MIT dft pdf for students. this has the pdf of dft.

This has the pdf of dft

1. 3.320: Lecture 5 (Feb 15 2005)
THE MANY THE MANY
--
BODY PROBLEM BODY PROBLEM
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

2. When is a particle like a wave ? Wavelength •
m
omentum = Planck
↕
λ
•p
=
h
( h = 6.6 x 10
-34
J s )
)
,
(
t
rr
Ψ
=
Ψ
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

3. Time-dependent Schrödinger’s equation
(Newton’s 2
nd
law for quantum objects)
t
t
r
i
t
r
t
r
V
t
r
m
∂
Ψ
∂
=
Ψ
+
Ψ
∇
−
)
,
(
)
,
(
)
,
(
)
,
(
2
2
2
r
h
r
r
r
h
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

4. Stationary Schrödinger’s Equation (I)
t
t
r
i
t
r
t
r
V
t
r
m
∂
Ψ
∂
=
Ψ
+
Ψ
∇
−
)
,
(
)
,
(
)
,
(
)
,
(
2
2
2
r
h
r
r
r
h
*
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

5. Stationary Schrödinger’s Equation (II)
)
(
)
(
t
f
E
t
f
dtd
i
=
h
)
(
)
(
)
(
2
2
2
r
E
r
r
V
m
r
r
r
h
ϕ
ϕ
=
⎥⎦⎤
⎢⎣⎡
+
∇
−
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

6. )
(
)
(
t
f
E
t
f
dtd
i
=
h
⎟⎠⎞
⎜⎝⎛
−
=
t
E
i
t
f
h
exp
)
(
Free particle
Ψ
(x,t)=
φ
(x)f(t)
)
(
)
(
2
2
2
x
E
x
m
ϕ
ϕ
=
∇
−
h
⎟⎟⎠⎞
⎜⎜⎝⎛
=
x
mE
i
x
h
2
exp
)
(
ϕ
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

7. Interpretation of the Quantum Wavefunction (Copenhagen)
2
(,
)
xt
Ψ
is the probability of finding an electron
in
x
and
t
2
2
(
)
e
xp(
)
(
)
i
xE
t
x
ϕϕ
−=
h
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

8. A Traveling “Plane”
W
ave
(,
)
e
x
p
[
(
)
]
xt
i
k
x
t
ω
Ψ∝
−
Diagram of plane wave removed for copyright reasons.
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

9. Metal Surfaces (I)
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

10. Metal Surfaces (II)
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

11. Infinite Square Well
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
1
6
1
4
1
2
1
0 8 6 4 2 0
n
=
1
n
=
2
n
=
3
n
=
4
-
a
0
a
x
y
1
y
2
y
3
y
4
y
(
x
)
8
m
a
2
p
2
h
2
E
Figure by MIT OCW.

12. Finite Square Well
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
a
y
2
(
x
)
a
y
3
(
x
)
a
y
4
(
x
)
a
y
1
(
x
) 1
-
1
1
0
.
5
-
0
.
5
-
2
-
1
2

x
/
a
0
1
-
1
1
0
.
5
-
2
-
1
2

x
/
a
0
1
-
1
1
0
.
5
-
2
-
1
2
0
1
-
1
1
-
2
-
1
2

x
/
a

x
/
a
0
Figure by MIT OCW.

13. A Central Potential (e.g. the Nucleus)
22
2
2
22
22
2
ˆ
()
2
HV
r
mx
y
z
∂
∂∂
=−
∇
+
∇
=
+
+
∂
∂∂
h
2
2
2
22
2
2
2
11
1
ˆ
sin
(
)
2s
i
n
s
i
n
H
rV
r
mr
r
r
r
r
ϑ
ϑϑ
ϑ
ϑ
ϕ
⎡⎤
∂∂
∂
∂
∂
⎛⎞
⎛
⎞
=−
+
+
+
⎢⎜
⎟
⎜
⎟
⎥
∂∂
∂
∂
∂
⎝⎠
⎝
⎠
⎣⎦
h
)
,
(
)
(
)
(
ϕ
ϑ
ψ
lm
Elm
Elm
Y
r
R
r
=
r
22
2
22
2(
1
)
()
()
()
22
El
El
dd
l
l
Vr
R
r
E
R
r
m
d
r
r
dr
r
µ
⎡⎤
⎛⎞
+
−+
+
+
=
⎢⎥
⎜⎟⎝⎠
⎣⎦
hh
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

14. Solutions in a Coulomb Potential:
the Periodic Table
http://www.orbitals.com/orb/orbtable.htm
Court
e
sy of Da
vi
d
Manthey. Used wi
th permi
s
sion.
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari
___________________________________________________________ _

15. Orthogonality, Expectation Values,
and Dirac’s
<
bra|kets>
ψ
ψ
ψ
=
=
)
(
rr
ij
j
i
j
i
r
d
r
r
δ
ψ
ψ
ψ
ψ
=
=
∫
r
r
r
)
(
)
(
*
i
i
i
i
i
E
H
r
d
r
r
V
m
r
=
=
⎥⎦⎤
⎢⎣⎡
+
−
∫
ψ
ψ
ψ
ψ
ˆ
)
(
)
(
2
)
(
2
*
r
r
r
h
r
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

16. Matrix Formulation (I)
ψ
ψ
ψ
ψ
E
H
r
E
r
H
=
⇔
=
ˆ
)
(
)
(
ˆ
r
r
{
}
functions
orthogonal
k
,
1
n
k
n
n
n
c
ϕ
ϕ
ψ
∑
=
=
ψ
ϕ
ψ
ϕ
m
m
E
H
=
ˆ
m
n
k
n
m
n
Ec
H
c
=
∑
=
ϕ
ϕ
ˆ
,
1
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

17. Matrix Formulation (II)
m
n
m
k
n
n
Ec
H
c
=
∑
=
ϕ
ϕ
ˆ
,
1
m
k
n
n
mn
Ec
c
H
=
∑
=
,
1
⎟⎟⎟⎟⎟⎟ ⎠⎞
⎜⎜⎜⎜⎜⎜ ⎝⎛
=
⎟⎟⎟⎟⎟⎟ ⎠⎞
⎜⎜⎜⎜⎜⎜ ⎝⎛
⋅
⎟⎟⎟⎟⎟⎟ ⎠⎞
⎜⎜⎜⎜⎜⎜ ⎝⎛
k
k
kk
k
k
cc
E
cc
H
H
H
H
...
...
......
.
.
.
.
.
.
......
1
1
1
1
11
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

18. Variational
P
rinciple
[]
ˆ
||
|
H
E
<
ΦΦ
>
Φ=
<
ΦΦ
>
If , then
Φ
is the ground
state wavefunction, and viceversa…
[]
0
EE
Φ≥
[]
0
EE
Φ= Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

19. Energy of an Hydrogen Atom
ˆ
H
E
α
α
α
αα
Ψ
Ψ
=
ΨΨ
(
)
exp
Cr
α
α
Ψ=
−
22
2
2
3
2
11
,
22
CC
C
r
αα
α
α
α
α
ππ
π
α
αα
Ψ
Ψ
=
Ψ
−
∇
Ψ=
Ψ
−
Ψ=
−
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

20. Two-electron atom
)
,
(
)
,
(
|
|
1
21
21
2
1
2
1
2
1
2
1
22
21
r
r
E
r
r
r
r
rZ
rZ
el
r
r
r
r
r
r
ψ
ψ
=
⎥⎦⎤
⎢⎣⎡
−
+
−
−
∇
−
∇
−
Many-electron atom
2
11
11
(
,
...,
)
(
,
...,
)
2|
|
in
e
l
n
ii
i
j
i
ii
j
Z
rr
E
r
r
rr
r
ψψ
>
⎡⎤
−∇
−
+
=
⎢⎥
−
⎢⎥⎣⎦
∑∑
∑
∑
rr
rr
rr
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

21. Energy of a collection of atoms
N
e
N
N
e
e
N
e
V
V
V
T
T
H
−
−
−
+
+
+
+
=
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
•T
e
: quantum kinetic energy of the electrons
•V
e-
e
: electron-electron interactions
•V
N-N
: electrostatic nucleus-nucleus repulsion
•V
e-N
: electrostatic electron-nucleus attraction
(electrons in the field of all the nuclei)
()
∑∑
∑∑
∑
>
−
−
−
=
⎥⎦⎤
⎢⎣⎡
−
=
∇
−
=
ii
j
j
i
e
e
ii
I
i
I
N
e
i
e
r
r
V
r
R
V
V
T
|
|
1
ˆ
ˆ
21
ˆ
2
r
r
r
r
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

22. Electrons and Nuclei
)
,...,
,
,...,
(
)
,...,
,
,...,
(
ˆ
1
1
1
1
N
n
tot
N
n
R
R
r
r
E
R
R
r
r
H
r
r
r
r
r
r
r
r
ψ
ψ
=
•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
Born-
Oppenheimer
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

23. 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
b
e
rather crude to restrict to 10 the number of values of each variable at which to tabulate th
is function, but even so, full
tabulation would require 10
78
entries.”
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

24. Mean-field approach
•
I
ndependent 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
•
T
his 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

25. Hartree
Equations
The Hartree
e
quations 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)
)
(
)
(
)
(
)
,...,
(
2
2
1
1
1
n
n
n
r
r
r
r
r
r
L
r
r
r
r
ϕ
ϕ
ϕ
ψ
=
)
(
)
(
|
|
1
|
)
(
|
)
(
21
2
2
i
i
i
i
j
I
i
j
i
j
j
j
i
I
i
r
r
r
d
r
r
r
r
R
V
r
r
r
r
r
r
r
r
ϕ
ε
ϕ
ϕ
=
⎥⎥⎦⎤
⎢⎢⎣⎡
−
+
−
+
∇
−
∑∑
∫
≠
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

26. The self-consistent field
•
T
he single-particle Hartree
operator is self-
consistent ! I.e., it depends in itself on the orbitals
that are the solution of all other
Hartree
e
quations
•
W
e have
n
simultaneous integro-differential
equations for the
n
orbitals
•
S
olution is achieved iteratively
)
(
)
(
|
|
1
|
)
(
|
)
(
21
2
2
i
i
i
i
j
I
i
j
i
j
j
j
i
I
i
r
r
r
d
r
r
r
r
R
V
r
r
r
r
r
r
r
r
ϕ
ε
ϕ
ϕ
=
⎥⎥⎦⎤
⎢⎢⎣⎡
−
+
−
+
∇
−
∑∑
∫
≠
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

27. Iterations to self-consistency
•
I
nitial guess at the orbitals
•
Construction of all the operators
•
S
olution of the single-particle pseudo-
Schrodinger
equations
•
W
ith this new set of orbitals, construct the
Hartree
operators again
•
I
terate the procedure until it (hopefully)
converges
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

28. Differential Analyzer
Vann
evar
Bush and th
e Differen
t
ial Analyzer.
Court
e
sy of the MIT

Mu
seum. Used
wi
th permission.
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

29. What’s missing
•
I
t does not include correlation
•
T
he wavefunction is not antisymmetric
•
I
t does remove
nl
accidental degeneracy of
the hydrogenoid
a
toms
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

30. Spin-Statistics
•
A
ll 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
•
F
or bosons it is symmetric
)
,...,
,...,
,...,
,
(
)
,...,
,...,
,...,
,
(
2
1
2
1
n
j
k
n
k
j
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
ψ
ψ
−
=
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

31. Slater determinant
•
A
n antisymmetric
w
avefunction is constructed via a
Slater determinant of the individual orbitals
(instead
of just a product, as in the Hartree
a
pproach)
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
!
1
)
,...,
,
(
2
2
2
1
1
1
2
1
n
n
n
n
r
r
r
r
r
r
r
r
r
n
r
r
r
r
L
r
r
M
O
M
M
r
L
r
r
r
L
r
r
r
r
r
ν
β
α
ν
β
α
ν
β
α
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ψ
=
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

32. Pauli
p
rinciple
•
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

33. Hartree-Fock
Equations
The Hartree-Fock
equations are, again, obta
ined from the variational
p
rinciple: we
look for the minimum of the
many-electron Schroedinger
e
quation in the class of all
wavefunctions that are written as
a single Slater determinant
)
(
)
(
)
(
|
|
1
)
(
)
(
)
(
|
|
1
)
(
)
(
)
(
21
*
*
2
i
i
j
j
i
j
j
i
j
j
i
j
j
i
i
I
I
i
r
r
r
d
r
r
r
r
r
r
d
r
r
r
r
r
r
R
V
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
λ
µ
µ
λ
µ
λ
µ
µ
µ
λ
ϕ
ε
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
=
⎥⎥⎦⎤
⎢⎢⎣⎡
−
−
⎥⎥⎦⎤
⎢⎢⎣⎡
−
+
⎥⎦⎤
⎢⎣⎡
−
+
∇
−
∑
∫
∫
∑
∑
Slater
r
r
n
=
)
,...,
(
1
r
r
ψ
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

34. Shell structure of atoms
•
S
elf-interaction free
•
G
ood for atomic properties
•
S
tart higher-order perturbation theory
•
E
xchange is in, correlation still out
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

35. Faster, or better
•
T
he exchange integrals are the hidden cost
(fourth power). Linear-scaling efforts underway
•
S
emi-empirical methods (ZDO, NDDO,
INDO, CNDO, MINDO): neglect certain multi-center integrals
•
Configuration interaction, M
ǿ
ller-Plesset
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

36. Restricted vs. Unrestricted
•
S
pinorbitals
i
n the Slater determinant:
spatial orbital times a spin function
•
U
nrestricted: different orbitals
for different
spins
•
R
estricted: same orbital part
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

37. Koopmans’
T
heorems
•
T
otal energy is invariant under unitary
transformations
•
I
t is not the sum of the canonical MO orbital
energies
•
I
onization energy, electron affinity are
given by the eigenvalue
o
f the respective
MO, in the frozen orbitals
approximation
Feb 15 2005 3.320 Atomistic Modeling of Materials -- Gerbrand Ceder and Nicola Marzari

38. 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

39. Software
•
G
aussian (
http://www.gaussian.com
)
•
C
rystal (
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