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Unformatted text preview: Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 17, October 21, 2009 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 96-105.) Recapitulation of the Schrdinger Equation and its Eigenfunctions and Eigenvalues The operator that returns the system energy, E , as an eigenvalue is called the Hamiltonian operator, H . Thus, we write H = E (17-1) which is the Schrdinger equation. The typical form of the Hamiltonian operator with which we will be concerned takes into account five contributions to the total energy of a system (from now on we will say molecule, which certainly includes an atom as a possibility): the kinetic energy of the electrons and nuclei, the attraction of the electrons to the nuclei, and the interelectronic and internuclear repulsions. In more complicated situations, e.g., in the presence of an external electric field, in the presence of an external magnetic field, in the event of significant spin-orbit coupling in heavy elements, taking account of relativistic effects, etc., other terms are required in the Hamiltonian. We will consider some of these at later points, but we will not find them necessary for this lecture. Casting the Hamiltonian into mathematical notation (avoiding atomic units, for the moment, to ensure maximum clarity), we have H = " h 2 2 m e i # $ i 2 " h 2 2 m k k # $ k 2 " e 2 Z k 4 %& r ik k # i # + e 2 4 %& r ij i < j # + e 2 Z k Z l 4 %& r kl k < l # (17-2) where i and j run over electrons, k and l run over nuclei, h is Plancks constant divided by 2 , m e is the mass of the electron, m k is the mass of nucleus k , 2 is the Laplacian operator, e is the charge on the electron, Z is an atomic number, 4 is the permittivity of free space, and r ab is the distance between particles a and b . Note that is thus a function of 3 n coordinates where n is the total number of particles (nuclei and electrons), e.g., the x , y , and z cartesian coordinates specific to each particle. If we work in cartesian coordinates, the Laplacian has the form ! i 2 = " 2 " x i 2 + " 2 " y i 2 + " 2 " z i 2 (17-3) 17-2 In general, eq. 17-1 has many acceptable eigenfunctions for a given molecule, each characterized by a different associated eigenvalue E . That is, there is a complete set (perhaps infinite) of i with eigenvalues E i . For ease of future manipulation, we may assume without loss of generality that these wave functions are orthonormal, i.e., for a one particle system where the wave function depends on only three coordinates, ! """ i * ! j dx dy dz = # ij (17-4) where ij is the Kronecker delta (equal to one if i = j and equal to zero otherwise)....
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