Transcript PhysChem 728342 Introduction to Quantum Theory

Physical Chemistry III (728342)
Chapter 4:
Molecular
Structure
Piti Treesukol
Kasetsart University
Kamphaeng Saen Campus
The Molecular Structure

Valence-Bond Theory

Molecular Orbital Theory
• Shared electron pair
•  and  bonding
• Hybridization
• Linear Combination of Atomic Orbitals
(LCAO)
2
The Born-Oppenheimer
Approximation
 Nuclei are much heavier than
electrons so they move relatively
slowly and may be treated as
stationary while the electrons move
in the nuclei-field.
• Nuclei is fixed at arbitrary
locations.
Electrons adjust themselves to the
• The Schrödinger equation
the
wave
optimum positionfor
around
the nuclei.
 Once
the nucleiis
move,
Electron
functionBof electron
alone
being
readjust themselves instantaneously.
solved.
A
C
3
Born-Oppenheimer
Approximation
• The Schrödinger
equations for electrons at
different nuclear
separations can be
solved.

Molecular Potential
Curve:
How the energy of the
system varies with bond
length (geometry)
0

Energy
Molecular Potential Curve
RE
R
DE
A
B
• No kinetic energy of the
4
Valence-Bond Theory



The valence-bond (VB) theory was the
first quantum mechanical theory of
bonding to be developed.
The essential feature of VB theory are
the pairing of the electrons and the
accumulation of electron density in the
+
inter-nuclear region that stems form the
paring.
Using general chemistry language
• Spin pairing
5
The Hydrogen Molecule


The simplest molecule; H2
The spatial wave function
   H1s (r1 ) H1s (r2 )  A(1)B(2)
A
B
• electron 1 is on atom A and electron 2
is on atom B
• electrons are indistinguishable so it’s
 to
A(1know
) B(2)  Awhether
(2) B(1)
not possible
it’s
electron 1 that is on atom A or electron
  A(1) B(2)  A(2) B(1)
  A(1) B(2)  A(2) B(1)
2.
• the• Lower
superposition
of the
wave
energy
• Higher
energy functions
• Ground state of H2
• Excited state of H2
6

The formation of
the bond in H2 can
be pictured as due
to the high
probability that the
two electrons will
be found between
two nuclei
+ and
hence bind them
together.
0.12
0.1
0.08
0.06
0.04
0.02
0
-80
-60
- 40
-20
0
20
40
60
80
-60
- 40
-20
0
20
40
60
80
-60
- 40
-20
0
20
40
60
80
0.12
0.1
0.08
0.06
0.04
0.02
0
-80
0.12
0.1
0.08
0.06
0.04
0.02
Two electrons from different atoms
join the same orbital
0
-80
The plot between the distance between two H atom and
its wave function
7
Electron Spin

The total VB wave function for H2 is
 (1,2)  A(1) B(2)  A(2) B(1) (1,2)
• Using the Pauli principle, the total
wave function needs to be asymmetric.
 (2,1)  A(2) B(1)  A(1) B(2) (2,1)
• The spatial part is symmetric so the
spin part needs to be asymmetric. The
asymmetric
spin
for
2-electron
system
  (1,2)   1/ 2  (1) (2)   (2) (1)
is
8
Homonuclear Diatomic
Molecules
 Diatomic molecule with the same
atom types
2s 2 2 p1x 2 p1y 2 p1z
• N2 (N:
)
N
Z
N
2s
2px
2py
2pz
+
+
nodal plane
 bonding
 bonding
9
Polyatomic Molecules



Each sigma bond is formed by the
spin pairing of electrons in atomic
orbitals with cylindrical symmetry
about the relevant internuclear axis.
Each pi bond is formed by paring
electrons that occupy atomic orbitals
of the appropriate symmetry.
Examples
2s 2 p 2 p 2 p
+
2
• H2O2
2
x
1
y
1
z
1s1
10
Promotion

An apparent deficiency of valencebond theory is its inability to account
for carbon’s tetravalence.
2s 2 p 2 p
2
1
x
1
y
• According to VB theory, carbon can form
only 2 bonds.
2s 2 p 2 p 2 p
• This can be overcome
by allowing for
promotion, the excitation of an electron to
an orbital of higher energy.
1
1
x
1
y
1
z
 Now carbon atom is ready to form 4
bonds.
 This promotion requires an investment
11
Hybridization



The description of the bonding in
CH4 is still not complete.
The electron density distribution in
the promoted atom is equivalent to
the electron density.
Each electron occupies a hybrid
orbital formed by interference
between the 2s and 2p orbitals.
12
Linear Combination for
Hybrid
Orbitals
 sp hybridization
http://csi.chemie.tu-darmstadt.de/ak/immel/
1
s  p x 
2
1
s  p x 
h2 
2
h1 

sp2 hybridization
1
s
3
1
h2 
s
3
1
h3 
s
3
h1 

1
1
px 
py
6
2
1
1
px 
py
6
2
2
px
6
sp3 hybridization
1
s
4
1
h2 
s
4
1
h3 
s
4
1
h4 
s
4
h1 
1
4
1
4
1
4
1
4
1
4
1
px 
4
1
px 
4
1
px 
4
px 
1
4
1
py 
4
1
py 
4
1
py 
4
py 
pz
pz
pz
pz
13
Directions of Hybrid Orbitals
sp
sp2
sp3
sp2d
sp3d
sp3d2
14

CH4

C2 H4
Bonding in VBT
 H2O
15
Molecular Orbital Theory

Basic Ideas

Electrons in a molecule can be
described by their wave function or
specified by their orbitals.
The Molecular Orbital can be
derived from

• Electrons can spread throughout the
entire molecule.
• Electrons should not be regarded as
belonging to particular bonds.
• Atomic orbitals
• Schrödinger equation
16
The Hydrogen-Molecule Ion

The simplest molecule with one
electron is H2+
• The Hamiltonian for the single electron
 1

1 1
+
H 
   
 
in H2 is
2m
r
r
R
rA1
rB1
R
2
2
1
e

A1
B1

where rA1 and rB1 are the distances of
the electron (1) from the two nuclei (A
and B) and R is the distance between
the two nuclei. H  E
• The one-electron wave functions
17
Linear Combinations of Atomic
Orbitals
 The wave function of electrons in a
molecule is a superposition of the
  N (atomic

) orbitals.
N ( A  B)
comprising

H 1s A
H 1sB
• The molecular orbitals (MO) can be
written as the linear combinations of
atomic orbitals (AO).
• An approximate molecular orbital
formed from a linear combination of
atomic orbitals is called an LCAO-MO.
18
Bonding Orbitals

  N (is
A  B)
If the molecular orbital
the probability density corresponding to
the wave
 N ( A function
 B  2 AB) +

2

2
2
2
0.12
0.1
0.08
0.06
• A2 and B2 are the probabilities
if the electron were confined to
the atomic orbitals A and B
• 2AB is the overlap density
0.04
0.02
0
-50

-40
-30
-20
-10
0
10
20
30
40
50
The overlap density represents an
enhancement of the probability of
finding the electron in the internucler
region.
19
Antibonding Orbitals


The linear combination - corresponds to
a higher energy than +.
The probability density of - is
 2  N 2 ( A2  B2  2 AB)

0.08
There is a reduction in probability
density between the nuclei due to
the -2AB term which is destructive
interference where the two atomic
orbitals overlap.
The antibonding electron is excluded from the
internuclear region and it can pulls the nuclei
apart.
0.03
-0.02-50
-40
-30
-20
-10
0
10
20
30
40
50
-0.07
-0.12

20

Types of Bonding
Sigma bonding ()
A

+
B
A
B

Pi bonding ()
A
+
B
A
B
Sigma antibonding (*) Pi antibonding (*)
A
+
B
A
B
A
+
B
A
B
21

The Electronic Structure of
Diatomic Molecules
Using the building-up principle to
deduce the ground electronic
configurations
• Construct molecular orbitals by
combining the available atomic
orbitals.
• Accommodate the electrons supplied
by the atoms in the molecular orbitals
to achieve the lowest energy.
 Pauli exclusion principle
 Hund’s maximum multiplicity rule
22
The Hydrogen and Helium
Molecules
 H2
H2 2*
H 1s
H 1s

He2
H2 1
He2 2*
He 1s
He 1s
He2 1
He2 has one bond and one antibond. An
antibond is slightly more antibonding than a
bond is bonding. The He2 molecule has a
higher energy than the separated atoms so it
is unstable relatively to the individual atoms
23
Bond Order

A measure of the net bonding in a
diatomic molecule is its bond order,
1
b;
b  (n  n*)
2
»n= number of electrons in bonding
orbitals
»n= number of electrons in antibonding
orbitals
• The greater the bond order, the shorter
the bond
• The greater the bond order, the
24
Period 2 Diatomic Molecules

General form of the sigma orbitals
  cA2s A2s  cB2s B2s  cA2 p  A2 p  cB2 p  B2 p
z
z
z
z
• From the four atomic orbitals we can
form four molecular orbitals of sigma
symmetry by an appropriate choice of
the coefficients c.
• Four orbitals can be separated into two
sets according
their
energy
levels
  1  c A2 sto

c

A2 s
B2s B2s
  2  c A 2 p  A 2 p  cB 2 p  B 2 p
z
z
z
z
25
LCAO-MO

The forming of the LCAO-MO
depends on
• Energy level of the atomic orbitals
• Symmetry of the atomic orbitals
y
z
x
2s orbital
26
Molecular Orbital
Diagram
2
2
E-configuration: ( ) ( *) (
)2 ( )4(
2s
2px
2p
Energy
4(
2
*)
*)
2p
2px
2s
27
MO Diagram of Diatomic
Molecules

O
(2
x 6 valence
 N2 (2 x 5 valence
2
electrons)
electrons)
• O = 1s2 2s2 2p4
• Bond order = 2
and B2’s diagrams
MO of N2
(2s)2(2s*)2 (2p)4(2px)2
Energy
• N = 1s2 2s2 2p3
• Bond order = 3
• Exception Similar to C2
MO of O2
(2s)2(2s*)2(2px)2(2p)4(2p*)2
28
Overlap Integral
 *d    A  B   A  B d
*

  A2 d   B2 d  2 *A B d
Overlap integral, S, is the extent to which two
atomic orbitals on different
atoms
overlap.
*
S   A B d
A
B
small overlapping
A
1
0.8
B
large overlapping
S
0.6
• S is small if two orbitals are on 0.4atoms that
are far apart.
0.2
• If A and B are simultaneously large
in
the
0
0
2
4
6
same region, then S may be large.
R/a
• If the two normalized atomic orbitals
are
The overlap integral between two H1s
identical then S = 1.
orbitals as a function of their separation.
0
29
Heteronuclear Diatomic
Molecules
 Heteronuclear diatomic molecule
is a
diatomic molecule formed from atoms of
two different elements
• The electron distribution in the
covalent bond between the atoms is
not evenly shared.
• Polar bond: a covalent bond in which
the+electron
pair is shared
H
+
F unequally
H
H
*
*
by the two atoms.
H-1s
H1sA
H
H
Homonuclear Diatom
H1sB

d
H
F
d-

F-2p
Heteronuclear Diatom
30
Bondings

A bond consists of two electrons in
an orbital of the
form
 c  c 
A

A
B
B
• ci is the coefficent of the wave
function i.
The proportions of the atomic
orbitals A and B in the bond are
|cA|2 and |cB|2, respectively.
• Non polar bond: |cA|2 = |cB|2
• Polar bond: |cA|2  |cB|2
• Ionic bond: |ci|2 = 0 (i = A or B)
31

A bond with |cA|2 
|cB|2
Polar Bonding
The atomic orbital energy levels of H and
F atoms and the molecular orbitals of HF
molecule.
• The atomic orbital
with the lower
energy makes the 13.6 eV
large contribution to
the bonding
molecular orbital.
• One with the higher
energy makes the
large contribution to
the antibonding
molecular orbital.
• Electrons prefer
residing on the
*
13.4 eV
H1s
18.8 eV

F2p
18.6 eV
  0.19 H 1s  0.98 F 2 p
 *  0.98 H 1s  0.19 F 2 p
32
The Variational Principle


The coefficients in the linear combinations
used to build molecular orbitals can be
provided by the variation principle.
The Variational Priniciple:
If an arbitrary wave function (trial wave
function) is used to calculate the
energy, the value calculated is never
less than the true energy.
• The best coefficients
in
the
trial
wave
trial wave functions
function will provide the lowest energy
  energy
cAA  cBcan

B
but the lower
be 
achieved by
using a morecoefficients
complicated trail wave
function.
33

The energy of an molecule can be
determined from
its
wave
function:
*
E    H d 

The trial wave function  is not
normalized because its coefficients (ci)
*
are arbitrary at this
thus the

Hstage
d

E
*
energy is in the
form
of


d


• Using the variation principle, we
search for the
that
E coefficients
E
value
0
0
minimize the
of
c A
cB the energy
34
*
2


d




 d
  c A A  cBB  d
2
 c A2   A2 d  cB2  B2 d  2c AcB   AB d
 c A2  cB2  2c AcB S
*

 Hd   Hd
  c A A  cBB H c A A  cBB d
 c A2   A H A d  cB2  B HB d  c AcB   A HB d  c AcB  B H A d
 c A2 A  cB2 B  c AcB  AB  c AcB  BA
 c A2 A  cB2 B  2c AcB  AB
35

The energy can be determined from
when
cA2 A  cB2 B  2cAcB  AB
E
cA2  cB2  2cAcB S
 A    A H A d
 B    A H A d

coulomb integral
 AB    A HB d   B H A d
resonance integral
S    AB d
overlap integral
We want to solve for the coefficients
that minimize the energy E.
36
Molecular Orbitals for
Polyatomic
Systems
 The molecular orbtials of polyatomic
molecules are
built
in
the
same
way:
  c

i
i
i

when i
is an atomic orbital and
the sum extends overall orbitals of
all the atoms in the molecule.
Using the variation
   c principle with trial
wave functions
i i
i
• A complete set of i is call the basis set
(a set of atomic orbitals)
• The coefficients can be solved in the
37

Schrödinger equation
H  E 
   c j j
A MO is represented by
LCAO of n basis function
j
H  c j j  E  c j j
j
j
multiply by i and integrate over all
space,   H  c  d  E    c  d
i
j
j
i
j
H c
ij
j

j
j
j
j
 E  Sij c j
j
There are n equations (with different
38
The Matrix Formulation

For a two-atom system, the secular
equations have the form
H AA  ESAA cA  H AB  ESAB cB  0
H BA  ESBA cA  H BB  ESBB cB  0
H AA    A H Ad
S AA    A Ad
H AB    A HB d
S AB    AB d
• Using the Matrix notation
 H AA H AB 
 S AA S AB 
 cA 
 S  
 c   
H  
 H BA H BB 
 S BA S BB 
 cB 
Hc  ScE or  H  ES c  0
39



If there are N orbitals in the basis
set, then there are N eigenvalues (E)
and N corresponding column
vectors.
Ei
There would Hc
bei NScisecular
equations
to be solved.
c be written
 ccan
 E 0  in the
N equations
 E  

C  
c
c 
0 E 


matrix form by introducing the
matrices
1, A
2, A
1, B
2, B
1
2
HC  SCE
40

If we use an orthogonal basis set;
   d  1 if i  j
i
j
0
if i  j
we can neglect the matrix S (it can
be either 1 or 0), then
HC  CE
• The eigenvalue E can be solved by
C 1 HC  E
41
Schrödinger equation for
H  E
MO
H (c1 1  c2 2    cn n )  E (c1 1  c2 2    cn n )
• for each MO
 H11

 H 21
 

H
 n1
H12
H 22
H n2
• for all MOs
HC  C E
 H11

 H 21
H


H
 n1
H12
H 22
H n2
 H1n  c1 
 c1 
 
 
H 2 n  c2 
 c2 

E
 Hc  Ec








 
 
c 
H nn  c3 
 3
2  c1(2)1  c2(2) 2   cn(2) n  E2
 H1n 
 c1(1)
 (1)

H 2n 
 c2
C

 





 c (1)
H nn 
 n
c1( 2)  c1( n ) 
 E1


( 2)
(n)
c2
c2 
0
E





0
cn( 2)
cn( n ) 

0
E2
0
0

0



En 

Matrix S is neglected where Sii = 1 and Sij = 0
42
Secular Equation for Diatomic
Molecules
 Consider the molecular orbital built
from two AOs
• The coefficients are given by the
  ESequations
  E c   of
   ES
 c 
ES cthe
 0 two
solutions
secular

   0
A
A
A
B
  ES cA   B  E cB  0
AA
 BA  ESBA
AB
AB
A
 B  ESBB  cB 
• The secular equations have a solution
  E   ES
 zero:
   E   E   is
ES   0
if the secular
determinant
 ES   E
2
A
A
B
B
 A  B
• For homonuclear
 E     ES  molecule
0  E 
2
2
 
1 S
43
Homoatomic Molecule
For homoatomic molecule
 
E 
1 S
 and  are

The lower energy corresponds to the
 
1
bondingE orbital

c c 



negative
1 S
A
B
21  S 1/ 2
energy

1
The higher
corresponds
to the
E 
c  c 
S
21  S 
antibonding1 orbital

 
A
 A  B
B
 
1/ 2
 A  B
21and
21  S wave
 S  antibonding
The bonding
1/ 2
1/ 2
44
Heteroatomic Molecule
For heteroatomic molecule
If we approximate that S=0 (a simple
case)
E   energy
  tan corresponds
  A cos  B sin to the
 The lower
bonding orbital

A
E   B   tan


   A sin   B cos
The higher energy
corresponds
to the
2
1
 arctan
antibonding orbital
2
 
B
A
and
• If the difference between A and B are
45
The Hückel Approximation

Consider conjugated molecules (Alternation
of single and double bonds)
• Pi orbitals are treated separately from the sigma
orbitals.
• Consider sigma bonds to be fixed and
concentrate on finding the energies of the pi
bonds and antibonds.

The Pi molecular orbital energy level
diagrams of conjugated molecules can be
constructed using a set of approximations
suggested by Erich Hückel.
• All overlap integrals (S) are set equal to zero.
= 45 = 56 = 
• All resonance
integrals
() between
12 = 23 = 34non-neighbors
2
6
4
are set equal
to
zero.
13 = 14 = 15 = 16 = 24 … = 0
3
5
1
• All remaining resonance integrals are set equal.
46
Ethene – Hückel Approx.

The Pi orbitals of ethene can be
expressed by as LCAOs of the C-2p
orbitals.
  c A A  cB B
and B are the C-2p orbitals on atoms A
  E   ES
and B.
0


ES


E
• Using the variational principle (A= B = )
•
A
 E

2
   E    2  0

 E
• Using Hückel approximation
E    
47

Two possible energy levels are
corresponding to the bonding (+)
E (-).
and antibonding


-
The ground stateconfiguration
is
E     2 
12.
2*
C2pA
C2pB
1

*
+
The excitation energy from 1 to 2*
is 2.
48
Butadiene – Hückel Approx.

The Hückel approximation for
HC  CE
butadiene
B
D
A
C
H ij   i H j d
1  C2 p , A 2  C2 p , B 3  C2 p ,C 4  C2 p , D
 H11 H12 H13 H14 
  0 0 




H
H
H
H



0



22
23
24 
H   21

0   
H
H 32 H 33 H 34 
 31





H

0 0  
 41 H 42 H 43 H 44 
0
0
0
   1.62



0


0
.
62

0
0


E

0
0
  0.62
0



0
0
0
  1.62 

0.602
0.602  0.372
 0.372


0
.
602
0
.
372

0
.
372
0
.
602


C
0.602  0.372  0.372  0.602


 0.372  0.602

0
.
602
0
.
372


49

Pi-MOs of butadiene
E1    1.62
 1  0.372 1  0.602 2  0.602 3  0.372 4
E2    0.62
 2  0.602 1  0.372 2  0.372 3  0.602 4
E3    0.62
 3  0.602 1  0.372 2  0.372 3  0.602 4
E4    1.62
 4  0.372 1  0.602 2  0.602 3  0.372 4
4*
3*
2


1
Excitation energy: E 2  3 *  2 0.62
50
Delocalization Energy and
Stability
 -electron binding energy: The sum
of the
energies of each  electron.
 Ethene (1 -bond)
E   
• Energy levels
E  2     
• -electron binding energy


Butadiene (2 -bonds)
E    1.62 ,   0.62
• Energy levels
E  2    1.62   2    0.62 
• -electron binding energy

 4  4.48
• The energy of the butadiene molecule is 0.48
more stable than the sum of two individual 
bonds.

Comparison between the -electron binding
energies of ethene and butadiene shows the
extra stabilization of a conjugated system
51
Benzene

1
2
6
3
4
5
Benzene – Aromatic
Stability
  0 0 0  
   0 0 0 


0    0 0
H 

0 0    0
0 0 0    


  0 0 0   
E    2 ,    ,   
  2
 
b2u
e2u
e1g
a2u
 
  2
E  2    2   4     
 6  8
b2u
e2u
e2u
The -bond formation energy: 8
The delocalization energy: 2
a2u
e1g
e1g
52
Extended Hückel Theory

The Hückel theory has been modified
to make it applicable to more
sophisticate molecules.
• Include both  and  orbitals




R
R


An and
overlap between
H1s orbitals
• CalculateS overlap
store
them
 e
 1    integrals
 

as a matrix S;a for aexample
2
 R / a0
1
3
0
0
• Set the  terms to the negative of ionization
energies of1 each orbital
K is a constant (equal to 1.75)
Hij  2 KSij (Hii  H jj )
• Set the  terms to the values calculated from
HC  SCE
• The Schrödinger
equation can be solved by
S 1 HC  CE
C 1 S 1 HC  E
53

H3
+
Using the extended Hückel theory with
1 S S 
H3+


H
• The overlap integral matrix
is
S
S  S
 S
1
KS KSmatrix
 

 1 KS
• The Hamiltonian
is



H  KS
 KS     KS
KS KS
 KS
 
1
KS
S
1 
H
H
KS 
KS 
1 
• The eigenvalues
are
0.679
, 0.679, and
0.724 0.577
 0.816
1.281. C   0.408  0.036 0.577 c s are not normalized.
  0.408 matrix

0.688 0.577is
• The eigenvector
ij
  N0.577 1sA  0.577 1sB  0.577 1sC 
• An example of the molecular orbitals (non-
54
Mulliken Population Analysis

Consider a two-atom molecular orbital
  cA A  cB B
• The probability density for an electron that
occupies this orbital is
2  cA A  cB B   cA2 A2  cB2 B2  2cAcB A B
2
• The total probability is
1  cA2  cB2  2cAcB S A B
and the atomic population (i) and the
overlap population (ij) are defined as
i  ci2 ij  2ci c j Sij

In general cases, one-half of the sum of
overlap populations is the gross orbital
population.
55
Self-Consistent Field
Calculations
Hartree Method

The wave function of n-electron system
can be approximated as a product of n
hydrogenic orbitals (trial wave function
with adjustable parameters), called a
many-electron atom
   (1) (2) (n)
Hartree product.
1
2
n
 Equation

• The Hartree





V
1

 1    1
2
 2me
2
1
Kinetic energy
1

1
1 1
Potential energy
Orbital energy
(in the electrostatic field
from other particles)
i by
i
 Each orbital is solved iteratively
using
the Variational
to get a better
E   principle
  J
wave function (
).
• Total Energy
Coulomb integral
i
i
i
j i
ij
57
Slater Determinants

Slater Determinant of an atomic orbital
• The many-electron wave function is a
product of one-electron wave functions
 orbitals):
 (1) (1) (2) (2) (3) (3)... ( N ) ( N )
(spin
a
a
b
n
 This wave function doesn’t satisfy the Pauli
principle!
 The acceptable
thewave
function can
 (1)form
 (1of
) 
(1)
(2)  (2) determinant
  ( 2)
be written
in
a
matrix’s
form (Slater
1 





N!
determinant).
a
a
n
a
a
n
 a ( N )  a ( N )   n ( N )
A compact form:  
1
1
 a (1) a (2)  n ( N ) 
 a (1) a (2)  n ( N )
N!
N!
58

Slate determinant of He



1 1s(1) 1s(1)
1

1s(1)1s(2)
N! 1s(2) 1s(2)
N!

1
1s(1)1s(2)  1s(2)1s(1)
N!

Slater determinant of Li
1s (1) 1s (1) 2s (1)
1
1

1s(2) 1s (2) 2s (2) 
1s (1)1s (2)2s (3)
N!
N!
1s (3) 1s(3) 2s(3)

 


1
1s (1) 1s(2)2s (3)  2s (2)1s (3)  1s (1)2s (2)1s(3)  1s (2)2s (3)   2s (1) 1s (2)1s (3)  1s(2)1s(3)
N!

 These wave functions satisfies the Pauli
1
principle.
1s(1)1s(2)2s(3)  
N!
1
1
1s(2)1s(1)2s(3) 
1s(2)1s(3)2s(1)
N!
N!
59
The Hartree-Fock Method

The wave function can be written in the
Slater determinant1form of each orbital.
A compact form:  
N!
 a (1) a (2)  n ( N )
• Each of the electron wave function must
satisfy the Hartree-Fock
equations:
f  (1)    (1)
1
 Fock operator:
 Core

i

f1  h1   2 J j (1)  K j (1)
j
Z ne2
2 2
h1  
1  
Hamiltonian:
2me
n 4 0 rn1
 *
 e2  

d   (1)
J j (1)J:
 (1)   j ( 2) j ( 2)
 Coulomb operator,
 4 0 r12  

 *
 e2  
d  j (1)
K j (1) K:
(1)   j (2)  (2)
 Exchange operator,
4

r
0 12 



An exchange term is a result from electron-interchanging (the use of Slater determinant).
60
The Roothaan Equations

Express the wave function as a linear
   c  of a basis function (
combination
)in the Hartree-Fock
f  c  1   method
 c  1
N
i
j 1
ji
j
N
1
N
i 
i 1


i 
i 1
*
j
• Multiply from
the
over the
1dr by
c  left
    and
c  1integrate
dr
 f 
coordinates of electron 1
 c   (1) f  (1)dr   c   (1) (1)dr
N
*
j 1
N
i 
i 1
*
j 
j
N
i 1
j
N
*
j
i
N
c  F
i 1
i 
i 1
i
1 j
1

i 1
i
*
j
i
1
N
ji
    ci S ji
i 1
 SC
• The Roothaan FC
equations
(in Matrix form)
Fij    *j (1) f1 j (1)d
Sij    *j (1)i (1)d
61
Basis Functions

Basis functions: a complete set of known
functions.
• Any functions of interest can be
represented by a linear combination of
basis functions.
• A simplest set of basis functions is a set of
occupied AO wave function.




H  H-1s
He  He-1s
C  C-1s C-2s C-2px C-2py C-2pz
etc.
• Unoccupied orbitals may be required to
create MOs
• A larger setHof basis functions
H is generally
+
used to make the orbital more realistic, for
62
Semi-empirical & ab initio
Methods
 To solve the Schrödinger equation,
a
large number of integrals need to be
calculated.
• ab initio Method: all integrals are calculated
• Semi-empirical Method: many of the
integrals are estimated by appealing to
experimental data and some are neglected.
63
Density Functional theory


If the system comprises of n electrons, the
wave function is a function of 3n parameters.
Solving the equation is painful work.
DFT proposes that the energy of the system
is a function of electron density, which is a
E   E  E  EThus the electron
function of 3 parameters.
 the wave
 (r )    rof
density can be used, instead
functions, to represent the system.
K
P
XC
n
i 1
2
i
2
2
2
N


Z
e


(
r
)
e


j
2
 The density
can
be
calculated
by
2




dr

V
(
r
)


1
2
XC 1  i ( r1 )   i i ( r1 )

2mewave
 0 rj1
4 0can
r12

j 1 4function

and the
be solved
from
Kohn-Sham equation.
64