Lecture 1 - Bonding and Hybridization

||2 = probability density
(2) Wavefunctions Different systems have different wavefunctions. If the system is in a stationary state, then its probability density does not change in time. The 1s state of the hydrogen atom is such a state. This means that if I prepare a lot of identical 1s H atoms and wait a while before measuring the position of their electrons, then the probability distribution I get doesn't depend on how long I wait. If the H atoms are in a superposition of these stationary states, then the distribution will change with time in a complicated way. So what are these stationary states?
E. Kwan Hydrogenlike Atoms (3) Quantum Numbers
Lecture 1: Bonding and Hybridization E
all positive energies allowed
0 ..........n=4 and up........... n=3 n=2 2,0,0 2s 2,1,-1 2,1,0 2,1,1 2p 3s 3p 3d
r
(n, l, m)
-13.6 eV
n=1 1,0,0 1s
n,l,m Actually, not all the energies are quantized. Zero energy is defined as the electron and the proton at an infinite distance where they don't interact. If the electron has more than zero energy, then you can interpret that as an ionized atom: the electron has escaped the potential of the hydrogen atom, and is free to be anywhere it wants, with any energy (more or less). Notice that the levels get closer and closer together as n increases, but the s/p/d/f energies remain degenerate on each level. This is only true for hydrogenlike atoms.
I thank Professor Frank Weinhold (Wisconsin) for many helpful discussions in the preparation of this lecture.
E. Kwan
Lecture 1: Bonding and Hybridization
In this state, we will find the electron at a radius of <r> on average (but not every time). In quantum mechanics, we postulate that there is an abstract mathematical object called the wavefunction, which, when squared, gives the probability distribution (from which one can calculate the radial distribution function);
Here, we imagine the nucleus is fixed at the origin. In spherical coordinates, the position of the electron is (r, , ). Here are the key facts about the hydrogenlike atoms: (1) Radial Distribution Functions Q: Where is the electron? A: Don't ask. The electron isn't anywhere exactly, any more than the sound wave from a violin is anywhere specific. More precisely, if we prepare a bunch of hydrogen atoms in exactly the same way, and then we measure the position of the electron, then we will not get the same answer every time. Rather, we will get some distribution of answers. This is a probability distribution. To make this more concrete, here is an example. One of the states you can prepare the hydrogen atom in is the "1s state," which is its lowest energy, or ground, state. The "radial probability distribution" plots the probability of finding the electron in a spherical shells extending out from the nucleus.
Helpful References 1. Coulson's Valence, 3rd ed. McWeeny, R. Oxford: Oxford University Press, 1979.
2. Quantum Chemistry, 5th ed. Levine, I.N. Upper Saddle River, By the end of the course, I would like you to be able to think and write intelligently about any organic reaction. New Jersey: Prentice-Hall, Inc., 2000. 3. Valency and Bonding: A Natural Bond Orbital Donor-Acceptor Perspective, Cambridge, Cambridge University Press, 2005.
Scope of Lecture shape of lone pairs equivalent vs. inequivalent hybridization
multielectron atoms
the variational principle
bonding and hybridization
the virial theorem basis functions vs. orbitals
onding and Hybridization Bonding and Hybridization
Eugene E. Kwan Sept. 5, 2012.
Chem 106
Course Details: Instructor: Dr. Eugene Kwan (ekwan@) TF: Dr. Filippo Sladojevich (sladojevich@) Textbook/Readings: course notes only
1s radial distribution
Chem 106
Hydrogenlike Atoms Let us start at the beginning, with the hydrogen atom that you (hopefully) have come to know and love. The hydrogen atom has one proton and one electron. In hydrogenlike atoms, there are Z protons and one electron: r +Z protons This includes H, He+, Li2+, etc. one electron
Chem 106
As it turns out, these stationary states can't be just anything. In fact, they are labelled by three quantum numbers, which have to be integers. Because these numbers can only take on certain discrete values, the stationary states are said to be quantized. principal quantum number: n = 1, 2, 3... energies: E 1 / n2 angular momentum quantum number: l = 0, 1, 2, ..., n-1 l=0 (s); l=1 (p); l=2 (d); l=3 (f); ... magnetic quantum number: m = -l, -l+1, ..., 0, ..., l-1, l - essentially give different orientations of similar orbitals - for example, for n=2, there are three possible solutions: (n, l, m) (2, 0, 0) = 2s (2, 1, -1), (2, 1, 0), (2, 1, 1): linear combinations are the 2px, 2py, and 2pz (4) Energies Notice that the energy depends only on n. That means the 2s and 2p wavefunctions (any one electron wavefunction is called an orbital) are of the same energy. The 3s, 3p, and 3d states are also of the same energy (but don't have the same energy as the 2s/2p). En = -13.6 eV / n2 (for Z=1)