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Computational Chemistry

Computational Chemistry. G. H. CHEN Department of Chemistry University of Hong Kong. Beginning of Computational Chemistry. In 1929, Dirac declared, “The underlying physical laws necessary for the mathematical theory of ...the whole of chemistry are thus completely know, and

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Computational Chemistry

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  1. Computational Chemistry G. H. CHEN Department of Chemistry University of Hong Kong

  2. Beginning of Computational Chemistry In 1929, Dirac declared, “The underlying physical laws necessary for the mathematical theory of ...the whole of chemistry are thus completely know, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.”

  3. Computational Chemistry Quantum Chemistry Molecular Mechanics Bioinformatics Create & Analyse Bio-information SchrÖdinger Equation F = M a

  4. Nobel Prizes for Computational Chemsitry Mulliken,1966 Fukui, 1981 Hoffmann, 1981 Pople, 1998 Kohn, 1998

  5. Computational Chemistry Industry Company Software Gaussian Inc. Gaussian 94, Gaussian 98 Schrödinger Inc. Jaguar Wavefunction Spartan Q-Chem Q-Chem MSI InsightII, Discovery 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

  6. Quantum Chemistry Methods • Ab initio molecular orbital methods • Semiempirical molecular orbital methods • Density functional method

  7. SchrÖdinger Equation Hy = Ey Wavefunction Hamiltonian H = (-h2/2ma)2 - (h2/2me)ii2 +  ZaZbe2/rab - i Zae2/ria + ije2/rij Energy

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

  9. C60 and Superconductor What is superconductor? Electrical Current flows for ever ! Soccer Ball Applications: Magnet, Magnetic train, Power transportation

  10. Crystal Structure of C60 solid Crystal Structure of K3C60

  11. K3C60 is a Superconductor (Tc = 19K) Power Transportation Magnetic Train Superconducting Magnet Erwin & Pickett, Science, 1991

  12. The mechanism of superconductivity in K3C60 was discovered using computational chemistry methods Varma et. al., 1991; Schluter et. al., 1992; Dresselhaus et. al., 1992 Chen & Goddard, 1992 Mechanism of Superconductor Vibration of Atoms Effective Attraction ! Vibration Spectrum of K3C60 GH Chen, Ph.D. Thesis, Caltech (1992)

  13. Carbon Nanotubes (Ijima, 1991)

  14. Calculated STM Image of a Carbon Nanotube (Rubio, 1999) STM Image of Carbon Nanotubes (Wildoer et. al., 1998)

  15. Computer Simulations (Saito, Dresselhaus, Louie et. al., 1992) Carbon Nanotubes (n,m): Conductor, if n-m = 3I I=0,±1,±2,±3,…;or Semiconductor, if n-m  3I Metallic Carbon Nanotubes: Conducting Wires Semiconducting Nanotubes: Transistors Molecular-scale circuits ! 1 nm transistor! 30 nm transistor! 0.13 µm transistor!

  16. Experimental Confirmations: Lieber et. al. 1993; Dravid et. al., 1993; Iijima et. al. 1993; Smalley et. al. 1998; Haddon et. al. 1998; Liu et. al. 1999 Wildoer, Venema, Rinzler, Smalley, Dekker, Nature 391, 59 (1998)

  17. Science 9th November, 2001 Logic gates (and circuits) with carbon nanotuce transistor Science 7th July, 2000 Carbon nanotube-Based nonvolatile RAM for molecular computing

  18. Large Gear Drives Small Gear G. Hong et. al., 1999

  19. Computer-Aided Drug Design Human Genome Project GENOMICS Drug Discovery

  20. ALDOSE REDUCTASE Diabetic Complications Diabetes Sorbitol Glucose

  21. Inhibitor Aldose Reductase Design of Aldose Reductase Inhibitors

  22. Database for Functional Groups Structure-activity-relation Prediction: Drug Leads LogIC50: 0.6861,0.88 LogIC50: 0.6382,1.0

  23. Prediction Results using AutoDock LogIC50: 0.77,1.1 LogIC50: -1.87,4.05 LogIC50: -2.77,4.14 LogIC50: 0.68,0.88

  24. Computer-aided drug design Chemical Synthesis Screening using in vitro assay Animal Tests Clinical Trials

  25. Bioinformatics • Improve content & utility of bio-databases • Develop tools for data generation, capture & annotation • Develop tools for comprehensive functional studies • Develop tools for representing & analyzing sequence similarity & variation

  26. Computational Chemistry • Increasingly important field in chemistry • Help to understand experimental results • Provide guidelines to experimentists • Application in Materials & Pharmaceutical industries • Future: simulate nano-size materials, bulk materials; replace experimental R&D Objective: More and more research & development to be performed on computers and Internet instead in the laboratories

  27. Quantum Chemistry G. H. Chen Department of Chemistry University of Hong Kong

  28. Contributors: Hartree, Fock, Slater, Hund, Mulliken, Lennard-Jones, Heitler, London, Brillouin, Koopmans, Pople, Kohn Application: Chemistry, Condensed Matter Physics, Molecular Biology, Materials Science, Drug Discovery

  29. Emphasis Hartree-Fock method Concepts Hands-on experience Text Book “Quantum Chemistry”, 4th Ed. Ira N. Levine http://yangtze.hku.hk http://ecourse.hku.hk:8900/public/CHEM3504

  30. Contents 1. Variation Method 2. Hartree-Fock Self-Consistent Field Method 3. Perturbation Theory 4. Semiempirical Methods

  31. The Variation Method The variation theorem Consider a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E1. If f is any normalized, well- behaved function that satisfies the boundary conditions of the problem, then  f* Hf dt >E1

  32. Proof: Expand f in the basis set { yk} f = kakyk where {ak} are coefficients Hyk = Ekyk then f* Hf dt = kjak* aj Ej dkj = k | ak|2Ek> E 1k | ak|2 = E1 Since is normalized, f*f dt = k | ak|2 = 1

  33. i. f : trial function is used to evaluate the upper limit of ground state energy E1 ii. f= ground state wave function,  f* Hf dt = E1 iii. optimize paramemters in f by minimizing  f* Hf dt / f* f dt

  34. Application to a particle in a box of infinite depth l 0 Requirements for the trial wave function: i. zero at boundary; ii. smoothness  a maximum in the center. Trial wave function: f = x (l - x)

  35. * H  dx = -(h2/82m)  (lx-x2) d2(lx-x2)/dx2 dx = h2/(42m)  (x2 - lx)dx = h2l3/(242m) * dx =  x2 (l-x)2 dx = l5/30 E = 5h2/(42l2m)  h2/(8ml2) = E1

  36. Variational Method (1) Construct a wave function (c1,c2,,cm) (2) Calculate the energy of : E E(c1,c2,,cm) (3) Choose {cj*} (i=1,2,,m) so that Eis minimum

  37. Example: one-dimensional harmonic oscillator Potential: V(x) = (1/2) kx2 = (1/2) m2x2 = 22m2x2 Trial wave function for the ground state: (x) = exp(-cx2) * H  dx = -(h2/82m)  exp(-cx2) d2[exp(-cx2)]/dx2 dx + 22m2  x2 exp(-2cx2) dx = (h2/42m) (c/8)1/2 + 2m2 (/8c3)1/2 * dx =  exp(-2cx2) dx = (/2)1/2 c-1/2 E= W = (h2/82m)c + (2/2)m2/c

  38. To minimize W, 0 = dW/dc = h2/82m - (2/2)m2c-2 c = 22m/h W= (1/2) h

  39. Extension of Variation Method    . . . E3y3 E2y2 E1y1 For a wave function f which is orthogonal to the ground state wave function y1, i.e. dtf*y1 = 0 Ef = dtf*Hf / dtf*f>E2 the first excited state energy

  40. The trial wave function f: dtf*y1 = 0 f = k=1 akyk dtf*y1 = |a1|2 = 0 Ef = dtf*Hf / dtf*f = k=2|ak|2Ek / k=2|ak|2 >k=2|ak|2E2 / k=2|ak|2 = E2

  41. Application to H2+ e + + y1 y2 f = c1y1 + c2y2 W = f*H f dt / f*f dt = (c12H11 + 2c1 c2H12+ c22H22 ) / (c12 + 2c1 c2S + c22 ) W (c12 + 2c1 c2S + c22) = c12H11 + 2c1 c2H12+ c22H22

  42. Partial derivative with respect to c1(W/c1 = 0) : W (c1 + S c2) = c1H11 + c2H12 Partial derivative with respect to c2(W/c2 = 0) : W (S c1 + c2) = c1H12 + c2H22 (H11 - W) c1 + (H12 - S W) c2 = 0 (H12 - S W) c1 + (H22 -W) c2 = 0

  43. To have nontrivial solution: H11 - W H12 - S W H12 - S W H22 -W For H2+,H11 = H22; H12 < 0. Ground State: Eg = W1 = (H11+H12) / (1+S) f1= (y1+y2) / 2(1+S)1/2 Excited State: Ee = W2 = (H11-H12) / (1-S) f2= (y1-y2) / 2(1-S)1/2 = 0 bonding orbital Anti-bonding orbital

  44. Results: De = 1.76 eV, Re = 1.32 A Exact: De = 2.79 eV, Re = 1.06 A 1 eV = 23.0605 kcal / mol

  45. 2p 1s Further Improvements H p-1/2exp(-r) He+ 23/2p-1/2exp(-2r) Optimization of 1s orbitals Trial wave function: k3/2p-1/2exp(-kr)  Eg = W1(k,R) at each R, choose kso thatW1/k = 0 Results: De = 2.36 eV, Re = 1.06 A Resutls: De = 2.73 eV, Re = 1.06 A Inclusion of other atomic orbitals

  46. a11x1 + a12x2 = b1 a21x1 + a22x2 = b2 (a11a22-a12a21) x1 = b1a22-b2a12 (a11a22-a12a21) x2 = b2a11-b1a21 Linear Equations 1. two linear equations for two unknown, x1 and x2

  47. Introducing determinant: a11 a12 = a11a22-a12a21 a21 a22 a11 a12b1 a12 x1 = a21 a22 b2 a22 a11 a12a11 b1 x2 = a21 a22a21 b2

  48. Our case: b1 = b2 = 0, homogeneous   1. trivial solution: x1 = x2 = 0   2. nontrivial solution:   a11 a12 = 0 a21 a22 n linear equations for n unknown variables a11x1 + a12x2 + ... + a1nxn= b1 a21x1 + a22x2 + ... + a2nxn= b2 ............................................ an1x1 + an2x2 + ... + annxn= bn

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