The Hydrogen Atom and Atomic Orbitals

The simplest atomic system that we can consider is the hydrogen atom. To obtain the Hamiltonian operator for this three-dimensional system, we must replace the operator d2/dx2 by the partial differential operator 

The potential energy of the electron is given by V = e2l(x2 + y2 + z2)l/2, where (x2 + y2 + z2)l/2 is the radial distance from the nucleus. So the Schrodinger equation for the hydrogen atom is 

For an s orbital the angular part of the wave function is constant 

A plot of the angular function of 0 against the value of 9 in a plane through the z axis has the form of a plot of cos 9 and consists of two circles with a node in a line along the x axis 

Other atoms with only a single electron (He+, Li2+, etc.) are known as hydrogen-like atoms. The Schrodinger equation for such a system is the same as that for the hydrogen atom 

With minor adjustments, the same result holds when the system consists of two masses oscillating about their center of mass, like two atoms joined by a chemical bond which is a spring with force constant equal to k (see Chapter 2). 

The s are the allowed electronic energies, and the corresponding wavefunctions give the probability of finding the electron at any point in space. 

The solution of equation 3.25 requires a few pages of algebra and will not be given in detail here. The conceptual framework of the procedure is as follows. 

Equation 3.25 is transformed from cartesian to polar coordinates, so that electronic states are described by wavefunctions ilf(r, 0, ( )). 

Since each term in the hamiltonian depends either on the radial coordinate r or on the angular coordinates 0 and ( ), the total wavefunction can be factorized into a product of a radial part, R f), that depends only on the nucleus-electron distance, and an angular part, 5(0, ( )), that depends only on angular coordinates  

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For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

The hydrogen atom orbitals give us the numbers 2, 8, 18, and 32—the numbers we find separating the specially stable electron populations of the inert gases. It was necessary to multiply n2 by two—an important factor that could not have been anticipated. Furthermore, it will be necessary to find an explanation for the occurrence of eight-electron differences both at neon and at argon and eighteen-electron differences both at krypton and at xenon. 

In NH and NFS, three p orbitals are involved in the bonding [see representation (30)]. Figure 16-10 shows the spatial arrangement implied by assuming persistence of the hydrogen atom orbitals after bonding. We expect, then, that ant-... 

There is a similar effect in the length of C-H bonds, but this is less dramatic, primarily because the hydrogen atomic orbital involved (1 ) is considerably smaller than any of the hybrid orbitals we are considering. Nevertheless, C-H bonds involving sp-hybridized carbon are shorter than those involving i /t -hybridized carbon, and those with -hybridized carbon are the longest. 

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. 

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. 

Table 1.1 The radial and angular components of the hydrogenic atomic orbitals with distinct normalization constants for the radial and angular functions. The parameter, p = extends the application of the functions in the table entries for non-hydrogen one-electron atomic species. Remember that the solutions to the angular equation in are exp(-f / — im0) and the real forms given are obtained by taking the sums and differences of the expansions of the complex exponentials and then applying equations 1.1 to 1.3 to these results. The column headed -I-/- indicates the particular choices of sum when relevant. ...

Cells A 14 to B 15 are the linked sums of the kinetic and potential energy terms over the Slater functions defined in columns basisl D and E. For later consistency, these sums of the energy terms over the bases, in the evaluation of the Schrodinger equation, for the hydrogen atom orbitals are identified as the Fock matrix elements on this worksheet. 

A similar relation then holds between hybridized orbital

hydrogen atomic orbital on the atom at the left hand position of Be. 

Figure 3.2 shows the radial distribution functions for the hydrogen 2s and 2p orbitals, from which it can be seen that the 2s orbital has a considerably larger probability near the nucleus than the 2p orbital. When an electron in a polyelectronic atom occupies the n = 2 level, it would be more stable in the 2s orbital than in the 2p orbital. In the 2s orbital it would be nearer the nucleus and be more strongly attracted than if it were to occupy the 2p sub-set. 

For H, we write the MO as the sum of one-electron wavefunctions on each of the two protons A and B. The ground state is going to reflect contributions mainly from the ground state atomic wavefunctions, the hydrogen Is and Isg orbitals, where Is is centered on nucleus A and Isg on nucleus B, so our first molecular orbitals will be sums of these two functions. 

In Chapter 9 we used the hydrogenic atomic orbitals and the building-up principle to deduce the ground electronic configurations of many-electron atoms. Here we use the same procedure for many-electron diatomic molecules (such as Hj with two electrons and even Brj with 70), but using the HJ molecular orbitals as a basis. 

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