3.012 Fund of Mat Sci: Bonding – Lecture 11
BONDING IN MOLECULES
The future of electronics ? A pentacene molecule (left) deposited on SiO2 as a thin film (right)
Image removed for copyright reasons.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Wed Oct 19
• Study: 24.2, 24.4-6
• Read math supplement of Engel-Reid (A.7
and A.8, working with determinants and
working with matrices)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time:
1. Stability determined by the interplay of n-n, ee-, e-n interactions and the quantum kinetic
energy
2. Many-electron wavefunction as product of
single-particle orbitals (each one LCAO)
3. Many-atom Hamiltonian
4. sp, sp2 and sp3 hybridizations – bond lengths
and bond energies
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Complexity of the many-body Ψ
r
r
ψ = ψ (r1 ,..., rn )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Mean-field approach
• Independent particle model (Hartree): each
electron moves in an effective potential,
representing the attraction of the nuclei and
the average effect of the repulsive
interactions of the other electrons
• This average repulsion is the electrostatic
repulsion of the average charge density of
all other electrons
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hartree Equations
•The Hartree equations can be obtained directly from the
variational principle, once the search is restricted to the many-body
wavefunctions that are written – as above – as the product of single
spin-orbitals (i.e. we are working with independent electrons)
r
r
r
r
r
ψ (r1 ,..., rn ) = ϕ1 (r1 ) ϕ 2 (r2 ) Lϕ n (rn )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hartree Equations
r r
⎡ 1 2
r 2 1
r⎤ r
r
⎢ − ∇i + ∑ I V ( RI − ri ) + ∑ ∫ ϕ j (rj ) r r drj ⎥ ϕi (ri ) = εϕi (ri )
| rj − ri | ⎥⎦
j ≠i
⎢⎣ 2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
The self-consistent field
• The single-particle Hartree operator is selfconsistent ! I.e., it depends in itself on the
orbitals that are the solution of all other
Hartree equations
• We have n simultaneous integro-differential
equations for the n orbitals
• Solution is achieved iteratively
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Differential Analyzer
Vannevar Bush and the Differential Analyzer.
Courtesy of the MIT Museum. Used with permission.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Spin-Statistics
• All elementary particles are either fermions
(half-integer spins) or bosons (integer)
• A set of identical (indistinguishable)
fermions has a wavefunction that is
antisymmetric by exchange
r r
r
r
r
r r
r
r
ψ (r1 , r2 ,..., rj ,..., rk ,..., rn ) = −ψ (r1 , r2 ,..., rk ,..., rj ,..., rn )
• For bosons it is symmetric
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Slater determinant
• An antisymmetric wavefunction is constructed via a
Slater determinant of the individual orbitals (instead
of just a product, as in the Hartree approach)
r
r
r
ϕα (r1 ) ϕ β (r1 ) L ϕν (r1 )
r
r
r
r r
r
1 ϕα (r2 ) ϕ β (r2 ) L ϕν (r2 )
ψ (r1 , r2 ,..., rn ) =
M
M
O
M
n!
r
r
r
ϕα (rn ) ϕ β (rn ) L ϕν (rn )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Pauli principle
• If two states are identical, the determinant
vanishes (i.e. we can’t have two electrons in
the same quantum state)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hartree-Fock Equations
•The Hartree-Fock equations are, again, obtained from the variational principle: we
look for the minimum of the many-electron Schrödinger equation in the class of all
wavefunctions that are written as a single Slater determinant
r r ⎤
r
⎡ 1 2
⎢ − 2 ∇ i + ∑ V ( RI −ri ) ⎥ ϕλ (ri ) +
I
⎣
⎦
⎡
r 2 1
r⎤
r
⎢ ∑ ∫ ϕ µ (rj ) r r drj ⎥ ϕλ (ri ) −
| rj − ri | ⎥⎦
⎢⎣ µ
⎡ * r
1
r
r r⎤
r
∑µ ⎢ ∫ ϕµ (rj ) | rr − rr | ϕµ (ri ) ϕλ (rj )drj ⎥ = εϕλ (ri )
⎢⎣
⎥⎦
j
i
r
r
ψ (r1 ,..., rn ) = Slater
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Example: two electrons in H2
r
r
1 ϕα (r1 ) ϕ β (r1 )
1
r r
r
r
r
r
⎡
(
r
)
(
r
)
(
r
)
(
r
=
−
ψ (r1 , r2 ) =
ϕ
ϕ
ϕ
ϕ
r
r
α 1
β 2
α 2
β 1 )⎤
⎣
⎦
2 ϕα (r2 ) ϕ β (r2 )
2
r
ϕα , β (r ) = full solution of (integro-differentail) Hartree-Fock equations, or
r
r r
r r
ϕα (r ) = c1Ψ1s (r − RA ) + c2 Ψ1s (r − RB ) × ( spin − up )
r
r r
r r
ϕ β (r ) = c1Ψ1s (r − RA ) + c2 Ψ1s (r − RB ) × ( spin − down )
(
(
)
)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
H2 and He2
1σu*
1s
1σu*
1s
1s
1s
1σg
H2
1σg
He2
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetries
• Rotation along molecular axis → σ
• Nodal plane ┴ molecular axis → π
• Parity upon inversion:
r
r
r
r
r → −r ⇒ Ψ ( r ) = Ψ ( −r )
r
r
Ψ ( r ) = −Ψ ( − r )
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Formation of a π Bonding Orbital
See animation at
http://winter.group.shef.ac.uk/orbitron/MOs/N2/2px2px-pi/index.html
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Symmetries
Contour plots of several bonding and antibonding orbitals of H2+.
Images removed for copyright reasons.
See p. 528, figure 24.4 in Engel, T., and P. Reid. Physical Chemistry. Single volume ed. San Francisco, CA: Benjamin Cummings, 2005.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homonuclear Diatomic Levels (I)
Diagram of Orbital Regions for 2p atomic orbitals and LCAO molecular orbitals made from them removed for copyright reasons.
See p. 667, figure 18.11 in Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homonuclear Diatomic Levels (II)
π∗g2px σ∗u2pz π∗g2py
σg2pz
2pxA 2pyA 2pzA
πu2px πu2py
2pzB 2pxB 2pyB
Orbital Energy
σ∗u2s
2sA
2sB
σg2s
σ∗u1s
1sA
Atomic
Orbitals
σg1s
Molecular
Orbitals
1sA
Atomic
Orbitals
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Fluorine dimer F2
3σ*
u
1πg*
2p
_
+
2p
_
+
+
+
_
_
1πu
_
+
_
+
+
+
_
3σg
2σ*
u
2s
2s
2σg
+
+
Figure by MIT OCW.
_
+
2σu*
2σg
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
+
1πg*
_
+
3σg
_
_
1πu
Homonuclear Diatomic Levels (III)
Li2
Be2
B2
C2
N2
O2
F2
σ
σ∗
Figure by MIT OCW.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Bond Order
Graph of bond order, bond energy, bond length, and force constant against
number of electrons. Removed for copyright reasons.
See p. 535, figure 24.11 in Engel, T., and P. Reid. Physical Chemistry.
Single volume ed. San Francisco, CA: Benjamin Cummings, 2005.
Bond order = ½ of
[bonding electrons antibonding electrons]
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Advanced Reading Material
•
•
•
•
Further readings (from less to more advanced):
Atkins Physical Chemistry Freeman & Co (2001)
Thaller Visual Quantum Mechanics Telos (2000)
Bransden & Joachain Quantum Mechanics (2nd
ed) Prentice Hall (2000)
Bransden & Joachain Physics of Atoms and
Molecules (2nd ed) Prentice Hall (2003)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)