Kinetic energy term

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

We want to derive an expression for the rate of change of the energy with time, dE/dl. Firs we differentiate the kinetic energy term with respect to time ... 

Replacing the nuclear and electronic momenta by the modifications shown above in the kinetic energy terms of the full electronic and nuclear-motion hamiltonian results in the following additional factors appearing in H ... 

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

If we scale time as t = xr, then the frst term in (5.52) decreases as l/>/, while the other two are independent of friction. Therefore, at large rj the second derivative term in (5.52), as well as the kinetic energy term in the action, can be neglected, and the entire effect of friction is to change the timescale. That is, the solution to (5.52) is Q x) = Q x/ri) where Q is a function independent of rj. The instanton velocity is scaled as Q cc and the action (5.38) grows linearly with r, ... 

Equation 6-10 is the macroscopic energy balance equation, in which potential and kinetic energy terms are neglected. From tliermodynamics, the enthalpy per unit mass is expressed as... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

In order to obtain the kinetic energy term for use in the energy balance equation, it is necessary to obtain the average kinetic energy per unit mass in terms of the mean velocity. 

The power requirement is then that for compression of the gas from pressure Pi to P3 and for imparting the necessary kinetic energy to it. Under normal conditions, however, the kinetic energy term is negligible. Thus for an isothermal efficiency of compression 17. the power required is ... 

Kinetic energy term. Its expression is straightforward to write ... 

We show next that the parity operator IT commutes with the Hamiltonian operator H if the potential energy F(q) is an even function of q. The kinetic energy term in the Hamiltonian operator is given by... 

For liquids having a velocity of less than 7 ft/sec (2 m/sec) the kinetic energy term p v2/g can usually be ignored. The liquid horsepower is obtained from the pressure drop ... 

The blower calculations are similar to those for the pumps. The optimum velocity for air is around 75ft/sec and the pressure drop is about 0.2 psi per 100ft of piping. At this velocity the kinetic energy term in the pressure-drop equation cannot be ignored. The pressure drop can be approximated if a 14 in. duct is specified and Figure 8-7b is used. 

For a many-electron molecule, the Hamiltonian operator can thus be written as the sum of the electrons kinetic energy term, which in turn is the sum of individual electrons ... 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

---