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PH575 Spring 2019 Lecture #4 - Homo- & heteronuclear diatomic molecule: Sutton Ch. 2 pp 25-31
nm nmi i
i∑ = 1
β = 1 H 2
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H2 : a homonuclear diatomic molecule
http://wps.prenhall.com
2 H atoms, 1s orbitals only
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1,1s 2,1s
Goal: find molecular orbitals, and their energies, E
Ψ
Ψ = 1,1s Ψ 1,1s + 2,1s Ψ 2,1sΨ = c1 1,1s + c2 2,1s
We know these atomic orbitals, and their energies. These are the basis kets. Assume orthogonal (not quite true, but OK) 1,1s 2,1s = 0
H Ψ = E Ψ
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Overlap of atomic orbitals on different atoms
http://wps.prenhall.com
+ + + +
----
A large overlap integral requires spatial overlap, but spatially overlapping orbitals can still have zero overlap if the signs are appropriately arranged. e.g 2s is orthogonal to 2p on same atom. 4
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We want to find molecular orbitals, and their energies, E
Ψ = c1 1,1s + c2 2,1s
H Ψ = E Ψ
H c1 1,1s + c2 2,1s!" #$ = E c1 1,1s + c2 2,1s!" #$
Project onto basis ket #1
c1 1,1s H 1,1s + c2 1,1s H 2,1s
= E c1 1,1s 1,1s?
+ c2 1,1s 2,1s?
!
"##
$
%&&
On-site matrix element H11=E0 (assume known)
Hopping matrix element H12=β(assume known)
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On-site energy:
1 H 1 = 1 − 2
2m∇2 +V1 r( ) +V2 r( ) 1
E0 = E1,atomic + 1V2 r( ) 1On-site energy is approximately the atomic energy, but small correction due to other potential. WF 1 is small where V2 is large and vice versa (note: e-e interactions neglected).
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1a 2a 3a 4a 5axV0
0 V2V1
x 2x 1 1 2
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Hopping energy:
1 H 2 = 1 − 2
2m∇2 +V1 r( ) +V2 r( ) 2
β = E1,atomic 1 2 + 1V2 r( ) 2β = 1V2 r( ) 2
1a 2a 3a 4a 5axV0
0
V2V1
x 2x 1 1 2
Hopping energy is integral of individual potential and both atomic wave functions. Could have additional contribution if there is n-n overlap.
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c1E0 + c2β = Ec1c1β + c2E0 = Ec2
Project onto basis kets
E0 − E β
β E0 − E= 0 Linear algebra => quadratic eqn
Ea = E0 − βEb = E0 + β
β = 1,1s V2 (r) 2,1s < 0 For overlapping s orbitals, β is negative. This is because s orbitals are +ve everywhere and V2 < 0 (attractive potential)
We want to find molecular orbitals, and their energies, E
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We want to find molecular orbitals, and their energies, E
c1E0 + c2β = Ec1c1β + c2E0 = Ec2
Now we can find the c's for each of Eb and Ea, and hence the corresponding Ψb and Ψa.
Ψa =121,1s − 2,1s( )
Ψb =121,1s + 2,1s( )
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H2 molecule
http://wps.prenhall.com
electron density between nuclei
electron density outside of nuclei
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H2 molecule
http://wps.prenhall.com
Bonding is a quantum mechanical phenomenon that results from the interference of quantum waves! To view bonding as the "sharing" of electrons, we can show (Sutton p33-31) that an electron oscillates from atom #1 to atom #2 at a frequency 2β/h. How does it overcome the large ionization potential? It tunnels! Again a quantum mechanical phenomenon.
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σ bonding with p-orbitals
http://wps.prenhall.com
Ψa =121, p + 2, p( )
Ψb =121, p − 2, p( )
Why is the –ve sign associated with the bonding orbital in this example? 12
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π bonding with p-orbitals
http://wps.prenhall.com13
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MO diagram for p-orbitals
http://wps.prenhall.com14
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• H2 molecule has 2 atoms and the 2 molecular orbitals are (relatively) close in energy, one higher and one lower than the energy of the atomic orbital
• The hopping term represents tunneling of electrons across the potential barrier between atoms (see Sutton). The faster the tunneling, the stronger the interaction, and the bigger the splitting.
• The orbitals are bonding (electron density between nuclei tending to draw nuclei together), or antibonding (electron density on opposite sides of nuclei, causing nuclear repulsion)
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Atom A Atom B
A, sB, sp
A H B = B H A = β
Off-site integral
B H B = EB A H A = EA
On-site integrals
EA
EB
Ψ = cA A + cB B
Heteronuclear diatomic molecule
General form of MO
H Ψ = E Ψ Schrödinger eigenvalue equation 16
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Want to find these molecular orbitals, and their energies, E
cAEA + cBβ = EcAcAβ + cBEB = EcB
Project onto basis kets
EA − E β
β EB − E= 0 Linear algebra => quadratic eqn
Ea = ε + Δ2 + β 2( )1/2
Eb = ε − Δ2 + β 2( )1/2
EA
EB
εΔ
Δ
Ea
Eb 17
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Want to find these molecular orbitals, and their energies, E
Now we can find the c's for each of Eb and Ea, and hence the corresponding Ψb and Ψa.
cA,a2
cB,a2 =
1
1+ 2 Δ / β( )2 − 2 Δ / β( ) 1+ Δ / β( )2( )1/2
cA,b2
cB,b2 =
1
1+ 2 Δ / β( )2 + 2 Δ / β( ) 1+ Δ / β( )2( )1/2
cAEA + cBβ = EcAcAβ + cBEB = EcB
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bonding
antibondingcA2 cB
2
cA,a2
cB,a2 =
1
1+ 2 Δ / β( )2 − 2 Δ / β( ) 1+ Δ / β( )2( )1/2
cA,b2
cB,b2 =
1
1+ 2 Δ / β( )2 + 2 Δ / β( ) 1+ Δ / β( )2( )1/2
covalent
B-A+
B+A-
ionic
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Electronegativity
http://wps.prenhall.com
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sp3-hybridization
http://wps.prenhall.com
has same sign as
has opposite sign to
This commonly shown picture is actually inaccurate. It shows 1s-2p hybridization. One always sees 2s-2p hybridization. 2s electron distribution is different from 1s!
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HF and HCl
http://wps.prenhall.com
Overlap of the bonding sp3 hybrid orbital of F and Cl with H in HF and HCl. Overlap of the 1s orbital of hydrogen is better with the smaller 2sp3 hybrid orbital of fluorine than with the larger 3sp3 hybrid orbital of chlorine, resulting in a shorter, stronger bond in HF than in HCl.
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http://www.sparknotes.com/chemistry/bonding/molecularorbital/section1.html 23