1

•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

2

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

3

Bra and ket notation

xpsh 2

11

4

(**) HW

zyx

zyx

zyx

zyx

pppsh

pppsh

pppsh

pppsh

2

12

12

12

1

4

3

2

1

5

Wave Functions of Hydrogen Atom

zy

x

p

p

s

s

,2

2

2

1

PFrRr ,,

6

Atomic Wave Function Orthonormality

,0

,0

,1

,1

ddrdnpns

ji

jiddrdnpnp

ddrdnsns

i

ji

djiddrdrr ji ,,,,

(*)HW

7

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

8

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

12

Wave Functions

Analysis

13

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).