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Various characteristics of mesomorphism can be explained using intermolecular interactions between a pair of liquid crystalline molecules. The intermolecular interactions have been calculated considering multipole-multicentere expansion method and modified by second order perturbation treatments. For calculation of multipole i.e. charge, dipole, etc. at each atomic center of molecules, para-butyl-p’-cyano-biphenyl, GAMESS, an ab initio program, with 6-31G^{*} basis set has been used. The stacking, in-plane and terminal interaction energies explain the liquid crystalline behaviour of the system.

There are certain substances which do not directly pass from a crystalline solid to an isotropic liquid state and vice versa, rather adopt an intermediate structure which flows like liquids but still possesses the anisotropic physical properties similar to crystalline solids. In view of the wide-spread use of liquid crystals from industrial and technological developments to biomedical applications and display devices, the subject of liquid crystal science has attracted increasing interest of the researchers from different disciplines [1-4]. The peculiar changes— characteristics of mesomorphic behaviour which occur at phase transition, are primarily governed by the nature and strength of intermolecular interactions acting between sides of planes and ends of a pair of molecules [

In the light of the above facts, intermolecular interaction energy studies in case of some mesogens have been carried out in our laboratory and efforts have been made to explain mesomorphism [12-17]. In continuation of our earlier studies on thermotropic liquid crystals, the present paper embodies the results of both stacking and in-plane intermolecular interactions in case of p-butyl-p’-cyanobiphenyl (4CB) which is a lower homologue of p-alkylp’-cyano-biphenyl series. Thermodynamic parameters reveal that 4CB shows crystal to nematic transition at 48˚C and passes to an isotropic melt state at 16.5˚C [

According to the energy decomposition obtained by perturbation treatment, the total interaction energy () between two molecules is expressed as [

where and represent electrostatic, polarization, dispersion and repulsion energy components respectively. The formulae for various energy terms are given as under.

According to the multicentered-multipole expansion method [

where_{,} and etc. represent monopolemonopole, monopole-dipole, dipole-dipole and interaction energy terms consisting of multipoles of higher orders respectively. However, consideration upto the first three terms are found to be sufficient for most of the molecular interaction problems [

where q_{i} and q_{j} are the monopoles at each of the atomic centre of the interacting molecules i and j; is the inter-atomic distance. The constant, C is a conversion factor, approximately equal to 332 which expresses the energy in kcal/mole of the dimer.

The monopole-dipole energy term is expressed as

and the dipole-dipole interaction term is given by

where µ_{i} and µ_{j} represent atomic dipoles, the subscript of r has been removed without any change in its meaning.

The polarization energy of some molecule (say, s) is obtained as a sum of the polarization energies for the various bonds:

where is the electric field created at the bond u by all surrounding molecules and is the polarizability tensor of this bond. is the vector joining the atom λ in molecule (t) to the centre of polarizable charge on bond u of molecule (s).

Dispersion and repulsion terms are calculated together using Kitaigorodskii type of formula as given below [22-24]:

where

and

where and are the van der Waals radii of atoms λ and ν respectively. The parameters A, B and γ do not depend on the atomic species: this necessary dependence is brought about by and the factors and which allow the energy minimum to have different values according to the atomic species involved [

Molecular geometry of 4CB has been constructed using crystallographic data from literature [^{*} basis set. The energy minimization has been carried out for both stacking and in-plane interactions separately.

One of the interacting molecules is kept fixed throughout the process while both lateral and angular variations are introduced in the other in all respects relative to the fixed one. The first molecule has been assumed to be in the X-Y plane with X-axis lying along the long molecular axis while origin is chosen approximately at the centre of mass of the molecule. The second molecule has been translated initially along the Z-axis (perpendicular to the molecular plane) and subsequently along Xand Y-axes. Variation of interaction energy with respect to rotation about Z-axis has been examined in the range of ±60˚. Accuracies up to 0.1 Å in sliding (translation) and 1˚ in rotation have been achieved [12,13, 28].

The schemes of three modes of interactions of a molecular pair are shown in

interaction namely stacking, in-plane and terminal interactions have been shown in Figures 1(b)-(d) respectively. The molecular geometry of 4CB as optimized by GAMESS with 6-31G^{*} basis set is shown in

The variation of interaction energy with respect to sliding (translation) of one of the stacked molecules along the long molecular axis (X-axis) corresponding to four fixed rotations about the Z-axis, namely Z, Z, Z and Zhas been shown in

within electrostatic terms in comparison to monopoledipole and dipole-dipole interactions. Polarization term is found to be insignificant in stabilizing the molecules in the crystals. Again dispersion components are mainly responsible for the attractions between the pairs of 4CB molecules though the exact optimum point is always located by Kitaigorodskii energy curve which has a gross similarity with the total energy curve. It is interesting to note here that for translation in the range of ±2.0 Å, minor variation in the energy (less than 1.0 kcal/mole) is observed which implies that in the stacked pair of 4CB, molecules can slide one above the other in a range of ±2.0 Å without any significant change in the energy (

The angular dependence of stacking energy components (

The energy corresponding to the optimum angle located at 0˚ has been refined with accuracies 1˚ in rotation and 0.1 Å in translation. The final lowest energy stacked geometry, thus obtained, has been shown in

The intermolecular interaction energy calculations may reasonably be correlated with the mesomorphic behaviour of the system. When the solid crystals of 4CB molecules are heated, thermal vibrations disturb the molecular ordering of the strongly packed geometrical arrangement of 4CB molecules. Consequently, attractions between the pair of molecules which largely comprise of dispersion forces tend to get weaker at higher temperatures and hence translational freedom along the long molecular axis (

molecular pair (

The length of molecule is approximately 15 Å, to investigate the terminal interactions away from the van der Waals contacts, the interacting molecule has been shifted along the axis by ±20 Å with respect to fixed one and allowed to rotate along the Xand Y-axis. The energies at such point having examined and found terminal interaction of a pair of 4CB molecule with minimum energy of −2.81 kcal/mole with interplaner separation 4.55 Å.

The minimum energy configuration in case of terminal interaction has been shown in

The interaction energy calculation can be correlated with the mesomorphic behavior of the system. When solid crystals of 4CB are heated, thermal vibrations disturb the molecular order of the strongly packed 4CB molecules. Consequently, the attraction within a pair of molecules, largely comprising the dispersion forces, tend to get weaker at higher temperatures, and hence the possibility of relative movement within a molecular pair along the long molecular axis is considerable enhanced. The freedom of molecule in a pair to slide along an axis perpendicular to long molecular axis (Y-axis) is energetically restricted. While terminal interactions, are quite insignificant. The results favour the nematic behaviour of the system. At very high temperature breaking of all dispersion forces results and possible stacking geometry even perpendicular stacking become equally probable which ultimately causes the system to become an isotropic melt.

The most prominent energy minima of above mentioned interactions are refined, and values thus obtained are listed in _{1} is maximum and ultimate magnitude of stacking is larger than in-plane and terminal interactions. Further, all possible geometrical arrangements between a molecular pair during stacking, in-plane and terminal interactions have been considered.

It may, therefore, be concluded that intermolecular interaction energy calculations are helpful in analyzing the liquid crystallinity in terms of molecular forces. Results favour the nematic behaviour of the system at higher temperatures because the molecules of 4CB are capable of sliding along the long molecular axis with a simultaneous relative orientation of 40˚. At very high temperatures, an all-round breaking of dispersion forces results and all possible stacking geometries (even perpendicular stacking) are almost equally favoured, which ultimately cause the system to pass on to an isotropic melt state.

One of us Dr. Manoj Kumar Dwivedi is thankful to UGC, New Delhi for providing financial assistant in the form of JRF/SRF. Authors are also thankful to Dr. Mark S.

Gordon for providing an ab initio program GAMESS.