ЖУРНАЛ ФИЗИЧЕСКОЙ ХИМИИ, 2010, том 84, № 2, с. 280-285
СТРОЕНИЕ ВЕЩЕСТВА И КВАНТОВАЯ ХИМИЯ
ORDERING IN HOMOLOGOUS SERIES OF 4'-«-ALKYL-4-CYANOBIPHENYL (яСВ) - A COMPARATIVE COMPUTATIONAL STUDY
© 2010 P. L. Praveen, N. Ajeetha, and D. P. Ojha
Liquid Crystal Research Laboratory, Postgraduate Department of Physics, Andhra Loyola College, Vijayawada, 520 008, A.P., India E-mail: firstname.lastname@example.org Received October 29, 2008
Abstract — A comparative computational analysis of molecular ordering in homologous series of monotropic nematic liquid crystal 4'-n-alkyl-4-cyanobiphenyl (nCB) molecules with alkyl groups, butyl (4CB), pentyl (5CB), hexyl (6CB), heptyl (7CB), has been carried out with respect to translational and orientational motion. The atomic net charge and dipole moment at each atomic center were evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh—Schrodinger perturbation theory and the multicenter—multipole expansion method were employed to evaluate long-range intermolecular interactions, while a 6-exp potential function was assumed for short-range interactions. Various possible geometrical arrangements of molecular pairs with regard to different energy components were considered, and the most favorable configuration was found. A comparative picture of molecular parameters, such as total energy, binding energy, and total dipole moment of 4CB, 5CB, 6CB and 7CB, is given. The results are discussed in the light of other theoretical observations.
The driving force for the recent renewal of interest in the chemistry and physics of liquid crystals comes mainly from the large number of possible applications involving this intermediate state of matter . Many of these applications have advantages that are unique to the liquid crystal field [2—4]. The proper understanding of liquid crystalline behaviour requires an adequate theoretical background as a precursor to application of new developments and accounting for abnormal properties of the materials . The potential energy of interaction of two molecules is considered as a prime requirement in theoretical investigation of molecular interactions. This interaction determines the physical properties of liquid crystals, as well as the type of kinetics of physical and physicochemical processes in these substances .
The role of molecular interactions in mesogenic compounds has attracted attention of several workers based on the Rayleigh—Schrodinger perturbation theory [7, 8]. These studies were aimed at computing the interaction energy of a molecular pair depending on the angle and distance, but efforts were directed toward explaining the aligned structure or, at best, correlating the minimum energy with the observed crystal structures. It has been observed that the interaction energy of a pair of mesogens indicates the preference of a particular configuration, depending on the relative energies.
The homologous series «CB molecules chosen for the present investigation are widely used in electro optical display devices because of their strong positive di-
electric anisotropy. The stability of liquid crystal phases can often be enhanced by increasing the length and polarizability of the molecule or by adding a terminal cyano group promoting polar interactions in the molecular pairs.
This paper deals with the computation of pair interaction energies for 4CB, 5CB, 6CB and 7CB molecular pairs but detailed results are reported only for 4CB at an intermediate distance of 10 Á for stacking and 8 Á for in-plane interactions. Similarly, a distance of 22 Á was set for terminal interactions. However, the salient features of the residual molecules (5CB, 6CB and 7CB) are shown in Fig. 1. The intermolecular separation was selected to eliminate the possibility of van der Waals contacts completely and keep the molecules within the range of short and medium-range interactions. Further, instead of finding the exact minimum energy configuration, we considered the general behaviour of the molecules surrounding a fixed molecule in a particular frame of reference.
SIMPLIFIED FORMULA AND COMPUTATIONAL DETAILS
The geometry of «CB molecules (4CB, 5CB, 6CB and 7CB) was constructed on the basis of the published crystallographic data with standard values of bond lengths and bond angles .
In order to find the interaction energy of two molecules, it is necessary to compute atomic net charges and dipole moments with an all valence electron method. Hence, in the present work, the CNDO/2 method  was used to compute the net atomic
Fig. 1. Geometry of 4CB, 5CB, 6CB, and 7CB molecules.
charges and dipole moments at each atomic centre of the molecule because using only the Mulliken atomic net charges (given, for example, by an ab initio program) for the calculation of an electrostatic interaction would be in correct for two reasons: (i) the atomic dipoles must be taken into account, and (ii) the homopolar dipoles must be taken into account, or else atomic net charges obtained according to Lowdin's procedure should be used. An ab initio program which gives only Mulliken net charges is therefore quite misleading as concerns a reasonable representation of the molecular charge distribution in terms of charges and dipoles [11-13].
A detailed computational scheme based on simplified formula given by Claverie  for evaluating the interaction energy of a molecular pair was used when calculating the energy at fixed configurations. According to the second order of the perturbation theory is modified for intermediate range interactions , the
total pair interaction energy of molecules ( Upair) is represented as a sum of several terms contributing to the total energy:
^pair + Upol + ^disp + ^rep,
where Uel, Upol, Udisp, and Urep are the electrostatic, polarization, dispersion and repulsion energy terms respectively.
In turn, electrostatic term is expressed as
Uel = UQQ + UQMI + UMIMI + —,
where Uqq, Uqmi, and UMIMI etc. are monopolemonopole, monopole—dipole, and dipole—dipole terms respectively. In fact, the inclusion of higher order multipoles does not affect significantly the electrostatic interaction energy and the calculation involving only the above terms gives satisfactory results . The evaluation of electrostatic term was, therefore, restricted by the dipole—dipole energy term.
A comparative picture of molecular parameters, such as total energy (Et), binding energy (Eb) and total dipole moment (p.)
Compound —Et, amu —Eb, amu D
4CB 145.3 18.56 2.56
5CB 154.03 19.85 2.58
6CB 162.72 21.09 2.65
7CB 171.42 22.35 2.64
In the present work, the dispersion and short-range repulsion terms are considered together because several semiemperical approach, viz. the Lennard-Jones or Buckingham approach, actually proceed in this way. Kitaygorodsky introduced a Buckingham formula whose parameters were later modified by Kitaygorodsky and Mirskay for hydrocarbon molecules and several other molecules, which finally gave the expression :
Ujjsp + Uep = XX ,
U(X,v) = KxKv (- A//Z + Be yZ),
where Z = RXv/<v ; < = [(2<)(2<)]1/2, R^ and
R^ are the van der Waals radii of X and v atoms respectively. A, B and y parameters are independent ofpartic-ular species. But RXXv and KXKV factor, which determine the energy minimum, have different values according to the atomic species involved. The necessary formulae may be found elsewhere .
In this case, the origin was set on an atom close to the centre of mass of the molecule. The x-axis was directed along a bond parallel to the long molecular axis, while the y-axis lied in the plane of the molecule, and the z-axis was normal to the molecular plane.
RESULTS AND DISCUSSION
The molecular geometries of 4CB, 5CB, 6CB and 7CB are shown in Fig. 1. A comparative picture of molecular parameters, such as total energy, binding energy and total dipole moment of 4CB, 5CB, 6CB, and 7CB is given Table. As evident from Table, the total energy and binding energy of these molecules are arranged in a series
7CB > 6CB > 5CB > 4CB while the dipole moments change as follows: 6CB > 7CB > 5CB > 4CB.
The results of interaction energy calculations with regard to the different modes of interactions are discussed below.
One of the interacting molecules is fixed in the x—y plane in such a manner that the x-axis is directed along a bond parallel to the long molecular axis, while the other molecule is kept at a distance of 10 A along the z-axis from the fixed one. Rotations about the z-axis with respect to the configuration x (0°) y (0°) were considered with a step of 10°, and the interaction energies calculated at all points are given in Fig. 2. Evidently, the dispersion energy is mainly responsible for the attraction between 4CB molecules, although the exact minimum is estimated always from the Kitaygor-odsky energy curve, which is generally similar to the total energy curve.
The nematic character of liquid crystals is manifested in their translational mobility along the long molecular axis. Therefore, translations were considered at a step of 0.2 A, and the corresponding changes in various interaction energy components are reported in Fig. 3. All components increase with an increase in overlapping, the increase being smaller in the case of electrostatic and polarization energies. Evidently, molecules in a stacked pair can slide in a range of 2.0 ± 0.4 A with no significant change in the energy and, hence, molecular order can be kept up to 2. 8 A at the thermal agitation.
An interacting molecule was kept at a distance of 8 A along the y-axis from the fixed other one to avoid van der Waals contacts. Similar calculations were carried out for in-plane interactions. Again, rotations about the y and x-axes were considered, and the energy was minimized with respect to the translation and rotation about x,
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