Orientation-dependent Force Fields

Molecules with Orientation-dependent Force Fields. In 1941, Stockmayer showed how to modify the Lennard-Jones 12-6 potential to allow for orientation-dependent forces that result from the presence of permanent dipoles of moment p. embedded at molecular centres  

From statistical mechanics, the version of equation (25) that allows for orientation-dependent forces is 

Buckingham and Pople gave a more complete treatment than Stockmayer s of the interaction between two molecules having non-central force fields. Their treatment allows not only for dipole-dipole interaction but also for dipole-quadrupole, quadrupole-quadrupole, dipole-induced dipole, steric, anisotropic, and other effects. An important feature of the Buckingham-Pople procedure is that the various integrals needed can be calculated from a master table that they give. Buckingham and Pople applied their method to four polar compounds for which values of the dipole moment and the polarizability were known, and used B(T) to 

Perez Masii and M. Dfaz Pefia, Anales de Quim., 1959, 55B, 229. 

and MeOH, and a related procedure has been used to determine the octupole moments of CH4, CF4, and CMe4, A value of 0 for benzene has been derived with the aid of a spherical-shell potential.  

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The idea that unsymmetrical molecules will orient at an interface is now so well accepted that it hardly needs to be argued, but it is of interest to outline some of the history of the concept. Hardy [74] and Harkins [75] devoted a good deal of attention to the idea of force fields around molecules, more or less intense depending on the polarity and specific details of the structure. Orientation was treated in terms of a principle of least abrupt change in force fields, that is, that molecules should be oriented at an interface so as to provide the most gradual transition from one phase to the other. If we read interaction energy instead of force field, the principle could be reworded on the very reasonable basis that molecules will be oriented so that their mutual interaction energy will be a maximum. 

On the other hand, polar molecules create a force field around them that is attractive or repulsive, depending on the relative orientation of the neighboring polar molecule. In this case, the spectrum of molecular arrangements actually explored by an ensemble of strongly polar molecules is severely restricted. It follows that these molecules display a more marked tendency to give a dimensionally unlimited ordered molecular arrangement and a limited mutual solubility with apolar solvents. 

Structure, then for every tetrahedron there is another tetrahedron which has the exact opposite orientation the electric fields of the dipoles compensate each other. If, however, all tetrahedra have the same orientation or some other mutual orientation that does not allow for a compensation, then the action of all dipoles adds up and the whole crystal becomes a dipole. Two opposite faces of the crystal develop opposite electric charges. Depending on the direction of the acting force, the faces being charged are either the two faces experiencing the pressure (longitudinal effect) or two other faces in a perpendicular or an inclined direction (transversal effect). 

After their creation, positive ions are accelerated through a voltage difference V. They thus acquire a velocity t> that depends on their mass m. Following acceleration, the ions enter a transversal magnetic field of intensity B. The orientation of this field does not modify the ions velocity but forces them on a circular trajectory that is a function of their m/z ratio. The fundamental relationship of dynamics F = ma (a designates acceleration), applied to ions of mass m on which a Lorentz force F = qv A B is exerted leads to the following relationship ... 

Both A A and cis trans equilibria of siderophore complexes can exist in solution. The chirality of the ligand can impose a preferred metal-center chirality. In addition, the degree of this preference depends on the stereochemical rigidity of the ligand. In principle, the magnitude of the molar circular dichroism can be used as a measure for diastereoisomeric equilibria based on a comparison of the solid-state and solution ellipticity. Nevertheless, predictions of metal-center chiralities require theoretical calculations. For example, empirical-force-field calculations of iron(III) enterobactin show that the A orientation at the metal center is more stable than the A by 0.5 kcalmoH, which is consistent with the CD spectra. ... 

Scheme J is an inverted orientation of the holder the holder rotates about its own axis and revolves around the central axis of the centrifuge at the same angular velocity in the same direction. This produces a quite different pattern of the centrifugal force field compared to scheme I, displaying either a rotating or an oscillating pattern depending on the location of the point on the holder.

Molecular dynamic calculations (on the basis of the semiempirical MNDO force field) were performed for two mutual orientations of a fullerene molecule and the direction of the implanting atom (Fig. 4a). Fig. 4b shows the dependence of the threshold energy of formation of the endohedral complex C C6o (vertical axis) for Orientation I. The obtained results show that the process of formation of endohedral complex proceeds as follows. The implanting atom passes a central part of a five-member (or six-member) ring, cavity of cluster, and is reflected from an opposite side of a cluster. This process continues since the basic part of kinetic energy is not transferred into the vibrational energy of a molecule. The vertical lines of Fig. 4 correspond to the head-on collisions of the implanting atom with the atoms of the five-member face of a molecule. At the top level of Fig. 4 the isolines of the same surface are presented. 

The range of systems that have been studied by force field methods is extremely varied. Some force fields have been developed to study just one atomic or molecular species under a wider range of conditions. For example, the chlorine model of Rodger, Stone and Tildesley [Rodger etal. 1988] can be used to study the solid, liquid and gaseous phases. This is an anisotropic site model, in which the interaction between a pair of sites on two molecules depends not only upon the separation between the sites (as in an isotropic model such as the Lennard-Jones model) but also upon the orientation of the site-site vector with respect to the bond vectors of the two molecules. The model includes an electrostatic component which contains dipole-dipole, dipole-quadrupole and quadrupole-quadrupole terms, and the van der Waals contribution is modelled using a Buckingham-like function. 

In Eqs. [15] and [16], D is the effective dielectric constant and is generally preset to a low value (1-1.5) to simulate the gas phase. However, most force fields optionally allow for a user-defined dielectric constant. In some force fields a distance-dependent dielectric constant (function) is used. In Eq. [16], and a describe the orientation of the bond dipoles, and x, and are the magnitudes of interacting dipoles i and /, respectively.22... 

Due to the mechanical moment, free dipoles orient along the field, and then M = 0. Dipoles turn quite quickly, we can assume that they are always focused along the field. The force of magnetic interaction depends not only on change of magnetic field, but also variations of magnetic moment of the carrier. If these values are constant in space then F = 0. Hence, to obtain the required values of the forces, the magnetic field should vary in space. 

Gas density enters the question of interaction forces indirectly by modifying the particle s trajectory in the potential field of the matter it interacts with. There are highly approximative means of dealing with this for spheres where the Knudsen number regime is well defined. In the case of other shapes such as rods which may span several ranges of Kn, the effect can become impossibly complicated when orientation dependency of the forces is included. 

For all but spherically symmetrical molecules, van der Waals forces are anisotropic. The polarizabihties of most molecules are different in different molecular directions because the response of electrons in a bond to an external field will usually be anisotropic. A consequence of this effect is that the dispersion force between two molecules will depend on their relative molecular orientation. In nonpolar liquids, the effect is of minor importance because the molecules are essentially free to tumble and attain whatever orientation is energetically favorable. However, in sohds, hquid crystals, and polar media, the effect can be important in determining the relative fixed orientation between molecules, thereby affecting or controlling specific conformations of polymers or proteins in solution, critical transition temperatures in liquid crystals and membranes, and so on. Repulsive forces in polar molecules are also orientation dependent, and are often of greater importance in controlling conformations and orientations. 

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