Bonding in H2+: MOTheory

Jan. 28, 2014

H2+ is the simplest molecular species

The Hamiltonian is

H = p2

2m e

2

rA e

2

rB+e2

R

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Approx. wavefunction for H2+

guessed from R =

Two equally likely dissociation pathways for H2+

1 H2+ HA + HB+

2 H2+ HB + HA+

When R =, has significant amplitude only near A andnear B

1 Electron near A = = cA1sA2 Electron near B = = cB1sB

Guess that = cA1sA + cB1sB everywhere in spaceMatter wave constructed by superposition of H AO

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Indistinguishability restricts cA and cB

Physically measurable quantities unaffected on interchangingparticle labels

||2 is unaffected by particle interchange

|(A,B)|2 = |(B,A)|2

= |cA|2 = |cB|2 = |c|2

For real values, cA = cB and cA = cB or

= c(1sA 1sB)

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Normalization used to find c.

||2d = c2

|1sA 1sB|2d

1 = c2(|1sA|2d

1sA1sBd

1sA1s

Bd +

|1sB|2d

)When R,

1sA1sBd =

1sA1s

Bd = 0.

1 = c2(1 + 0 + 0 + 1)

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At infinite internuclear distanceE(H+2 ) = EH(1s)

E() =

all spaceHd

=

near A

1s(A)2HA

1s(A)2d

+

near B

1s(B)2HB

1s(B)2d

=1

2EH(1s) +

1

2EH(1s) = EH(1s)

Define this as the zero of energy

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At finite R, is approximately a sumof atomic orbitals

+ = c0 (1s(A) + 1s(B))

= c1 (1s(A) 1s(B))

Linear Combination of Atomic Orbitals (LCAO)1 Maximum simplicity with reasonable accuracy2 Natural and intuitive appeal3 Simplifies calculations - atomic orbitals already determined

to high accuracy4 Wavefunctions around atoms in molecules like those

around free atomsBond formation a small perturbation to atomic structure

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Normalization constant depends onthe overlap integral

|+|2d = 1 = c20

(|1s(A)|2 +

|1s(B)|2d

+

1s(A)1s(B)d +

1s(A)1s(B)d

)Defining S =

1s(A)1s(B)d , we find

|+|2d = 2c20(1 + S)

S is zero when A and Bare infinitely far apart

S has a large contributionfrom regions where 1s(A)and 1s(B) are bothappreciable

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Electron density between the nuclei ismore in +

The electron density in the + MO is

|+|2 =|1s(A)|2 + |1s(B)|2 + 2(1s(A)1s(B))

2(1 + S)

S reduces the electron density everywhere.Between the nuclei 2(1s(A)1s(B)) is dominant and

= |+|2 > |1s(A)|2 + |1s(B)|2

Constructive interference of the two atomic orbitals between thenuclei in +.

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There are 0 and 1 nodes in the twolowest MOs

Dotted - non-interacting,dashed - LCAO, solid - exact

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Two lowest MOs show interferenceBonding MO Antibonding MO

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Forces responsible for binding

Potential energy - electrostatic

+ describes a bonding state

Kinetic energy - delocalization

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The approximate energy of + isgiven by

+H+d

E+(R) =1

2(1 + S)

(1s(A)H1s(A)d +

1s(A)H1s(B)d+

1s(B)H1s(A)d +

1s(B)H1s(B)d)

Define an integral

Hij =

iHjd

E+(R) =HAA +HAB +HBA +HBB

2(1 + S)

E+(R) =HAA +HAB

(1 + S)

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The atomic integral HAA has aCoulomb term

HAA =

1s(A)

(p2

2m e

2

rA

)1s(A)d +

e2

R

1s(A)1s(A)d

1s(A)1s(A)e2

rBd

= EH(1s) +e2

R+ J

where

J =

e2

rB1s(A)1s(A)d

is the Coulombic interaction between the orbital around oneatom and the other nucleus.

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The resonance integral HAB has anexchange term

HAB =

1s(A)

(p2

2m e

2

rB

)1s(B)d +

e2

R

1s(A)1s(B)d

1s(A)1s(B)e2

rAd

= S

(EH(1s) +

e2

R

)+K

where

K =

e2

rA1s(A)1s(B)d

has no classical interpretation.

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In the GS the electron has a netbonding effect

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Exchange integral is important forbinding

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The LCAO method can begeneralized

If = c1f1 + c2f2 then E =Hd is

E(c1, c2) =c21H11 + 2c1c2H12 + c

22H22

c21S11 + 2c1c2S12 + c22S22

At the minimum of the energy

E

c1=E

c2= 0

c1(H11 ES11) + c2(H12 ES12) = 0

c1(H21 ES21) + c2(H22 ES22) = 0

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The LCAO method can be extended

If =

i cifi then E =Hd is

E(c1, c2) =

i c

2iHii +

i

j>i 2cicjHij

i c2iSii +

i

j>i 2cicjSij

At the minimum of the energy

E

ci= 0

c1(H11 ES11) + c2(H12 ES12) + = 0

c1(H21 ES21) + c2(H22 ES22) + = 0

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