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. One of the most common and useful of these is the orbital approximation. We must therefore rely on approximations to solutions of the many- electron Schrodinger equation. 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. The wave function for an atom simultaneously depends on (describes) all the electrons in the atom. The last formula, as is easy to prove, holds for the atomic orbitals of hydrogen atoms (see p. Earlier, people thought the orbitals decay as exp(-2c,-r), where c,- is the orbital energy expressed in atomic units. It has been proved that each of the Hartree-Fock orbitals has the same asymptotic dependence on the distance from the molecule (N.C. 23 Molecular orbital of the hydrogen molecule,, composed of the atomic orbitals of hydrogen atoms. 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. We begin our consideration of chemical bonding by looking at the simplest possible molecule, H2. Three integers are required for a complete specification. The atomic orbitals of hydrogen are labeled by quantum numbers. Positive phase for wave functions (orbitals) shown red, negative as white. įigure 7.1 Overlap of the atomic orbitals of hydrogen. However, the number of states associated with each combination of (n,, m) is twice as large because of the two values for m. The allowed quantum numbers follow the same pattern as those for the hydrogen atom. The first three quantum numbers describe its spatial distribution and the fourth specifies its spin state. Each electron in an atom has associated with it a set of four quantum numbers (n,, m, mfj. 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. 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. Their angular dependence is identical to that of the hydrogen orbitals, so quantum numbers and m are associated with each atomic orbital. These Hartree orbitals resemble the atomic orbitals of hydrogen in many ways. At first sight, the bond is a simple one, compounded of the Is and the atomic orbitals of hydrogen and halogen respectively. 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. These units are actually derived from the solution of the H these quantities are used to define the units of angular momentum, mass and charge respectively. In au ft = 1, tWe = 1 and the electron charge e = I, i.e. 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. To calculate wavefunctions for any stationary state of an H-like atom we would like solutions to the Schrbdinger equation Įquation (A9.1) assumes nothing about the units of the quantities used. For a qualitative insight into chemical bonding, these can be reintroduced later. This avoids the complication of considering electron-electron interactions. īy H-like we mean that a solitary electron moves in the field of a positively charged nucleus. We will also take the opportunity to cover some basic ideas in quantum mechanics. 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. The discussion of chemical bonding in the main text depends on the description of molecular orbitals as linear combinations of atomic orbitals.
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