3502_35_dece09_LG

3502_35_dece09_LG - Chem 3502/5502 Physical Chemistry...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 35, December 09, 2009 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 319-331.) Solved Homework Since molecular hydrogen only has two electrons, there is only one Coulomb integral deriving from evaluation of the generic expression for interelectronic repulsion " 1 r ij i < j # " i = 1, j = 2 = " r 1 , r 2 ( ) 2 r 12 $$ d r 1 d r 2 $$ d % 1 d % 2 where recall that the integration over spin coordinates ω 1 and ω 2 will give 2 for the singlet spin function. For closed-shell hydrogen we can call the above integral J σσ . For the closed-shell wave function of H 2 in eq. 35-8, J σσ is J "" = " 1 ( ) 2 " 2 ( ) 2 r 12 ## d r 1 d r 2 = 1 4 1s a 1 ( ) + b 1 ( ) [ ] 2 a 2 ( ) + b 2 ( ) [ ] 2 r 12 ## d r 1 d r 2 = 1 4 aaaa ( ) + 2 aaab ( ) + aabb ( ) + 2 abaa ( ) + 4 abab ( ) + 2 abbb ( ) + bbaa ( ) + 2 bbab ( ) + bbbb ( ) $ % & & ( ) ) where we’ve used the shorthand notation for Coulomb integrals and stopped writing 1s everywhere. Now, note that since both a and b are 1s orbitals, and since order of indices doesn’t matter, there are really only 4 kinds of integrals: all 4 indices identical, 3 of one index and one of the other, 2 of each index in each case paired on the same side, and 2 of each index with both indices appearing once on each side. That is we can simplify to J "" = 1 2 ( ) + ( ) + 1 2 ( ) + abab ( )
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35-2 Now we’ll do the same for the open-shell wave function of eq. 35-9. In that case we have J OSS = 1 2 1s a 1 ( ) b 2 ( ) + b 1 ( ) a 2 ( ) [ ] 2 r 12 d r 1 d r 2 "" = 1 2 aabb ( ) + 2 abba ( ) + bbaa ( ) [ ] = ( ) + abab ( ) Notice that this is the expected result for the case of two electrons in two orbitals, namely, J ab + K ab . Now, when the two atoms are very far apart, the only integral that won’t be zero will be (aa|aa). So, only the closed-shell wave function still has a Coulomb repulsion at long distance (this makes intuitive sense—in the open-shell wave function there is one electron on each atom, so if they are far apart, how can there be Coulomb repulsion?) This is another illustration of why RHF dissociates incorrectly, because of the contribution of the ionic term that places both electrons on one atom. At short distance (i.e., bonding distance), both wave functions have one Coulomb contribution (ab|ab), so there’s no distinction between them there. The open-shell case has a full (aa|bb) and the closed-shell case only 1/2 (aa|bb). However, the closed-shell case also has 1/2 (aa|aa) and it is obvious that this must be bigger than 1/2 (aa|bb) for every interatomic distance (until the two nuclei fuse into one!) So, even neglecting the extra repulsion from the 2(aa|ab) in the closed-shell wave function, it clearly has more Coulomb repulsion at bonding distance than the open-shell wave function. So, why does H 2 form a bonded closed-shell ground state? The answer must be that the attraction to the nuclei is much larger when both electrons are in the σ orbital than is the case when each is forced to separately occupy a single 1s orbital. Just looking at the shapes of the orbitals this seems reasonable. The σ orbital has most of its amplitude “sandwiched” between the two nuclei, so it’s close to both at the same time. The
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3502_35_dece09_LG - Chem 3502/5502 Physical Chemistry...

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