Hydrogenlike orbitals, table

We can use the quantum mechanical model of the atom to show how the electron arrangements in the hydrogenlike atomic orbitals of the various atoms account for the organization of the periodic table. Our main assumption here is that all atoms have the same type of orbitals as have been described for the hydrogen atom. As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these hydrogenlike orbitals. This is called the aufbau principle. 

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. 

The orbital has two nodal cones. The orbital has two nodal planes. Note that the view shown is not the same for the various orbitals. The relative signs of the wave functions are indicated. The other three real M orbitals in Table 6.2 have the same shape as the 3d -y orbital but have different orientations. The 3djcy orbital has its lobes lying between the x and y axes and is obtained by rotating the 3d -f orbital by 45° about the z axis. The 3dy and 3dx orbitals have their lobes between the y and z axes and between the X and z axes, respectively. (Online three-dimensional views of the real hydrogenlike orbitals are at www.falstad.com/qmatom these can be rotated nsing a mouse.)... 

Normalized radial functions for a hydrogenlike atom are given in Table A 1.1 and plotted graphically in Fig. A 1.1 for the first ten combinations of n and /. It will be seen that the radial functions for Is, 2p, 3d, and 4f orbitals have no nodes and are everywhere of... 

Verify that the 2)dxy orbital given in Table 7.1 is a normalized eigenfunction of the hydrogenlike Schrddinger equation. 

Table 1.4 summarizes the relationship between quantum numbers and hydrogenlike atomic orbitals. When Z = 0, (2/ + 1) = 1 and there is only one value of m/, so we have an s orbital. When / = 1, (2/ + 1) = 3, so there are three values of m , giving rise to three p orbitals, labeled p, Py, and p. When / = 2, (2Z + 1) = 5, so there are five values of mi, and the corresponding five d orbitals are labeled with more elaborate subscripts. In the following sections we discuss the s, p, and d orbitals separately. 

The formulas for the hydrogenlike ion solutions (in atomic units) of most interest in quantum chemistry are listed in Table 4-2. The tabulated functions are all in real, rather than complex, form. Problems involving atomic orbitals are generally far easier to solve in atomic units. 

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