Theoretical Study on Second Hyperpolarizabilities of Intramolecular

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Cite This: ACS Omega 2019, 4, 2741−2749

Theoretical Study on Second Hyperpolarizabilities of Intramolecular Pancake-Bonded Diradicaloids with Helical Scaffolds Shota Takamuku† and Masayoshi Nakano*,†,‡,§,∥ †

Department of Materials Engineering Science, Graduate School of Engineering Science, ‡Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, and §Quantum Information and Quantum Biology Division, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Toyonaka, Osaka 560-8531, Japan ∥ Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan

ACS Omega 2019.4:2741-2749. Downloaded from pubs.acs.org by 95.85.71.171 on 02/06/19. For personal use only.

S Supporting Information *

ABSTRACT: Using the spin-unrestricted density functional theory method, we have investigated a novel class of open-shell singlet fused-ring molecules with helical scaffolds toward the development of highly efficient third-order nonlinear optical (NLO) systems. It turns out that the helical molecules possess pancake bonding with intermediate diradical character and thereby exhibit large enhancement of orientationally-averaged second hyperpolarizability γs values (third-order NLO properties at the molecular scale) as compared to their closed-shell analogues. It is also found that the asymmetric charge distribution between the cofacial radical sites located on both the terminal rings of the helical scaffold leads to further enhancement of the γ value in the intermediate open-shell character region as compared to the open-shell analogues without asymmetric charge distributions. These results indicate that the helical systems with intermediate diradical character are potential candidates for the building block of highly efficient openshell singlet NLO materials, which are superior to the conventional closed-shell NLO materials. experimental researchers.14−19 In previous studies, we have theoretically proposed a novel class of highly efficient NLO systems based on open-shell singlet molecules.13,20,21 We introduce the diradical character y, which is a quantumchemically well-defined index and takes a value between 0 (closed-shell electronic structure) and 1 (pure open-shell (diradical) electronic structure), to quantitatively evaluate the instability of the chemical bond22,23 and have revealed the relationship between diradical character y and second hyperpolarizability γ, which is the third-order NLO properties at the molecular scale.13,20,21,24−26 From this theory (see the Supporting Information for details), the γ in the valenceconfiguration-interaction model with two electrons in two active orbitals is expressed by21

1. INTRODUCTION Helicene-like molecules, that is, polycyclic aromatic compounds with nonplanar screw-shaped scaffolds, have been receiving increasing attention due to their unique structure characterized by helicity and their unusual physicochemical properties1 like second-order nonlinear optical (NLO) properties,2 two-photon circular dichroism,3,4 circularly polarized luminescence,5 and so on. Recent rapid advances in synthetic technology have realized a variety of molecules with helical scaffolds,1,6−12 and investigations of exploring their mechanisms as well as of materials design are intensively conducted toward the realization of future photonics devices based on such helicene-like molecules. On the other hand, a new design concept, open-shell singlet character-based design principle, has recently emerged for developing highly efficient quantum functional materials since the tuning of the open-shell character tends to cause unique electronic structures and physicochemical properties.13,14 Also, since this character is based on the effective bond nature characterized by bonding and antibonding frontier molecular orbitals in the ground-state molecular systems and thus is closely related to conventional chemical concepts like resonance structure and aromaticity, the investigations of practical chemical guidelines for tuning the open-shell character as well as of design realistic open-shell singlet systems have attracted much attention from theoretical and © 2019 American Chemical Society

γ 4 RBA

( )

8q 4

=− (1 +

U3

+

i 1 − q2 )2 jjjj1 − 2rK + k 4 4q

(1 − 2rK +

1 1 − q2

)2 (

1 1 − q2

yz z 2 z 1−q z {

3

1

)

(1)

Received: December 24, 2018 Accepted: January 21, 2019 Published: February 6, 2019 2741

DOI: 10.1021/acsomega.8b03619 ACS Omega 2019, 4, 2741−2749

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Here RBA is an effective diradical length; U is an effective Coulomb repulsion; rK is the ratio of twice the exchange integral (K) to U (rK = 2K/U); q is the effective bond order defined by q = 1 − y.16 From this equation, it is found that molecules with intermediate diradical character y exhibit significant enhancement of γ values as compared to their closed-shell analogues. On the basis of this theoretical prediction, we have proposed various types of molecules and their aggregates with intermediate open-shell characters as potential candidates for third-order NLO systems.13 Although open-shell singlet systems are usually considered to be unstable due to the existence of radical electrons, it is recently reported that several stable molecules with intermediate diradical character are synthesized by using protecting groups for the radical sites15,18,24,27,28 and create thermally stable crystals.29−31 Furthermore, several diradical molecules are found to be stable enough to undergo the measurement of NLO properties and to exhibit large third-order NLO properties in solution24,32 and in the solid state.33 Thus, these experimental results have confirmed the validity of our theoretical prediction of NLO properties of open-shell singlet systems. A variety of open-shell singlet systems including polycyclic hydrocarbons, transition metal−metal bonded systems, and heavy main group compounds have been so far investigated theoretically and experimentally.13,34,35 Recently, as a novel class of open-shell singlet systems, pancake-bonded systems have been proposed in molecular crystals composed of open-shell monomers with radical sites.29−31,36,37 These systems have covalent-like through-space π-stacking multicenter interactions and tend to exhibit the intermediate diradical character values in the equilibrium intermolecular distance region.38−41 It has been found that several pancake-bonded systems with intermediate open-shell character show larger third-order NLO properties in the stacking direction than the π-stacked closed-shell analogues.42,43 There are investigations of designing helical molecular actuators triggered by redox processes of intramolecular pancake bonding with helical structure44 and of tuning the diradical character by modification of the terminal rings and/or helical structure, where the degree of cofacial matching between the radical sites in both the terminal rings is found to be essential for tuning the diradical character.45 On the basis of these results, we expect that the intermediate diradical systems with relevant helical scaffolds become novel candidates for building blocks of highly efficient open-shell NLO materials. In general, the NLO properties of molecules are known to be affected by the intermolecular interaction, vibrational effect, and environmental effects,45,46 which are not included in quantum chemical calculations of a single molecule in vacuo. On the other hand, the exploration of the mechanism of NLO phenomena and design of efficient NLO substances mostly start from the evaluation of hyperpolarizabilities at the molecular level and then go to the investigation at the molecular aggregate level or in the solid state to clarify the intermolecular interaction and environmental effects on the hyperpolarizabilities.47−50 Namely, it is important at first to find the molecules with large hyperpolarizabilities as well as to extend the search area of candidates for efficient NLO materials. Furthermore, the static hyperpolarizabilities, which are focused in the present study, are expected to be useful for a relative comparison of the NLO properties among different molecular species since the static hyperpolarizabilities are good approximation to the dynamic (frequency-dependent) hyper-

polarizabilities in the off-resonant region. Indeed, the applicability of our design guideline (y−γ correlation) for static γ in open-shell singlet systems to dynamic γ (third harmonic generation (THG)) spectra has already been investigated in our previous paper,51 and even in the resonant region, we have observed a similar y−γ correlation, that is, bellshape behavior of the THG γ as a function of y, to that obtained in the static case though the THG intensities are resonantly enhanced. Note here that although the zero-point vibrational average (ZPVA) could also affect the THG response, the ZPVA correction is usually known to be about 1 order of magnitude smaller than that of electronic static hyperpolarizabilities.52 In this study, therefore, we investigate the relationships between diradical character, static second hyperpolarizability γ, and the helical structures in intramolecular pancake-bonded diradicaloids with helical scaffolds using the spin-unrestricted density functional theory (DFT) method. The present results are expected to contribute to constructing novel design guidelines for highly efficient openshell singlet NLO materials composed of helical pancakebonded systems.

2. MODELS AND CALCULATION METHODS Figure 1 shows the helical model systems examined in this study. We choose three helical model systems with

Figure 1. Model systems examined. 8CTR, 7AZ, and 13C2NHTCN are diradicaloids, whereas 10helicene, 7helicene, and 13C2NH are their closed-shell analogues, respectively. Coordinate axes are also shown.

intermediate diradical characters (8CTR, 7AZ, and 13C2NHTCN), which are investigated by the previous theoretical and experimental studies.53−56 As references, we have also investigated closed-shell analogues (10helicene, 7helicene, and 13C2NH), which have the same number of rings as those in the intermediate diradical counterparts (8CTR, 7AZ, and 13C2NHTCN), respectively. 8CTR indicates [8]cethrene, which substitutes two phenalenyl rings for two terminal benzene rings of [8]helicene-like [5]cethrene. 56 7AZ is an azulene-like [7]helicene analogue terminated by five- and seven-membered rings. 13C2NHTCN and 13C2NH indicate [13]-C2NH helicenes (composed of pyrroles) with and without tetracyano groups, respectively, and resemble C2S helicenes composed of thiophenes.57 2742


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The open-shell singlet nature is evaluated by using diradical character y, which is obtained at the spin-projected unrestricted HF (PUHF) level of approximation by58−60 2T y=1− 1 + T2

Table 1. Mean Distances between the Ring Centers of the Stacked Rings, D, and Diradical Characters y

(2)

Here T is the orbital overlap between the corresponding orbital pairs and can also be represented by the occupation numbers of UHF natural orbitals, that is, the occupation number of the highest occupied natural orbital (HONO) (nHONO) and that of the lowest unoccupied natural orbital (LUNO) (nLUNO), as follows − nLUNO n T = HONO (3) 2

a b

* (r) ϕ * (r) ϕ d(r) = y[ϕLUNO (r) + ϕHONO (r)] LUNO HONO (4)

The static second hyperpolarizability (γijkl) values in the B convention62 are calculated by the finite-field approach,63 in which the γ value is calculated using the fourth-order numerical differentiation of the energy under the applied electric field along each (x, y, z) axis (F) 1 ∂ 4E , (i , j , k , l = x , y , z ) 6 ∂Fi ∂Fj ∂Fk ∂Fl

(5)

Here the line connecting the centers of the cofacial stacked rings is set as the z-axis, whereas the line connecting the midpoint between the cofacial stacked ring centers and the center of the ring (for odd-numbered ring systems) or the midpoint of the bond (for even-numbered ring systems) in the opposite side to the cofacial stacked rings is set as the y-axis. To compare the NLO properties of systems having different orientations of spin polarization, we also evaluate the orientationally averaged second hyperpolarizability γs, expressed as γs =

D (Å)a

y (-)b

10helicene 8CTR 7helicene 7AZ 13C2NH 13C2NHTCN

3.155 3.068 2.903 2.860 3.293 3.025

0.000 0.833 0.097 0.721 0.004 0.556

Calculated at the (U)M05-2X/6-31G(d) level of approximation. Calculated at the PUHF/6-31+G(d) level of approximation.

rings, D (at the (U)M05-2X/6-31G(d) level of approximation) and diradical characters y (at the PUHF/6-31+G(d) level of approximation). 8CTR is found to have 0.09 Å shorter D than the closed-shell reference 10helicene, which has the same number of rings as 8CTR, due to the covalent-like interaction between the cofacial phenalenyl radical units. 13C2NHTCN is also found to have 0.26 Å shorter D than the closed-shell reference 13C2NH, where the change in D is more significant than that between 8CTR and 10helicene because 13C2NHTCN has better spatial matching between the radical sites and thereby exhibits stronger covalent-like interaction than 8CTR. These structural trends are also verified by the diradical character values, that is, the diradical character of 13C2NHTCN is 0.3 Å smaller than that of 8CTR. In contrast, 7AZ does not show the same trend in the relationship between y and D: 7AZ shows a smaller y than 8CTR, whereas the difference of D between 7AZ and 7helicene is smaller than that between 8CTR and 10helicene. This smaller difference of D (0.043 Å) between 7AZ and 7helicene is found to be caused by the smaller inter-ring distance at the inner sites and larger inter-ring distance at the outer sites in 7AZ due to the bonding interaction at the inner sites and antibonding interaction at the outer sites (see Figure 2). This feature was also discussed in detail in our previous paper.56

In this study, the diradical character values are calculated at the PUHF/6-31+G(d) level of approximation.60 The odd electron density analysis61 is also used for investigating the spatial distribution of odd (unpaired) electrons (d(r)) concerning the HONO and LUNO at the PUHF/6-31G(d) level of approximation

γijkl = −

system

1 (γ + γxxzz + γyyzz + 2γxxxx + 2γyyyy + 2γzzzz) 5 xxyy (6)

These γ values in eqs 5 and 6 are evaluated by using the Romberg differential scheme.64,65 The density analysis for hyperpolarizability is also conducted to clarify the spatial contribution of electrons to γ values γijkl = −

1 6

∫ dr riρjkl (r),

(i , j , k , l = x , y , z )

Figure 2. Highest occupied molecular orbital (HOMO) of 7AZ at the RM05-2X/6-31G(d) level of approximation. Isosurfaces are drawn at ±0.03 au.

3.2. Second Hyperpolarizability. Figure 3 shows the orientationally averaged second hyperpolarizability γs and the diradical character y in each system. Table 2 lists γ components and γs values of the model systems. It is found that all of the model systems with intermediate diradical characters (7AZ, 8CTR, and 13C2NHTCN) exhibit larger γs values than the closed-shell reference systems (7helicene, 10helicene, and 13C2NH), respectively, by a factor of 1.8−8.2, although there are large differences in γs between those open-shell singlet systems. 7AZ shows the largest γs value in those systems, which are 7.5 times as large as that of the closed-shell reference

(7a)

where γ density ρjkl(r) is defined as61,66 ρjkl (r) =

∂ 3ρ(r) ∂Fj ∂Fk ∂Fl

, (j , k , l = x , y , z ) F=0

(7b)

3. RESULTS AND DISCUSSION 3.1. Geometric and Electronic Structure. Table 1 shows the mean distance between the ring centers of the stacked 2743


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than that of 8CTR (y = 0.833). The γyyyy component of 13C2NHTCN is also found to be ∼4 times as large as that of 13C2NH due to the intramolecular charge transfer between the terminal dicyano groups and the annulated pyrrole rings. Figure 4 shows the charge distributions of 13C2NH and 13C2NHTCN. The summations of Hirshfeld charges of the middle pyrrole ring are −0.001 au (13C2NH) and −0.003 au (13C2NHTCN), respectively. Although the averaged values of summations of Hirshfeld charges of the terminal pyrrole rings are (0.001 + 0.003)/2 = 0.002 au (13C2NH) and (0.229 + 0.229)/2 = 0.229 au (13C2NHTCN), the averaged value of the summations of Hirshfeld charges of dicyano groups is (−0.406 − 0.405)/2 = −0.406 au (13C2NHTCN). This feature represents large intramolecular charge transfer between the dicyano groups and terminal pyrrole rings in 13C2NHTCN, resulting in more significant enhancement of the γ value67,68 as compared to 8CTR. In contrast, 7AZ shows a much larger enhancement in γzzzz, which leads to the largest γs values in the present systems. To clarify the origin of such larger enhancement in each component of γ, we perform the odd electron and γ density analyses. Figure 5a−d shows the odd electron densities (top and side views) of 7AZ, the γzzzz density (side view) of 7AZ, and γzzzz density (side view) of 7helicene, respectively. As shown in Figure 5a,b, the odd electron densities are primarily localized at the radical sites (cyclopentadienyl and tropylium units) in 7AZ. Figure 5c indicates that the γ densities are also localized at the same positions as those with significant odd electron densities. Furthermore, as seen from the comparison of the γ densities between the intermediate open-shell system (7AZ) (Figure 5c) and the closed-shell reference system (7helicene, Figure 5d), the intermediate open-shell systems have much larger amplitudes of γ density distributions than their closedshell analogues. This clearly indicates that the diradical electrons are the origin of γ enhancement of the intermediate open-shell systems. Next, we move on to the interesting feature that the γzzzz of 7AZ is 43.8 times larger than that of 7helicene and is also significantly (1−2 orders) larger than those of the other model systems. In the previous study, we have revealed that asymmetric charge distribution with the intermediate diradical character leads to further enhancement of γ values as compared to symmetric diradicaloids.25 Since the 7AZ system exhibits asymmetric charge distribution between radical sites due to its asymmetric molecular structure between cyclopentadienyl (the summation of Hirshfeld charges = −0.039 au) and tropylium (0.039 au) as shown in Figure 5b, the 7AZ system is expected to meet the conditions for the asymmetric open-shell NLO systems with significant enhancement of γ values (charge difference between the cofacial rings = 0.078 au and an intermediate diradical character y = 0.721). Indeed, we observe

Figure 3. Diradical character y (left vertical axis and red bar) (PUHF/ 6-31+G(d)) and orientationally averaged second hyperpolarizability γs (right vertical axis and blue stripe bar) (LC-(U)BLYP/6-31+G(d)) in each model system.

system (7helicene). From these results, the intermediate diradical characters are found to lead to such enhancements of γs. It is also found that the differences in γs between the intermediate open-shell singlet systems (γs = (37.6−125.5) × 103 au) are significantly larger than those between the closedshell systems (γs = (16.8−21.2) × 103 au) (see Table 2). To reveal the origin of such differences of γs values between intermediate open-shell and closed-shell systems, we investigate each γ component of those systems (Table 2). 8CTR shows 1.8 times enhancement of γs as compared to its closed-shell analogue (10helicene). This is the smallest enhancement of γs for diradicaloid as compared to that for its corresponding closed-shell analogue among the present systems. The γzzzz (in the π-stacking direction) and γyyyy of 8CTR also show similar enhancement ratios (∼2 and ∼3 times, respectively) as compared to those of 10helicene. The enhancement of γyyyy of 8CTR as compared to that of 10helicene is predicted to be related to the π-conjugation size difference between them (9.30 Å in 8CTR and 7.33 Å in 10helicene), that is, the larger π-conjugation leads to enhancement of the γ value due to the reduction of the excitation energy contributing to γ.47 On the other hand, among the intermediate diradicaloids (8CTR, 7AZ, and 13C2NHTCN), 8CTR shows the smallest γs value since 8CTR presents a smaller γzzzz than those of 7AZ and 13C2NHTCN due to its nearly pure diradical nature (y = 0.833). 13C2NHTCN shows ∼8 times enhancement of γzzzz as compared to the closed-shell reference 13C2NH. This γzzzz enhancement ratio is shown to be larger than that (∼2) of 8CTR as compared to 10helicene due to the more intermediate diradical character of 13C2NHTCN (y = 0.556)

Table 2. γijkl (×103 au) and Orientationally Averaged γs (×103 au) at the LC-(U)BLYP/6-31+G(d) Level of Approximation γxxxx γyyyy γzzzz γxxyy γxxzz γyyzz γs

10helicene

8CTR

7helicene

7AZ

13C2NH

13C2NHTCN

21.45 18.48 20.54 7.02 8.37 7.38 21.20

20.78 62.55 41.91 9.23 8.35 13.86 37.63

16.35 20.54 9.93 6.59 5.54 6.22 16.70

27.21 31.72 434.63 7.79 51.42 7.84 125.53

25.71 24.21 11.35 49.58 6.42 4.96 16.81

21.02 98.13 84.53 51.10 1.75 3.91 63.44

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Figure 4. Hirshfeld charges of 13C2NH (a) and 13C2NHTCN (b) at the LC-(U)BLYP/6-31+G(d) level of approximation.

Figure 5. Odd electron densities of 7AZ (top view (a) and side view (b) with a contour value of 0.1 au) at the PUHF/6-31+G(d) level of approximation; γzzzz densities (yellow (positive) and blue (negative)) of 7AZ (side view) with a contour value of ±2000 au (c) and γzzzz densities of 7helicene ±20 au (d) at the LC-(U)BLYP/6-31+G(d) level of approximation.

a large dipole moment difference (Δμz) in the terminal-ringstacking direction between the ground and the first excited states of 7AZ (Δμz = 6.848 D at the spin-flip (SF)-timedependent (TD)DFT (PBE50)/6-31G(d) level of approximation (see Table 3)), which contributes to the enhancement of γzzzz,25 in contrast to 7helicene (Δμz = 0.080 D) (see Table 3 for others). This prediction is also substantiated by the fact that the 7AZ system has 5.14 times larger γzzzz value than

13C2NHTCN, which has a negligible charge difference between the cofacial rings (0.000 au) in the ground state and a small dipole moment difference (Δμz = −0.229 D) although it lies in the intermediate y region (y = 0.556). Here we discuss the contribution of through-bond or through-space interaction to the enhancement of γ values. The variation trend of diradical character as a function of the number of rings connecting to the radical units56 shows that the molecules without the pancake bonding interaction show an almost pure open-shell nature (y ∼ 1) except for the small ring numbers, in which the molecules have only through-bond interaction. This indicates that intermediate diradical character in the molecules with the pancake bonding interaction originates in the through-space interaction. For NLO diradicaloids with intermediate y values, the γ is known to be enhanced in the direction connecting the primary distribution positions of radical electrons. As shown in the above discussion, the enhancement for the z-axis (pancake bonding direction) is predicted to be caused by the intermediate diradical character of the pancake bonding, whereas the enhancement for the y-axis is caused by the size effect or intramolecular charge transfer in the through-bond interaction.

Table 3. Excitation Energies E (eV), Oscillator Strengths f (-), and Dipole Moment Differences Δμz (μz(excited state) − μz(ground state)) [D] for the Present Systems at the Spin-Flip-TDDFT (PBE50)/6-31G(d) Level of Approximation system

E (eV)

f (-)

Δμz (D)

10helicene 8CTR 7helicene 7AZ 13C2NH 13C2NHTCN

3.41 1.70 4.05 1.38 5.63 2.14

0.1244 0.1008 0.3341 0.1198 0.0421 0.2157

0.000 −0.054 0.080 6.848 0.915 −0.229 2745


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ring organic molecules, which have intramolecular pancake bonds, using the spin-unrestricted density functional theory method. It is found that the model systems with intermediate diradical characters show 1.8−7.5 times larger orientationally averaged second hyperpolarizability γs values than their closedshell analogues. 8CTR (y = 0.833) shows the smallest enhancement of γs in the model systems due to their nearly pure open-shell nature. It turned out that 7AZ exhibits the largest enhancement (7.5 times) of γs due to the asymmetric charge distribution (causing charge transfer excitation) between the cofacial terminal cyclopentadienyl and tropylium units. From these results, we present the design guidelines of a new class of helical molecules for building blocks of highly efficient third-order NLO materials: (i) intermediate diradical character by using pancake-bonding, which can be achieved by putting radical units at appropriate cofacial ring sites, for example, using large-sized helical structures because such a long helical structure like 13C2NHTCN tends to reduce distortion, resulting in relevant cofacial radical-site interaction showing its intermediate diradical character, and (ii) asymmetric charge distributions between the radical sites of terminal cofacial rings as shown in 7AZ. Furthermore, such helical molecules with intermediate diradical characters are expected to attract much attention from the viewpoint of highly efficient open-shell magneto-optical materials.3,71 The investigation toward this direction is in progress in our laboratory.

Judging from these features, we can conclude that the intermediate diradical character enhances γzzzz due to the through-space contribution. Finally, we discuss the stability and the availability of those helical systems as the NLO materials. The pancake-bonded systems show the stability because the radical sites are protected by the pancake-bonding.31,36,37 Furthermore, the crystal of [7]helicene shows the π−π stacking structures between molecules,69 which suggests that the helical molecules with the intramolecular pancake-bonding also tend to take the stacking structures between the molecules and to have covalent-like intermolecular interactions, like other pancakebonded systems.29−31,36 Therefore, there is a possibility of creating stable crystals of helical pancake-bonded molecules. From the viewpoint of availability of those compounds as NLO substances, we investigate herein their optically-allowed first excitation energies by spin-flip (SF) time-dependent (TD) DFT calculations. The excitation energies (E) are important quantities to consider the application to optical devices because the excitation energies are related to the available light frequency region for various NLO processes.70 Table 3 lists the excitation energies as well as the oscillator strengths of these molecules for the transition in the π−π stacking direction. 7AZ (y = 0.721) is shown to have a primary single excitation configuration HOMO → LUMO for the first excitation with the excitation energy of E = 1.38 eV, which is 0.34 times as small as that of 7helicene (y = 0.097). 8CTR (y = 0.833) is also shown to have a primary single excitation configuration HOMO → LUMO in the first excited state with the excitation energy of E = 1.70 eV, which is 0.50 times as small as that of 10helicene (y = 0.000). 13C2NHTCN is shown to have a primary single excitation configuration HOMO − 2 → LUMO for the first excited state with the excitation energy of E = 2.14 eV, which is 0.38 times as small as that of 13C2NH (y = 0.004). Note that the excited state of 13C2NH is not composed of the orbitals of the terminal rings because the distance between the terminal rings is too far to interact with each other. Such reduction of the first excitation energies in open-shell singlet systems with intermediate diradical character is predicted to contribute to the enhancement of their γ amplitudes, but the y−γ correlation is more complicated as shown in eq 1 and our previous papers.21,25 Although such reduction of the first excitation energies in these open-shell singlet systems has the possibility of causing resonance enhancement of the frequency-dependent NLO properties, the y−γ correlation obtained for the present systems is predicted to be retained even in the resonance region, as seen from our previous results on the real and imaginary NLO spectra using the two-site diradical model.51 The excitation energies of the present molecules with intermediate diradical characters are 0.85−1.51 times as large as those of the typical organic diradicaloids like dicyclopenta[b,g]naphthaleno[1,2,3-cd;6,7,8-c′d′]diphenalene (E = 1.42 eV and y = 0.854) and s-indaceno[1,2,3-cd;5,6,7-c′d′]diphenalene (E = 1.62 eV and y = 0.770), which are known to exhibit much larger third-order NLO properties than closed-shell analogues of similar size in two-photon absorption measurment.24 Thus, the present molecules with intermediate diradical characters are also expected to become candidates for NLO materials.

5. COMPUTATIONAL DETAILS All of the structures were optimized using the DFT method, the (U)M05-2X/6-31G(d), which is a hybrid functional with 52% of Hartree−Fock exchange72 and was previously found to well reproduce the geometries in conjugated systems with pancake bonds.40,72 We have confirmed that all of the states giving minima of total energies have only real vibrational frequencies. The calculations for the γ values and their densities are performed with LC-(U)BLYP (with the range separating parameter μ = 0.33 bohr−1), which is known to semiquantitatively reproduce γ values of organic delocalized open-shell molecules evaluated using the strong-correlated wavefunction methods73 with the 6-31+G(d) basis set. Note here that diffuse functions are needed to quantitatively reproduce γ because the lack of diffuse functions is known to underestimate γ for the open-shell singlet p-quinodimethane model.74 Furthermore, we calculated the excitation energies for these systems using the spin-flip TDDFT (SF-TDDFT) method with the PBE50 (50% Perdew−Burke−Ernzerhof (PBE) and 50% Hartree−Fock exchange and 100% PBE correlation) functional and 6-31G(d) basis set to discuss their availability as NLO materials, since the first excitation energies for open-shell singlet diradicaloids calculated with this method are known to well reproduce those obtained by the highly correlated ab initio method like the EOM-SF-CCSD(dT) method.75 The geometry optimization and the calculation of γ values were performed out using the Gaussian 09 program package,76 and the excitation energies were calculated using the Q-Chem program package.77 ChemCraft78 and MacMolPlt79 were used for visualization of structures, orbitals, odd electron densities, and γ densities.

4. CONCLUSIONS In this study, we investigate the relationship between diradical character y and second hyperpolarizability γ in helical fused2746


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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/acsomega.8b03619. Romberg differentiation procedure; y−γ correlation in two-site diradical model; excitation energies and oscillator strengths of calculated systems; and Cartesian coordinates of calculated systems (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +81-6-68506265. ORCID

Shota Takamuku: 0000-0002-8823-7496 Masayoshi Nakano: 0000-0002-3544-1290 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant no. JP18H01943 in Scientific Research (B), Grant no. JP17H05157 in Scientific Research on Innovative Areas “πSystem Figuration”, and Grant no. JP26107004 in Scientific Research on Innovative Areas “Photosynergetics”, and the JSPS Research Fellowship for Young Scientists (JP18J10067). This work was also supported by the Interactive Materials Science Cadet program, Japan. Theoretical calculations were partly performed using the Research Center for Computational Science, Okazaki, Japan.

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REFERENCES

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Theoretical Study on Second Hyperpolarizabilities of Intramolecular

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