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Unformatted text preview: Molecular Modeling: SemiEmpirical Methods
C372 Introduction to Cheminformatics II Kelsey Forsythe Recall
Molecular Models Empirical/Molecular Modeling Semi Empirical Neglect Core Electrons Approximate/parameterize HF Integrals Ab Initio/DFT Neglect Electrons Full Accounting of Electrons Huckel Theory Assumptions Atomic basis set parallel 2p orbitals No overlap between orbitals, ( S ij = ij ) 2p Orbital energy equal to ionization potential of methyl radical (singly occupied 2p orbital) The stabilization energy is the difference between the 2p parallel configuration and the 2p perpendicular configuration
E stabilization = 2 E p  E = 2  E E = H ij 2 Nonnearest interactions are zero = Ex. Benzene (C3H6)
One porbital per carbon atom asis size = 6 Huckel determinant is  0 0 0 0 E E 0 0 0 0 0 0 0 0 0 0 E 0 E 0 0 0 = 0, < 0 0 E Ex. Allyl (C3H5)
One porbital per carbon atom basis size = 3 Huckel matrix is  E 0 E = 0, < 0 0 E Resonance stabilization same for allyl cation, radical and anion (NOT found experimentally) E = + 2 , ,  2 Huckel Theory Molecular Orbitals? # MO = #AO Procedure to find coefficients in LCAO expansion 3 methylene radical orbitals as AO basis set Substitute energy into matrix equation Matrix multiply Solve resulting Nequations in Nunknowns Huckel Theory Substitute energy into matrix equation:  E1 b 0 a1 b a  E1 b a2 0 = 0 b a  E1 a3 E1 = a + 2b  (a + 2b ) b a 0 a1 b a  (a + 2b ) b 0 a2 = a3 0 b a  (a + 2b ) Huckel Theory Matrix multiply :  (a + 2b ) b 0 a1 b a  (a + 2b ) b 0 a2 = a3 0 b a  (a + 2b ) Doing the math! (a  (a + 2b )) * a1 + b * a2 + 0 * a3 = 0 b * a1 + (a  (a + 2b )) * a2 + b * a3 = 0 0 * a1 + b * a2 + (a  (a + 2b )) * a3 = 0 3  equations,3  unknowns  NOT Huckel Theory Matrix multiply : 2b * a1 + b * a2 = 0 b * a1 + a2 =  2a1 (1) (2) (3) 2b * a2 + b * a3 = 0 a2 =  2a3 b * a2 + 2b * a3 = 0 (1) and (3) reduce problem to 2 equations, 3 unknowns (1)  (3) a1 = a3 a 2 =  2a1 Need additional constraint! Huckel Theory Normalization of MO: Y * Y dr = 1 M (a M1*f * 1 * + a 2 * f 2 + a3 * f 3* *(a1* f 1 + a 2 * f 2 + a3 * f 3 )dr = 1 ) If M * f j = d ij f i* (a1) 2 + ( a 2) 2 + (a3) 2 = 1 Gives third equation, NOW have 3 equations, 3 unknowns.
1 = 3
2 2 ( 2 1) + 2(1) + (1) = 1 (1) 2 + (2) 2 + (3) 2 = 1 2 = 21 1 = 1 1 = 2 2 1 2 a3 = , 2 = 2 2 Huckel Theory Have 3 equations, 3 unknowns. ( ) 2 + (2) 2 + ( 3) 3 = 1 1 = 3 1
( ) 2 + 2( ) 2 + ( ) 2 = 1 1 1 1 ( ) 2 + (2) 2 + ( 3) 2 = 1 1 2 = 2 1 = 1 1 1 = 2 2 1 2 a3 = , 2 = 2 2 Huckel Theory 1 2 1 1 = * 1 + * 2 + 3 2 2 2
2p orbitals Aufbau Principal Fill lowest energy orbitals first Follow Pauliexclusion principle Carbon Aufbau Principal Fill lowest energy orbitals first Follow Pauliexclusion principle  2
Allyl + 2 Huckel Theory For lowest energy Huckel MO
1 2 1 a1 = ,a2 = ,a3 = 2 2 2 Bonding orbital Huckel Theory Next two MO NonBonding orbital
a1 = 2 2 ,a2 = 0,a3 = 2 2 Antibonding orbital
1 2 1 a1 = ,a2 = ,a3 = 2 2 2 Huckel Theory Electron Density qr =
j n j *a 2 jr j=1 The rth atom's electron density equal to product of # electrons in filled orbitals and square of coefficients of ao on that atom Ex. Allyl 2 1 q1 = 2 * = .5, qc 2 =1 qc 3 = .5 2 Total = 2 Electron density on c1 corresponding to the occupied MO is 2 1 MM 1 = 2 * M M 2 MM Central carbon is neutral with partial positive charges on end carbons Huckel Theory Bond Order n j *a jr * a js
j prs = Ex. Allyl Bond order between c1c2 corresponding to the occupied MO is
1 = 2 11
2 2 1 12 coefficient of AO for HOMO on c2 = 2 Corresponds to partial pi bond 2 1 due to delocalization over three 12 = 2 * * = .707 carbons (Gilbert) 2 2 coefficient of AO for HOMO on c1 = Beyond OneElectron Formalism HF method hi = hartree hamiltonian 1 2 =  Mi 2
Vi {j}= j
jj i Ignores electron correlation Effective interaction potential Hartree Product H = hi separability
i Zk Mr + Vi {j} k =1 ik
dr = j
jj i M jr r j ij j y 2 j rij dr = N y
i i Fock introduced exchange (relativistic quantum mechanics) HFExchange For a two electron system = y a (1)a (1) *y b (2)a (2) M P Permutivity operator M 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) M 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) M P = (2) (2) * (1) (1)  (1) (1) * (2) (2)
a b a b =  a (1) (1) b (1) (1) = Slater Determinant a (2) (2) b (2) (2) 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 metals SemiEmpirical 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 SemiEmpirical 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) What is Differential Overlap? When solving HF equations we integrate/average the potential energy over all other electrons
hi = hartree hamiltonian 1 =  Mi2 2
Vi {j}= j jj i Zk Mr + Vi {j} k =1 ik
dr = j
jj i M jr r j ij j y 2 j rij dr Computing HFmatrix introduces 1 and 2 electron integrals Estimating Energy
v ( r ) =
F v c m * F m (r ) d<E > v =0 dc v v v F * ( r )F j ( r )dr ) = 0 " m 1 mi 4 2 4 43 4
S mj m Want to find c's so that
i v v v c m ( F * ( r ) HF j ( r )dr  E 1 m4 4 2 4 4 3
H mj v M c = 0 } M ij = v v F * ( r ) HF j ( r )  E i v v F * ( r )F j ( r ) i M ij = H ij  E Sij Non  trivial solutions for det(M) = 0 Estimating Energy
v v v c m ( F * ( r ) HF j ( r )dr  E 1 m4 4 2 4 4 3
H mj i v v v F * ( r )F j ( r )dr ) = 0 " m 1 mi 4 2 4 43 4
S mj M 1 electron
k=1 * fm Zk f u dr rik
2  electron
* f mVi { j}f n dr 1 * * f m (1)f n (1) f l (2)f s (2) dr r j 1 i 4 4 4 4 4 2 4 ij4 4 4 4 3
Differential Overlap Complete Neglect of Differential Overlap (CNDO) Basis set from valence STOs S Neighbor AO overlap integrals, S, are assumed zero or = Twoelectron terms constrain atomic/basis orbitals (But atoms may be different!) ( ) = ( ) Reduces from N4 to N2 number of 2electron integrals Integration replaced by algebra IP from experiment to represent 1, 2electron orbitals Complete Neglect of Differential Overlap (CNDO) Overlap ( )= ( ) Methylene radical (CH2*) Interaction same in singlet (spin paired) and triplet state (spin parallel)
2 ( )MOverlap of electrons paired in sp
= Parallel : Paired : ( )MOverlap of electrons parallel, one in sp
1 in p  orbital = 2 Intermediate Neglect of Differential Overlap (INDO) Pople et al (1967) Use different values for interaction of different orbitals on same atom s.t. Paired : Methylene (mmll )j Overlap of electrons paired in sp 2 = g sp 2 , sp 2 (mmll )j Parallel : Overlap of electrons parallel, one in sp 2 1 in p  orbital = g sp 2 , p j g sp 2 , sp 2 Neglect of Diatomic Differential Overlap (NDDO) Most modern methods (MNDO, AM1, PM3) All twocenter, twoelectron integrals allowed iff: and are on same atomic center and are on same atomic center and Recall, CINDO = and = Allow for different values of depending on orbitals involved (INDOlike) Neglect of Diatomic Differential Overlap (NDDO) MNDO Modified NDDO AM1 modified MNDO PM3 larger parameter space used than in AM1 Sterics MNDO overestimates steric crowding, AM1 and PM3 better suited but predict planar structures for puckered 4, 5 atom rings Transition States MNDO overestimates (see above) Hydrogen Bonding Both PM3 and AM1 better suited Aromatics too high in energy (~4kcal/mol) Radicals overly stable Charged species AO's not diffuse (only valence type) Performance of Popular Methods (C. Cramer) Summarize Molecular Models Empirical/Molecular Modeling Semi Empirical Ab Initio/DFT Neglect Electrons Neglect Core Electrons Approximate/parameterize HF Integrals Full Accounting of Electrons HT EHT CNDO MINDO NDDO MNDO HF Completeness ...
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 Spring '11
 YoonSupLee

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