Quantum ESPRESSO Workshop
June 25-29, 2012
The Pennsylvania State University
University Park, PA
Density Functional Theory
An introduction
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
•
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
– 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
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, n(r), 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:
E = E(W) º Etotal (W)
dE
dW
dP
d 2E
B = -W
=W
dW
dW2
P=-
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:
s ab = •
1 ¶E
W ¶uab
uαβ is the symmetric stress tensor that defines the deformation
Example: stress in Si as a function of strain along the (100) direction:
Electronic ground state
•
•
Equilibrium atomic geometries and atomic vibrations: simply obtained by the
electronic ground state
Given a geometrical configuration of nuclei:
E = E({R I })
FI = -
dE
dR I
dFI
d 2E
CI,J = =dR J
dR I dR J
•
•
Where FI is the force on nucleus I and CIJ 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
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 manybody system:
Electronic structure methods deal with solving this formidable problem starting
from the fundamental equation for a system of electrons ({ri}) and nuclei ({RI})
2
2
Z
e
1
e
I
Hˆ = Ñ 2i + å
+ å
å
2me i
2 i ¹ j | ri - r j |
i,I | ri - R I |
2
2
1
e
-å
ÑI2 + å
2 I ¹J | R I - R J |
I 2MI
2
The many-body problem
•
Electronic terms:
ZI e 2
2
1
e
Ñ 2i + å
+ å
å
2me i
2 i ¹ j | ri - r j |
i,I | ri - R I |
2
•
Nuclear terms:
2
1
e
-å
ÑI2 + å
2 I ¹J | R I - R J |
I 2MI
2
•
•
•
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)
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
Hˆ = Tˆ + Vˆext + Vˆint + EII
where T is the kinetic energy of the electrons, Vext is the potential acting on the
electrons due to the nuclei
Vˆext = åVI (| ri - R I |)
i,I
Vint is the electron-electron interaction term and EII is the classical energy term of
the system of ionic point charges
(Here we take =me=1)
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
Y = Y(r1,r2 ,...,rN ;s1,s2 ,...,sN ) = Y({ri ;si })
where
| Y({ri ;si }) |2 dr N
•
is the quantum mechanical probability of finding the systems of electrons with
coordinates within {r,r+drN} and spin sN
The many-body wavefuntion for the electrons can be obtained solving the
Schroedinger equation for the system:
ˆ = EY
HY
where E is the ground state energy of the system in the external potential of the ions.
The many-body electron wavefunction
•
•
The fundamental problem of electronic structure theory is the evaluation of the manybody electron wavefunction
Knowledge of Ψ allows us to evaluate all the fundamental properties of the system as
expectation values of quantum mechanical operators
O=
•
Y Oˆ Y
Y Y
ˆ
d r d r d r Y ({r})OY({r})
ò
º
ò d r d r d r Y({r})
3
3
1
*
3
N
2
3
3
1
2
3
N
2
For example, a quantity of great relevance in the description of the electronic system
is the density of particles (electron density)
n(r) =
ˆ Y
Y n(r)
Y Y
º
3
3
d
r
d
ò 2 r3
d 3rN å Y({r})
3
3
ò d r1d r2
s
d 3rN Y({r})
that is the expectation value of the density operator
ì 1 if r = ri
ˆ = å d (r - ri ) = ïí
n(r)
i
ïî0 if r ¹ ri
2
2
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):
E=
•
•
Y Hˆ Y
Y Y
º Hˆ = Tˆ + Vˆint + ò d 3rVext (r)n(r)
The ground state wavefunction Ψ0 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 éë Y ùû ³ E0
E0 = min E éë Y ùû
Y
Ground state properties
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•
•
•
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
¶E
¶E
¶Hˆ
¶Y ˆ
¶Y
FI = =- Y
Y H Y - Y Hˆ
- II
¶R I
¶R I
¶R I
¶R I
¶R I
•
Since the middle terms cancel at the ground state (by the definition of ground state
wavefunction):
FI = -
¶V (r) ¶E
¶E
= - ò d 3rn(r) ext - II
¶R I
¶R 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 C = Vint + E CC where E CC = EHartree + ò d 3rVext (r)n(r) + EII
•
EHartree is the self-interaction energy of the electron density, treated as a classical
charge density
EHartree =
•
1 3 3 n(r)n(r ')
d rd r '
ò
2
|r -r '|
Vint 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
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
2
é
ù s
2
s
ˆ
Heff y (r) = ê Ñ +Veff (r)ú y i (r) = e isy is (r)
ë 2me
û
s
i
•
•
•
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
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)
Hˆ eff = å Hˆ atom + DU(r)
•
•
•
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
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
æ y 11 y 12
1
ç 2
2
Y=
det
y
y
1
ç 1
2
(N !) 2
ç
è
•
•
ö
÷
÷
÷
ø
Use of this wavefunction ansatz gives rise to equations of the form of noninteracting 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)
“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
é 1
ù
Y Hˆ Y = å ò dry is* (r) ê - Ñ 2 +Vext (r) úy is (r) + EII
i,s
ë 2
û
1
1
s *
s
s *
+ å ò drdr 'y i i (r)y j j (r ')
y isi (r)y j j (r ')
2 i, j ,si ,s j
|r -r '|
- å ò drdr 'y is* (r)y s*
(r ')
j
i, j,s
•
•
1
y sj (r)y is (r ')
|r - r '|
Direct term
Exchange term
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
“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, Ec 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 manybody electronic wavefunction Ψ can be substituted by a much simpler quantity, that is
the electronic density
n(r) =
•
•
ˆ Y
Y n(r)
Y Y
º
3
3
ò d r2d r3
3
3
d
r
d
ò 1 r2
d 3rN å Y({r})
2
s
d 3rN Y({r})
2
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
3
n(r)d
r =N
ò
where N is the total number of electrons in the system
Definitions
Function: a prescription which maps one or more numbers to another
number:
y = f (x) = x2
Operator: a prescription which maps a function onto another function:
¶2
¶2
O = 2 so that Of (x) = 2 f (x)
¶x
¶x
Functional: A functional takes a function as input and gives a number as
output:
F[f (x)] = y
Here f(x) is a function and y is a number.
An example is the functional to integrate x from –∞ to ∞:
¥
F[f (x)] =
ò f (x)dx
-¥
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 h3 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
1 n(r)n(r ') 3 3
d rd r '
òò
2
|r - r '|
Local exchange
ETF éën(r)ùû = C1 ò n 3 (r)d 3r + ò Vext (r)n(r)d 3r + C2 ò n 3 (r)d 3r +
5
Kinetic energy
•
•
•
4
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)
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
Vext(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 Vext(r), that
potential is determined uniquely, except for a constant, by the ground state
density n0(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 n0(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 Vext(r). For any particular
Vext 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
1
Proof of Theorem 1: suppose there were two different external potentials Vext
2
and Vext
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:
E (1) = Y(1) Hˆ (1) Y(1) < Y(2) Hˆ (1) Y(2)
which leads to
(1)
(2)
E (1) < Y(2) Hˆ (1) Y(2) = E (2) + Y(2) Hˆ (1) - Hˆ (2) Y(2) = E (2) + ò d 3r {Vext
(r) -Vext
(r)}n(r)
But changing the labelling we can equally say that
(2)
(1)
E (2) < E (1) + ò d 3r {Vext
(r) -Vext
(r)}n(r)
Summing the above expression we get the absurd result
E(1)+ E(2)< E(2)+ E(1)
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 Vext(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
EHK [n] = T[n] + Eint [n] + ò d 3rVext (r)n(r) + EII
º FHK [n] + ò d 3rVext (r)n(r) + EII
•
FHK[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 FHK[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.
•
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 manybody 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
1
HKS (r) = - Ñ2 +VKS (r)
2
through the solution of the corresponding Schroedinger equation for N
independent electrons (Kohn-Sham equations)
Kohn and Sham ansatz
Non-interacting
auxiliary
particles in an
effective potential
Interacting electrons
+ real potential
•
The density of this auxiliary system is then:
n(r) = å å | y is (r) |2
•
s i =1,N
The kinetic energy is the one for the independent particle system:
1
1
s
2
s
Ts = - å å y i (r) Ñ y i (r) = å å | Ñy is (r) |2
2 s i =1,N
2 s i =1,N
•
We define the classic electronic Coulomb energy (Hartree energy) as usual:
EHartree [n] =
1
n(r)n(r ')
3
3
d
rd
r
'
2 òò
|r -r '|
Kohn and Sham equations
•
Finally, we can rewrite the full H-K functional as
EKS [n] = Ts [n] + ò d 3rVext (r)n(r) + EHartree [n] + EII + E xc [n]
•
All many body effects of exchange and correlation are included in Exc
E xc [n] = FHK [n] - (Ts [n] + EHartree [n]) =
Tˆ - Ts [n] + Vˆint - EHartree [n]
•
•
•
So far the theory is still exact, provided we can find an “exact” expression for
the exchange and correlation term
If the universal functional Exc[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 Exc[n] describes the true exchangecorrelation energy, the Kohn-Sham method provides a feasible approach to
calculating the ground state properties of the many-body electron system.
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 HKS 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): Exc[n] is a sum of contributions from each
point in space depending only upon the density at each point independent on
other points
LDA
3
E xc [n] = ò d rn(r)e xc (n(r))
•
•
•
where e xc (n) is the exchange and correlation energy per electron.
e xc (n) 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
0.458
Exchange as e x (n) = where rs is defined as the average distance between
rs
4p
1
electrons at a given density n :
•
3
rs3 =
n
Correlations from exact Monte Carlo calculations (Ceperley, Alder, 1980)
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 Vxc
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 KohnSham 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
•
•
•
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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
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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)
LSDA
E xc
[n- ,n¯ ] = ò d 3rn(r)e xc (n- (r),n¯ (r))
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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
Local Density Approximation
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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
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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 Fxc is a dimensionless function and εxhom 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
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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
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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
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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 Exc
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
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Orbital dependent functionals - expressing Exc explicitly in terms of the independent
particle orbitals, naturally implies that Exc 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
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Hybrid functionals - combination of orbital-dependent Hartree-Fock and an explicit
density functional. Simplest form “half-and-half”:
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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.