Mutual potential energy

Examination of Fig. VII-4 shows that the mutual interaction between a pair of planes occurs once if the planes are a distance a apart, twice for those la apart and so on. If the planes are labeled by the index I, as shown in the figure, the mutual potential energy is... 

Just as it is useful to replace the force between two point charges by their mutual potential energy U. so we can replace the electric field by a more general quantity called the electmstatic potential . This is related to E in the same way that U is related to F... 

Likewise to find the mutual potential energy of two charge distributions Pa(fa) and Pb(fb) we would have to evaluate the integral... 

In order to determine the operator, we first write down the classical energy expression in terms of the coordinates and momenta. For the electron in a hydrogen atom, the classical energy is the sum of the kinetic energy and the mutual potential energy of the eleetron and the nucleus (a proton)... 

This gives us the general rule for relating force to mutual potential energy you differentiate the force with respect to the coordinates of the particle of interest. 

Some force fields make special provision for the mutual electrostatic potential energy of pairs of atoms that have different electronegativities. If atom A has a formal charge of i2a and atom B (distant J ab from Qa) has a formal charge of (2b, then their mutual potential energy is... 

The state of any particle at any instant is given by its position vector q and its linear momentum vector p, and we say that the state of a particle can be described by giving its location in phase space. For a system of N atoms, this space has 6iV dimensions three components of p and the three components of q for each atom. If we use the symbol F to denote a particular point in this six-dimensional phase space (just as we would use the vector r to denote a point in three-dimensional coordinate space) then the value of a particular property A (such as the mutual potential energy, the pressure and so on) will be a function of r and is often written as A(F). As the system evolves in time then F will change and so will A(F). 

In Chapter 1,1 discussed the concept of mutual potential energy and demonstrated its relationship to that of force. So, for example, the mutual potential energy of the diatomic molecule discussed in Section 1.1.2 is... 

This is a general rule we differentiate U by the coordinates of the particle in question in order to recover the force on that particle, from the expression for the mutual potential energy. Knowing the force F, we can use Newton s second law... 

Consider the two-dimensional box in Figure 2.5. If we know the positions of each of the N particles in the square and their pair potential, then we can calculate the total mutual potential energy... 

We do this as follows. The N particles are placed in a starting configuration, for example a regular lattice. Each particle is then tentatively moved at random. For each move, we calculate the change in the mutual potential energy, AU. If At/ is negative, then we allow the move. If At/ is positive, we allow the move with a probability of expi—U/kaT). 

Iti Chapter 1, we dealt at length with molecular mechanics. MM is a classical model where atoms are treated as composite but interacting particles. In the MM model, we assume a simple mutual potential energy for the particles making up a molecular system, and then look for stationary points on the potential energy surface. Minima correspond to equilibrium structures. 

The second two-electron integral is a little more difficult to understand formally, it represents the mutual potential energy of the overlap charge distribution -eiAff(r]) s(ri) due to electron 1 with an identical density -eifRi 2) l s0 2) due to electron 2. 

The spheres represent (roughly) the 2p atomic orbitals on C and N, and half an electron resides in each sphere. The mutual potential energy of this charge distribution can be easily calculated from elementary electrostatics. For small distances, a polynomial fit was used instead. 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

When a molecule is dissociated into a pair of ions, the work done is stored as the mutual potential energy of the ions. In deriving equation (3), we considered a similar process, in which charges of one sign were separated and conveyed across from one condenser plate to the other the work done in this process is stored as the mutual potential energy of... 

On the other hand, when a molecule in a vacuum is broken up into a positive ion and a negative ion, the mutual attraction, instead of disappearing rapidly, falls off slowly. At large separations of the ions the mutual potential energy — e2/r approaches zero as 1/r. In Fig. 8a the... 

Let the initial distance between the particles AH and B be denoted by r. The mutual potential energy of the two charged particles is —c2/r, as in the simple ionic dissociation depicted in Fig. 8a. If the value of r is sufficiently great, the energy associated with the electrostatic fields will not depend appreciably on r. For the proton transfer there is thus a characteristic quantity similar to D or. 

Complete and Incomplete Ionic Dissociation. Brownian Motion in Liquids. The Mechanism of Electrical Conduction. Electrolytic Conduction. The Structure of Ice and Water. The Mutual Potential Energy of Dipoles. Substitutional and Interstitial Solutions. Diffusion in Liquids. 

For two permanent dipoles the mutual potential energy is a minimum when the axes of the dipoles lie along the line joining their centers, as... 

When each of two small dipoles has a dipole moment n, and the distance between their centers, r, is large compared with the length of either dipole, their mutual potential energy in the position of Fig. 216 is1... 

This is the mutual electrostatic potential energy when the dipole lies as shown in Fig. 22. We see that the value is more than four times as large as the value obtained above for the mutual potential energy of two H20 dipoles in their most favorable position. 

In the case of a singly charged atomic ion in aqueous solution we have estimated the mutual potential energy between the ion and an adjacent water molecule when they are of nearly the same size, and have found the value to be about four times as great as the mutual potential energy of two adjacent water molecules. We conclude then that in the vicinity of an atomic ion the water structure will have to build itself round the ion, insofar as this is possible. 

Using the method given in Note 2 of the Appendix, obtain an expression for the mutual potential energy of the two dipoles shown in Fig. 21... 

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