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3502_32_dece02_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 32, December 2, 2006 (Some material in this lecture has been adapted from Cramer, C. J. Essentials of Computational Chemistry , Wiley, Chichester: 2002; pp. 40-45; 181-182.) Solved Homework In cartesian coordinates (Å), our water geometry placed oxygen at position (0.000, 0.000, 0.116), one hydrogen at (0.000, 0.751, 0.465) and the other hydrogen at (0.000, 0.751, 0.465). To work in a.u., we need to convert these coordinates to bohr using 1 bohr = 0.529 Å. In that case, the coordinates for water are oxygen at position (0.000, 0.000, 0.219), one hydrogen at (0.000, 1.420, 0.879) and the other hydrogen at (0.000, 1.420, 0.879). From Mulliken population analysis, we had charges of q = –0.372 on oxygen and 0.186 on each hydrogen Thus, we have x = x i q i i = 1 3 " = 0.0 # 0.372 ( ) + 0.0 0.186 ( ) + 0.0 0.186 ( ) = 0.00 y = y i q i i = 1 3 " = 0.0 # 0.372 ( ) + 1.420 0.186 ( ) + # 1.420 0.186 ( ) = 0.00 z = z i q i i = 1 3 " = 0.219 # 0.372 ( ) + # 0.879 0.186 ( ) + # 0.879 0.186 ( ) = # 0.41 Recalling μ = x 2 + y 2 + z 2 we see that < μ > = 0.41 a.u. From lecture 14, we know that 1 a.u. of dipole moment is 2.542 debye. So, the dipole moment from the Mulliken charges in units of debye is 2.542 x 0.61 = 1.04 D. This result is qualitatively in agreement with the experimental value of 1.8 D, suggesting that Mulliken charges have physically realistic magnitudes, at least based on our water molecule example. One might go on, then, to make chemical interpretations based on variations in Mulliken charges. For instance, if we compute methanol and acetic acid to have oxygen-
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32-2 bound proton Mulliken charges of 0.125 and 0.441, respectively, we might interpret this to imply that methanol will be a weaker acid than water and acetic acid a stronger one. Such a simple predictive tool might be quite useful in judging relative acidity for less obvious cases. Other Computed Properties—Spatial Extent The dipole moment (and higher moments) describe directional polarizations in the charge distribution of a molecule. Such polarizations are very important in understanding how molecules interact with external electric fields, to include playing a role in absorption spectroscopy, as we’ve previously seen. The dipole involves the expectation values of cartesian coordinates x , y , and z to first power, so it reflects an oriented distribution, positive vs. negative. The quadrupole moment, on the other hand, involves contributions from terms including x 2 , y 2 , and z 2 , and these measure displacement from a center along cartesian axes rather than bias from one side to another. Note that if we take the sum of the three cartesian displacements squared we have r 2 = x 2 + y 2 + z 2 (32-1) If we evaluate < r 2 > only for the electronic part of the wave function, this defines what is called the “electronic spatial extent”. That is, it is a measure of how “big” the molecule is, since it reports how far out the electronic density extends with significant probability.
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