Combined waves

One starts with the Hamiltonian for a molecule H r, R) written out in terms of the electronic coordinates (r) and the nuclear displacement coordinates (R, this being a vector whose dimensionality is three times the number of nuclei) and containing the interaction potential V(r, R). Then, following the BO scheme, one can write the combined wave function [ (r, R) as a sum of an infinite number of terms... 

It is also possible to have a combined wave, which has both gradual and abrupt parts. The general rule for an isothermal, trace system is that in passing from the initial condition to the feed point in the isotherm plane, the slope of the path must not decrease, if it does, then a shock chord is taken for that part of the path. Referring to Fig. 16-19, for a transition from (0,0) to (1,1), the dashes indicate shock parts, which are connected by a simple wave part between points Pi and Pg. 

FIG. 16-19 Path in isotherm plane for a combined wave (After Tudge). 

Thus, laser femtosecond pulses allow us to combine wave (optical) and corpuscular (ion electron) microscopy. This kind of microscopy is based on... 

For nonconvex equilibrium functions, combined wave solutions are possible which consist in parts of spreading waves and in others of constant pattern waves. In the equilibrium diagram combined wave solutions represent the convex hull of the equilibrium line, as illustrated in Fig. 5.5(c). 

Fig. 5.5. Construction of wave solutions in the scalar case, (a) Constant pattern wave (b) spreading wave (c) combined wave solution.

New phenomena compared to nonreactive Langmuir systems are the same as in the binary case - that is, the existence of combined waves due to the occurrence of inflection points of the equilibrium functions y(x) or Y(X) and limitations on feasible product composition due to adsorptivity reversal similar to azeotropic distillation. Nonreactive examples for the latter were treated in Refs. [6 - 8], reactive examples will be discussed in the next section. 

Atomic orbitals are wave functions and the different wave functions can be combined together rather in, the way waves combine. You may be already familiar with the ideas of combining waves— they can add together constructively (in-phase) or destructively (out-of-phase). 

We learned in Chapter 5 that each solution to the Schrodinger equation, called a wave function, represents an atomic orbital. The mathematical pictures of hybrid orbitals in valence bond theory can be generated by combining the wave functions that describe two or more atomic orbitals on a single atom. Similarly, combining wave functions that describe atomic orbitals on separate atoms generates mathematical descriptions of molecular orbitals. 

Grant, W. D., Madsen, O. S., 1979. Combined wave and current interaction with a rough bottom. 

The valence bond and molecular orbital theories differ in how they use the orbitals of two hydrogen atoms to describe the orbital that contains the electron pair in H2. Both theories assume that electron waves behave much like more familiar waves, such as sound and light waves. One property of waves that is important here is called interference in physics. Constructive interference occurs when two waves combine so as to reinforce each other ( in phase ) destructive interference occurs when they oppose each other ( out of phase ) (Figure 1.15). In the valence bond model constructive interference between two electron waves is seen as the basis for the shared electron-pair bond. In the molecular orbital model, the wave functions of molecules are derived by combining wave functions of atoms. 

FIGURE 6-5 Superposition of two waves of different frequencies but identical amplitudes (a) wave 1 with a period of lA i (b) wave 2 with a period of 1/t (j j 1.25t, ) (c) combined wave pattern. Note that superposition of r, and V2 produces a beat pattern with a period of 1/A/ where - U l i. ... 

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