21 - Chem 3502/5502 Physical Chemistry II (Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2010 Laura Gagliardi Lecture 21, November 03, 2010 (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 α = ( /2) β (eq. 20-19) For the operation on β we have S x β = 2 0 1 1 0 0 1 = 2 1 0 = 1 2 α Working with the S y operator involves S y α = 2 0 i i 0 1 0 = 2 0 i = 1 2 i β and S y β = 2 0 i i 0 0 1 = 2 i 0 = 1 2 i α For the raising operator we have
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21-2 S + α = 0 1 0 0 1 0 = 0 and S + β = 0 1 0 0 0 1 = 1 0 = α Similarly S α = 0 0 1 0 1 0 = 0 1 = β and S β = 0 0 1 0 0 1 = 0 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 Ψ 1,2 ( ) = a 1 ( ) b 2 ( ) a 2 ( ) b 1 ( ) (21-1) Note that a different, but completely equivalent way to write this is Ψ 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 Ψ . We have
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21-3 Ψ * 1,2 ( ) −∞ Ψ 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 0+1=2. So, the normalized form for eq 21-2 is Ψ 1,2 ( ) = 1 a 1 ( ) b 1 ( ) a 2 ( ) b 2 ( ) (21-4) Now, consider a case of more than two electrons. We might make a determinant along the lines of Ψ 1,2, , N ( ) = a 1 ( ) b 1 ( ) n 1 ( ) a 2 ( ) b 2 ( ) n 2 ( ) a N ( ) b N ( ) n N ( ) (21-5) Note that swapping the coordinates of any two electrons is equivalent to swapping two rows in the determinant (e.g., we could switch rows 1 and 2 and the effect would be that, in every term in the determinant, electrons 1 and 2 would have been swapped). It is a known feature of determinants that when two rows are swapped, the value of the determinant changes sign. So, this satisfies antisymmetry perfectly.
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21 - Chem 3502/5502 Physical Chemistry II (Quantum...

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