Computational Chemistry
Download
1 / 171

Computational Chemistry - PowerPoint PPT Presentation


  • 138 Views
  • Uploaded on

Computational Chemistry. G. H. CHEN Department of Chemistry University of Hong Kong. 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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Computational Chemistry' - plato-diaz


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Computational Chemistry

G. H. CHEN

Department of Chemistry

University of Hong Kong


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.”

Dirac


Computational Chemistry

Quantum Chemistry

Molecular Mechanics

SchrÖdinger Equation

F = M a


Nobel Prizes for Computational Chemsitry

Mulliken,1966

Fukui, 1981

Hoffmann, 1981

Pople, 1998

Kohn, 1998


Computational Chemistry Industry

Company

Software

Gaussian Inc. Gaussian 94, Gaussian 98

Schrödinger Inc. Jaguar

Wavefunction Spartan

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


Carbon Nanotubes (Ijima, 1991)


Calculated STM Image

of a Carbon Nanotube

(Rubio, 1999)

STM Image of Carbon Nanotubes (Wildoer et. al., 1998)


Computer Simulations (Saito, Dresselhaus, Louie et. al., 1992)

Carbon Nanotubes (n,m):

Conductor, if n-m = 3I I=0,±1,±2,±3,…;or

Semiconductor, if n-m  3I

Metallic Carbon Nanotubes: Conducting Wires

Semiconducting Nanotubes: Transistors

Molecular-scale circuits ! 1 nm transistor!

30 nm transistor!

0.13 µm transistor!


Experimental Confirmations: 1992)

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)


R 1992)L 7.39 kΩ

L 16.6 pH

Rc 6.45 kΩ (0.5g0-1)

C 0.073 aF

g0=2e2/h

L ~

~

≈ 18.8 pH

Yam, Mo, Wang, Li, Chen, Zheng, Goddard (2008)


Microelectromechanical Systems (MEMS) 1992)

Micro-Electro-Mechanical Systems (MEMS) is the integration of mechanical elements, sensors, actuators, and electronics on a common silicon substrate through microfabrication technology. While the electronics are fabricated using integrated circuit (IC) process sequences (e.g., CMOS, Bipolar, or BICMOS processes), the micromechanical components are fabricated using compatible "micromachining" processes that selectively etch away parts of the silicon wafer or add new structural layers to form the mechanical and electromechanical devices.

Nanoelectromechanical Systems (NEMS)

K.E. Drexler, Nanosystems: Molecular Machinery,

anufacturing and Computation (Wiley, New York, 1992).


Large Gear Drives Small Gear 1992)

G. Hong et. al., 1999


Nano-oscillators 1992)

Nanoscopic Electromechanical Device

(NEMS)

Zhao, Ma, Chen & Jiang, Phys. Rev. Lett. 2003


Computer-Aided Drug Design 1992)

Human Genome Project

GENOMICS

Drug Discovery


ALDOSE REDUCTASE 1992)

Diabetic

Complications

Diabetes

Sorbitol

Glucose


Design of Aldose Reductase Inhibitors 1992)

Inhibitor

Aldose Reductase

Hu, Chen & Chau, J. Mol. Graph. Mod. 24 (2006)


Prediction Results using AutoDock 1992)

LogIC50: 0.77,1.1

LogIC50: -1.87,4.05

LogIC50: -2.77,4.14

LogIC50: 0.68,0.88

Hu, Chen & Chau, J. Mol. Graph. Mod. 24 (2006)


Computer-aided drug design 1992)

Chemical Synthesis

Screening using in vitro assay

Animal Tests

Clinical Trials


Quantum Chemistry 1992)

G. H. Chen

Department of Chemistry

University of Hong Kong


Contributors: 1992)

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 1992)

Hartree-Fock method

Concepts

Hands-on experience

Text Book

“Quantum Chemistry”, 4th Ed.

Ira N. Levine

http://yangtze.hku.hk/lecture/chem3504-3.ppt


Quantum Chemistry Methods 1992)

  • Ab initio molecular orbital methods

  • Semiempirical molecular orbital methods

  • Density functional method


Schr 1992)Ödinger Equation

Hy = Ey

Wavefunction

Hamiltonian

H = (-h2/2ma)2 - (h2/2me)ii2

+  ZaZbe2/rab - i Zae2/ria

+ ije2/rij

Energy


Contents 1992)

1. Variation Method

2. Hartree-Fock Self-Consistent Field Method

3. Beyond Hartree-Fock

4. Perturbation Theory

5. Molecular Dynamics


The Variation Method 1992)

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* Hf dt >E1


Proof: 1992)

Expand f in the basis set { yk}

f = kakyk

where

{ak} are coefficients

Hyk = Ekyk

then

f* Hf dt = kjak* aj Ej dkj

= k | ak|2Ek> E 1k | ak|2 = E1

Since is normalized, f*f dt = k | ak|2 = 1


i. 1992)f : trial function is used to evaluate the upper limit

of ground state energy E1

ii. f= ground state wave function,  f* Hf dt = E1

iii. optimize paramemters in f by minimizing

 f* Hf dt / f* f dt


Application to a particle in a box of infinite depth 1992)

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)


1992)* 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


Variational Method 1992)

(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


Example: one-dimensional harmonic oscillator 1992)

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) d2[exp(-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


To minimize W 1992),

0 = dW/dc = h2/82m - (2/2)m2c-2

c = 22m/h

W= (1/2) h


Extension of Variation Method 1992)

   .

.

.

E3y3

E2y2

E1y1

For a wave function f which is orthogonal to

the ground state wave function y1, i.e.

dtf*y1 = 0

Ef = dtf*Hf / dtf*f>E2

the first excited state energy


The trial wave function 1992)f: dtf*y1 = 0

f = k=1 akyk

dtf*y1 = |a1|2 = 0

Ef = dtf*Hf / dtf*f = k=2|ak|2Ek / k=2|ak|2

>k=2|ak|2E2 / k=2|ak|2 = E2


Application to H 1992)2+

e

+ +

y1 y2

f = c1y1 + c2y2

W = f*H f dt / f*f dt

= (c12H11 + 2c1 c2H12+ c22H22 )

/ (c12 + 2c1 c2S + c22 )

W (c12 + 2c1 c2S + c22) = c12H11 + 2c1 c2H12+ c22H22


Partial derivative with respect to 1992)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: 1992)

H11 - W H12 - S W

H12 - S W H22 -W

For H2+,H11 = H22; H12 < 0.

Ground State: Eg = W1 = (H11+H12) / (1+S)

f1= (y1+y2) / 2(1+S)1/2

Excited State: Ee = W2 = (H11-H12) / (1-S)

f2= (y1-y2) / 2(1-S)1/2

= 0

bonding orbital

Anti-bonding orbital


Results: 1992)De = 1.76 eV, Re = 1.32 A

Exact: De = 2.79 eV, Re = 1.06 A

1 eV = 23.0605 kcal / mol


2 1992)p

1s

Further Improvements

H p-1/2exp(-r)

He+ 23/2p-1/2exp(-2r)

Optimization of 1s orbitals

Trial wave function: k3/2p-1/2exp(-kr) 

Eg = W1(k,R)

at each R, choose kso 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


a 1992)11x1 + 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 1992)determinant:

a11 a12

= a11a22-a12a21

a21 a22

a11 a12b1 a12

x1 =

a21 a22 b2 a22

a11 a12a11 b1

x2 =

a21 a22a21 b2


Our case: b 1992)1 = 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


a 1992)11 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


inhomogeneous 1992) case: bk = 0 for at least one k

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)


homogeneous 1992) 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

l=1

where, Clk is called cofactor


Trial wave function 1992)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 i=1,2,...,n

k=1

Sikdtfi fk

Hikdtfi H fk

W  dt f Hf /  dt f f


( 1992)i) W1W2 ... Wnare 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): 1992)

 = c11 + c22

  ( 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  = c11 + c22 + ...... + cnn

linear combination of atomic orbitals

n

 ( Hik - SikW ) ck = 0 i = 1, 2, ......, n

k=1

Hikdti* H k Sikdti*k Skk = 1


The Born-Oppenheimer Approximation 1992)

Hamiltonian

H =(-h2/2ma)2 - (h2/2me)ii2

+  ZaZbe2/rab - i Zae2/ria

+ ije2/rij

H y(ri;ra) = E y(ri;ra)


The Born-Oppenheimer Approximation: 1992)

  • (1) y(ri;ra) = yel(ri;ra) yN(ra)

  • (2) Hel(ra )= - (h2/2me)ii2- i Zae2/ria

  • + ije2/rij

  • VNN = b ZaZbe2/rab

  • Hel(ra)yel(ri;ra) = Eel(ra)yel(ri;ra)

  • (3) HN =(-h2/2ma)2 +U(ra)

  • U(ra) = Eel(ra) + VNN

  • HN(ra)yN(ra) = E yN(ra)


Assignment 1992)

Calculate the ground state energy and bond length of H2

using the HyperChem with the 6-31G

(Hint: Born-Oppenheimer Approximation)


Hydrogen Molecule H 1992)2

e

+ +

e

two electrons cannot be in the same state.

The Pauli principle


Wave function: 1992)

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) = c2f(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


Wave function 1992)f of H2 :

y(1,2) = 1/2! [f(1)a(1)f(2)b(2) - f(2)a(2)f(1)b(1)]

f(1)a(1) f(2)a(2)

= 1/2!

f(1)b(1) f(2)b(2)

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


Energy: 1992)E

  • E=2dt1 f*(1) (Te+VeN) f(1) + VNN

  • + dt1 dt2 |f2(1)| e2/r12 |f2(2)|

  • = i=1,2 fii + J12 + VNN

  • To minimize Eunder the constraintdt |f2|= 1,

  • useLagrange’s method:

  • L = E - 2 e [dt1 |f2(1)|- 1]

  • dL = dE - 4 e dt1 f*(1)df(1)

    = 4dt1 df*(1)(Te+VeN)f(1)

    • +4dt1 dt2 f*(1)f*(2) e2/r12 f(2)df(1)

  • - 4 e dt1 f*(1)df(1)

  • = 0


[ 1992)Te+VeN +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


f 1992)(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;

eis the energy of orbital f.

LCAO-MO: f = c1y1 + c2y2

Multiple y1 from the left and then integrate :

c1F11 + c2F12 = e (c1 + S c2)


Multiple 1992)y2from the left and then integrate :

c1F12 + c2F22 = e (S c1 + c2)

where,

Fij = dt yi*( f + J) yj = Hij + dt yi*Jyj

S = dt y1y2

(F11 - e) c1 + (F12 - S e) c2 = 0

(F12 - S e) c1 + (F22 -e) c2 = 0


Secular Equation: 1992)

F11 - eF12 - S e

= 0

F12 - SeF22 -e

bonding orbital: e1 = (F11+F12) / (1+S)

f1= (y1+y2) / 2(1+S)1/2

antibonding orbital: e2 = (F11-F12) / (1-S )

f2= (y1-y2) / 2(1-S)1/2


Molecular Orbital Configurations of 1992)

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: 1992)

one-half the difference between the number of bonding and antibonding electrons


f 1992)1(1)a(1) f2(1)a(1)

y(1,2) = 1 /2

f1(2)a(2) f2(2)a(2)

= 1/2 [f1(1) f2(2) - f2(1)f1(2)] a(1) a(2)

---------------- f1

---------------- f2


E 1992)y = dt1dt2 y* H y

= dt1dt2 y* (T1+V1N+T2+V2N+V12+VNN) y

= <f1(1)| T1+V1N|f1(1)>

+ <f2(2)| T2+V2N|f2(2)>

+ <f1(1) f2(2)| V12 | f1(1) f2(2)>

- <f1(2) f2(1)| V12 | f1(1) f2(2)> + VNN

= i<fi(1)| T1+V1N |fi(1)>

+ <f1(1) f2(2)| V12 | f1(1) f2(2)>

- <f1(2) f2(1)| V12 | f1(1) f2(2)> +VNN

= i=1,2 fii + J12 -K12 + VNN


Average Hamiltonian 1992)

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)


Hartree-Fock Equation 1992):

[ 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


Summary 1992)

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.


4. 1992)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. 1992)Fock operator (or, average Hamiltonian) consists

of one-electron operators f(i) and Coulomb

operators Jj(i)and exchange operators Kj(i)














N 1992)aelectrons spin up and Nbelectrons spin down. 

Fock matrix for an electron 1with spin up:

Fa(1) = f a(1) + j [ Jja(1) - Kja(1) ] + jJjb(1)

j=1,Na

j=1,Nb

Fock matrix for an electron 1 with spin down:

Fb(1) = f b(1) + j [ Jjb(1) - Kjb(1) ] + jJja(1)

j=1,Nbj=1,Na


  • Energy = 1992)jafjja+jbfjjb+(1/2) iaja ( Jijaa- Kijaa )

  • + (1/2) iajb ( Jijbb- Kijbb ) + iajbJijab

    + VNN

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)

i=1,Naj=1,Nb


f 1992)jjfjja  <fja| f |fja>

JijJijaa  <faj(2)| Jia(1)|faj(2)>

KijKijaa  <faj(2)| Kia(1)|faj(2)>

JijJijab  <fbj(2)| Jia(1)|fbj(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: ( Na= Nb= n/2 )


Hartree-Fock Method 1992)

1. Many-Body Wave Function is approximated

by Slater Determinant

2. Hartree-Fock Equation

Ffi = ei fi

FFock operator

fi the i-th Hartree-Fock orbital

ei the energy of the i-th Hartree-Fock orbital


3. 1992)Roothaan Method (introduction of Basis functions)

fi= k ckiyk LCAO-MO

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

4. Hartree-Fock-Roothaanequation

j ( Fij - ei Sij ) cji = 0

Fij  < i| F | j > Sij  < i| j >

5. Solve the Hartree-Fock-Roothaan equation

self-consistently


The Condon-Slater Rules 1992)

<fa(1)fb(2)fc(3)...fd(n) | f(1) | fe(1)ff(2)fg(3)...fh(n)>

= <fa(1) | f(1) | fe(1)> < fb(2)fc(3)...fd(n) | ff(2)fg(3)...fh(n)>

= <fa(1) | f(1) | fe(1)>

ifb=f, c=g, ..., d=h; 0, otherwise

<fa(1)fb(2)fc(3)...fd(n) | V12 | fe(1)ff(2)fg(3)...fh(n)>

= <fa(1) fb(2) | V12 | fe(1) ff(2)> < fc(3)...fd(n) | fg(3)...fh(n)>

= <fa(1) fb(2) | V12 | fe(1) ff(2)>

ifc=g, ..., d=h; 0, otherwise


------- 1992)

the lowest unoccupied molecular orbital  -------

the highest occupied molecular orbital  -------

-------

LUMO

HOMO

Koopman’s Theorem

The energy required to remove an electron from a

closed-shell atom or molecules is well approximated

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

which the electron is removed.


# 1992)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


Slater-type orbitals (STO) 1992)

nlm = Nrn-1exp(-r/a0) Ylm(,)

x the orbitalexponent

* is used instead of  in the textbook

Basis Set i = p cip p

Gaussian type functions

gijk = N xi yj zk exp(-ar2)

(primitive Gaussian function)

p = u dupgu

(contracted Gaussian-type function, CGTF)

u = {ijk} p = {nlm}


Basis set of GTFs 1992)

STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G**

-------------------------------------------------------------------------------------

complexity & accuracy

Minimal basis set: one STO for each atomic orbital (AO)

STO-3G: 3 GTFs for each atomic orbital

3-21G: 3 GTFs for each inner shell AO

2 CGTFs (w/ 2 & 1 GTFs) for each valence AO

6-31G: 6 GTFs for each inner shell AO

2 CGTFs (w/ 3 & 1 GTFs) for each valence AO

6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows

6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen

Polarization

Function


Diffuse Basis Sets: 1992)

For excited states and in anions where electronic density

is more spread out, additional basis functions are needed.

Diffuse functions to 6-31G basis set as follows:

6-31G* - adds a set of diffuse s & p orbitals to atoms

in 1st & 2nd rows (Li - Cl).

6-31G** - adds a set of diffuse s and p orbitals to atoms

in 1st & 2nd rows (Li- Cl) and a set of diffuse

s functions to H

Diffuse functions + polarisation functions:

6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets.

Double-zeta (DZ) basis set:

two STO for each AO


6-31 1992)G 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 1992):

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 1992)

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: 1992)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)


Ab Initio Molecular Orbital Calculation: H 1992)2O

(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.


8. 1992)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”.


e 1992)-

e-

+

+

The Hartree-Fock treatment of H2


The Valence-Bond Treatment of H 1992)2

f1 = 1(1) 2(2)

f2 = 1(2) 2(1)

 = c1 f1 + c2 f2

H11 - W H12 - SW

H21 - SW 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} [a(1)b(2)-a(2)b(1)]/2

= 0


Comparison of the HF and VB Treatments 1992)

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 1992)

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)



Assignment 1992)

8.4, 8.10, 8.12b, 8.40, 10.5, 10.6, 10.7, 10.8,

11.37, 13.37


Electron Correlation 1992)

Human Repulsive Correlation


Electron Correlation: 1992) 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


- 1992)e

-e r12

r2 r1

+2e

H = - (h2/2me)12 - 2e2/r1 - (h2/2me)22 - 2e2/r2 + e2/r12

H10 H20 H’


H 1992)0 = H10 + H20

y(0)(1,2) = F1(1) F2(2)

H10 F1(1) = E1 F1(1)

H20 F2(1) = E2 F2(1)

E1 = -2e2/n12a0n1 = 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) = <y(0)(1,2)| H’ |y(0)(1,2)> = 5e2/4a0

EE(0) + E(1) = -108.8 + 34.0 = -74.8 (eV)

[compared with exp. -79.0 eV]


Nondegenerate Perturbation Theory 1992)

(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


Introducing a parameter 1992)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) + l2En(2) + ... + lkEn(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, < yn(0) | yn(j) > = 0, j=1,2,...,k,...


  • H 1992)0yn(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)


Multiplied 1992)ym(0) from the left and integrate,

<ym(0) | H0 | yn(1) > + < ym(0) | H' | yn(0) > = < ym(0)|yn(1) >En(0) + En(1)mn

<ym(0)|yn(1) > [Em(0)- En(0)] + < ym(0) | H' | yn(0) > = En(1)mn

The first order:

For m = n,

En(1) = < yn(0) | H' | yn(0) > Eq.(1)

For m  n, <ym(0)|yn(1) > = < ym(0) | H' | yn(0) > / [En(0)- Em(0)]

If we expand yn(1) =  cnm ym(0),

cnm = < ym(0) | H' | yn(0) > / [En(0)- Em(0)] for m  n;

cnn = 0.

yn(1) = m < ym(0) | H' | yn(0) > / [En(0)- Em(0)] ym(0) Eq.(2)


The second order 1992):

<ym(0)|H0|yn(2) > + < ym(0)|H'|yn(1) > = < ym(0)|yn(2)

>En(0) + < ym(0)|yn(1) >En(1) + En(1)mn

Set m = n, we have

En(2) = m  n |<ym(0) | H' | yn(0) >|2 / [En(0)- Em(0)] Eq.(3)


Discussion: (Text Book: page 522-527) 1992)

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 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.


Moller-Plesset (MP) Perturbation Theory 1992)

The MP unperturbed Hamiltonian H0

H0 = mF(m)

whereF(m)is the Fock operator for electron m.

And thus, the perturbation H’

H’=H - H0

Therefore, the unperturbed wave function is

simply the Hartree-Fock wave function .

Ab initio methods: MP2, MP4


Example One: 1992)

Consider the one-particle, one-dimensional system

with potential-energy function

V = b for L/4 < x < 3L/4,

V = 0 for 0 < x  L/4 & 3L/4  x < 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)1/2sin(x/L) and 2 = (2/L)1/2sin(2x/L),

respectively.


Example Two 1992):

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+.


Perturbation Theory for a Degenerate Energy Level 1992)

Hydrogen Atom

n=3 3s, 3px ,3py , 3pz , 3d1-5

n=2 2s, 2px ,2py , 2pz

  n=1 1s

B / 

H = H0 + H’

H0yn(0) = Ed(0) yn(0) n=1,2,...,d

H’ is small compared with H0


(1) 1992)Apply the results of nondegenerate perturbation theory

cnm = < ym(0) | H' | yn(0) > / [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. 


When 1992)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) + l2En(2) + ... + lkEn(k) + ...

fn(0) = kckyk(0)


  • H 1992)0yn(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,

    <ym(0) | H0 | yn(1) > + < ym(0) | H' | fn(0) >

    = < ym(0)|yn(1) >Ed(0) + En(1)<ym(0)| fn(0) >

    <ym(0)|yn(1) > [Em(0)- Ed(0)] + < ym(0) | H' | fn(0) > = En(1)< ym(0)| fn(0) >

  • For 1  m  d,

    • n [< ym(0) | H' | yn(0) > -Em(1)mn] cn= 0

  • Em(1) = < fm(0) | H' | fm(0) >

Assignment 2: 9.2, 9.4a, 9.9, 9.18, 9.24





Singly Excited Configuration Interaction (CIS): 1992)

Changes only the excited states

+


Doubly Excited CI (CID): 1992)

Changes ground & excited states

+

Singly & Doubly Excited CI (CISD):

Most Used CI Method


Full CI (FCI): 1992)

Changes ground & excited states

+

+

+ ...


Coupled-Cluster Method 1992)

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

=


Coupled-Cluster Doubles (CCD) Method 1992)

yCCD= eT2 y(0)

y(0): Hartree-Fock ground state wave function

yCCD: Ground state wave function

T2 : two electron excitation operator

T2

=


Complete Active Space SCF (CASSCF) 1992)

Active space

All possible configurations


Density-Functional Theory (DFT) 1992)

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 |i2|i >-  drZae2(r) /r1a

+ (1/2)   dr1 dr2e2/r12 + Exc[(r)]

Kohn-Sham EquationGround State: Phys. Rev. 140, A1133 (1965) FKSyi = ei yi

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

Vxc dExc[(r)] / d(r)

A popular exchange-correlation functional Exc[(r)]: B3LYP


180 1992)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 references

for same compound are less than 1 kcal/mol; and

error bars are all less than 1kcal/mol


Hu, Wang, Wong & Chen, J. Chem. Phys. (Comm) (2003) 1992)

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) 1992)

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 ?


  • Mezey (1999)

r(r)

  • Fournais (2002)

rD(r) r(r)system properties

D

  • Holographic electron density theorem for time-independent systems

Analytical continuation


r 1992)D(r,t) v(r,t)system properties

Holographic electron density theorem

r(r,t)

D

  • Holographic electron density theorem for time-dependent systems

It is difficult to prove the analyticity for r(r,t) rigorously!

Zheng, Wang, Yam, Mo & Chen, PRB (2007)


The electron density distribution of the reduced system determines all physical properties or processes of the entire system!

Existence of a rigorous TDDFT for Open System


A Grain of Sand determines all physical properties or processes of the entire system!

William BlakeTo see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour. 

一粒沙子

威廉.布莱克

从一粒沙子看到一个世界,从一朵野花看到一个天堂,把握在你手心里的就是无限,永恒也就消融于一个时辰。


Transient current (red lines) & applied bias voltage (green lines) for the Al-CNT-Al system. (a) Bias voltage is turned on exponentially, Vb = V0 (1-e-t/a) with V0 = 0.1 mV &a = 1 fs. Blue line in (a) is a fit to transient current, I0(1-e-t/τ) with τ = 2.8 fs & I0 =13.9 nA. (b) Bias voltage is sinusoidal with a period of T = 5 fs. The red line is for the current from the right electrode & squares are the current from the left electrode at different times.

Vb = V0 (1-e-t/a)

V0 = 0.1 mV &a = 1 fs

Switch-on time: ~ 10 fs


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 Geometry Size Consistent

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 Geometry Size Consistent

(1) Neglect or incomplete treatment of electron correlation

(2) Incompleteness of the Basis set

(3) Relativistic effects

(4) Deviation from the Born-Oppenheimer approximation


Extended Huckel MO Method Geometry Size Consistent

(Wolfsberg, Helmholz, Hoffman)

Independent electron approximation

Schrodinger equation for electron i

Hval = iHeff(i)

Heff(i) = -(h2/2m) i2 + Veff(i)

Heff(i) i = i i

Semiempirical Molecular Orbital Calculation


LCAO-MO: Geometry Size Consistent

fi= r criyr

s (Heffrs- eiSrs ) csi = 0

Heffrs < r|Heff| s >Srs< r| s >

  • Parametrization:

  • Heffrr < r|Heff| r >

  • = minus the valence-state ionization

  • potential (VISP)


Atomic Orbital Energy VISP Geometry Size Consistent

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

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

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

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

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

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


CNDO, INDO, NDDO Geometry Size Consistent

(Pople and co-workers)

Hamiltonian with effective potentials

Hval = i [ -(h2/2m) i2 + Veff(i) ] + ij>i e2 / rij

two-electron integral:

(rs|tu) = <r(1) t(2)| 1/r12 | s(1) u(2)>

CNDO: complete neglect of differential overlap

(rs|tu) = rs tu (rr|tt) rs tu rt


INDO: intermediate neglect of differential overlap Geometry Size Consistent

(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, INDOare parametrized so that the overall

results fit well with the results of minimal basis ab

initio Hartree-Fock calculation.

CNDO/S, INDO/S are parametrized to predict

optical spectra.


MINDO, MNDO, AM1, PM3 Geometry Size Consistent

(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


Molecular Mechanics (MM) Method Geometry Size Consistent

F = Ma

F : Force Field


Molecular Mechanics Force Field Geometry Size Consistent

  • Bond Stretching Term

  • Bond Angle Term

  • Torsional Term

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


Bond Stretching Potential Geometry Size Consistent

Eb = 1/2 kb (Dl)2

where, kb : stretch force constant

Dl : difference between equilibrium

& actual bond length

Two-body interaction


Bond Angle Deformation Potential Geometry Size Consistent

Ea = 1/2 ka (D)2

where, ka : angle force constant

D : difference between equilibrium

& actual bond angle

Three-body interaction


Periodic Torsional Barrier Potential Geometry Size Consistent

Et = (V/2) (1+ cosn )

where, V : rotational barrier

t: torsion angle

n : rotational degeneracy

Four-body interaction


C Geometry Size Consistent2H3Cl


Non-bonding interaction Geometry Size Consistent

van der Waals interaction

for pairs of non-bonded atoms

Coulomb potential

for all pairs of charged atoms


MM Force Field Types Geometry Size Consistent

  • MM2 Small molecules

  • AMBER Polymers

  • CHAMM Polymers

  • BIO Polymers

  • OPLS Solvent Effects


CHAMM FORCE FIELD FILE Geometry Size Consistent


/( Geometry Size Consistentkcal/mol)

/Ao


/( Geometry Size Consistentkcal/mol/Ao2)

/Ao


/ Geometry Size Consistentdeg

/(kcal/mol/rad2)


/( Geometry Size Consistentkcal/mol)

/deg


Algorithms for Molecular Dynamics Geometry Size Consistent

Euler method:

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)+s1t/2]

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

s4 = dx/dt [w/ t=t+t, x = x(t)+s3t]

Very accurate but slow!


Algorithms for Molecular Dynamics Geometry Size Consistent

Verlet Algorithm:

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

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

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

Efficient & Commonly Used!


Calculated properties
Calculated Properties Geometry Size Consistent

  • Structure, Geometry

  • Energy & Stability

  • Vibration Frequency & Mode

  • Real Time Dynamics


Dynamics simulation for Protein folding Geometry Size Consistent

Simulation Time: 100ps

Temperature: 300K

RIBOSOMAL PROTEIN (C-TERMINAL DOMAIN)

PDB code: 1CTF (68 amino acid)


Summary Geometry Size Consistent

Hamiltonian

H = (-h2/2ma)2 - (h2/2me)ii2 +  ZaZbe2/rab - i Zae2/ria + ije2/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* Hf dt / f* f dt>E1

Variational Method

(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


Extension of Variation Method Geometry Size Consistent

Slater determinant of H2 :

y(1,2) = 1/2! [f(1)a(1)f(2)b(2) - f(2)a(2)f(1)b(1)]

f(1)a(1) f(2)a(2)

= 1/2!

f(1)b(1) f(2)b(2)

For a wave function f which is orthogonal to the ground state wave function y1, i.e.

dtf*y1 = 0

Ef = dtf*Hf / dtf*f>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.


Hartree-Fock Equation Geometry Size Consistent:

[ 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


Basis set of GTFs Geometry Size Consistent

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


Beyond the Hartree-Fock Geometry Size Consistent

Configuration Interaction (CI)*

Perturbation theory*

Coupled Cluster Method

Density functional theory

En(1) = < yn(0) | H' | yn(0) >

Moller-Plesset (MP) Perturbation Theory

The MP unperturbed Hamiltonian H0

H0 = mF(m)

whereF(m)is the Fock operator for electron m.

And thus, the perturbation H’

H’=H - H0


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%  


Molecular Mechanics Force Field Geometry Size Consistent

  • Bond Stretching Term

  • Bond Angle Term

  • Torsional Term

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

F = Ma

F : Force Field


ad