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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
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 " \$ % & ( )
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

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