Instantaneous dipoles

The average cloud is spherically synnnetric with respect to the nucleus, but at any instant of time there may be a polarization of charge givmg rise to an instantaneous dipole moment. This instantaneous dipole induces a corresponding instantaneous dipole in the other atom and there is an interaction between the instantaneous dipoles. The dipole of either atom averages to zero over time, but the interaction energy does not because the instantaneous and induced dipoles are correlated and... 

Fi E- Zi eri F2 >. Here erj is the one-eleetron operator deseribing the interaetion of an eleetrie field of magnitude and polarization E with the instantaneous dipole moment... 

This model was later expanded upon by Lifshitz [33], who cast the problem of dispersive forces in terms of the generation of an electromagnetic wave by an instantaneous dipole in one material being absorbed by a neighboring material. In effect, Lifshitz gave the theory of van der Waals interactions an atomic basis. A detailed description of the Lifshitz model is given by Krupp [34]. 

To understand the origins of dispersion forces, let us consider two Bohr atoms, each of which consists of an electron orbiting around a nucleus comprised of a proton, having a radius ao, often referred to as the first Bohr radius . It is obvious that a Bohr atom has no permanent dipole moment. However, the Bohr atom can be considered to have an instantaneous dipole moment given by... 

The dispersion (London) force is a quantum mechanieal phenomenon. At any instant the electronic distribution in molecule 1 may result in an instantaneous dipole moment, even if 1 is a spherieal nonpolar moleeule. This instantaneous dipole induces a moment in 2, which interacts with the moment in 1. For nonpolar spheres the induced dipole-induced dipole dispersion energy function is... 

Both attractive forces and repulsive forces are included in van der Waals interactions. The attractive forces are due primarily to instantaneous dipole-induced dipole interactions that arise because of fluctuations in the electron charge distributions of adjacent nonbonded atoms. Individual van der Waals interactions are weak ones (with stabilization energies of 4.0 to 1.2 kj/mol), but many such interactions occur in a typical protein, and, by sheer force of numbers, they can represent a significant contribution to the stability of a protein. Peter Privalov and George Makhatadze have shown that, for pancreatic ribonuclease A, hen egg white lysozyme, horse heart cytochrome c, and sperm whale myoglobin, van der Waals interactions between tightly packed groups in the interior of the protein are a major contribution to protein stability. 

An instantaneous dipole moment on one molecule distorts the electron cloud on a neighboring molecule and gives rise to a dipole moment on that molecule the two instantaneous dipoles attract each other. An instant later, the swirling electron cloud of the first molecule will give rise to a dipole moment in a different direction,... 

FIGURE 5.7 (a) The instantaneous dipole moments in two neighboring rod-shaped molecules tend to be close together and interact strongly over a relatively broad region of the molecule, (b) Those on neighboring spherical molecules tend to be farther apart and interact weakly over only a small region of the molecule. 

Van der Waals-London interactions are due to fluctuations in electron distribution as the electrons circulate within their orbits. These instantaneous dipoles are usually weak, but are, regardless, the most common interaction resulting in adsorption.31 Stronger interactions result from charge transfer. 

In order to arrive at a mathematical relationship to describe London forces, we will use an intuitive approach. First, the ability of the electrons to be moved within the molecule is involved. Atoms or molecules in which the electrons are highly localized cannot have instantaneous dipoles of any great magnitude induced in them. A measure of the ability of electrons in a molecule to be shifted is known as the electronic polarizability, a. In fact, each of the interacting molecules has a polarizability, so the energy arising from London forces, Ei, is proportional to a2. London forces are important only at short distances, which means that the distance of separation is in the denominator of the equation. In fact, unlike Coulomb s law, which has r2 in the denominator, the expression for London forces involves r6. Therefore, the energy of interaction as a result of London forces is expressed as... 

The discussion thus far has focused on the forces between an array of atoms connected together through covalent bonds and their angles. Important interactions occur between atoms not directly bonded together. The theoretical explanation for attractive and repulsive forces for nonbonded atoms i and j is based on electron distributions. The motion of electrons about a nucleus creates instantaneous dipoles. The instantaneous dipoles on atom i induce dipoles of opposite polarity on atom j. The interactions between the instantaneous dipole on atom i with the induced instantaneous dipole on atom j of the two electron clouds of nonbonded atoms are responsible for attractive interactions. The attractive interactions are know as London Dispersion forces,70 which are related to r 6, where r is the distance between nonbonded atoms i and j. As the two electron clouds of nonbonded atoms i and j approach one another, they start to overlap. There is a point where electron-electron and nuclear-nuclear repulsion of like charges overwhelms the London Dispersion forces.33 The repulsive... 

The physical origin of the dispersion interaction is often described in terms of a quasi-classical induced-dipole-induced-dipole picture. The quantum-mechanical fluctuations of the electronic distribution about its spherically symmetric average can be pictured as leading to an instantaneous (snapshot) dipole /za(mst) on monomer a, which in turn induces an instantaneous dipole tb(mst) on b. Thus, if the dipole fluctuations of the two monomers are properly correlated, a net attraction of the form (5.25) results. As remarked by Hirschfelder et al,28... 

Dispersion forces. These are weak attractions caused by instantaneous fluctuation of the electron distribution in molecules and even atoms. They were first posed by Fritz London whose focus was on helium liquefaction. Such London forces fall off with the sixth power of the distance of separation. Any individual fluctuation creates a +/— local charge and that instantaneous dipole can interact with other such instantaneous dipoles nearby. The important... 

Figure 11.7 van der Waals bond caused by the creation of an instantaneous dipole. Momentary variations in the electron charge distribution of an atom causes a momentary dipole attraction between the asymmetric negative charge and the positive nuclear charge of another atom. 

These are the weakest of all intermolecular bonds. They result from the random movement of electrons within an atom or molecule. This movement can result in a separation of charge across the atom or molecule (an instantaneous dipole Fig. 11.7). This small separation of charge (indicated by <5+ and 8 ) will then influence neighboring atoms or molecules, and cause an induced dipole. These van der Waals bonds (sometimes known as London forces) occur between nonpolar molecules or atoms such as I2, 02, H2, N2, Xe, Ne, and between the aliphatic chains of lipids (see below). 

Noble gases and non-polar molecules such as COz and CH4 do not have dipoles. In these molecules, the movement of electrons results in nonpolar molecules becoming temporarily polar an instantaneous dipole is formed. The molecule which becomes momentarily polar then causes its neighboring molecule to become polar. Thus a weak attraction occurs between the molecules. This attraction is named the van der Waals force. 

As well as these permanent dipole moments, random motion of electron density in a molecule leads to a tiny, instantaneous dipole, which can also induce an opposing dipole in neighbouring molecules. This leads to weak intermolecu-lar attractions which are known as dispersive forces or London forces, and are present in all molecules, ions and atoms - even those with no permanent dipole moment. Dispersive forces decrease rapidly with distance, and the attractions are in proportion to 1/r6, where r is the distance between attracting species. 

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