Transcript Document

Advanced Physical Chemistry
G. H. CHEN
Department of Chemistry
University of Hong Kong
Quantum Chemistry
G. H. Chen
Department of Chemistry
University of Hong Kong
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
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
Quantum Chemistry Methods
• Ab initio molecular orbital methods
• Semiempirical molecular orbital methods
• Density functional method
SchrÖdinger Equation
Hy=Ey
Wavefunction
Hamiltonian
H = (-h2/2m)2 - (h2/2me)ii2
+  ZZe2/r - i  Ze2/ri
+ i j e2/rij
Energy
Contents
1. Variation Method
2. Hartree-Fock Self-Consistent Field Method
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, wellbehaved function that satisfies the boundary
conditions of the problem, then
 f* H f dt > E1
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
Proof:
Expand f in the basis set { yk}
f = k kyk
where
{k} are coefficients
Hyk = Ekyk
then
 f* H f dt = k j k* j Ej dkj
= k | k|2 Ek > E 1 k | k|2 = E1
Since is normalized,
 f*f dt = k | k|2 = 1
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
Application to a particle in a box of infinite depth
0
l
Requirements for the trial wave function:
i. zero at boundary;
ii. smoothness  a maximum in the center.
Trial wave function: f = x (l - x)
 f* H f dx = -(h2/82m)  (lx-x2) d2(lx-x2)/dx2 dx
= h2/(42m)  (x2 - lx) dx
= h2l3/(242m)
 f*f dx =  x2 (l-x)2 dx = l5/30
Ef = 5h2/(42l2m)  h2/(8ml2) = E1
Variational Method
(1) Construct a wave function f(c1,c2,,cm)
(2) Calculate the energy of f:
Ef  Ef(c1,c2,,cm)
(3) Choose {cj*} (i=1,2,,m) so that Ef is
minimum
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:
f(x) = exp(-cx2)
 f* H f 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
 f*f dx =  exp(-2cx2) dx = (/2)1/2 c-1/2
Ef = W = (h2/82m)c + (2/2)m2/c
To minimize W,
0 = dW/dc = h2/82m - (2/2)m2c-2
c = 22m/h
W = (1/2) h
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 f*y1 = 0
Ef = dt f*Hf / dt f*f > E2
the first excited state energy
The trial wave function f:
dt f*y1 = 0
f = k=1 ak yk
dt f*y1 = |a1|2 = 0
Ef = dt f*Hf / dt f*f = k=2|ak|2Ek / k=2|ak|2
> k=2|ak|2E2 / k=2|ak|2 =
E2
Application to H2+
e
f=c y +c y
1 1
2 2
+
+
y
1
y
2
W =  f*H f dt /  f*f 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
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
=
0
H22 - W
For H2+, H11 = H22; H12 < 0.
Ground State: Eg = W1 = (H11+H12) / (1+S)
f1 = (y1+y2) / 2(1+S)1/2
bonding orbital
Excited State: Ee = W2 = (H11-H12) / (1-S)
f2 = (y1-y2) / 2(1-S)1/2
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
Further Improvements
Optimization of 1s orbitals
H
-1/2 exp(-r)
He+
23/2 -1/2 exp(-2r)
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
Inclusion of other atomic orbitals
1s
2p
Resutls: De = 2.73 eV, Re = 1.06 A
Linear Equations
1. two linear equations for two unknown, x1 and x2
a11x1 + a12x2 = b1
a21x1 + a22x2 = b2
(a11a22-a12a21) x1 = b1a22-b2a12
(a11a22-a12a21) x2 = b2a11-b1a21
Introducing determinant:
a11 a12
= a11a22-a12a21
a21 a22
a11 a12
b1
a12
a21 a22
b2
a22
a11 a12
a11 b1
x1 =
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
a11 a12
a21 a22
det(aij) xk= .
.
an1 an2
...
...
...
...
a1,k-1
a2,k-1
.
an,k-1
...
...
...
...
a1n
a2n
.
ann
where,
det(aij) =
a11 a12
a21 a22
.
.
an1 an2
b1 a1,k+1
b2 a2,k+1
.
.
b2 an,k+1
... a1n
... a2n
... .
... ann
inhomogeneous case: bk = 0 for at least one k
a11 a12
a21 a22
.
.
an1 an2
...
...
...
...
a1,k-1
a2,k-1
.
an,k-1
xk =
det(aij)
b1
b2
.
b2
a1,k+1 ...
a2,k+1 ...
. ...
an,k+1 ...
a1n
a2n
.
ann
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
l=1
where, Clk is called cofactor
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
k=1
Sik  dt fi fk
Hik   dt fi H fk
W   dt f H f /  dt f f
i=1,2,...,n
Linear variational theorem
(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
Molecular Orbital (MO):
f = c1y1 + c2y2
( H11 - W ) c1 + ( H12 - SW ) c2 = 0
S11=1
( H21 - SW ) c1 + ( H22 - W ) c2 = 0
S22=1
Generally : yi a set of atomic orbitals, basis set
LCAO-MO
f = c1y1 + c2y2 + ...... + cnfn
linear combination of atomic orbitals
n
 ( Hik - SikW ) ck = 0
k=1
Hik   dt yi* H yk
i = 1, 2, ......, n
Sik  dt yi*yk
Skk = 1
The Born-Oppenheimer Approximation
Hamiltonian
H = (-h2/2m)2 - (h2/2me)ii2
+  ZZe2/r - i  Ze2/ri
+ i j e2/rij
H y(ri;r) = E y(ri;r)
The Born-Oppenheimer Approximation:
(1) y(ri;r) = yel(ri;r) yN(r)
(2) Hel(r )= - (h2/2me)ii2 - i Ze2/ri
+ ij e2/rij
VNN =  ZZe2/r
Hel(r) yel(ri;r) = Eel(r) yel(ri;r)
(3) HN = (-h2/2m)2 + U(r)
U(r) = Eel(r) + VNN
HN(r) yN(r) = E yN(r)
Assignment
Calculate the ground state energy and bond length of H2
using the HyperChem with the 6-31G
(Hint: Born-Oppenheimer Approximation)
Hydrogen Molecule H2
e
+
+
e
The Pauli principle
two electrons cannot be in the same state.
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
Therefore: c1 = c2 = 1
c2c1 = 1
According to the Pauli principle,
c1 = c2 =- 1
The Pauli principle (different version)
the wave function of a system of electrons must
be antisymmetric with respect to interchanging
of any two electrons.
Wave function f of H2 : Slater Determinant
y(1,2) = 1/2! [f(1)(1)f(2)(2) - f(2)(2)f(1)(1)]
=
1/2!
f(1)(1)
f(2)(2)
f(1)(1)
f(2)(2)
Energy: Ey
Ey=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 Ey under the constraint  dt |f2| = 1,
use Lagrange’s method:
L = Ey - 2 e [ dt1 |f2(1)| - 1]
dL = dEy - 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
[ Te+VeN + dt2 f*(2) e2/r12 f(2) ] f(1) = e f(1)
Average Hamiltonian
Hartree-Fock equation
(f+J)f=ef
f(1) = Te(1)+VeN(1)
J(1) = dt2 f*(2) e2/r12 f(2)
one electron operator
two electron Coulomb operator
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)
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
Secular Equation:
F11 - e F12 - S e
F12 - Se F22 - e
bonding orbital:
= 0
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
Homo nuclear Diatomic Molecules H2, Li2, O, He2, etc
Moecule
H2+
H2
He2+
He2
Li2
Be2
C2
N2+
N2
O2+
O2
Bond order

1

0
1
0
2

3
2
2
De/eV
2.79
4.75
1.08
0.0009
1.07
0.10
6.3
8.85
9.91
6.78
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
----------------
f1
----------------
f2
f1(1)(1)
f2(1)(1)
y(1,2) = 1 /2
f1(2)(2)
f2(2)(2)
= 1/2 [f1(1) f2(2) - f2(1) f1(2)] (1) (2)
Ey =  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
Particle One:
Particle Two:
f(1) + J2(1) - K2(1)
f(2) + J1(2) - K1(2)
f(j)  -(h2/2me)j2 -  Z/rj
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:
[ 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)
F(2)  f(2)+ J1(2) - K1(2)
Fock operator for 1
Fock operator for 2