Post on 20-Dec-2015
Computational Chemistry
G. H. CHENDepartment of Chemistry
University of Hong Kong
In 1929, Dirac declared, “The underlying physical laws necessary for the mathematical theory of ...thewhole 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.”
Beginning of Computational Chemistry
Dirac
Computational Chemistry
Quantum Chemistry
Molecular Mechanics
Bioinformatics
Create & Analyse Bio-information
SchrÖdinger Equation
F = M a
Mulliken,1966 Fukui, 1981 Hoffmann, 1981
Pople, 1998 Kohn, 1998
Nobel Prizes for Computational Chemsitry
Computational Chemistry Industry
Company Software
Gaussian Inc. Gaussian 94, Gaussian 98Schrödinger Inc. Jaguar Wavefunction SpartanQ-Chem Q-ChemAccelrys InsightII, Cerius2
HyperCube HyperChemCelera Genomics (Dr. Craig Venter, formal Prof., SUNY, Baffalo; 98-01)
Applications: material discovery, drug design & research
R&D in Chemical & Pharmaceutical industries in 2000: US$ 80 billionBioinformatics: Total Sales in 2001 US$ 225 million
Project Sales in 2006 US$ 1.7 billion
LODESTAR v1.02--Localized Density Matrix: STAR performer
http://yangtze.hku.hk
Software Development at HKU
Quantum Chemistry Methods
• Ab initio molecular orbital methods
• Semiempirical molecular orbital methods
• Density functional method
H E
SchrÖdinger Equation
HamiltonianH = (h2/2m
h2/2me)ii2
+ ZZeri e2/ri
ije2/rij
Wavefunction
Energy
Vitamin CC60
Cytochrome c
heme
OH + D2 --> HOD + D
energy
C60 and Superconductor
Applications: Magnet, Magnetic train, Power transportation
What is superconductor? Electrical Current flows for ever !
Crystal Structure of C60 solid
Crystal Structure of K3C60
K3C60 is a Superconductor (Tc = 19K)
Erwin & Pickett, Science, 1991 GH Chen, Ph.D. Thesis, Caltech (1992)
Vibration Spectrum of K3C60
Effective Attraction !
The mechanism of superconductivity in K3C60 was discovered using com-putational chemistry methodsVarma et. al., 1991; Schluter et. al., 1992; Dresselhaus et. al., 1992;Chen & Goddard, 1992
Carbon Nanotubes (Ijima, 1991)
STM Image of Carbon Nanotubes (Wildoer et. al., 1998)
Calculated STM Image of a Carbon Nanotube (Rubio, 1999)
Computer Simulations (Saito, Dresselhaus, Louie et. al., 1992)
Carbon Nanotubes (n,m):Conductor, if n-m = 3I I=0,±1,±2,±3,…;orSemiconductor, if n-m 3I
Metallic Carbon Nanotubes: Conducting WiresSemiconducting Nanotubes: Transistors
Molecular-scale circuits ! 1 nm transistor!
0.13 µm transistor!
30 nm transistor!
Wildoer, Venema, Rinzler, Smalley, Dekker, Nature 391, 59 (1998)
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
Science 9th November, 2001Logic gates (and circuits) with carbon nanotuce transistor
Science 7th July, 2000Carbon nanotube-Based nonvolatile RAM for molecular computing
Nanoelectromechanical Systems (NEMS)
K.E. Drexler, Nanosystems: Molecular Machinery, Manufacturing and Computation (Wiley, New York, 1992).
Large Gear Drives Small Gear
G. Hong et. al., 1999
Nano-oscillators
Zhao, Ma, Chen & Jiang, Phys. Rev. Lett. 2003
Nanoscopic Electromechanical Device (NEMS)
0 500 1000 1500 2000
-30
-20
-10
0
10
20
30
40
Rel
ativ
e di
stan
ce (
Ang
stro
m)
Time (ps)
(5,0)@(14,0)55A @ 70A, 500K
Hibernation Awakening
Oscillation
Quantum mechanical investigation of the field emission from Quantum mechanical investigation of the field emission from the tips of carbon nanotubesthe tips of carbon nanotubes
Zettl, PRL 2001Zheng, Chen, Li, Deng & Xu, Phys. Rev. Lett. 2004
Computer-Aided Drug Design
GENOMICS
Human Genome Project
ALDOSE REDUCTASE
O
HO OH
HO OH
HO
glucose
HO
HO OH
HO OH
HO
sorbitol
Aldose Reductase
NADPH NADP
Diabetes DiabeticComplications
Glucose Sorbitol
Design of Aldose Reductase Inhibitors
Aldose Reductase
Inhibitor
Hu, Chen & Chau, J. Mol. Graph. Mod. 24 (2006)
Database of Function Group Ionic Potential(eV) Electronic Affinity(eV) Volume(cm**3/mol)2-Thienyl 9.38414881 -0.026661055 61.9533-Cl-C6H4 8.749035019 -0.706309133 73.0334-C6H5-C6H4 8.407247908 -1.270557034 133.8164-Cl-C6H4 8.746870744 -0.76855741 75.6984-NO2-C6H4 9.217321674 -0.160533773 81.5724-OCH3-C6H4 8.242902306 -1.182299665 88.289C2H3 8.260300105 -1.561982616 28.975C2H5 7.780243105 -2.573416988 40.829C3H7 6.81804497 -2.444365755 45.863C4H9 6.670585636 -2.356589514 60.671C5H11 6.598274428 -2.296956191 84.137C6H5-CH2 6.902499381 -1.683414531 88.078C6H5-CH2CH2 5.575403173 -2.024168064 86.479C6H5 12.8343891 -1.226684615 49.929CF3 8.758052916 -0.370141762 31.487CH3 8.935004666 -2.507617614 29.582Cl 14.52590929 2.123404035 23.704H 13.55771463 -2.062392576 8.547SCH3 10.38525408 0.363370563 38.7083-4-Methylenedioxo 6.761934134 1.946070446 36.34Br 13.24832638 1.933493026 23.893CN 16.12363076 2.202225384 24.052F 11.71218241 -2.221199855 10.315NH2 19.92831772 -0.3938852 22.034NHAC 8.159304652 -1.024625532 64.609NO2 9.624434885 0.930074828 27.083OCF3 10.21475618 0.064048786 31.575OCH2CH2CH3 6.173209263 -0.238732972 54.174OCH2CHCH2 7.543726177 0.023658445 55.761OCH3 8.206405401 -0.98717652 25.579OH 15.02158915 -1.515044158 15.942SO2CH3 8.676981042 1.277315225 50.894COOH 7.70352688 -0.44795408 28.721CH2-COOH 9.59715824 -0.29063744 46.036CH2-C6H4-OMe 6.1379112 -1.86670608 90.357C6H4-34OH 8.37323712 -1.09197392 88.207N-2CH3 8.9735112 -2.00731648 45.5332CH2-C6H4-OH 5.72024976 -2.0214088 107.122C6H5-OH 9.89372528 -1.1650712 78.805CH2-C10H8 6.89898896 -1.66132432 121.142CH2-C6H4-OH 6.30185104 -1.8973088 87.058br-C6H4 8.711011456 -0.72427344 90.4682C6H5-CH 6.52476048 -1.38209456 145.166CH2-C6H3-34OH 6.49713888 -1.72708304 90.4693CH2-C6H4-OH 6.3709472 -2.19444704 116.4473CH2-O-pH 6.93279584 -2.16905312 97.131COO-t-Bu 11.29505024 1.3880432 73.451O-3CH2-OCH2pH 7.26343904 -0.59915888 137.142O-3CH2-OH 6.79831088 -0.59987968 55.06OCO-2CH2-COCH2Me 6.4562192 1.8424872 86.718OCO-2CH2-COO-CHMePh 6.5978496 -6.5978496 154.362OCO-3CH2-CH2Me 6.60117072 1.32051376 111.819OCO-N-Hex 6.64197888 1.33964352 140.502
Database for Functional Groups
2.5 3.0 3.5 4.0 4.5 5.0
2.5
3.0
3.5
4.0
4.5
5.0
NH
NMe
NH
HN
O
O
O
R1
R2
R3
R4
Sup
ervi
se V
alue
s
Exp. Values (logIC50 nm)[Three Hidden Neurons]
NH
NMe
NH
HN
O
O
OC6H5
C6H5
NO2
NH
NMe
NH
HN
O
O
OC6H5
C6H5
F
LogIC50: 0.6382,1.0 LogIC50: 0.6861,0.88
Prediction: Drug Leads
Structure-activity-relation
LogIC50: 0.77,1.1
LogIC50: -1.87,4.05
LogIC50: -2.77,4.14 LogIC50: 0.68,0.88
Prediction Results using AutoDock
Hu, Chen & Chau, J. Mol. Graph. Mod. 24 (2006)
Computer-aided drug design
Chemical Synthesis
Screening using in vitro assay
Animal Tests
Clinical Trials
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
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 R&D
Objective:More and more research & development to be performed on computers and Internet instead in the laboratories
Quantum Chemistry
G. H. ChenDepartment of Chemistry
University of Hong Kong
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
Emphasis Hartree-Fock methodConcepts Hands-on experience
Text Book “Quantum Chemistry”, 4th Ed. Ira N. Levine
http://yangtze.hku.hk/lecture/chem3504-3.ppt
Contents 1. Variation Method2. Hartree-Fock Self-Consistent Field Method3. Perturbation Theory4. Semiempirical Methods
The Variation Method
Consider a system whose Hamiltonian operatorH is time independent and whose lowest-energy eigenvalue is E1. If is any normalized, well-
behaved function that satisfies the boundary conditions of the problem, then
* H dE1
The variation theorem
Proof:Expand in the basis set { k}
= k kk
where {k} are coefficients
Hk = Ekk
then* H dk j k
*j Ej kj
= k |k|2 Ek E 1 k |k|
2 = E1
Since is normalized, *dk |k|
2 = 1
i. : trial function is used to evaluate the upper limit of ground state energy E1
ii. = ground state wave function, * H dE1
iii. optimize paramemters in by minimizing * H d * d
Requirements for the trial wave function: i. zero at boundary; ii. smoothness a maximum in the center. Trial wave function: = x (l - x)
Application to a particle in a box of infinite depth
0 l
* H dx = -(h2/82m) (lx-x2) d2(lx-x2)/dx2 dx = h2/(42m) (x2 - lx) dx = h2l3/(242m)
* dx = x2 (l-x)2 dx = l5/30
E = 5h2/(42l2m) h2/(8ml2) = E1
(1) Construct a wave function (c1,c2,,cm)
(2) Calculate the energy of :
E E(c1,c2,,cm)
(3) Choose {cj*} (i=1,2,,m) so that E is minimum
Variational Method
Example: one-dimensional harmonic oscillator Potential: V(x) = (1/2) kx2 = (1/2) m2x2 = 22m2x2
Trial wave function for the ground state:
(x) = exp(-cx2)
* H dx = -(h2/82m) exp(-cx2) d2[exp(-cx2)]/dx2
dx + 22m2 x2 exp(-2cx2) dx = (h2/42m) (c/8)1/2 + 2m2 (/8c3)1/2
* dx = exp(-2cx2) dx = (/2)1/2 c-1/2
E = W = (h2/82m)c + (2/2)m2/c
To minimize W,
0 = dW/dc = h2/82m - (2/2)m2c-2
c = 22m/h
W = (1/2) h
...
E3 3
E2 2
E1 1
Extension of Variation Method
For a wave function which is orthogonal to the ground state wave function 1, i.e.
d *1 = 0
E = d *H/ d * > E2
the first excited state energy
The trial wave function : d *1 = 0 k=1 ak k
d *1 = |a1|2 = 0
E = d *H/ d * = k=2|ak|
2Ek / k=2|ak|2
> k=2|ak|2E2 / k=2|ak|
2 = E2
e
+ +
1
2
c1
1 + c
2
2
W = H d d= (c1
2 H11 + 2c1 c2 H12 + c22 H22 )
/ (c12 + 2c1 c2 S + c2
2 )
W (c12 + 2c1 c2 S + c2
2) = c12 H11 + 2c1 c2 H12 + c2
2 H22
Application to H2+
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
To have nontrivial solution:
H11 - W H12 - S W
H12 - S W H22 - W
For H2
+, H11 = H22; H12 < 0.
Ground State: Eg = W1 = (H11+H12) / (1+S)
= () / 2(1+S)1/2
Excited State: Ee = W2 = (H11-H12) / (1-S)
= () / 2(1-S)1/2
= 0
bonding orbital
Anti-bonding orbital
Results: De = 1.76 eV, Re = 1.32 A
Exact: De = 2.79 eV, Re = 1.06 A
1 eV = 23.0605 kcal / mol
Trial wave function: k3/2 -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
1s 2pInclusion of other atomic orbitals
Further Improvements H -1/2 exp(-r)He+ 23/2 -1/2 exp(-2r)
Optimization of 1s orbitals
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
(a11a22-a12a21) x1 = b1a22-b2a12
(a11a22-a12a21) x2 = b2a11-b1a21
Linear Equations
1. two linear equations for two unknown, x1 and x2
Introducing determinant:
a11 a12
= a11a22-a12a21
a21 a22
a11 a12 b1 a12
x1 =
a21 a22 b2 a22
a11 a12 a11 b1
x2 =
a21 a22 a21 b2
Our case: b1 = b2 = 0, homogeneous
1. trivial solution: x1 = x2 = 0
2. nontrivial solution: a11 a12
= 0 a21 a22
n linear equations for n unknown variables
a11x1 + a12x2 + ... + a1nxn= b1
a21x1 + a22x2 + ... + a2nxn= b2
............................................an1x1 + an2x2 + ... + annxn= bn
a11 a12 ... a1,k-1 b1 a1,k+1 ... a1n
a21 a22 ... a2,k-1 b2 a2,k+1 ... a2n
det(aij) xk= . . ... . . . ... .
an1 an2 ... an,k-1 b2 an,k+1 ... ann
where,
a11 a12 ... a1n
a21 a22 ... a2n
det(aij) = . . ... .
an1 an2 ... ann
a11 a12 ... a1,k-1 b1 a1,k+1 ... a1n
a21 a22 ... a2,k-1 b2 a2,k+1 ... a2n
. . ... . . . ... . an1 an2 ... an,k-1 b2 an,k+1 ... ann
xk =
det(aij)
inhomogeneous case: bk = 0 for at least one k
(a) travial case: xk = 0, k = 1, 2, ... , n
(b) nontravial case: det(aij) = 0
homogeneous case: bk = 0, k = 1, 2, ... , n
For a n-th order determinant, n
det(aij) = alk Clk
l=1
where, Clk is called cofactor
Trial wave function is a variation function which is a combination of n linear independent functions { f1 , f2 , ... fn},
c1f1 + c2f2 + ... + cnfn
n [( Hik - SikW ) ck ] = 0 i=1,2,...,n
k=1
Sik d fi fk
Hik d fi H fk
W dH d
(i) W1 W2 ... Wn are n roots of Eq.(1),
(ii) E1 E2 ... En En+1 ... are energies
of eigenstates; then, W1 E1, W2 E2, ..., Wn En
Linear variational theorem
Molecular Orbital (MO): = c11 + c22
( H11 - W ) c1 + ( H12 - SW ) c2 = 0
S11=1
( H21 - SW ) c1 + ( H22 - W ) c2 = 0
S22=1
Generally : i a set of atomic orbitals, basis set
LCAO-MO = c11 + c22 + ...... + cnn
linear combination of atomic orbitals
n
( Hik - SikW ) ck = 0 i = 1, 2, ......, nk=1
Hik d i* H k Sik d i
*k Skk = 1
Hamiltonian
H = (h2/2mh2/2me)ii
2 + ZZeri e2/ri
ije2/rij
H ri;rri;r
The Born-Oppenheimer Approximation
ri;relri;rNr
el(r )= h2/2me)ii2
ie2/ri
ije2/rij VNN = ZZer
Hel(r) elri;rel(r)elri;r
(3) HN = (h2/2m U(r)
U(r) = el(r) + VNN
HN(r) NrNr
The Born-Oppenheimer Approximation:
Assignment
Calculate the ground state energy and bond length of H2
using the HyperChem with the 6-31G(Hint: Born-Oppenheimer Approximation)
e + +
e
two electrons cannot be in the same state.
Hydrogen Molecule H2
The Pauli principle
Since two wave functions that correspond to the same state can differ at most by a constant factor = c2 abc1ab=c2ab+c2c1ab
c1 = c2 c2c1 = 1Therefore: c1 = c2 = 1According to the Pauli principle, c1 = c2 =1
Wave function:= ab+ c1ab= ab+ c1ab
Wave function f H2 : ! [
!
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
E=2 dTe+VeN) + VNN
+ dde2/r12 | = i=1,2 fii + J12 + VNN
To minimize Eunder the constraint d|use Lagrange’s method: L = E dL = E d
4 dTe+VeN) +4 dde2/r12
d
Energy: E
[ Te+VeN +de2/r12
f + Jf(1) = Te(1)+VeN(1) one electron
operator
J(1) =de2/r12 two electron
Coulomb operator
Average Hamiltonian
Hartree-Fock equation
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; is the energy of orbital LCAO-MO: c11 + c22
Multiple 1 from the left and then integrate :
c1F11 + c2F12 = (c1 + S c2)
Multiple 2 from the left and then integrate :
c1F12 + c2F22 = (S c1 + c2) where,
Fij = di* ( f + J ) j = Hij + di
* J j
S = d1 2
(F11 - ) c1 + (F12 - S ) c2 = 0
(F12 - S ) c1 + (F22 - ) c2 = 0
Secular Equation: F11 - F12 - S F12 - S F22 -
bonding orbital: 1 = (F11+F12) / (1+S)
= () / 2(1+S)1/2
antibonding orbital: 2 = (F11-F12) / (1-S )
= () / 2(1-S)1/2
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.
Bond Order:one-half the difference between the number of bonding and antibonding electrons
---------------- 1
---------------- 2
12 12 = 1/2 [122 1
ddH
dd(T1+V1N+T2+V2N+V12
+VNN)
1 T1+V1N|12 T2+V2N|2 + 12 V12 1212 V12 12 +
VNN
= i i T1+V1N |i+ 12 V12 1212 V12 12 + VNN
= i=1,2 fii + J12 K12 + VNN
Particle One: f(1) + J2(1)
K2(1)Particle Two: f(2) + J1(2)
K1(2)
f(j) h2/2me)j
2 Zrj
Jj(1) drj
* e2/r12j
Kj(1) j drj*
e2/r12
Average Hamiltonian
f(1)+ J2(1) K2(1)1(1)11(1)f(2)+ J1(2) K1(2)2(2)22(2)
F(1) f(1)+ J2(1) K2(1) Fock operator for 1F(2) f(2)+ J1(2) K1(2) Fock operator for 2
Hartree-Fock Equation:
Fock Operator:
1. At the Hartree-Fock Level there are two possible Coulomb integrals contributing the energy betweentwo 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.
Summary
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.
6. Fock operator (or, average Hamiltonian) consists of one-electron operators f(i) and Coulomb operators Jj(i) and exchange operators Kj(i)
Nelectrons spin up and Nelectrons spin down.
Fock matrix for an electron 1 with spin up:
F(1) = f (1) + j [ Jj(1) Kj
(1) ] + j Jj(1)
j=1,N j=1,N
Fock matrix for an electron 1 with spin down: F(1) = f (1) + j [ Jj
(1) Kj(1) ] + j
Jj(1)
j=1,Nj=1,N
f(1) h2/2me)12 N ZNr1N
Jj(1) drj
e2/r12j
Kj(1) j
drj
*e2/r12
Energy = j fjj
+j fjj
+(1/2) i j
( Jij Kij
)
+ (1/2) i j
( Jij Kij
) + i j
Jij
+ VNN
i=1,Nj=1,N
fjjfjj
jf j
JijJij
j(2)Ji
j(2)Kij
Kij
j(2)Ki
j(2)
JijJij
j(2)Ji
j(2) F(1) = f (1) + j=1,n/2 [ 2Jj(1) Kj(1) ] Energy = 2 j=1,n/2 fjj + i=1,n/2 j=1,n/2 ( 2Jij Kij ) +VNN
Close subshell case: ( N= N= n/2 )
1. Many-Body Wave Function is approximated by Slater Determinant
2. Hartree-Fock EquationF 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 cki k LCAO-MO
{k } is a set of atomic orbitals (or basis functions)
4. Hartree-Fock-Roothaan equation j ( Fij - i Sij ) cji = 0
Fij iF j Sij ij
5. Solve the Hartree-Fock-Roothaan equation self-consistently
abcdnf(1) efghnaf(1)
ebcdnfghn= af(1) eif b=f, c=g, ..., d=h; 0, otherwise abcdnV12 |efghnabV12
efcdnghn= abV12 efif c=g, ..., d=h; 0, otherwise
The Condon-Slater Rules
-------the lowest unoccupied molecular orbital -------
the highest occupied molecular orbital ------- -------
The energy required to remove an electron from aclosed-shell atom or molecules is well approximatedby minus the orbital energy of the AO or MO fromwhich the electron is removed.
HOMO
LUMO
Koopman’s Theorem
# HF/6-31G(d) Route section water energy Title
0 1 Molecule Specification O -0.464 0.177 0.0 (in Cartesian coordinatesH -0.464 1.137 0.0H 0.441 -0.143 0.0
Slater-type orbitals (STO) nlm = N rn-1exp(r/a0) Ylm(,)
the orbitalexponent* is used instead of in the textbook
Gaussian type functionsgijk = N xi yj zk exp(-r2)
(primitive Gaussian function)p = u dup gu
(contracted Gaussian-type function, CGTF)u = {ijk} p = {nlm}
Basis Set i = p cip 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 orbital3-21G: 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 rows6-31G**: 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 densityis 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
6-31G for a carbon atom: (10s4p) [3s2p]
1s 2s 2pi (i=x,y,z)
6GTFs 3GTFs 1GTF 3GTFs 1GTF
1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)
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
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(DZ+P, 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
Assignment: Calculate the structure, groundstate 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)
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.
Ab Initio Molecular Orbital Calculation: H2O
(using HyperChem)
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 “Conv=YES” 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”.
The Hartree-Fock treatment of H2
+
e-
+
e-
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 = <1(1) 2(2)|H|1(1) 2(2)>
H12 = H21 = <1(1) 2(2)|H|1(2) 2(1)>
S = <1(1) 2(2)|1(2) 2(1)> [ = S2 ]
The Heitler-London ground-state wave function
{[1(1) 2(2) + 1(2) 2(1)]/2(1+S)1/2} [(1)(2)(2)(1)]/2
= 0
The Valence-Bond Treatment of H2
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
At large distance, the system becomes H ............ HMO: 50% H ............ H 50% H+............ H
VB: 100% H ............ H
The VB is computationally expensive and requireschemical intuition in implementation.
The Generalized valence-bond (GVB) method is avariational method, and thus computationally feasible.(William A. Goddard III)
)1()2(
)2()1(
2
1
f
f2211 fcfc
022
12
21
11
WH
SWH
SWH
WH
22121
21212112
21212211
)1()2()2()1(
)1()2()2()1(
)2()1()2()1(
SS
HHH
HHH
The Heitler-London ground-state wave function
2/)1()2()2()1()1(2/)1()2()2()1( 2121 S
R
R
Assignment 8.4, 8.10, 8.12b, 8.40, 10.5, 10.6, 10.7, 10.8, 11.37, 13.37
Electron Correlation
Human Repulsive Correlation
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-FockConfiguration Interaction (CI)*Perturbation theory*Coupled Cluster MethodDensity functional theory
-e -e r12
r2 r1
+2e
H = - (h2/2me)12 - 2e2/r1 - (h
2/2me)22 - 2e2/r2 + e2/r12
H10 H2
0 H’
H0 = H10 + H2
0
(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, ...
(0)(1,2) = (1/2a0)3/2exp(-2r1/a0) (1/2a0)
3/2exp(-2r1/a0)
E(0) = 4e2/a0
E(1) = <(0)(1,2)| H’ |(0)(1,2)> = 5e2/4a0
E E(0) + E(1) = -108.8 + 34.0 = -74.8 (eV) [compared with exp. -79.0 eV]
Ground state wave function
H = H0 + H’H0n
(0) = En(0)n
(0)
n(0) is an eigenstate for unperturbed system
H’ is small compared with H0
Nondegenerate Perturbation Theory (for Non-Degenerate Energy Levels)
H( = H0 + H’Hn = Ennnn
nn
kn(k)
nnn
nkn
(k)
the original Hamiltonian
Introducing a parameter
nnn
nn
(k)nn
nn
n(k)
Where, < nn
(j) > = 0, j=1,2,...,k,...
Hn = En
n
solving for Enn
HnH’n
= Enn
nn
solving for Enn
HnH’n
= Enn
nn
nn
solving for Enn
Multiplied m
(0) from the left and integrate,<m
(0) Hn(1) > + <m
(0) H'n(0) > = <m
(0)n(1) >En
Enmn
<m(0)n
(1) > [EmEn
+ <m(0) H'n
(0) > = Enmn
For m = n,
For m n, <m(0)n
(1) > = <m(0) H'n
(0) > /
[EnEm
If we expand n(1) = cnmm
(0),
cnm = <m(0) H'n
(0) > / [EnEm
for m n;
cnn = 0.
n(1) = m <m
(0) H'n(0) > / [En
Emm
(0) Eq.(2)
The first order:
En<n
(0) H'n(0) > Eq.(1)
The second order:
<m(0)Hn
(2) > + <m(0)H'n
(1) > = <m(0)n
(2)
>En<m
(0)n(1) >En
Enmn
Set m = n, we have
En= m n |m
(0) H'n(0) >|2 / [En
Emq.(3)
a. Eq.(2) shows that the effect of the perturbationon the wave function n
(0) is to mix in
contributions from the other zero-th order states m
(0) mn. Because of the factor 1/(En(0)-Em
(0)),
the most important contributions to the n(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 H’nn;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.
Discussion: (Text Book: page 522-527)
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
Example One:Consider the one-particle, one-dimensional systemwith potential-energy function V = b for L/4 < x < 3L/4,V = 0 for 0 < x L/4 & 3L/4 x < Land 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 groundand 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(2x/L),
respectively.
Example Two:As the first step of the Moller-Plesset perturbation theory, Hartree-Fock method gives the zeroth-orderenergy. 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
+.
Perturbation Theory for a Degenerate Energy Level
B /
Hydrogen Atom
n=3 3s, 3px , 3py , 3pz , 3d1-5
n=2 2s, 2px , 2py , 2pz
n=1 1s
H = H0 + H’H0n
(0) = Ed(0)n
(0)
n=1,2,...,dH’ is small compared with H0
cnm = <m(0) H'n
(0) > / [EnEm
for 1 m, n d
WRONG ! something very different !
(1)Apply the results of nondegenerate perturbation theory
(2) What happened ?
c11(0) + c22
(0) + ... + cdd(0) is an eigenstate for H0
There are infinite number of such states that are degenerate.
When H’ is switched on, these states are no longerdegenerate, and nondegenerate eigenstates of H0 + H’ appear !
Therefore, even for zero-th order of eigenstates, there are sudden changes !
(3) Introducing a parameter H( = H0 + H’Hn = Ennthe original Hamiltoniannn
nn
kn
(k)nd
nn
kn
(k)n
kck k(0)
HnH’n
= Edn
nn
solving for Enn
n
Multiplied m(0) from the left and integrate,
<m(0) Hn
(1) > + <m(0) H'n
(0) >
= <m(0)n
(1) >EdEn
<m(0)n
(0) >
<m(0)n
(1) > [EmEd
+ <m(0)
H'n(0) > = En
<m(0)n
(0) >
For 1 m d, n<m
(0) H'n(0) > Em
mn] cn
Em
<m(0) H'm
(0) >Assignment 2: 9.2, 9.4a, 9.9, 9.18, 9.24
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
++
+ ...
= eT(0)
(0): Hartree-Fock ground state wave function: Ground state wave functionT = T1 + T2 + T3 + T4 + T5 + …Tn : n electron excitation operator
Coupled-Cluster Method
=T1
CCD = eT2(0)
(0): Hartree-Fock ground state wave functionCCD: Ground state wave functionT2 : two electron excitation operator
Coupled-Cluster Doubles (CCD) Method
=T2
Complete Active Space SCF (CASSCF)
Active space
All possible configurations
Density-Functional Theory (DFT)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)E0 = h2/2me)i <i |i
2 |i > dr e2(r) /
r1 dr1 dr2 e2/r12 + Exc[(r)]
Kohn-Sham Equation Ground State: Phys. Rev. 140, A1133 (1965)
FKS i = i i
FKS h2/2me)ii2 e2 / r1jJj + Vxc
Vxc Exc[(r)] / (r)
A popular exchange-correlation functional Exc[(r)]: B3LYP
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
180 small- or medium-size organic molecules:
1. C.L. Yaws, Chemical Properties Handbook, (McGraw-Hill, New York, 1999)2. D.R. Lide, CRC Handbook of Chemistry and Physics, 3rd ed. (CRC Press, Boca Raton, FL, 2000)3. J.B . Pedley, R.D. Naylor, S.P. Kirby, Thermochemical data of organic compunds, 2nd ed. (Chapman and Hall, New York, 1986)
Differences of heat of formation in three referencesfor same compound are less than 1 kcal/mol; and error bars are all less than 1kcal/mol
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
Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003)
Ground State Excited State CPU Time Correlation Geometry Size Consistent (CHNH,6-31G*)HFSCF 1 0 OK
DFT ~1
CIS <10 OK
CISD 17 80-90% (20 electrons)CISDTQ very large 98-99%
MP2 1.5 85-95% (DZ+P)MP4 5.8 >90% CCD large >90%
CCSDT very large ~100%
Relativistic Effects
Speed of 1s electron: Zc / 137
Heavy elements have large Z, thus relativistic effects areimportant.
Dirac Equation:Relativistic Hartree-Fock w/ Dirac-Fock operator; orRelativistic Kohn-Sham calculation; orRelativistic effective core potential (ECP).
(1) Neglect or incomplete treatment of electron correlation
(2) Incompleteness of the Basis set
(3) Relativistic effects
(4) Deviation from the Born-Oppenheimer approximation
Four Sources of error in ab initio Calculation
Semiempirical Molecular Orbital Calculation
Extended Huckel MO Method (Wolfsberg, Helmholz, Hoffman)
Independent electron approximation
Schrodinger equation for electron i
Hval = i Heff(i)
Heff(i) = -(h2/2m) i2 + Veff(i)
Heff(i) i = i i
LCAO-MO: i = r cri r
s ( Heff
rs - i Srs ) csi = 0
Heffrs rHeff s Srs
rs Parametrization: Heff
rr rHeff r minus the valence-state ionization potential (VISP)
Atomic Orbital Energy VISP--------------- e5 -e5
--------------- e4 -e4
--------------- e3 -e3
--------------- e2 -e2
--------------- e1 -e1
Heff
rs = ½ K (Heffrr + Heff
ss) Srs K:
13
CNDO, INDO, NDDO(Pople and co-workers)
Hamiltonian with effective potentialsHval = i [ -(h
2/2m) i2 + Veff(i) ] + ij>i e
2 / 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
INDO: intermediate neglect of differential overlap(rs|tu) = rs tu (rr|tt) when r, s, t & u not on same atom;
(rs|tu) 0 when r, s, t and u are 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 overallresults fit well with the results of minimal basis abinitio Hartree-Fock calculation.
CNDO/S, INDO/S are parametrized to predict optical spectra.
PRDDOH = i [ -(h
2/2m) i2 + Veff(i) ] + ij>i e
2 / rij
Basis set: the minimum basis set (STO-3G) PRDDO: partial retention of diatomic differential overlap
(rs|tu) = 0 if r and s (and t and u) are different basis functions.
MINDO, MNDO, AM1, PM3(Dewar and co-workers, University of Texas, Austin) MINDO: modified INDOMNDO: modified neglect of diatomic overlap AM1: Austin Model 1PM3: MNDO parametric method 3MINDO, MNDO, AM1 & PM3: *based on INDO & NDDO *reproduce the binding energy
2
1PHF core
Key: How to approximate ?
Fock Matrix
MNDO-PM3
cdabdcba
(using NDDO)
ab
ba
abb
a PPUVF,
, 2
1
ab
b
ab
b
abb
a
PF
PPUF
,
,,
2
1
32
1
Semiempirical M.O. Method
Where, 2/ S
aaa
a
ZIV 1*
aI : the ionization potential
One centre integrals: (given)
spsphppppG
ppppGppssGssssG
sppp
pspss
,
'',, 2
Core-electron attraction: (given)
l l
llm
B
m
A
m MM1 2
21
,
RfMM iji j
ll
B
ml
A
ml
l l
l
1 2
2121
2
1
2
11
2
2, 2
12
1 21
2
B
l
A
lijij RRf
l
lD
,, , bb UU
:characteristic of monopole, dipole, quadrupole
:charge separations
Molecular Mechanics (MM) Method
F = MaF : Force Field
Molecular Mechanics Force Field
• Bond Stretching Term
• Bond Angle Term
• Torsional Term
• Non-Bonding Terms: Electrostatic Interaction & van der Waals Interaction
C2H3Cl
Bond Stretching PotentialEb = 1/2 kb (l)2
where, kb : stretch force constantl : difference between equilibrium & actual bond length
Two-body interaction
Bond Angle Deformation PotentialEa = 1/2 ka ()2
where, ka : angle force constant
: difference between equilibrium & actual bond angle
Three-body interaction
Periodic Torsional Barrier PotentialEt = (V/2) (1+ cosn )where, V : rotational barrier
: torsion angle n : rotational degeneracy
Four-body interaction
Non-bonding interaction
van der Waals interactionfor pairs of non-bonded atoms
Coulomb potential
for all pairs of charged atoms
MM Force Field Types
• MM2 Small molecules
• AMBER Polymers
• CHAMM Polymers
• BIO Polymers
• OPLS Solvent Effects
######################################################## ## ## ## TINKER Atom Class Numbers to CHARMM22 Atom Names ## ## ## ## 1 HA 11 CA 21 CY 31 NR3 ## ## 2 HP 12 CC 22 CPT 32 NY ## ## 3 H 13 CT1 23 CT 33 NC2 ## ## 4 HB 14 CT2 24 NH1 34 O ## ## 5 HC 15 CT3 25 NH2 35 OH1 ## ## 6 HR1 16 CP1 26 NH3 36 OC ## ## 7 HR2 17 CP2 27 N 37 S ## ## 8 HR3 18 CP3 28 NP 38 SM ## ## 9 HS 19 CH1 29 NR1 ## ## 10 C 20 CH2 30 NR2 ## ## ## ########################################################
CHAMM FORCE FIELD FILE
atom 1 1 HA "Nonpolar Hydrogen" 1 1.0081atom 2 2 HP "Aromatic Hydrogen" 1 1.0081atom 3 3 H "Peptide Amide HN" 1 1.0081atom 4 4 HB "Peptide HCA" 1 1.0081atom 5 4 HB "N-Terminal HCA" 1 1.0081atom 6 5 HC "N-Terminal Hydrogen" 1 1.0081atom 7 5 HC "N-Terminal PRO HN" 1 1.0081atom 8 3 H "Hydroxyl Hydrogen" 1 1.0081atom 9 3 H "TRP Indole HE1" 1 1.0081atom 10 3 H "HIS+ Ring NH" 1 1.0081atom 11 3 H "HISDE Ring NH" 1 1.0081atom 12 6 HR1 "HIS+ HD2/HISDE HE1" 1 1.0081
################################ ## ## ## Van der Waals Parameters ## ## ## ################################
vdw 1 1.3200 -0.0220vdw 2 1.3582 -0.0300vdw 3 0.2245 -0.0460vdw 4 1.3200 -0.0220vdw 5 0.2245 -0.0460vdw 6 0.9000 -0.0460vdw 7 0.7000 -0.0460vdw 8 1.4680 -0.0078vdw 9 0.4500 -0.1000vdw 10 2.0000 -0.1100
/Ao /(kcal/mol)
################################## ## ## ## Bond Stretching Parameters ## ## ## ##################################
bond 1 10 330.00 1.1000bond 1 11 340.00 1.0830bond 1 12 317.13 1.1000bond 1 13 309.00 1.1110bond 1 14 309.00 1.1110bond 1 15 322.00 1.1110bond 1 17 309.00 1.1110bond 1 18 309.00 1.1110bond 1 21 330.00 1.0800
/(kcal/mol/Ao2) /Ao
################################ ## ## ## Angle Bending Parameters ## ## ## ################################
angle 3 10 34 50.00 121.70angle 13 10 24 80.00 116.50angle 13 10 27 20.00 112.50angle 13 10 34 80.00 121.00angle 14 10 24 80.00 116.50angle 14 10 27 20.00 112.50angle 14 10 34 80.00 121.00angle 15 10 24 80.00 116.50angle 15 10 27 20.00 112.50angle 15 10 34 80.00 121.00angle 16 10 24 80.00 116.50angle 16 10 27 20.00 112.50
/(kcal/mol/rad2) /deg
############################ ## ## ## Torsional Parameters ## ## ## ############################torsion 1 11 11 1 2.500 180.0 2torsion 1 11 11 11 3.500 180.0 2torsion 1 11 11 22 3.500 180.0 2torsion 2 11 11 2 2.400 180.0 2torsion 2 11 11 11 4.200 180.0 2torsion 2 11 11 14 4.200 180.0 2torsion 2 11 11 15 4.200 180.0 2torsion 2 11 11 22 3.000 180.0 2torsion 2 11 11 35 4.200 180.0 2torsion 2 11 11 36 4.200 180.0 2torsion 11 11 11 11 3.100 180.0 2torsion 11 11 11 14 3.100 180.0 2torsion 11 11 11 15 3.100 180.0 2torsion 11 11 11 22 3.100 180.0 2torsion 11 11 11 35 3.100 180.0 2torsion 11 11 11 36 3.100 180.0 2
/(kcal/mol) /deg
Algorithms for Molecular Dynamics
x(t+t) = x(t) + (dx/dt) t
Fourth-order Runge-Kutta method:
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)+s1t/2] s3 = dx/dt [w/ t=t+t/2, x = x(t)+s2t/2] s4 = dx/dt [w/ t=t+t, x = x(t)+s3 t]
Very accurate but slow!
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
• Structure, Geometry
• Energy & Stability
• Vibration Frequency & Mode
• Real Time Dynamics
SummaryHamiltonianH = (h2/2m
h2/2me)ii2 +
ZZeri e2/riije2/rij
Consider a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E1. If is any well-behaved function that satisfies the boundary conditions of the problem, then * H d * dE1
The variation theorem
(1) Construct a wave function (c1,c2,,cm)
(2) Calculate the energy of : E E(c1,c2,,cm)
(3) Choose {cj*} (i=1,2,,m) so that E is minimum
Variational Method
Extension of Variation Method
For a wave function which is orthogonal to the ground state wave function 1, i.e.
d *1 = 0
E = d *H/ d * > 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.
Slater determinantf H2 : ! [!
f(1)+ J2(1) K2(1)1(1)11(1)f(2)+ J1(2) K1(2)2(2)22(2)F(1) f(1)+ J2(1) K2(1) Fock operator for 1F(2) f(2)+ J1(2) K1(2) Fock operator for 2
Hartree-Fock Equation:
Fock Operator:
LCAO-MO: c11 + c22
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
Basis set of GTFs STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G**------------------------------------------------------------------------------------- complexity & accuracy
# HF/6-31G(d) Route section water energy Title
0 1 Molecule Specification O -0.464 0.177 0.0 (in Cartesian coordinatesH -0.464 1.137 0.0H 0.441 -0.143 0.0
Gaussian 98 Input file
Comparison of the HF and VB Treatments
Electron Correlation
Beyond the Hartree-Fock
Configuration Interaction (CI)*
Perturbation theory*
Coupled Cluster Method
Density functional theory
En<n
(0) H'n(0) >
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
Ground State Excited State CPU Time Correlation Geometry Size Consistent (CH3NH2,6-31G*)HFSCF 1 0 OK
DFT ~1
CIS <10 OK
CISD 17 80-90% (20 electrons)CISDTQ very large 98-99%
MP2 1.5 85-95% (DZ+P)MP4 5.8 >90% CCD large >90%
CCSDT very large ~100%