First-order hyperpolarizabilities of propellanes: elucidating structure–property relationships

Abstract

Following recent experimental work demonstrating strong nonlinear optical properties, namely second harmonic generation of light, in crystals composed of 16,20-dinitro-(3,4,8,9)-dibenzo-2,7-dioxa-5,10-diaza[4.4.4]propellane molecules [A. Miniewicz, S. Bartkiewicz, E. Wojaczyńska, T. Galica, R. Zaleśny and R. Jakubas, J. Mater. Chem. C, 2019, 7, 1255–1262] in this paper we aim to investigate “structure–property” relationships for a series of 16 propellanes presenting a wide palette of substituents with varying electron-accepting/donating capabilities. To that end, we use electronic- and vibrational-structure theories and a recently developed generalized few-state model combined with a range-separated CAM-B3LYP functional to analyze electronic and vibrational contributions to the first hyperpolarizability for the whole series of molecules. The variations in computed properties are large among the studied set of substituents and can reach an order of magnitude. It has been demonstrated that the maximum values of frequency-independent first hyperpolarizability are expected for strong electron-accepting NO₂ substituents, but only at the preferred position with respect to the electronegative oxygen atom in the 1,4-oxazine moiety. This holds for electronic as well as vibrational counterparts.

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1 Introduction

Propellanes are molecular systems owing their name to the propeller-like structure. In the case of smaller members of the family, the three rings connected through a common covalent bond cause very strained and unstable geometries. Owing to this, propellanes undergo polymerization to produce interesting structures such as staffanes. Snatzke and Zanati were the first to synthesize propellane in 1965.¹ Propellanes, however, have been presumably obtained in some earlier preparations.^2,3 The name “propellane” was first used by Ginsberg in 1966.⁴ Since then, several research groups have synthesized and studied the properties of different propellanes including heteropropellanes – where the ring structures are not necessarily hydrocarbons.^5–11

Apart from their strained geometry, propellanes are popular due to their diverse applications in synthetic chemistry ranging from materials to natural products.⁵ It is important to note that although the research on propellanes started about 80 years ago, these systems are very common in natural products. Propellanes containing larger rings are stable enough to provide a platform for designing desired molecular structures such as for highly efficient organic light-emitting diodes.¹² The common covalent bond in propellanes can easily be broken and the resultant valency can be satisfied by different groups. In this way, the unstrained propellane moiety can be attached to different types of molecular entities (e.g. during polymerization). This provides a number of possibilities in synthetic chemistry.^13,14 The preparation of staffanes and various dendrimers essentially involves the breaking of the said common covalent bond.^5,15 These experimental advances were supported, in parallel, by the theoretical studies^5,16–19 on propellanes. However, these studies were limited to strain energy, the mechanism of the polymerization reaction of propellane, and singlet–triplet transition, to name a few.

In spite of numerous applications of various propellanes in organic synthesis, there are very few studies focusing on non-linear optical properties of such compounds. Very recently, some of the present authors have synthesized a heteropropellane-based molecule, 16,20-dinitro-(3,4,8,9)-dibenzo-2,7-dioxa-5,10-diaza[4.4.4]propellane (DDDDP) and studied its second-order nonlinear optical (NLO) properties.¹¹ It was found, using 1064 nm nanosecond laser pulses, that the crystalline powder of DDDDP displays a very high second harmonic generation (SHG) efficiency.¹¹ This was the first SHG study for a propellane-based system. These promising experimental results for a single propellane and the on-going quest for second-order NLO materials²⁰ motivated the present theoretical work, in which we wish to study the molecular first hyperpolarizability for larger sets of propellane derivatives. In particular, we aim to establish “structure–property” relationships using a palette of substituents at different positions. To that end, we employed several state-of-the-art theoretical approaches combined with electronic- and vibrational-structure calculations. In fact, computer simulations provide an invaluable tool in the analysis of second-order nonlinear optical properties of molecules and their assemblies.^21–29

2 Theory and computational methods

In the presence of an electric field (F), the Cartesian component of the total dipole moment μ_i may be expressed as a Taylor series:³⁰where μ⁰_a is the a-th component of the permanent dipole moment; α_ab, β_abc and γ_abcd are the components of the molecular polarizability, and first and second hyperpolarizability tensors, respectively. ω_σ is the sum of the interacting fields with frequencies ω_i; and K⁽²⁾ and K⁽³⁾ are factors ensuring that all hyperpolarizabilities of the same order have the same static limit. Using perturbation theory, the (hyper)polarizabilities can be expressed as a sum-over all the states of the system (in general rovibronic). In this work, we are interested in the first hyperpolarizability, whose general sum-over-states expression is given as³¹where represents the summation over all the permutations of the pairs (a, −ω_σ), (b, ω₁), and (c, ω₂) and E_i = ħω_i (i = 1, 2, σ). The prime over the double summation in eqn (2) represents a restricted summation where none of the indices could label the ground state. E_mn and μ^mn_a stand for the excitation energy and ath component of the transition dipole moment for m → n transition, respectively. Note that [small mu, Greek, macron]^mn = [small mu, Greek, macron]^mn − μ⁰⁰. The static limit of the tensor components can be obtained by setting all three angular frequencies to zero. The commonly used first hyperpolarizability, β_‖, can be calculated from the β_abc components using the following relation:³²

In the above equations [small mu, Greek, macron]⁰⁰_a stands for the ground-state dipole unit in the direction of the ground-state dipole moment (μ⁰⁰_a/|μ⁰⁰|).

Within the Born–Oppenheimer (BO) approximation, the molecular (hyper)polarizabilities defined by eqn (1) may be separated into pure electronic (P^e) and pure vibrational (P^v) contributions, as well as the zero-point vibrational averaging (ZPVA) correction:³²

P = P^e + P^v + P^ZPVA (4)

where P = α, β, γ. Alternatively, one can divide the property P into electronic, nuclear relaxation (P^nr) and curvature (P^curv) contributions:³²

P = P^curv + P^nr + P^curv (5)

The latter separation is employed in this study and we only focus on the nuclear relaxation contribution to first hyperpolarizability. The P^nr terms are usually larger than P^curv terms. In order to treat the effect of molecular vibrations on electric properties, Bishop and Kirtman proposed a double (electrical and mechanical) perturbation theory (BKPT) treatment.³³ The P^nr contributions are given by the leading terms of each type of “square bracket” term of BKPT pure vibrational contributions. Within this approach, the nuclear relaxation first and second hyperpolarizability may be expressed in the “square bracket” notation as:

β^nr = [μα]^(0,0) + [μ³]^(1,0) + [μ³]^(0,1) (6)

Each square bracket involves products of the normal coordinate derivatives of the electronic electrical properties indicated, as well as harmonic vibrational frequencies and anharmonic force constants. Each term is labeled by a pair of superscripts (p, q) denoting the order in electrical and mechanical anharmonicities, respectively. There is an efficient finite-field nuclear relaxation method (FF-NR)^34,35 for computing nuclear relaxation hyperpolarizabilities. However, in this study the field-induced coordinates (FICs) approach is used^36,37 to determine the anharmonicity contributions to nuclear relaxation first hyperpolarizability. FICs are linear combinations of field-free normal coordinates associated with the change in equilibrium geometry induced by a static electric field. The displacement of the i-th field-free normal coordinate at the field-relaxed geometry is given by:³⁶

where,and

The indices i, j and k stand for normal coordinates whereas a and b label the Cartesian directions. a^ij…ab…_nm involves the n-th and m-th derivatives of the potential energy V (Q, F) with respect to normal coordinates and field components, respectively. The first-order FICs:

are sufficient to compute the static nuclear relaxation contribution to first hyperpolarizability:³⁶ indicates the sum over all permutations of the indices a, b,…

Electronic and vibrational structure calculations, assuming C₂ symmetry point group for all compounds but one, were performed with the Gaussian suite of programmes.³⁸ To that end we used the range-separated CAM-B3LYP functional³⁹ and the aug-cc-pVDZ basis set.⁴⁰ The choice of this functional is dictated by its satisfactory performance in predicting electronic and vibrational contributions to the electric properties of various orders.^41–44 The results of these calculations were used for computing vibrational hyperpolarizabilities using custom computer routines. Additionally, the components of static electronic β and different transition dipole moments were calculated using the LSDALTON programme package using the CAM-B3LYP/6-31+G level of theory to study the convergence of β with the number of electronic states based on the generalized few-state model (GFSM).⁴⁵ The GFSM calculations were performed using in-house code.

3 Results and discussion

The structures of all the sixteen compounds studied in this work are given in Fig. 1. All these molecules are propellane derivatives with three wings: one consisting of unsubstituted cyclohexane, and the other two are identical benzo-oxazine moieties. The two benzo-oxazine wings are substituted with various groups such as –NH₂, –CF₃ (mono- and di-), –SCH₃, –SO₂CH₃, –SCF₃, –OCF₃, dioxolane, –CH₂OCH₃, and –NO₂ yielding a set composed of 16 compounds (Fig. 1). The methodology of preparation of this type of compound, based on the reaction of cyclic 1,2-diketones and ortho-aminophenols has been already developed and it can be adapted for the synthesis of most promising derivatives. The whole series of compounds, except 5, can be divided into three major categories covering a wide range of Hammett substituent constants:⁴⁶ (a) Singly-substituted compounds (at each wing) in the para position in relation to the ether oxygen; their order in the series can be established according to the electron-accepting capabilities of the substituent: 13 (–NO₂, substituent constant is 0.78) 4 (–SO₂CH₃, 0.72), 3 (–CF₃, 0.54), 7 (–SCF₃, 0.50), 8 (–OCF₃, 0.35), 10 (–CH₂OCH₃, 0.01), 2 (–SCH₃, 0.00). (b) Singly-substituted compounds (at each wing) in positions other than para: 11 (m-NO₂, 0.71), 12, and 14; 15 (m-NH₂, −0.16 and m-NO₂, 0.71) and 16 (m-NH₂, −0.16). (c) Doubly-substituted compounds (on each wing): 1 (p-CH₃, −0.17 and m-CF₃, 0.43), 6 (p-SCH₃, 0.00 and m-CH₃, −0.07), 9 (two m-CF₃ groups, 0.43). It is worth pointing out that the oxygen atom was chosen as a “standard” because its electronegativity is higher than that of nitrogen (NH group). Note that the experimentally studied DDDDP molecule¹¹ is also included in the set shown in Fig. 1 (as compound 11).

Fig. 1 Compounds studied in this work and their labelling.

The values of electronic first hyperpolarizability (β_‖^e) for all the compounds calculated using the aug-cc-pVDZ basis set and CAM-B3LYP functional are presented in Table 1. We will start the discussion with the purely electronic counterpart as it contributes significantly to second-order processes involving both static and dynamic fields. Compound 11 has the maximum value for β_‖^e and compound 16 has the least value (in magnitude). The ratio between the maximum and minimum values of β_‖^e is 54 : 1. The compound with the second largest β_‖^e is 15, which is the only compound asymmetrically substituted (–NO₂ and –NH₂) at para positions. We note that the replacement of one electron-accepting group with an electron-donating one, while preserving the positions of the substituents, does not lead to deterioration of the electronic first hyperpolarizability. It is interesting to note that compounds 1, 3 and 9 have almost the same values for β_‖^e indicating that there is almost no effect of adding an extra –CF₃ group or changing the position of the –CF₃ group in the benzo-oxazine wings. Similarly, compounds 2 and 6 have similar values for β_‖^e indicating that there is no effect of adding an extra methyl group in the benzo-oxazine ring. On replacing –SCH₃ by –CF₃, i.e., on moving from compound 2 to 3, there is an increase in β_‖^e value. Replacing –CF₃ by –OCF₃ (compound 3 to compound 8) leaves the β_‖^e values practically unchanged and replacing –CF₃ by –SCF₃, i.e., on moving from compound 3 to 7, the β_‖^e values are almost doubled. This is probably because of the available vacant d-orbital on the S-atom. On replacing –SCH₃ (compound 2) by –SO₂CH₃ (compound 4) there is about a three-fold increase in the β_‖^e values and replacing –SCH₃ (compound 2) by –SCF₃ (compound 7) there is a corresponding 2.5-fold increase. Compounds 12 and 14 show a variation in β_‖^e which emphasizes the importance of substituent placement. Finally, let us recall that compound 16 has two electron-donating substituents at the meta position and exhibits the smallest β_‖^e value. Such substituent placement maximizes β_‖^e but only for strong electron-donating groups.

Table 1 Vectorial electronic and harmonic vibrational contributions to the first hyperpolarizability (in au) computed at the CAM-B3LYP/aug-cc-pVDZ level

Compound	β ‖ ^e	[μα]‖^(0,0)	Sum
1	196	3053	3249
2	218	1814	2032
3	246	3291	3537
4	−521	−5997	−6518
5	112	538	650
6	247	1442	1689
7	534	3307	3841
8	212	2839	3051
9	251	3358	3609
10	449	−1481	−1032
11	1198	7330	8528
12	368	3060	3428
13	710	5862	6572
14	214	−226	−12
15	1087	6896	7983
16	22	−2117	−2095

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In order to shed light on large values of β_‖^e found for 11, we employed the corresponding generalized few-state model (GFSM) developed by some of the current authors.⁴⁵ Within GFSM, the expression for β_‖^e is given as⁴⁵

whereand for the static case,and θ^cd_ab is the angle between transition dipole moments μ^ab and μ^cd. The prime in eqn (12) indicates a restricted summation, where the ground state is not included. Other terms have their usual meanings. Each β^GFSM_mn can either be positive or negative depending on the values of A_mn, i.e., the relative orientation of the involved transition dipole moment vectors. We have included up to 50 excited states in GFSM calculations. The convergence of β_‖^e with the number of states involved in GFSM for 11 is shown in Fig. 2. The corresponding data for some other studied propellanes are presented in Fig. S1 in the ESI† file and allow for drawing a conclusion that β_‖^e is slowly convergent with respect to the number of electronic excited states, thus indicating that there are no key electronic states determining the first hyperpolarizability. In contrast, the results shown in Fig. 2 for 11 indicate that GFSM is quite successful in the quantitative prediction of β_‖^e values, including its sign.

Fig. 2 Convergence of the electronic first hyperpolarizability for 11 with the number of electronic states (N-state model accounts also for electronic ground state). The inset shows the electronic density difference (Δρ (r)) between electronic singlet state ρ (r, S_i) and the ground electronic state ρ (r, S₀). The contour value of density was set at 0.005 au and positive (negative) values of Δρ (r) are represented by blue (red) color. Calculations were performed at the CAM-B3LYP/6-31+G level.

We have noticed that involving all the 50 excited states gives β_‖^GFSM = 1642 au, 75% of which is contributed by the top five positive and the top five negative terms. β_mn as well as the corresponding dipole (D_mn), angle (A_mn), and energy (E_mn) terms are given in Table 2.

Table 2 Largest β^GFSM_mn terms for compound 11 computed at the CAM-B3LYP/6-31 + G level

m, n	β mn = βnm	D mn	A mn	5.0 × Emn
3, 3	333	6.3544	1.0	0.0191
3, 4	327	6.4129	0.98	0.0192
4, 4	477	3.0659	3.0	0.0193
3, 11	−40	1.4270	−0.7	0.0251
4, 11	−70	1.8760	−0.94	0.0252
4, 12	−51	0.4353	−3.0	0.0255
8, 8	81	0.6591	3.0	0.0245

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It is interesting to note that the large positive terms, i.e., β₃₃, β₃₄, and β₄₄ have the largest dipole contributions. The first two have an angle term ∼1, whereas the last one has an angle term equal to 3 (which is maximum). The dipole contribution in β₄₄ is less than half of those in either β₃₃ or β₃₄, but the corresponding maximum angle contribution makes it the largest contributing β_mn. The character of two key electronic states (S₃ and S₄) is shown in the inset in Fig. 2 – there is a symmetric density shift from 1,4-oxazine moieties towards nitro groups. The angles between each pair of contributing transition dipole moments are presented in Table 3, which clearly indicates that in β₄₄ and β₈₈ all the transition dipole moments are aligned parallel to each other making the angular contribution the maximum. In the group of negative β_mn contributors, one of the pairs of transition dipole moments makes an angle larger than 90 leading to an overall negative term. Among all the major negative contributors, the dipole contributions are very small and the corresponding energy contributions are slightly higher than the positive β_mn and hence for compound 11β_‖^e is the largest. Let us close this section highlighting that although the GFSM analysis was performed using 6-31+G basis, the data in Table S1 in the ESI† file indicate that there is a satisfactory agreement with property values obtained with the larger aug-cc-pVDZ basis set.

Table 3 Angle between transition dipole moments involved in major β_mn terms

β mn	a	b	c	d	θ ^cd ab	β mn	a	b	c	d	θ ^cd ab
β 33	0	0	0	3	90	β 311	0	11	3	11	90
	0	0	3	3	0		0	3	3	11	90
	0	3	3	3	90		0	11	0	0	90
β 34	0	4	3	4	90		0	3	0	11	45
	0	3	3	4	12		0	0	3	11	180
	0	0	0	4	0	β 411	0	11	4	11	160
	0	3	0	4	90		0	4	4	11	90
	0	0	3	4	90		0	4	0	11	90
β 44	0	0	4	4	0		0	0	4	11	90
	0	4	4	4	0	β 412	0	12	4	12	180
β 88	0	0	0	8	0		0	4	4	12	180
	0	0	8	8	0		0	12	0	0	0
	0	8	8	8	0		0	4	0	12	0
							0	0	4	12	180

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We will now turn to the analysis of vibrational contributions to the first hyperpolarizability. In order to gain an insight into their importance to the static property, we have computed the purely harmonic term, as defined by eqn (6), for 1–16. The results are assembled in Table 1 and demonstrate that the [μα]_‖^(0,0) term prevails over the electronic counterpart and in many cases the ratio reaches up to the factor of 10. Similarly to what has been demonstrated for the purely electronic β_‖^e term, the largest harmonic vibrational counterpart is found for 11. On passing from electronic contributions to the sum of electronic and vibrational terms, the property-based ordering of compounds changes, albeit the largest value is found in both cases for 11. It should be highlighted that we consider here frequency-independent first hyperpolarizability and the vibrational contributions become less significant for dynamic fields. However, if at least one static field is involved, they might add a substantial contribution to the overall second-order nonlinear optical response.⁴⁷ In what follows we will analyze in more detail compounds 11 as well 12–14 with the NO₂ group placed at various positions. Table 4 and Fig. 3 show lowest-order anharmonic terms contributing to β_‖. Three major conclusions can be drawn from these data, namely: (i) both electrical and mechanical anharmonicities make a substantial contribution to the vibrational counterpart (e.g., for compound 11 the absolute value of the (1,0) term is 40% of (0,0)), (ii) anharmonic corrections may significantly reduce the purely harmonic term, as they are of opposite signs (see e.g.11), and (iii) there is no clear pattern of the electrical and mechanical anharmonicities for compounds 11–14. It should be thus concluded that for static hyperpolarizability these terms should be evaluated explicitly as they might have a significant effect on overall trends in β_‖.

Table 4 Breakdown of vibrational first hyperpolarizability (in au) into anharmonic terms

Compound	[μ^3]‖^(1,0)	[μ^3]‖^(0,1)	[μ^3]‖^(1,0) + [μ^3]‖^(0,1)
11	−2959	−1258	−4217
12	−655	1148	493
13	808	−2078	−1270
14	1052	−622	430

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Fig. 3 Breakdown of β_‖ for 11–14 into electronic and vibrational contributions.

In what follows we will analyze in more detail the origins of large harmonic vibrational hyperpolarizability for compound 11. Fig. 4 shows the orientation of compound 11 in the cartesian coordinate system (note that the two-fold symmetry axis is parallel to the cartesian z direction). The breakdown of [μα]_‖^(0,0) into symmetry-unique non-vanishing cartesian components is presented in Fig. 5. As seen, the cartesian zzz component parallel to the symmetry axis is by far the largest. In order to link this component with the vibrational structure of 11 we performed the sum-over-normal modes calculations and the results are shown in Fig. 6. It is clear that the first normal mode with wavenumber 17 cm⁻¹ makes the dominant contribution (91% of [μα]_zzz^(0,0)). Atomic displacements for the normal mode in question show the symmetric vibrations of the propellane wings (see Fig. 4). For propellane molecule 11, in a noncentrosymmetric crystal structure with the z-axis parallel to the crystal polar axis,¹¹ the existence of such “butterfly motion” polar mode could be beneficial for the crystal electrooptic properties, especially in the THz frequency range.⁴⁸ Strong second-order nonlinear optical properties of organic crystals, e.g., such as found for DAST crystals,^49,50 can be exploited for both generation and detection of THz radiation. Organic crystals possessing high THz-generation efficiencies are highly desired for THz spectroscopy, imaging and analysis due to unique penetration properties of THz radiation.^51,52

Fig. 4 The orientation of compound 11 in the cartesian coordinate system. Atomic displacements corresponding to the lowest-frequency normal mode are indicated by arrows.

Fig. 5 Cartesian components of harmonic vibrational hyperpolarizability (in au) for compound 11.

Fig. 6 Convergence of the zzz component of harmonic vibrational hyperpolarizability (in au) for compound 11.

4 Conclusions

Aiming to establish “structure–property” relations, in this work we have employed computational quantum chemistry methods to study first order hyperpolarizabilities of 16 propellanes. In particular, we considered a wide palette of symmetrically-placed substituents (–CF₃, –SCH₃, –SO₂CH₃, –SCF₃, –OCF₃, –CH₂OCH₃, dioxolane and –NO₂) which cover a wide range of values of Hammett substituent constants. It has been demonstrated that the maximum values of frequency-independent first hyperpolarizability can be obtained for strong electron-accepting NO₂ substituents but only at the preferred positions with respect to the electronegative oxygen atom in the 1,4-oxazine moiety. This holds for electronic as well as vibrational counterparts. However, asymmetric substitution with NO₂ and NH₂ groups at these preferred positions (compound 15) leads to equally high electronic and vibrational hyperpolarizability values. The variations in computed properties are large among the studied set of substituents and can reach an order of magnitude. In order to shed light on these findings, we employed a generalized few state model to study the electronic counterpart. It turned out that the overall convergence of the first hyperpolarizability with respect to the number of electronic states is slow, indicating a complicated origin of this property. Some of the studied herein propellanes (7, 10, 13), due to similar molecular structure with compound 11 and large values of their first hyperpolarizabilities can be considered as promising candidates for synthesis, crystal growth and the evaluation of their second order NLO properties, providing that they will crystallize in noncentrosymmetric point groups.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors cordially thank Miss Elizaveta F. Petrusevich for the preparation of the graphical abstract. The authors acknowledge the Wroclaw Center for Networking and Supercomputing for granting computational resources. MMA acknowledges the research initiation grant from IIT Bhilai.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2cp00381c

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