us important insight into the nonradiative decay process associated with the solvent reorganization, because energetically favorable pathways along the solvent fluctuation coordinates can be found. The D− S−A type molecular probes were analyzed with our approach, and their off−on switching mechanism was investigated. The compound 2-H shows the turn-on response by the UV irradiation, but compound 1-H does not change its fluorescent property upon the UV irradiation (it is always fluorescent). Obtained free energy surfaces revealed that 1-H has the relatively high CT surface, and the large energy barrier exists to reach the intersection of CT and ES. On the other hand, surfaces of 2-H indicated the presence of decay pathways through the CT surface after vertical excitation, and its energy barrier is much smaller than that of 1-H in the ”off” state. After dissociation of the acceptor unit, the fluorophore emits light again, and hence, the mechanism of the turn-on response was elucidated. We also confirmed that our treatment of multiple solvent configurations is important to understand the solvent relaxation and fluctuation. Electronic structures of acceptors of 1-H and 2-H were compared, and the height of the CT surface can be determined by the energy level of their LUMOs. By our

− (⟨ΦSDΦSA |Ĥ |ΦSDΦSA ⟩ + Δμ4 )

(A.6)

ΔF6 = (⟨ΨSDΨSA|Ĥ |ΨSDΨSA⟩ + ΔFα(x , y)) − (⟨ΦSDΦSA |Ĥ |ΦSDΦSA ⟩ + ΔF(x , y))

(A.7)

Ĥ is the Hamiltonian in vacuum. Ψ and Φ are wave functions at real and hypothetical states, respectively, and they are characterized by four subscripts, G (Gas phase), S (Solution), D (Donor), and A (Acceptor). Δμ2 and Δμ4 are equilibrium solvation free energies. ΔFSTn 0 , ΔF(x,y) and ΔFα are composed as follows: r ΔF0STn = ΔFcav + Δμ2

(A.8)

h ΔF(x , y) = ΔFcav (x , y) + Δμ4

(A.9)

ΔFα(x , y) = ΔFeh → r(x , y) + ΔF(x , y)

(A.10)

The energy to combine two cavities which are obtained from separated donor and acceptor pair is defined by ΔFrcav for the real and ΔFhcav for the hypothetical state. ΔFhe →r corresponds to the energy to move the electron density distribution against the electric field from the solvent, and this process returns the electron density distribution from the hypothetical to the real state: F


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ΔFeh → r(x , y) = Vx , y{(1 − x − y)dST1 + x dST 2 + ydST 3 − dSTn} (N = 1, 2, 3)

(A.11)

where Vx,y is the electric potential from the solvent at the hypothetical state. Equation A.11 is obviously a product of electric potential and varied electron density distribution from that of STn.

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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.jpcb.6b02298. Figures of active orbitals selected for CASSCF calculations, geometrical parameters of optimized geometries at GS, ES, CT, and radial distribution functions between atomic sites of solvent and solute (BODIPY), and table of TD-DFT calculation (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Research Center for Computational Science, Okazaki, Japan, for providing computational resources. This work was supported by Program for Leading Graduate Schools “Integrative Graduate Education and Research in Green Natural Sciences”, MEXT, Japan, and by a CREST grant from JST. D.Y. thanks the support by the Grant-in-Aid for Young Scientists B (No. 24750015) and Scientific Research (C) (No. 15K05385). S.I. was partially supported by a CREST grant from JST.

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REFERENCES

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