3.012 Fund of Mat Sci: Bonding – Lecture 9
VARIATIONS
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Homework for Fri Oct 14
• Study: 21.4, 23.3
• Read: 23.4, 24.1, 24.2
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Last time:
1. Screening, and coupled, self-consistent
Hartree equations for many-electron atoms
2. 4th quantum number: spin
3. Filling (auf-bau) of the periodic table
4. Physical trends on sizes, IP, EA. (e.g., why He
is smaller than H)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Variational Principle
E [Ψ ] =
Ψ Ĥ Ψ
Ψ Ψ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Variational Principle
E [Ψ ] =
E [ Ψ ] ≥ E0
Ψ Ĥ Ψ
Ψ Ψ
If E [ Ψ ] = E0 , then Φ is the ground
state wavefunction, and viceversa…
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Good Quantum Numbers
d Aˆ
dt
=
d Ψ Aˆ Ψ
dt
1 ⎡ˆ ˆ⎤
,
=
A
H
⎦
ih ⎣
If A commutes with the Hamiltonian, its
expectation value does not change with time
(it’s a constant of motion – if we are in an
eigenstate, that quantum number will
remain constant)
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Atomic Units
• me=1, e=1, a0 (Bohr radius)=1, h = 1
1
ε0 =
4π
1 Z2
Energy of 1s electron= −
2 n2
(1 atomic unit of energy=1 Hartree=2 Rydberg=27.21 eV
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Energy of an Hydrogen Atom
E=
Ψ Ĥ Ψ
Ψ Ψ
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Energy of an Hydrogen Atom
Eα =
Ψα Ĥ Ψα
Ψα Ψα
Ψα = C exp ( −α r )
Ψα Ψα = π
C2
α
3
,
1 2
C2
Ψα − ∇ Ψα = π
2
2α
1
C2
Ψα − Ψ α = −π 2
r
α
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hydrogen Molecular Ion H2+
⎡
⎛
1
⎢− ∇2 + ⎜ r 1 r − 1 r − 1 r
⎜ RA − RB r − RA r − RB
⎢ 2
⎝
⎣
⎞⎤ r
⎟ ⎥ψ (r ) = Eψ (rr )
⎟⎥
⎠⎦
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Linear Combination of Atomic Orbitals
• Most common approach to find out the groundstate solution – it allows a meaningful definition
of “hybridization”, “bonding” and “anti-bonding”
orbitals.
• Also knows as LCAO, LCAO-MO (for molecular
orbitals), or tight-binding (for solids)
• Trial wavefunction is a linear combination of
atomic orbitals – the variational parameters are the
coefficients:
Ψ trial = c1Ψ1s
(
r r
r r
r − RA + c2 Ψ1s r − RB
)
(
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
)
Linear Combination of Atomic Orbitals
Ψ trial = c1Ψ1s
(
r r
r r
r − RA + c2 Ψ1s r − RB
)
(
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
)
Bonding and Antibonding (I)
Image of the orbital region for LCAO molecular orbitals removed for copyright reasons.
See Mortimer, R. G. Physical Chemistry. 2nd ed. San Diego, CA: Elsevier, 2000, p. 657, figure 18.7.
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)
Hydrogen Molecular Ion H2+
• Born-Oppenheimer approximation: the
electron is always in the ground state
corresponding to the instantaneous ionic
positions
3.012 Fundamentals of Materials Science: Bonding - Nicola Marzari (MIT, Fall 2005)