Potential energy of attraction

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

Table 1 Changes in hypothetical potential energies of attraction of oppositely charged spherical ions based on reductions in dielectric constant in passing from water to methanol and ethanol, and the computed changes in binding constant assuming the changes in potential energy are translated into free energies of binding3...

These results shown in Figures 1 and 2 demonstrate the similarity of the effects of short-range forces on the properties of nonelectrolytes and concentrated electrolytes. One finds both positive and negative deviations from ideality and these effects may be ascribed to the difference between the intermolecular potential energy of attraction of unlike species to the mean of the corresponding potentials for pairs of like molecules. Previous discussion of these systems has focused on the hydration of the positive ion as the dominant effect, but we see in Figure 1 that... 

Notice that, although we focused on the /th electron, the subscript of h refers to the particular orbital (i.e., the spatial characteristics of the orbital). Integration over the coordinates of any electron with the same distribution would have produced the same result. The quantity h, is the sum of the kinetic energy and the potential energy of attraction to all of the nuclei of any electron which has the distribution . 

The second dipole acts on the original dipole in a similar fashion, giving a second contribution to the interaction energy that is identical to Equation (10) except that the subscripts are interchanged. The total potential energy of attraction is the sum of these two contributions ... 

That is, the potential energy of attraction is identical in the two cases. This is an important result as far as the extension of molecular interactions to macroscopic spherical bodies is concerned. What it says is that two molecules, say, 0.3 nm in diameter and 1.0 nm apart, interact with exactly the same energy as two spheres of the same material that are 30.0 nm in diameter and 100 nm apart. Furthermore, an inspection of Equation (49) reveals that this is a direct consequence of the inverse sixth-power dependence of the energy on the separation. Therefore the conclusion applies equally to all three contributions to the van der Waals attraction. Precisely the same forces that are responsible for the association of individual gas molecules to form a condensed phase operate —over a suitably enlarged range —between colloidal particles and are responsible for their coagulation. 

We shall not present the detailed analysis of this complication. In essence, it involves the time-dependent Schrodinger equation rather than the time-independent equation that resulted in Equation (22). Casimir and Polder have investigated this situation. They found that for values of r > X, the potential energy of attraction according to the modified London treatment is given by... 

Equation (60) may now be integrated over values of z between the distance of closest approach d and infinity. The result of this integration gives the potential energy of attraction per unit area between two blocks of infinite extension ... 

The Hamaker constant has energy units since (3 has the units energy length6 and the term in parentheses in Equation (62) has the units (volume-1)2. The potential energy of attraction between blocks as calculated by Equation (63) is expressed per unit area of the facing surfaces. Note also that this attraction grows weaker as the distance increases. This is different from the... 

TABLE 10.4 Potential Energy of Attraction Between Two Particles with the Indicated Geometries... 

The first term on the right is the operator for the electrons kinetic energy the second term is the operator for the potential energy of attraction between the electrons and the nucleus (r, being the distance between electron i and the nucleus) the third term is potential energy of repulsion between all pairs of electrons ru being the distance between electrons / and j) the last term is the spin-orbit interaction (discussed below). In addition, there are other relativistic terms besides spin-orbit interaction, which we neglect. 

We integrate the potential energy of attraction of each nucleus for an infinitesimal portion of the charge cloud and sum for all the nuclei. If we know p0 the integrals to be summed are readily calculated. 

In these expressions written with use of so-called atomic units (elementary charge, electron mass and Planck constant are all equal to unity) RQs stand as previously for the spatial coordinates of the nuclei of atoms composing the system r) s for the spatial coordinates of electrons Mas are the nuclear masses Zas are the nuclear charges in the units of elementary charge. The meaning of the different contributions is as follows Te and Tn are respectively the electronic and nuclear kinetic energy operators, Vne is the operator of the Coulomb potential energy of attraction of electrons to nuclei, Vee is that of repulsion between electrons, and Vnn that of repulsion between the nuclei. Summations over a and ft extend to all nuclei in the (model) system and those over i and j to all electrons in it. 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

There are three energy contributions that must be considered in the description of the helium atom (1) the kinetic energy of the electrons as they move around the nucleus, (2) the potential energy of attraction between the nucleus and the electrons, and (3) the potential energy of repulsion between the two electrons. 

In chapter 1.3 a number of examples of elaborations have already been given, mostly using lattice statistics. All of them Involve a "divide and rule" strategy, in that the system (i.e. the adsorbate) is subdivided into subsystems for which subsystem-partition functions can be formulated on the basis of an elementary physical model. For instance, in lattice theories of adsorption one adsorbed atom or molecule on a lattice site on the surface may be such a subsystem. In the simplest case the energy levels, occurring in the subsystem-partition function consist of a potential energy of attraction and a vibrational contribution, the latter of which can be directly obtained quantum mechanically. Having... 

The variation of the potential energy of interaction between colloidal particles and sohd surfaces can be also succeeded by the addition of a detergent to the suspending medium, which leads to a decrease in the Hamaker constant and, consequently, in the potential energy of attraction. 

The total potential energy of interaction V is the sum of the potential energy of attraction VA and that of repulsion Vr ... 

The potential energy of attraction in a vacuum for similar spherical particles of radius a whose centers are separated by a distance R is given by the expression... 

Potential Energy of Attraction—Interaction—Repulsion See Gibbs Energy of... 

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