20_01fig_PChem.jpg

Hydrogen Atom

M

m

r

Potential Energy2

( )4 4

e n

o o

q q ZeV r

r r

+

Kinetic Energy

ˆ ˆ ˆn eK K K

R

C2 2

2 2ˆ2 2R rKM m

22ˆ ( )

2 rK rm

M m m

20_01fig_PChem.jpg

Hydrogen Atom2 2

2ˆ ˆ ˆ( ) ( )2 4r

o

ZeH K r V r

m r

22 2

2 2 2 2 2

1 1 1sin

sin sinr

d d d d dr

r dr dr r d d r d

22 2

2 2

1 1ˆ sinsin sin

d d dL

d d d

2 2 22

2 2

ˆˆ

2 2 4 o

d d L ZeH r

mr dr dr mr r

Radial Angular Coulombic

20_01fig_PChem.jpg

Hydrogen Atom( , , )r will be an eigenfunction of 2ˆ ˆ ˆ, & zH L L

.( , , ) ( ) ( , )n l mr R r Y Separable

ˆ ( , , ) ( , , )H r E r

2 2 22

. .2 2

ˆ( ) ( , ) ( ) ( , )

2 2 4 n l m n n l mo

d d L Zer R r Y E R r Y

mr dr dr mr r

2 22

. .2 2

( 1)( , ) ( ) ( , ) ( )

2 2l m n l m n

d d l lY r R r Y R r

mr dr dr mr

2

. .( , ) ( ) ( ) ( , )4l m n n n l m

o

ZeY R r E R r Y

r

2 2 22

2 2

( 1)( ) 0

2 2 4 n no

d d l l Zer E R r

mr dr dr mr r

20_01fig_PChem.jpg

Hydrogen Atom

22

2 2 2 2

21 ( 1)( ) 0

2n

no

E md d l l Zmer R r

r dr dr r r

2 22 2

2 2 2 2

1 1 22

d d d d d dr r r

r dr dr r dr dr r dr dr

Recall

2

2

40.0529o

oa nmme

Bohr Radius

2

2 2 2

22 ( 1) 2( ) 0n

no

E md d l l ZR r

r dr dr r a r

20_01fig_PChem.jpg

Hydrogen Atom

21

2 2 2

22 ( 1) 20r

o

E md d l l Ze

r dr dr r a r

Assume1( ) 0 asR r r

1( ) rR r e Let’s try

2 12 2

22 ( 1) 20r

o

E ml l Ze

r r a r

2 12 2

21 2 1( 1) 2 0

o

E mZl l

r a r

2 12

22( 1) 0; 2 0; & 0

o

E mZl l

a

It is a ground state as it has no nodes

20_01fig_PChem.jpg

Hydrogen Atom

2 2

10; ; &2o

Zl E

a m

2 12

22( 1) 0; 2 0; & 0

o

E mZl l

a

22 2 2 2 2

1 2 202 2 4 o

Z Z meE

a m m

2 4

2 24 2o

Z me

0

1

0,( , , ) ( ) ( , ) ( )

Zr

aln l mr R r Y CR r Ce

The ground state as it has no nodes n=1, and since l=0 and m = 0, the wavefunction will have no angular dependence

2 2

1 202

ZE

a m

20_01fig_PChem.jpg

Hydrogen Atom

In general:

Laguerre Polynomials

11

12

33

1 23

33

55

1 0 ( ) 1

2 0 ( ) (2!)(2 )

1 ( ) (3!)

3 0 ( ) (3!) 3 3 0.5

1 ( ) (4!)(4 )

2 ( ) (5!)

n l L x

n l L x x

l L x

n l L x x x

l L x x

l L x

0

2Zrx

na

0

32 1

4 3 30 0 0

4 ( 1)! 2 2( )

[( )]

l Zr

nal ln n l

Z n l Zr ZrR r e L

n a n l na na

2 1

0

2ln l

ZrL

na

1S- 0 nodes

2S- 1 node

3S-2 nodes

Energies of the Hydrogen Atom

In general:

4 2

2 2 2

1

24n

o

me ZE

n

2 2

20

1

24 o

e Z

a n

2

22

Z

n

2

0

27.24H

o

eE eV

a

Hartrees

kJ/mol

627.51 / 2625.5 /kcal mol kJ mol

Wave functions of the Hydrogen Atom

In general:

0

32 1

4 3 30 0 0

4 ( 1)! 2 2( )

[( )]

l Zr

nal ln n l

Z n l Zr ZrR r e L

n a n l na na

,

1( , ) (cos( ))

2mm im

l m l lY C P e

,( , , ) ( ) ( , )ln l mr R r Y

Z=1, n = 1, l = 0, and m = 0:

00 (cos( )) 1P 0

0

1

2C

11

0

21

rL

a

01 0,0( , , ) ( ) ( , )r R r Y

001 3

0

2( )

r

aR r ea

0,0

1( , )

2Y

0 0

3 30 0

2 1 1

2

r r

a ae ea a

Z=1, n = 2, l = 0, and m = 0:

12

0 0

2! 2r r

La a

0,0

1( , )

2Y

02

300

122 2

r

ae r

aa

0202 3

00

1( ) 1

22

r

a rR r e

aa

02 0,0( , , ) ( ) ( , )r R r Y

Wave functions of the Hydrogen Atom

Hydrogen AtomZ=1, n = 2, l = 1

022,1,0 3

0 0

1 2( , , ) cos

8

r

arr e

a a

m = 0: m = +1/-1:

022,1, 1 3

0 0

1 1( , , ) sin

8

r

arr e e

a a

022,1, 2,1, 1 2,1, 1 3

0 0

1 1 1( , , ) ( , , ) ( , , ) sin cos

2 8

r

ax

rr r r e

a a

022,1, 2,1, 1 2,1, 1 3

0 0

1 1 1( , , ) ( , , ) ( , , ) sin sin

2 8

r

ay

rr r r e

i a a

+

_

-+-+

+-

-+

+- -

+

20_06fig_PChem.jpg

* 2, , , ,( ) ( , , ) ( , , ) sinn l m n l mP R r r r drd d

* * 2, ,( ) ( , ) ( ) ( , ) sinl l

n l m n l mR r Y R r Y r drd d * 2 *

, ,( ) ( ) ( , ) ( , )sinl ln n l m l mR r R r r dr Y Y d d

For radial distribution functions we integrate over all angles only

2* 2 *

, ,

0 0

( ) ( ) ( ) ( , ) ( , )sinl ln n l m l mP r R r R r r Y Y d d

* 2( ) ( )l ln nR r R r r Prob. density as a function of r.

Radial Distribution Functions

20_09fig_PChem.jpg

Radial Distribution Functions0

22

30

4r

are

a

0* 0 21,0,0 1 1( ) ( ) ( )P r R r R r r00

1 30

2( )

r

aR r ea

0202 3

00

1( ) 1

22

r

a rR r e

aa

0* 0 22,0,0 2 2( ) ( ) ( )P r R r R r r

0 3 42

30 0 02 4

r

ae r rr

a a a

20_08fig_PChem.jpg

* 2, , , ,( , , ) ( , , ) ( , , ) sinn l m n l mP r r r r

* 2 *, , , ,( ) ( ) ( , ) ( , )sinl l

n n l m l mR r R r r Y Y

, ,( ) ( , )n l l mP r P

X

Y

Z

Probability Distributions

0

42

2,1 1,0 50

( ) ( , ) cos sin32

r

arP r P e

a

, ( )n lP r, ( , )l mY

20_12fig_PChem.jpg

Atomic Units

Set:2

2

41 . .o

oa a ume

2 4 2

2 22 2 24 2n

o

Z me ZE

nn

2 2 22

2 2

ˆˆ

2 2 4e e o

d d L ZeH r

m r dr dr m r r

2

0

1, 1, & 14

e

em

Hartrees

22

2 2

ˆ1

2 2

d d L Zr

r dr dr r r

2

2

Z

r

a.u.

Much simpler forms.

0

3 3

1 30

Zr

a Zrs

Z Ze e

a

023 3 2

2 300

1 12 22 22 2

r ra

s

Z e r Z e r

aa

AtomsPotential Energy

2

( )4 4

e nen i

o i o i

q q ZeV r

r r

Kinetic Energy2 2

2 2ˆ2 2 iR r

i e

KM m

22ˆ ( )

2 ii ri i e

K rm

C

me

me

2

( )4 4

i jee

o ij o ij

q q eV r

r r

=r12

M

2

2i

i

i

Z

r

1

ijr

1

1( ) ( )en i ee ij

i i j i iji ij

ZV V r V r

r r

Helium Atom

C

me

me

=r12

M

2 1ˆ ˆ ˆ2i

i i j ii ij

ZH K V

r r

2 1

2i

i i ji ij

Z

r r

,

1ˆi

i i i j ij

Hr

1 212

1ˆ ˆH Hr

Cannot be separated!!!

2

2i

ii

ZH

r

Hydrogen like 1 e’ Hamiltonian

i.e. r12 cannot be expressed as a function of just r1 or just r2

What kind of approximations can be made?

Ground State Energy of Helium Atom

Eo

E1

E2

I1 = 24.587 ev

Eo

E1

E2

I2 = 54.416 ev

Ionization Energy of He

EFree

Eo=- 24.587 - 54.416 ev

=- 79.003 ev =- 2.9033 Hartrees

Perturbation Theory1 2

12

1ˆ ˆ ˆH H Hr

01 2

ˆ ˆ ˆH H H 1

12

1H

r 0 0 0

1 2 1 2( , ) ( ) ( )r r r r

0 0 01 1 1 1 1 1

ˆ ( ) ( )H r E r 101

1(1 ) rs e

1

20 11

12r Z

Hr

2 20 11 2 2

1

22

2 2 1

ZE

n

Ground State Energy of Helium Atom0

1 2ˆ ˆ ˆH H H 0 0 0

1 2 1 2( , ) ( ) ( )r r r r

0 0 0 01 2 1 2 1 2

ˆ ˆ ˆ( , ) ( ) ( )H r r H H r r 0 0 0 0

1 1 2 2 1 2ˆ ˆ( ) ( ) ( ) ( )H r r H r r

0 0 0 02 1 1 1 2 1

ˆ ˆ( ) ( ) ( ) ( )r H r r H r

0 0 0 02 1 1 1 2 1( ) ( ) ( ) ( )r E r r E r

1 2 0 00 0 2 1( ) ( )E E r r

0 01 2( , )E r r

0 0 01 2 2 2 4E E E H

20 011 22

1

22

ZE E

n

Not even close.Off by 1.1 H, or3000 kJ/mol

Therefore e’-e’ correlation, Vee, is very significant

Ground State Energy of Helium Atom

0

12

1ˆ ˆH Hr

1

12

1H

r

0 0 01 2 1 2( , ) ( ) ( )r r r r

0 01 1 1 1 1 1

ˆ ( ) ( )H r E r

01 2

ˆ ˆ ˆH H H

0 1 0 0 0 1 01 2 1 2 1 2

ˆ, ,E E E E E r r H r r

1 2

0 1 0 0* 01 2 1 2 1 2 1 2 1 2

12

1ˆ, , , ,S S

r r H r r r r r r dV dVr

1 2 1 2ˆ ( , ) ( , )H r r E r r 0 1

1 2 1 2 1 2( , ) ( , ) ( , )r r r r r r

Ground State Energy of Helium Atom

30 (1 ) iZri

Zs e

1 2

1 0 1 0 0* 01 2 1 2 1 2 1 2 1 2

12

1ˆ, , , ,S S

E r r H r r r r r r dV dVr

1 2

0* 0* 0 01 2 1 2 1 2

12

1

S S

r r r r dV dVr

1 2 1 2

2 22 2

1 2 1 2 1 2 1 2 1 2120 0 0 0 0 0

2 2 2 2 1 2 2 2 2sin sinZr Zr Zr Zre e e e r r dr dr d d d d

r

12

1 5 51 (1)1 (2) 1 (1)1 (2)

8 4

Zs s s s

r

0 1 54 2.75H

4E E E Closer but still far off!!!

1

0

1.2531.5%

4

E

E

Perturbation is too large for PT to be accurate, much higher corrections would be required

Variational Method

0ˆ

exact exactH E

i ii

c i i iH E

The wavefunction can be optimized to the system to make it more suitable

Consider a trail wavefunction t and exact

Is the true wavefunction, where:

Then

0

ˆt t

t t

HE

The exact energy is a lower bound

,n exactis a complete set

Assume the trial function can be expressed in terms of the exact functions

0 0ˆ ˆ 0t t t t t tH E H E

We need to show that

texact

t

Variational Method

0 0ˆ ˆ

t t i i j ji j

H E c H E c

*0

ˆi j i j

i j

c c H E *

0ˆ

i j i j i ji j

c c H E *

0i j i ij iji j

c c E E

*0 0i i i

i

c c E E *

00 & 0i i ic c E E

Since

0

ˆt t

t t

HE

Variational Energy

var

ˆ( ) ( )( )

( ) ( )t t

t t

HE

E0

Evar()

var ( ) 0d

Ed

min

2

var2( ) 0

dE

d

Variational Method For He Atom3

1,0,01 ( ) ( ) iZri

Zs i e

r

Let’s optimize the value of Z, since the presence of a second electrons shields the nucleus, effectively lowering its charge.

1 2 1 2var

1 2 1 2

ˆ( , ) ( , )

( , ) ( , )

HE

r r r r

r r r r

33

1,0,01 ( ) ( ) eff ieff Z r

i

Zs i e

r 1 2( , ) 1 (1)1 (2)s s r r

1 2 1 2( , ) ( , ) 1 (1) 1 (1) 1 (2) 1 (2) 1s s s s r r r r

var 1 212

1ˆ ˆ1 (1)1 (2) 1 (1)1 (2)E s s H H s sr

Variational Method For He Atom

var 1 212

1ˆ ˆ1 (1)1 (2) 1 (1)1 (2)E s s H H s sr

1 2

12

ˆ ˆ1 (1)1 (2) 1 (1)1 (2) 1 (1)1 (2) 1 (1)1 (2)

11 (1)1 (2) 1 (1)1 (2)

s s H s s s s H s s

s s s sr

1 2

12

ˆ ˆ1 (1) 1 (1) 1 (2) 1 (2) 1 (2) 1 (2) 1 (1) 1 (1)

11 (1)1 (2) 1 (1)1 (2)

s H s s s s H s s s

s s s sr

1 212

1ˆ ˆ1 (1) 1 (1) 1 (2) 1 (2) 1 (1)1 (2) 1 (1)1 (2)s H s s H s s s s sr

Variational Method For He Atom

31

3

1 (1) effeff Z rZs e

var 1 212

1ˆ ˆ1 (1) 1 (1) 1 (2) 1 (2) 1 (1)1 (2) 1 (1)1 (2)E s H s s H s s s s sr

1 1 11 1 1 1

ˆ ˆ ˆ1 (1) 1 (1) 1 (1) 1 (1) 1 (1) 1 (1)eff effZ ZZ Zs H s s K s s K s

r r r r

11 1

1ˆ1 (1) 1 (1) 1 (1) 1 (1)effeff

Zs K s Z Z s s

r r

1 11

ˆ ˆ effZH K

r

2

1ˆ 1 (1) 1 (1)

2effZ

H s s 2

22eff

n

ZE

n

2

11 1

1ˆ1 (1) 1 (1) 1 (1) 1 (1)2eff

eff

Zs H s Z Z s s

r

Variational Method For He Atom2

11 1

1ˆ1 (1) 1 (1) 1 (1) 1 (1)2eff

eff

Zs H s Z Z s s

r

3 31 1

3 322

1 1 1 1 11 10 0 0

1 11 (1) 1 (1) sineff effeff effZ r Z rZ Zs s e e r drd d

r r

31

3 22

1 1 1 1 1

0 0 0

sineffZ reffZre dr d d

3123

1 1

0

4 effZ r

effZ re dr

200

1( 1)au auue du au e

a

23 0

2

14 2 1 2 0 1

2

effZ aeff eff eff

eff

Z Z e Z eZ

32

14 1

4eff effeff

Z ZZ

Variational Method For He Atom2

11 1

1ˆ1 (1) 1 (1) 1 (1) 1 (1)2eff

eff

Zs H s Z Z s s

r

2

2eff

eff eff

ZZ Z Z

2

2ˆ1 (2) 1 (2)

2eff

eff eff

Zs H s Z Z Z

12

1 51 (1)2 (2) 1 (1)2 (2)

8 effs s s s Zr

Similarly

Recall from PT

var 1 212

1ˆ ˆ1 (1) 1 (1) 1 (2) 1 (2) 1 (1)1 (2) 1 (1)1 (2)E s H s s H s s s s sr

2 2

5

2 2 8eff eff

eff eff eff eff eff

Z ZZ Z Z Z Z Z Z

Variational Method For He Atom

2 2var

52

8eff eff eff effE Z Z ZZ Z

2 52

8eff eff effZ ZZ Z

var

5 52 2 0

8 16eff effeff

dE Z Z Z Z

dZ

5 272

16 16effZ

2

var

27 27 5 272(2) 2.8479H

16 16 8 16E

Much closer to -2.9033 H

(E= 0.055 H = kJ/mol error)

Variational Method For He Atom

271.69

16effZ 3 27

161 271 ( )

16

ir

s i e

1 2

3 27 27

16 161 2

1 27( , ) 1 (1)1 (2)

16

r rs s e e

r r

Optimized wavefunction

Variational Method For He Atom

27

16effZ 3 27

161 271 ( )

16

ir

s i e

1 2

3 27 27

16 161 2

1 27( , ) 1 (1)1 (2)

16

r rs s e e

r r

Optimized wavefunction

1 2 1 2

3

2

1 2( , ) Z r Z r Z r Z rZ Ze e e e

r r

1.19 & 2.18Z Z var 2.8757HE

Other Trail Functions

(E= 0.027 H = kJ/mol error)

Optimizes both nuclear charges simultaneously

Variational Method For He Atom

1 2( )1 2 12

1( , ) (1 )Z r re br

N r r

1.849 & 0.364Z b var 2.8920HE

Other Trail Functions

(E= 0.011 H = kJ/mol error)

Z’, b are optimized. Accounts for dependence on r12.

In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required.

The more accurate the energy the more parameters that must be optimizedthe more demanding the calculation.

Variational Method For He Atom

In M.O. calculations the wavefunction used are designed to give the most accurate energies for the least computational effort required.

The more accurate the energy the more parameters that must be optimizedthe more demanding the calculation.

-2.862879 H

-2.862871 H

-2.84885 H

Experimental -79.003 ev -2.9003 H

The H2+ Molecule

2

( )4

i ine i

o i i

Z e ZV r

r r

2 2 2 22 2 2 2ˆ ˆ ˆ

2 2 2 2A B A Bnuclear electronic R R r rA B e e

K K KM M m m

One electron problem

Two nuclei

Define electron position, i..e. internal coordinates, w.r.t. nuclear positions.

2

( )4

A B A Bnn AB

o AB AB

Z Z e Z ZV R

R R

2 2 2 21 1 1 1

2 2 2 2A B A BR R r rA BM M

The H2+ Molecule

1 1 1

AB A BR r r

2 2 2 21 1 1 1

2 2 2 2A B A BR R r rA BM M

ˆ ˆ ˆH K V

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )N A N B e A e BK K R K R K r K r

ˆ ˆ ˆ ˆ( ) ( ) ( )nn AB en A en BV V R V r V r

Since ZA=1 and ZB=1

2 2 2 21 1 1 1 1 1 1ˆ2 2 2 2A B A BR R r r

A B AB A B

HM M R r r

The H2+ Molecule

2 2 2 21 1 1 1 1 1 1ˆ2 2 2 2A B A BR R r r

A B AB A B

HM M R r r

Nuclear Electronic

( , ) ( ; ) ( )R r R r R

The nuclear positions determine the electronic wavefunction

Assume electronic motion is much faster than nuclear motion, implies that the nuclear positions are essentially static

ˆ ˆ ˆ( ) ( ; )N eH H H R R rThe electronic part is determined by the nuclear positions

Separable??

ˆ ( , ) ( , )H R r W R r W- Total Energy

The H2+ Molecule

ˆ ˆ ˆ( , ) ( ) ( ; ) ( ; ) ( ) ( ; ) ( )N e TH R r H H R R E R R r r R r R

ˆ ˆ ˆ ˆ ( ; ) ( ) ( ; ) ( )N nn e ne TK V K V R E R r R r R

ˆ ˆ ( ) ( )N nn NK V E R R

ˆ ˆ ( ; ) ( ; )e ne eK V R E R r r

Potential energy surface.

Of primary interest

Nuclear

Electronic

ˆ ˆ ( ) ( )N nn e TK V E E R R

Linear Variational WFctns.

( ) ( )i ii

c r r

Suppose the trial wavefunction can be expressed in terms of an expansion of an appropriate set of functions, not necessarily othonormal

2 1ii

c 1

i j ijij

i jS

S i j

var

ˆ( ) ( )( )

( ) ( )

HE

r r

rr r

ˆ ˆi i j j i j i j

i j i j

i j i ji i j j i j

i j

c H c c c H

c cc c

i j iji j

i j iji j

c c H

c c S

Linear Variational WFctns.

var i j ij i j iji j i j

E c c S c c H

var 0i

dE

dc

2i i j ij i j ijj ji i

d dE c c S c c H

dc dc

2i i j ij i j ijj j

E c c S c c H

For each ci

Find the optimum coefficients, that minimize Evar.

ii j ij i i j ij i j ij

j j ji i i

dE d dc c S E c c S c c H

dc dc dc

0j ij j ij ij

c H c S E

1 1

Linear Variational WFctns.

11 12 12 1 1 1 1 1

21 21 22 2 2 2 2 2

1 1 2 2

1 1 2 2

i i j i j n i n i

i i j i j n i n i

j i j j i j jj i jn i jn ij

n i n n i n nj i nj nn i in

H E H E S H E S H E S c

H E S H E H E S H E S c

H E S H E S H E H E S c

H E S H E S H E S H E c

0

ˆ ˆi jH H

* * * *1 2

ˆ wherei i i i i i ij inH E c c c c

0j ij ij ij

c H S E 0i iE H S

1i j

ij

i j

S i j

S

Need to diagonalize matrix, to find eigenvalues and eigen vectors:

Linear Combination of Atomic Orbitals.Lets use the 1s Hydrogen like orbitals to be a basis for a trial function and apply variational theory to find the best approximate wavefunction

( ) 1 ( ) 1 ( )i iA A iB Bc s c s r r r

1 ( ) & 1 ( )A Bs sr rWhere are Hydrogen like wavefunction with n=1, l=0, centred in nucleus a and b, resp.

2 21 1 1 1ˆ2 2A Br r

A B

Hr r

ˆi i iH E

( ) 1 ( ) 1 ( )i iA A iB Bc s c s r r r

Linear Combination of Atomic Orbitals.

2 21 1 1 1ˆ ˆ1 1 1 12 2A Bk l k r r l kl

A B

s H s s s Hr r

H

0i iE H S

* *i iA iBc c

11 1k l

lk

l ks s

S l k

S

0AA i AB i AB iA

BA i BA BB i iB

H E H E S c

H E S H E c

ˆ 1 1 1 1iA A iB B i iA A iB BH c s c s E c s c s

Linear Combination of Atomic Orbitals.

2 21 1 1 1ˆ1 1 1 1 12 2A Bkk k k k r r k

A B

H s H s s sr r

2

1 1 1 1 13

Rkl k l l k lk

RS s s s s S e R

0AA i AB i AB iA

AB i AB AA i iB

H E H E S c

H E S H E c

1ˆ ˆ1 1 1 1 12

Rkl k l kl l kH s H s S R e s H s

21 1 11

2Re

R R

AA BBH H

AB BAH H

AB BAS S

Linear Combination of Atomic Orbitals.

0AA i AB i AB

AB i AB AA i

H E H E S

H E S H E

2 2

0AA i AB i ABH E H E S

2 2 2 2(1 ) 2 ( ) 0i AB i AB AB AA AA ABE S E H S H H H

2 2 2 2

2

2( ) 4( ) 4(1 )

2(1 )

AA AB AB AA AB AB AB AA AB

AB

H H S H H S S H HE

S

2 2 22( ) 2 2

2(1 )(1 )AA AB AB AB AB AA AB AA AB

AB AB

H H S H H H S H S

S S

( ) ( )

(1 )(1 ) (1 )AA ABAA AB AB AB AA AB

AB AB AB

H HH H S H H S

S S S

Prediction of the Bond

(1 )AA AB

AB

H HE

S

2

2

1 ( 1) ( 1) 1

21 1

3

R R

g

R

R e R R eE

RR R e

Bonding and Antibonding Orbitals of H2+

23_09fig_PChem.jpg

Density Difference Between MO’s and 1s O’s

23_11fig_PChem.jpg

Electron Densities of Sigma and Pi M.O’.s

1 1( ) 1 ( ) 1 ( )

2 2g A Bs s r r r* 1 1( ) 1 ( ) 1 ( )

2 2u A Bs s r r r

*, ,

1 1( ) 2 ( ) 2 ( )

2 2g x A x Bp p r r r , ,

1 1( ) 2 ( ) 2 ( )

2 2g x A x Bp p r r r

Bonding

BondingAntibonding

Antibonding

g=gerade (same) u=ungerade

(opposite)

-13.6 e.v.

-19.6 e.v.

-18.6 e.v.

Electron population on F is larger, ie. bond in polarized to F, ie. shows the F is more electronegative.

(0.345)1 (0.840)2H zFs p

Other Types of M.O.’s

23_13fig_PChem.jpg

MO’s for the Diatomics

23_02tbl_PChem.jpg

Energy Level Diagram For the Diatomics

Electron Configuration for H2 &He2

23_17fig_PChem.jpg

Electron Configuration of N2

23_16fig_PChem.jpg

Electron Configuration of F2

23_18fig_PChem.jpg

Electron Configurations of the Diatomics

23_20fig_PChem.jpg

Bonding in HF