DFT Notes Cocula Carter

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Introduction to Solid State Physics, Band Structure Theory and

Periodic Density-Functional Theory

Vincent Cocula & Emily A. Carter

Contents

I. Free Electron Gas 3

A. Energy levels in one dimension 3

B. Fermi energy 4

C. Free electron gas in three dimensions 4

D. Density of states 6

II. Band Structure I : A “Chemical” Approach 7

A. Linear chain of hydrogen atoms 7

B. Translational invariance - Bloch’s theorem 10

III. Band Structure II : A “Physical” Approach 10

A. Reciprocal lattice - Brillouin zones 10

B. Bloch’s theorem - Generalization 14

IV. Planewave density-functional theory 15

A. Periodic Calculations 15

B. The Hohenberg-Kohn theorems 17

C. The Kohn-Sham equations 19

D. The exchange-correlation potential 20

E. Electron-ion Interaction 22

F. Self-Consistent Field 23

G. Planewave basis convergence and k-point sampling 25

H. Successive improvement of the trial wavefunction 27

I. Electronic Temperature and Fermi surface smearing 28



2

Reference material:

• C. Kittel, Introduction to Solid State Physics (Ed. Wiley & Sons, 1986)

• Ashcroft & Mermin, Solid State Physics (Saunders College Publishing, 1976)

• A. Sutton, Electronic Structure of Materials (Oxford Science Publications, 1996)

• P. D. Haynes, Linear-scaling methods in ab initio quantum-mechanical calculations

(PhD Thesis, Cambridge, 1998)

• Britney’s Guide to Semiconductor Physics (http://britneyspears.ac/lasers.htm)



3

I. FREE ELECTRON GAS

A. Energy levels in one dimension

Let’s consider a free electron (i.e. no forces) confined in an infinite square-well potential.

In one dimension, the potential can be defined as:

V (x) = 0 0 < x < L

V (x) = ∞ elsewhere

The wavefunction ψn(x) of the electron is a solution to the one-dimensional Schrodinger

equation

Hψn(x) = − h2

2md2

dx2ψn(x) = nψn(x)

where n is the energy of the electron occupying the orbital ψn.

The wavefunction should vanish at the boundaries, so that ψn(0) = ψn(L) = 0, and the

general solution of the eigen-equation should be an imaginary exponential-like function:

ψn(x) = A exp(ikx)

One can notice that the solution is simply a planewave with wave-vector k, consistent

with the fact that our electron is treated as a free particle. Also one can take a particular

solution of the Schrodinger equation to be a sine function

ψn(x) = A sin(kx)

where A is a constant.

Solving for k, the electronic wavefunction becomes

ψn(x) = A sin

nπ

L x

where n is an integer number of half-wavelengths between 0 and L.

The orbital energies are easily evaluated and take on the values

n = h2k2

2m =

h2

2m

nπ

L

2



4

FIG. 1: Energy levels in one-dimension (taken from Kittel)

B. Fermi energy

Let’s concider now that we have N electrons confined in our one-dimensional potential.

The Fermi energy F is defined as the energy of the topmost filled level in the ground state

of the N electron system. Since electrons have to obey the Pauli Principle, each orbital can

only be filled by 2 electrons at most. The Fermi level corresponds then to the level with

nF = N/2 and the Fermi energy is

F =

h2

2mN π

2L2

C. Free electron gas in three dimensions

The generalization of the problem from one to three dimensions is rather straightforward.

We now consider a free-electron gas of N particles confined in a cubic box of edge L. The

Schrodinger equation now becomes

− h2

2m∇2ψk(r) = k ψk(r)

for which the wavefunctions are required to be periodic in the three directions x, y and

z , so that

ψk(x + L, y,z ) = ψk(x,y,z )

and similarly for the y and z directions. The wavefunctions satisfying those conditions

are again the free particle planewaves, thus



5

FIG. 2: 3D Fermi sphere within the k-space (taken from Britney Spears)

ψk(r) = exp(ik · r)

with wave-vector

k2 = k2x + k2

y + k2z

for which the components of the wave-vector satisfy

kx = 0; ±2π

L ; ±

4π

L ; ...

Similarly, the energy k of the orbital with wave-vector k is then



6

FIG. 3: Density of states (taken from Britney Spears)

k = h2

2mk2 =

h2

2m(k2

x + k2y + k2

z)

This has an important consequence, and gives rise to the Fermi sphere. In the ground

state of the N -electron system, the orbitals may be represented as points within the 3-

dimensional k-space. The occupied orbitals are included in a sphere, and the energy at the

surface of this sphere is the Fermi energy. The sphere is called the Fermi sphere, and the

Fermi energy is such that

F = h2

2mk2F

D. Density of states

The total number of electrons contained in the Fermi sphere within the volume element

2πL

3is given by:



7

2 4πk3

F /3

(2π/L)3 =

V

3π2k3F = N

which gives us an expression for the three-dimensional Fermi vector

kF =

3π2N

V

1/3

,

and thus to the Fermi energy

F = h2

2m

3π2N

V

2/3

.

We define the density of states D() as the number of electrons per unit energy range,

and is given by

D() = dN

d =

V

2π2

2m

h2

3/21/2.

II. BAND STRUCTURE I : A “CHEMICAL” APPROACH

We now go back to a one-dimensional system. In the previous section, we have just

been solving the Schrodinger equation for an electron in a finite system of arbitrary length

L, supposed small. This cannot be a good approximation of an extended solid, but it is

reasonable to suppose that a solid could be modelled by an infinite collection of such systems

in 3-dimensional space.

In this section, we will take the fictitious linear chain of hydrogen atoms as an example

to model our solid system, and we will show that this gives rise to a band structure. We

will also introduce very briefly the ideas underlying Bloch’s Theorem.

A. Linear chain of hydrogen atoms

A solid is nothing but a collection of atoms arranged within some symmetrical order. In

one dimension, one can model such a system by, i.e., a linear chain of N hydrogen atoms.

Each hydrogen atom is associated with an s state, and our aim will be to find the expression

for the molecular orbitals ψ for the chain. In the LCAO picture, the total wavefunctions

will be



8

FIG. 4: Energy spectrum for the linear chain of hydrogen atoms (taken from Sutton)

|ψ =

N

j=1 c j | j

where | j stands for the s wavefunction of the j th atom in the chain. The task is then to

find the molecular coefficients c j and the energy of the molecular orbital. The Schrodinger

equation to be solved is

H |ψ = E |ψ

which, in terms of the atomic orbitals, becomes

N j=1

c j H | j = E N

j=1

c j | j

We have to solve for the Hamiltonian matrix elements using the secular equation

N j=1

c j p| H | j = E N

j=1

c j p | j

where p is again one of the wavefunction on one of the sites. We now make the (violent!)

assumptions in order to simplify the problem, where the overlap matrix will be taken such

that p | j = δ pj, and the Hamiltonian matrix elements will be all taken equal to zero, except

for the on-site elements p| H | p = α and for the case of neighboring sites p| H | p ± 1 = β .

This is often referred to as the ”orthogonal type binding” ir ”orthogonal H uckel theory”.

One then obtains a system of N equations:



9

FIG. 5: Broadening of the energy levels into bands (taken from Kittel)

αc1 + βc2 = Ec1

βc1 + αc2 + βc3 = Ec2

........................βcN −2 + αcN −1 + βcN = EcN −1

βcN −1 + αcN = EcN

We can solve the system of coupled linear equations (for the complete derivation, see

Sutton), and show that the allowed energies of this electronic system can be written as

E m = α + 2β cos

mπ

N + 1

where m = 1, 2, 3,...,N . It is very instructive to plot those allowed energies as the length

of the chain (e.g. N ) increases. We can see that even if it gives rise to discrete energy levels

in the case when N stays reasonably small, the levels then pile up to form a continuum when

N becomes large. For an extended solid, we achieve the limit where N is infinite and the

spectrum of energy then becomes a continuum of indistinguishable (although quantifiable)

levels. We call this an energy band.



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B. Translational invariance - Bloch’s theorem

The result (and its consequences) shown above was derived using an infinite number of

hydrogen atoms along a one-dimensional chain. It is interesting to note that similar results

could be derived for rings of such atoms, and other systems, provided that the number of

sites N becomes large. But most interestingly, we assumed all the atoms to be identical to

one another, so that nothing physically distinguishes a particular atomic site from another.

We can reasonably assume then that in both the cases of a linear chain, or of a ring, the

observable quantities must be invariant as we go along the chain/ring. The density for

example should be identical from one site to another, and more generally, if the atomic

spacing is denoted as a, one should always have the translational symmetry relation:

|ψ(x + a)|2 = |ψ(x)|2

and for those two wavefunctions to satisfy this condition, one can see that they should

only differ by a phase factor, meaning that

ψ(x + a) = eiφa ψ(x)

This is Bloch’s Theorem, and we will see that in fact this phase factor is the product of

the quantum number k times the lattice spacing a.

III. BAND STRUCTURE II : A “PHYSICAL” APPROACH

A. Reciprocal lattice - Brillouin zones

With the results of the previous section in mind, let’s go back to our free electron-system,

and built a one-dimensional crystal made of an infinite number of atoms, each separated

by a lattice constant a. Since we should also have translational invariance in this case, the

statement made before for the wavefunction still holds, and we should have

ψ(x + a) = ψ(x)

Remembering that the solution to the Schrodinger equation for this system is planewave-

like, we will assume the wavefunction to be a travelling wave so that



11

FIG. 6: E vs. k for the free electron gas and for the nearly free electron-gas (taken from Kittel)

ψ(x) = exp(igxx)

and thus the condition to be satisfied by the wavefunction now becomes

eigxa = 1

and the phase factor should then take the values

gx = 0; 2π

a ; ...

The generalization in three-dimensions is straightforward, and gives us the values of the

reciprocal lattice vectors G:

G

gx = 2πa

nx

gy = 2πa

ny

gz = 2πa nz

We saw that in our one-dimensional system, the solution to the Schrodinger equation was

a wavefunction depending on the quantum number kx so that

ψk(x) = eikxx

for which the phase factor can now be expressed in terms of the reciprocal lattice param-

eter



12

FIG. 7: Construction of the first Brillouin zone (taken from Kittel)

kx = ±

nπ

a = ±

1

2 gx

The zone in k-space between −π/a and +π/a is the first Brillouin zone of the lattice.

For the nearly-free electron gas in a periodic potential of atoms, the energy (k) does not

resemble the parabola given in the free-electron case. Instead, an energy gap arises at the

limits of the Brillouin zones to avoid crossing, as shown in Fig. 2. This is due to the fact that

for those specific values of the phase factor, one gets destructive interference between waves

traveling to the “right” and to the “left”. In between, the energies are allowed to form a



13

FIG. 8: Band structure for the nearly free electron gas in one-dimension (taken from Ashcroft)

continuum, a band. The perturbation due to the weak potential from the neighboring sites,

as well as the construction of the one-dimensional band structure is nicely shown in Fig.

9.4.



14

B. Bloch’s theorem - Generalization

It is very interesting to realize that one can have a pretty good picture of the electronic

structure of an infinite solid, by just considering the solid as an infinite chain of isolated

particles - or at least slightly perturbed. The only important consequence of it is the

broadening of the energy spectrum to form energy bands. But the real reason why all this

could be achieved is because we have realized something important here: the translational

invariance of the system as we go along the chain and/or the solid. This is the idea behind

all the solid state physics theory, this is the foundation of Bloch’s theorem, this is the reason

why it becomes so easy to work in Fourier-space instead of the more familiar real-space.

Bloch’s theorem is thus based on this idea of translational invariance among the different

atomic sites and/or primitive cells of the crystal, allowing us to consider only one of those

repeating units, thus tremendously simplifying the problem to be tackled.

Let’s now consider a real three-dimensional crystal. For simplicity (but this is almost

always the case when one solves in practice those equations), we will consider the nuclei

to be static, and the electrons to be non-interacting particles moving in a static potential

V (r). The perfect crystal constitutes a periodic array of lattice vectors R. The potential

is periodic and invariant under translation, so that we always have

V (r + R) = V (r)

and hence the Hamiltonian operator should be periodic, i.e.

H (r + R) = H (r).

The game would then be to find a translation operator T R for each lattice vector R which

could act on any function f (r) so that

T Rf (r) = f (r + R).

One can show (see Haynes) that such an operator acting onto the wavefunction would

yield the following identity relation

T R ψ(r) = ψ(r + R) = exp(ik · R) ψ(r).

If we now consider the function u(r) = exp(−ik · r)ψ(r), we can see that



15

u(r + R) = exp(−ik · [r + R]) ψ(r + R)

= exp(−ik · r) exp(−ik · R) exp(ik · R) ψ(r)

= exp(−ik · r) ψ(r)

thus the function u(r) also has the periodicity of the lattice and is cell-periodic since we

saw that u(r + R) = u(r), and it makes it possible to express the eigenstates of the full

Hamiltonian as

ψnk(r) = unk(r) eik·r

Since now we have taken care of the translational invariance, Bloch’s theorem allows us

to simplify the problem: instead of having to solve for the entire infinite space, we are only

left with solving for the wavefunction within a single cell (in fact the first Brillouin zone),

but at an infinite number of k-points. Outside of this cell, the values of the wavefunctions

are identical by translational symmetry so that:

ψn,k+G(r) = ψn,k(r)

n,k+G = n,k

IV. PLANEWAVE DENSITY-FUNCTIONAL THEORY

A. Periodic Calculations

The simulation of a crystal may appear to be a formidable task, as one usually under-

stands a crystal as an infinitely extended system. Of course, it is well known that except for

a few cases, crystals also exhibit a high degree of symmetry. Using this property, Bravais

defined classes of crystals, all differing by the symmetry of the repeating unit composing thenetwork of atoms. Those repeating units, called unit cells, are repeated uniformly in the

3-dimensional space to form the infinitely extended solid. It becomes then natural that the

theoretical simulation of the extended crystal lattice only consists of a three-dimensional

periodic boundary condition problem. Fig. 9 schematically represents the simulated unit

cell of a body-centered cubic (bcc) lattice, isolated, and then rendered as repeated in the

three dimensions in space.



16

FIG. 9: Schematic representation of a simulation unit cell, and its 3-dimensional periodic images.

The task of simulating an infinitely extended system like a crystal, often reduces to the

simulation of a simple unit cell containing only a few elements. This of course dramatically

simplifies the problem to be solved, and is a rigorous and accurate representation of the

solid, as it implicitly models the infinitely extended system by means of three-dimensional

periodic boundary conditions. It should be noted that one other way to simulate an extended

lattice would be to model a cluster large enough so that the atoms at its center behave as in

the real bulk solid. Nevertheless, this is not always very accurate, and introduces inherent

errors such as edge effects and spatial anisotropy of the system modelled, both of which

are undesirable features when one wants to simulate crystals. Nonetheless, this does not

mean that the method cannot as well be used to simulate systems that do not necessarily

exhibit translational invariance with respect to the three spacial coordinates. As an extreme

example, one can use the method to even simulate isolated atoms, and the principle is

schematically shown in Fig. 10. The supercell simply contains the isolated atom, surrounded

by vacuum, and is repeated in the three-dimensional space by means of periodic boundary

conditions. The cell is then chosen big enough so that the centered atom is not perturbed

by its periodic images; in other words, the amount of vacuum surrounding the atom should

be chosen large enough so that the atomic charge densities between periodic atomic images

do not overlap in space.

Following this general idea, it the becomes possible to use the method to generate all new

classes of systems, such as point defects, voids and dislocations in the crystalline lattice; one-

dimensional systems or quasi one-dimensional systems such as atomic wires and nanotubes

or two-dimensional systems such as surfaces and interfaces. For example, a surface is usually



17

FIG. 10: Schematic representation of the simulation supercell (projected onto two dimensions) of

an isolated atom. The atom is placed at the center of the cell, surrounded by vacuum. The amount

of vacuum present should be large enough so that periodic images of the atom do not interact with

each other.

modelled by a slab. Again, the surfaces should not interact with their periodic images, so

that the amount of vacuum should be chosen large enough so that the charge density at

the surface normal direction of the slab dies off well before the boundary of the supercell.

Also, in order to accurately represent surface effects, the slab should be chosen thick enough

so that the surface atomic layers behave as if they were subject to a bulk-like environmentwithin the center of the slab.

B. The Hohenberg-Kohn theorems

Density Functional Theory (DFT) was proposed by Hohenberg and Kohn (1964) and is

nowadays one of the major methods used for electronic structure calculations by physicists

as well as chemists. Hohenberg and Kohn proved formally that the ground state total

energy of an electron gas of density n in the presence of an external potential vext can be

calculated exactly if its density is known. The total ground state energy E of the electron gas,

including exchange and correlation, is given as a unique functional (function of a function)

of the density:

E ≡ E [n(r)] . (1)



18

The Hamiltonian for the system of N electrons subject to an external potential is

H = T + V e-e +N i=1

vext(ri) (2)

where T is the kinetic energy operator, and V e-e is the electron-electron interaction opera-

tor. In our case, the external potential will simply be taken as the sum of all the ion-electron

interactions in our solid, where ”ion” is typically taken to mean a species comprised of either

a bare nucleus or a nucleus and associated core electrons.

We can define a universal functional F of the density, such that

F [n] = min|ψ|2→n

ψ|T + V e-e|ψ (3)

is independent of choice of system and external potential. The Hohenberg-Kohn theorems

of the density functional theory tell us that

E [n] = F [n] +

vext(r)n(r)dr ≥ E GS (4)

where E GS is the ground-state total energy of the electronic system; and that for its

corresponding ground-state density nGS(r), and only for it, we have the exact equality

F [nGS] +

vext(r)nGS(r)dr = E GS. (5)

The density functional theory is an exact theory in principle, and provides us with a simple

framework for determining the ground state properties of our electron gas (distribution).

Given an external potential, and a known functional F [n] of the electron density, one needs

only to variationally minimize the value of E [n] in Eq. (4) to obtain all necessary information

about the ground state of the system. The formidable advantage of the DFT is that one only

deals with the density n(r), a function of only three space variables, instead of the usual 3 N

variables associated with the many-electron wavefunction. This variable reduction provides

a substantial simplification of the problem compared to quantum chemistry methods. At the

same time, the DFT only provides us with an exact way to compute ground-state properties

and is not suitable for the study of excited states - unlike quantum chemistry methods

(though response methods, such as time-dependent DFT, can yield excited states energies).

Before the density functional theory becomes applicable for electronic structure calculations,

one needs to determine the expression for the functional F [n]. This functional must include



19

the kinetic energy of the electrons T s[n], as well as their Hartree classical Coulomb repulsion

energy E H[n], their exchange and correlation energies E xc[n] (we will discuss in further detail

those two properties in a subsequent section). One can write the following expression for

the functional F [n]:

F [n] = T s[n] + E H[n] + E xc[n]. (6)

Although the evaluation of the Hartree energy is quite straightforward, its corresponding

potential being computed by

V H(r) = E H[n(r)]

δn(r) =

e2

2

n(r)

|r − r|dr, (7)

the expressions for the kinetic energy and exchange-correlation functionals are unknown.

Much of the basic research effort nowadays focuses on the development of those functionals,

with some success for E xc[n], but the challenge remains for the kinetic energy one, making

the applicability of the density-functional theory in the Hohenberg-Kohn form most of the

time impractical.

C. The Kohn-Sham equations

In 1965, Kohn and Sham showed that it is possible to reduce the many-body quan-

tum mechanical problem to an exactly equivalent set of one-electron equations, solved self-

consistently. Inspired by the idea of the Hohenberg-Kohn theorems, the Kohn-Sham to-

tal energy functional of an electron gas subject to an external potential (including kinetic,

Hartree, exchange-correlation energy) for a set of doubly occupied orbitals ψi (a single Slater

determinant) is rewritten as

E [ψi] = 2[ h2

2m]i

ψi∇

2ψidr + E H[n] + E xc[n] +

vextn(r)dr (8)

where the kinetic energy functional has been replaced by its exact analytical expression

acting on the set of orbitals ψi. The electron density n(r) is then given as

n(r) = 2i

|ψi(r)|2. (9)



20

In the case of a crystalline solid, the external potential is simply equal to the ionic

interaction between the nuclei and the electrons,

vext

≡ V ion

. (10)

As for the DFT, only the minimum value of the Kohn-Sham energy functional has a

physical meaning, and in order to determine the ground state total energy of the electronic

system, one then needs to determine the set of orbitals ψi(r) that minimizes it. This is done

by self-consistently solving the Kohn-Sham equations:

−

h2

2me∇2 + V ion(r) + V H (r) + V xc(r)

ψi(r) = εi ψi(r), (11)

where εi is the Kohn-Sham eigenvalue associated with electronic state i, V H is the Hartree

potential given by

V H (r) = e2

ρ(r)

|r − r| d3r, (12)

V ion is the ionic potential describing the attractive interaction between electrons and nuclei,

and V xc is the exchange-correlation potential given by the functional derivative

V xc(r) = δ E xc [n(r)]

δ n(r)

. (13)

The Kohn-Sham equations are in fact a reformulation of the Hohenberg-Kohn problem,

where one maps the many-electron system onto a system of noninteracting particles (the

single determinant ansatz) under an effective potential due to the other electrons and their

interaction with the nuclei.

D. The exchange-correlation potential

Although DFT is formally an exact theory, unfortunately one doesn’t know the exact

expression for the kinetic energy functional nor for the exchange-correlation functional in

terms of the density. A great deal of work has been and is still being done in order to

derive reliable expressions for those functionals. Particularly, one distinguishes two different

formulations for the exchange-correlation functional: the Local Density Approximation, or

LDA, and the Generalized Gradient Approximation, or GGA.



21

The anti-symmetry of the wavefunction requires that electrons with equivalent spins

occupy different quantum states. This is more commonly known as the Pauli exclusion

principle. This results in a spatial separation of those two particles, and this ’exchange’

energy contributes to lowering the total energy of the electron gas. Furthermore, electrons

are equivalently charged particles and therefore repel each other. One of the major con-

sequences of this is that the movement in space of a specific particle cannot be entirely

decoupled from the collective movement of the others surrounding it. The movement of the

electrons is ”correlated”. This, for example, naturally forbids two electrons to be present at

the exact same point in space at the exact same time; which is not forbidden in the Hartree-

Fock theory since it does not include correlation effects. In density-functional theory, both

exchange and correlation effects are included, all within the so-called exchange-correlation

functional F xc[n].

In the LDA formulation, the exchange-correlation energy per electron at a point r in space

is assumed to be the exchange-correlation energy per electron in a homogeneous electron

gas (εhomxc ) which has the same density as the electron gas considered at the same point in

space. Its analytic expression is given by

E xc [n(r)] = εhomxc (r) n(r) d3r. (14)

where

E LDAxc [n(r)] =

n(r) eLDA

xc [n] dτ (15)

The exact exchange-correlation energy for the homogeneous electron gas was computed

by Cperley and Alder using the Quantum Monte Carlo method, which was then parameter-

ized in a functional form by Perdew and Zunger. The local density approximation, originally

proposed by Kohn and Sham because of its formal simplicity, is clearly wrong, due to thesimple fact that the electron density around an atom cannot be assumed to be homoge-

neous. Nevertheless, it is amazing how much success the method has had in describing

condensed phase systems. The method not only has proven to be simple in formalism, but

also useful and very powerful in describing many properties of many systems. Of course,

the approximation shows serious breakdown when a system exhibits substantial electronic

density spatial fluctuations. It is therefore necessary to take into account the gradient of



22

the density as a new parameter for the construction of the functional F xc[n], which generally

comes at a non-negligible cost in terms of formalism complexity, and numerical difficulties.

In the GGA approach, one tries to correct the LDA approximation by introducing a

dependence on the gradient of the density, in order to take into account the possible inho-

mogeneity of the electron gas. Many different formulations and parameterizations have been

developed so far to correct the LDA, but those details are left to the interested reader.

E. Electron-ion Interaction

Planewave DFT methods can be divided into all-electron (AE) and pseudopotential (PsP)

theories. AE methods usually employ a mixed basis set, where the planewaves are aug-

mented locally around each atomic site with radial solutions to the Schrodinger equation in

order to accurately describe the oscillatory behavior of the wavefunction in the core region.

The PsP method is an approximation to the AE method, and is based on the well-known

fact that the valence electrons are responsible for most chemical and physical properties of

molecules and solids. It is therefore tempting to simplify the description of atoms to those of

pseudo-atoms, in which only the valence electrons are explicitly treated in the self-consistent

calculation, thereby dramatically reducing the computational cost. Quantum mechanically,

the electronic states of an atom are generally well separated energetically into discrete core

and valence electronic states. It can be shown that it is possible to create a new set of basis

functions for the solution of the atomic Schrodinger equation so that one can calculate the

valence states of the atom without explicitly taking the core electrons into account. These

new functions are the so-called pseudo-wavefunctions, which give rise to the so-called pseu-

dopotential, which is a screened potential experienced only by the valence electrons. In order

to ensure that the pseudo and all-electron (AE) wavefunctions and potentials are exactly

identical in the valence region of the atom, both pseudo-wavefunctions and pseudopotentials

are set to match the AE wavefunctions and potentials beyond some core radius rc.

The main advantage of the pseudopotential approach is its lower cost compared to AE meth-

ods. It is computationally far less demanding, and with the ever increasing computational

power available, it allows first principles simulations of quite realistic and fairly large samples

of condensed matter. The key to the reduced expense is that the pseudopotential replaces

the true singular Coulombic potential by a much smoother one near the nucleus, and the



23

pseudo-wavefunctions are by construction smooth and nodeless within the core radius rc.

This is crucial when the electronic wavefunction is expanded in the usual solid-state basis

of planewaves, as an enormous number of planewaves would be required to describe the

oscillatory behavior of the AE valence wavefunction in the core region. Instead, by using

pseudopotentials, a much smaller basis set is required to achieve basis set convergence.

F. Self-Consistent Field

Density Functional Theory, like the Hartree-Fock method, is a self-consistent field (SCF)

method. This means that the calculation runs in cycles, and convergence is achieved when

the results given by solving the SCF equations are consistent with the assumptions made at

the beginning of the cycle. In the particular case of DFT, the principal quantity computed

and checked for self-consistency, is the electronic density. Fig. 11 schematically illustrates

the major steps of a density-functional self-consistent loop.

Prior to entering the self-consistent loop per se, preliminary computation is usually per-

formed. For example, in the case of a calculation employing pseudopotentials to represent

the electrostatic interaction between the valence electrons and nuclei plus core electrons, one

usually performs their generation once and for all at the beginning of the DFT calculation

for all ionic species. This step does not need to be performed for an all-electron calculations.

The next step is usually to build the basis set required to expand the density-functional

quantities. We give here the example of a calculation using a planewave basis, where one

needs to choose a kinetic energy cutoff for the planewave functions to be inserted in the

basis.

Just before entering the SCF loop, one builds a trial electronic density as a first guess.

This is usually done by using a superposition of atomic-like densities localized on each atomic

site present. This is a totally acceptable first guess for the total density since one expects

the surrounding atoms to perturb only slightly the atomic density of the ions, as it would

be assumed in a tight-binding picture. Using this trial density, we can now build the total

density-functional Hamiltonian H [n(r)], including ionic, Hartree and exchange-correlation

potentials and self-consistently solve the DFT equations. After the Hamiltonian is built,

it is diagonalized in order to compute the energies. The basis set coefficients are then

updated and a new charge density n(r) can be computed. At that point, one checks for self-



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FIG. 11: Schematic representation of the self-consistent algorithm for a density-functional based

calculation. Some steps may slightly vary depending on the type of implementation is used, for

our case, we have deliberately chosen the more specific case of a planewave-based DFT.

consistency of the answer, and if this new updated charge density agrees numerically (within

some user-defined numerical threshold) with the density used to build the Hamiltonian at

the beginning of the SCF cycle, we have reached the end point of the loop. We then exit,

and compute all the desired converged quantities, such as the total crystal energy, the band

structure, density of states, atomic populations, etc... On the other hand, if our new density



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n(r) does not agree with the starting density n(r), one generates a new input density and

starts another SCF cycle: build the new density-dependent Hamiltonian, solve, compute the

density, and check for self-consistency.

There have been many efforts to create reliable schemes for the generation of this new

density, mostly because the effectiveness and the numerical stability as a whole of the SCF

algorithm relies on it. It is very common nowadays to employ some type of ’density mixing’,

where the new input density is usually taken as a mix between the old input density n(r)

and the newly computed one n(r):

nnew(r) = αn(r) + (1 − α)n(r) (16)

where α is a density mixing coefficient. Much attention has been taken to improve onthe general concept of the density-mixing scheme and optimizations of the value for α, so

that the method ensures one with optimum convergence behavior.

G. Planewave basis convergence and k-point sampling

In principle, in an effectively infinite system like a crystal, an infinite number of electronic

wavefunctions have to be calculated. Those electronic states are allowed on a set of k-points

determined by the boundary conditions of the system studied. Bloch’s Theorem reduces

the problem of computing an infinite number of electronic wavefunctions to one where one

needs only to compute a finite number of wavefunctions, but at an infinite number of k-

points. In fact, for close values of k-points, the fluctuations of the wavefunctions can be

neglected, so that it may be possible to solve for only a set of “particular” points in k-

space. Practically this is done by k-point sampling where one chooses a finite number of

k-points, but dense enough such that we can reasonably represent the possible variations

of the electronic wavefunctions. We then restrict our effort to calculating the occupied

eigenstates of the Hamiltonian for those specific k-vectors that lie within the first Brillouin

zone. The volume of the Brillouin zone ΩBZ is related to the volume of the original (real)

cell Ωcell by

ΩBZ = (2π)3

Ωcell



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0 2 4 6 8 10 12 14 16 18 20

Number of k-points

-865.8

-865.7

-865.6

-865.5

-865.4

-865.3

-865.2

-865.1

E t o t

( e V / a t o m )

FIG. 12: Bulk diamond Si: Etotal vs. # of k-points

so that for large primitive cells, the Brillouin zone is very small and only a few special k-

points are needed to describe accurately the variations of the wavefunction, and consequently

those of the density.

The Bloch theorem is also important in that it allows us to expand the crystal wavfunction

in terms of a discrete set of planewaves. The completeness of the basis set would require usto include an infinite number of such functions, which is of course impractical. However, for

a planewave function of the form

f i(r) = ci,k+Ge−i(k+G)·r, (17)

the coefficients ci,k+G’s are typically more important for planewaves of small kinetic energy

rather than for ones with large kinetic energy, where the kinetic energy of the planewave is

given by

E kin = f i(r)

−

h2

2m∇2

f i(r) =

h2

2m|k + G|2. (18)

It becomes possible to truncate the basis set required for basis convergence, by including

only the functions with a small kinetic energy, up to some chosen kinetic energy cutoff. The

kinetic energy cutoff is usually chosen so that the value of the crystal total energy does not

vary if one increases the planewave expansion by choosing a larger kinetic energy cutoff.



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relation

ψ|H |ψ ≥ εo, (21)

where we have the equality if and only if the trial wavefunction is indeed equal to theground state wavefunction. The main idea of the algorithm is therefore to self-consistently

update the starting trial wavefunction to converge towards the real ground state wavefunc-

tion and thus determine within some numerical threshold the value for the ground state

energy o. Arbitrarily, the wavefunction can always be expanded as a linear combination of

basis functions, so that we have

ψ =n cnφn, (22)

so that Eq. 21 gives

n,m

c∗ncmψn|H |ψm ≥ εo, (23)

with the normalization constraint on the coefficients so that the wavefunction satisfies

Eq. 19,

n

|cn|2 = 1. (24)

Practically, during the self-consistent DFT loop, those coefficients are updated so as to

converge the total crystal wavefunction to the ground-state wavefunction. It is important to

emphasize that due to the inevitable incompleteness of the basis set chosen, the converged

crystal wavefunction can never be exactly equal to the true ground-state function. There-

fore, the converged eigen-energies εo also represent an upper limit to the true ground state

expectation value.

I. Electronic Temperature and Fermi surface smearing

In order to accurately and efficiently compute band structure energies, partial band occu-

pancies are introduced. Although their physical significance are debatable, they have shown

to dramatically increase electronic convergence rates. While computing the band energies in



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reciprocal space one must perform the integration over the first Brillouin zone of the filled

bands, so that

n

1

ΩBZ ΩBZ

εnk θ(ε

nk −

F)dk, (25)

where the index n runs over all bands, εnk is the energy of band n at the k-point k and

ΩBZ is the volume of the first Brillouin zone. θ(ε) is the Heaviside step distribution function

associated with the Fermi energy F. As we previously discussed, it is in principle necessary

to compute a finite number of electronic wavefunctions at an infinite number of k-points,

but this is in practice done using a finite set of special k-points which ensures satisfactory

convergence of the integration over the first Brillouin zone. Numerically, this means that

the integration of Eq. (25) reduces to a sum over all available k-points so that the equationreduces to

n

k

wkεnk θ(εnk − F), (26)

where wk is the weight associated with k-point k. This summation converges increasingly

slowly with the number of k-points, but for cases where bands are completely filled, such

as insulators or semi conductors, this integral can be accurately be computed even for a

small set of special k-points. By contrast, metallic phases pose a challenge. Because we

use partial occupancies, the convergence of the sum can become very cumbersome. In order

to overcome those difficulties, one usually replaces the Heaviside step function θ(ε) by a

smoother function near the Fermi energy. This very much resembles the shape of the Fermi-

Dirac surface distribution function for temperatures other than 0K. This method of Fermi

smearing replaces the step function by some other function, for example the Fermi-Dirac

function

f nk

εnk − F

kBT

=

1

exp[ εnk−FkBT

] − 1, (27)

for temperature T .

The introduction of this ’fictitious’ temperature in our electronic system has the conse-

quence that the total electronic energy is not variational any more. Instead, the total energy

E must be replaced by the free energy F to account for entropic effects so that



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F = E −nk

wk kBT S (f nk). (28)

Nonetheless, it is always possible to accurately extrapolate this formula for T → 0 K

from

E (T → 0K) = 1

2(E + F ), (29)

and therefore still obtain physically meaningful quantities at 0 K.

Practically, when one performs calculations on metallic systems, it is highly desirable

to use this Fermi smearing. One usually introduces a finite fictitious temperature for the

electronic system, which can be reduced as the algorithm reaches convergence. This tem-

perature must always be monitored so that it does not give raise to a significant entropy

term.

DFT Notes Cocula Carter

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