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Unformatted text preview: Chem 120A Approximation methods: Perturbation Theory 04/03/06 Spring 2006 Lecture 28 READING: Engel (CS): Continue with Ch. 10-11 (we will return to these later) The course notes! Additional optional reading: McQuarrie and Simon, Physical Chemistry , Ch. 7-8 See website for additional supplemental reading. Why do we care about perturbation theory? In this class, you have seen several simple systems for which we can solve the Schr ´ ’odinger equation exactly, including the free particle, the particle in a box, the harmonic oscillator, and the hydrogenic (one electron) atoms. In each of these systems, we are concerned with a single particle that experiences some potential, V , which is a function of the particle’s position. It is instructive now to consider a problem where we have two particles in some potential that depends on their positions, and that these particles also interact with each other. The simplest example of this in physical chemistry is the helium atom. The helium atom consists of 3 particles, the nucleus and two electrons. The Hamiltonian for He will now have terms for the kinetic energy of each of the three particles, the Coulomb potential between each electron and the nucleus, and the Coulomb potential between the two electrons. In atomic units, the Hamiltonian is ˆ H = − ∇ 2 N 2 M − ∇ 2 R 1 2 m − ∇ 2 R 2 2 m − 2 R 1 − 2 R 2 + 1 | R 1 − R 2 | , (1) where N labels the nucleus kinetic energy operator, M is the mass of the nucleus, m is the electron mass, and the numbers 1 and 2 label operators for each of the two electrons. We can simplify this slightly by converting to the center of mass frame and considering the motion of only the electrons in the center of mass frame: ˆ H = − ∇ 2 r 1 2 m − ∇ 2 r 2 2 m − 2 r 1 − 2 r 2 + 1 | r 1 − r 2 | , (2) where ~ r is the coordinate for an electron in the center of mass frame of reference (see Lecture 20). This Hamiltonian now resembles the hydrogen atom Hamiltonian, but with one key difference: the interaction term for the electron-electron repulsion. This term depends on the coordinates of both electrons, which makes the Hamiltonian inseparable, and makes it impossible to solve the Schr¨odinger equation exactly . In Lecture 24 you saw how the orbital approximation can be applied to the helium atom. This is just one example of how approximation methods are useful in quantum mechanics. Another example of a powerful approximation is the hybridization model that you saw last week in lecture, which is commonly applied to understand the geometry of organic compounds. The main point here is that once there is more than one interacting particle, one can no longer exactly solve the Schr¨odinger equation, but most of the interesting problems in chemistry involve more than one particle!...
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