Potential, intermolecular moments

V Pore volume of sorbent filled at P and T Vq Total pore volume of sorbent a Polarizability r Sorbate concentration Sorption potential q Initial sorption potential Dipole moment < ) Intermolecular potential Vq Characteristic vibrational frequency... 

The water molecule has a dipole moment, fx, of 1.84 X 10 esu-cm (esu = electrostatic unit). If water dipoles in ice are separated by 2.76 A, estimate the potential intermolecular attraction energy (in kcal/mole) due to oriented dipole-dipole interaction. Compare this quantity with the enthalpy of sublimation of ice (11.3 kcal/mole), which represents the amount of energy needed to break these bonds. Why might this calculation underestimate the actual energy of water molecule interaction ... 

Keywords Atoms in molecules - Chemical bonding - Force fields - Intermolecular potential - Multipole moments - Quantum chemical topology - Electrostatic potential Coulomb interaction - Reduced density matrix - Polarisation - Quantum mechanics Convergence... 

The only feasible procedure at the moment is molecular dynamics computer simulation, which can be used since most systems are currently essentially controlled by classical dynamics even though the intermolecular potentials are often quantum mechanical in origin. There are indeed many intermolecular potentials available which are remarkably reliable for most liquids, and even for liquid mixtures, of scientific and technical importance. However potentials for the design of membranes and of the interaction of fluid molecules with membranes on the atomic scale are less well developed. 

The inner and outer potential differ by the surface potential Xa — (fa — ipa- This is caused by an inhomogeneous charge distribution at the surface. At a metal surface the positive charge resides on the ions which sit at particular lattice sites, while the electronic density decays over a distance of about 1 A from its bulk value to zero (see Fig. 2.1). The resulting dipole potential is of the order of a few volts and is thus by no means negligible. Smaller surface potentials exist at the surfaces of polar liquids such as water, whose molecules have a dipole moment. Intermolecular interactions often lead to a small net orientation of the dipoles at the liquid surface, which gives rise to a corresponding dipole potential. 

In this expression, the dipole dipole interactions are included in the electrostatic term rather than in the van der Waals interactions as in Eq. (9.43). Of the four contributions, the electrostatic energy can be derived directly from the charge distribution. As discussed in section 9.2, information on the nonelectrostatic terms can be deduced indirectly from the charge density. The polarizability a, which occurs in the expressions for the Debye and dispersion terms of Eqs. (9.41) and (9.42), can be expressed as a functional of the density (Matsuzawa and Dixon 1994), and also obtained from the quadrupole moments of the experimental charge density distribution (see section 12.3.2). However, most frequently, empirical atom-atom pair potential functions like Eqs. (9.45) and (9.46) are used in the calculation of the nonelectrostatic contributions to the intermolecular interactions. 

Spectroscopic measurement. Specifically, if the induced dipole moment and interaction potential are known as functions of the intermolecular separation, molecular orientations, vibrational excitations, etc., an absorption spectrum can in principle be computed potential and dipole surface determine the spectra. With some caution, one may also turn this argument around and argue that the knowledge of the spectra and the interaction potential defines an induced dipole function. While direct inversion procedures for the purpose may be possible, none are presently known and the empirical induced dipole models usually assume an analytical function like Eqs. 4.1 and 4.3, or combinations of Eqs. 4.1 through 4.3, with parameters po, J o, <32, etc., to be chosen such that certain measured spectral moments or profiles are reproduced computationally. 

Highly developed quantum chemical methods exist to compute with an ever increasing precision molecular and supermolecular properties from first principles. For example, attempts to compute intermolecular interaction potentials and, more recently, induced dipole moments, are well known for the simpler atomic and molecular systems. 

It is clear that intermolecular force and induced dipole function arise from the same physical mechanisms, electron exchange and dispersion. Since at the time neither intermolecular potentials nor the overlap-induced dipole moments were known very dependably, direct tests of the assumptions of a proportionality of force and dipole moment were not possible. However, since the assumption was both plausible and successful, it was widely accepted, even after it was made clear that for an explanation of... 

In other words, for tetrahedral molecules, these relationships differ from the ones used for the linear molecules, especially Eq. 4.18. As a consequence, we must rederive the relationships for the spectral line shape and spectral moments. If the intermolecular interaction potential may be assumed to be isotropic, the line shape function Vg(a> T), Eq. 6.49, which appears in the expression for the absorption coefficient a, Eq. 6.50, may still be written as a superposition of individual profiles,... 

Fig. 12.5 Illustration of the orientation angles used in the Stockmayer intermolecular potential. Molecule j consists of atoms A and B, and molecule i consists of atoms C and D. The vector ry runs from the center of mass of molecule i to the center of mass of molecule j. The vector JTJ gives the orientation and magnitude of the dipole moment of molecule i, with a similar definition for JTj. A ghost copy of molecule j is shifted to left to more easily visualize the orientation angle ifr. See Eqs. 12.11 to 12.13 and accompanying text for definition and description of these angles.

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

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... 

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