Bra-ket notation Quantum states representations Homo-nuclear diatomic molecule
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•Bra-ket notation•Quantum states representations•Homo-nuclear diatomic molecule•Hetero-nuclear diatomic molecule•Bond energy
The Diatomic Molecule MATS-535 Electronics and Photonics Materials
Dr. Vladimir GavrilenkoNorfolk State University
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Bra and ket notation
A wave function is a representation of the quantum state in real space. The is called a ‘ket’. At each point r in space the quantum state is represented by the function .
r r
The quantum state could be expanded in a set of ortho-normal basis states:
C
Where C’s are called expansion coefficients
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Bra and ket notation
xpsh 2
11
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(**) HW
zyx
zyx
zyx
zyx
pppsh
pppsh
pppsh
pppsh
2
12
12
12
1
4
3
2
1
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Wave Functions of Hydrogen Atom
zy
x
p
p
s
s
,2
2
2
1
PFrRr ,,
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Atomic Wave Function Orthonormality
,0
,0
,1
,1
ddrdnpns
ji
jiddrdnpnp
ddrdnsns
i
ji
djiddrdrr ji ,,,,
(*)HW
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The Homonuclear Diatomic Molecule
1 2
Schrodinger equations for isolated H-atoms
22
,11
2
1
f
f
EH
EH
Full wave function of the H-molecule ,21 21 CC
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The Electronic Structure
2121
,
2121 CCECCH
EH
212212
,211211
2121
2121
CCECCH
CCECCH
Schrodinger equation
Projection onto basis set
Orthogonality conditions:
01221
,12211
dd
dd
2112
02211
2222211
1122111
,
,
HH
EHH
ECHCHC
ECHCHC
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The Secular Equation
0
,0
201
210
CEEC
CCEE
Secular equation
0
0
220
0
0
EE
EE
EE
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Solutions of the Secular Equation
Solutions
0
0
EE
EE
a
b Bonding (b) and antibonding (a) molecular orbital energies
212
1
212
1
a
b
Normalized eigen states
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Electron Energy Structure and Wave Functions of Hydrogen Molecule
LUMO – Lowest Unoccupied Molecular Orbital
HOMO – Highest Occupied Molecular Orbital
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Wave Functions
Analysis
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Wave Functions
Analysis
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Dependence on Time
Hdt
di Time dependent Schrodinger equation
Substitute: ,21 21 CC
1221
2211
2112
02211
HHHH
EHHHH
2221212
2121111
CHCHdt
dCi
CHCHdt
dCi
2012
2101
CECdt
dCi
CCEdt
dCi
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Dependence on Time
First order differential equations with constant coefficients are solved by exponential functions:
,22
11
ti
ti
eAtC
eAtC
titi
titi
ab
ab
beaetC
beaetC
2
1 ,
where
00 ,EE
ab
Boundary conditions: at t=0 molecule is in state 1. Therefore:
2/1
00,10 21
ba
CC
tSinetC
tCosetC
tEi
tEi
0
0
2
1 ,The probability that the molecule is in state 1 or 2:
tSintC
tCostC
22
2
22
1 ,
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The Heteronuclear Diatomic Molecule
A B
Schrodinger equations for isolated H-atoms
BEBH
AEAH
B
A
,
Assume: BA EE
Full wave function of the H-molecule ,BCAC BA
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The Electronic Structure
Schrodinger equation: BCACEBCACH
EH
BABA
,
Projection onto basis set
BCACEBBCACHB
BCACEABCACHA
BABA
BABA
,
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The Secular Equation
0
,0
BBA
BAA
CEEC
CCEE
Secular equation
0
0
2
EEEE
EE
EE
BA
B
A
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The Secular Equation
Substitution:
BA
BA
EE
EE
2
1
,2
1 Average on-site energy
Solution: 2/122
2/122
a
b
E
E
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Charge Redistribution
2/122
2/122
a
b
E
EInsert
0
,0
BBA
BAA
CEEC
CCEE
Obtain for:
x
.1221
1
,1221
1
2/122
2
2/122
2
xxxC
C
xxxC
C
B
A
B
A
For the bonding state
For the antibonding state
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The Charge Transfer in Heteronuclear Diatomic Molecule
A B
1. For: 1,02
B
A
C
Cx
The homonuclear case: no charge transfer
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The Charge Transfer in Heteronuclear Diatomic Molecule
A B
2. For:
1
0
,1 2
2
B
A
B
A
C
Cgantibondin
C
Cbonding
x
BA EE
1. Bonding state: charge is transferred to the B-molecule (lower on-site energy)2. Antibonding state: charge is transferred to the A-molecule (higher on-site energy)
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The Ionic Bond Parameters
Polarity:
2/121 x
xp
Covalency: 2/121
1
xc
122 cp
0x
x Completely ionic limit
Completely covalent limit
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Problems:
1. Using solutions of the secular equation for homonuclear diatomic molecule obtain orthonormal wave functions (see slide 10)
2. Show that wave functions of hydrogen atom are mutually orthogonal (problem marked by(*)) (slide 6).
3. Assuming mutual ortho-normality of atomic s- and p-functions show ortho-normality of the sp3 hybrides (problem marked by(**)) (slide 4).
4. Obtain conditions for eigen function coefficients corresponding to bonding and antibonding states for heteronuclear diatomic molecule (slide 22).