Lobe Equation

The torque is obtained from Equation 4-162, assuming an efficiency of 0.80 for the 1 2 lobe profile motor. This is... 

The minimum value of the critical pressure will occur when the number of lobes is 2, and substituting this value into equation 13.49 gives ... 

The normalization factors are included in equations 4.11 and 4.12 and the minus sign in equation 4.12 means combine the 2s orbital with the 2p orbital after it has been reversed so that its positive lobe is directed in the opposite direction from that in equation 4.1 V. 

The Schrodinger equation is a differential equation, which means that solutions of it are themselves equations. The solutions, however, are not differential equations, but simple equations for which graphs can be drawn. Such graphs, which are three-dimensional pictures that show the electron density, are called orbitals or electron clouds. Most students are familiar with the shapes of the s and p atomic orbitals (Figure 1.1). Note that each p orbital has a node—a region in space where the probability of finding the electron is extremely small.2 Also note that in Figure 1.1 some lobes of the orbitals are labeled + and others -. ... 

In addition to transforming the radial equation to the Numerov form, the transformation also changes the scale to one more nearly matching the frequency of the oscillations of the radial function. For bound functions, the wavelength of the wavefunction increases with r, and the substitution x = n(r) keeps the number of points per lobe closer to being constant. 

Since from Equation (9.72), the STFT of a sum of sinusoids is a sum of shifted and scaled versions of W (ffl), the two-lobe case of Figure 9.24 can be simply extended to the case where there are L overlapping lobes. Specifically, a relation can be written which reflects the linear dependence of the STFT on all L lobes [Quatieri and Danisewicz, 1990],... 

Here, A and B are the fine-structure and the rotational constants of OH and is the azimuthal angle out of the rotation plane with

internuclear vector, cos2 and sin2 represent, respectively, lobes in the rotation plane and perpendicular to it. The Cj approach the value 2 in the limit of large j which has the consequence that p(A ) is preferentially oriented in the rotation plane while p A") is preferentially oriented perpendicular to the plane of rotation. Equation (11.15) gives the relative weights of the in-plane and the out-of-plane contributions of the real A-doublet states. They play a vital role in the photolysis of water through the AxBi state (see Section 11.2.2). 

As discussed in Section 6.1.3, we could replace tp2Py with — stationary state (see Equation 6.9). So there is nothing wrong with writing the negative lobe on top and the positive on the bottom. 

Surfaces may be drawn to enclose the amplitude of the angular wave function. These boundary surfaces are the atomic orbitals, and lobes of each orbital have either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. 

The asymmetry of orbitals arises because of a property of orbitals that we have not yet considered. When the wave equation for a p orbital is solved, the two lobes have opposite algebraic signs, and -. Thus, when a p orbital hybridizes with an s orbital, the positive p lobe adda to the s orbital, but the negative p lobe aubtracis from the a orbital. The resultant hybrid orbital is unsymmotrical about the nucleus and is strongly oriented in one direction, as shown in Figure Lll. 

In the FMO diagrams shown above (Figs. 8.28-8.31) for cycloaddition reactions the orbital lobes are shown with the same sizes (or coefficients), but usually they are of different sizes (Fig. 8.32). Coefficients are derived from the wave equations for the tr-orbitals. The unsymmetrical substitution of a diene or dienophile perturbs the orbital coefficients in an... 

The result of the calculations is the set of atomic orbitals familiar to all chemists. Figure 2-7 shows diagrams of s, p, and d orbitals and Figure 2-8 shows lines of constant electron density in several orbitals. The different signs on the wave functions are shown by different shadings of the orbital lobes in Figure 2-7, and the outer surfaces shown enclose 90% of the total electron density of the orbitals. The orbitals we use are the common ones used by chemists others that are also solutions of the Schrodinger equation can be chosen for special purposes. 

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