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Density Functional Theory

An introduction

Quantum ESPRESSO Workshop


June 25
-
29, 2012

The Pennsylvania State University

University Park, PA

Marco Buongiorno Nardelli

Department of Physics and Department of Chemistry

University of North Texas

and

Oak Ridge National Laboratory

Overview


Properties of matter naturally fall into two main categories determined , respectively,
by the
electronic ground state

and the
electronic excited states


Electronic ground state determines equilibrium properties such as:


cohesive energy, equilibrium crystal structure, phase transitions between structures,
elastic constants, charge density, magnetic order, static dielectric and magnetic
susceptibilities, nuclear vibration and motion, etc.


Electronic excited states determine properties such:


low
-
energy excitations in metals, optical properties, transport, etc.


In our overview of electronic structure methods we will focus mostly on ground state
properties, and we will cover the basic principles underlying the computational
approaches, and we will learn how to compute some of the above properties using a
state
-
of
-
the
-
art scientific software package:
Quantum
-
ESPRESSO!

Electronic ground state


Closed
-
shell systems: rare gases and
molecular crystals. They remain atom
-
like
and tend to form close
-
packed solids


Ionic systems: compound formed by
elements of different electronegativity.
Charge transfer between the elements
thus stabilizes structures via the strong
Coulomb (electrical) interaction between
ions


Covalent bonding: involves a complete
change of the electronic states of the
atoms with pair of electrons forming
directional bonds


Metals: itinerant conduction electrons
spread among the ion cores. Electron

gas


as electronic glue of the system


Stable structure of solids are classified on the basis of their electronic ground
state, which determines the
minimum energy equilibrium structure
, and thus the
characteristics of the bonding between the nuclei

Electronic ground state


From the above discussion it starts to appear clearly the fundamental role
played by the electrons, and in a broader sense, by the

electron density

, in
determining the properties of real materials.


The electron density, , can be measured experimentally, providing support
for the bonding picture in different materials.


Since the electron density determines the ground state properties of the
material, its knowledge determines also the stable structure of the system:


Knowledge of the stable structure of the system as a function of pressure or
temperature is perhaps the most fundamental property of condensed matter: the
equation of state


Electronic structure theory is able to predict the electronic density that
corresponds to the minimum energy of the system as a function of volume
(
Ω
)
, so, in particular, it is straightforward to compute:





n
(
r
)

E

E
(

)
º
E
t
o
t
a
l
(

)
P


d
E
d

B



d
P
d



d
2
E
d

2
E = total energy of
the ground state

P = pressure

B = bulk modulus


Electronic ground state


Large variations in volume (thus in pressure) can give rise to phase transitions in
materials: at a given pressure a different structural phase becomes more stable than
the

natural


one.


Predictive power of electronic structure calculations in finding new structures of
matter under different external conditions: the quantity to compute then becomes the
enthalpy,
H=E+
PV
.


Example: Si and
Ge

phase diagram. Upon increasing pressure Si(
Ge
) changes its
equilibrium structure from diamond to
β
-
tin.

Electronic ground state


Elasticity: stress
-
strain relations in materials depend on the electronic ground state
and can be obtained via electronic structure methods


Variation of the total energy with respect to specific deformation of the shape of the
materials gives direct information on elasticity properties:





u
αβ
is
the symmetric stress tensor that defines the deformation



Example: stress in Si as a function of strain along the (100) direction:



















1


E

u


Electronic ground state


Equilibrium atomic geometries and atomic vibrations: simply obtained by the
electronic ground state


Given a geometrical configuration of nuclei:









Where
F
I

is the force on nucleus I and
C
IJ

are the force constants for lattice
dynamics


Knowing the force on each nucleus for any configuration, allows us to search for


The ground state of the complete (
electrons+nuclei
) system


The dynamical evolution at finite temperature (through a molecular
dynamics simulation)


The vibrational spectrum of the system






E

E
(
{
R
I
})
F
I


d
E
d
R
I
C
I
,
J


d
F
I
d
R
J


d
2
E
d
R
I
d
R
J
The many
-
body problem


How do we solve for the electronic ground state?

Solve a
many
-
body problem
:
the study of the effects of interaction between bodies, and the behavior of a
many
-
body system


The collection of nuclei and electrons in a piece of a material is a formidable
many
-
body problem, because of the intricate motion of the particles in the many
-
body system:






Electronic structure methods deal with solving this formidable problem starting
from the fundamental equation for a system of electrons ({
r
i
}) and nuclei ({
R
I
})





ˆ
H


2
2
m
e
Ñ
i
2

Z
I
e
2
|
r
i

R
I
|

1
2
i
,
I
å
e
2
|
r
i

r
j
|
i
¹
j
å
i
å

2
2
M
I
Ñ
I
2

I
å
1
2
e
2
|
R
I

R
J
|
I
¹
J
å
The many
-
body problem


Electronic terms:





Nuclear terms:





Electrons are fast (small mass, 10
-
31
Kg)
-

nuclei are slow (heavy mass, 10
-
27
Kg)


natural separation of variables


In the expression above we can ignore the kinetic energy of the nuclei, since it is
a small term, given the inverse mass of the nuclei


If we omit this term then the nuclei are just a fixed potential (sum of point
charges potentials) acting on the electrons: this is called the





Born
-
Oppenheimer approximation


The last terms remains to insure charge neutrality, but it is just a classical term
(
Ewald

energy)






2
2
m
e
Ñ
i
2

Z
I
e
2
|
r
i

R
I
|

1
2
i
,
I
å
e
2
|
r
i

r
j
|
i
¹
j
å
i
å





2
2
M
I
Ñ
I
2

I
å
1
2
e
2
|
R
I

R
J
|
I
¹
J
å
The electronic Hamiltonian


The Born
-
Oppenheimer approximation justifies the separation of electronic and ionic
variables due to the different time
-
scales of the relative motion


Electrons remain in their ground state as ions move:


Ions are responsible for the fixed external potential in which electrons move




where
T

is the kinetic energy of the electrons,
V
ext

is the potential acting on the
electrons due to the nuclei





V
int

is the electron
-
electron interaction term and
E
II

is the classical energy term of
the

system of ionic point charges


(Here we take =
m
e
=1)



ˆ
H

ˆ
T

ˆ
V
e
xt

ˆ
V
i
n
t

E
I
I



ˆ
V
e
xt

V
I
(
|
r
i

R
I
|
)
i
,
I


The electronic Hamiltonian


In quantum mechanical terms, the system of the electrons in the external potential of
the atoms is described by the many
-
body
wavefunction

of the system





where





is
the quantum mechanical probability of finding the systems of electrons with
coordinates within {
r
,
r
+
d
r
N
} and spin
s
N


The many
-
body
wavefuntion

for the electrons can be obtained solving the
Schroedinger

equation for the system:





where E is the ground state energy of the system in the external potential of the ions.









(
r
1
,
r
2
,
.
.
.
,
r
N
;
s
1
,
s
2
,
.
.
.
,
s
N
)


(
{
r
i
;
s
i
})



|

(
{
r
i
;
s
i
})
|
2
d
r
N


ˆ
H


E

The many
-
body electron wavefunction


The fundamental problem of electronic structure theory is the evaluation of the many
-
body electron
wavefunction


Knowledge of
Ψ

allows us to evaluate all the fundamental properties of the system as
expectation values of quantum mechanical operators






For example, a quantity of great relevance in the description of the electronic system
is the density of particles (electron density)






that is the expectation value of the density operator



n
(
r
)


ˆ
n
(
r
)




d
3
r
2
d
3
r
3
d
3
r
N

(
{
r
})
2
s


d
3
r
1
d
3
r
2
d
3
r
N

(
{
r
})
2





O


ˆ
O




d
3
r
1
d
3
r
2
d
3
r
N


(
{
r
})

ˆ
O

(
{
r
})
d
3
r
1
d
3
r
2
d
3
r
N

(
{
r
})
2


ˆ
n
(
r
)


(
r

r
i
)

1

i
f

r

r
i
0

i
f

r
¹
r
i

ì
í
ï
î
ï
i
å
The many
-
body electron wavefunction


Main quantity is indeed the ground state energy E that is calculated as the
expectation value of the Hamiltonian (it follows from the
Schroedinger

equation):





The ground state
wavefunction

Ψ
0

is the one that corresponds to the state with the
lowest energy that obeys all symmetries of particles and conservation laws


It allows us to introduce a

variational

principle


for the ground state:




E


ˆ
H




ˆ
H

ˆ
T

ˆ
V
i
n
t

d
3
r
V
e
xt
(
r
)
n
(
r
)



E

é
ë
ù
û
³
E
0
E
0

m
i
n

E

é
ë
ù
û
Ground state properties


Ground state properties, determined by the knowledge of the ground state
wavefunction
, include total energy, electron density and correlation function for the
system of the electrons in the external potential of the atoms


In the limit of small perturbations, also excited state properties can be derived, using
what, in quantum mechanics, is called

perturbation theory



For instance, small ionic displacements around the equilibrium positions will give us
information on the forces acting on the atoms, or more in general, on the vibrational
properties of the system


Force theorem

(aka Hellman
-
Feynman theorem), one of the most fundamental
theorems in quantum mechanics





Since the middle terms cancel at the ground state (by the definition of ground state
wavefunction
):





F
I



E

R
I




ˆ
H

R
I





R
I
ˆ
H



ˆ
H



R
I


E
I
I

R
I



F
I



E

R
I


d
3
r
n
(
r
)

V
e
xt
(
r
)

R
I



E
I
I

R
I
Forces depend on the ground state electron density!

How do we solve the electronic
structure problem
?


Solving for
Ψ

is a formidable problem
-

electron
-
electron interactions, that is
long
-
range Coulomb forces, induce correlations that are basically impossible to
treat exactly
-

independent electrons approximations


To appreciate the origin of this point of view, it is helpful to separate the different
Coulombic

contributions (classical and interacting) to the description of the
electronic system




E
Hartree

is the self
-
interaction energy of the electron density, treated as a classical
charge density





V
int

is the difficult part for which approximations are needed. In independent
electron approximations, this part is most often included as an effective potential
fitted to other more accurate data





E
C

V
i
n
t

E
C
C

w
h
e
r
e

E
C
C

E
H
a
r
t
r
e
e

d
3
r
V
e
xt
(
r
)
n
(
r
)

E
I
I




E
H
a
r
t
r
e
e

1
2
d
3
r
d
3
r
'
n
(
r
)
n
(
r
'
)
|
r

r
'
|

Electronic structure methods


In independent electron approximations, the electronic structure problem involves the
solution of a
Schroedinger
-
like equation for each of the electrons in the system





In this formalism, the ground state energy is found populating the lowest
eigenstates

according to the Pauli exclusion principle


Central equation in electronic structure theory. Depending on the level of
approximation we find this equation all over:


Semi
-
empirical methods (empirical pseudopotentials, tight
-
binding)


Density Functional Theory


Hartree
-
Fock and beyond


Mathematically speaking, we need to solve a generalized eigenvalue problem using
efficient numerical algorithms





ˆ
H
e
f
f

i
s
(
r
)


2
2
m
e
Ñ
2

V
e
f
f
s
(
r
)
é
ë
ê
ù
û
ú

i
s
(
r
)


i
s

i
s
(
r
)
The tight
-
binding method


Solution of an effective Hamiltonian obtained as a superposition of Hamiltonians
for isolated atoms plus corrections coming from the overlap of the
wavefuctions

(atomic orbitals)













Very efficient from a computational point of view


can handle reasonably large systems (between ab initio and atomistic)


Needs parameters form experiments or ab initio calculations





ˆ
H
e
f
f

ˆ
H
a
t
o
m


U
(
r
)
å
Hartree
-
Fock methods


Standard method for solving the many
-
body
wavefunction

of an electronic
system starting from a particular
ansatz

for the expression of
Ψ


A convenient form is to write a properly
antisymmetrized

(to insure the Pauli
principle is satisfied) determinant
wavefunction

for a fixed number of electrons
with a given spin (Slater determinant), and find the single determinant that
minimizes the total energy for the full interacting Hamiltonian






Use of this
wavefunction

ansatz

gives rise to equations of the form of non
-
interacting electrons where the effective potential depends upon the particular
electronic state


Methodologies to solve these equations have been developed mostly in the
framework of quantum chemistry calculations (J.
Pople
, Nobel prize for
Chemistry, 1998
-

GAUSSIAN: quantum chemistry code,
http://
www.gaussian.com
)






1
(
N
!
)
1
2
d
e
t

1
1

2
1

1
2

2
2
æ
è
ç
ç
ç
ö
ø
÷
÷
÷

Exchange


and

Correlation



The basic equations that define the Hartree
-
Fock method are obtained plugging the
Slater determinant into the electronic Hamiltonian to derive a compact expression for
its expectation value










Direct term is essentially the classical
Hartree

energy (acts between electrons with
different spin states (
i
=j

terms cancel out in the direct and exchange terms)


Exchange term acts only between same spin electrons, and takes care of the energy
that is involved in having electron pairs with parallel or anti
-
parallel spins together
with the obedience of the Pauli exclusion principle



ˆ
H


d
r

i
s

(
r
)

1
2
Ñ
2

V
e
xt
(
r
)
é
ë
ê
ù
û
ú
ò
i
,
s
å

i
s
(
r
)

E
I
I

1
2
d
r
d
r
'
ò
i
,
j
,
s
i
,
s
j
å

i
s
i

(
r
)

j
s
j

(
r
'
)
1
|
r

r
'
|

i
s
i
(
r
)

j
s
j
(
r
'
)

d
r
d
r
'
ò
i
,
j
,
s
å

i
s

(
r
)

j
s

(
r
'
)
1
|
r

r
'
|

j
s
(
r
)

i
s
(
r
'
)
Direct term

Exchange term


Exchange


and

Correlation



Exchange term is a two
-
body interaction term: it takes care of the many
-
body
interactions at the level of two single electrons.


In this respect it includes also correlation effects at the two
-
body level: it neglects all
correlations but the one required by the Pauli exclusion principle


Since the interaction always involve pairs of electrons, a two
-
body correlation term is
often sufficient to determine many physical properties of the system


In general terms it measures the joint probability of finding electrons of spin
s

at point
r

and of spin
s


at point
r



Going beyond the two
-
body treatment of Hartree
-
Fock introduces extra degrees of
freedom in the
wavefunctions

whose net effect is the reduction of the total energy of
any state


This additional energy is termed the

correlation


energy,
E
c

and is a key quantity for
the solution of the electronic structure problem for an interacting many
-
body system


Towards Density Functional Theory


The fundamental tenet of Density Functional Theory is that the complicated many
-
body electronic
wavefunction

Ψ

can be substituted by a much simpler quantity, that is
the electronic density






This means that a scalar function of position,
n
(
r
), determines all the information in
the many
-
body
wavefunction

for the ground state and in principle, for all excited
states


n
(
r
) is a simple non
-
negative function subject to the particle conservation sum rule





where
N

is the total number of electrons in the system


n
(
r
)


ˆ
n
(
r
)




d
3
r
2
d
3
r
3
d
3
r
N

(
{
r
})
2
s


d
3
r
1
d
3
r
2
d
3
r
N

(
{
r
})
2


n
(
r
)
d
3
r

N

Definitions

Function
: a prescription which maps one or more numbers to another
number:



Operator
: a prescription which maps a function onto another function:




Functional
: A functional takes a function as input and gives a number as
output:



Here
f
(
x
) is a function and
y

is a number.

An example is the functional to integrate
x

from


to
∞:



y

f
(
x
)

x
2


O


2

x
2

so

t
h
a
t

O
f
(
x
)


2

x
2
f
(
x
)


F
[
f
(
x
)
]

y


F
[
f
(
x
)
]

f
(
x
)
d
x




Towards Density Functional Theory


Density Functional Theory (DFT) is based on ideas that were around since the
early 1920

s: Thomas
-
Fermi theory of electronic structure of atoms (1927)


Electrons are distributed uniformly in the 6
-
dimensional space (3 spatial
coordinates x 2 spin coordinates) at the rate of 2 electrons per h
3

of volume


There is an effective potential fixed by the nuclear charges and the electron
density itself


Energy functional for an atom in terms of the electron density alone






Need approximate terms for kinetic energy and electronic exchange
-

no
correlations


Kinetic energy of the system electrons is approximated as an explicit functional
of the density, idealized as non
-
interacting electrons in a homogeneous gas with
density equal to the local density at any given point.


Local exchange term later added by Dirac (still used today)





E
T
F
n
(
r
)
é
ë
ù
û

C
1
n
5
3
(
r
)
d
3
r

V
e
xt
(
r
)
n
(
r
)
d
3
r

C
2
n
4
3
(
r
)
d
3
r
ò

1
2
n
(
r
)
n
(
r
'
)
|
r

r
'
|
ò
ò
ò
ò
d
3
r
d
3
r
'
Kinetic energy

Local exchange

Thomas
-
Fermi method


The ground state density and energy can be found by minimizing the functional
E[n] for all possible n(
r
) subject to the constraint on the total number of electrons




Using the method of Lagrange multipliers, the solution can be found by an
unconstrained minimization of the functional




where µ (Lagrange multiplier) is the Fermi energy


For small variations of the density
δn
(
r
), the condition for a stationary point leads
to the following relation between density and total potential




with


Only one equation for the density! remarkably simpler than the full many
-
body
Schroedinger

equation with 3N degrees of freedom for N electrons

The Hoenberg
-
Kohn theorems


The revolutionary approach of
Hohemberg

and Kohn in 1964 was to formulate DFT
as an
exact theory of a many
-
body system


The formulation applies to any system of interacting particles in an external potential
V
ext
(
r
), including any problem of electrons and fixed nuclei, where the
hamiltonian

can
be written





Foundation of Density Functional Theory is in the celebrated
Hoenberg

and Kohn
theorems

Hohenberg
-
Kohn theorems



DFT is based upon two theorems:


Theorem 1
: For any system of electrons in an external potential
V
ext
(
r
), that
potential is determined uniquely, except for a constant, by the ground state
density
n
0
(
r
)


Corollary 1
: Since the Hamiltonian is thus fully determined it follows that the
many
-
body
wavefunction

is determined. Therefore, all properties of the
system are completely determined given only the ground state density
n
0
(
r
)


Theorem 2
: A universal functional of the energy
E
[
n
] can be defined in terms
of the density n(
r
), valid for any external potential
V
ext
(
r
). For any particular
V
ext

the exact ground state of the system is determined by the global
minimum value of this functional


Corollary 2
: The functional
E
[
n
] alone is sufficient to determine the ground
state energy and density. In general, excited states have to be determined
by other means.


The exact functionals are unknown and must be very complicated!

Hohenberg
-
Kohn theorems


Proofs of H
-
K theorems are exceedingly simple, and both based on a simple
reduction ad absurdum

argument


Proof of Theorem 1: suppose there were two different external potentials


and with same ground state density,
n
(
r
).


The two potentials lead to two different Hamiltonians with different
wavefunctions
, that are hypothesized to lead to the same density. Then:




which leads to




But changing the
labelling

we can equally say that




Summing the above expression we get the absurd result

E
(1)+
E
(2)
<
E
(2)+
E
(1)



V
e
xt
1


V
e
xt
2

E
(
1
)


(
1
)
ˆ
H
(
1
)

(
1
)


(
2
)
ˆ
H
(
1
)

(
2
)

E
(
1
)


(
2
)
ˆ
H
(
1
)

(
2
)

E
(
2
)


(
2
)
ˆ
H
(
1
)

ˆ
H
(
2
)

(
2
)

E
(
2
)

d
3
r

{
V
e
xt
(
1
)
(
r
)

V
e
xt
(
2
)
(
r
)
}
n
(
r
)

E
(
2
)

E
(
1
)

d
3
r

{
V
e
xt
(
2
)
(
r
)

V
e
xt
(
1
)
(
r
)
}
n
(
r
)
Hohemberg
-
Kohn theorems


There cannot be two different external potentials differing by more than a constant
which give rise to the same non
-
degenerate ground state charge density.


The density uniquely determines the external potential to within a constant.


Then the
wavefunction

of any state is determined by solving the
Schroedinger

equation with this Hamiltonian.


Among all the solutions which are consistent with the given density, the unique
ground state
wavefunction

is the one that has the lowest energy.



BUT: we are still left with the problem of solving the many
-
body problem in the
presence of
V
ext
(
r
)


EXAMPLE: electrons and nuclei
-

the electron density uniquely determines the
positions and types of nuclei, which can easily be proven from elementary quantum
mechanics, but we still are faced with the original problem of many interacting
electrons moving in the potential due to the nuclei

Hohenberg
-
Kohn theorems


Theorem 2 gives us a first step towards an operative way to solve the problem


Theorem 2 can be proved in a very similar way, and the demonstration leads to
a general expression for the universal functional of the density in DFT






F
HK
[
n
] is a universal functional of the density that determines all the many
-
body
properties of the system



PROBLEM: we do not know what is this functional!


We only know that:


is a functional of the density alone


is independent on the external potential (thus its universality)


It follows that if the functional F
HK
[n] were known, then by minimizing the total
energy of the system with respect to variations in the density function n(
r
), one
would find the exact ground state density and energy.


E
H
K
[
n
]

T
[
n
]

E
i
n
t
[
n
]

d
3
r
V
e
xt
(
r
)
n
(
r
)

E
I
I


F
H
K
[
n
]

d
3
r
V
e
xt
(
r
)
n
(
r
)

E
I
I

Hohenberg
-
Kohn extensions


Hohenberg
-
Kohn theorems can be generalized to several types of particles


special role of the density and the external potential in the
Hohenberg
-
Kohn
theorems is that these quantities enter the total energy explicitly only through the
simple bilinear integral term



If there are other terms in the Hamiltonian having this form, then each such pair
of external potential and particle density will obey a
Hohenberg
-
Kohn theorem


For example, Spin Density Functional Theory: Zeeman term that is different for
spin up and spin down fermions in external magnetic fields


All argument above can be generalized to include two types of densities, the
particle density and the spin density




with a density functional




In absence of magnetic fields, the solution can still be polarized (as in
unrestricted Hartree
-
Fock theory)

Kohn and Sham ansatz


H
-
K theory is in principle exact (there are no approximations, only two elegant
theorems) but impractical for any useful purposes


Kohn
-
Sham
ansatz
:
replace a problem with another
, that is the original many
-
body problem with an auxiliary independent
-
particle model


Ansatz
: K
-
S assume that the ground state density of the original interacting
system is equal to that of some chosen non
-
interacting system that is exactly
soluble, with all the difficult part (exchange and correlation) included in some
approximate functional of the density.


Key assumptions:


The exact ground state density can be represented by the ground state
density of an auxiliary system of non
-
interacting particles. This is called

non
-
interacting
-
V
-
representability

;


The auxiliary Hamiltonian contains the usual kinetic energy term and a local
effective potential acting on the electrons


Actual calculations are performed on this auxiliary Hamiltonian




through the solution of the corresponding
Schroedinger

equation for N
independent electrons (Kohn
-
Sham equations)


H
KS
(
r
)


1
2
Ñ
2

V
KS
(
r
)
Kohn and Sham ansatz


The density of this auxiliary system is then:




The kinetic energy is the one for the independent particle system:





We define the classic electronic Coulomb energy (
Hartree

energy) as usual:








n
(
r
)

|

i
s
(
r
)
|
2
i

1
,
N

s




T
s


1
2

i
s
(
r
)
Ñ
2

i
s
(
r
)

i

1
,
N
å
s
å
1
2
|
Ñ

i
s
(
r
)
|
2
i

1
,
N
å
s
å



E
H
a
r
t
r
e
e
[
n
]

1
2
d
3
r
d
3
r
'
n
(
r
)
n
(
r
'
)
|
r

r
'
|


Interacting electrons


+ real potential

Non
-
interacting
auxiliary

particles in an
effective potential

Kohn and Sham equations


Finally, we can rewrite the full H
-
K functional as




All many body effects of exchange and correlation are included in
E
xc







So far the theory is still exact, provided we can find an

exact


expression for
the exchange and correlation term


If the universal functional
E
xc
[n] were known, then the exact ground state energy
and density of the many
-
body electron problem could be found by solving the
Kohn
-
Sham equations for independent particles.


To the extent that an approximate form for
E
xc
[n] describes the true exchange
-
correlation energy, the Kohn
-
Sham method provides a feasible approach to
calculating the ground state properties of the many
-
body electron system.


E
KS
[
n
]

T
s
[
n
]

d
3
r
V
e
xt
(
r
)
n
(
r
)

E
H
a
r
t
r
e
e
[
n
]

E
I
I


E
xc
[
n
]


E
xc
[
n
]

F
H
K
[
n
]

(
T
s
[
n
]

E
H
a
r
t
r
e
e
[
n
]
)

ˆ
T

T
s
[
n
]

ˆ
V
i
n
t

E
H
a
r
t
r
e
e
[
n
]
Kohn and Sham equations



The solution of the Kohn
-
Sham auxiliary system for the ground state can be
viewed as the problem of minimization with respect to the density n(
r
) that can
be done varying the
wavefunctions

and applying the chain rule to derive the
variational

equations:




subject to the
orthonormalization

constraint



Since



One ends up with a set of
Schroedinger
-
like equations



where H
KS

is the effective Hamiltonian



with



Kohn and Sham equations


The great advantage of recasting the H
-
K functional in the K
-
S form is that
separating the independent particle kinetic energy and the long range
Hartree

terms, the remaining exchange and correlation functional can be reasonably
approximated as a local or nearly local functionals of the electron density


Local Density Approximation

(LDA):
E
xc
[n] is a sum of contributions from each
point in space depending only upon the density at each point independent on
other points



where is the exchange and correlation energy per electron.



is a universal functional of the density, so must be the same as for a
homogeneous electron gas of given density
n


The theory of the homogeneous electron gas is well established and there are
exact expression (analytical or numerical) for both exchange and correlation
terms


Exchange as




Correlations from exact Monte Carlo calculations (
Ceperley
, Alder, 1980)


E
xc
L
D
A
[
n
]

d
3
r
n
(
r
)

xc
(
n
(
r
)
)




xc
(
n
)



xc
(
n
)


























x
(
n
)


0
.
4
5
8
r
s

w
h
e
r
e

r
s

i
s
d
e
f
i
n
e
d

a
s
t
h
e

a
ve
r
a
g
e

d
i
st
a
n
ce

b
e
t
w
e
e
n

e
l
e
ct
r
o
n
s
a
t

a

g
i
ve
n

d
e
n
si
t
y
n
:

4

3
r
s
3

1
n
Kohn and Sham equations


The eigenvalues are not the energies to add or subtract electrons from the interacting
many
-
body system


Exception: highest eigenvalue in a finite system is minus the ionization energy,


-
I. No other eigenvalue is guaranteed to be correct by the Kohn
-
Sham construction.


However, the eigenvalues have a well
-
defined meaning within the theory and they
can be used to construct physically meaningful quantities


perturbation expressions for excitation energies starting from the Kohn
-
Sham
eigenfunctions

to obtain new functionals


explicit many
-
body calculation that uses the Kohn
-
Sham
eigenfunctions

and
eigenvalues as input. Commonly done in Quantum Monte Carlo simulations


In rigorous terms, the eigenvalues in the KS theory have a well defined mathematical
meaning: derivative of the total energy with respect to the occupation of a state

Kohn and Sham equations


The previous result, trivial in the non
-
interacting case, raises interesting issues in the
KS case


Given the expression for the exchange and correlation energy, one can derive the
expression for the exchange and correlation potential
V
xc





It can be shown that the response part of the potential (the derivative of the energy
wrt

the density) can vary discontinuously between states giving rise to discontinuous
jumps in the eigenvalues:

band
-
gap discontinuity



Critical problem of the gap in an insulator: the eigenvalues of the ground state Kohn
-
Sham potential should not be the correct gap, at least in principle.


Indeed, it is well known that most known KS functionals underestimate the gap of
insulators, however, this is an active field of research and new developments are
always possible.

Kohn and Sham equations


Finally, the set of K
-
S equations with LDA for
exchange and correlation give us a
formidable theoretical tool to study ground
state properties of electronic systems


Set of
self
-
consistent

equations that have to
be solved simultaneously until convergence is
achieved


Note:

K
-
S eigenvalues and energies are
interpreted as true electronic
wavefunction

and electronic energies (electronic states in
molecules or bands in solids)


Note:

K
-
S theory is a ground
-
state theory and
as such is supposed to work well for ground
state properties or small perturbations upon
them


Extremely successful in predicting materials
properties
-

golden standard in research and
industry


Local Density Approximation


Although it might seem counterintuitive, solids can be often considered as close
to the limit of the homogeneous electron gas = electron gas immersed in a
uniformly positive charge background (true for metals, increasingly less true for
very inhomogeneous charge distributions such as in nanostructures and isolated
molecules)


In this limit it is known that exchange and correlation (
x
-
c
) effects are local in
character and the
x
-
c

energy is simply the integral of the
x
-
c

energy density at
each point in space assumed to be the same as a homogeneous electron gas
with that density


Generalizing to the case of electrons with spin (spin
-
polarized or unrestricted),
we can introduce the Local Spin Density Approximation (LSDA)




Most general local expression for the exchange and correlation energy


Ultimately, the validity of LDA or LSDA approximations lies in the remarkably
good agreement with experimental values of the ground state properties for
most materials


Can be easily improved upon without loosing much of the computational appeal
of a local form


E
xc
L
SD
A
[
n

,
n

]

d
3
r
n
(
r
)

xc
(
n

(
r
)
,
n

(
r
)
)

Local Density Approximation


The rationale for the local approximation is that for the densities typical of those found
in solids, the range of the effects of exchange and correlation is rather short
-
range =
the exchange and correlation hole is well localized


However, there is no rigorous proof of this, only actual observations and one should
test different cases individually


Problem of self
-
interactions: in the Hartree
-
Fock approximation the unphysical self
term in the
Hartree

interaction (the interaction of an electron with itself) is exactly
cancelled by the non
-
local exchange interaction.


In the local approximation to exchange, the cancellation is only approximate and
there remain spurious self
-
interaction terms that are negligible in the homogeneous
gas but large in confined system such as atoms (need of Self
-
Interaction Corrections
or SIC)


However, in most known cases LSDA works remarkably well, due to the lucky
occurrence that the exchange and correlation hole, although approximate, still
satisfies all the sum rules.

Generalized Gradient Approximations


The first step beyond the L(S)DA approximation is a functional that depends
both on the magnitude of the density
n
(
r
) and of its gradient |

n
(
r
)|: Generalized
Gradient Approximations (GGA

s) where higher order gradients are used in the
expansion:




where
F
xc

is a dimensionless function and
ε
x
hom

is the exchange energy of the
uniform electron gas.


Gradients are difficult to work with and often can lead to worse results. There
are however consistent ways to improve upon L(S)DA using gradient
expansions


Most common forms differ by the


choice of the F function:


PW91, PBE, BLYP,…


Beyond GGA


Beyond GGA

s
:


Non
-
local density functionals: functionals that depends on the value of the density
around the point
r
(Average Density and Weighted Density Approximations)




where



Orbital dependent functionals: mostly useful for systems where electrons tend to be
localized and strongly interacting


SIC
-

self
-
interaction corrected functionals


LDA+U
-

local functional + orbital
-
dependent interaction for highly localized
atomic orbitals (Hubbard U)


EXX (exact exchange)
-

functionals that include explicitly the exact exchange
integral of Hartree
-
Fock


Hybrid functionals (B3LYP)
-

combination of orbital
-
dependent Hartree
-
Fock and
explicit DFT. Most accurate functional on the market
-

most preferred for
chemistry calculations


Beyond GGA


SIC
-

methods that use approximate functionals and add

self
-
interaction
corrections


to attempt to correct for the unphysical self
-
interaction in many
functionals for exchange and correlation
E
xc


Old approach proposed first by
Hartree

himself to compute the electronic
properties of atoms: different potential for each occupied state by subtracting a
self
-
interaction term due to the charge density of that state.


In extended system such a simple approach does not work and one has to
resort to more sophisticated ways to subtract the spurious interaction.


Most useful for describing magnetic order and magnetic states in transition
metal oxides and similar.



LDA+U
-

LDA or GGA type calculations coupled with an additional orbital
dependent interaction, usually considered only for highly localized atomic
-
like
orbitals on the same site, as the U interaction in Hubbard models.


The overall effect is to shift the energy of the atomic
-
like orbitals
wrt

all the other
levels


The

U


parameter is often taken from

constrained density functional


calculations so that the theories do not contain adjustable parameters.


Mostly useful in transition metal systems



Beyond GGA


Orbital dependent functionals
-

expressing
E
xc

explicitly in terms of the independent
particle orbitals, naturally implies that
E
xc

has discontinuities at filled shells
-

essential
for a correct description of the energy gap in insulators


Search for Optimized Effective Potentials (OEP): mainly applied to the Hartree
-
Fock
exchange functional, which is straightforward to write in terms of the orbitals, which is
called

exact exchange


or

EXX

.

Beyond GGA


Hybrid functionals
-

combination of orbital
-
dependent Hartree
-
Fock and an explicit
density functional. Simplest form

half
-
and
-
half

:




More sophisticated forms involve mixing of different exchange and correlation
models, as in B3LYP, where




and coefficients are fitted to atomic and molecular data. Certain degree of empirical
fitting is required
-

determines the accuracy of the model.