Potentials angular-dependent

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Finally, the assumed spherical synnnetry of the interactions implies that the volume element r 2 is dri2- For angularly-dependent potentials, the second virial coefficient... 

Let us consider a one-component system of hard spheres of unit diameter (7=1 with angular-dependent associative potential. The nonassociative potential is thus given by Eq. (35), whereas the associative forces are described by Eq. (61) with d = a = I, a = 1.05, and 6 = 21°. 

We report here some results for a simple model of a one-component fluid interacting via a slightly modified Lennard-Jones potential, with angular-dependent associative forces. The model is considered in contact with the adsorbing surface. The principal aim of the simulation is to investigate the... 

Fig. 1.22. Angular-dependent potential U for one-dimensional libration over barriers of height U0. The arrows show the way of libration below barriers and random translations from one well to another due to high-energy fluctuations. The broken line presents the approximation of the parabolic well valid at the bottom.

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

Analyzing orientational structures of adsorbates, assume that the molecular centers of mass are rigidly fixed by an adsorption potential to form a two-dimensional lattice, molecular orientations being either unrestricted (in the limit of a weak angular dependence of the adsorption potential) or reduced to several symmetric (equivalent) directions in the absence of lateral interactions. In turn, lateral interactions should be substantially anisotropic. 

Theoretical treatment of such structures is favored by the fact that the azimuthal component of the angular dependence of the adsorption potential is determined by the arrangement symmetry for the substrate atoms closest to the adsorbate, i.e., by the substrate crystal lattice. As a result, an isolated molecule can have several... 

The angular dependence of lateral interactions for nonpolar molecules (including quadrupole-quadrupole and Van der Waals dipole-dipole interactions as well as major terms of the power series expansion of repulsive atom-atom potentials in terms of the molecular linear dimension to intermolecular distance ratio) can be represented in a unified form 47 52... 

A specific role of adsorption potentials U j,angular dependences, they also... 

A theoretical treatment of the effect caused by the competition between the sine-like angular-dependent component of the adsorption potential and dipole lateral interaction demonstrated that the values 6 are the same in the ground state and at the phase transition temperature.81 Study of the structure and dynamics for the CO monolayer adsorbed on the NaCl(lOO) surface using the molecular dynamics method has also led to the inference that angles 0j are practically equalized in a wide temperature range.82 That is why the following consideration of orientational structures and excitations in a system of adsorbed molecules will imply, for the sake of simplicity, the constant value of the inclination angle ty =0(see Fig. 2.14) which is due to the adsorption potential u pj,q>j). 

The function u g>j J describes the azimuthal angular dependence of the adsorption potential u pj,[Pg.31]

Finally, Fig. 5.13 shows the adiabatic bending potential for nonlinear deformations. The angular dependence appears similar to a cosine-like (or dipole-dipole) behavior near equilibrium, but departs conspicuously from this mathematical form at larger deformation angles. The potential shows the strong propensity for linear F H F H-bonding arrangements consistent with maximization of n-o donor-acceptor overlap. 

There should exist a correlation between the two time-resolved functions the decay of the fluorescence intensity and the decay of the emission anisotropy. If the fluorophore undergoes intramolecular rotation with some potential energy and the quenching of its emission has an angular dependence, then the intensity decay function is predicted to be strongly dependent on the rotational diffusion coefficient of the fluorophore.(112) It is expected to be single-exponential only in the case when the internal rotation is fast as compared with an averaged decay rate. As the internal rotation becomes slower, the intensity decay function should exhibit nonexponential behavior. 

In molecular crystals, the relative importance of the electrostatic, repulsive, and van de Waals interactions is strongly dependent on the nature of the molecule. Nevertheless, in many studies the lattice energy of molecular crystals is simply evaluated with the exp-6 model of Eq. (9.45), which in principle accounts for the van der Waals and repulsive interaction only. As underlined by Desiraju (1989), this formalism may give an approximate description, but it ignores many structure-defining interactions which are electrostatic in nature. The electrostatic interactions have a much more complex angular dependence than the pairwise atom-atom potential functions, and are thus important in defining the structure that actually occurs. 

These relations are valid only for force laws and potential functions which are functions of die intermolecular separation alone. For an angular dependent potential, the force on molecule is different and, in addition, there is a torque tending to rotate the molecule (Ref 8e, p 22)... 

An expression of the type (7.101), which gives the bond order explicitly in terms of the positions of the neighbouring atoms, is called a bond order potential (BOP). Angularly dependent bond order potentials were first derived heuristically for the elemental semiconductors by TersofF (1988). We will see in the next chapter that a many-body expansion for the bond order may be derived exactly within the model. 

If molecules are involved, isotropic potential functions are in general not adequate and angular dependences reflecting the molecular symmetries may have to be accounted for. In general, up to five angular variables may be needed, but in many cases the anisotropies may be described rigorously by fewer angles. We must refer the reader to the literature for specific answers (Maitland et al., 1981) and mention here merely that much of what will interest us below can be modeled in the framework of the isotropic interaction approximation. 

After a system had equilibrated it was followed for an additional 600 steps or for 3 x 10"12 s. During this production phase of the calculations the velocities were not changed but the temperatures were continually monitored. The random translational and rotational temperature fluctuations that occurred during this phase are illustrated by Figure 2. In this particular instance, the rotational and translational temperature fluctuations are out of phase with one an other. This behavior is typical of a system with a strong angular dependent potential. The distribution of the x... 

The intermolecular potential consists of the sum of Eqs.(176) and (177). This simulation was done for 216 and 512 molecules. However, only the autocorrelation functions from the 512 molecules case are discussed here. The small dipole moment of carbon monoxide makes the orientational part of this potential so weak that molecules rotate essentially freely, despite the fact that this calculation was done at a liquid density. The results for the Stockmayer simulation serve the purpose of providing a framework for contrasting results from more realistic, stronger angular-dependent potentials. 

The intermolecular potential consists of the sum of Eqs. (176), (177), (178), and (179). This simulation was done for 216 and 512 molecules but again only the autocorrelation functions for 512 molecules are discussed here. This potential is the strongest angular dependent potential we considered. The results from this potential indicate that it is slightly stronger than that in real liquid carbon monoxide. For example the mean square torque/TV2), for this simulation is 36 x 10-28 (dyne-cm)2 51 and the experimental value is 21 x 10-28 (dyne-cm)2. If this potential is taken seriously, then it should be pointed out that this small discrepancy in torques could be easily removed by using a smaller quadrupole moment. This would be a well justified step since experimental quadrupole moments for carbon monoxide range from 0.5 x 10-26 to 2.43 x 10-26 esu.49... 

Fig. 2.2. Schematic diagrams of the angular dependence of the S, and S2 potentials along 6 from the initial to the final state of adiabatic photoreactions.37...

Similar calculations enable one to define more precisely the parameters of atom-atom potentials. Hutson [1988, 1990] parametrized the inter-molecular potential using the measured angular dependences of energy... 

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