Slide 1
Quantum Mechanics: the Practice
Slide 2
Reminder: Electrons As Waves
Wavelength • momentum = Planck
?
λ•p=h
( h = 6.6 x 10-34 J s )
The wave is an excitation (a vibration): We
need to know the amplitude of the excitation
at every point and at every instant
r
Ψ = Ψ (r , t )
Slide 3
Wave Mechanics
r
r
r
h2 2 r
∂Ψ (r , t )
−
∇ Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) = ih
2m
∂t
r
r
Ψ (r , t ) = ψ ( r ) f (t )
r
r 
1  h2 2 r
1 ∂f (t )
∇ ψ (r ) + V (r )ψ (r ) = ih
r −
f ∂t

ψ (r )  2m
 h2 2
r  r
r
∇ + V (r )ψ (r ) = Eψ (r )
−
 2m

Slide 4
Stationary Schroedinger’s equation
(a)
(b)
 h2 2
r  r
r
∇ + V (r )ψ (r ) = Eψ (r )
−
 2m

2
h
Hˆ = Tˆ + Vˆ = −
∇ 2 + Vˆ
2m
r
r
Hˆ ψ (r ) = Eψ (r )
•It’s not proven – it’s postulated, and it is confirmed experimentally
•It’s an eigenvalue equation
•Boundary conditions (and regularity) must be specified
Slide 5
Interpretation of the Quantum Wavefunction
r
Ψ (r , t )
2
is the probability of finding an electron
in r and t
r
r
r
If V=V(r) , it’s separable: Ψ ( r , t ) = ψ ( r ) f (t ) = ψ ( r ) exp(−
i
Et )
h
Remember the free particle, and the principle of indetermination:if
the momentum is perfectly known, the position is perfectly unknown
r r
r
Ψ (r , t ) = A exp[i (k • r − ωt )]
Slide 6
Infinite Square Well
Slide 7
Finite Square Well
Slide 8
A Central Potential (e.g. the Nucleus)
r
h2 2
∇ + V (r )
Hˆ = −
2µ
∇2 =
∂2
∂2
∂2
+
+
∂x 2 ∂y 2 ∂z 2
1
1
h2  1 ∂  2 ∂ 
∂ 
∂ 
∂2 
Hˆ = −
r
+ 2
 sin ϑ
+ 2 2
 2
 + V (r )
2 µ  r ∂r  ∂r  r sin ϑ ∂ϑ 
∂ϑ  r sin ϑ ∂ϕ 2 
r
ψ Elm (r ) = RElm (r )Ylm (ϑ , ϕ )
 h 2  d 2 2 d  l (l + 1)h 2

 2 +
 +
+ V (r ) REl (r ) = E REl (r )
−
2
r dr 
2 µr
 2µ  dr

Slide 9
Solutions in a
Coulomb
Potential: the
Radial
Wavefunctions
Slide 10
Solutions in a Coulomb Potential:
the Periodic Table
http://www.orbitals.com/orb/orbtable.htm
5 d orbital
Slide 11
<Bra|kets>
r
ψ = ψ (r ) =| ψ >
r
r r
< ψ i | ψ j >= ∫ψ i* (r )ψ j (r ) dr = δ ij
r  h2
r  r r
< ψ i | Hˆ | ψ i >= ∫ψ i* (r ) −
+ V (r )ψ i (r ) dr = Ei
 2m

Slide 12
Variational Principle
< φ | Hˆ | φ >
E [φ ] =
< φ |φ >
E [φ ] ≥ E0
Slide 13
Electrons and Nuclei
Hˆ = Tˆe + Vˆe −e + VˆN − N + Vˆe − N
r
r
r
r
Hˆ ψ (r1 ,..., rn ) = Etotψ (r1 ,..., rn )
•We treat only the electrons as quantum particles, in the
field of the fixed (or slowly varying) nuclei
•This is generically called the adiabatic or BornOppenheimer approximation
Slide 14
Two-electron atom
 1 2 1 2 Z Z
r r
1  r r
− ∇1 − ∇ 2 − − + r r ψ (r1 , r2 ) = Eψ (r1 , r2 )
2
r1 r2 | r1 − r2 | 
 2
Slide 15
Energy of a collection of atoms
• VN-N: electrostatic nucleus-nucleus repulsion
• Te: quantum kinetic energy of the electrons
• Ve-N: electrostatic electron-nucleus attraction
(electrons in the field of all the nuclei)
• Ve-e: electron-electron interactions
1
Tˆe = − ∑ ∇ i2
2 i
(
)
r r

Vˆe − N = ∑ ∑ V RI − ri 
i  I

1
Vˆe −e = ∑∑ r r
i j >i | ri − r j |
Slide 16
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
Slide 17
Hartree Equations
r r
 1 2
r 2
r r
r
1
 − ∇ i + ∑ I V ( RI − ri ) + ∑ ∫ | ϕ j (r j ) | r r dr j ϕ i (ri ) = ε ϕ i (ri )
| r j − ri | 
j ≠i
 2
r
r
r
r
r
ψ (r1 ,..., rn ) = ϕ1 (r1 ) ϕ 2 (r2 ) Lϕ n (rn )
•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
orbitals (i.e. we are working with independent electrons)
Slide 18
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
Slide 19
Iterations to self-consistency
• Initial guess at the orbitals
• Construction of all the operators
• Solution of the single-particle pseudoSchrodinger equations
• With this new set of orbitals, construct the
Hartree operators again
• Iterate the procedure until it (hopefully)
converges
Slide 20
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
Slide 21
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
r
ψ (r1 , r2 ,..., rn ) =
r
r
ϕα (r1 ) ϕ β (r1 ) L ϕν (r1 )
r
r
r
1 ϕα (r2 ) ϕ β (r2 ) L ϕν (r2 )
n!
M
M
O
M
r
r
r
ϕα (rn ) ϕ β (rn ) L ϕν (rn )
Slide 22
Pauli principle
• If two states are identical, the determinant
vanishes (I.e. we can’t have two electrons in
the same quantum state)
Slide 23
Hartree-Fock Equations
•The Hartree-Fock equations are, again, obtained from the variational principle: we
look for the minimum of the many-electron Schroedinger 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 ) ϕ λ (ri ) +
I



r r
r
1
* r
 ∑ ∫ ϕ µ (rj ) r r ϕ µ ( rj ) drj ϕ λ (ri ) −
| rj − ri |
 µ


1
r
r
r
r
r
∑µ ∫ ϕ µ (r ) | rr − rr | ϕ λ (r )dr ϕ µ (r ) = ε ϕ λ (r )

*
j
j
j
r
i
r
j

ψ (r1 ,..., rn ) = Slater
i
i
Slide 24
Density-functional Theory
• Conceptually very different from Hartree-Fock –
variational principle on the charge density
• In practice, equations have the same form, but for
the exchange energy – obtained from the density,
not the wavefunctions
• It’s exact in principle, but approximate in practice:
different forms for the exchange-correlation
density: LDA, GGA, hybrids (Hartree-Fock
exchange + density-functional correlations)