Hydrogen atom orbital shapes

The solutions of the angular dependent part are the spherical harmonics, Y, known to most chemists as the mathematical expressions describing shapes of (hydrogenic) atomic orbitals. It is noted that Y is defined only in terms of a central field and not for atoms in molecules. 

There are several discrete atomic orbitals available to the electron of a hydrogen atom. These orbitals differ in energy, size, and shape, and exact mathematical descriptions for each are possible. Following is a qualitative description of the nature of some of the hydrogen atomic orbitals. 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

So far we have talked about the shapes of the hydrogen atomic orbitals but not about their energies. For the hydrogen atom the energy of a particular orbital is determined by its value of n. Thus all orbitals with the same value of n have the same energy—they are said to be degenerate. This feature... 

The sizes and shapes of the hydrogen atom orbitals are important in chemistry because they provide the foundations for the quantum description of chemical bonding and the molecular shapes to which it leads. Sizes and shapes of the orbitals are revealed by graphical analysis of the wave functions, of which the first few are given in Table 5.2. Note that the radial functions are written in terms of the dimensionless variable a, which is the ratio of Zr to ao- For Z = 1, a- = 1 at the radius of the first Bohr orbit of the hydrogen atom. 

For the hydrogen atom, we can solve the Schrodinger equation exactly to obtain the allowed energy levels and the hydrogen atomic orbitals. The sizes and shapes of these orbitals tell us the probability distribution for the electron in each quantum state of the atom. We are led to picture this distribution as a smeared cloud of electron density (probability density) with a shape that is determined by the quantum state. 

For all other atoms, we have to generate approximations to solve the Schrodinger equation. The Hartree orbitals describe approximately the amplitude for each electron in the atom, moving under an effective force obtained by averaging over the interactions with all the other electrons. The Hartree orbitals have the same shapes as the hydrogen atomic orbitals—but very different sizes and energy values—and thus guide us to view the probability distribution for each electron as a smeared cloud of electron density. 

The Hartree orbitals have the same shapes as the corresponding hydrogen atomic orbitals, but their sizes are quite different. 

Figure 7.18 shows boundary surface diagrams for the li, 2s, and 3s hydrogen atomic orbitals. All s orbitals are spherical in shape but differ in size, which increases as the principal quantum number increases. Although the details of electron density variation within each boundary surface are lost, there is no serious disadvantage. For us the most important features of atomic orbitals are their shapes and relative sizes, which are adequately represented by boundary surface diagrams. 

FIGURE 6.13 Shapes of some hydrogen-atom orbitals. 

It is fortunate that whenever the SCF method is employed for any element or ion having more than one electron, the resulting wave functions always tend to resemble those of the hydrogen atom. Hence, the probability electron densities for the other elements can be compared to the hydrogen/c orbital shapes. However, these wave functions are not identical to those of hydrogen, and the following differences should be noted ... 

It is easy to see that the full shape of the orbital is better represented by the sum of these two Gaussians, especially at the tail of the cur ve where chemical bonding takes place, than it is by one Gaussian. When we run an STO-2G ab initio calculation on the hydrogen atom using the GAUSSIAN stored parameters rather than supplying oirr own, the input file is... 

Split valence basis sets allow orbitals to change size, but not to change shape. Polarized basis sets remove this limitation by adding orbitals with angular momentum beyond what is required for the ground state to the description of each atom. For example, polarized basis sets add d functions to carbon atoms and f functions to transition metals, and some of them add p functions to hydrogen atoms. 

Remember that the molecular shape ignores the lone pair. The hydronium ion has a trigonal pyramidal shape described by the three s p hybrid orbitals that form bonds to hydrogen atoms. 

Quantum numbers and shapes of atomic orbitals Let us denote the one-electron hydrogenic Hamiltonian operator by h, to distinguish it from the many-electron H used elsewhere in this book. This operator contains terms to represent the electronic kinetic energy ( e) and potential energy of attraction to the nucleus (vne),... 

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