3502_21_octo30_LG

3502_21_octo30_LG - Chem 3502/5502 Physical Chemistry II...

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Unformatted text preview: Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 21, October 30, 2009 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 514-518.) Solved Homework We already showed that S x = ( h /2) (eq. 20-19) For the operation on we have S x ! = h 2 1 1 " # $ % & 1 " # $ % & = h 2 1 " # $ % & = 1 2 h ( Working with the S y operator involves S y ! = h 2 " i i # $ % & ( 1 # $ % & ( = h 2 i # $ % & ( = 1 2 i h ) and S y ! = h 2 " i i # $ % & ( 1 # $ % & ( = h 2 " i # $ % & ( = " 1 2 i h ) For the raising operator we have 21-2 S + ! = h 1 " # $ % & 1 " # $ % & = and S + ! = h 1 " # $ % & 1 " # $ % & = h 1 " # $ % & = h ( Similarly S ! " = h 1 # $ % & ( 1 # $ % & ( = h 1 # $ % & ( = h ) and S ! " = h 1 # $ % & ( 1 # $ % & ( = Spin-Free Many-Electron Wave Functions and Antisymmetry We have already seen that if we have two electrons in two orbitals a and b , then an acceptable antisymmetric wave function is ! 1,2 ( ) = a 1 ( ) b 2 ( ) " a 2 ( ) b 1 ( ) (21-1) Note that a different, but completely equivalent way to write this is ! 1,2 ( ) = a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) (21-2) If the orbitals a and b are orthonormal, let's consider what needs to be done to normalize 21-2 S + ! = h 1 " # $ % & 1 " # $ % & = and S + ! = h 1 " # $ % & 1 " # $ % & = h 1 " # $ % & = h ( Similarly S ! " = h 1 # $ % & ( 1 # $ % & ( = h 1 # $ % & ( = h ) and S ! " = h 1 # $ % & ( 1 # $ % & ( = Spin-Free Many-Electron Wave Functions and Antisymmetry We have already seen that if we have two electrons in two orbitals a and b , then an acceptable antisymmetric wave function is ! 1,2 ( ) = a 1 ( ) b 2 ( ) " a 2 ( ) b 1 ( ) (21-1) Note that a different, but completely equivalent way to write this is ! 1,2 ( ) = a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) (21-2) If the orbitals a and b are orthonormal, let's consider what needs to be done to normalize . We have 21-3 ! * 1,2 ( ) "# # $ ! 1, 2 ( ) "# # $ dr 1 dr 2 = a * 1 ( ) "# # $ b * 2 ( ) "# # $ a 1 ( ) b 2 ( ) dr 1 dr 2 " a * 1 ( ) "# # $ b * 2 ( ) "# # $ a 2 ( ) b 1 ( ) dr 1 dr 2 " a * 2 ( ) "# # $ b * 1 ( ) "# # $ a 1 ( ) b 2 ( ) dr 1 dr 2 + a * 2 ( ) "# # $ b * 1 ( ) "# # $ a 2 ( ) b 1 ( ) dr 1 dr 2 (21-3) Since a and b are orthogonal, anytime we have an integral over the coordinates of either electron 1 or electron 2 (or both) that involves the product a * b , it will be zero. On the other hand, since a and b are normalized, if the only products in the integrals are a * a or b * b , they will be equal to one. Thus, the value of eq. 21-3 is 1 0+1=2....
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3502_21_octo30_LG - Chem 3502/5502 Physical Chemistry II...

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