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

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

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

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

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

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

Scanning Tunneling Microscope Manipulating Atoms by Hand

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

Calculated Electron distribution at equator

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

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

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

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

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

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

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

MM Force Field Types MM2Small molecules AMBERPolymers CHAMMPolymers BIOPolymers OPLSSolvent Effects

CHAMM FORCE FIELD FILE

/A o /(kcal/mol)

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

/(kcal/mol/rad 2 ) /deg

/(kcal/mol)/deg

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!

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

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

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

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

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

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

 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-

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)

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

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)

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

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

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

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

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*, G*, 6-31+G** and G** basis sets. Double-zeta (DZ) basis set: two STO for each AO

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)

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

Configuration Interaction (CI) + + …

Single Electron Excitation or Singly Excited

Double Electrons Excitation or Doubly Excited

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

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

Full CI (FCI): Changes ground & excited states

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

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

 = 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

 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

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

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)

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

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)

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

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

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.

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

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).

Ground State: ab initio Hartree-Fock calculation

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

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

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

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

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

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

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)

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

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

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

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

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 

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

PPP Semiempirical Hamitonian Polyacetylene

Liang, Yokojima & Chen, JPC, 2000

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

Liang, Yokojima & Chen, JPC, 2000

Flat Panel Display

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

Energy Intensity electron hole

Carbon Nanotube

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

Surprising! DFT: no or very small gap

Absorption Spectra of (9,0) SWNTs

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

Tang et. al, 2000

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

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

GENOMICS Human Genome Project

Design of Aldose Reductase Inhibitors Aldose Reductase

Goddard, Caltech