1

Quantum Mechanics and the Chemical Bond

vrije Universiteit amsterdam

Pyrite FeS2

Quartz SiO2

n=1

n=4n=3

n=2

n=5

1,2,3.......1

, 1, 2,....

nl nm l l l l

=≤ −= − − −

The s-wave (l =0) and p-waves (l =1) of the H-atom

+- +

+ --

The d-waves (l =2) of the H atom

+ ++- -

-+

++- -

+

+- -+

+-

-

2

E&R 391

p-states; l=1

d-states; l=2

Chemical bonding in H2+

rL

d

rR

The H2+ molecule-ion (E&R 505-514)

( ) ( )2

, , , ,4 o

e x y z E x y zdψ ψ

πε+ =Proton-proton

repulsion

( ) ( )2 2

, , , ,4 4o L o R

e ex y z x y zr rψ ψ

πε πε− −Proton-electron

attraction

( )2 2 2 2

2 2 2 , ,2

x y zm x y z

ψ⎛ ⎞∂ ∂ ∂

− + +⎜ ⎟∂ ∂ ∂⎝ ⎠Kinetic energy of the electron rL

d

rR

The H2+ molecule-ion

( )

( ) ( )

( ) ( )

2 2 2 2

2 2 2

2 2

2

, ,2

, , , ,4 4

, , , ,4

o L o R

o

x y zm x y z

e ex y z x y zr r

e x y z E x y zd

ψ

ψ ψπε πε

ψ ψπε

⎛ ⎞∂ ∂ ∂− + +⎜ ⎟∂ ∂ ∂⎝ ⎠

− −

+ =

H2+ molecule-ion: simpler notation

L R( )

( ) ( )

( ) ( )

2 2 2 2

2 2 2

2 2

2

, ,2

, , , ,4 4

, , , ,4

o L o R

o

x y zm x y z

e ex y z x y zr r

e x y z E x y zd

ψ

ψ ψπε πε

ψ ψπε

⎛ ⎞∂ ∂ ∂− + +⎜ ⎟∂ ∂ ∂⎝ ⎠

− −

+ =

L R ppT V V V Eψ ψ ψ ψ ψ− − + = L R ppT V V V Eψ ψ ψ ψ ψ− − + =

H2+ molecule-ion: simpler notation

L R

1 1

a H sa b H sbc c

aL bRψ φ φ= +

= +

Probability density cannot depend on what is chosen as left and right

2 2

2 2 2 2

2 2 2 2

22

aL bR

a L abLR b Rb L baLR a R

ψ = +

= + +

= + +2 2 or a b a b a b= ⇒ = = −

3

L R ppT V V V Eψ ψ ψ ψ ψ− − + =

H2+ molecule-ion: normalisation

L R

( )( )

g g

u u

c L R

c L R

ψ

ψ

= +

= −

( )( )

2 2 2 2

2

2

1 2 1 1

d c L LR R d

c S

ψ τ τ= ± +

= ± + =

∫ ∫

1 1 ; 2 2 2 2g uc c

S S= =

+ −

L R ppT V V V Eψ ψ ψ ψ ψ− − + =

H2+ molecule-ion: bonding state

( )g gc L Rψ = +

( )L R ppT V V V Eψ ψ− − + =

( )*L R pp

*

T V V V d

E d E

ψ ψ τ

ψ ψ τ

− − + =

= =

∫∫

As the wave function is normalised

we need only to calculate

( )*L R ppT V V V dψ ψ τ− − +∫ with

H2+ molecule-ion: bonding state

( )g gc L Rψ = +( )*L R ppT V V V dψ ψ τ− − +∫ with

( )( )( )2L R ppT V V VgE c L R L R dτ= + − − + +∫

( )( )( )( )( )( )( )

L R pp

L R pp

L R pp

T V V V

T V V V

T V V V

L R L R d

L R Ld

L R Rd

τ

τ

τ

+ − − + + =

+ − − + +

+ − − + +

∫∫∫

( )( ) ( )( )( )( )( )( )( )

L R pp

1

R

pp

T V V V

V

V

s

L R Ld L R Ld

L R E Ld

L R Ld

L R Ld

τ τ

τ

τ

τ

= + − + + − + =

= + +

+ + −

+ +

∫ ∫∫∫∫

H2+ molecule-ion: bonding state

( )( )L R ppT V V VL R Ldτ+ − − +∫

( )

( )

1

pp

1

V 1

sS EJ K

S

+

− −

+ +

H2+ molecule-ion: bonding state

( )( )( ) ( )

L R pp

1 pp

T V V V

1 V 1s

L R Ld

S E J K S

τ+ − − + =

= + − − + +∫

( ) ( )

( ) ( )

21 pp

1 pp

1 pp

2 1 V 1

1 1 V 11

V1

bonding g s

s

bonding g s

E c S E J K S

S E J K SS

J KE E ES

⎡ ⎤= + − − + + =⎣ ⎦

⎡ ⎤= + − − + + =⎣ ⎦++

= = + −+

Definitions

1 ppV1bonding sJ KE E

S+

= + −+

2

ppV4 o

edπε

= R LS dφ φ τ= ∫

2

4R Ro L

eJ dr

φ φ τπε−

= −∫Coulomb integral

2

4R Lo L

eK dr

φ φ τπε−

= −∫

Exchange integral

4

H2+ molecule-ion: antibonding state

( )( )( ) ( )

L R pp

1 pp

T V V V

1 V 1s

L R Ld

S E J K S

τ+ − − + =

= − − + + −∫

( ) ( )

( ) ( )

21 pp

1 pp

1 pp

2 1 V 1

1 1 V 11

V1

antibonding u s

s

antibonding u s

E c S E J K S

S E J K SS

J KE E ES

⎡ ⎤= − − + + − =⎣ ⎦

⎡ ⎤= − − + + − =⎣ ⎦−−

= = + −−

H2+ molecule-ion: compensation

1 pp

1 pp

V1 1

V1 1

antibonding s

bonding s

J KE ES S

J KE ES S

= + − +− −

= + − −+ +

L R

Proton-proton repulsion is

approx. compensated by proton-electron

attraction

1

1

s

s

E K

E K

≅ +

≅ −

E1s

( )12

L R−

1sE K≅ +

( )12

L R+

1sE K≅ −

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ

Φ=exp(-sqrt( (x-1)^2+y^2))+exp(-sqrt( (x+1)^2+y^2))

Y

X

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ

Φ=exp(-sqrt( (x-1)^2+y^2))-exp(-sqrt( (x+1)^2+y^2))

Y

X

( )12

L R−

( )12

L R+

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ2

IΦI2=(exp(-sqrt( (x-1)^2+y^2))+exp(-sqrt( (x+1)^2+y^2)))^2

Y

X

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ2

IΦI2=(exp(-sqrt( (x-1)^2+y^2))-exp(-sqrt( (x+1)^2+y^2)))^2

Y

X

( )212

L R−

( )212

L R+

Wave function Probability density

E&R page 512

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ

Φ=exp(-sqrt( (x-1)^2+y^2))+exp(-sqrt( (x+1)^2+y^2))

Y

X

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ

Φ=exp(-sqrt( (x-1)^2+y^2))-exp(-sqrt( (x+1)^2+y^2))

Y

X

( )12

L R−

( )12

L R+

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ2

IΦI2=(exp(-sqrt( (x-1)^2+y^2))+exp(-sqrt( (x+1)^2+y^2)))^2

Y

X

-20

2

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6

-2

0

2

Φ2

IΦI2=(exp(-sqrt( (x-1)^2+y^2))-exp(-sqrt( (x+1)^2+y^2)))^2

Y

X

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 1 2 3x

0

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 1 2 3x

Closer look at the probability density

5

E&R 512-513

Wave function

Probability amplitude

-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-20

24

-2.0

-1.5

-1.0

-0.5

0.0

Pote

ntia

l

Y

X

-10 -8 -6 -4 -2 0 2 4 6 8 10-4

-20

24

-2.0

-1.5

-1.0

-0.5

0.0

Pote

ntia

l

Y

X

( )12

L R−

( )12

L R+

Effect of proton-proton separation

2K

( )12

L R−

( )12

L R+

1sE K≅ −

1sE K≅ +

Effect of proton-proton separation

Atomic separation

4 8

Ene

rgy

Energy level separation as a function of lattice spacing

2K

bonding

antibonding

E1s

+ _

+

( )12

L R−

1sE K≅ +

( )12

L R+

1sE K≅ −

E1s

+ _ _

++

( )12

L R−

( )12

L R+

6

Es

( )12 L Rs s−

( )12 L Rs s+

Bonding and antibonding states built up from an s-orbital

Epx

( )12 x L xRp p+

( )12 xL xRp p−

Bonding and antibonding states built up from a px-orbital

Epz

( )12 zL zRp p−

( )12 zL zRp p+

Bonding and antibonding states built up from a pz-orbital

Homonuclear diatomic molecules

n εn (eV) l=0 l=1 l=2

n

.. …..

4 ε4= -0.85 4s 4p 4d

3 ε3= -1.51 3s 3p

3d

2 ε2= -3.40 2s 2p

1 ε1= -13.6 1s

O1s22s22p4

E2s

The O2 molecule

O O

2

-2

2

4-2E2p

7

The O2 molecule is magneticThe O2molecule is magnetic

E2s

E2p

n εn (eV) l=0 l=1 l=2

n

.. …..

4 ε4= -0.85 4s 4p 4d

3 ε3= -1.51 3s 3p

3d

2 ε2= -3.40 2s 2p

1 ε1= -13.6 1s

F1s22s22p5

The F2 molecule

2

-2

2

4-4

F F

E2p

E2s

n εn (eV) l=0 l=1 l=2

n

.. …..

4 ε4= -0.85 4s 4p 4d

3 ε3= -1.51 3s 3p

3d

2 ε2= -3.40 2s 2p

1 ε1= -13.6 1s

Ne1s22s22p6

The Ne2 molecule

Ne Ne

does not exist

2

-2

24

-4 -2

E2p

E2s

A little complication

8

n εn (eV) l=0 l=1 l=2

n

.. …..

4 ε4= -0.85 4s 4p 4d

3 ε3= -1.51 3s 3p

3d

2 ε2= -3.40 2s 2p

1 ε1= -13.6 1s

N1s22s22p3

Atomic radii (pm) The N2 molecule is not like this

E2p

The N2molecule is not

like this

E2s

2

-2

2

4

N N

The N2 molecule The N2molecule

N N E2s

E2p

2

-2

42

E2s

E2p E2pE2p

E2sE2s

The N2 molecule

right wrong

9

Heteronuclear diatomic molecules

Es

( )12 L Rs s−

( )12 L Rs s+

Homonuclear Heteronuclear

( )0.2 0.98L Rs s+

( )0.98 0.2L Rs s−

EL

ER

Polar bonding and antibonding states

( )0.2 0.98L Rs s+

( )0.98 0.2L Rs s−

EL

ER

E1sHydrogen

E2pFluor

H F

Ionic bond

H 1s

1s-13.6 eV

-0.7 eV

The electron affinity of H is

0.7 eV

F1s22s22p5

1s

2s 2p-18.6 eV

-3.3 eV

The electron affinity of F is

3.3 eV

10

F1s22s22p5H 1s

1s

2s 2p

1s-13.6 eV-18.6 eV

F1s22s22p5H 1s

1s

2s 2p

1s-13.6 eV-18.6 eV

-3.3 eV

The transfer of an electron from H to F costs 10.3 eV

F1s22s22p5H 1s

1s

2s 2p

1s-13.6 eV-18.6 eV

F1s22s22p5H 1s

1s

2s 2p

1s-13.6 eV-18.6 eV

The transfer of an electron from F to H costs 17.9 eV

-0.7 eV

1H

2He

3Li

4Be

5B

6C

7N

8O

9F

10Ne

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89Ac57La

58Ce

59Pr

60Nd

61Pm

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

71Lu

1H

2He

3Li

4Be

5B

6C

7N

8O

9F

10Ne

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89Ac57La

58Ce

59Pr

60Nd

61Pm

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

71Lu

11

http://www.webelements.com/webelements/scholar/properties/image-line/

1H

2He

3Li

4Be

5B

6C

7N

8O

9F

10Ne

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89Ac57La

58Ce

59Pr

60Nd

61Pm

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

71Lu

http://www.webelements.com/webelements/scholar/properties/image-line/

1H

2He

3Li

4Be

5B

6C

7N

8O

9F

10Ne

11Na

12Mg

13Al

14Si

15P

16S

17Cl

18Ar

19K

20Ca

21Sc

22Ti

23V

24Cr

25Mn

26Fe

27Co

28Ni

29Cu

30Zn

31Ga

32Ge

33As

34Se

35Br

36Kr

37Rb

38Sr

39Y

40Zr

41Nb

42Mo

43Tc

44Ru

45Rh

46Pd

47Ag

48Cd

49In

50Sn

51Sb

52Te

53I

54Xe

55Cs

56Ba

57La

72Hf

73Ta

74W

75Re

76Os

77Ir

78Pt

79Au

80Hg

81Tl

82Pb

83Bi

84Po

85At

86Rn

87Fr

88Ra

89Ac57La

58Ce

59Pr

60Nd

61Pm

62Sm

63Eu

64Gd

65Tb

66Dy

67Ho

68Er

69Tm

70Yb

71Lu

2

,0

2

0

4

4

i ji j ij

d

e aE Q Qa r

eMa

πε

πε

=

= −

∑

For NaCl structure Md=1.7476

For CsCl structure Md=1.7627

Formation of an ionic solid

anearest neighbour distance

NaCl (ionic radii)

NaCl crystalas in the book

Ionic radii

12

NaCl with Na in an octahedral hole Ionic radii

CsCl with Cl in a cubic hole Ionic radii

CaF2 with F in a tetrahedral hole

jiji ij

QQraDE ∑=

,2

2

2

Formation of an ionic solid

Positive ion

Negative ion