Kinetic energy effects

In the previous section we examined the variational result of the two-term wave function consisting of the covalent and ionic functions. This produces a 2 x 2 Hamiltonian, which may be decomposed into kinetic energy, nuclear attraction, and electron repulsion terms. Each of these operators produces a 2 x 2 matrix. Along with the overlap matrix these are 

As we discussed above, the two functions have the same charge density, and this implies that Tn = Tec and Vu = Vcc, but we expect Gu Gec-The normalization of the wave function requires 

Therefore, the principal role of the inclusion of the ionic term in the wave function is the reduction of the kinetic energy from the value in the purely covalent wave function. Thus, this is the delocalization effect alluded to above. We saw in the last section that the bonding in H2 could be attributed principally to the much larger size of the exchange integral compared to the Coulomb integral. Since the electrical effects are contained in the covalent function, they may be considered a first order effect. The smaller added stabilization due to the delocalization when ionic terms are included is of higher order in VB wave functions. 

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The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

McLuckey, S.A. Glish, G.L. Cooks, R.G. Kinetic Energy Effects in Mass Spectrometry/Mass Spectrometry Using a Sector/Quadrapole Tandem Instra-ment. Int. J. Mass Spectrom. Ion Phys. 1981, S9, 219-230. 

Kv is the viscosimetric constant and equals about 3 X 10 7 m2 s 2 for the viscosimeter used. KV is a correction factor for kinetic energy effects during the outflow. Its value is about 250 sec3 for viscosimeters as used in routine assays (49), but it can be neglected for our viscosimeter at outflow times above 20 sec. 

The name steric or Pauli repulsion for AEPauli already suggests it is repulsive (positive, antibonding), in spite of the negative contribution AVPauli. The repulsive character is due to the strongly positive AT0 (see Figure 2). Steric repulsion is evidently a kinetic energy effect and may also be appropriately called kinetic repulsion. 

Thermodynamic Governing Equations. Derivation of the expression for entropy production arising from mess transfer requires application of the fundamental balance equations. Potential and kinetic energy effects as well as momentum effects are neglected. With these assumptions the governing equations are given as follows ... 

Hydrogen at a temperature of 20°C and an absolute pressure of 1380 kPa enters a compressor where the absolute pressure is increased to 4140 kPa. If the mechanical efficiency of the compressor is 55 percent on the basis of an isothermal and reversible operation, calculate the pounds of hydrogen that can be handled per minute when the power supplied to the pump is 224 kW. Kinetic-energy effects can be neglected. 

Substituting into the energy equation and assuming horizontal flow and negligible kinetic energy effects yields ... 

Figure 18. The Oxy plane of linear configurations (above) and the Oyz plane of isoceles configurations in the A-IA2A3 problem. The potential function is deeper for darker gray. The lines cross at the center and at the equilibrium points. The white dashed line is the reaction path, if there were no inertial or nondiagonal kinetic energy effects.

The application of the method to the determination of bond dissociation energies in polyatomic molecules has been carried out, at least at first, in a less rigorous manner. The results are justified mainly by their self-consistency and also by their often remarkably close agreement with results obtained by different methods. For example, D(CH -H) was determined in two separate ways by electron impact, and the agreement between them was taken to show that errors due to kinetic energy effects were unlikely to be present. 

Further examples of the use of electron impact measurements to the determination of dissociation energies in polyatomic molecules are given in the discussion of particular values of dissociation energies in Chap. 9. It may be said here that in most cases the results are consistent with other information, suggesting that kinetic energy effects are often unimportant. 

A flow undergoes sudden contraction and expansion upon entering and leaving a capillary. Thus the velocity of the flow is less near the ends than that in the middle point of the capillary. This effect may be considered as equivalent to an increase in the effective length of the capillary. It is frequently called the Couette correction. The Hagen-Poiseuille equation, corrected for both the end and kinetic energy effects, can then be written as ... 

Zero-Point Energy and Kinetic Energy Effects in Quantum Clusters and Ultracold Clouds... 

If the system is adiabatic, 8q = 0 if no shaft work is done, SW, = 0 if the kinetic energy effects are negligible, d(U I2gc) = 0 and, if the elevation above the datum plane is constant, dZ = 0. Hence only the enthalpy term remains... 

The most notable aspect of these arrest results is the fact that the arrest values are only approximately 17 % of the initiation values as a result of the unstable crack growth. These low values are most likely due to the extreme time dependence of the chosen adhesive, along with artifacts associated with kinetic energy effects due to the rapid crack growth. Crack. jump distances as great as 150 mm were observed in static DCB testing, although 40-60 mm jumps were more common. These are comparable to stick-slip results collected by Blackman et al.[7]. who show Jumps of up to 100 mm. 

T. O. Tiernan and R. E. Marcotte, Collision-induced dissociation of NO and 02 at low kinetic energies. Effects of internal ionic excitation, J. Chem. Phys. 53, 2107-2122 (1970). 

The situation is far more complicated for non-spherical or more complex solvent molecules. In the first place, the very concept of a hard-core diameter is not a well-defined quantity. For water, for instance, one may conveniently choose the effective diameter of the water molecule as the location of the first peak of the radial distribution function g R) for pure water. If we adopt this definition, we find that there exists a small positive temperature dependence of the molecular diameter of water. The rationale for this behavior is quite simple. In liquid water at room temperature, most of the water molecules are engaged in hydrogen bonds. The optimal distance for a hydrogen bond is about 2.76 A, well within the effective hard-core diameter assigned to a water molecule, about 2.8 A. As the temperature increases, one should consider at least two competing effects. On the one hand, we have the kinetic energy effect that was described above, which tends to decrease the effective diameters of the free water molecules. On the other hand, hydrogen-bonded pairs are broken as we increase the temperature hence. 

S. A. McLuckey, G. L. Glish, and R. G. Cooks, Kinetic energy effects in mass spectrometry/mass spectrometry using a sector/quadrupole tandem instrument, Int. J. Mass Spectrom. Ion Rhys. 39, 219-230 (1981). 

Viscosities were measured in capillary viscometers of the suspended level Ubbelohde type (Cannon Instrument Co.) using a Wescan Automatic Viscosity Timer. Corrections for kinetic energy effects were not necessary since flow times were well over 100 seconds in all cases. Viscosities at 25 C of at least four concentrations were measured and extrapolated to infinite dilution using the usual relations ... 

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