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Computer Modeling

Computer Modeling

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Computer Modeling

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  1. Computer Modeling Dr. GuanHua CHEN Department of Chemistry University of Hong Kong http://yangtze.hku.hk/lecture/comput06-07.ppt

  2. Computational Chemistry • Quantum Chemistry SchrÖdinger Equation H = E • Molecular Mechanics F = Ma F : Force Field • Bioinformatics

  3. Computational Chemistry Industry Company Software Gaussian Inc. Gaussian 94, Gaussian 98 Schrödinger Inc. Jaguar Wavefunction Spartan Q-Chem Q-Chem Accelrys InsightII, Cerius2 HyperCube HyperChem Informatix Celera Genomics Applications: material discovery, drug design & research R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billion Bioinformatics: Total Sales in 2001 US$ 225 million Project Sales in 2006 US$ 1.7 billion

  4. Vitamin C C60 energy heme OH + D2 --> HOD + D Cytochrome c

  5. Quantum Chemistry Methods • Ab initio Molecular Orbital Methods Hartree-Fock, Configurationa Interaction (CI) MP Perturbation, Coupled-Cluster, CASSCF • Density Functional Theory • Semiempirical Molecular Orbital Methods Huckel, PPP, CNDO, INDO, MNDO, AM1 PM3, CNDO/S, INDO/S

  6. SchrÖdinger Equation Hy = Ey Wavefunction Hamiltonian H = (-h2/2ma)2 - (h2/2me)ii2 - i Zae2/ria (+  ZaZbe2/rab ) + ije2/rij Energy One-electron terms: (-h2/2ma)2 - (h2/2me)ii2 - i Zae2/ria Two-electron term: ije2/rij

  7. Hartree-Fock Method Orbitals 1. Hartree-Fock Equation Ffi = ei fi FFock operator fi the i-th Hartree-Fock orbital ei the energy of the i-th Hartree-Fock orbital

  8. 2. Roothaan Method (introduction of Basis functions) fi= k ckiyk LCAO-MO { yk }is a set of atomic orbitals (or basis functions) 3. Hartree-Fock-Roothaan equation j ( Fij - ei Sij ) cji = 0 Fij  < i|F | j > Sij  < i| j > 4. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

  9. Graphic Representation of Hartree-Fock Solution 0 eV Electron Affinity Ionization Energy

  10. A Gaussian Input File for H2O # HF/6-31G(d) Route section water energy Title 0 1 Molecule Specification O -0.464 0.177 0.0 (in Cartesian coordinates H -0.464 1.137 0.0 H 0.441 -0.143 0.0 Basis Set i = p cip p { yk }is a set of atomic orbitals (or basis functions) STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G** ------------------------------------------------------------------------------------- complexity & accuracy

  11. Gaussian type functions gijk = N xi yj zk exp(-ar2) (primitive Gaussian function) yp = u dupgu (contracted Gaussian-type function, CGTF) u = {ijk} p = {nlm}

  12. Electron Correlation: avoiding each other The reason of the instantaneous correlation: Coulomb repulsion (not included in the HF) Beyond the Hartree-Fock Configuration Interaction (CI) Perturbation theory Coupled Cluster Method Density functional theory

  13. Configuration Interaction (CI) + + …

  14. Single Electron Excitation or Singly Excited

  15. Double Electrons Excitation or Doubly Excited

  16. Singly Excited Configuration Interaction (CIS): Changes only the excited states +

  17. Doubly Excited CI (CID): Changes ground & excited states + Singly & Doubly Excited CI (CISD): Most Used CI Method

  18. Full CI (FCI): Changes ground & excited states + + + ...

  19. Perturbation Theory H = H0 + H’ H0yn(0) = En(0) yn(0) yn(0) is an eigenstate for unperturbed system H’ is small compared with H0

  20. Moller-Plesset (MP) Perturbation Theory The MP unperturbed Hamiltonian H0 H0 = mF(m) whereF(m)is the Fock operator for electron m. And thus, the perturbation H’ H’=H - H0 Therefore, the unperturbed wave function is simply the Hartree-Fock wave function . Ab initio methods: MP2, MP3, MP4

  21. Coupled-Cluster Method y= eT y(0) y(0): Hartree-Fock ground state wave function y: Ground state wave function T = T1 + T2 + T3 + T4 + T5 + … Tn : n electron excitation operator T1 =

  22. Coupled-Cluster Doubles (CCD) Method yCCD= eT2 y(0) y(0): Hartree-Fock ground state wave function yCCD: Ground state wave function T2 : two electron excitation operator T2 =

  23. Complete Active Space SCF (CASSCF) Active space All possible configurations

  24. Density-Functional Theory (DFT) Hohenberg-Kohn Theorem: Phys. Rev. 136, B864 (1964) The ground state electronic density (r) determines uniquely all possible properties of an electronic system (r) Properties P (e.g. conductance), i.e. PP[(r)] Density-Functional Theory (DFT) E0 = - (h2/2me)i <i |i2|i >-  drZae2(r) /r1a + (1/2)   dr1 dr2e2/r12 + Exc[(r)] Kohn-Sham EquationGround State: Phys. Rev. 140, A1133 (1965) FKSyi = ei yi FKS- (h2/2me)ii2-  Zae2 /r1a + jJj + Vxc Vxc dExc[(r)] / d(r) A popular exchange-correlation functional Exc[(r)]: B3LYP

  25. Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003) B3LYP/6-311+G(d,p) B3LYP/6-311+G(3df,2p) RMS=21.4 kcal/mol RMS=12.0 kcal/mol RMS=3.1 kcal/mol RMS=3.3 kcal/mol B3LYP/6-311+G(d,p)-NEURON & B3LYP/6-311+G(d,p)-NEURON: same accuracy

  26. Time-Dependent Density-Functional Theory (TDDFT) Runge-Gross Extension: Phys. Rev. Lett. 52, 997 (1984) Time-dependent system (r,t) Properties P (e.g. absorption) TDDFT equation: exact for excited states Isolated system Open system Density-Functional Theory for Open System ??? Further Extension: X. Zheng, F. Wang & G.H. Chen (2005) Generalized TDDFT equation: exact for open systems

  27. Ground State Excited State CPU Time Correlation Geometry Size Consistent (CHNH,6-31G*) HFSCF   1 0 OK  DFT   ~1   CIS   <10 OK  CISD   17 80-90%   (20 electrons) CISDTQ   very large 98-99%   MP2   1.5 85-95%   (DZ+P) MP4   5.8 >90%   CCD   large >90%   CCSDT   very large ~100%  

  28. Search for Transition State Transition State: one negative frequency k   e-DG/RT DG Reactant Product Reaction Coordinate

  29. Gaussian Input File for Transition State Calculation #b3lyp/6-31G opt=qst2 testthe first is the reactant internal coordinate0 1OH 1 oh1 H 1 oh1 2 ohh1oh1 0.90ohh1 104.5The second is the product internal coordinate0 1OH 1 oh2H 1 oh3 2 ohh2oh2 0.9oh3 10.0ohh2 160.0

  30. Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman) Independent electron approximation Schrodinger equation for electron i Hval = iHeff(i) Heff(i) = -(h2/2m) i2 + Veff(i) Heff(i) i = i i Semiempirical Molecular Orbital Calculation

  31. LCAO-MO: fi= r criyr s (Heffrs- eiSrs ) csi = 0 Heffrs < r|Heff| s >Srs< r| s > • Parametrization: • Heffrr < r|Heff| r > • = minus the valence-state ionization • potential (VISP)

  32. Atomic Orbital Energy VISP --------------- e5 -e5 --------------- e4 -e4 --------------- e3 -e3 --------------- e2 -e2 --------------- e1 -e1 Heffrs = ½ K(Heffrr + Heffss) SrsK: 13

  33. CNDO, INDO, NDDO (Pople and co-workers) Hamiltonian with effective potentials Hval = i [ -(h2/2m) i2 + Veff(i) ] + ij>i e2 / rij two-electron integral: (rs|tu) = <r(1) t(2)| 1/r12 | s(1) u(2)> CNDO: complete neglect of differential overlap (rs|tu) = rs tu (rr|tt) rs tu rt

  34. INDO: intermediate neglect of differential overlap (rs|tu) = 0 when r, s, t and u are not on the same atom. NDDO: neglect of diatomic differential overlap (rs|tu) = 0 if r and s (or t and u) are not on the same atom. CNDO, INDOare parametrized so that the overall results fit well with the results of minimal basis ab initio Hartree-Fock calculation. CNDO/S, INDO/S are parametrized to predict optical spectra.

  35. MINDO, MNDO, AM1, PM3 (Dewar and co-workers, University of Texas, Austin) MINDO: modified INDO MNDO: modified neglect of diatomic overlap AM1: Austin Model 1 PM3: MNDO parametric method 3 *based on INDO & NDDO *reproduce the binding energy

  36. Relativistic Effects Speed of 1s electron: Zc / 137 Heavy elements have large Z, thus relativistic effects are important. Dirac Equation: Relativistic Hartree-Fock w/ Dirac-Fock operator; or Relativistic Kohn-Sham calculation; or Relativistic effective core potential (ECP).

  37. Four Sources of error in ab initio Calculation (1) Neglect or incomplete treatment of electron correlation (2) Incompleteness of the Basis set (3) Relativistic effects (4) Deviation from the Born-Oppenheimer approximation

  38. Quantum Chemistry for Complex Systems

  39. Quantum Mechanics / Molecular Mechanics (QM/MM) Method Combining quantum mechanics and molecular mechanics methods: QM MM

  40. Hamiltonian of entire system: H = HQM +HMM +HQM/MM Energy of entire system: E = EQM(QM) + EMM(MM) + EQM/MM(QM/MM) EQM/MM(QM/MM) = Eelec(QM/MM) + Evdw(MM) + EMM-bond(MM) EQM(QM) + Eelec(QM/MM) = <| Heff |> Heff = -1/2 ii2 + ij 1/rij - ai Za/ria - bi qb/rib + gi Vv-b(ri) + ad Za Zd/rad + ba Zaqb/rba QM MM

  41. Quantum Chemist’s Solution Linear-Scaling Method: O(N) Computational time scales linearly with system size Time Size

  42. Linear Scaling Calculation for Ground State Divide-and-Conqure (DAC) W. Yang, Phys. Rev. Lett. 1991

  43. York, Lee & Yang, JACS, 1996 Superoxide Dismutase (4380 atoms) AM1 Strain, Scuseria & Frisch, Science (1996): LSDA / 3-21G DFT calculation on 1026 atom RNA Fragment

  44. Linear Scaling Calculation for Excited State Liang, Yokojima & Chen, JPC, 2000

  45. Fast Multiple Method LDM-TDDFT: CnH2n+2

  46. LODESTAR: Software Package for Complex Systems Characteristics: O(N) Divide-and-Conquer O(N) TDHF (ab initio & semiemptical) O(N) TDDFT Light Harvesting System Nonlinear Optical CNDO/S-, PM3-, AM1-, INDO/S-, & TDDFT-LDM

  47. Photo-excitations in Light Harvesting System II strong absorption: ~800 nm generated by VMD generated by VMD

  48. Carbon Nanotube