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Quantum Electronic Structure

Electronically Excited Solute Described by RISM Approach Coupled with Multi-Reference Perturbation Theory: Vertical Excitation Energies of Bio-Imaging Probes Ryosuke Yale Shimizu, Takeshi Yanai, Yuki Kurashige, and Daisuke Yokogawa J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00599 • Publication Date (Web): 09 Oct 2018 Downloaded from http://pubs.acs.org on October 12, 2018

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Journal of Chemical Theory and Computation

Electronically Excited Solute Described by RISM Approach Coupled with Multi-Reference Perturbation Theory: Vertical Excitation Energies of Bio-Imaging Probes Ryosuke Y. Shimizu,† Takeshi Yanai,†,‡,¶ Yuki Kurashige,§ and Daisuke Yokogawa∗,†,‡ †Department of Chemistry, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan ‡Institute of Transformative Bio-Molecules (WPI-ITbM), Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan ¶Japan Science and Technology Agency, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan §Department of Chemistry, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan E-mail: [email protected] Phone: +81 52 789 2851

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Abstract For theoretically studying molecules with fluorescence in the near-infrared region, high-accuracy determination of state energy level is required for meaningful analyses since the spectra of interest are of very narrow energy range. In particular, these molecules are in many cases handled in solution; therefore, consideration of the solvation effect is essential upon calculation together with the electronic structure of the excited state. Earlier studies showed that they cannot be described with conventional methods such as PCM-TD-DFT, yielding results far from experimental data. Here, we have developed a new method by combining a solvation theory based on statistical mechanics (RISM) and a multi-reference perturbation theory (CASPT2) with the extension of the density matrix renormalization group (DMRG) reference states for calculating the photochemical properties of near-infrared molecules and have obtained higher-accuracy prediction.

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Introduction Development of new fluorescent molecules have enabled the visualization of various bio-

logical events and have supported advances in the field of bio-imaging. 1–3 Ideal fluorescence for these probes have been within the visible to near-infrared (NIR) region since biological tissue can be efficiently penetrated, allowing deep tissue imaging in many higher organisms. 4 While conventional NIR molecules have had fluorescence in the so-called “first NIR window,” approximately 700-1000 nm, recently, molecules with fluorescence in the “second NIR window,” approximately 1000-1700 nm, have captured attention; these molecules are expected to further improve the performance of bio-imaging, yet are underexplored. 5,6 Considering the difficulty of measuring transition energies experimentally, theoretical calculations hold a great advantage in accelerating the study of these molecules by providing complementary or direct description of the excitation. Nonetheless, since these molecules possess photochemical properties in a narrow energy range, highly accurate calculations are required for 2


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meaningful analyses. Accomplishing this task requires achieving high accuracy in the calculation of both the electronic structure of the excited solute and the solvation effect. One of the most widely used methods for calculating excited states is the time-dependent density functional theory (TD-DFT). 7,8 Although TD-DFT can evaluate large systems at an affordable cost, it often has uncontrollable difficulties in correctly assessing the electronic structure of long π-conjugated systems, 9–11 a general characteristic of the fluorescent molecules in focus. In terms of solvation, the polarizable continuum model (PCM) 12,13 is widely employed. Again, its computational cost is reasonable; however, this method is known to be unable to illustrate local solvation structures such as hydrogen bonding well. 14 Although the combination of these methods, PCM-TD-DFT, 15 is widely employed for calculating excited states in solution, it is reportedly incapable of providing the desired accuracy of the aforementioned type of molecules. 16,17 Hence, there has been a marked challenge in developing a method to rectify the limitations of the conventional approaches and acquire higher accuracy with reasonable computational cost. In this article, we present the development of a new method by combining the complete active space second-order perturbation theory (CASPT2) 18,19 and the reference interaction site model (RISM) 20 for the models of excitation and solvation, respectively. The CASPT2 method is a multi-reference perturbation theory, which is capable of accurately describing correlated many-electron wave functions at the second-order perturbation level using the multi-configurational states determined by a complete active space self-consistent field method (CASSCF) 21 calculation as references. Limitations in choosing the active space for CAS calculations have recently been greatly eased with the use of the density matrix renormalization group (DMRG) in DMRG-CASSCF 22–24 and DMRG-CASPT2. 25,26 The RISM method is a solvation theory of a statistical mechanical approach, in which the solvation structure is represented by correlation functions. Since the correlation functions are obtained by solving the RISM equations analytically, the calculation is efficient compared to methods

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such as QM/MM-MD which demands large computational cost for sampling based on MD simulations. Furthermore, information of the structure of the solvent molecule is preserved in RISM, facilitating the description of local interactions such as hydrogen bonding. 27,28 These advantages have prompted the combination of RISM with various QM approaches (RISM-SCF 29–33 and RISM-SCF-SEDD 34–36 ).

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Method In combining RISM and CASPT2, we first defined the Helmholtz free energy of the solute

in solution calculated with the CASPT2 method as, 31,32

ˆ |Ψ⟩ + ∆µ + E2 ACASPT2 ≡ ⟨Ψ| H = ACASSCF + E2 ,

(1)

ˆ is the Hamiltonian constructed in the gas phase, Ψ is the CAS reference where the operator H function, ∆µ is the excess chemical potential, and E2 is the second-order energy. Singer and Chandler’s hypernetted-chain (HNC) closure relation 37 was employed for the excess chemical potential term given as follows: [ ] ∫ 1 2 1 1∑ ρs dr cαs (r) − hαs (r) + hαs (r)cαs (r) . ∆µ = − β α,s 2 2

(2)

Greek subscripts refer to the interaction sites of the solute and the Roman subscripts refer to the sites of the solvent, ρs is the number density of the solvent, and β is 1/kB T , where kB is the Boltzmann constant and T is temperature. The functions hαs and cαs are the total and direct correlation functions, respectively, and are obtained by solving the RISM equations. 20 The Helmholtz free energy ACASSCF defined in eq 1 is regarded as a functional of the correlation functions hαs , cαs , and tαs = hαs − cαs , and the CAS reference function Ψ.

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This free energy is minimized by defining the following Lagrangian: 31

L = A [c, h, t, Ψ] − E (⟨Ψ|Ψ⟩ − 1) .

(3)

Taking the variation of the Lagrangian with respect to the functions yields, [ ] ∫ 1∑ ∂∆µ ∂∆µ ∂∆µ ρs dr δL = − δtαs (r) + δhαs (r) + δcαs (r) β α,s ∂tαs ∂hαs ∂cαs ˆ + Vt ∂p − E |Ψ⟩ = 0, + ⟨δΨ| H ∂ ⟨Ψ| where Vi = −

∑ s

and

∫ ∫ qs ρs

hαs (r)fi (r′ ) drdr′ ′ |r − r |

(4)

(i ∈ α),

∑ ∂p [Ξ + (m − 1)Γ]−1 R′pq a†p aq . = ∂ ⟨Ψ| pq

(5)

(6)

The operators a†p and aq are creation and annihilation operators respectively, f is the auxiliary basis set (ABS), and the remaining matrices are as defined in previous work. 17,38 From the Ornstein-Zernike equation, HNC closure relation, and the RISM equation, the terms in the square brackets in eq 4 become 0. Hence, eq 4 simplifies to,

ˆ solv − E |Ψ⟩ = 0, ⟨δΨ| H

(7)

ˆ solv as, where we have defined the solvated Hamiltonian H ˆ solv = H ˆ + Vt ∂p . H ∂ ⟨Ψ|

5


(8)


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RISM calculations are performed with the electron density distribution on each atomic site. The distributions are obtained from the one-particle reduced density matrix defined as, γpq = ⟨Ψ| a†p aq |Ψ⟩ .

(9)

For molecules with large active spaces which require the DMRG, the reference wave function is simply replaced by the active-space DMRG wave function |ΨDMRG ⟩. 25,26 The absorption energy ω a was then calculated as the difference of the excited state energy [E]

[G]

ECASPT2 and the free energy at the ground state ACASPT2 : 17 [E]

[G]

ω a = ECASPT2 − ACASPT2 ,

(10)

where [G] ˆ [G] |Ψ⟩ + E2[G] + ∆µ[G] , ACASPT2 = ⟨Ψ| H

(11)

[E] ˆ [E] |Ψ⟩ + E2[E] + ∆µ[G] + (V[G] )t (p[E] − p[G] ). ECASPT2 = ⟨Ψ| H

(12) [G]

The superscripts G and E represent the ground and excited states, respectively. Vi

is

the electrostatic potential on the i-th solute site induced by the ground state solvation structure at the ground state, and p[G] and p[E] are the matrices including the expansion coefficients of the ABS for approximating the electron density of the ground and excited states, respectively. 17,38 For the absorption process, it is assumed that the relaxation of the solvation structure does not occur at the excited state following the Franck-Condon principle. 17 This can be observed in eq 12, where the sum of the first and second term is the energy of the solute at the excited state in the potential induced by the ground state solvation structure. Although it is well known that consideration of vibronic effects is important in computing absorption maxima, 39,40 we plan to implement these effects into our method in the future, and limit the present study to computing only vertical excitation energies.

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The sketch of our implementation of the method is depicted in Scheme 1. For the ground state calculation, first Ψ is determined from ground state-specific (DMRG-)CASCI and orbital optimization steps. Then, the one-particle reduced density matrix γpq , formed from Ψ, is passed to the RISM code for computing the electrostatic potential Vi on each solute site induced by the solvation structure. These electrostatic potentials are used to construct the ˆ solv for the successive iteration. Once Ψ converges, the obtained solsolvated Hamiltonian H vated Hamiltonian is used in ground state-specific (DMRG-)CASPT2 to apply perturbation in solution. In computing the absorption energy, the potentials obtained from the ground state calculation, V[G] , are used to construct the solvated Hamiltonian for determining Ψ at the excited state; this corresponds to the calculation of the excited state with the solvation structure fixed at the ground state. Finally, perturbation is applied in solution for the excited state via state-specific (DMRG-)CASPT2 with the obtained solvated Hamiltonian. Since it is unnatural to include solvation effects on state-averaged electron density, all calculations are conducted state-specifically. START

START V[G]

,V

(DMRG-)CASCI

(DMRG-)CASCI

Orbital Optimization


RISM Calculation

Update V ---

converged?

converged?

ˆ solv H

ˆ solv H

(DMRG-)CASPT2

(DMRG-)CASPT2

END

END

Scheme 1: Flowchart of the RISM-CASPT2 method for the ground (left) and excited (right) state in calculating absorption energies.

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Computational Details The Lennard-Jones parameters used for the RISM calculations were taken from OPLS-

AA 41 and are provided in the supporting information together with solvent parameters. The solvent data required for the RISM calculations were computed using ex-RISM. 42,43 The basis sets employed were aug-cc-pVDZ 44 for nitrogen and oxygen, and cc-pVDZ 45 for the other atoms. The optimized geometries used for calculations involving RISM were computed using the CAM-B3LYP level of theory with symmetry employed at 300 K in respective solutions with RISM using the gamess program package 46,47 in which our method has been implemented in the past. The same program package was used for calculating absorption energies with RISM-TD-DFT. Calculations involving PCM employed optimized geometries computed with the CAM-B3LYP level of theory using PCM in respective solutions. The PCM-TD-DFT calculations were conducted with the gaussian program package 48 and PCM-MRMP2 calculations were conducted with the gamess program package. As written in the previous section, the reference wave functions for RISM-CASPT2 were determined with DMRG-CASSCF approach when using large active spaces. In this study, the active-space DMRG calculations were performed with 256 spin-adapted renormalized basis states. This size of renormalized basis produces near convegent DMRG results when accounting for quasi-one dimensional π conjugations, such as the systems considered in this study. The fourth-order cumulants were neglected in an approximate manner in the solution of the CASPT2 equation, for which the reduced density matrices of up to the third order were calculated based on the DMRG algorithm. 26 All calculations involved in the RISM-CASPT2 method and its extension with the active-space DMRG reference states were conducted with the orz program package 49 in which we have newly implemented our method into. Symmetry was not employed for all absorption energy calculations.

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Results and Discussion As an assessment of the newly developed method, we have calculated the absorption

energies of the molecules given in Scheme 2. To evaluate the reliability of the solvation theory of choice and the accuracy of the electronic structure, we have selected MQZ for a molecule with high solvatochromism in absorption, 4AP as a molecule with absorption in the near UV region, and CY3 and ICG were chosen for long π-conjugated systems, related to bio-imaging probes. 3,5 All π and π ∗ orbitals were included in the active space. The size of the active space was (12e,11o) for MQZ and (14e,12o) for 4AP. For CY3 and ICG, the active spaces were set to (20e,19o) and (32e,31o), respectively; their configuration space exceeds the capability of the conventional CASSCF/CASPT2 method, but were allowed to be handled by its DMRG extension. The localized form of 31 active orbitals used as DMRG sites for ICG is displayed in the supporting information. All experimental absorption energies given refer to the absorption maxima of the molecules which are results under the presence of molecular vibrations; however, these effects are neglected in our calculations. Note that when using DMRG reference wave functions, RISM-CASPT2 calculations are denoted as RISM-DMRG-CASPT2.

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O -O

NH N+

H2N O

N-Methyl-6-oxyquinolone (MQZ)

N+

4-Aminophthalimide (4AP)

N

Modeled indocarbocyanine (CY3)

N+

N

Modeled indocyanine green (ICG)

Scheme 2: Target molecules.

Calculated absorption energies of MQZ in water (WAT), methanol (MET), acetonitrile (ACN), and dimethyl sulfoxide (DMSO) are summarized in Table 1. For comparison, the results of PCM-TD-DFT calculations using the CAM-B3LYP 50 and PBE0 51 functionals are included. Comparing the TD-DFT level results with and without PCM, the solvation effect in the absorption energies was shown to be insensitive to the solvent polarity, constantly shifting gas-phase energies by an energy of approximately 1.0 eV and 0.9 eV for CAM-B3LYP and PBE0, respectively. Hence, the PCM-TD-DFT predictions showed no solvatochromism in absorption and were generally quantitatively away from the experimental absorption energies. The effect of the solvation model can be checked by using RISM in place of PCM at the TD-DFT level; this level of theory is referred to as RISM-TD-DFT. 17 We confirmed that the solvation energies obtained by RISM-TD-DFT vary depending on the solvents, showing an appreciable solvatochromic behavior in absorption. The use of RISM greatly improved the absorption energies with an error of less than 0.3 eV relative to the experimental data. 10


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The RISM-TD-DFT results with the PBE0 functional especially showed good agreement with the experimental data; however, this is not always the case, and we will address on this later. In terms of the results from calculations employing multi-reference perturbation theories, using PCM as the solvation model again showed constant shifting of the gas-phase energies, not reproducing the solvatochromism in absorption. On the other hand, the RISM-CASPT2 method successfully illustrated solvatochromism in the absorption energies of MQZ, which were accurately predicted with an error of less than 0.2 eV relative to the experimental data. It was thus shown that the CASPT2 treatment generally further reduced the error in prediction, compared to those of the RISM-based TD-DFT results. Table 1: Calculated absorption energies of MQZ in various solutions (eV)

PCM-TD-DFT CAM-B3LYP PBE0 RISM-TD-DFT CAM-B3LYP PBE0 PCM-MRMP2 RISM-CASPT2 Exp.a a

Gas

ACN DMSO MET WAT

2.30 2.15

3.34 3.02

3.33 3.01

3.34 3.02

3.38 3.06

2.30 2.15 2.14 2.04 –

2.68 2.44 2.44 2.38 2.45

2.70 2.46 2.44 2.38 2.44

2.94 2.66 2.43 2.73 2.88

3.23 2.91 2.45 3.21 3.03

References 52–54.

Indeed, vibronic effects hold an important role in predicting absorption maxima. 39,40 However, these effects can be ignored to some extent for molecules with absorption in the near UV region where the vibronic contribution becomes relatively smaller when compared to the total absorption energy. Calculating the absorption energies of 4AP with the RISMCASPT2 method in the same solvents as MQZ showed similar accuracy with the error within approximately 0.2 eV relative to the experimental results (Table 2). These results provided further evidence of the accuracy of the method and proved that agreement with

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the experimental results was not due to compensation of errors. Table 2: Calculated absorption energies of 4AP with RISM-CASPT2 in various solutions (eV)

RISM-CASPT2 Exp.a

Gas ACN DMSO MET WAT 3.80 3.62 3.55 3.54 3.40 3.67 3.49 3.32 3.36 3.36

a

Reference 55. The experimental value in hexane is used as a comparison to the calculations in the gas-phase.

One of the advantages of RISM is its ability to describe local solvation structures. To illuminate this point, we have computed the charge density ρq of the solvent around the solute site α by the following equation:

ρqα (r) = 4πr2

∑

qs ρs gαs (r),

(13)

s

where gαs (r) is the radial distribution function representing the solvation structure. The charge density of the solvents around O- resulting from RISM-CASPT2 calculations of MQZ have been summarized in Figure 1. This solute site was chosen to focus on the degree of hydrogen bonding of each solvent. Results showed high positive charges present in the proximity of the O- site of MQZ in water. This indicates that there was strong hydrogen bonding between the solute site and solvent. Similarly, methanol also showed positive charges around the solute site, but was less than that of water. This corresponds to the solvation energy in methanol being less than that of water. Acetonitrile and DMSO showed nearly identical charge densities, and almost no positive charges existed around the solute site, which resulted in the small and identical solvation energies for the calculations in these solvents.

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WAT MET ACN DMSO

Figure 1: The charge density of the solvents, water, methanol, acetonitrile, and DMSO, around O- from the absorption energy calculations of MQZ by RISM-CASPT2.

As mentioned earlier, calculating absorption energies within the NIR region requires highly-accurate theoretical treatment. In order to highlight this point, the error in the calculated absorption energies compared to the experimental values 56 of CY3 and ICG, in DMSO and water, respectively, have been summarized in Figure 2. The absorption energies calculated by both PCM-TD-DFT and RISM-TD-DFT with the PBE0 functional had large errors, approximately 0.65 eV. They exceed acceptable errors for reliably studying the molecules of interest in the “second NIR window” (1000-1700 nm), 5 which spans for 0.5 eV. On the other hand, the RISM-DMRG-CASPT2 method predicted the absorption energies with the error reduced to within 0.2 eV, which seems to be satisfactory. These results illustrated the high applicability of RISM-CASPT2 in combination with the DMRG technique describing highly correlated π electronic states in solution.

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Figure 2: Error in calculated absorption energies of CY3 and ICG by PCM-TD-DFT, RISMTD-DFT, and RISM-DMRG-CASPT2. The PBE0 functional was employed for TD-DFT calculations. Experimental values from Reference 56. The consideration of the multi-reference character seemingly played a key role in delivering the high accuracy. The natural orbital occupation number analysis based on the RISM-CASSCF wave function showed that the occupations of HOMO and LUMO digress from a simple high-spin open-shell character as the molecules become larger in size (Table 3). This means that the single-reference (or mean-field) treatment of excited states was acceptable for small systems such as MQZ, but inadequate in accounting for the energy levels of the larger systems such as CY3 and ICG. Table 3: Natural occupation numbers of HOMO and LUMO from RISM-CASSCF molecule HOMO LUMO MQZ 0.94 1.07 CY3 1.04 0.98 ICG 1.24 0.78

5

Conclusions To summarize, we have presented a new method for calculating excited states in solu-

tion by combining RISM and (DMRG-)CASPT2. Absorption energy calculations of MQZ, 14


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CY3, and ICG with our new method successfully reproduced experimental measurements with satisfactory accuracy which suffices analysis of these bio-imaging related systems. The improvement was made by ameliorating the treatments of the excited electronic structure and the solvation effect simultaneously. The newly developed RISM-CASPT2 method and its extension with the DMRG reference are highly applicable for the theoretical study of the fluorescent molecules of interest within the “second NIR window.”

Acknowledgement This work was partly supported by JSPS KAKENHI Grant Numbers JP15K05385, JP16H06353, for Daisuke Yokogawa, JSPS KAKENHI Grant Numbers JP16H04101, JP15H01097 and JST PRESTO Grant Number JP17937609, for Takeshi Yanai, and JSPS KAKENHI Grant Number JP16H00855, for Yuki Kurashige.

Supporting Information Available The following files are available free of charge. • Lennard-Jones parameters and solvent parameters. • Localized active orbitals used as DMRG sites for ICG.

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Graphical TOC Entry

23


START 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


START

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V[G]

,V

(DMRG-)CASCI

(DMRG-)CASCI



RISM Calculation

Update V ---

converged?

converged?

ˆ solv H

ˆ solv H

(DMRG-)CASPT2

END


(DMRG-)CASPT2

END

-

OChemical Theory and Computation Journal of

Page 25 of 27 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

N+

N-Methyl-6-oxyquinolone (MQZ)

N+

N

Modeled indocarbocyanine (CY3)

N+

N ACS Paragon Plus Environment

Modeled indocyanine green (ICG)

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

WAT MET ACN DMSO


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Electronically Excited Solute Described by RISM ... - ACS Publications

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