3.320: Lecture 5 (Feb 15 2005)
THE MANY-BODY PROBLEM
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
When is a particle like a wave ?
Wavelength • momentum = Planck
↕
λ • p = h ( h = 6.6 x 10-34 J s )
r
Ψ = Ψ (r , t )
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Time-dependent Schrödinger’s equation
(Newton’s 2nd law for quantum objects)
r
r
r
r
h
∂Ψ (r , t )
2
−
∇ Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) = ih
2m
∂t
2
1925-onwards: E. Schrödinger (wave equation), W. Heisenberg
(matrix formulation), P.A.M. Dirac (relativistic)
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Stationary Schrödinger’s Equation (I)
r
r
r
r
∂Ψ (r , t )
h
2
−
∇ Ψ ( r , t ) + V ( r , t ) Ψ ( r , t ) = ih
2m
∂t
2
*
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Stationary Schrödinger’s Equation (II)
d
ih f (t ) = E f (t )
dt
⎡ h2 2
r ⎤ r
r
∇ + V (r )⎥ϕ (r ) = Eϕ (r )
⎢−
⎣ 2m
⎦
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
d
ih f (t ) = E f (t )
dt
⎛ E ⎞
f (t ) = exp⎜ − i t ⎟
⎝ h ⎠
Free particle Ψ(x,t)=φ(x)f(t)
h2 2
−
∇ ϕ ( x ) = Eϕ ( x )
2m
Feb 15 2005
⎛ 2mE
ϕ ( x) = exp⎜⎜ i
h
⎝
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
⎞
x ⎟⎟
⎠
Interpretation of the Quantum
Wavefunction (Copenhagen)
Ψ ( x, t )
2
is the probability of finding an electron
in x and t
2
i
ϕ ( x) exp(− Et ) = ϕ ( x)
h
Feb 15 2005
2
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
A Traveling “Plane” Wave
Ψ ( x, t ) ∝ exp[i (kx − ωt )]
Diagram of plane wave removed for copyright reasons.
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Metal Surfaces (I)
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Metal Surfaces (II)
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Infinite Square Well
8ma2
π2h2
ψ(x)
E
16
ψ4
n=4
14
12
10
ψ3
n=3
8
6
4
2
ψ2
n=2
ψ1
n=1
0
-a
0
a
Figure by MIT OCW.
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
x
Finite Square Well
aψ1(x)
aψ2(x)
1
1
0.5
-2
-1
0
-0.5
0.5
x/a
1
-2
2
-1
-1
aψ3(x)
aψ4(x)
1
1
-1
0
1
0.5
1
2
-2
-1
-1
0
x/a
1
-1
Figure by MIT OCW.
Feb 15 2005
2
-1
x/a
-2
0
x/a
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
2
A Central Potential (e.g. the Nucleus)
2
h
∇ 2 + V (r )
Hˆ = −
2m
2
2
2
∂
∂
∂
∇2 = 2 + 2 + 2
∂x ∂y
∂z
2
2
⎡
⎤
1
1
1
h
∂
∂
∂
∂
∂
⎛
⎞
⎛
⎞
2
ˆ
H =−
+ V (r )
⎢ 2 ⎜r
⎟+ 2
⎜ sin ϑ
⎟+ 2 2
2⎥
2m ⎣ r ∂r ⎝ ∂r ⎠ r sin ϑ ∂ϑ ⎝
∂ϑ ⎠ r sin ϑ ∂ϕ ⎦
r
ψ Elm (r ) = RElm (r )Ylm (ϑ , ϕ )
⎡ h 2 ⎛ d 2 2 d ⎞ l (l + 1)h 2
⎤
+ V (r ) ⎥ REl (r ) = E REl (r )
⎢−
⎜ 2+
⎟+
2
r dr ⎠
2µ r
⎣ 2m ⎝ dr
⎦
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Solutions in a Coulomb Potential:
the Periodic Table
http://www.orbitals.com/orb/orbtable.htm
____________________________________________________________
Courtesy of David Manthey. Used with permission.
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Orthogonality, Expectation Values,
and Dirac’s <bra|kets>
r
ψ = ψ (r ) = ψ
r
r r
∫ψ (r )ψ j (r ) dr = ψ i ψ j = δ ij
*
i
2
⎡
r
r ⎤ r r
h
*
ˆψ =E
ψ
r
V
r
ψ
r
d
r
=
ψ
H
(
)
(
)
(
)
−
+
i
i
i
⎥ i
∫ i ⎢⎣ 2m
⎦
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Matrix Formulation (I)
r
r
Hˆ ψ (r ) = Eψ (r ) ⇔
ψ =
∑c
n =1, k
n
ϕn
Hˆ ψ = E ψ
{ ϕ } k orthogonal functions
n
ϕ m Hˆ ψ = E ϕ m ψ
ˆ
c
ϕ
H
∑ n m ϕn = Ecm
n =1, k
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Matrix Formulation (II)
ˆ
c
ϕ
H
∑ n m ϕ n = Ecm
n =1, k
∑H
n =1, k
⎛ H11
⎜
⎜ .
⎜ .
⎜
⎜ .
⎜H
⎝ k1
......
......
Feb 15 2005
c = Ecm
mn n
H1k ⎞ ⎛ c1 ⎞
⎛ c1 ⎞
⎟ ⎜ ⎟
⎜ ⎟
. ⎟ ⎜ .⎟
⎜ .⎟
. ⎟⋅⎜ . ⎟ = E ⎜ . ⎟
⎟ ⎜ ⎟
⎜ ⎟
. ⎟ ⎜ .⎟
⎜ .⎟
⎟
⎜
⎟
⎜
⎟
H kk ⎠ ⎝ ck ⎠
⎝ ck ⎠
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Variational Principle
< Φ | Hˆ | Φ >
E [Φ ] =
<Φ|Φ >
E [ Φ ] ≥ E0
If E [ Φ ] = E0 , then Φ is the ground
state wavefunction, and viceversa…
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Energy of an Hydrogen Atom
Eα =
Ψα Ĥ Ψα
Ψα Ψα
Ψα = C exp ( −α r )
Ψα Ψα = π
C2
α
3
,
Feb 15 2005
1 2
C2
Ψα − ∇ Ψα = π
2
2α
1
C2
Ψα − Ψ α = −π 2
r
α
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Two-electron atom
⎡ 1 2 1 2 Z Z
r r
1 ⎤ r r
⎢− ∇1 − ∇ 2 − − + r r ⎥ψ (r1 , r2 ) = Eelψ (r1 , r2 )
2
r1 r2 | r1 − r2 | ⎦
⎣ 2
Many-electron atom
⎡ 1
Z
1
2
⎢ − ∑ ∇ i − ∑ + ∑∑ r r
i ri
i j > i | ri − rj
⎢⎣ 2 i
Feb 15 2005
⎤ r
r
r
r
⎥ψ (r1 ,..., rn ) = Eelψ (r1 ,..., rn )
| ⎥⎦
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Energy of a collection of atoms
Hˆ = Tˆe + TˆN + Vˆe −e + VˆN − N + Vˆe − N
•
•
•
•
Te: quantum kinetic energy of the electrons
Ve-e: electron-electron interactions
VN-N: electrostatic nucleus-nucleus repulsion
Ve-N: electrostatic electron-nucleus attraction
(electrons in the field of all the nuclei)
1
Tˆe = − ∑ ∇ i2
2 i
Vˆe − N
Feb 15 2005
(
)
r r⎤
⎡
= ∑ ⎢∑ V RI − ri ⎥
i ⎣ I
⎦
1
Vˆe −e = ∑∑ r r
i j >i | ri − r j |
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Electrons and Nuclei
r
r
r
r
r
r
r
r
Hˆ ψ (r1 ,..., rn , R1 ,..., RN ) = Etotψ (r1 ,..., rn , 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
•Adiabatic means that there is no coupling between
different electronic surfaces; B-O no influence of the
ionic motion on one electronic surface.
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Complexity of the many-body Ψ
“…Some form of approximation is essential, and this would
mean the construction of tables. The tabulation function of one
variable requires a page, of two variables a volume and of three
variables a library; but the full specification of a single wave
function of neutral iron is a function of 78 variables. It would be
rather crude to restrict to 10 the number of values of each
variable at which to tabulate this function, but even so, full
tabulation would require 1078 entries.”
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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 the product of single 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 )
r r
⎡ 1 2
r 2
r⎤ r
r
1
⎢ − ∇ i + ∑ I V ( RI − ri ) + ∑ ∫ | ϕ j (rj ) | r r drj ⎥ϕ i (ri ) = ε ϕ i (ri )
| rj − ri | ⎥⎦
j ≠i
⎢⎣ 2
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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
r r
⎡ 1 2
r 2
r⎤ r
r
1
⎢ − ∇ i + ∑ I V ( RI − ri ) + ∑ ∫ | ϕ j (rj ) | r r drj ⎥ϕ i (ri ) = ε ϕ i (ri )
| rj − ri | ⎥⎦
j ≠i
⎢⎣ 2
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Differential Analyzer
Vannevar Bush and the Differential Analyzer.
Courtesy of the MIT Museum. Used with permission.
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
What’s missing
• It does not include correlation
• The wavefunction is not antisymmetric
• It does remove nl accidental degeneracy of
the hydrogenoid atoms
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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 )
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Pauli principle
• If two states are identical, the determinant
vanishes (i.e. we can’t have two electrons in
the same quantum state)
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
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
ψ (r1 ,..., rn ) = Slater
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
⎥⎦
⎡ * r
r r⎤
r
r
1
∑µ ⎢∫ ϕ µ (rj ) | rr − rr | ϕ λ (rj )drj ⎥ϕ µ (ri ) = ε ϕ λ (ri )
⎢⎣
⎥⎦
j
i
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Shell structure of atoms
•
•
•
•
Self-interaction free
Good for atomic properties
Start higher-order perturbation theory
Exchange is in, correlation still out
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Faster, or better
• The exchange integrals are the hidden cost
(fourth power). Linear-scaling efforts
underway
• Semi-empirical methods (ZDO, NDDO,
INDO, CNDO, MINDO): neglect certain
multi-center integrals
• Configuration interaction, Mǿller-Plesset
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Restricted vs. Unrestricted
• Spinorbitals in the Slater determinant:
spatial orbital times a spin function
• Unrestricted: different orbitals for different
spins
• Restricted: same orbital part
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Koopmans’ Theorems
• Total energy is invariant under unitary
transformations
• It is not the sum of the canonical MO orbital
energies
• Ionization energy, electron affinity are
given by the eigenvalue of the respective
MO, in the frozen orbitals approximation
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Atomic Units and Conversion Factors
(see handout)
1 a.u. = 2 Ry = 1 Ha
1 Ry = 13.6057 eV
1 eV = 23.05 kcal/mol
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
Software
• Gaussian (http://www.gaussian.com)
• Crystal (http://www.cse.clrc.ac.uk/cmg/CRYSTAL/,
http://www.theochem.unito.it/)
References
• F. Jensen, Introduction to Computational Chemistry
• J. M. Thijssen, Computational Physics
• B. H. Bransden and C. J. Joachim, Quantum
Mechanics, and also Physics of Atoms and Molecules
Feb 15 2005
3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari