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C372_L5 - Molecular Modeling SemiEmpirical Methods C372...

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Unformatted text preview: Molecular Modeling: SemiEmpirical Methods C372 Introduction to Cheminformatics II Kelsey Forsythe Semi-Empirical Methods Advantage Faster than ab initio Less sensitive to parameterization than MM methods Disadvantage Accuracy depends upon parameterization Semi-Empirical Methods Ignore Core Electrons Approximate part of HF integration Estimating Energy Recall Eclassical <E>quantal v ) v v (r )O (r )dr < O >= v v average / expectation value of observable O * v ( r ) ( r ) dr * v v v * (r ) x (r )dr < x >= v v v average / expectation value of x * v ( r ) ( r ) dr Approximate Methods SCF (Self Consistent Field) Method (a.ka. Mean Field or Hartree Fock) Pick single electron and average influence of remaining electrons as a single force field (V0 external) Then solve Schrodinger equation for single electron in presence of field (e.g. Hatom problem with extra force field) Perform for all electrons in system Combine to give system wavefunction and energy (E) Combine to give system wavefunction and energy (E Repeat to error tolerance (Ei+1Ei) Estimating Energy v x v v Y * (r ) HY (r )dr x < E >= v v x average / expectation value of energy * v xY (r )Y (r )dr F v v Y ( r ) = x cm * F m ( r ) m Want to find c's so that x i d<E> v =0 dc v x v v v v v cm ( x * (r ) HF j (r )dr - E x * (r )F j (r )dr ) = 0 " m Fm F mi 1442443 1 4 4 2 4 43 H mj v v x v M x c = 0 } ij = x * (r ) HF j (r ) - E M Fi M ij = H ij - E S ij Non - trivial solutions for det(M) = 0 v v F * ( r )F j ( r ) x i S mj Estimating Energy i v x v v cm ( * (r ) H j (r )dr - E 1m 4 2 4 4 3 4 H mj v v v * (r ) j (r )dr ) = 0 m 1 mi 4 2 4 43 4 S mj xM 11M 12M 13.........M 1N x x x F simulataneous xM 21M 22 x x equations gives a xM 31 M 33 x x xM x O matrix equation xM x M NN x x N1 v M c =0 v x v v v M ij = * (r ) H j (r ) - E * (r ) j (r ) i i M ij = H ij - E S ij Non - trivial solutions for det(M) = 0 M ij x i th row, j th column Matrix Algebra Finding determinant akin to rotating M = 0(i j) matrix until diagonal ( ) ij 11M 12M 13.........M 1N M M 21M 22 31 M 33 M M O M N1 M NN M ij i th row, j th column ) Matrix Algebra 11 12 det 21 22 det 1 3 2 4 11 12 = ( 11 22) - ( 12 21) 21 22 = (1* 4) - (2 * 3) = -2 Huckel Theory Assumptions Atomic basis set parallel 2p orbitals No overlap between orbitals, Sij = ij ) ( 2p Orbital energy equal to ionization potential of methyl radical (singly occupied 2p orbital) The stabilization energy is the difference between the 2pparallel configuration and the 2p perpendicular configuration E stabilization = 2 E - E = 2 - E p = Nonnearest interactions are zero E = H ij 2 Ex. Allyl (C3H5) One porbital per carbon atom basis size = 3 Huckel matrix is - E 0 -E = 0, < 0 0 -E E = + 2 , , 2 Resonance stabilization same for allyl cation, radical and anion (NOT found experimentally) Ex. Allyl (C3H5) Huckel matrix (determinant form)no resonance -E 0 0 0 a - E 0 = 0,b < 0 0 0 a -E E = a ,a ,a - E 0 -E = 0, < 0 0 -E E = + 2 , , Huckel matrix (determinant form)resonance (beta represents overlap/interaction between orbitals) In matrix (determinant form) Energy of three isolated methylene sp2 orbitals Overlap between orbital 1 and orbital 2 (hence matrix element H12) 2 Energy of resonance system. Note the lowest energy is less than the isolated orbital/AO due (this is resonance stabilization) Includes nonnearest neighbor orbital Extended Huckel Theory (aka Tight Binding Approximation) interactions Experimental Valence Shell Ionization Potentials used to model matrix elements Generally applicable to any element Useful for calculating band structures in solidstate physics Beyond One-Electron Formalism HF method hi = hartree hamiltonian 1 2 =- xi 2 Vi {j}= v jv i Ignores electron correlation Effective interaction potential Hatree Product H = hi separability i Zk x r + Vi {j} k =1 ik dr = v jv i M vr r j ij v y 2 j rij dr = N y i i Fock introduced exchange (relativistic quantum mechanics) HF-Exchange For a two electron system = y a (1)a (1) *y b (2)a (2) x P Permutivity operator x Py a (1)a (1) *y b (2)a (2) = y a (2)a (2) *y b (1)a (1) NO CHANGE IN SIGN Fock modified wavefunction = y a (1)a (1) *y b (2)a (2) -y a (2)a (2) *y b (1)a (1) x P = y (2)a (2) *y (1)a (1) -y (1)a (1) *y (2)a (2) a b a b = - Slater Determinants Ex. Hydrogen molecule = a (1) (1) * b (2) (2) a = a (1) (1) * b (2) (2) - a (2) (2) * b (1) (1) x P = (2) (2) * (1) (1) - (1) (1) * (2) (2) a b a b = - a (1) (1) b (1) (1) 4 = Slater Determinant a (2) (2) b (2) (2) Beyond One-Electron Formalism HF method hi = hartree hamiltonian 1 2 =- xi 2 Vi {j}= v jv i Ignores electron correlation Effective interaction potential Hatree Product H = hi separability i Zk x r + Vi {j} k =1 ik dr = v jv i M vr r j ij v y 2 j rij dr = N y i i Fock introduced exchange (relativistic quantum mechanics) Neglect of Differential Overlap (NDO) CNDO (1965, Pople et al) MINDO (1975, Dewar ) MNDO (1977, Thiel) INDO (1967, Pople et al) ZINDO SINDO1 STObasis (/Sspectra,/2 dorbitals) /1/2/3, organics /d, organics, transition metals Organics Electronic spectra, transition metals 13 row binding energies, photochemistry and transition Semi-Empirical Methods SAM1 Closer to # of ab initio basis functions (e.g. d orbitals) Increased CPU time Extrapolated ab initio results for organics "slightly empirical theory"(Gilbertmore ab initio than semiempirical in nature) G1,G2 and G3 Semi-Empirical Methods AM1 PM3 (1989, Stewart) Modified nuclear repulsion terms model to account for Hbonding (1985, Dewar et al) Widely used today (transition metals, inorganics) Larger data set for parameterization compared to AM1 Widely used today (transition metals, inorganics) General Reccommendations More accurate than empirical methods Less accurate than ab initio methods Inorganics and transition metals Pretty good geometry OR energies Poor results for systems with diffusive interactions (van der Waals, Hbonded, radicals etc.) Complete Neglect of Differential Overlap (CNDO) Vi {j}= v jv i vr r j ij dr = v jv i v y 2 j rij dr Overlap integrals, S, is assumed zero Neglect of Differential Overlap (NDO) Vi {j}= v jv i vr r j ij dr = v jv i v y 2 j rij dr Gives rise to overlap between electronic basis functions of different types and on different atoms Complete Neglect of Differential Overlap (CNDO) Oneelectron overlap integral for different electrons is zero (as in Huckel Theory) Twoelectron integrals are zero if basis functions not identical Intermediate Neglect of Differential Overlap (CNDO) Overlap integrals, S, is assumed zero Eigenvalue Equation Matrix * Vector = Matrix (diagonal) * Vector Schrodinger's equation! HY = EY v ( r ) = F F v c m * F m (r ) F m v H c m * F m (r ) = E m v c m * F m (r ) m M ij = H ij - E j Sij Solutions for det(M) = 0 The solutions to this differential equation are equal to the solutions to the matrix ...
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