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Unformatted text preview: CHEM 120A Problem Set 8 Solutions David Hoffman , Doran Bennett , Tara Yacovitch 4/12/2010 1. Well begin this problem by noting that the easiest way to solve this problem is to make use of matrices. Our plan of action will be to form a Hamiltonian matrix using  cov i =  AB i +  BA i and  ion i =  AA i +  BB i 1 as our basis kets (vectors). We can diagonalize H whose matrix elements will be functions of R and find two eigenvalues (which will also be functions of R ) and two eigenkets (eigenvectors) whose coefficients in our original basis will be functions of R . Lets break this problem into smaller pieces. We know that H can be written as, H = T (1) + T (2) + V (1) + V (2) + V (1 , 2) From symmetry we know that T (1) = T (2) and V (1) = V (2) so we can determine the matrix elements of T (1), V (1) and V and then sum them to find H . Before we start determining matrix elements it is important that we first normalize our kets. h cov  cov i = ( h AB  + h BA  )(  AB i +  BA i ) = h AB  AB i + h AB  BA i + h BA  AB i + h BA  BA i = 2 + 2 e 2 R/a (1) Similarly we find that, h ion  ion i = 2 + 2 e 2 R/a (2) This means that we can divide our entire matrix by 2 + 2 e 2 R/a after determining the matrix elements. Finding the matrix elements is a task that is simple in principle but can get confusing in practice. All four matrix elements for V (1 , 2) will be derived because they are the most complicated. The matrix elements of T and V will be left as an exercise for the reader. 1 Im using the notation that  AB i is equivalent to  A (1) B (2) i . 1 Starting with the on diagonal terms, h cov  V  cov i = ( h AB  + h BA  ) V (  AB i +  BA i ) = h AB  V  AB i + h AB  V  BA i + h BA  V  AB i + h BA  V  BA i Here its important to note that h AB  V  BA i = h AA  V  BB i = 2 1 + 1 1 + R/a v 2 e 2 R/a (3) h ion  V  ion i = ( h AA  + h BB  ) V (  AA i +  BB i ) = h AA  V  AA i + h AA  V  BB i + h BB  V  AA i + h BB  V  BB i = 2 2 + e 2 R/a 1 + R/a v 2 (4) Now on to the off diagonal terms. Remember that H is hermitian so we only need to calculate one of the elements because the other will be the same. h cov  V  ion i = ( h AB  + h BA  ) V (  AA i +  BB i ) = h AB  V  AA i + h AB  V  BB i + h BA  V  AA i + h BA  V  BB i = 4 v 2 e R/a (5) Finally we have, H ( R ) = 2 T ( R ) + 2 V ( R ) + V ( R ) 2 + 2 e 2 R/a (6) Where the bold indicates that we are talking about matrices and not operators. For the rest of this question youll need either a math program like Mathematica, Maple or MatLab or a spreadsheet program like Excel. Examples of how to solve this question with Excel and Mathematica are posted with this solution set....
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 Spring '10
 CHandler

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