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

•
Ö
H


•
•

Computational Chemistry Industry
Company
Gaussian Inc.
Schrödinger Inc.
Wavefunction
Q-Chem
Accelrys
HyperCube
Informatix
Celera Genomics
Software
Gaussian 94, Gaussian 98
Jaguar
Spartan
Q-Chem
InsightII, Cerius 2
HyperChem
Applications: material discovery, drug design & research
R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billion
Bioinformatics: Total Sales in 2001
Project Sales in 2006
US$ 225 million
US$ 1.7 billion

heme
Cytochrome c
C60 energy
Vitamin
OH + D
2
--> HOD + D
C

•
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

Ö
H
 =
E

Wavefunction
Hamiltonian
H =


(
h 2 /2m
-  i
+  i


 j

)
Z
 e 2 /r i  e 2 /r ij

2 - ( h 2 /2m e
(
+




Z

Z

)
 e
2 i

/ r i
2

)
Energy
One-electron terms:


(
h 2 /2m

)

2
Two-electron term:
 i
 j e 2 /r ij
- ( h 2 /2m e
)
 i
 i
2
-  i


Z
 e 2 /r i 

Hartree-Fock Method
Orbitals
1. Hartree-Fock Equation
F f i
= e i f i
F Fock operator f i the ith Hartree-Fock orbital e i the energy of the ith Hartree-Fock orbital

2. Roothaan Method (introduction of Basis functions) f i
=
{
 k
 k c ki
 k
LCAO-MO
} is a set of atomic orbitals (or basis functions)
3. Hartree-Fock-Roothaan equation
 j
( F ij
e i
S ij
) c ji
= 0
F ij
 <  i
|
F
|  j
>
S ij
 <  i
|  j
>
4. Solve the Hartree-Fock-Roothaan equation self-consistently (HFSCF)

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

A Gaussian Input File for H
2
O
# 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 f i
=
 p c ip
 p
{
 k
} is a set of atomic orbitals (or basis functions)
STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G**
------------------------------------------------------------------------------------ complexity & accuracy

Gaussian type functions 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}

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

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) n
(0)
E n
(0)
 n
(0)
H’
H 0

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

T
1

= e T

(0)

(0)

T = T
1
+ T
2
+ T
3
+ T
4
+ T
5
+ …
T n

T
2

CCD
= e T
2

(0)

(0)

CCD
T
2

Active space
All possible configurations

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)
E
0
=
- ( h 2 /2m e
)
 i
<
+ (1/2)   dr
1
 dr
2 i
|
 i
2 e 2 /r
|

12 i
+
>
- 
E xc

 dr
[

(r)]
Z
 e 2

(r) / r
1

Kohn-Sham Equation Ground State :
Phys. Rev. 140, A1133 (1965)
F
KS
V xc
 - ( h 2 /2m
 d
E xc
F
KS e

)
 i i

= e i
2
[

(r)] / d
(r)
 i
- 
 i
Z
 e 2 / r
1

+  j
J j
+ V xc
A popular exchange-correlation functional E xc
[

(r)]: B3LYP

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

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

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

Search for Transition State
Transition State: one negative frequency k
  e -
D
G/RT
D
G
Reactant
Product
Reaction Coordinate

Gaussian Input File for Transition State Calculation
#b3lyp/6-31G opt=qst2 test the first is the reactant internal coordinate
0 1
O
H 1 oh1
H 1 oh1 2 ohh1 oh1 0.90
ohh1 104.5
The second is the product internal coordinate
0 1
O
H 1 oh2
H 1 oh3 2 ohh2 oh2 0.9
oh3 10.0
ohh2 160.0

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

LCAO-MO: f
 i s
=
H eff
 r c ri
( H eff rs
 r
rs
 <  r e i
S rs
) c si
|
H eff
| 
= 0 s
>
S rs
 <  r
|  s
>
Parametrization:
H eff rr
 <  r
|
H eff
|  r
>
= minus the valence-state ionization potential (VISP)

---------------
---------------
---------------
---------------
---------------
Atomic Orbital Energy e
5 e
4 e
3 e
2 e
1
H eff rs
= ½
K ( H eff rr
+ H eff ss
) S rs
VISP
-e
5
-e
4
-e
3
-e
2
-e
1
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) = <
 r
(1)
 t
(2)| 1/r
12
|
 s
(1)
 u
(2)>
CNDO: complete neglect of differential overlap
(rs|tu) = d rs d tu
(rr|tt)
 d rs d 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

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).

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

QM
MM

Hamiltonian of entire system:
H = H
QM
+H
MM
+H
QM/MM
Energy of entire system:
E = E
QM
(
QM
) + E
MM
(
MM
) + E
QM/MM
(
QM/MM
)
E
QM/MM
(
QM/MM
) = E elec
(
QM/MM
) + E vdw
(
MM
) + E
MM-bond
(
MM
)
E
QM
(
QM
) + E elec
H eff
= -
1/2
 i
+


 i

V i
2
(
QM/MM
) = <

| H eff v-b
+
 ij
(r i
1/r
) +

 ij

d


 i
Z

|

/r i 
>
-



Z

Z d
/r
d
+



 i q

/r i 
Z
 q

/r

QM
MM

Linear-Scaling Method: O(N)
Computational time scales linearly with system size
Time
Size

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

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

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

Fast Multiple Method
LDM-TDDFT: C n
H
2n+2

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

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

Carbon Nanotube

Quantum mechanical investigation of the field emission from the tips of carbon nanotubes
Zheng, Chen, Li, Deng & Xu, Phys. Rev. Lett. 2004
Zettl, PRL 2001

•
Bond Stretching Term
•
Bond Angle Term
•
Torsional Term
•
Electrostatic Term
Molecular Mechanics
F = Ma
F : Force Field

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

Bond Angle Deformation Potential
E a
= 1/2 k a
(
D
) 2 where, k a
D 
: 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

•
•
MM2
AMBER
Molecules
Polymers
•
CHAMM Polymers
•
BIO Polymers
•
OPLS Solvent Effects

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

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

Multiple Scale Simulation

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

Nanoscopic Electromechanical Device
(NEMS)
Zhao, Ma, Chen & Jiang, Phys. Rev. Lett. 2003

ALDOSE REDUCTASE
Diabetes
OH HO
O
HO OH HO
Aldose Reductase
HO
NADP
HO glucose
Glucose
OH
NADPH
HO
HO sorbitol
Sorbitol
OH
Diabetic
Complications

Inhibitor
Aldose Reductase
Design of Aldose Reductase Inhibitors

Descriptors:
Electron negativity
Volume
Database for Functional Groups

5.0
4.5
4.0
3.5
3.0
X
6'
7'
O
5'
HN
NH
NMe
O
8'
N
H
O
2.5
2.5
3.0
3.5
4.0
4.5
Fig 3 QSAR OF INHIBITOR CONCENTRATION OF INHIBITING AR Log(IC
50
)

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
X
6'
7'
O
5'
HN
NH
NMe
O
8'
N
H
O
-0.1
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Fig 2 QSAR OF LOWER THE SCIATIC NERVE SORBITOL LEVEL(%)
Possible drug leads: ~ 350 compounds

CYS298
TYP219
LEU300
Cerius2 LigandFit NADPH
TRP111
PHE122
HIS110
TRP20
TYR48
VAL47
LYS77
TRP79

To further confirm the AR-ARI binding,
We perform QM/MM calculations on drug leads.
CHARMM
5'-OH, 6'-F, 7'-OH
X
6'
7'
O
5'
HN
NH
NMe
O
8'
N
H
O
Binding energy is found to be –45 kcal / mol

Aldose reducatse
Inhibitor
(4R)-6’-fluoro-7’-hydroxyl-8’-bromo-3’-methylspiro-
[imidazoli-dine-4,4’(1’H)-quinazoline]-2,2’,5(3’H)-trione
Cerius2 LigandFit Hu & Chen, 2003

Interaction energy between ligand and protein
Quantum Mechanics/Molecular Mechanics
(QM / MM)
Hu & Chen, 2003

O
5'
HN
X
6'
7'
8'
N
H
NH
NMe
O
O a:Inhibitor concentration of inhibit Aldose Reductase; b: the percents of lower sciatic nerve sorbitol levels c: interaction with AR in Fig. 4

QSAR determination & prediction (Neural Network)
Docking (Cerius2)
QM / MM (binding energy)
?

Software in Department
1. Gaussian
2. Insight II
CHARMm: molecular dynamics simulation, QM/MM
Profiles-3D: Predicting protein structure from sequences
SeqFold: Functional Genomics, functional identification of protein w/ sequence and structure comparison
NMR Refine: Structure determination w/ NMR data
3. Games
4. HyperChem
5. AutoDock (docking)
6. MacroModel
6. In-House Developed Software
LODERSTAR
Neural Network for QSAR
Monte Carlo & Molecular Dynamics

Lecture Notes for Physical Chemistry
Year 2
1.
Intermediate Physical Chemistry (CHEM2503) (Powerpoint format .ppt)
Year 3
1.
Advanced Physical Chemistry (Powerpoint format .ppt)
2.
Electronic Spectroscopy (Powerpoint format .ppt)
3.
Electronic Spectroscopy (assignment) (rar file)
Postgraduate Course
1.
Research Techniques in Chemistry (Powerpoint format .ppt)
Course Work Download Molecule
M.Sc Course
1.
Computational Modeling of Macromolecular Systems
(Powerpoint format .ppt) Download Molecule

HYPERCHEM Exercise
Part A: Study the electronic structure and vibrational spectrum of formaldehyde
Step 1: Build up the structure of the formaldehyde.
1.
Run HYPERCHEM software in the start menu.
2.
Double click the drawing tool to open the elements table dialogue box and select carbon atom.
Close the element table.
(Drawing tool)
3.
L-click the cursor on the workspace. A carbon atom will appear.
(Make sure drawing tool is selected. R-click on the atom if you want to delete it)
4.
Repeat (2) and choose oxygen instead of carbon. Move the cursor to the carbon centre and drag the
Formaldehyde
O
C
H
5.
L-click the bond between carbon and oxygen to create a double bond.
6.
L-click on Build in the menu bar and switch on ‘ add H & model build’ (i.e. make sure a tick appeared on the left of this function.).
Step 2: Optimize the structure using RHF and 6-31G* basis set.
7.
L-Click on Setup in the menu bar and L-click ab Initio;
L-Click on 6-31G*; then, L-Click on Options button;
Select RHF, set Charge to 0 and Multiplicity to 1 (default for charge 0);
L-Click OK buttons after modifications were done.
8.
L-Click on Compute in the menu bar and select Geometry Optimization;
Select Polak-Ribiere and set RMS gradient to 0.05 and max cycles to 60;
L-Click
OK button (The calculation will be started. Repeat the step till “Conv=YES” appears in the status line.).
Record the energy appeared in the status line
9.
L-Click on Compute in the menu bar and select Orbitals.
Record energy levels and point groups of required molecular orbitals (MO)
(Optional: You can draw the contour plot of the selected orbital and visualize the shape of the orbital.)
10. L-Click on Compute in the menu bar and select Vibrations.
11. L-Click on Compute in the menu bar and select Vibrational Spectrum.
Record the frequencies of different vibrational modes and their corresponding oscillator strengths.
(Optional: You can turn on animate vibrations, select any vibrational modes, and L-Click on OK button. The molecule begins to vibrate. To suspend the animation, L-Click on Cancel button.)

Part B: Molecular Dynamics of Tetrapeptide
1.
L-click Databases on the menu bar. Choose Amino Acids.
2.
Select Beta sheet.
3.
L-click Ala, Tyr, Asp and Gly to create tetrapeptide Ala-Tyr-Asp-Gly.
4.
L-click on rotate-out-of-plane tool and use it to rotate the molecule to a proper angle for observation and measurements.
(Rotate-out-of-plane tool)
5.
L-Click on Setup in the menu bar and L-click Molecular Mechanics;
L-Click on MM+;
L-Click OK buttons after modifications were done.
6.
L-Click on Compute in the menu bar and select Geometry Optimization;
7.
Record the total energies.
8.
L-Click on Compute in the menu bar and L-click Molecular Dynamics;
Run molecular dynamics at 0K and 300K with constant temperature.
Simulation Time: 1ps
9.
Record the total energies.
Part C: Molecular Dynamics of Ribosomal Protein
Procedures:
10.
Use a web-browser and Go to http://yangtze.hku.hk/lecture_notes.htm
.
11.
R-click the title labeled “Download molecule” and save it in a folder in your local disk (C:\).
12.
L-click on File in the menu bar and select open to load in the molecule.
(You should notify that this file has extension filename .ENT and is in PDB format.)
13.
L-click on rotate-out-of-plane tool and use it to rotate the molecule to a proper angle for observation and measurements.
(Rotate-out-of-plane tool)
14.
L-Click on Setup in the menu bar and L-click Molecular Mechanics;
L-Click on MM+;
L-Click OK buttons after modifications were done.
15.
L-Click on Compute in the menu bar and L-click Molecular Dynamics;
Run molecular dynamics at 300K with constant temperature.
Simulation Time: 1ps
16.
Record the total energy.