2. Computational Modeling of Macromolecular Systems Dr. GuanHua CHEN Department of Chemistry University of Hong Kong

3. Computational Chemistry Quantum Chemistry Schr Ö dinger Equation H  = E  Molecular Mechanics F = Ma F : Force Field

4. Computational Chemistry Industry CompanySoftware Gaussian Inc.Gaussian 94, Gaussian 98 Schrödinger Inc.Jaguar WavefunctionSpartanQ-Chem Molecular Simulation Inc. (MSI)InsightII, Cerius2, modeler HyperCubeHyperChem Applications: material discovery, drug design & research R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billion Sales of Scientific Computing in 2000: > US$ 200 million

5. Cytochrome c (involved in the ATP synthesis) heme Cytochrome c is a peripheral membrane protein involved in the long distance electron transfers 1997 Nobel Prize in Biology: ATP Synthase in Mitochondria

6. Simulation of a pair of polypeptides Duration: 100 ps. Time step: 1 ps (Ng, Yokojima & Chen, 2000)

7. Protein Dynamics Theoretician leaded the way ! (Karplus at Harvard U.) 1. Atomic Fluctuations 10 -15 to 10 -11 s; 0.01 to 1 A o 2. Collective Motions 10 -12 to 10 -3 s; 0.01 to >5 A o 3. Conformational Changes 10 -9 to 10 3 s; 0.5 to >10 A o

8. Scanning Tunneling Microscope Manipulating Atoms by Hand

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

10. Calculated Electron distribution at equator

11. The electron density around the vitamin C molecule. The colors show the electrostatic potential with the negative areas shaded in red and the positive in blue. Vitamin C

12. Molecular Mechanics (MM) Method F = Ma F : Force Field

13. Molecular Mechanics Force Field Bond Stretching Term Bond Angle Term Torsional Term Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction

14. Bond Stretching Potential E b = 1/2 k b (  l) 2 where, k b : stretch force constant  l : difference between equilibrium & actual bond length Two-body interaction

15. Bond Angle Deformation Potential E a = 1/2 k a (  ) 2 where, k a : angle force constant   : difference between equilibrium & actual bond angle  Three-body interaction

16. Periodic Torsional Barrier Potential E t = (V/2) (1+ cosn  ) where, V : rotational barrier  : torsion angle n : rotational degeneracy Four-body interaction

17. Non-bonding interaction van der Waals interaction for pairs of non-bonded atoms Coulomb potential for all pairs of charged atoms

18. MM Force Field Types MM2Small molecules AMBERPolymers CHAMMPolymers BIOPolymers OPLSSolvent Effects

19. CHAMM FORCE FIELD FILE

21. /A o /(kcal/mol)

22. /(kcal/mol/A o2 ) /Ao/Ao

23. /(kcal/mol/rad 2 ) /deg

24. /(kcal/mol)/deg

25. Algorithms for Molecular Dynamics Runge-Kutta methods: x(t+  t) = x(t) + (dx/dt)  t Fourth-order Runge-Kutta x(t+  t) = x(t) + (1/6) (s 1 +2s 2 +2s 3 +s 4 )  t +O(  t 5 ) s 1 = dx/dt s 2 = dx/dt [w/ t=t+  t/2, x = x(t)+s 1  t/2] s 3 = dx/dt [w/ t=t+  t/2, x = x(t)+s 2  t/2] s 4 = dx/dt [w/ t=t+  t, x = x(t)+s 3  t] Very accurate but slow!

26. Algorithms for Molecular Dynamics Verlet Algorithm: x(t+  t) = x(t) + (dx/dt)  t + (1/2) d 2 x/dt 2  t 2 +... x(t -  t) = x(t) - (dx/dt)  t + (1/2) d 2 x/dt 2  t 2 -... x(t+  t) = 2x(t) - x(t -  t) + d 2 x/dt 2  t 2 + O(  t 4 ) Efficient & Commonly Used!

27. Calculated Properties Structure, Geometry Energy & Stability Mechanic Properties: Young’s Modulus Vibration Frequency & Mode

28. Crystal Structure of C 60 solid Crystal Structure of K 3 C 60

29. Vibration Spectrum of K 3 C 60 GH Chen, Ph.D. Thesis, Caltech (1992)

30. 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

31. H  E  Schr Ö dinger Equation Hamiltonian H =   (  h 2 /2m      h 2 /2m e )  i  i 2 +     Z  Z  e   r   i     e 2 /r i   i  j  e 2 /r ij Wavefunction Energy

32.  f(1)+ J 2 (1)  K 2 (1)  1 (1)  1  1 (1)  f(2)+ J 1 (2)  K 1 (2)  2 (2)  2  2 (2) F(1)  f(1)+ J 2 (1)  K 2 (1) Fock operator for 1 F(2)  f(2)+ J 1 (2)  K 1 (2) Fock operator for 2 Hartree-Fock Equation: Fock Operator: + e-e- e-e-

33. f(1)   h 2 /2m e )  1 2   N Z N  r 1N one-electron term if no Coulomb interaction J 2 (1)  dr   2   e 2 /r 12  2  Ave. Coulomb potential on electron 1 from 2 K 2 (1)    2  dr   2 *  e 2 /r 12    Ave. exchange potential on electron 1 from 2 f(2)   h 2 /2m e )  2 2   N Z N  r 2N J 1 (2)  dr   1   e 2 /r 12  1  K 1 (2)  1  dr   1 *  e 2 /r 12  Average Hamiltonian for electron 1 F(1)  f(1)+ J 2 (1)  K 2 (1) Average Hamiltonian for electron 2 F(2)  f(2)+ J 1 (2)  K 1 (2)

34. 1. Many-Body Wave Function is approximated by Single Slater Determinant 2. Hartree-Fock Equation F  i =  i  i F Fock operator  i the i-th Hartree-Fock orbital  i the energy of the i-th Hartree-Fock orbital Hartree-Fock Method

35. 3. Roothaan Method (introduction of Basis functions)  i =  k c ki  k LCAO-MO {  k } is a set of atomic orbitals (or basis functions) 4. Hartree-Fock-Roothaan equation  j ( F ij -  i S ij ) c ji = 0 F ij  i  F  j  S ij  i  j  5. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

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

37. The energy required to remove an electron from a closed-shell atom or molecules is well approximated by minus the orbital energy  of the AO or MO from which the electron is removed. Koopman’s Theorem

38. Slater-type orbitals (STO)  nlm = N r n-1 exp(  r/a 0 ) Y lm ( ,  )  the orbital  exponent Gaussian type functions (GTF) g ijk = N x i y j z k exp(-  r 2 ) (primitive Gaussian function)  p =  u d up g u (contracted Gaussian-type function, CGTF) u = {ijk}p = {nlm} Basis Set  i =  p c ip  p

39. Basis set of GTFs STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G** -------------------------------------------------------------------------------------  complexity & accuracy Minimal basis set: one STO for each atomic orbital (AO) STO-3G: 3 GTFs for each atomic orbital 3-21G: 3 GTFs for each inner shell AO 2 CGTFs (w/ 2 & 1 GTFs) for each valence AO 6-31G: 6 GTFs for each inner shell AO 2 CGTFs (w/ 3 & 1 GTFs) for each valence AO 6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows 6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen Polarization Function

40. Diffuse Basis Sets: For excited states and in anions where electronic density is more spread out, additional basis functions are needed. Diffuse functions to 6-31G basis set as follows: 6-31G* - adds a set of diffuse s & p orbitals to atoms in 1st & 2nd rows (Li - Cl). 6-31G** - adds a set of diffuse s and p orbitals to atoms in 1st & 2nd rows (Li- Cl) and a set of diffuse s functions to H Diffuse functions + polarisation functions: 6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets. Double-zeta (DZ) basis set: two STO for each AO

41. 6-31G for a carbon atom:(10s12p)  [3s6p] 1s2s2p i (i=x,y,z) 6GTFs 3GTFs 1GTF3GTFs 1GTF 1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s)(s) (s) (p) (p)

42. Electron Correlation: avoiding each other Two reasons of the instantaneous correlation: (1) Pauli Exclusion Principle (HF includes the effect) (2) Coulomb repulsion (not included in the HF) Beyond the Hartree-Fock Configuration Interaction (CI)* Perturbation theory* Coupled Cluster Method Density functional theory

43. Configuration Interaction (CI) + + …

44. Single Electron Excitation or Singly Excited

45. Double Electrons Excitation or Doubly Excited

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

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

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

49. H = H 0 + H’ H 0  n (0) = E n (0)  n (0)  n (0) is an eigenstate for unperturbed system H’ is small compared with H 0 Perturbation Theory

50. Moller-Plesset (MP) Perturbation Theory The MP unperturbed Hamiltonian H 0 H 0 =  m F(m) where F(m) is the Fock operator for electron m. And thus, the perturbation H ’ H ’ = H - H 0 Therefore, the unperturbed wave function is simply the Hartree-Fock wave function . Ab initio methods: MP2, MP3, MP4

51.  = e T  (0)  (0) : Hartree-Fock ground state wave function  : Ground state wave function T = T 1 + T 2 + T 3 + T 4 + T 5 + … T n : n electron excitation operator Coupled-Cluster Method = T1T1

52.  CCD = e T 2  (0)  (0) : Hartree-Fock ground state wave function  CCD : Ground state wave function T 2 : two electron excitation operator Coupled-Cluster Doubles (CCD) Method = T2T2

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

54. Density-Functional Theory (DFT) Hohenberg-Kohn Theorem: 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) E 0 =  h 2 /2m e )  i    dr   e 2  (r) / r 1    dr 1 dr 2 e 2 /r 12 + E xc [  (r) ] Kohn-Sham Equation: F KS  i =  i  i F KS   h 2 /2m e )  i  i 2     e 2 / r 1   j  J j + V xc V xc   E xc [  (r) ] /  (r)

55. Semiempirical Molecular Orbital Calculation Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman) Independent electron approximation Schrodinger equation for electron i H val =  i H eff (i) H eff (i) = -(h 2 /2m)  i 2 + V eff (i) H eff (i)  i =  i  i

56. LCAO-MO:  i =  r c ri  r  s ( H eff rs -  i S rs ) c si = 0 H eff rs  r  H eff  s  S rs  r  s  Parametrization: H eff rr  r  H eff  r   minus the valence-state ionization  potential (VISP)

57. Atomic Orbital Energy VISP ---------------e 5 -e 5 ---------------e 4 -e 4 ---------------e 3 -e 3 ---------------e 2 -e 2 ---------------e 1 -e 1 H eff rs = ½ K (H eff rr + H eff ss ) S rs K:1  3

58. CNDO, INDO, NDDO (Pople and co-workers) Hamiltonian with effective potentials H val =  i [ -(h 2 /2m)  i 2 + V eff (i) ] +  i  j>i e 2 / r ij two-electron integral: (rs|tu) = CNDO: complete neglect of differential overlap (rs|tu) =  rs  tu (rr|tt)   rs  tu  rt

59. 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, INDO are 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.

60. 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

61. 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).

62. Ground State: ab initio Hartree-Fock calculation

63. Computational Time: protein w/ 10,000 atoms ab initio Hartree-Fock ground state calculation: ~20,000 years on CRAY YMP

65. In 2010: ~24 months on 100 processor machine One Problem: Transitor with a few atoms Current Computer Technology will fail !

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

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

68. Density-Matrix Minimization (DMM) Method Li, Nunes and Vanderbilt, Phy. Rev. B. 1993 Minimize the Energy or the Grand Potential:  = Tr [ (3  2 - 2  3 ) (H-  I) ]

69. Orbital Minimization (OM) Method Mauri (1993), Ordejon (1993), Galii (1994), Kim (1995) Minimize the Energy or the Grand Potential:  = 2  n  ij c n i (H-  I) ij c n j -  nm  ij c n i (H-  I) ij c m j  l c n l c m l

70. Fermi Operator Expansion (FOE) Method Goedecker & Colombo (1994) Expand Density Matrix in Chebyshev Polynomial:  (H) = c 0 I + c 1 H + c 2 H 2 + … = c 0 I / 2 +  c j T j (H) + … T 0 (H) = I T 1 (H) = H T j+1 (H) = 2HT j (H) - T j-1 (H)

71. Superoxide Dismutase (4380 atoms) York, Lee & Yang, JACS, 1996 

72. Linear Scaling First Principle Method Two-electron integrals : V abcd =  a  b  e 2 / r 12  d  c  Coulomb Integrals: Fast Multiple Method (FMM) Exchange-Correlation (XC): Use of Locality Strain, Scuseria & Frisch, Science (1996): LSDA / 3-21G DFT calculation on 1026 atom RNA Fragment

73. Linear Scaling Calculation for Ground State Yang, Phys. Rev. Lett. 1991 Li, Nunes & Vanderbilt, Phy. Rev. B. 1993 Baroni & Giannozzi, Europhys. Lett. 1992. Gibson, Haydock & LaFemina, Phys. Rev. B 1993. Aoki, Phys. Rev. Lett. 1993. Cortona, Phys. Rev. B 1991. Galli & Parrinello, Phys. Rev. Lett. 1992. Mauri, Galli & Car, Phys. Rev. B 1993. Ordej ó n et. al., Phys. Rev. B 1993. Drabold & Sankey, Phys. Rev. Lett. 1993.

74. Linear Scaling Calculation for EXCITED STATE ? A Much More Difficult Problem !

76. Localized-Density-Matrix (LDM) Method  ij (0) = 0 r ij > r 0  ij = 0 r ij > r 1 Yokojima & Chen, Phys. Rev. B, 1999    Principle of the nearsightedness of equilibrium systems (Kohn, 1996) Linear-Scaling Calculation for excited states  t 

77. Heisenberg Equation of Motion Time-Dependent Hartree-Fock Random Phase Approximation

78. PPP Semiempirical Hamitonian Polyacetylene

79. Liang, Yokojima & Chen, JPC, 2000

80. Yokojima, Zhou & Chen, Chem. Phys. Lett., 1999

81. Liang, Yokojima & Chen, JPC, 2000

82. Flat Panel Display

83. Cambridge Display Technology Weight: 15 gram Resolution: 800x236 Size: 45x37 mm Voltage: DC, 10V

85. Energy Intensity electron hole

88. Carbon Nanotube

90. Liang, Wang, Yokojima & Chen, JACS (2000)

91. Surprising! DFT: no or very small gap

92. Absorption Spectra of (9,0) SWNTs

93. Smallest SWNT: 0.4 nm in diameter Wang, Tang & etc., Nature (2000) Three possibilities: (4,2), (3,3) & (5,0) SWNTs

94. Tang et. al, 2000

95. Absorption of SWNTs (4,2), (3,3) & (5,0)  C 332 H 12  C 420 H 12  C 330 Liang, & Chen (2001)

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

97. GENOMICS Human Genome Project

98. Design of Aldose Reductase Inhibitors Aldose Reductase

99. Goddard, Caltech