Particles potential energy

For N bound electrons the sum of the particle potential energy given by Equation (17) and the electron repulsion terms are to be taken into account for the total potential energy of the system. An extremely good description of the general properties of strongly coupled plasma is given by Ichimaru [66]. We will adopt the atomic unit (au) for further theoretical descriptions and calculation purpose unless otherwise stated. 

The first term on the right-hand side of (110) is the gradient of the particle potential energy and the second term the force induced by the imposed gradient of the overall flow. 

The quantum mechanical Lagrangian density for a system of many particles interacting via a many-particle potential energy operator V is... 

Using the form of the energy operators, the Schrodinger equation can be established immediately. Thus, for the translational motion of a free particle (potential energy zero) along the x axis, we have... 

Fig. 34a-c. Results of a model calculation of a Xe atom adsorbed on a high electron density jellium surface (e.g. aluminium), (a) Contours of constant electron density in a cut perpendicular to the surface through the center of the Xe atom, (b) Xe valence p-elechon density vs. distance (difference density between metal-adatom system and sum of clean metal plus single Xe atom except 5p level), (c) Effective single particle potential energy contributions due to electrostatic dipole, Ves, and the exchange-correlation interaction, Vxc, respectively, [82Lan],... 

The function measures the particle potential energy a distance r from the centre of the cell. The time-averaged properties of the dispersion are calculated via the configurational partition function,... 

C A system of oppositely charged particles. Potential energy is gained when the charges are separated. It is converted to kinetic energy as the attraction pulls the charges together. 

If a one-dimensional problem is being solved, only the first term of eq. (7.3.6) is used. The term U(x, y, z) is the particular particle potential energy in the force field the information on the particular type of problem is concentrated exactly in this term. 

In order to solve a quantum mechanical problem, we should substitute an analytical expression for the particle potential energy in the given force field, U(x, y, z), in eq. (7.3.5) find the values of parameter E at which the Schrodinger equation allows solutions and calculate the wavefunction ij/ x, y, z) as a solution of this equation. The parameter E stands out as the particle energy. 

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

We conclude this section by discussing an expression for the excess chemical potential in temrs of the pair correlation fimction and a parameter X, which couples the interactions of one particle with the rest. The idea of a coupling parameter was mtrodiiced by Onsager [20] and Kirkwood [Hj. The choice of X depends on the system considered. In an electrolyte solution it could be the charge, but in general it is some variable that characterizes the pair potential. The potential energy of the system... 

The total potential energy of A particles in a given configuration (r. 

McMillan-Mayer theory of solutions [1,2], which essentially seeks to partition the interaction potential into tln-ee parts that due to the interaction between the solvent molecules themselves, that due to die interaction between the solvent and the solute and that due to the interaction between the solute molecules dispersed within the solvent. The main difference from the dilute fluid results presented above is that the potential energy u(r.p is replaced by the potential of mean force W(rp for two particles and, for particles of solute in the solvent, by the expression... 

Here we have separated temis in the potential energy which involve tlie extra test particle, n Hesf... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

Here is the original, many-body potential energy fiinction, while Vq is a sum of single-particle spring potentials proportional to As X —> 0 the system becomes a perfect Einstein crystal, whose free energy... 

At larger particle separation, a second minimum may occur in tire potential energy. In many cases, tliis minimum is too shallow to be of much significance. For larger particles, however, tire minimum may become of order kT. Aggregation in tliis minimum is referred to as secondary minimum flocculation. 

In this section, we extend the above discussion to the isotopomers of X3 systems, where X stands for an alkali metal atom. For the lowest two electronic states, the permutational properties of the electronic wave functions are similar to those of Lij. Their potential energy surfaces show that the baniers for pseudorotation are very low [80], and we must regard the concerned particles as identical. The Na atom has a nuclear spin " K, and K have nuclear... 

---