lecture31_umn

lecture31_umn - Lecture 31 Other Computed Properties...

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Lecture 31 November 30, 2009 Other Computed Properties Partial Atomic Charges Multipole Moments Molecular Electrostatic Potential Frontier Molecular Orbital Reactivity
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Charges on Atoms From Hartree-Fock: quantitative charges on atoms. Occupied orbitals made up of basis functions on different atoms . If basis functions on O are used more than basis functions on C for the occupied orbitals : more charge on O than C . Dividing the electrons up and assigning them to specific atoms :"population analysis" One of the first such schemes: proposed by R. S. Mulliken in 1955
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Mulliken Population Analysis Electrons are divided up amongst the atoms according to the degree to which different AO basis functions contribute to the overall wave function . In restricted Hartree-Fock total number of electrons N (31-1) each normalized, occupied MO ψ contains two electrons. Replace each ψ by its linear expansion in AO basis function N = 2 j occupied " # j * r j ( ) $ j r j ( ) d r j
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Total Number of Electrons (31-2) Divide the total number of electrons up into two sums : squares of single AO basis functions products of two different AO basis functions . Electrons associated with only a single basis function : they belong entirely to the atom on which that basis function resides. N = 2 j occupied " c jr # r r j ( ) r " $ % ( ) * c js s r j ( ) s " $ % ( ) d r j = 2 j occupied " r , s " c jr r r j ( ) * c js s r j ( ) d r j = 2 j occupied " c jr 2 + c jr c js S rs r + s " r " $ % ( )
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Electrons shared between basis functions Second term: electrons ‘shared’ between basis functions . Divide these up evenly between the two atoms on which basis functions r and s reside . Follow this prescription and divide the basis functions up over atoms k so as to compute the atomic population N k (31-3) Insert definition of density matrix (eq. 29-3) (31-4) N k = 2 j occupied " c jr 2 + c jr c js S rs r , s # k , r $ s " + c jr c js S rs r # k , s % k " r # k " ( ) * + N k = P rr + P rs S rs r , s " k , r # s $ + P rs S rs r " k , s % k $ r " k $
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lecture31_umn - Lecture 31 Other Computed Properties...

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