Orbitals hydrogenlike

In dealing with molecules, the real hydrogenlike orbitals are more useful than the complex ones. For example, we shall see in Section 15.6 that the real atomic orbitals 2p, 2py, and 2p of the oxygen atom have the proper symmetry to be used in constructing a wave function for the H2O molecule, whereas the complex 2p orbitals do not. 

The hydrogenlike wave functions are one-electron spatial wave functions and so are hydrogenlike orbitals (Section 6.5). These functions have been derived for a one-electron atom, and we cannot expect to use them to get a truly accurate representation of the wave function of a many-electron atom. The use of the orbital concept to approximate many-electron atomic wave functions is discussed in Chapter 11. For now we restrict ourselves to one-electron atoms. 

There are two fundamentally different ways of depicting orbitals one way is to draw graphs of the functions a second way is to draw contour surfaces of constant probability density. 

FIGURE 6.10 Polar graphs of the 0 factors in the s and hydrogen-atom wave functions. 

Sifi is negative for jir 0 ir. Strictly speaking, in graphing cos 6 we only get the upper circle, which is traced out twice to get two tangent circles, we must graph cos 0.  

First consider drawing graphs. To graph the variation of lA as a function of the three independent variables r, 6, and (f , we need four dimensions. The three-dimensional nature of our world prevents us from drawing such a graph. Instead, we draw graphs of the 

Now consider drawing contour surfaces of constant probability density. We shall draw surfaces in space, on each of which the value of the probability density, is constant. Naturally, if i/r is constant on a given surface, lA is also constant on that surface. The contour surfaces for i/rpandfor i/r are identical. 

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The Fourier transforms of the hydrogenlike orbitals were shown by Fock [18] to be expressible in terms of 4-dimensional hyperspherical harmonics when momentum space is mapped onto the surface of a 4-dimensional unit hypersphere by the transformation ... 

With this transformation, as Fock was able to show, the Fourier-transformed hydrogenlike orbitals become ... 

Aufbau principle the principle stating that as protons are added one by one to the nucleus to build up the elements, electrons are similarly added to hydrogenlike orbitals. (12.13) Autoionization the transfer of a proton from one molecule to another of the same substance. (7.2)... 

One of the very early triumphs of quantum theory was the exact solution of the wave equation for hydrogenlike atoms. It was therefore natural to try to use hydrogenlike orbitals to build up solutions to the Hartree-Fock equations for... 

The reader will recognize that this is just the wave equation obeyed by the familiar hydrogenlike orbitals, except that Z/n has been replaced by the constant k. Thus, if we start with a hydrogenlike orbital and replace Z/n everywhere by the constant k, we will have generated a set of Coulomb Sturmians. They have the form... 

The reader may verify that these become the familiar hydrogenlike orbitals if k is replaced by Z n, where Z is the nuclear charge and n is the principal quantum number. It can be shown [19] that the Coulomb Sturmians obey a set of potential-weighted orthonormality relations of the form ... 

The general discussion of Section 30, which is essentially a perturbation calculation, is not capable of very high accuracy, especially since it is not ordinarily practicable to utilize any central field except the coulombic one leading to hydrogenlike orbital functions. In this section we shall consider the application of the variation method (Sec. 26) to low-lying states of simple atoms such as lithium and beryllium. This type of treatment is much more limited than that of the previous section, but for the few states of simple atoms to which it has been applied it is more accurate. 

The function fa composed of Is hydrogenlike orbital wave functions with effective nuclear charge 2e leads to a minimum in the energy curve at r, = 1.01 A and the value 2.9 v.e. for the energy of dissociation D, into He + He+. A more accurate treatment1 can be made by minimizing the energy for each value... 

This picture of polyelectronic atoms leads to hydrogenlike orbitals for these atoms. They have the same general shapes as the orbitals for hydrogen, but their sizes and energies are different. The differences occur because of the interpiay between nuclear attraction and the electron repulsions. 

Some hydrogenlike orbital surfaces are shown in Rg. 6.13. The 2s orbital has a... 

The hydrogen-atom potential-energy function is even, and the hydrogenlike orbitals can be chosen to have definite parity (Problems 7.17 and 7.23). 

Let /i be a normalized Is hydrogenlike orbital for nuclear charge occupied by electron 1. Let/2 be the same function for electron 2 ... 

We chose to use the real 2p hydrogenlike orbitals, rather than the complex ones. 

Accurate representation of a many-electron atomic orbital (AO) requires a linear combination of several Slater-type orbitals. For rough calculations, it is convenient to have simple approximations for AO s- We might use hydrogenlike orbitals with effective nuclear charges, but Slater suggested an even simpler method to approximate an AO by a single function of the form (11-14) with the orbital exponent C taken as... 

The orbital concept and the Pauli exclusion principle allow us to understand the periodic table of the elements. An orbital is a one-electron spatial wave function. We have used orbiteils to obteiin approximate wave functions for many-electron atoms, writing the wave function as a Slater determinant of one-electron spin-orbitals. In the crudest approximation, we neglect all interelectronic repulsions and obtain hydrogenlike orbitals. The best possible orbitals are the Heu tree-Fock SCF functions. We build up the periodic table by feeding electrons into these orbitals, each of which can hold a pair of electrons with opposite spin. 

Discuss the similarities and differences between a li and a 2s hydrogenlike orbital. 

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