Molecular Bonding Molecular Schrödinger equation r - nuclei s - electrons r 1 2 1 s 2 W V 0 j i 2 m0 i 1 j 1
Download ReportTranscript Molecular Bonding Molecular Schrödinger equation r - nuclei s - electrons r 1 2 1 s 2 W V 0 j i 2 m0 i 1 j 1
Molecular Bonding
Molecular Schrödinger equation r - nuclei s - electrons
j r
1 1
M j
2
j
1
m
0
i s
1
i
2
2 2
W
V
0
M j m
0 = mass of j th nucleus = mass of electron Laplace operator for nuclei
j
Laplace operator for electrons
i V
e
2 4
0
r ii
j
4
0
r jj
2 2
4
0
r ij
electron-electron repulsion nuclear-nuclear repulsion Coulomb potential electron-nuclear attraction
Copyright – Michael D. Fayer, 2007
Born-Oppenheimer Approximation Electrons very light relative to nuclei they move very fast.
In the time it takes nuclei to change position a significant amount, electrons have “traveled their full paths.” Therefore, Fix nuclei - calculate electronic eigenfunctions and energy for fixed nuclear positions.
Then move nuclei, and do it again.
The resulting curve is the energy as a function of internuclear separation.
If there is a minimum – bond formation.
Copyright – Michael D. Fayer, 2007
Born-Oppenheimer Approximation Separation of total Schrödinger equation into an electronic equation and a nuclear equation is obtained by expanding the total Schrödinger equation in powers of
M – average nuclear mass, m 0 - electron mass.
/
1/4 1) Not exact 2) Good approximation for many problems 3) Many important effects are due to the “break down” of the Born-Oppenheimer approximation.
Copyright – Michael D. Fayer, 2007
Born-Oppenheimer Approximate wavefunction
n
x
,
n
x
, electronic coordinates nuclear coordinates electronic wavefunction depends on electronic quantum number, n nuclear wavefunction depends on electronic quantum number, n and nuclear quantum number,
Copyright – Michael D. Fayer, 2007
Electronic wavefunction
n
depends on fixed nuclear coordinates,
.
Obtained by solving “electronic Schrödinger for fixed nuclear positions,
.
equation” No nuclear kinetic energy term.
S
ii
i
2
n
2
M
0 2
U n
n
0 The energy
U n
depends on the nuclear coordinates and the electronic quantum number.
The potential function
complete potential function for fixed nuclear coordinates.
Solve, change nuclear coordinates, solve again.
Copyright – Michael D. Fayer, 2007
Solve electronic wave equation Nuclear Schrödinger equation becomes
j r
1 1
M j
2
j
n
2 2
E n
,
U n n
( )
0
U n
the electronic energy as a function of nuclear coordinates,
, acts as the potential function.
Copyright – Michael D. Fayer, 2007
Before examining the hydrogen molecule ion and the hydrogen molecule need to discuss matrix diagonalization with non-orthogonal basis set.
No interaction
H
0
A
E A A
0
H B
E B B
States have same energy:
E A
E B
E
0 Degenerate With interaction of magnitude
H A
E A
0
B H B
E B
0
A
The matrix elements are
A H A B H A A H B
E
0
B H B
E
0
H AA H BA H AB H BB
Copyright – Michael D. Fayer, 2007
Hamiltonian Matrix
H
A B
0
B
E
0
H AA H BA H AB H BB
Matrix diagonalization form secular determinant
E
0
E
0
0 Energy Eigenvalues
E E
0 0
Eigenvectors
1 2
A
1 2
B
1 2
A
1 2
B
Copyright – Michael D. Fayer, 2007
Matrix Formulation - Orthonormal Basis Set eigenvalues
j N
1
a ij
ij
u j
0
i
1, 2
N
vector representative of eigenvector This represents a system of equations
a
11
u
1
2
3
1
a
22
u
2
1
3
2
a
33
u
3
0
0 only has solution if
0
a
11
a
21
a
31
a
22
a
12
a
32
a
13
a
33
a
23
0
Copyright – Michael D. Fayer, 2007
Basis Set Not Orthogonal Basis vectors not orthogonal
i j
ij
0 overlap In Schrödinger representation
ij
j d
0
j N
1
a ij ij
u j
0
i
1, 2
N
system of equations only has solution if ( a ( a
a 11 21
31
21 31
)
)
( a 12 ( a
a 32 22
12
)
32
) ( a 13
( a 23
a
33
13
) 23
)
0
Copyright – Michael D. Fayer, 2007
For a 2×2 matrix with non-orthogonal basis set
H H BA AA
E
E H AB H BB
E
E
0
E
eigenvalues overlap integral 0, recover standard 2×2 determinant for orthogonal basis.
E S
H AA
1
H
AB E A
H AA
1
H
AB
S
A
1 2 2 1
A
B
A
B
Copyright – Michael D. Fayer, 2007
Hydrogen Molecule Ion - Ground State A simple treatment
e–
r
A
r
B H + A
r
AB + H B
Born-Oppenheimer Approximation electronic Schrödinger equation
2
2
m
0 2
E
e
2 4
0
r A
e
2 4
0
r B
e
2 4
0
r AB
0
2 - refers to electron coordinates electron kinetic energy Have multiplied through by
2m 0 2
Copyright – Michael D. Fayer, 2007
Large nuclear separations
r
AB
System looks like H atom and H + ion Energy
E
E H
Rhc
13.6 eV Ground state wavefunctions
U
1
s A U
1
s B
Either H atom at A in 1s state or H atom at B in 1s state
U
1
s A
and degenerate
B
Copyright – Michael D. Fayer, 2007
Suggests simple treatment involving
U
1
s A
as basis functions and
U
1
s B
not orthogonal Form 2×2 Hamiltonian matrix and corresponding secular determinant.
H H BA AA
E
E H AB H BB
E
E
0
E
eigenvalues overlap integral
H AA
U
1 *
s A H U
1
s A d
H BA
U
* 1
s B H U
1
s A d
U U A
1
s B d
H AA
H BB H AB
H BA
Copyright – Michael D. Fayer, 2007
Energies and Eigenfunctions
E S
H AA
1
H
AB E A
H AA
1
H
AB
S
2 1
U
1
s A
U
1
s B
A
2 1
U
1
s A
U
1
s B
S - symmetric (+ sign ) A - antisymmetric (- sign)
Copyright – Michael D. Fayer, 2007
Evaluation of Matrix Elements Need H AA , H AB , and
H AA
U
* 1
s A H U
1
s A d
H
2
2
m
0
e
2 4
0
r A
e
2 4
0
r B
e
2 4
0
r AB
Part of H looks like Hydrogen atom Hamiltonian
2 2
m
0
e
2 4
0
r A
U
1
s A
E U H
1
s A
These terms operating on can be set equal to ,
A E U
1
s A E H
- energy of 1s state of H atom.
Copyright – Michael D. Fayer, 2007
Then
H AA
*
U
1
s A
E H
e
2 4
0
r B
e
2 4
0
r AB
U
1
s A d
H AA
E H e
2 4
0
a D
0
J
*
U
1
s A
e
2 4
0
r B
U
1
s A d
(Coulomb Integral)
J
e
2 4
0
a
0
1
D
e
2
D
1
1
D
D
r AB a
0
a
0
0
h e
2 2 distance in units of the Bohr radius
Copyright – Michael D. Fayer, 2007
H BA
U
1 *
s B
E H
e
2 4
0
r B
e
2 4
0
r AB
U
1
s A d
E H
e
2 4
0
K
*
U
1
s B
e
2 4
0
r B
U
1
s A d
(Again collecting terms equal to the H atom Hamiltonian.) (Exchange integral)
K
e
2 4
0
a
0
e
D
1
D
e
D
1 1 3
D
2
(K is a negative number) J - Coulomb integral - interaction of electron in 1s orbital around A with a proton at B.
K - Exchange integral – exchange (resonance) of electron between the two nuclei.
Copyright – Michael D. Fayer, 2007
These results yield
E S
E H
e
2 4
0
E A
E H
e
2 4
0
J
K
1
J
K
1
(K is a negative number) The essential difference between these is the sign of the exchange integral, K.
Also consider
E N
H AA
E H
e
2 4
0
a
0
e
2
D
1
1
D
Classical no exchange Interaction of hydrogen 1s electron charge distribution at A with a proton (point charge) at B.
Electron fixed on A.
Copyright – Michael D. Fayer, 2007
classical – no exchange
-0.8
repulsive at all distances anti-bonding M.O.
-0.9
E A
E
e
2 8
0
a
0
-1.0
-1.1
E N
D e
E S
-1.2
0 2
r AB
4
D =
r AB a
o 6
r AB
equilibrium bond length
D e
dissociation energy bonding M.O.
8 10
H 1s energy
E H
1.0
8
e
2 0
a
0
Copyright – Michael D. Fayer, 2007
This Calc.
Dissociation Energy
D e
1.77 eV (36%) Exp.
2.78 eV Variation in Z' 2.25 eV (19%) Equilibrium Distance
r AB
1.32 Å (25%) 1.06 Å 1.06 Å (0%)
Copyright – Michael D. Fayer, 2007
Hydrogen Molecule
e
2
r
12 e 1
r A
2
r B
2
r A
1 H
A
+
r A B r B
1 H
B
+
In Born-Oppenheimer Approximation the electronic Schrödinger equation is
2 1
2 2
2
m
0 2
E
e
2 4
0
r A
1
e
2 4
0
r B
1
e
2 4
0
r A
2
e
2 4
0
r B
2
e
2 4
e
2 4
0
r AB
0
Copyright – Michael D. Fayer, 2007
r AB
, system goes to 2 hydrogen atoms Use product wavefunction as basis functions
I
U
1
s A
(1)
U
1
s B
(2)
II
U
1
s A
(2)
U
1
s B
(1) These are degenerate.
Copyright – Michael D. Fayer, 2007
Form Hamiltonian Matrix
H H I I
H II I H I II H II II
H I I H II I
E
2
E H I II H II II
2
E
E
0 secular determinant
H I I
I
*
H
I
2
H I II
*
I H
II
2
I II
2
H I I
H II II H I II
H II I
Copyright – Michael D. Fayer, 2007
Diagonalization yields
E S
H I I
H I II
1
2
E A
H I I
H I II
1
2
S
2 1 2
U
1
s A U
1
s B
A
2 1 2
U
1
s A U
1
s B
U
1
s B U
1
s A
U
1
s B U
1
s A
S - symmetric orbital wavefunction A - antisymmetric orbital wavefunction
Copyright – Michael D. Fayer, 2007
Evaluating attraction to nuclei
H I I
U
* 1
s A
(1)
U
* 1
s B
(2) 2
E H
e
2 4
0
r B
1
e
2 4
0
r A
2 nuclear-nuclear repulsion Two kinetic energy terms and two electron-nuclear attraction terms comprise 2 H atom Hamiltonians.
e
2 4
e
2 4
0
r AB
U
1
s A
(1)
U
1
s B
(2)
2 electron-electron repulsion
H I I
2
E H
2
J
J
e
2 4
0
r AB
J same as before.
J
e
2 4
0
U
1
s A
(1)
U
1
s B
(2)
r
12 2
1 2
J
e
2 4
0
a
0
1
D
e
2
D
1
D
11 8
3 4
D
1 6
D
2
D
r AB
/
a
0
Copyright – Michael D. Fayer, 2007
H I II
U
1 *
s A
(1)
U
* 1
s B
(2) 2
E H
e
2 4
0
r B
1
e
2 4
0
r A
2
e
2 4
e
2 4
0
r AB
U
1
s A
(2)
U
1
s B
(1)
1 2
H I II
2
E H K K
2
e
2 4
0
r AB K
e
2 4
0
U
1
s A
(1)
U
1
s B
(2)
r
12
U
1
s A
(2)
U
1
s B
(1)
K
e
2 20
0
a
0
e
2
D
25 8
23
D
3
D
2
4 1 3
D
3
K and
same as before.
1 2
D
r AB
/
a
0
6
D
2
1og
D
2
Ei
4
D
Ei
2
D
0.5772
(Euler's constant)
e D
1 1 3
D
2
E i
- integral logarithm math tables, approx., see book.
Copyright – Michael D. Fayer, 2007
J and K - same physical meaning as before.
J - Coulomb integral. Attraction of electron around one nucleus for the other nucleus.
K - Corresponding exchange integral.
J' - Coulomb integral. Interaction of electron in 1s orbital on nucleus A with electron in 1s orbital on nucleus B.
K' - Corresponding exchange or resonance integral.
Copyright – Michael D. Fayer, 2007
Putting the pieces together
E S
2
E H
e
2 4
0
r AB
2
J
J
1
2
K K
E A
2
E H
e
2 4
0
r AB
2
J
J
1
2
K K
K' is negative.
The essential difference between these is the sign of the K' term.
E N
H I I
2
E H
2
J
J
e
2 4
0
r AB
Energy of
I
U
1
s A U
1
s B
No exchange.
Classical interaction of spherical H atom charge distributions.
Charge distribution about H atoms A and B.
Electron 1 stays on A.
Electron 2 stays on B.
Copyright – Michael D. Fayer, 2007
classical – no exchange
-1.6
-1.7
-1.8
E A
repulsive at all distances anti-bonding M.O.
-1.9
This Calc.
-2
E N
-2.1
D e
E S
-2.2
r AB
-2.3
0.5
1.5
2.5
D =
r AB a
o 3.5
Dissociation Energy
D e
3.14 eV (33%) bonding M.O.
4.5
5.5
2 × H 1s energy
E H
1.0
8
e
2 0
a
0 Equilibrium Distance 0.80 Å (8%)
r AB
Exp.
4.72 eV 0.74 Å Variation in Z' 3.76 eV (19%) 0.76 Å (3%) Vibrational Frequency - fit to parabola, 3400 cm -1 ; exp, 4318 cm -1
Copyright – Michael D. Fayer, 2007