Conformational Dependence of the First Hyperpolarizability of the Li@B₁₀H₁₄ in Solution

Abstract

Using the ASEC-FEG approach in combination with atomistic simulations, we performed geometry optimizations of a C_s conformer of the lithium decahydroborate (Li@B₁₀H₁₄) complex in chloroform and in water, which has been shown to be the most stable in the gas phase and calculated its first hyperpolarizability. At room temperature, ASEC-FEG calculations show that this conformer is stable only in chloroform. However, it is found that the nonlinear response of the C_s conformer in chloroform is mild, and the result for the hyperpolarizability is markedly decreased in comparison with the result of the C_2v conformer.

1. Introduction

Among the molecular systems with nonlinear optical (NLO) responses, electrides have excess electrons that are not located on any specific atom [1,2]. Several theoretical investigations have demonstrated that electride-like structures can be designed by doping an alkali metal atom into organic complexants and can exhibit extraordinary NLO properties relative to the corresponding undoped organic ones [3,4,5,6,7,8,9,10,11]. However, organic electrides may have limited practical applications due to their thermal instability at room temperature and reactivity with air [1,12].

Electrides with relatively good thermal stability and NLO properties have been designed by the addition of lithium atoms to inorganic borane clusters [13,14,15]. In particular, Muhammad et al. [13] showed, in the gas phase, that an inorganic electride obtained from adding a Li atom to the cavity of decaborane (B₁₀H₁₄) can have a remarkable first hyperpolarizability. In that study, the authors considered an arrangement in which an added Li atom interacts with terminal H atoms on the open face of the basket-shaped molecule (C_2v symmetry). Based on this geometric structure, we carried out an investigation of the stability of this molecular shape of Li@B₁₀H₁₄ in chloroform and water. We showed, at room temperature, that the geometric changes induced in solution in the C_2v conformer of Li@B₁₀H₁₄ result in a stable structure only in chloroform [16]. Recently, Medved et al. [17] have shown, in the gas phase, based on the global energy minimum analysis, that C_s conformers (Li atoms close to the decaborane basket from the side or the side-bottom) of Li@B₁₀H₁₄ can be significantly more stable than the C_2v structure, with possible consequences on their NLO responses. Thus, from the most recent theoretical results, there is a possibility that Li@B₁₀H₁₄ conformers containing a loosely bound excess electron may be stable in the solution and present different NLO responses.

In the present study, we report the results on the stability and first hyperpolarizability of a C_s conformer of Li@B₁₀H₁₄ (Figure 1) in chloroform and water solvents, which has been shown to be the most stable in the gas phase [17]. This represents an extension of a previous calculation on the stability and first hyperpolarizability of the C_2v conformer of Li@B₁₀H₁₄ (Figure 1) in solution [16]. The absorption spectrum is also presented, considering that the addition of an excess electron can affect the electronic excitation energies of this inorganic electride. The equilibrium structure of the C_s conformer in each medium, at temperature and pressure under ambient conditions, was obtained using the ASEC-FEG scheme [18,19,20,21,22,23,24,25] by employing iteratively the sequential quantum mechanics/molecular mechanics (S-QM/MM) method [26,27]. The S-QM/MM approach has been successfully applied to reliably describe the effects of solvent on the electronic properties of solute molecules [28,29,30,31,32].

2. Computational Details

We have used the ASEC-FEG methodology to optimize the geometry of the Li@B₁₀H₁₄ system in the solution. The methodology is described in detail in previous papers [22,23,24,25]. Briefly, it consists in applying iteratively the Sequential QM/MM methodology [26,27] combined with the Average Solvent Electrostatic Configuration (ASEC) mean field technique [21] and the Free Energy Gradient (FEG) method [18,19,20] to provide free energy gradients and hessians to feed a quasi-newton optimization procedure in the search for the nearest local minimum in the free energy hypersurface.

During the classical simulations, all solvent molecules were kept rigid, and the temperature and pressure were kept at ambient conditions. The molecular properties of the solution were obtained with the unrestricted second-order Møller–Plesset Perturbation Theory (UMP2) using the ASEC mean field [21], where the solvent molecules were treated as point charges. Following previous work [16], boron and hydrogen atoms were described with the cc-pVDZ basis set. For the lithium atom, the aug-cc-pVDZ basis set was used because it is more appropriate to describe the loosely bound valence electron. Thus, for the ASEC-FEG geometry optimizations of the C_s conformer of Li@B₁₀H₁₄, such basis sets were used, denoted as (Li-aug)-cc-pVDZ. For water, the potential used was the extended simple point charge (SPC/E) model [33], because it leads to very good agreement for the critical point of water [34]. For chloroform, we have used the OPLS force field [35]. The atomic charges were obtained by performing an electrostatic potential fit (Merz-Kollman) [36] to the MP2/aug-cc-pVDZ charge density.

Here we report the permanent dipole moment (μ) and the first hyperpolarizability properties that can be compared with experimental results of hyper-Rayleigh scattering (HRS) in the solution [37]. The static first hyperpolarizability components (β_ijk) were obtained as numerical differentiation of the energy calculated in the presence of different values of the applied electric field, as implemented in the Gaussian 16 package [38]. Both μ and β_ijk were calculated at the MP2/aug-cc-pVDZ level of theory. The choice of this basis set is supported by previous tests and ensures a good compromise between computational cost and accuracy [39,40,41].

In solution, the NLO quantities that can be compared with HRS measurements are the HRS first hyperpolarizability (β_HRS) and the associated depolarization ratio (DR), given respectively by:

$${\mathsf{\beta }}_{\mathrm{HRS}}=\sqrt{\langle {\mathsf{\beta }}_{ZZZ}^2\rangle +\langle {\mathsf{\beta }}_{ZXX}^2\rangle }$$

(1)

and

$$\mathrm{DR}=\frac{\langle {\mathsf{\beta }}_{ZZZ}^2\rangle }{\langle {\mathsf{\beta }}_{ZXX}^2\rangle },$$

(2)

whose expressions for $\langle {\mathsf{\beta }}_{ZZZ}^2\rangle$ and $\langle {\mathsf{\beta }}_{ZXX}^2\rangle$ are given in Ref. [42].

The vertical electronic excitation energies were calculated using the TD-DFT method with different exchange–correlation functionals based on the generalized gradient approximation (GGA): LC-BLYP [43], CAM-B3LYP [44], M05-2X [45], and M06-2X [46] functionals. The aug-cc-pVDZ basis has been adopted in all TD-DFT calculations. All QM calculations were performed using unrestricted methods, and the spin contamination was found to be very small, less than 2% from the exact expected value. The classical Monte Carlo simulations were performed with the DICE program [47] and the QM calculations with the Gaussian 16 program [38].

3. Results and Discussion

Table 1. MP2/(Li-aug)-cc-pVDZ results for bond lengths (Å) of the C_s conformer in the gas phase and solution. In the case of chloroform, the values are convergent within an error bar of 0.001 Å.

Length	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)
B1-Li	2.256	2.295	3.150
B2-Li	2.244	2.281	3.230
B3-Li	2.256	2.295	3.228
H9-Li	2.021	2.051	2.679
H10-Li	1.957	1.980	2.764
H11-Li	2.021	2.051	2.981
H23–Li	2.043	2.169	2.819

shows the MP2/(Li-aug)-cc-pVDZ results for bond lengths between the Li atom and closest B and H atoms of the optimized geometry of C_s conformer of Li@B₁₀H₁₄ in the gas phase, chloroform, and water. The converged values of maximum and RMS forces are below 6.5 × 10⁻⁴ and 2.5 × 10⁻⁴ hartree/bohr, respectively, which are reasonable values considering fluctuations due to liquid dynamics [22]. For the C_s conformer in chloroform, there is a slight increase in the bond lengths B₁–Li, B₂–Li, and B₃–Li relative to the gas phase results, but these are smaller than the corresponding closest distances of the C_2v conformer [16]. This indicates that the C_s conformer of Li@B₁₀H₁₄ is stable at room temperature in chloroform. Differently, the electride-like structure of the C_s conformer is unstable in water at room temperature, and during the simulations, the lithium ion dissociates, as previously shown for the C_2v conformer [16]. Nevertheless, we compute the properties of interest using the in-water geometry from the last step of the iterative process.

Having obtained the optimized geometry of the C_s conformer in chloroform, the relative stability of the C_s and C_2v conformers is evaluated at the MP2/aug-cc-pVDZ level. Our energy calculations indicate that the C_s conformer in chloroform is more stable than the C_2v conformer by an amount of 21 kcal/mol. This finding is in agreement with the previous study by Medved et al. [17], which showed that the C_s conformer is the most stable form of Li@B₁₀H₁₄ in the gas phase. It should be emphasized that the inclusion of the solute-solvent interaction contribution in the energy calculations of the conformers gives a marked decrease in the energy difference between the conformers, which is 35 kcal/mol in the gas phase. Following previous work [16], we have also calculated the vertical ionization potential (VIP) of the C_s conformer in the gas phase and solution at the MP2/aug-ccpVDZ level, used as an indicator of thermal stability. It is found that the formation of excess electrons is slightly more stable in chloroform (6.27 eV) than in the gas phase (6.06 eV). In chloroform, however, our previous result indicated that the excess electron is slightly more stable for the C_2v conformer (6.66 eV) [16] than for the C_s one. In water, the VIP value of 6.05 eV for the C_s conformer also suggests relative stability for the excess electron on the decaborane cluster. For comparison, the VIP value for the C_s conformer in chloroform is significantly larger than some electrides formed by doping the lithium atom in organic compounds, such as Li@calyx[4]pyrrole (4.16 eV) [3], Li-[15]aneN₅ (2.32 eV) [6], and (Li⁺@n⁶adz)Li⁻ (2.88 eV) [48].

Figure 2 illustrates the behavior of the dipole moment of the C_s conformer in chloroform solution during the iterative process, indicating a rapid convergence pattern. The MP2/aug-cc-pVDZ results for µ of the C_s conformer in the gas phase and solution are quoted in

Table 2. MP2/aug-cc-pVDZ results for the dipole moment (in a.u.) of the C_s conformer in the gas phase and solution. For comparison, the corresponding results for the C_s and C_2v conformers are included.

Conformer	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)
Cs	2.53 (2.55 2)	2.91	6.56
C2v	3.59 1 (3.72 2)	4.59 1	9.42 1

¹ Results taken from Ref. [16]. ² Results taken from Ref. [17].

. The corresponding results for the C_2v conformer are also reported in this table. For the C_s conformer in chloroform, partial negative charges are distributed on the boron and hydrogen atoms closest to the lithium atom, and the polarization effect induces a slight increase in q(Li), with the in-chloroform value of 0.81 e when compared with the gas phase result of 0.77 e. This leads to the in-chloroform result for µ of 2.91 a.u., which is increased by 15% when compared to the gas phase result of 2.53 a.u., but it is 37% smaller than the in-chloroform result for the C_2v conformer of 4.59 a.u. [16]. In water, the excess charge of the C_s conformer is delocalized over the decaborane cluster. The in-water q(Li) charge is found to be 0.97 e, close to +1 e, which shows that the Li atom has been ionized to form a cation and the B₁₀H₁₄ anion, and the in-water μ value is increased by 59% relative to the gas phase value. In the gas phase, our MP2/aug-cc-pVDZ results for µ are in good agreement with the MP2/6-31+G(d)//MP2/aug-cc-pVDZ results of Medved et al. [17], which are also quoted in Table 2, whose differences do not reach 4%.

MP2/aug-cc-pVDZ results for the static HRS first hyperpolarizability and depolarization ratio of the C_s conformer obtained in the gas phase and solution are gathered in

Table 3. MP2/aug-cc-pVDZ results for the static HRS first hyperpolarizability (in a.u.) and depolarization ratio of the C_s conformer. For comparison, the corresponding results for the C_2v conformer are included.

Conformer	βHRS (a.u.)			DR
	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)
Cs	504	212	154	4.94	4.02	7.23
C2v 1	6214	2786	64	6.20	6.52	3.57

¹ Results taken from Ref. [16].

. The corresponding results for the C_2v conformer are also reported in this table. One can see that the polarization effect of the excess electron for β_HRS of the Li@B₁₀H₁₄ complex is particularly affected by the conformational structure, as shown by Medved et al. [17]. In the gas phase, the result for β_HRS of the C_s conformer (504 a.u.) is markedly reduced by 92% when compared with that of the C_2v conformer (6214 a.u.). As shown in previous work [16], however, the electride-like characteristic gives a very large hyperpolarizability for the C_2v conformer.

Our MP2/aug-cc-pVDZ results for β_HRS of the C_s conformer are very sensitive to the solvent effects, with significant reductions in chloroform and in water. For this conformer in chloroform, the result of 212 a.u. for β_HRS has a significant decrease (58%) relative to the gas phase result of 504 a.u., indicating that the contribution of the excess electron for β_HRS is mild. In water, the MP2 model predicts for β_HRS of the C_s conformer a result of 154 a.u. which is 27% smaller than the result in chloroform. Despite the structure of the C_s conformer being unstable in water, the first hyperpolarizability is not significantly affected in relation to the value in chloroform. Notice that solvent effects significantly impact the depolarization ratio (Table 3) of the C_s conformer in water, and the DR value of 7.23 indicates a marked dipolar contribution to β_HRS. In chloroform, however, there is a less pronounced dipolar contribution to β_HRS, whose DR value of 4.02 for the C_s conformer is 38% smaller than the corresponding value of the C_2v conformer.

A qualitative interpretation of the solvent dependence of the hyperpolarizability of the C_s conformer can be obtained from the two-level model [49], where β_HRS is proportional to the difference between the crucial excited state permanent dipole moment and the ground state permanent dipole moment (Δμ) and the oscillator strength (f₀) and inversely proportional to the third power of the transition energy (ΔE):

$${\mathsf{\beta }}_{\mathrm{HRS}}\propto \mathsf{\Delta }\mu f_0/\mathsf{\Delta }{E}^3$$

(3)

In this model, the transition energy plays an important role in determining the first hyperpolarizability. For the C_s conformer, the first absorption band is dominated by a highest occupied orbital molecular (HOMO) → lowest unoccupied orbital molecular (LUMO) excitation. However, TD-DFT results indicate that the first transition has a very small oscillator strength, regardless of the environment.

Figure 3 shows the absorption spectra of the C_s conformer in the gas phase and the solution obtained at the CAM-B3LYP/aug-cc-pVDZ level. The corresponding spectra for the C_2v conformer are also reported in this figure. TD-DFT/aug-cc-pVDZ results for the excitation energy and oscillator strength of the more intense electronic transition of the C_s conformer are quoted in

Table 4. TD-DFT results for excitation energy (in eV) and (oscillator strength) of the more intense electronic transition (considering the first 10 excited states) of the C_s conformer in gas phase and in solution. For comparison, the results for the first electronic transition of the C_2v conformer are included. All results were obtained using the aug-cc-pVDZ basis set.

	Cs Conformer			C2v Conformer 1
Method	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)	Gas-Phase	Chloroform (Converged Value)	Water (Last Step Value)
LC-BLYP	4.58 (0.0302)	4.48 (0.0439)	4.90 (0.0723)	2.23 (0.0504)	2.59 (0.0505)	3.81 (0.0069)
M05-2X	4.10 (0.0300)	4.21 (0.0331)	4.58 (0.0773)	2.28 (0.0503)	2.64 (0.0494)	3.55 (0.0065)
M06-2X	3.65 (0.0211)	3.95 (0.0225)	4.30 (0.0351)	1.92 (0.0573)	2.43 (0.0522)	3.44 (0.0011)
CAM-B3LYP	4.38 (0.0319)	4.22 (0.0201)	4.64 (0.0744)	1.87 (0.0578)	2.25 (0.0553)	3.63 (0.0061)

¹ Results taken from Ref. [16].

. The corresponding results for the first electronic transition C_2v conformer are also reported in this table. One can see that the absorption spectra of the C_s conformer, not only in chloroform but also in water, are characterized by more intense transitions with energies larger than 4 eV. Thus, there is a considerable difference between the crucial electronic transition energies for C_s and C_2v conformers in both chloroform and water by a factor around 2, and, therefore, the main factor for the marked decrease of β_HRS of the C_s conformer is related to the significant increases of the crucial transition energy (Equation (3)). In addition, similar energy values for the more intense transition in chloroform and water, suggest that the β_HRS of the C_s conformer does not change significantly in any environment.

4. Conclusions

We have more recently shown that the C_2v conformer of the Li@B₁₀H₁₄ complex, which had its NLO properties studied in the gas phase by Muhammad et al., is a stable structure in chloroform at room temperature. However, Medved et al. have recently shown that C_s conformers of the Li@B₁₀H₁₄ complex in the gas phase can be significantly more stable than the C_2v conformer, with a marked effect on the second-order NLO properties. In this work, we report an investigation on the stability and first hyperpolarizability of a C_s conformer in aprotic and protic solvents using the sequential QM/MM methodology in combination with the ASEC-FEG model, which has been shown to be the most stable in the gas phase. ASEC-FEG calculations show that the solvent-induced geometric changes in the C_s conformer at room temperature result in a stable structure in chloroform and an unstable structure in water. Our MP2 results show that the C_s conformer in chloroform is more stable than the C_2v conformer by an amount of 21 kcal/mol, which confirms the finding of Medved et al. However, the relative stability of the conformers is significantly reduced due to the solute-solvent interaction energy when compared with the gas phase result, which is 35 kcal/mol. Polarization effects due to the electrostatic embedding lead to a reduction of 58% of the hyperpolarizability of the C_s conformer in chloroform in comparison with the gas result of 504 a.u. In chloroform, the contribution of the excess electron to the first hyperpolarizability of the C_s conformer is mild, and the result for β_HRS of 212 a.u. is markedly decreased by a factor of 92% in comparison with the corresponding result for the C_2v conformer of 2786 a.u. Therefore, the small first hyperpolarizability of the C_s conformer suggests that this inorganic electride-type system should not present significant second-order nonlinear responses in any environment. This finding also illustrates that the influence of the environment must be carefully considered in the design of new molecules with excess electrons that are stable at room temperature for aiming applications in NLO.