Kinetic energy, neglection

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... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

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, ... 

Neglecting changes in kinetic energy and potential energy produces ... 

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 ... 

Neglecting the mass polarization and reintroducing the kinetic energy operator gives... 

The energy of a Is-electron in a hydrogen-like system (one nucleus and one electron) is —Z /2, and classically this is equal to minus the kinetic energy, 1/2 mv, due to the virial theorem E — —T = 1/2 V). In atomic units the classical velocity of a Is-electron is thus Z m= 1). The speed of light in these units is 137.036, and it is clear that relativistic effects cannot be neglected for the core electrons in heavy nuclei. For nuclei with large Z, the Is-electrons are relativistic and thus heavier, which has the effect that the 1 s-orbital shrinks in size, by the same factor by which the mass increases (eq. (8.2)). 

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

In the derivation above, we have included the kinetic energy of the nuclei in the Hamiltonian and considered a stationary state. In Eq. II.3, this term has been neglected, and we have instead assumed that the nuclei have given fixed positions. It has been pointed out by Slater34 that, if the nuclei are not situated in the proper equilibrium positions, the virial theorem will appear in a slightly different form. (A variational derivation has been given by Hirschfelder and Kincaid.11)... 

Wigner s formula is open to criticism also on another point, since he assumes the existence of a stationary electron state where the density is so low that the kinetic energy may be neglected. This is in contradiction to the virial theorem (Eq. 11.15), which tells us that the kinetic energy can never be neglected in comparison to the potential energy and that the latter quantity is compensated by the former to fifty per cent. A reexamination of the low density case would hence definitely be a problem of essential interest. 

Because the pipes are long, the kinetic energy of the fluid and minor losses at the entry to the pipes may be neglected. 

In the last column of Table 4.2, values are given for G /A calculated by ignoring the effect of changes in the kinetic energy of the gas, that is by neglecting the log term in equation 4.57. It will be noted that the effects are small ranging from about 2 per cent for P2a = 9 MN/m2 to about 13 per cent for P2a = 4 MN/m2. 

Using the value of (0) obtained by neglecting the kinetic energy ... 

What error would be introduced if the change in kinetic energy of the gas as a result of expansion were neglected ... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

A gas will obey the ideal gas equation whenever it meets the conditions that define the ideal gas. Molecular sizes must be negligible compared to the volume of the container, and the energies generated by forces between molecules must be negligible compared to molecular kinetic energies. The behavior of any real gas departs somewhat from ideality because real molecules occupy volume and exert forces on one another. Nevertheless, departures from ideality are small enough to neglect under many circumstances. We consider departures from ideal gas behavior in Chapter if. 

Pressure drop in the transmission pipes is a combination of pressure losses in the pipes and pipe fittings7. Pipe fittings include bends, isolation valves, control valves, orifice plates, expansions, reductions, and so on. If the fluid is assumed to be incompressible and the change in kinetic energy from inlet to outlet is neglected, then ... 

The derivative (nonadiabatic) coupling, ffy, is the term neglected in the Bom-Oppenheimer approximation that is responsible for nonadiabatic transitions between different states I and. /. It originates from the nuclear kinetic energy operator operating on the electronic wavefunctions ijf] and is given by... 

If the logarithmic term in the denominator (which comes from the change in kinetic energy of the gas) is neglected, the resulting equation is called the Weymouth equation. Furthermore, if the average density of the gas is used in the Weymouth equation, i.e.,... 

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