lecture32_umn - Lecture 32 Other Computed Properties...

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Lecture 32 December 2, 2009 Other Computed Properties Spatial Extent Ionization Potential and Electron Affinity Geometry Optimization
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Spatial Extent Multipole moments: directional polarizations in the charge distribution of a molecule. Dipole moment : expectation values of x , y , z to first power, so it reflects an oriented distribution, positive vs. negative. Quadrupole moment : contributions from x 2 , y 2 , z 2 displacement from a center along cartesian axes rather than bias from one side to another. Sum of the three cartesian displacements squared (32-1) r 2 = x 2 + y 2 + z 2
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Electronic Spatial Extent If < r 2 > only for the electronic part of the wave function. A measure of how big the molecule is: how far out the electronic density extends. For water in STO-3G calculation, < r 2 > = 17.74 a.u. r 2 >: atomic units of distance (bohr) and is ca. 4 a.u. Electron density being roughly equally distributed inside and outside a shell of 2.1 Å centered on the molecular origin. Distance consistent with the van der Waals radius of oxygen. Water radical anion: < r 2 > = 29.75 a.u. The extra electron is held weakly to the water molecule. Loosely held electrons localize at large distances from the nuclei: large values for < r 2 >. < r 2 > is a simple, single, quantum descriptor associated with molecular size.
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Ionization Potentials Total energy for a HF Slater determinant (32-2) ε i energy of a single electron in orbital i J and K Coulomb and exchange integrals . Now remove one electron from the molecule The minimum energy for that removal: ionization potential (IP). IP is the negative of the binding affinity for the least tightly held electron (i.e., one of the electrons in the HOMO). E HF = " HF H " HF = 2 # i i occupied $ % 2 J ij % K ij ( ) i , j occupied $
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Koopmans’ work Koopmans proved that: use only the occupied molecular orbitals of the neutral molecule as a basis set from which to form molecular orbitals for the radical cation , then the optimal MOs for the radical cation put exactly two electrons in the identical MOs as for the neutral and one electron in the identical HOMO as for the neutral .
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