3502_HWkey6 - Chem 3502/5502 Physical Chemistry II(Quantum...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2006 Laura Gagliardi Answers to Homework Set 6 From lecture 21: Evaluate < β (1) β (2) | S 2 | β (1) β (2) > . Remember to use eq. 20-26 to expand the operator When we expand the two-electron S 2 operator into its individual components, using atomic units to avoid repeatedly writing h we have 1 ( ) " 2 ( ) S 2 " 1 ( ) " 2 ( ) = " 1 ( ) " 2 ( ) # S 2 1 ( ) " 1 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) + " 1 ( ) " 2 ( ) # " 1 ( ) S 2 2 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # S x 1 ( ) " 1 ( ) S x 2 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # S y 1 ( ) " 1 ( ) S y 2 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # S z 1 ( ) " 1 ( ) S z 2 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) = " 1 ( ) " 2 ( ) # 1 2 1 2 + 1 % & ( ) * " 1 ( ) " 2 ( ) d $ 1 ( ) d $ 2 ( ) + " 1 ( ) " 2 ( ) # " 1 ( ) 1 2 1 2 + 1 % & ( ) * " 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # 1 2 + 1 ( ) 1 2 + 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # i 2 + 1 ( ) i 2 + 2 ( ) d $ 1 ( ) d $ 2 ( ) + 2 " 1 ( ) " 2 ( ) # 1 2 " 1 ( ) 1 2 " 2 ( ) d $ 1 ( ) d $ 2 ( ) = 1 2 1 2 + 1 % & ( ) * + 1 2 1 2 + 1 % & ( ) * + 0 + 0 + 1 2 = 2 Note how the terms deriving from S x and S y become zero since these operators transform the β spin function to the α spin function, and an integration over α ( i ) β ( i ) d ω ( i ) gives zero owing to the orthogonality of the spin functions.
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HW6-2 From lecture 22: Consider the fluorine atom. Its ground state is represented by the determinant 2 1s 2 2s 2 2p x 2 2p y 2 2p z . Although fluorine is the most electronegative element, by applying a high enough voltage one can still strip an electron from it to generate F + . If two electrons are always kept in the 1s orbital, what are all possible determinants that can be made by
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This note was uploaded on 12/18/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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3502_HWkey6 - Chem 3502/5502 Physical Chemistry II(Quantum...

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