Download Density functional theory

Density functional theory
Seminar Ib - 1. letnik, II. stopnja
Author: Matic Pečovnik
Mentor: Doc. Dr. Tomaž Rejec
Faculty of Mathematics and Physics, University in Ljubljana
Abstract
Many-body problems in physics have been the focal point of quantum mechanics in the previous
decades. Such problems are found in solid state physics, complex molecules and so on. Most of the
theoretical work must be done numerically. Here we have an issue with the computational power of
computers as the solving of the Schroedinger many-body equation requires computational power that
scales exponential with the number of particles in the system. In this seminar I present to the reader the
need to transition from wave-centered to density-centered approach when performing theoretical analysis
on complex many-body systems. Then I present the basic concepts of the density functional theory (DFT)
first introduced by Kohn and Sham that offers a solution to problems presented above. The second half of
my seminar will be focused on simulating the behaviour of a two dimensional electron gas with the use of
density functional theory.
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Contents
1
Introduction to density functional theory
1.1 Wave-centered approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Density functional theory - A workaround the Van Vleck catastrophe . . . . . . . .
2
2
3
2
The self-consistent Kohn-Sham equations
2.1 Uniqueness of the external potential - Hohenberg-Kohn Theorem
2.2 Obtaining the ground state . . . . . . . . . . . . . . . . . . . . . .
2.3 Local density approximation . . . . . . . . . . . . . . . . . . . . .
2.4 Improvements of the local density approximation . . . . . . . . .
2.5 Numerical packages for DFT calculations . . . . . . . . . . . . . .
3
4
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6
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3
An application of DFT - 2D electron gas
4
Conclusion
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Introduction to density functional theory
In this first chapter I will illuminate the problem we encounter when dealing with the many-body
wave function. Then I will introduce a possible solution in the form of density functional theory
(DFT) which plays a major role in numerically simulating the behaviour of many-body systems in
modern day physics.
1.1.
Wave-centered approach
Let D be the dimensionality of our system. In wave-centered computation the wave function
derived from the Schroedinger’s equation is the main body of focus.
Ψ = Ψ(r1 , r2 , . . . , rN )
(1)
It contains N coordinates ri = ( xi , yi , zi ; σi ) so as a whole it contains ( D + 1) N parameters with
which we describe the system. It has to satisfy the normalization restraint:
Z
V
dr1
Z
V
dr2 . . .
Z
V
|Ψ(r1 , r2 , . . . , rN )|2 drN = 1
(2)
It has to also satisfy the antisymmetricity of a fermion function(Pauli principle):
Pi,j Ψ = −Ψ
(3)
Here Pi,j is the parity operator that switches the i-th and j-th fermion.
In systems with only a couple of particles the wave-centered approach performs really well at
predicting the results of experiments and has therefore become the pillar of quantum mechanics
but it faces huge problems with many-body systems such as we find in nature. Molecules, crystals,
DNA molecules and other complex matter are all described with the Schroedinger equation but as
we can not numerically solve it on a realistic time scale it becomes useless. Let us demonstrate
this on an example.
2
In this example we will consider a molecule with N interacting electrons. We will disregard all
spin degrees of freedom and ignore symmetries. In numerical calculations we normally minimize
a functional of energy in accordance to the Rayleigh-Ritz minimal principle so we obtain the
ground state of a system.
e | H |Ψ
ei
E = minΨe hΨ
(4)
As we ignore the spin variable of the electrons the wave function consists of 3N continuous
variables which we have to represent discretely by M parameters p. In this way the energy is a
function of the parameters E = E( p1 , p2 , . . . , p M ). In practical calculations it turns out that we
require 3 ≤ n ≤ 10 parameters per variable for the desired accuracy od δE ∼ 10−2 . We therefore
need:
M = n3N 3 ≤ n ≤ 10
(5)
parameters of the wave function. If M = 109 is the maximum computable number of parameters
and n = 3 then this means the maximum number of computable electrons is:
N̄ =
1 M .
ln
=6
3
n
(6)
With the same thinking we can compute that for a system with a 100 electrons we would need
M̄ = 3300 = 10150 which is more than the predicted baryons in the whole universe [1]. This is
what we call the Van Vleck catastrophe.
1.2.
Density functional theory - A workaround the Van Vleck catastrophe
The first density functional theory was published by Thomas [2] and Fermi [3] in the 1920s. In the
centre of this approach was the density of electrons n(r) that is also easily measurable by various
techniques. It was not accurate enough as it neglects the quantum corrections to the Couloumb
potential which I will later introduce as exchange and correlation. It proposed an expected value of
the kinetic term in the H operator that depends solely on the density. This approach is acceptable
only in high electron density metals where the quantum corrections to the Couloumb potential
can be neglected.
Only in the 1960s a new complete formalism was proposed by Kohn and Sham [4] that sparked
a new era in physics.
It provides many results that are in accordance to results from experiments in various fields
such as crystals, biophysics, molecular physics and many more. These systems would be otherwise
inaccessible by numerical computations with wave functions as these systems contain N 1
electrons.
2.
The self-consistent Kohn-Sham equations
In the centre of our approach will be the density of electrons n(r) as the ground state n(r) uniquely
(to a trivial constant) defines the external potential v(r). This means that if we know the external
potential we in theory know n(r) and vice-versa. In the case of systems with degenerate ground
states any degenerate ground state ni (r) uniquely describes the same v(r) - This is widely known
as the Hohenberg-Kohn Theorem [5].
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2.1.
Uniqueness of the external potential - Hohenberg-Kohn Theorem
Let n(r) be the ground-state of a system described by the hamiltonian H1 . It corresponds to a
ground-state wave function Ψ1 with energy E1 , which we assume is non-degenerate.
E1 = hΨ1 | H1 |Ψ1 i =
Z
V
v1 (r)n(r) dV + hΨ1 | T + U |Ψ1 i
(7)
Here U is the interaction between electrons and T is the kinetic term. Now lets assume that there
exists a second potential v2 (r) 6= v1 (r) + v0 with its own ground-state wave function Ψ2 6= expiθ Ψ1
that gives the same density n(r). We can write the energy as:
E2 = hΨ2 | H2 |Ψ2 i =
Z
V
v2 (r)n(r) dV + hΨ2 | T + U |Ψ2 i
(8)
Since the system is non-degenerate the Rayleigh-Ritz minimal principle applies.
E1 < hΨ2 | H1 |Ψ2 i =
Z
V
v1 (r)n(r) dV + hΨ2 | T + U |Ψ2 i = E2 +
Z
(v1 (r) − v2 (r))n(r) dV
(9)
(v2 (r) − v1 (r))n(r) dV
(10)
V
With the same reasoning we can also conclude:
E2 < hΨ1 | H2 |Ψ1 i =
Z
V
v2 (r)n(r) dV + hΨ1 | T + U |Ψ1 i = E1 +
Z
V
If we add these equations we get a contradiction.
E1 + E2 < E1 + E2
(11)
Therefore it must apply that v1 (r) = v2 (r). [1]
2.2.
Obtaining the ground state
In this subsection we will be considering an inhomogeneous gas of electrons without spin. We can
write the energy functional E[n(r)] for this system as:
E[n(r)] =
Z
V
v(r)n(r) +
1
2
Z Z
V
n(r)n(r‘)
dr dr‘ + F [n(r)]
|r − r‘|
(12)
The equation is written in standard atomic units, meaning the all distances are measured in Bohrs
radius (a0 ) and energies in Hartrees (Ha).
Ha = 2Ry =
m e e4
= 27.2eV
(2e0 h)2
a0 =
e0 h 2
= 53pm
πme e2
(13)
The first part of the functional presents the interactions of electrons and the external potential and
the second part is the classical Coulomb interaction between the electrons also called the Hartree
interaction. F [n(r)] = Ts [n(r)] + Exc [n(r)] is a functional uniquely described by the electron
density and is independent of the system that is defined by the external potential v(r). Ts [n(r)] is
the functional of kinetic energy for a system of non-interacting electrons with the same density
and Exc [n(r)] = Ex [n(r)] + Ec [n(r)] is by our definition the functional that describes exchange
Ex [n(r)] and correlations Ec [n(r)] between electrons and contains everything about not already
explicitly written out.
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We are already familiar with the exchange part from our classes in Condensed matter on the
matter of the Hartree-Fock approximation and we know the exchange comes from using only one
Slater determinant to describe our wave-function. The correlation part comes from using more
Slater determinants to describe the interaction between electrons.
δExc [n(r)]
Let f xc be the functional derivative of Exc . So f xc (r) = δn
. The functional Exc [n(r)] is not
(r)
analytically known and neither is f xc (r). Because we have written the functional in a way where
we separate the non-interacting kinetic term Ts [n(r)] from the interacting kinetic term which
is hidden in Exc [n(r)] we know that the minimum of the functional will be the solution of the
corresponding Schroedinger equation which is in this context knows as the Kohn-Sham equation.
Z
1
n(r‘)
− ∇2 + v ( r ) +
dr‘ + f xc (r) − ei ϕi (r) = 0
(14)
2
V | r − r‘|
ei is a Lagrange multiplication factor ensuring the conservation of particles. Because the effective
potential, in which the interacting kinetic energy is hidden, is dependent on n(r) we must solve
this equation self-consistently meaning that:
N
n (r ) =
Z
∑ | ϕi (r)|2
V
i =1
n(r) dr = N
(15)
Here the sum goes over all electrons in the system and ϕi are filled in the ascending order of their
respective energies ei . As of yet there is no clear indication of the meaning of the Kohn-Sham
orbitals ϕi and their energies ei .
2.3.
Local density approximation
To use this formalism effectively we must come by some approximation for the functional Exc [n(r)].
The first approximation is the so-called local density approximation or LDA made by Kohn and
Sham in their original paper [4]. It assumes that the density n(r) is slowly varying on the scale of
the Fermi wavelength that we derive from k F which corresponds to the radius of the Fermi sphere.
It also works very well if the density is very high because the Fermi vector is small in this regime
and the variations of density on this scale are therefore small. We can express the LDA in the
following way:
Z
LDA
Exc
[n(r)] =
V
n(r)(ex [n(r)] + ec [n(r)]) dr
(16)
exc [n(r)] = ex [n(r)] + ec [n(r)] is the exchange-correlation energy per particle of a uniform electron
gas of density n. This functional is well known. The exchange (ex ) can be analytically calculated
[6] and we can estimate the correlation (ec ) with an analytical fit obtained with interpolation. A
rough estimate is presented here:
ex = −
0.458
+...
rs
ec = −
0.43
+...
rs + 7.8
(17)
1
3
) 3 and is a
Here rs is defined as the radius of a sphere that contains one electrons rs = ( 4πn
dimensionless quantity. In recent decades a better approximation for the correlation per electron
was obtained by using an analytical fit for the data gathered by Monte-Carlo simulations that has
the required behaviour when rs 1 and rs 1 which are analytically known.
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2.4.
Improvements of the local density approximation
As we have seen in the previous section we obtain adequate results with the LDA approximation
if the system has a very high density or if the local fluctuations of density are very small. But in
realistic example this is rarely the case. So we need a new better approximation to describe these
systems.
At the centre of our approach in LDA was that we can write the energy functional simply as a
functional of density as we can see in equation 16 but we must realize this is only the first order of
expansion of the general functional Exc that describes effects of correlation and exchange between
electrons. We can get a better approximation with the next orders in the expansion series that also
include the change of density with ∇n(r). We can write the expansion to the second order as:
GGA
Exc
[n(r)]
=
Z
V
f (n(r), ∇n(r)) dr
(18)
This second order is called the Generalized Gradient Approximation (GGA for short). For it to be
of any use we must find a analytical fit to the f (n(r), ∇n(r)) function. The best choice is still a
matter of debate.
We can see the improvement that we obtain with the GGA approximation in Fig. 1.
Figure 1: Errors of bond lengths computation for a GGA functional and LDA functional. We can see that
GGA is better or comparable than LDA in these calculations. Taken from [7].
Also we can generalize the DFT by including a finite temperature. You can see the whole
approach in [4], while here I will only cite the results. The Kohn-Sham equation is the same only
how we self-consistently produce the density changes:
N
n (r ) =
∑ | ϕi (r)|2
i =1
1
exp
ei − µ
kT
(19)
+1
k is the Boltzmann constant, T is the temperature and µ is the chemical potential derived from the
number of particles by integrating the equation 19. We can see that the only change is the Fermi
function.
These approaches can also be further generalized to study excitation energies, photo-absorption
spectra and frequency responses using time-dependent external potentials like oscillating electric
fields and magnetic fields in a theory we call time-dependent density functional theory - TD-DFT
for short.
From experience it is known that the accuracy of modelling the energy of bonds in molecules
is about 10-20% and the sizes of the bonds is about 1%[1] as we can see in Fig. 1. We get the
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bond lengths by introducing these lengths as parameters into the energy functional E[n(r)] and
minimizing the corresponding energy function of parameters by using the Rayleigh-Ritz principle.
We can not make a general statement for the quality of the DFT approach for all systems. For
instance in semiconductor systems the DFT gives quantitatively wrong results for the energy gap
while accurately predicting the energy band structure. In some systems like two dimensional
electron gas in specific cases the LDA is good enough to get quantitative results. Also systems
with very strong electron-electron interactions are better described by dynamical mean-field theory
[8].
2.5.
Numerical packages for DFT calculations
There are many numerical packages that one can acquire for DFT calculations. To know which
package suits one best we should note the important differences between these packages.
• Electrons We can divide the programs by how they treat electrons far from the Fermi surface.
Some consider all electrons as equally important for understanding the behaviour of the
system. Others use pseudo-potentials. When using pseudo-potentials we consider only the
electrons near the Fermi surface accurately while the electrons far bellow the Fermi level are
considered only as a background that effects the electrons near the Fermi surface.
• Basis set The numerical calculations are done in some basis set of functions. There are two
widely used variants. The Gaussian basis set (SIESTA) is local and the functions differ if we
are in solids or molecules. The plane-wave basis set (Quantum-Espresso, VASP, ABINIT) is
non-local and the functions are universal in all systems.
3.
An application of DFT - 2D electron gas
We can fabricate a 2D electron gas at the junction of two semiconductors. One such example is at
the heterojunction of AlGaAs and GaAs. The semiconductors have a different energy gap between
the valence and conducting bands and also their chemical potentials are not equal. When we put
the semiconductors in contact a current starts to flow, because of different chemical potentials.
It flows until both chemical potentials are equal. The result is that the energy bands bend to
accommodate this requirement. In this process of bending a potential well about 10 nm wide forms
at the junction and some electrons get trapped as their movement in the direction perpendicular
to the junction gets quantized. The chemical potential is usually lower then the energy of the first
excited state so electrons get trapped in the ground state of the well. Therefore their movement is
reduced to two dimensions in which they are almost completely free. We can see the junction in
Fig. 2.
This system of electrons is of particular interest to scientist because of its high mobility in the
two dimensions in which the electrons are free as it is comparable or even higher than in most
metals.
We can study this system using a quantum point contact (QPC) which is a narrow constriction
made by the potential of two metallic gate electrodes that separate two electron reservoirs of 2D
electron gas as see in Fig. 3a. The gate electrodes lie about 100 nm above the electron gas and
are separated from it by an insulator so the entire effect on the electron gas is the potential the
electrodes form. With such a device we can measure the conductance G in the QPC, by measuring
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Figure 2: A junction of two semiconductors. On the top we can see the energy band structure before we
create the junction. The red lines represent their respective chemical potentials. In the lower part of the
figure we can see the energy band structure after we create the junction with the chemical potential in
equilibrium. We can see the trapped electrons depicted in blue.
the current through such a device, that is proportional to the conductance, while changing the
voltage: I = GV
(a)
(b)
Figure 3: (a) An image of a QPC through a microscope. (b) An ELS (highlighted with the white circle) that
formed in a QPC calculated by DFT using LSDA approximation. Taken from [9].
In the constriction a saddle in the potential forms. When the Fermi energy of the electrons
reaches the value of the potential saddle the density of electrons at the saddle is low and high in
the reservoirs.
Now lets look at a 2D jellium system, where the density of electrons is constant, in the 2D
Hartree-Fock approximation. Also let the electrons have spin. We can define the polarization of
n −n
such a system as ζ = n↑ +n↓ where n↑,↓ are the densities of electrons with spin up or down. We
↑
↓
can write the energy of such a system with the parameters rs and ζ [10] as:
E (r s , ζ ) =
√
3
3
1 + ζ2
4 2 (1 − ζ ) 2 + (1 + ζ ) 2
−
3π
rs
rs2
8
(20)
Lets look at both limits for polarization ζ = 0 and ζ = ±1. Their respective energies are:
√
√
1
8 21
2
16 2 1
E (r s , ζ = 0) = 2 −
E (r s , ζ = ±1) = 2 −
3π rs
3π rs
rs
rs
(21)
We can see that for low densities (rs 1) the system will prefer to be polarized because the energy
of the polarized state is lower.
Now lets return to the QPC. Even though this system is not consistent with the jellium model
as the density is not constant we expect that we should get qualitatively similar behaviour. We
said that when the Fermi energy approaches the potential of the saddle the density there will be
low so we expect that we will get a spin-polarized local state at the saddle.
We can study systems where spin is relevant with DFT using a generalization of LDA named
local spin density approximation or LSDA where we can write the exchange and correlation
energies as:
Z
LDA
Exc
[n↑ (r), n↓ (r)] =
V
n(r)exc (n↑ (r), n↓ (r)) dr
(22)
This DFT approximation also adds correlations previously mentioned in Sec. 2.3 to the HartreeFock approximation we studied earlier. An article [9] has reported that by using LSDA the
authors have found a single electron that gets caught in the constriction and so forms a localized
spin-polarized state often named emergent localized state - ELS. We can see such an ELS in Fig.
3b.
Now we can generalize our findings from this relatively simple system to a more complex
system. To get high values of conductivity in this system one of the mentioned semiconductors is
doped with a n-type material like Si. These dopants contribute even more electrons into the 2D
electron gas but it comes at a cost. The dopants form a random potential seen in Fig. 4. Such a
disordered potential also contains saddles on which ELS could form. We can see these ELS in Fig.
4.
Figure 4: Upper left: A random disordered potential. Other pictures: Polarization calculated with the
Octopus code using the LSDA approximation as a function of the space coordinates for various electron
densities represented by the number of electrons N on top of every picture. For different densities different
ELS are prevalent. This is because an ELS forms when the Fermi energy is the same as the energy of the
saddle in the potential. Taken from [11].
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Such systems are still under study as we want to understand how the disordered potential
influences the conductivity of the 2D electron gas.
4.
Conclusion
The transition from the wave-centered approach to the density approach is paramount to being able
to quantify complex systems such as crystals and molecules as it allows us to predict behaviours of
these complex systems. With this new-found understanding we can put these systems to use from
nano physics such as the conductance of a 2D electron gas on the border of two semiconductors
to various other examples not mentioned in this seminar. It is also very useful in chemistry and
biology.
Many improvements can still be made to the DFT formalism as the exact functional that
describes the kinetic energy of interacting particles and the exchange and correlation of these
particles is not known yet. It may never be known exactly but more elaborate methods of
approximating this functional are to be expected in the future as more precision is demanded of
the DFT formalism.
References
[1] W. Kohn, “Nobel lecture: Electronic structure of matter—wave functions and density functionals,” Reviews of Modern Physics, vol. 71, pp. 1253–1266, October 1999.
[2] L. H. Thomas, “The calculation of atomic fields,” Proceedings of the Cambridge Philosophical
Society, vol. 23, no. 5, p. 542–548, 1927.
[3] E. Fermi, “Un metodo statistico per la determinazione di alcune prioprietà dell’atomo,” Rend.
Accad. Naz. Lincei., vol. 6, pp. 602–607, 1927.
[4] W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation
effects,” Physical Review Letters, vol. 140, pp. 1133–1138, November 1965.
[5] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Physical Review Letters, vol. 136,
November 1964.
[6] G. D. Mahan, Many particle physics. Springer, 1981.
[7] M. Fuchs, “Comparison of exchange-correlation functionals: from lda to gga and beyond,” in
A Hands-On Computer Course, 2005.
[8] D. Vollhardt, K. Byczuk, and M. Kollar, Dynamical Mean-Field Theory. Springer, 2011.
[9] T. Rejec and Y. Meir, “Magnetic impurity formation in quantum point contacts,” Nature,
vol. 442, pp. 900–903, August 2006.
[10] B. Tanatar and D. M. Ceperley, “Ground state of the two-dimensional electron gas,” Physical
Review B, 1989.
[11] R. Levy, Spontaneous Formation of Magnetic Moments and Dephasing in Two-Dimensional Disordered Systems. PhD thesis, Ben-Gurion University of the Negev, 2013.
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