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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. Company.

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Computational modeling of macromolecular systems

Computational Modeling of Macromolecular Systems

Dr. GuanHua CHEN

Department of Chemistry

University of Hong Kong


Computational chemistry

Computational Chemistry

  • Quantum Chemistry

    SchrÖdinger Equation

    H = E

  • Molecular Mechanics

    F = Ma

    F : Force Field


Computational modeling of macromolecular systems

Computational Chemistry Industry

Company

Software

Gaussian Inc.Gaussian 94, Gaussian 98

Schrödinger Inc.Jaguar

WavefunctionSpartan

Q-ChemQ-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


Computational modeling of macromolecular systems

Cytochrome c(involved in the ATP synthesis)

heme

1997 Nobel Prize

in Biology:

ATP Synthase in

Mitochondria

Cytochrome c is a peripheral membrane protein

involved in the long distance electron transfers


Computational modeling of macromolecular systems

Simulation of a pair of polypeptides

Duration: 100 ps. Time step: 1 ps (Ng, Yokojima & Chen, 2000)


Computational modeling of macromolecular systems

Protein Dynamics

1. Atomic Fluctuations

10-15 to 10-11 s; 0.01 to 1 Ao

2. Collective Motions

10-12 to 10-3 s; 0.01 to >5 Ao

3. Conformational Changes

10-9 to 103 s; 0.5 to >10 Ao

Theoretician leaded the way ! (Karplus at Harvard U.)


Computational modeling of macromolecular systems

Nanotechnology

Scanning Tunneling Microscope

Manipulating Atoms by Hand


Computational modeling of macromolecular systems

Large Gear Drives Small Gear

G. Hong et. al., 1999


Computational modeling of macromolecular systems

Calculated Electron distribution at equator


Computational modeling of macromolecular systems

Vitamin C

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.


Computational modeling of macromolecular systems

Molecular Mechanics (MM) Method

F = Ma

F : Force Field


Computational modeling of macromolecular systems

Molecular Mechanics Force Field

  • Bond Stretching Term

  • Bond Angle Term

  • Torsional Term

  • Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction


Computational modeling of macromolecular systems

Bond Stretching Potential

Eb = 1/2 kb (Dl)2

where, kb : stretch force constant

Dl : difference between equilibrium

& actual bond length

Two-body interaction


Computational modeling of macromolecular systems

Bond Angle Deformation Potential

Ea = 1/2 ka (D)2

where, ka : angle force constant

D : difference between equilibrium

& actual bond angle

Three-body interaction


Computational modeling of macromolecular systems

Periodic Torsional Barrier Potential

Et = (V/2) (1+ cosn )

where, V : rotational barrier

t: torsion angle

n : rotational degeneracy

Four-body interaction


Computational modeling of macromolecular systems

Non-bonding interaction

van der Waals interaction

for pairs of non-bonded atoms

Coulomb potential

for all pairs of charged atoms


Computational modeling of macromolecular systems

MM Force Field Types

  • MM2Small molecules

  • AMBERPolymers

  • CHAMMPolymers

  • BIOPolymers

  • OPLSSolvent Effects


Computational modeling of macromolecular systems

CHAMM FORCE FIELD FILE


Computational modeling of macromolecular systems

/(kcal/mol)

/Ao


Computational modeling of macromolecular systems

/(kcal/mol/Ao2)

/Ao


Computational modeling of macromolecular systems

/deg

/(kcal/mol/rad2)


Computational modeling of macromolecular systems

/(kcal/mol)

/deg


Computational modeling of macromolecular systems

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) (s1+2s2+2s3+s4) t +O(t5)

s1 = dx/dt

s2 = dx/dt [w/ t=t+t/2, x = x(t)+s1t/2]

s3 = dx/dt [w/ t=t+t/2, x = x(t)+s2t/2]

s4 = dx/dt [w/ t=t+t, x = x(t)+s3t]

Very accurate but slow!


Computational modeling of macromolecular systems

Algorithms for Molecular Dynamics

Verlet Algorithm:

x(t+t) = x(t) + (dx/dt) t + (1/2) d2x/dt2t2 + ...

x(t -t) = x(t) - (dx/dt) t + (1/2) d2x/dt2t2 - ...

x(t+t) = 2x(t) - x(t -t) + d2x/dt2t2 + O(t4)

Efficient & Commonly Used!


Calculated properties

Calculated Properties

  • Structure, Geometry

  • Energy & Stability

  • Mechanic Properties: Young’s Modulus

  • Vibration Frequency & Mode


Computational modeling of macromolecular systems

Crystal Structure of C60 solid

Crystal Structure of K3C60


Computational modeling of macromolecular systems

Vibration Spectrum of K3C60

GH Chen, Ph.D. Thesis, Caltech (1992)


Quantum chemistry methods

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


Computational modeling of macromolecular systems

SchrÖdinger Equation

Hy = Ey

Wavefunction

Hamiltonian

H = (-h2/2ma)2 - (h2/2me)ii2

+  ZaZbe2/rab - i Zae2/ria

+ ije2/rij

Energy


Computational modeling of macromolecular systems

Hartree-Fock Equation:

[ f(1)+ J2(1) -K2(1)] f1(1) = e1 f1(1)

[ f(2)+ J1(2) -K1(2)] f2(2) = e2 f2(2)

Fock Operator:

F(1) f(1)+ J2(1) -K2(1) Fock operator for 1

F(2) f(2)+ J1(2) -K1(2) Fock operator for 2

e-

+

e-


Computational modeling of macromolecular systems

f(1) -(h2/2me)12 -N ZN/r1N

one-electron term if no Coulomb interaction

J2(1) dr2 f2*(2)e2/r12 f2(2)

Ave. Coulomb potential on electron 1 from 2

K2(1) f1(1)  f2(1)  dr2 f2*(2) e2/r12 f1(2)

Ave. exchange potential on electron 1 from 2

f(2) -(h2/2me)22 -N ZN/r2N

J1(2) dr1 f1*(1)e2/r12 f1(1)

K1(2) q(2)  f1(1)  dr1 f1*(1) e2/r12 q(1)

Average Hamiltonian for electron 1

F(1) f(1)+ J2(1) -K2(1)

Average Hamiltonian for electron 2

F(2) f(2)+ J1(2) -K1(2)


Computational modeling of macromolecular systems

Hartree-Fock Method

1. Many-Body Wave Function is approximated

by Single Slater Determinant

2. 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


Computational modeling of macromolecular systems

3. Roothaan Method (introduction of Basis functions)

fi= k ckiyk LCAO-MO

{ yk }is a set of atomic orbitals (or basis functions)

4. Hartree-Fock-Roothaan equation

j ( Fij - ei Sij ) cji = 0

Fij  < i|F | j >Sij  < i| j >

5. Solve the Hartree-Fock-Roothaan equation

self-consistently (HFSCF)


Computational modeling of macromolecular systems

Graphic Representation of Hartree-Fock Solution

0 eV

Electron

Affinity

Ionization

Energy


Computational modeling of macromolecular systems

Koopman’s Theorem

The energy required to remove an electron from a

closed-shell atom or molecules is well approximated

by minus the orbital energy e of the AO or MO from

which the electron is removed.


Computational modeling of macromolecular systems

Slater-type orbitals (STO)

nlm = Nrn-1exp(-r/a0) Ylm(,)

x the orbitalexponent

Basis Set i = p cip p

Gaussian type functions (GTF)

gijk = N xi yj zk exp(-ar2)

(primitive Gaussian function)

p = u dupgu

(contracted Gaussian-type function, CGTF)

u = {ijk}p = {nlm}


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

6-31G for a carbon atom:(10s12p)  [3s6p]

1s2s2pi (i=x,y,z)

6GTFs 3GTFs 1GTF3GTFs 1GTF

1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s)(s) (s) (p) (p)


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

Configuration Interaction (CI)

+

+ …


Computational modeling of macromolecular systems

Single Electron Excitation or Singly Excited


Computational modeling of macromolecular systems

Double Electrons Excitation or Doubly Excited


Computational modeling of macromolecular systems

Singly Excited Configuration Interaction (CIS):

Changes only the excited states

+


Computational modeling of macromolecular systems

Doubly Excited CI (CID):

Changes ground & excited states

+

Singly & Doubly Excited CI (CISD):

Most Used CI Method


Computational modeling of macromolecular systems

Full CI (FCI):

Changes ground & excited states

+

+

+ ...


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

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

=


Computational modeling of macromolecular systems

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

=


Computational modeling of macromolecular systems

Complete Active Space SCF (CASSCF)

Active space

All possible configurations


Computational modeling of macromolecular systems

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)

E0 = - (h2/2me)i <i |i2|i >-  drZae2(r) /r1a

+ (1/2)   dr1 dr2e2/r12 + Exc[(r)]

Kohn-Sham Equation:

FKSyi = ei yi

FKS- (h2/2me)ii2-  Zae2 /r1a + jJj + Vxc

Vxc dExc[(r)] / d(r)


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

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)


Computational modeling of macromolecular systems

Atomic Orbital Energy VISP

---------------e5-e5

---------------e4-e4

---------------e3-e3

---------------e2-e2

---------------e1-e1

Heffrs = ½ K(Heffrr + Heffss) SrsK:13


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

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.


Computational modeling of macromolecular systems

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

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).


Computational modeling of macromolecular systems

Quantum Mechanical Simulation of Nano-size Systems

Ground State:ab initio Hartree-Fock calculation


Computational modeling of macromolecular systems

Computational Time: protein w/ 10,000 atoms

ab initio Hartree-Fock ground state calculation:

~20,000 years on CRAY YMP


Computational modeling of macromolecular systems

In 2010:

~24 months on 100 processor machine

One Problem:

Transitor with a few atoms

Current Computer Technology will fail !


Computational modeling of macromolecular systems

Quantum Chemist’s Solution

Linear-Scaling Method: O(N)

Computational time scales linearly with system size

Time

Size


Computational modeling of macromolecular systems

Linear Scaling Calculation for Ground State

Divide-and-Conqure (DAC)

W. Yang, Phys. Rev. Lett. 1991


Computational modeling of macromolecular systems

Density-Matrix Minimization (DMM) Method

Minimize the Energy or the Grand Potential:

 = Tr [ (32 - 23) (H-I) ]

Li, Nunes and Vanderbilt,Phy. Rev. B. 1993


Computational modeling of macromolecular systems

Orbital Minimization (OM) Method

Minimize the Energy or the Grand Potential:

 = 2 nijcni (H-I)ijcnj

- nmijcni (H-I)ijcmj lcnlcml

Mauri (1993), Ordejon (1993), Galii (1994), Kim (1995)


Computational modeling of macromolecular systems

Fermi Operator Expansion (FOE) Method

Expand Density Matrix in Chebyshev Polynomial:

(H) = c0I + c1H + c2H2 + …

= c0I / 2 + cjTj(H) + …

T0(H) = I

T1(H) = H

Tj+1 (H) = 2HTj(H) - Tj-1(H)

Goedecker & Colombo (1994)


Computational modeling of macromolecular systems

York, Lee & Yang, JACS, 1996

Superoxide Dismutase (4380 atoms)

AM1


Computational modeling of macromolecular systems

Linear Scaling First Principle Method

Two-electron integrals :

Vabcd = <fa(1) fb(2) | e2 / r12 | fd(1) fc(2)>

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


Computational modeling of macromolecular systems

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.

Ordejón et. al., Phys. Rev. B 1993.

Drabold & Sankey, Phys. Rev. Lett. 1993.


Computational modeling of macromolecular systems

Linear Scaling Calculation for EXCITEDSTATE ?

A Much More Difficult Problem !


Computational modeling of macromolecular systems

Linear-Scaling Calculation for excited states

Localized-Density-Matrix (LDM) Method

r = r(0) + dr

E(t)

rij(0) = 0 rij > r0

drij = 0 rij > r1

Yokojima & Chen, Phys. Rev. B, 1999

Principle of the nearsightedness of

equilibrium systems (Kohn, 1996)


Computational modeling of macromolecular systems

Heisenberg Equation of Motion

Time-Dependent Hartree-Fock

Random Phase Approximation


Computational modeling of macromolecular systems

Polyacetylene

PPP Semiempirical Hamitonian


Computational modeling of macromolecular systems

Liang, Yokojima & Chen, JPC, 2000


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

Liang, Yokojima & Chen, JPC, 2000


Computational modeling of macromolecular systems

Flat Panel Display


Computational modeling of macromolecular systems

Cambridge Display Technology

Weight: 15 gram

Resolution: 800x236

Size: 45x37 mm

Voltage: DC, 10V


Computational modeling of macromolecular systems

Intensity

electron

hole

Energy


Computational modeling of macromolecular systems

Carbon Nanotube


Computational modeling of macromolecular systems

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


Computational modeling of macromolecular systems

Surprising!

DFT: no or very small gap

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


Computational modeling of macromolecular systems

Absorption Spectra of (9,0) SWNTs


Computational modeling of macromolecular systems

Smallest SWNT: 0.4 nm in diameter

Wang, Tang & etc., Nature (2000)

Three possibilities:

(4,2), (3,3) & (5,0) SWNTs


Computational modeling of macromolecular systems

Tang et. al, 2000


Computational modeling of macromolecular systems

Absorption of SWNTs (4,2), (3,3) & (5,0)

(4,2): C332H12

(3,3): C420H12

(5,0): C330

Liang, & Chen (2001)


Quantum mechanics molecular mechanics qm mm method

Quantum Mechanics / Molecular Mechanics (QM/MM) Method

Combining quantum mechanics and

molecular mechanics methods:

QM

MM


Computational modeling of macromolecular systems

Human Genome Project

GENOMICS

Drug Discovery


Computational modeling of macromolecular systems

Aldose Reductase

Design of Aldose Reductase Inhibitors


Computational modeling of macromolecular systems

Goddard, Caltech

Goddard, Caltech


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