Computational Chemistry

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Computational Chemistry
G. H. CHEN Department of Chemistry University of
Hong Kong
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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.
Theory is reality ! W.A. Goddard III
Dirac
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Computational Chemistry
Quantum Chemistry Molecular Mechanics Bioinform
atics Create Analyse Bio-information
SchrÖdinger Equation
F M a
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Nobel Prizes for Computational Chemsitry
Mulliken,1966
Fukui, 1981
Hoffmann, 1981
Pople, 1998
Kohn, 1998
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Computational Chemistry Industry
Company
Software
Gaussian Inc. Gaussian 94, Gaussian
98 Schrödinger Inc. Jaguar Wavefunction Spart
an Q-Chem Q-Chem Accelrys InsightII,
Cerius2 HyperCube HyperChem Celera Genomics
(Dr. Craig Venter, formal Prof., SUNY, Baffalo
98-01)
Applications material discovery, drug design
research
RD 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
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Software Development at HKU
LODESTAR v1.02 --Localized Density Matrix STAR
performer
http//yangtze.hku.hk
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Quantum Chemistry Methods

• Ab initio molecular orbital methods
• Semiempirical molecular orbital methods
• Density functional method

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SchrÖdinger Equation
H y E y
Wavefunction
Hamiltonian H ??(-h2/2ma)??2 - (h2/2me)?i?i2
???? ZaZbe2/rab - ?i ?? Zae2/ria ?i ?j
e2/rij
Energy
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Vitamin C
C60
energy
heme
OH D2 --gt HOD D
Cytochrome c
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C60 and Superconductor
What is superconductor? Electrical Current
flows for ever !
Soccer Ball
Applications Magnet, Magnetic train, Power
transportation
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Crystal Structure of C60 solid
Crystal Structure of K3C60
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K3C60 is a Superconductor (Tc 19K)
Vibration of Atoms
The mechanism of superconductivity in K3C60 was
discovered using com-putational chemistry methods
Varma et. al., 1991 Schluter et. al., 1992
Dresselhaus et. al., 1992Chen Goddard, 1992
Effective Attraction !
Vibration Spectrum of K3C60
Erwin Pickett, Science, 1991
GH Chen, Ph.D. Thesis, Caltech (1992)
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Carbon Nanotubes (Ijima, 1991)
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Calculated STM Image of a Carbon Nanotube
(Rubio, 1999)
STM Image of Carbon Nanotubes (Wildoer et. al.,
1998)
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Computer Simulations (Saito, Dresselhaus, Louie
et. al., 1992) Carbon Nanotubes
(n,m) Conductor, if n-m 3I
I0,1,2,3,or Semiconductor, if n-m ? 3I
Metallic Carbon Nanotubes Conducting
Wires Semiconducting Nanotubes Transistors Mole
cular-scale circuits ! 1 nm transistor!
30 nm transistor!
0.13 µm transistor!
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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)
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Science 9th November, 2001 Logic gates (and
circuits) with carbon nanotuce transistor
Science 7th July, 2000 Carbon nanotube-Based
nonvolatile RAM for molecular computing
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Nanoelectromechanical Systems (NEMS)
K.E. Drexler, Nanosystems Molecular Machinery,
Manufacturing and Computation (Wiley, New York,
1992).
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Large Gear Drives Small Gear
G. Hong et. al., 1999
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Nano-oscillators
Nanoscopic Electromechanical Device (NEMS)
Zhao, Ma, Chen Jiang, Phys. Rev. Lett. 2003
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Oscillation
Hibernation
Awakening
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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
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Computer-Aided Drug Design
Human Genome Project
GENOMICS
Drug Discovery
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ALDOSE REDUCTASE
Diabetic Complications
Diabetes
Sorbitol
Glucose
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Design of Aldose Reductase Inhibitors
Inhibitor
Aldose Reductase
Hu Chen, 2003
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Database for Functional Groups
Structure-activity-relation
Prediction Drug Leads
LogIC50 0.6861,0.88
LogIC50 0.6382,1.0
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Prediction Results using AutoDock
LogIC50 0.77,1.1
LogIC50 -1.87,4.05
LogIC50 -2.77,4.14
LogIC50 0.68,0.88
Hu Chen, 2003
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Computer-aided drug design
Chemical Synthesis
Screening using in vitro assay
Animal Tests
Clinical Trials
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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

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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 RD

Objective More and more research development
to be performed on computers and Internet instead
in the laboratories
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Quantum Chemistry
G. H. Chen Department of Chemistry University of
Hong Kong
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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
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Emphasis Hartree-Fock method Concepts
Hands-on experience
Text Book Quantum Chemistry, 4th Ed.
Ira N. Levine
http//yangtze.hku.hk/lecture/chem3504-3.ppt
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Contents  1. Variation Method 2. Hartree-Fock
Self-Consistent Field Method 3. Perturbation
Theory 4. Semiempirical Methods
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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 H f dt gt E1
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Proof Expand f in the basis set yk f
?k akyk where ak are coefficients Hyk
Ekyk then ? f H f dt ?k ?j ak aj Ej dkj
?k ak2 Ek gt E 1 ?k ak2 E1
Since is normalized, ? ff dt ?k ak2
1
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i. f trial function is used to evaluate the
upper limit of ground state energy E1 ii. f
ground state wave function, ? f H f dt
E1 iii. optimize paramemters in f by
minimizing ? f H f dt / ? f f dt
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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)
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? ? 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
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 Variational Method
(1) Construct a wave function ?(c1,c2,???,cm) (2)
Calculate the energy of ? E? ?
E?(c1,c2,???,cm) (3) Choose cj (i1,2,???,m)
so that E? is minimum
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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) d2exp(-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
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To minimize W, 0 dW/dc h2/8?2m -
(?2/2)m?2c-2 c 2?2?m/h W (1/2) h?
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Extension of Variation Method
   . . . E3 y3 E2 y2 E1 y1
For a wave function f which is orthogonal to the
ground state wave function y1, i.e. ?dt fy1
0 Ef ?dt fHf / ?dt ff gt E2 the first
excited state energy
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The trial wave function f ?dt fy1 0   f
?k1 ak yk   ?dt fy1 a12 0   Ef ?dt fHf
/ ?dt ff ?k2ak2Ek / ?k2ak2 gt
?k2ak2E2 / ?k2ak2 E2
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Application to H2

e    

 
y1
y2 f c1y1 c2y2
W ? fH f dt / ? ff dt (c12 H11 2c1
c2 H12 c22 H22 ) / (c12 2c1 c2 S
c22 )   W (c12 2c1 c2 S c22) c12 H11
2c1 c2 H12 c22 H22
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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
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To have nontrivial solution   H11 - W H12 - S
W H12 - S W H22 - W   For H2, H11 H22 H12 lt
0.   Ground State Eg W1 (H11H12) / (1S)
f1 (y1y2) / ?2(1S)1/2 Excited
State Ee W2 (H11-H12) / (1-S) f2
(y1-y2) / ?2(1-S)1/2
0
bonding orbital
Anti-bonding orbital
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Results De 1.76 eV, Re 1.32 A   Exact
De 2.79 eV, Re 1.06 A  
1 eV 23.0605 kcal / mol
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Further Improvements   H p-1/2 exp(-r) He
23/2 p-1/2 exp(-2r)
Optimization of 1s orbitals
Trial wave function k3/2 p-1/2 exp(-kr) 
Eg W1(k,R)   at each R, choose k so 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
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    a11x1 a12x2 b1 a21x1 a22x2
b2   (a11a22-a12a21) x1 b1a22-b2a12 (a11a22-a12a
21) x2 b2a11-b1a21
Linear Equations

1. two linear equations for two unknown, x1 and x2
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n linear equations for n unknown variables
a11x1 a12x2 ... a1nxn b1 a21x1 a22x2
... a2nxn b2 ..................................
.......... an1x1 an2x2 ... annxn bn
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a11 a12 ... a1,k-1 b1 a1,k1 ... a1n
a21 a22 ... a2,k-1 b2 a2,k1 ...
a2n det(aij) xk . . ... . .
. ... . an1 an2 ... an,k-1
b2 an,k1 ... ann     where,
a11 a12 ... a1n a21 a22 ... a2n det(aij)
. . ... . an1 an2 ... ann
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inhomogeneous case bk 0 for at least one k
  a11 a12 ... a1,k-1 b1 a1,k1 ... a1n
a21 a22 ... a2,k-1 b2 a2,k1 ... a2n .
. ... . . . ... .
an1 an2 ... an,k-1 b2 an,k1 ... ann xk
det(aij)  
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homogeneous case bk 0, k 1, 2, ... , n
(a) travial case xk 0, k 1, 2, ... , n (b)
nontravial case det(aij) 0  
For a n-th order determinant, n det(aij)
? alk Clk l1 where, Clk is
called cofactor
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Trial wave function f is a variation function
which is a combination of n linear independent
functions f1 , f2 , ... fn,   f c1f1 c2f2
... cnfn   n ? ? ( Hik - SikW )
ck 0 i1,2,...,n k1 Sik
? ?dt fi fk Hik ? ? dt fi H fk W ? ? dt f H f
/ ? dt f f
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  (i) W1 ? W2 ? ... ? Wn are n roots of
Eq.(1), (ii) E1 ? E2 ? ... ? En ? En1 ? ...
are energies of eigenstates then, W1 ?
E1, W2 ? E2, ..., Wn ? En
Linear variational theorem
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Molecular Orbital (MO) ? c1?1 c2?2  
( H11 - W ) c1 ( H12 - SW ) c2 0
S111 ( H21 - SW ) c1 ( H22 - W
) c2 0
S221 Generally ??i? a set of atomic
orbitals, basis set LCAO-MO ? c1?1 c2?2
...... cn?n linear combination of atomic
orbitals n ? ( Hik - SikW ) ck 0 i
1, 2, ......, n k1 Hik ? ? dt ?i H ?k Sik ?
?dt ?i?k Skk 1
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The Born-Oppenheimer Approximation
Hamiltonian H ??(-h2/2ma)??2 -
(h2/2me)?i?i2 ???? ZaZbe2/rab -
?i ?? Zae2/ria ?i ?j e2/rij
   H y(rira) E y(rira)
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The Born-Oppenheimer Approximation

• (1) y(rira) yel(rira) yN(ra)
• (2) Hel(ra ) - (h2/2me)?i?i2 - ?i?? Zae2/ria
• ?i?j e2/rij
• VNN ???b ZaZbe2/rab
• Hel(ra) yel(rira) Eel(ra) yel(rira)
• (3) HN ??(-h2/2ma)??2 U(ra)
• U(ra) Eel(ra) VNN
• HN(ra) yN(ra) E yN(ra)

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Assignment Calculate the ground state energy and
bond length of H2 using the HyperChem with the
6-31G (Hint Born-Oppenheimer Approximation)
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Hydrogen Molecule H2
e  

  e two electrons cannot be
in the same state.
The Pauli principle
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Wave function f(1,2) ja(1)jb(2) c1
ja(2)jb(1) f(2,1) ja(2)jb(1) c1 ja(1)jb(2)
Since two wave functions that correspond to the
same state can differ at most by a constant
factor   f(1,2) c2 f(2,1) ja(1)jb(2)
c1ja(2)jb(1) c2ja(2)jb(1) c2c1ja(1)jb(2)
c1 c2 c2c1 1 Therefore c1
c2 ? 1 According to the Pauli principle, c1
c2 - 1
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The Pauli principle (different version)
the wave function of a system of electrons must
be antisymmetric with respect to interchanging
of any two electrons.
Slater Determinant
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Energy E?

• E?2? dt1 f(1) (TeVeN) f(1) VNN
• ? dt1 dt2 f2(1) e2/r12 f2(2)
• ?i1,2 fii J12 VNN
•  
• To minimize E? under the constraint ? dt f2
  1,
• use Lagranges method
•   L E? - 2 e ? dt1 f2(1) - 1
• dL dE? - 4 e ? dt1 f(1)df(1)
• 4? dt1 df(1)(TeVeN)f(1)
• 4? dt1 dt2 f(1)f(2) e2/r12 f(2)df(1)
• - 4 e ? dt1 f(1)df(1)
• 0 

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TeVeN ? dt2 f(2) e2/r12 f(2) f(1) e
f(1)  
Average Hamiltonian
Hartree-Fock equation
( f J ) f e f f(1) Te(1)VeN(1) one
electron operator J(1) ? dt2 f(2) e2/r12 f(2)
two electron Coulomb operator
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f(1) is the Hamiltonian of electron 1 in the
absence of electron 2 J(1) is the mean
Coulomb repulsion exerted on electron 1 by
2 e is the energy of orbital f.
LCAO-MO f c1y1 c2y2   Multiple y1 from the
left and then integrate c1F11 c2F12 e (c1
S c2)
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Multiple y2 from the left and then integrate
  c1F12 c2F22 e (S c1 c2)   where,
Fij ? dt yi ( f J ) yj Hij ? dt yi J
yj S ? dt y1 y2 (F11 - e) c1 (F12
- S e) c2 0 (F12 - S e) c1 (F22 - e) c2 0
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bonding orbital e1 (F11F12) / (1S)
f1 (y1y2) / ?2(1S)1/2   antibonding
orbital e2 (F11-F12) / (1-S ) f2
(y1-y2) / ?2(1-S)1/2
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Molecular Orbital Configurations of Homo nuclear
Diatomic Molecules H2, Li2, O, He2, etc
Moecule Bond order De/eV H2
? 2.79
H2 1 4.75
He2 ?
1.08 He2 0
0.0009 Li2
1 1.07 Be2
0
0.10 C2 2
6.3 N2
? 8.85
N2 3
9.91 O2 2?
6.78 O2
2 5.21
The more the Bond Order is, the stronger the
chemical bond is.
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Bond Order one-half the difference between the
number of bonding and antibonding electrons
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--------?-------- f1  
--------?-------- f2
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Ey ? dt1dt2 y H y ? dt1dt2 y
(T1V1NT2V2NV12VNN) y ltf1(1)
T1V1Nf1(1)gt ltf2(2) T2V2Nf2(2)gt
ltf1(1) f2(2) V12 f1(1) f2(2)gt
- ltf1(2) f2(1) V12 f1(1) f2(2)gt VNN
?i ltfi(1) T1V1N fi(1)gt ltf1(1)
f2(2) V12 f1(1) f2(2)gt - ltf1(2)
f2(1) V12 f1(1) f2(2)gt VNN
?i1,2 fii J12 - K12 VNN
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Average Hamiltonian
Particle One f(1) J2(1) - K2(1) Particle
Two f(2) J1(2) - K1(2)   f(j) ?
-(h2/2me)?j2 - ?? Za/rja Jj(1) q(1) ? q(1) ? dr2
fj(2) e2/r12 fj(2) Kj(1) q(1) ? fj(1)? dr2
fj(2) e2/r12 q(2)
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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
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Summary
1. At the Hartree-Fock Level there are two
possible Coulomb integrals contributing the
energy between two electrons i and j Coulomb
integrals Jij and exchange integral
Kij   2. For two electrons with different spins,
there is only Coulomb integral Jij 3. For two
electrons with the same spins, both Coulomb and
exchange integrals exist.
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4. Total Hartree-Fock energy consists of the
contributions from one-electron integrals fii
and two-electron Coulomb integrals Jij and
exchange integrals Kij   5. At the
Hartree-Fock Level there are two possible
Coulomb potentials (or operators) between two
electrons i and j Coulomb operator and
exchange operator Jj(i) is the Coulomb
potential (operator) that i feels from j, and
Kj(i) is the exchange potential (operator) that
that i feels from j.
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6. Fock operator (or, average Hamiltonian)
consists of one-electron operators f(i) and
Coulomb operators Jj(i) and exchange
operators Kj(i)
?
? ?
? ?
?  
? ?
? ? ?
? ?
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Fock matrix for an electron 1? with spin
down   Fb(1?) f b(1?) ?j Jjb(1?) - Kjb(1?)
?j Jja(1?) j1?,Nb
j1?,Na 
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f(1) ? -(h2/2me)?12 - ?N ZN/r1N Jja(1) ? ? dr2
fja(2) e2/r12 fja(2) Kja(1) q(1) ? fja(1) ? dr2
fja(2) e2/r12 q(2)
i1,Na j1,Nb
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fjj ? fjja ? ltfja f fjagt Jij ? Jijaa ?
ltfaj(2) Jia(1) faj(2)gt Kij ? Kijaa ? ltfaj(2)
Kia(1) faj(2)gt Jij ? Jijab ? ltfbj(2) Jia(1)
fbj(2)gt
  F(1) f (1) ?j1,n/2 2Jj(1) - Kj(1)
  Energy 2 ?j1,n/2 fjj ?i1,n/2
?j1,n/2 ( 2Jij - Kij ) VNN
Close subshell case ( Na Nb n/2 )
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Hartree-Fock Method
1. Many-Body Wave Function is approximated by
Slater Determinant 2. Hartree-Fock Equation F
fi ei fi   F Fock operator fi the i-th
Hartree-Fock orbital ei the energy of the i-th
Hartree-Fock orbital
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3. Roothaan Method (introduction of Basis
functions) fi ?k cki yk LCAO-MO  
yk is a set of atomic orbitals (or basis
functions) 4. Hartree-Fock-Roothaan equation
?j ( Fij - ei Sij ) cji 0   Fij ? lt ?i F
?j gt Sij ? lt ?i ?j gt 5. Solve the
Hartree-Fock-Roothaan equation
self-consistently
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The Condon-Slater Rules
ltfa(1)fb(2)fc(3)...fd(n) f(1)
fe(1)ff(2)fg(3)...fh(n)gt ltfa(1) f(1) fe(1)gt
lt fb(2)fc(3)...fd(n) ff(2)fg(3)...fh(n)gt
ltfa(1) f(1) fe(1)gt if bf, cg, ...,
dh 0, otherwise ltfa(1)fb(2)fc(3)...f
d(n) V12 fe(1)ff(2)fg(3)...fh(n)gt ltfa(1)
fb(2) V12 fe(1) ff(2)gt lt fc(3)...fd(n)
fg(3)...fh(n)gt ltfa(1) fb(2) V12 fe(1)
ff(2)gt if cg, ..., dh 0, otherwise
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------- the lowest
unoccupied molecular orbital ? -------
the highest occupied molecular orbital ?
-------
-------
LUMO
HOMO
Koopmans 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.
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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
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Basis Set ?i ?p cip ?p
Gaussian type functions gijk N xi yj zk
exp(-ar2) (primitive Gaussian function) ?p ?u
dup gu (contracted Gaussian-type function,
CGTF) u ijk p nlm
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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
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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 6-31G, 6-31G,
6-31G and 6-31G basis sets. Double-zeta
(DZ) basis set two STO for each AO
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6-31G for a carbon atom (10s4p) ? 3s2p
1s 2s 2pi (ix,y,z) 6GTFs
3GTFs 1GTF 3GTFs 1GTF
1CGTF 1CGTF 1CGTF 1CGTF 1CGTF
(s) (s) (s) (p)
(p)
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Minimal basis set One STO for
each inner-shell and
valence-shell AO of each atom
example C2H2 (2S1P/1S)
C 1S, 2S, 2Px,2Py,2Pz
H 1S
total 12 STOs as Basis set Double-Zeta (DZ)
basis set two STOs for each and
valence-shell
AO of each atom example C2H2
(4S2P/2S) C two
1S, two 2S,
two 2Px, two 2Py,two 2Pz
H two 1S (STOs)
total 24 STOs as Basis set
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Split -Valence (SV) basis set Two
STOs for each inner-shell and valence-shell AO
One STO for each inner-shell
AO Double-zeta plus polarization set(DZP, or
DZP) Additional STO w/l quantum
number larger
than the lmax of the valence - shell
? ( 2Px, 2Py ,2Pz ) to H
? Five 3d Aos to Li - Ne , Na -Ar
? C2H5 O Si H3
(6s4p1d/4s2p1d/2s1p)
Si C,O H
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Assignment Calculate the structure, ground state
energy, molecular orbital energies, and
vibrational modes and frequencies of a water
molecule using Hartree-Fock method with 3-21G
basis set. (due 30/10)
97
Ab Initio Molecular Orbital Calculation H2O
(using HyperChem)
1. L-Click on (click on left button of Mouse)
Startup, and select and L-Click on
Program/Hyperchem. 2. Select Build and turn
on Explicit Hydrogens. 3. Select Display and
make sure that Show Hydrogens is on L-Click
on Rendering and double L-Click Spheres. 4.
Double L-Click on Draw tool box and double
L-Click on O. 5. Move the cursor to the
workspace, and L-Click release. 6. L-Click on
Magnify/Shrink tool box, move the cursor to the
workspace L-press and move the cursor
inward to reduce the size of oxygen atom. 7.
Double L-Click on Draw tool box, and double
L-Click on H Move the cursor close to
oxygen atom and L-Click release. A hydrogen
atom appears. Draw second hydrogen atom using
the same procedure.
98
8. L-Click on Setup select Ab Initio
double L-Click on 3-21G then L-Click on
Option, select UHF, and set Charge to 0 and
Multiplicity to 1.    9. L-Click
Compute, and select Geometry Optimization,
and L-Click on OK repeat the step till
ConvYES appears in the bottom bar. Record
the energy. 10.L-Click Compute and L-Click
Orbitals select a energy level, record
the energy of each molecular orbitals (MO), and
L-Click OK to observe the contour plots of
the orbitals. 11.L-Click Compute and select
Vibrations. 12.Make sure that
Rendering/Sphere is on L-Click Compute and
select Vibrational Spectrum. Note that
frequencies of different vibrational
modes. 13.Turn on Animate vibrations, select
one of the three modes, and L-Click OK.
Water molecule begins to vibrate. To suspend the
animation, L-Click on Cancel.
99
The Hartree-Fock treatment of H2
100
The Valence-Bond Treatment of H2
f1 ?1(1) ?2(2) f2 ?1(2) ?2(1) ? c1 f1
c2 f2   H11 - W H12 - S W H21 - S W H22 -
W   H11 H22 lt?1(1) ?2(2)H?1(1) ?2(2)gt H12
H21 lt?1(1) ?2(2)H?1(2) ?2(1)gt S lt?1(1)
?2(2)?1(2) ?2(1)gt S2 The Heitler-London
ground-state wave function ?1(1) ?2(2) ?1(2)
?2(1)/?2(1S)1/2 a(1)b(2)-a(2)b(1)/?2
0
101
Comparison of the HF and VB Treatments
HF LCAO-MO wave function for H2 ?1(1)
?2(1) ?1(2) ?2(2) ?1(1) ?1(2) ?1(1)
?2(2) ?2(1) ?1(2) ?2(1) ?2(2) H
- H H H H
H H H - VB wave function for
H2   ?1(1) ?2(2) ?2(1) ?1(2)   H
H H H
102
At large distance, the system becomes
H ............
H MO 50 H ............ H 50
H............ H- VB 100 H
............ H The VB is computationally
expensive and requires chemical intuition in
implementation.
The Generalized valence-bond (GVB) method is
a variational method, and thus computationally
feasible. (William A. Goddard III)
103
The Heitler-London ground-state wave function
104
Assignment 8.4, 8.10, 8.12b, 8.40, 10.5,
10.6, 10.7, 10.8, 11.37, 13.37
105
Electron Correlation
Human Repulsive Correlation
106
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
107
H - (h2/2me)?12 - 2e2/r1 - (h2/2me)?22 - 2e2/r2
e2/r12 H10
H20
H
108
H0 H10 H20 y(0)(1,2) F1(1) F2(2) H10 F1(1)
E1 F1(1) H20 F2(1) E2 F2(1) E1 -2e2/n12a0
n1 1, 2, 3, ... E2 -2e2/n22a0 n2 1, 2, 3,
...
Ground state wave function
y(0)(1,2) (1/p1/2)(2/a0)3/2exp(-2r1/a0)
(1/p1/2)(2/a0)3/2exp(-2r1/a0) E(0) -
4e2/a0   E(1) lty(0)(1,2) H y(0)(1,2)gt
5e2/4a0 E ? E(0) E(1) -108.8 34.0
-74.8 (eV) compared with exp. -79.0 eV
109
Nondegenerate Perturbation Theory (for
Non-Degenerate Energy Levels)
H H0 H H0yn(0) En(0) yn(0) yn(0) is an
eigenstate for unperturbed system H is small
compared with H0
110
Introducing a parameter l
H(l) H0 lH H(l) yn(l) En(l) yn(l) yn(l)
yn(0) l yn(1) l2 yn(2) ... lk yn(k)
... En(l) En(0) l En(1) l2 En(2) ... lk
En(k) ...
l 1, the original Hamiltonian
yn yn(0) yn(1) yn(2) ... yn(k) ... En
En(0) En(1) En(2) ... En(k) ...
Where, lt yn(0) yn(j) gt 0, j1,2,...,k,...
111

• H0yn(0) En(0) yn(0)
• solving for En(0), yn(0)

• H0yn(1) H yn(0) En(0) yn(1) En(1)yn(0)
• solving for En(1), yn(1)

H0yn(2) H yn(1) En(0) yn(2) En(1)yn(1)
En(2)yn(0) ? solving for En(2),yn(2)
112
  Multiplied ym(0) from the left and
integrate, ltym(0) H0 yn(1) gt lt ym(0) H'
yn(0) gt lt ym(0)yn(1) gtEn(0) En(1) ?mn
ltym(0)yn(1) gt Em(0)- En(0) lt ym(0) H'
yn(0) gt En(1) ?mn
The first order
For m n,
En(1) lt yn(0) H' yn(0) gt Eq.(1)
For m ? n, ltym(0)yn(1) gt lt ym(0) H' yn(0)
gt / En(0)- Em(0) If we expand yn(1) ? cnm
ym(0), cnm lt ym(0) H' yn(0) gt / En(0)-
Em(0) for m ? n cnn 0.
yn(1) ?m lt ym(0) H' yn(0) gt / En(0)-
Em(0) ym(0) Eq.(2)
113
The second order
ltym(0)H0yn(2) gt lt ym(0)H'yn(1) gt lt
ym(0)yn(2) gtEn(0) lt ym(0)yn(1) gtEn(1)
En(1) ?mn   Set m n, we have
En(2) ?m ? n ltym(0) H' yn(0) gt2 /
En(0)- Em(0) Eq.(3)
114
Discussion (Text Book page 522-527)
a. Eq.(2) shows that the effect of the
perturbation on the wave function yn(0) is to
mix in contributions from the other zero-th
order states ym(0) m?n. Because of the factor
1/(En(0)-Em(0)), the most important
contributions to the yn(1) come from the states
nearest in energy to state n. b. To evaluate the
first-order correction in energy, we need only
to evaluate a single integral Hnn to evaluate
the second-order energy correction, we must
evalute the matrix elements H between the n-th
and all other states m. c. The summation in
Eq.(2), (3) is over all the states, not the
energy levels.
115
Moller-Plesset (MP) Perturbation Theory The MP
unperturbed Hamiltonian H0 H0 ?m
F(m) where F(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, MP4
116
Example One Consider the one-particle,
one-dimensional system with potential-energy
function   V b for L/4 lt x lt 3L/4, V 0 for
0 lt x ? L/4 3L/4 ? x lt L and V ? elsewhere.
Assume that the magnitude of b is small, and can
be treated as a perturbation. Find the
first-order energy correction for the ground and
first excited states. The unperturbed wave
functions of the ground and first excited states
are ?1 (2/L)1/2 sin(?x/L) and ?2 (2/L)1/2
sin(2?x/L), respectively.
117
Example Two As the first step of the
Moller-Plesset perturbation theory, Hartree-Fock
method gives the zeroth-order energy. Is the
above statement correct?
Example Three Show that, for any perturbation
H, E1(0) E1(1) ? E1 where E1(0) and E1(1)
are the zero-th order energy and the first order
energy correction, and E1 is the ground state
energy of the full Hamiltonian H0 H.
Example Four Calculate the bond orders of Li2
and Li2.
118
Perturbation Theory for a Degenerate Energy Level
Hydrogen Atom n3 3s, 3px
, 3py , 3pz , 3d1-5 n2
2s, 2px , 2py , 2pz   n1 1s
B / ?
H H0 H H0yn(0) Ed(0) yn(0)
n1,2,...,d H is small compared with H0
119
(1)Apply the results of nondegenerate
perturbation theory
cnm lt ym(0) H' yn(0) gt / En(0)- Em(0) ?
? for 1 ? m, n ? d   WRONG ! something very
different !
(2) What happened ?
c1 y1(0) c2 y2(0) ... cd yd(0) is an
eigenstate for H0 There are infinite number of
such states that are degenerate. 
120
When H is switched on, these states are no
longer degenerate, and nondegenerate eigenstates
of H0 H appear ! Therefore, even for zero-th
order of eigenstates, there are sudden changes !
(3) Introducing a parameter l
H(l) H0 lH H(l) yn(l) En(l) yn(l) l 1,
the original Hamiltonian
yn(l) fn(0) l yn(1) l2 yn(2) ... lk
yn(k) ... En(l) Ed(0) l En(1) l2 En(2)
... lk En(k) ... fn(0) ?k ck yk(0)
121

• H0yn(1) H fn(0) Ed(0) yn(1) En(1)fn(0)
• solving for En(1), fn(0) , yn(1)
• Multiplied ym(0) from the left and integrate,
• ltym(0) H0 yn(1) gt lt ym(0) H' fn(0)
  gt
• lt ym(0)yn(1) gtEd(0) En(1) ltym(0) fn(0) gt
  
• ltym(0)yn(1) gt Em(0)- Ed(0) lt ym(0) H'
  fn(0) gt En(1) lt ym(0) fn(0) gt
• For 1 ? m ? d,
• ?n lt ym(0) H' yn(0) gt - Em(1)?mn cn 0
• Em(1) lt fm(0) H' fm(0) gt

Assignment 2 9.2, 9.4a, 9.9, 9.18, 9.24
122
Configuration Interaction (CI)


123
Single Electron Excitation or Singly Excited
124
Double Electrons Excitation or Doubly Excited
125
Singly Excited Configuration Interaction (CIS)
Changes only the excited states

126
Doubly Excited CI (CID) Changes ground
excited states

Singly Doubly Excited CI (CISD) Most Used
CI Method
127
Full CI (FCI) Changes ground excited states


...
128
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

129
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

130
Complete Active Space SCF (CASSCF)
Active space
All possible configurations
131
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 lt?i ?i2 ?i gt- ?? ? dr
Zae2?(r) / r1a (1/2) ? ? dr1 dr2
e2/r12 Exc?(r) Kohn-Sham Equation FKS
yi ei yi FKS ? - (h2/2me)?i?i2 - ?? Zae2 / r1a
?j Jj Vxc Vxc ? dExc?(r) / d?(r)
132
Ground State Excited State CPU Time
Correlation Geometry Size Consistent

(CHNH,6-31G) HFSCF ?
? 1 0
OK ? DFT
? ?
1 ?
? CIS ?
? lt10
OK ? CISD
? ?
17 80-90 ?
?
(20
electrons) CISDTQ ? ?
very large 98-99 ?
? MP2 ?
? 1.5
85-95 ? ?

(DZP) MP4
? ?
5.8 gt90 ?
? CCD ? ?
large gt90
? ? CCSDT ?
? very large
100 ? ?
133
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).
134
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
135
Semiempirical Molecular Orbital Calculation
136
LCAO-MO fi ?r cri yr   ?s ( Heffrs - ei
Srs ) csi 0   Heffrs ? lt ?r Heff ?s
gt Srs ? lt ?r ?s gt

• Parametrization
• Heffrr ? lt ?r Heff ?r gt
• minus the valence-state
  ionization
• potential (VISP)

137
Atomic Orbital Energy
VISP --------------- e5 -e5 --------------- e
4 -e4 --------------- e3 -e3 --------------
- e2 -e2 --------------- e1 -e1   Heffrs
½ K (Heffrr Heffss) Srs K 1?3
138
CNDO, INDO, NDDO (Pople and co-workers) Hamiltoni
an with effective potentials Hval ?i -(h2/2m)
?i2 Veff(i) ?i?jgti e2 / rij
two-electron integral (rstu) lt?r(1) ?t(2)
1/r12 ?s(1) ?u(2)gt   CNDO complete neglect of
differential overlap (rstu) ?rs ?tu (rrtt) ?
?rs ?tu ?rt
139
INDO intermediate neglect of differential
overlap (rstu) ?rs ?tu (rrtt) when r, s, t
u not on same atom (rstu) ? 0 when r, s, t and
u are on the same atom. NDDO neglect of
diatomic differential overlap (rstu) 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.
140
PRDDO H ?i -(h2/2m) ?i2 Veff(i) ?i?jgti
e2 / rij Basis set the minimum basis set
(STO-3G)   PRDDO partial retention of diatomic
differential overlap (rstu) 0 if r and s
(and t and u) are different basis functions.
141
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 MINDO, MNDO, AM1 PM3   based on INDO
NDDO reproduce the binding energy
142
Semiempirical M.O. Method
Fock Matrix
Key How to approximate ?
MNDO-PM3
(using NDDO)
143
Where,
the ionization potential
One centre integrals (given)
Core-electron attraction (given)
characteristic of monopole, dipole, quadrupole
charge separations
144
Molecular Mechanics (MM) Method F Ma F
Force Field
145
Molecular Mechanics Force Field

• Bond Stretching Term
• Bond Angle Term
• Torsional Term
• Non-Bonding Terms Electrostatic Interaction
  van der Waals Interaction

146
Bond Stretching Potential Eb 1/2 kb
(Dl)2 where, kb stretch force
constant Dl difference between equilibrium
actual bond length
Two-body interaction
147
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
148
Periodic Torsional Barrier Potential Et
(V/2) (1 cosn? ) where, V rotational
barrier t torsion angle n
rotational degeneracy
Four-body interaction
149
Non-bonding interaction van der Waals
interaction for pairs of non-bonded atoms
Coulomb potential for all pairs of charged
atoms
150
MM Force Field Types

• MM2 Small molecules
• AMBER Polymers
• CHAMM Polymers
• BIO Polymers
• OPLS Solvent Effects

151
Summary
Hamiltonian H ??(-h2/2ma)??2 - (h2/2me)?i?i2
???? ZaZbe2/rab - ?i ?? Zae2/ria ?i ?j e2/rij
The variation theorem
Consider a system whose Hamiltonian operator H is
time independent and whose lowest-energy
eigenvalue is E1. If f is any well-behaved
function that satisfies the boundary conditions
of the problem, then ? f H f dt / ? f f
dt gt E1
 Variational Method
(1) Construct a wave function ?(c1,c2,???,cm) (2)
Calculate the energy of ? E? ?
E?(c1,c2,???,cm) (3) Choose cj (i1,2,???,m)
so that E? is minimum
152
Extension of Variation Method
For a wave function f which is orthogonal to the
ground state wave function y1, i.e. ?dt fy1
0 Ef ?dt fHf / ?dt ff gt E2 the first
excited state energy
The Pauli principle
two electrons cannot be in the same state
the wave function of a system of electrons must
be antisymmetric with respect to interchanging of
any two electrons.
153
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
LCAO-MO f c1y1 c2y2
Molecule Bond order De/eV
H2 1/2 2.79
H2 1
4.75 He2 1/2
1.08 He2
0 0.0009 Li2
1 1.07
Be2 0
0.10 C2 2
6.3 N2
1/2 8.85
N2 3
9.91 O2
2 5.21
Express Hartree-Fock energy in terms of fi, Jij
Kij
154
Basis set of GTFs   STO-3G, 3-21G, 4-31G, 6-31G,
6-31G, 6-31G ----------------------------------
--------------------------------------------------
-? complexity
accuracy
Gaussian 98 Input file
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
Comparison of the HF and VB Treatments
Electron Correlation
155
Beyond the Hartree-Fock Configuration
Interaction (CI) Perturbation theory Coupled
Cluster Method Density functional theory
En(1) lt yn(0) H' yn(0) gt
Moller-Plesset (MP) Perturbation Theory The MP
unperturbed Hamiltonian H0 H0 ?m
F(m) where F(m) is the Fock operator for
electron m. And thus, the perturbation H
  H H - H0  
156
Ground State Excited State CPU Time
Correlation Geometry Size Consistent

(CH3NH2,6-31G) HFSCF ?
? 1 0
OK ? DFT
? ?
1 ?
? CIS ?
? lt10
OK ? CISD
? ?
17 80-90 ?
?
(20
electrons) CISDTQ ? ?
very large 98-99 ?
? MP2 ?
? 1.5
85-95 ? ?

(DZP) MP4
? ?
5.8 gt90 ?
? CCD ? ?
large gt90
? ? CCSDT ?
? very large
100 ? ?