This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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. 514518.) Solved Homework We already showed that S x α = ( h /2) β (eq. 2019) For the operation on β 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 212 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 # $ % & ’ ( = SpinFree ManyElectron 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 ( ) (211) Note that a different, but completely equivalent way to write this is ! 1,2 ( ) = a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) (212) If the orbitals a and b are orthonormal, let's consider what needs to be done to normalize Ψ 212 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 # $ % & ’ ( = SpinFree ManyElectron 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 ( ) (211) Note that a different, but completely equivalent way to write this is ! 1,2 ( ) = a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) (212) If the orbitals a and b are orthonormal, let's consider what needs to be done to normalize Ψ . We have 213 ! * 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 (213) 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. 213 is 1 − − 0+1=2....
View
Full
Document
This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
 Staff

Click to edit the document details