DFT-based Study of Electric Field Effect on the Polarizability of Three Ringed Nematic Liquid Crystal Molecules

Owing to its successful application to complex molecular systems, computational density functional theory (DFT) has been used to study the effect of an electric field on the molecular polarizability and HOMO–LUMO gap of 1-phenyl-4-{2[(1s,4r)-4-pentylcyclohexyl]ethyl}benzene (1) and its fluoro-, chloro-, and cyanoderivatives, namely, 1-fluoro-4-(4-{2[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl)benzene (2), 1-chloro-4-(4-{2-[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl)benzene (3), and 4-(4-{2-[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl)benzonitrile (4). These molecules belong to the family of nematic liquid crystals with three rings: two benzene and one cyclohexane. Furthermore, two DFT approaches, namely, B3LYP and M062X, have been used to examine the results obtained. This study reveals a remarkable feature: the polarizability of these molecules follows nearly a step function when varied with respect to the electric field. The 4-(4-{2[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl)benzonitrile (4) polarizes more than all other derivatives, whereas 1-fluoro-4-(4{2-[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl) benzene (2) has the widest stability region of them all. With the increase in the electric field, polarizability increases in a smooth manner until a point called here the shoot-up point at which polarizability switches to a higher value and remains nearly constant as the field increases further. However, beyond a certain value of the electric field, polarizability undergoes a steep fall. It is also found that the effective length (long molecular axis) of the molecule has a direct effect on its polarizability.

Introduction "Liquid crystal" or "mesophase" are the terms used to describe a state of matter that exhibits mechanical and symmetric properties between those of a viscous liquid and a crystal. Nematic liquid crystals consist of rod-like molecules that, on an average, line up parallel to a preferred direction. This orientation of molecules is modified when placed in an electric or magnetic field [1,2]. Research on liquid crystals is an ever-evolving field, with several novel and high-end applications of these molecules, but there is always scope to improve and enhance their performance. Novel molecules that are promising candidates for liquid crystal applications are a vital area of research. With the assistance of state-ofthe-art computational techniques, this research can be helpful in understanding the nature and properties of mesophase molecules.
The dielectric constants of liquid crystals are anisotropic. The relationship between the dielectric constant and molecular polarizability of liquid crystals can be described by the Maier-Meier equations [3]. Interactions between molecules and forces like dispersion are affected by polarizability and polarization caused by external and internal factors [4].
The effect of an electric field on liquid crystal molecules is an important area of investigation as most liquid crystal devices use electric fields, optical fields, or thermally induced refractive index changes to modulate light [5]. Upadhyay et al. [6] reported the effect of continually increasing the electric field on the polarizability of 5CB and its derivatives, which are room-temperature nematics with two benzene rings without any linker. Using a molecular dynamics simulation, coarse-grained dynamics simulation, and density functional theory (DFT), Ju et al. [7] have worked on 5CB to predict some of its physical properties. Similarly, Kumar et al. [8] used DFT method for investigating odd-even effects under the influence of an external applied electric field on N(p-n-heptyloxy-benzylidene)p-toluidine by increasing  [11] the length of an alkyl chain. Therefore, taking a hint from these studies in general and from Upadhyay et al. [6] in particular, it seems necessary to study effects on a different set of molecules, where the number of rings is greater than two so that it can be determined that previously observed effects apply to other nematics too.
This paper presents a comparative study of the effect of a continually increasing electric field on the molecular polarizability and HOMO  Table 1 gives a summary of the molecular structure, transition temperatures, and dielectric contents of the molecules studied.

Methods
The molecules under study were designed using Gauss View 5.0 and the structures of these molecules were optimized using computational DFT, which has delivered several promising results for characterization, explanation, and establishing a correlation of properties of various types of molecules [12,13]. Structural optimiza-tion and energy calculations were performed using the B3LYP [14,15] hybrid functional for Gaussian-type orbitals and the 6-31G** [16] basis set in the Gaussian 09 package [17]. The dihedral angle of the linker between cyclohexane and the benzene ring was varied from 0° to 350° to find the minimum energy configuration of the structures. The variation of molecular polarizability (α) on application of the electric field was calculated using the BL3YP and M062X [18] methods using a 6-31G** basis set with the help of the Gaussian 09 package.
Jensen [19] explained the calculation of polarizability as implemented in Gaussian 09. Starting with an initial value of 0.000 a.u., the field was varied through increments of 0.002 a.u. The trace of the polarizability tensor, thus, obtained was used to find the value of polarizability.   Figure 2). The HOMO-LUMO gap also varies with the electric field, as shown in Figure 2. It can be seen that the HOMO-LUMO gap achieves stabilization for the same value at which there is the stabilization of polarizability.

1-fluoro-4-(4-{2-[(1s,4r)-4-pentylcyclohexyl]ethyl} phenyl)benzene.
Substituting hydrogen with fluorine in the benzene ring at one end of the molecule ( Figure  1(b)) resulted in increased long molecular length (as shown in Figure 1), which in turn raised the zero-field polarizability by a small amount (approximately 273.2 and 270.5 a.u. for B3LYP and M062X, respectively). Polarizability sharply increases beyond an electric field of 0.018 a.u. and 0.022 a.u. for B3LYP and M062X (increments of 0.06 a.u. for each), respectively, up to polarizabilities of approximately 1324 and 1691 a.u., respectively. After remaining stable between electric fields of 0.020-0.046 a.u. for B3LYP and 0.024-0.046 a.u. for M062X, polarizability steeply falls. The variation in the HOMO-LUMO gap is modified in accordance with the electric field, as shown in Figure 3.

1-chloro-4-(4-{2-[(1s,4r)-4-pentylcyclohexyl]ethyl} phenyl)benzene.
When fluorine is replaced with another halogen chlorine, this results in a slight increase in effective length (long molecular axis) and higher zero-field polarizability (approximately 289 and 285 a.u. for B3LYP and M062, respectively) than the previous two cases. Although the shoot-up point for the electric field has been reduced by 0.04 a.u. (0.016 and 0.02 a.u., respectively), the stability region has narrowed by 0.06 a.u. for both B3LYP and M062X. The electric field values at which a sharp decrease occurs are 0.036 a.u. and 0.034 a.u. for B3LYP and M062X, respectively. Here, the HOMO-LUMO gap and molecular polarizability also become stabilized for the same set of electric field values.

4-(4-{2-[(1s,4r)-4-pentylcyclohexyl]ethyl}phenyl)
benzonitrile. Replacement by a cyano-group produces not only the longest effective length (long molecular length) but also the highest zero-field polarizability (approximately 298 and 293 a.u. for B3LYP and M062X, December 2020  Vol. 24  No. 4 respectively) of all cases. The shoot-up point (0.018 and 0.022 a.u., respectively) is 0.02 a.u. higher than that in the previous case and the point where a steep decrease begins is 0.04 and 0.034 a.u. for B3LYP and M062X, respectively. The polarizability in the stability region is highest (approximately 1492 and 1879 a.u. for B3LYP and M062X, respectively) among the previous cases. The corresponding HOMO-LUMO gap variation is shown in Figure 5.
Effect of the field on the charge distribution. The conjugation effect that arises because of the increasing electric field can be justified by the Mulliken charge distribution on the respective atoms of these molecules at three electric fields, which correspond to the initial lower value of polarizability (before the shoot-up point), region of stability, and region in which polarizability drops. The Mulliken charge values are provided in Tables S1 to S12 (Supporting Information).

Discussion
To ascertain the results obtained using the B3LYP exchange-correlation functional, it was decided to use another exchange-correlation functional, M062X. Since the study is based on theoretical calculations, it is important to validate the results obtained from one functional with those from another. Moreover, both functionals have a proven track record when applied to DFT calculations in complex molecules [18,20].
The patterns of variation of polarizability and of the HOMO-LUMO gap were found to be similar to those obtained by Upadhyay et al. [6] for two-ringed 5CB (along with its derivatives) but with greater sharpness at the points of abrupt change. The variations in the HOMO-LUMO gap and the polarizability with electric field have a common feature, i.e., there are two regions, upper and lower, in which polarizability remains almost stable with a shoot-up point between these two regions.
The HOMO-LUMO gap decreases in a uniform manner and then becomes stabilized. Compound 4 is polarized most compared with the others, whereas compound 2 has the widest upper stability region. The shoot-up point is the lowest for compound 1. Above this point, polarizability increases because, owing to the conjugation effect, which increases the number of mobile electrons, hydrogen atoms attached to the end carbon atoms of the alkyl chain start gaining negative charge, as can be seen in Figure 6. As the field is gradually increases, the polarizability achieves  Except for compound 1, all compounds have an electronegative functional group at one end. Compound 2, despite having the most electronegative atom, fluorine, at one end, becomes the least polarized (compared with compounds 3 and 4) in the electric field instability region and it is only slightly more polarized than compound 1. The effective length (long molecular axis) of the molecule has a direct effect on its polarizability. As seen from Figure 1, the molecule with a larger length has greater polarizability than the others, but also molecule 2 has the widest stability region.
On application of the electric field, both M062X and B3LYP start with nearly identical values of polarizability (slightly higher for B3LYP), but in the upper stability region, M062X gives a much higher value than B3LYP. In addition, the shoot-up point for M062X is higher than that for B3LYP. The upper stability region is wider for B3LYP than M062X. Moreover, the HOMO-LUMO gap is flat in the stability region of the respective molecules.  (4). ) are used in numerous applications such as LCDs and Spatial Light Modulators. Therefore, it is important to understand how electric fields affect liquid crystal molecules in terms of polarizability, as this is a deciding factor for dispersion forces and interaction energy in a bulk material. The study of polarizability is also important for electro-optic properties such as birefringence and the dielectric constant. The above studies showed that there is a certain range of applied electric fields for which polarizability and the HOMO-LUMO gap remain stable or constant. This study revealed a remarkable feature, namely, that the polarizability of these molecules follows nearly a step function. This can be attributed to the charge conjugation effect that comes into play as the magnitude of the electric field is increased. However, as the strength of the electric field crosses a certain high value, the availability of a large number of charge carriers causes a sudden drop in the polarizability, which is expected because a large number of mobile charge carriers makes the molecule less polarizable. The substitution of hydrogen with different groups, such as fluorine, chlorine, and cyano-affects the molecular length as well as the charge distribution, which has a direct effect on the magnitude of polarizability.

Conclusion
As no experimental data are available for these molecules and for such study, it cannot be determined precisely which model is closer to reality. M062X is accurate in predicting valence and Rydberg electronic excitation energies, including dispersion corrections and provides excellent results for aromatic-aromatic stacking interactions [6]. M06 functionals perform better than B3LYP for a model system with dispersion and ionic hydrogen-bonding interactions [20], whereas spectroscopic studies show that the B3LYP method is a better predictor of the experimental values than the M062X method [21].
However, this study established how the polarizability and HOMO-LUMO gap for these molecules vary with an externally applied electric field. Although the values obtained using the B3LYP and M062X functionals are different, the trends followed by both the polarizability and the HOMO-LUMO gap are similar in both cases. Therefore, this study established how the polarizability and the HOMO-LUMO gap of these molecules will vary with the intensity of the externally applied electric field.         1 C -0.067443 1 C -0.046667