The Atomic Orbitals of Hydrogen

The discussion of chemical bonding in the main text depends on the description of molecular orbitals as linear combinations of atomic orbitals. In this appendix we show how solutions of the Schrodinger equation for H-like atoms give us the atomic orbitals that are used as the building blocks in this approach. We will also take the opportunity to cover some basic ideas in quantum mechanics. 

By H-like we mean that a solitary electron moves in the field of a positively charged nucleus. This avoids the complication of considering electron-electron interactions. For a qualitative insight into chemical bonding, these can be reintroduced later. 

To calculate wavefunctions for any stationary state of an H-like atom we would like solutions to the Schrbdinger equation  

Equation (A9.1) assumes nothing about the units of the quantities used. If we switch to atomic units (au) then the manipulation and solution of this formula becomes clearer because we remove the clutter of the physical constants. In au ft = 1, tWe = 1 and the electron charge e = I, i.e. these quantities are used to define the units of angular momentum, mass and charge respectively. These units are actually derived from the solution of the H 

Molecular Symmetry David J. Willock 2009 John Wiley Sons, Ltd. ISBN 978-0-470-85347-4 

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The decrease in ionic character from hydrogen fluoride to hydrogen iodide is marked and due to increasing symmetry in the distribution of the bonding electrons. At first sight, the bond is a simple one, compounded of the Is and the atomic orbitals of hydrogen and halogen respectively. 

These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways. Their angular dependence is identical to that of the hydrogen orbitals, so quantum numbers and m are associated with each atomic orbital. The radial dependence of the orbitals in many-electron atoms differs from that of one-electron orbitals because the effective field differs from the Coulomb potential, but a principal quantum number n can still be defined. The lowest energy orbital is a Is orbital and has no radial nodes, the next lowest s orbital is a 2s orbital and has one radial node, and so forth. Each electron in an atom has associated with it a set of four quantum numbers (n, , m, mfj. The first three quantum numbers describe its spatial distribution and the fourth specifies its spin state. The allowed quantum numbers follow the same pattern as those for the hydrogen atom. However, the number of states associated with each combination of (n, , m) is twice as large because of the two values for m. ... 

Figure 7.1 Overlap of the atomic orbitals of hydrogen. Positive phase for wave functions (orbitals) shown red, negative as white...

The atomic orbitals of hydrogen are labeled by quantum numbers. Three integers are required for a complete specification. 

We begin our consideration of chemical bonding by looking at the simplest possible molecule, H2. The molecular orbitals derived for this system form the basis of the molecular orbitals for all other diatomic molecules, in much the same way that the atomic orbitals of hydrogen form the basis for all atomic orbitals. 

Fig. 23 Molecular orbital of the hydrogen molecule, , composed of the atomic orbitals of hydrogen atoms,...

It has been proved that each of the Hartree-Fock orbitals has the same asymptotic dependence on the distance from the molecule (N.C. Handy, M.T. Matron, HJ. Silverstone,PM x. Rev. 180 (1969) 45), i.e. const - exp( —V—2Ema. r), where Cma. is the orbital energy of HOMO. Earlier, people thought the orbitals decay as exp(—2c,-r), where c,- is the orbital energy expressed in atomic units. The last formula, as is easy to prove, holds for the atomic orbitals of hydrogen atoms (see p. 178). R. Ahlrichs,... 

The wave function for an atom simultaneously depends on (describes) all the electrons in the atom. The Schrodinger equation is much more complicated for atoms with more than one electron than for a one-electron species such as a hydrogen atom, and an explicit solution to this equation is not possible even for helium, let alone for more complicated atoms. We must therefore rely on approximations to solutions of the many-electron Schrodinger equation. One of the most common and useful of these is the orbital approximation. In this approximation, the electron cloud of an atom is assumed to be the superposition of charge clouds, or orbitals, arising from the individual electrons these orbitals resemble the atomic orbitals of hydrogen (for which exact solutions are known), which we described in some detail in the previous section. Each electron is described by the same allowed combinations of quantum numbers (w, m(, and /,)... 

The Atomic Orbitals of Hydrogen 367 The expectation for the average electron position in this orbital is then... 

Construct a concept map representing the atomic orbitals of hydrogen and their properties. 

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