Computational Implementations in Electronic Structure … · Computational Implementations in...

Computational Implementations in

Electronic Structure Theory

N. N. Lathiotakis

Theoretical and Physical Chemistry Institute

National Hellenic Research Foundation

Athens, Greece

OutlineOutline

1. Electronic structure methods

2. Introduction to density-functional theory (DFT)

• Fundamentals

• Flavors of DFT.

• Successes and failures

3. Computer implementations

• Gaussian basis sets

• Expansion in Plane waves

• APWs, wavelets, SWs, grid methods

4. Introduction to Quantum Espresso

• Basic input, capabilities

5. Some Results

• Graphene force fields

• Graphene allotropes under pressure

• Magnetic impurities in ZnO

BooksBooks

• Fundamentals:• Dreizler R.M., Gross E.K.U. (1990), Density functional theory.

• Parr R. G., Yang W. (1989), Density-Functional Theory of Atoms

and Molecules.

• Applications:• C. Fiolhais, F. Nogueira, M. Marques (Eds) (2003),

A Primer in Density Functional Theory.

• W. Koch, M. C. Holthausen (2000),

A Chemist’s Guide to Density Functional Theory

• Richard M. Martin (2004),

Electronic Structure: Basic theory and practical Methods

• E. Kaxiras (2003), Atomic and electronic structure of solids

• D.S Sholl, J. A. Steckel (2009), Density Functional Theory: a

practical introduction.

1. Electronic structure methods1. Electronic structure methods

The problem in electronic structure theoryThe problem in electronic structure theory

• To solve the quantum mechanical problem of a system of N particles

(e.g. electrons or nuclei) in the presence of external electric and

magnetic fields.

• Born Oppenheimer approximation: electronic and nuclear “motions”

are separated and could be determined independently: Fixed usually

classical nuclear motion.

• Without any applied fields the problem reduces to the quantum

mechanical description of N electrons “living” in the external potential

of the nuclei.

The many-body Schrödinger EquationThe many-body Schrödinger Equation

1 2 1 2ˆ ( , , , ) ( , , , )

N NH r r r E r r rΨ = Ψ

� � � � � �⋯ ⋯

ˆ ˆ ˆ ˆH T V W= + +The Hamiltonian:

We are looking for the solution 1 2( , , , )

Nr r rΨ� � �⋯ of

where

2

2

1 1 , 1

1ˆ ˆ ˆ, ( ), ( , )2 2

N N N

i i i j

i i i j

i j

T V V r W W r rm= = =

≠

= − ∇ = =∑ ∑ ∑ℏ � � �

For an atom:

2 2

( ) , ( , )i i j

i i j

Ze eV r W r r

r r r= − = +

−

� � ��

Equivalently: Minimization of|

HE

< Ψ Ψ >=

< Ψ Ψ >

How useful is Ψ?How useful is Ψ?

Silicon atom:

~1028 Kg100g/HD

~1029 HD’s1 TB / Hard Disc:

2.5 x 1041 bytes4 bytes / number:

1042 numbers10 entries / coordinate:

42 coordinates14 electrons:

Mass of the earth: 6 x 1026 Kg

• Ψ variation is restricted by constraints imposed by symmetry

but still the problem is impossible to solve.

• The full information contained in Ψ is not necessary for the

expectation values of most observables that correspond to

1- and 2-body operators.

Electronic Structure MethodsElectronic Structure Methods

• Determinantal methods:

• Hartree Fock (Ψ is expressed as a single Slater determinant)

• Configuration Interaction (CI) (Ψ is expressed as a linear

combination of all possible Slater determinants)

• Methods based on functionals of reduced density matrices (RDM)/densities:

• DFT

• Reduced Density Matrix Functional Theory (RDMFT)

Reduced density matrices (RDM)Reduced density matrices (RDM)

• The density matrix:

The expectation value of one-body observable:

•Reduced density matrices (of order p, 1 ≤ p < N ):

•The densities (diagonals of density matrices):

•The density (one electron density):

*

1 2 1 2 1 2 1 2( , , , , , , , ) ( , , , ) ( , , , )N N N N Nr r r r r r r r r r r rγ ′ ′ ′ ′ ′ ′= Ψ Ψ� � � � � � � � � � � �⋯ ⋯ ⋯ ⋯

ˆ ˆ{ }N

A Tr Aγ=

1 2 1 2

3 3

1 1 2 1 1 2 1

( , , , , , , , )

( , , , , , , , , , , , , , )

p p p

p N N p p N p p N

r r r r r r

Nd r d r r r r r r r r r r r

p

γ

γ+ + +

′ ′ ′ =

′ ′ ′

∫ ∫

� � � � � �⋯ ⋯

� � � � � � � � � � � �⋯ ⋯ ⋯ ⋯ ⋯

1 2 1 2 1 2( , , , ) ( , , , , , , , )p p p p p

r r r r r r r r rρ γ=� � � � � � � � �⋯ ⋯ ⋯

3 3 *

2 2 2( ) ( , , , ) ( , , , )N N N

r d r d r r r r r r rρ = Ψ Ψ∫ ∫� � � � � �

⋯ ⋯ ⋯ ⋯

The exact energy functional of γ2The exact energy functional of γ2

The one-body, reduced density matrix (1-RDM):

The one-body density (usual particle density):

3 3 *

1 2 2 2( , ) ( , , , ) ( , , , )N N Nr r d r d r r r r r r rγ ′ ′= Ψ Ψ∫ ∫� � � � � � � � � �

⋯ ⋯ ⋯

1 1( ) ( ) ( , )r r r rρ ρ γ= =� � � �

The exact total energy of an electronic system as a functional

of the 2-RDM, γ2 is:

1

3 3 2

1 2 1 1 1 1 1 1 1

3 3 3 2 1 21 2

1 2

1[ , , ] ( ) [ ( , )]

2

( , )( ) ( )

| |

rE E d r d r r r r r

r rd r V r r d r d r

r r

ρ γ ρ δ γ

ρρ

′ ′ ′= = − − ∇

+ +−

∫ ∫

∫ ∫ ∫

� � � � � �

� ��

� �

• In principle E can be minimized with respect to γ2.

• Unfortunately, the N-representability constraints for γ2 are complicated

and not completely known.

DFT/RDMFTDFT/RDMFT

• The N-representability constraints for ρ and γ1 are simple:

1. Every ρ integrating to N is N-representable.

2. For γ1 the necessary and sufficient conditions are:

where ni’s are the eigenvalues of γ1.

• Two approximate theories:

1. DFT: The total energy is a functional of ρ.

2. RDMFT: The total energy is a functional of γ1.

, 0 1,i i

i

n N n= ≤ ≤∑

DFT is the workhorse of present day electronic

structure calculations

DFT is the workhorse of present day electronic

structure calculations

2. Introduction to DFT2. Introduction to DFT

Hohenberg-Kohn theoremHohenberg-Kohn theorem

( )V r� Ψ ( )rρ

�

CD

C D

The map CD is invertible:

( )V r�

( )rρ�(1-1)

• Non degenerate ground state:

0 0[ ] [ ] | | [ ] .V

E T W Vρ ρ ρ=< Ψ + + Ψ >Due to D-1 map:

By virtue of Rayleigh-Ritz: E[ρ] is minimized by the density

ρ0 corresponding to V0:

0 00 0[ ] [ ]V V

E E Eρ ρ= ≤

Hohenberg-Kohn theoremHohenberg-Kohn theorem

3VΨ

C D• Degenerate ground state:

C is not a map, C-1 is. D is a map but not D-1: (Ψ[ρ] is not defined).

CD-1 is a map. Subsets in Ψ and ρ are disjoint.

1V

3V

2V

2VΨ

1VΨ

3Vρ

1Vρ

2Vρ

Although D-1 is not a map the universal functional

F[ρ] is still uniquely defined:

3| | ( ) ( )F T W E d r r V rρ=< Ψ + Ψ >= − ∫� � �

Coulomb potential

of the nuclei

Kohn-Sham theoremKohn-Sham theorem

The ground state density of the interacting system can be calculated

as ground state density of a non-interacting system with an effective

potential .( )sv r�

• Kohn-Sham system:

2

2

1

3

0

[ ]( ) ( ) ( )2

( ) ( )

( ')[ ]( ) ( ) ' [ ]( )

'

s j j j

occ

j

j

s xc

v r r r

r r

rv r v r d r v r

r r

ρ ϕ ε ϕ

ρ ϕ

ρρ ρ

=

∇− + =

=

= + +−

∑

∫

� � �

� �

��

� �

Hartree

potential

Exchange-correlation

potential:

( ) (universal)( )

xcxc

Ev r

r

δ

δρ=�

�

Kohn-Sham systemKohn-Sham system

• Transforms the problem into a problem of self-consistent

1-electron equations.

• The universal functional Exc should be approximated.

Approximations:

• Local Density Approximation (LDA):

• GGA and hybrid functionals

• Orbital functionals (OEP, OPM method)

•Meta-GGA: (use kinetic energy density also)

• Is there any meaning in the 1-electron energies, i.e. the

eigenvalues of the KS hamiltonian?

3 hom

XC xc ( ( ))E d r e rρ= ∫� �

3

XC ( , )E d r g ρ ρ= ∇∫��

XC XC 1 2[ , , , ]NE E ϕ ϕ ϕ= ⋯

3

XC ( , , )E d r g ρ ρ τ= ∇∫��

FunctionalsFunctionals

• LDA

• Slater Exchange ( )

• Corellation: Fit for the Homogeneous Electron Gas

Monte Carlo: Ceperley-Alder, PRL45, 566(1980)

Fit: Vosko, Wilk, Nusair, Can. J. of Phys. 58 1200 (1980)

Perdew-Wang, PRB45 13244 (1992)

• GGA (dependence on the density gradient)

1. Non-empirical

PW91 (PRB33, 8800)

PBE (PRL77, 3865)

2. Hybrid

Becke86 (JCP84, 4524)

LYP/BLYP (PRB37, 785)

B3LYP (JPC98, 11623):

• Orbital Functionals (OEP, OPM)

Exact exchange (EXX)

1/3ρ∼

B3LYP B88 LYP LDA

XC X X0 X C C(1 ) (1 )E a E aE bE cE a E= − + + + + −

Successes of LDASuccesses of LDA

<25%Cohesive energies

<2%Lattice constants of solids

Few %Bands of metals/Fermi

surface

<5%Equilibrium distances

<0.5%Atomic and molecular

ground state energies

Typical deviation from exp.Quantity

Failures of LDAFailures of LDA

• Too small ionization potentials

• Negative ions not bound (no electron affinity)

• Band gaps of semiconductors/insulators too small

• Some transition metal oxides predicted as metals

• Van der Walls interaction not described

LDA and GGALDA and GGA

• Both LDA and GGA underestimate the band gaps

of semiconductors/insulators.

more correctfavors close packingstructure

1%(longer)1% (shorter)Bond length

5%100% (too negative)Ec

0.5%5% (not negative enough)Ex

GGALDAProperty

The fundamental gapThe fundamental gap

The difference of the ionization potential and the electron affinity:

XC

KS

gap

)(2)1()1(

∆+=

−−++=−=

ε

NENENEAIE

where XC XCXC

( ) ( )N N

E E

r rδ δ

δ δ

δρ δρ+ −

∆ = −� �

∆XC is the discontinuity of the xc potential upon adding (removing)

an electron. It is zero for LDA, GGA.

LDA and GGA underestimate gaps by a factor of a 1/2

The fundamental gapThe fundamental gap

EXX results concern the KS gap only. If the discontinuity

is added EXX results are as bad as Hartree-FockStädele et al, PRB59 10031 (1999)

The gap for the HSE hybrid functionalThe gap for the HSE hybrid functional

HSE functional gives gaps close to experiment: Heyd et al JCP 123 174101 (2005)

The Fermi surface of CuThe Fermi surface of Cu

6.30.6700.712

6.50.7480.797

2.70.9730.947

10.50.2560.283

1.51.1661.183

1.41.3581.339

3.20.5920.573

Diff (%)Exp (A-1)LDA (A-1)vector

(1 )

(1 0 0 )Q

( 2 )

(1 0 0 )Q

(1 )

(1 1 0 )Q

( 2 )

(1 1 0 )Q( 3 )

(1 1 0 )Q

( 4 )

(1 1 0 )Q

(1 )

(1 1 1 )Q

LDA: Lathiotakis et al, JMMM 185, 293 (1998); Exp.: M.R. Halse, Phil. Tr. Roy. Soc. London, A 265, 507 (1969).

3. Computer Implementations3. Computer Implementations

Computer codesComputer codes

Codes targeting finite systems:

1. Orbitals expanded in Gaussian type orbitals

• Gaussian (commercial)

• Turbomole (commercial)

• molpro (commercial)

• ORCA (open source)

• GAMESS (free closed source)

• DMoL (commercial)

• Psi3 (open source)

• NWCHEM (open source)

2. Real-Space (grid) with pseudopotentials

• OCTOPUS (open source)


Codes targeting periodic systems (solids, surfaces, 2D

Systems, polymers:

1. Orbitals expanded in Gaussian type orbitals

• CRYSTAL (commercial)

2. Orbitals expanded in numerical atom centered basis

• SIESTA (open source)

3. Orbitals expanded in Plane Waves

• Quantum Espresso (open source)

• ABINIT (open source)

• CPMD (open source)

• VASP (commercial)

• CASTEP (commercial)


4. Orbitals expanded in Augmented Plane Waves (FP):

• ELK (open source)

• EXCITING (open source)

• Fleur (open source)

• Wien2k (commercial)

5. Orbitals expanded in Wavelets (Daubechies):

• BigDFT (open source)

6. Real space (grid):

• GPAW (open source)

4. Quantum Espresso4. Quantum Espresso

The QE projectThe QE project

• ESPRESSO stands for:

opEn Source Package for Research in Electronic Structure, Simulation, and Optimization.

• Open Source: GNU General Public License

• Developers :

DEMOCRITOS National Simulation Center (Trieste) and SISSA (Trieste)

CINECA National Supercomputing Center in Bologna

Ecole Polytechnique Fédérale de Lausanne

Princeton University

Massachusetts Institute of Technology

University of Oxford

QE capabilitiesQE capabilities

Ground-state calculations:

• Self-consistent calculation of total energies• Kohn-Sham orbitals obtained through minimization• Several k-point sampling, Gamma point calculations • Broadening schemes (Fermi-Dirac, Gaussian, Methfessel-Paxton, and

Marzari-Vanderbilt)• Separable norm-conserving and ultrasoft (Vanderbilt) pseudo-

potentials, PAW (Projector Augmented Waves)• Exchange-correlation functionals: from LDA to generalized-gradient

corrections (PW91, PBE, B88-P86, BLYP) to meta-GGA, exact exchange (HF) and hybrid functionals (PBE0, B3LYP, HSE);

• DFT+U• Polarization• Spin polarized calculations, Spin-orbit coupling and noncollinear

magnetism

QE can also calculateQE can also calculate

1) Geometry optimization and unit cell optimization

2) Ab-initio molecular dynamics: Car-Parrinello, microcanonical, canonical, additional constraints

3) Phonon frequencies and modes, full phonon dispersions, inter-atomic force constants

4) Electron-phonon interactions

5) Infrared and (non-resonant) Raman cross-sections, NMR chemical shifts

6) Quantum transport

QE runs in the large majority of platforms, has been ported to GPUs, parallelization with MPI or OpenMP.

Self-consistent cycle

Initial guess for KS orbitals

2

1

( ) ( )occ

j

j

r rρ ϕ=

=∑� �

3

0

( ')[ ]( ) ( ) ' [ ]( )

's xc

rv r v r d r v r

r r

ρρ ρ= + +

−∫�

� � � ��

2

{ [ ]( )} ( ) ( )2

s j j jv r r rρ ϕ ε ϕ∇

− + =� � �

convergenceNo

Yes

Exit electronic optimization

mixing

broadening

PseudopotentialsPseudopotentials

• Core electrons do not participate in chemical bond (remain atomic like)

• Core electron orbitals have enormous peaks close to the nuclei • Valence electrons are oscillating rapidly close to the nuclei

Core electrons are not included in the calculation but are

taken into account through pseudopotentials

Example Si-atom: (1s2 2s2 2p6) 3s2 3p2

Core electrons

Large variety of pseudopotentials in QE site: type and

functional in the filename

PseudopotentialsPseudopotentials

In the input file:

ATOMIC_SPECIES

Mg 24.305 Mg.pw91-np-van.UPF

Si 28.085 Si.pw91-n-van.UPF

O 15.999 O.pw91-van_ak.UPF

H 1.008 H.pw91-van_ak.UPF

Same exchange-correlation functional for all pseudopotentials

Typical input file (graphene)Typical input file (graphene)

&control

calculation = 'scf'

restart_mode='from_scratch',

prefix='graphene',

pseudo_dir = '/home/lathiot/espresso-4.3/pseudo/',

outdir='/home/lathiot/tmp/'

/

&system

ibrav= 0, celldm(1) =4.7375, nat= 2, ntyp= 1,

ecutwfc =70.0, ecutrho=560.0

occupations='smearing'

nbnd = 8

degauss=0.001

/

&electrons

diagonalization='david'

electron_maxstep = 100

mixing_mode = 'plain'

mixing_beta = 0.7

conv_thr = 1.0d-8

/

ATOMIC_SPECIES

C 12.000 C.pbe-rrkjus.UPF

CELL_PARAMETERS

1.0 0.0 0.0

0.5 0.8660254 0.0

0.0 0.0 12.0

ATOMIC_POSITIONS alat

C 0.00 0.00 0.00

C 1.0 0.5773503 0.00

K_POINTS automatic

10 10 1 0 0 0

calculation = ‘scf’

‘nscf’

‘relax’

‘vc_relax’

ecutwfc, ecutrho:Kinetic energy

cutoffs for wavefunction and

density

occupations=‘smearing’

‘fixed’

degauss: smearing width (Ryd)

nbnd: number of bands, i.e.

eigenvalues to solve.

Open in browser file:///home/lathiot/espresso-5.0.1/PW/Doc/INPUT_PW.html

Typical input file (graphene)Typical input file (graphene)

&control

calculation = 'scf'

restart_mode='from_scratch',

prefix='graphene',

pseudo_dir = '/home/lathiot/espresso-4.3/pseudo/',

outdir='/home/lathiot/tmp/'

/

&system

ibrav= 0, celldm(1) =4.7375, nat= 2, ntyp= 1,

ecutwfc =70.0, ecutrho=560.0

occupations='smearing'

nbnd = 8

degauss=0.001

/

&electrons

diagonalization='david'

electron_maxstep = 100

mixing_mode = 'plain'

mixing_beta = 0.7

conv_thr = 1.0d-8

/

ATOMIC_SPECIES

C 12.000 C.pbe-rrkjus.UPF

CELL_PARAMETERS

1.0 0.0 0.0

0.5 0.8660254 0.0

0.0 0.0 12.0

ATOMIC_POSITIONS alat

C 0.00 0.00 0.00

C 1.0 0.5773503 0.00

K_POINTS automatic

10 10 1 0 0 0

diagonalization: type of minimization

(david, cg)

mixing_mode, mixing_beta:

mixing parameters for scf convergence

conv_thr: Convergence criterion total energy

CELL_PARAMETERS: Bravais lattice vectors.

, , , ,k G k G i k G i k i k G

G

H C Cε′ ′+ + + +′

=∑

ATOMIC_POSITIONS : atomic positions in

unit cell

K_POINTS: reciprocal lattice mesh

in the Brillouin zone.

(In the example 10x10x1 mesh)

They can also be specified explicitly.

Convergence with k-point sampling must be checked

Finite structures (Gamma point)Finite structures (Gamma point)

Can we calculate non periodic systems, e.g. a molecule?

Yes: Set lattice vectors to large numbers and

K_POINTS gamma, i.e. only 1 k-point at (0,0,0)

5. Results5. Results

Graphene under strain

● Uniaxial:

● Hydrostatic:

Stress-strain plot

● Uniaxial

Young Modulus & Poisson ratio

● Young modulus:

Our 0.96 TPa

Other theoretical (0.91-1.09) TPa

Experiment 1.0 ± 0.1 TPa

Poisson ratio:

Our 0.22

Other theoretical (0.15-0.28)

Stress-strain plot

● Hydrostatic

Fitting of empirical potential:

A) Bond stretching


B) Angle Bending


C) Dihedral Angle Bending

Electronic Structure of Pentaheptite

● It occurs through Stone-Wales transformations

● Energetically 0.25 eV/atom above Graphene

● Pentaheptite:

● Pentaheptite is a metal:


● Fermi Surface


4) Tetraoctite:3) Pentaheptite B:

2) Pentaheptite A:1) Graphene

Graphene and sp2 allotropes under extreme strain

● Strain vertical to the armchair:



● Strain vertical to zig-zag:

Magnetism in doped ZnO

• ZnO based materials: Large Technological importance:

• Spintronics

• LED

• Detectors/sensors

• Spintronic Applications:

• Exploiting electron spin in electronic transfer to carry information

• ‘Spin’: an extra degree of freedom in improvement of devices

• Dilute Magnetic Semiconductors: semiconducting materials that have

acquired Ferromagnetic properties.

• Possible path: doping of with Transition Metal atoms

• Increase of FM properties with co-doping

• The Failure of LDA/GGA in the Gap of ZnO:

Experimental Gap: 3.4 eV

LDA/GGA calculated gap: 0.75 - 0.85 eV

• Two additive reasons:

1) LDA and GGA are known to underestimate the Gap by ~50%

2) The d-band of Zn is centered in a false position within LDA/GGA:

enhanced artificial hybridization of Zn-d with O-p

Zn-O Dos

Zn-O Dos

Cobalt doped ZnO

• There are experimental data on the position of Co-d impurity band:

Cobalt doped ZnO

• There are experimental data on the position of Co-d impurity band:

• With LDA+U we can reproduce this result

Calculation Details

• 3x3x3 supercell (108 atoms), 2 k-points.

Calculation Details

• 3x3x3 supercell (108 atoms) with 2 k-points.• 2 Co impurities (4% at)

• Target: To find structures that are Ferromagnetic

• Energy difference of FM from the AFM configuration.

FM

AFM

Structures for Co-doped ZnO

C2:C1:

D: E:

FM vs AFM configuration

0.400.80D

0.010.03E

14.640.72C2

45.3442.50C1

FM-AFM (GGA +U)

(meV)

FM-AFM (GGA)

(meV)

• Two Co impurities in ZnO are AFM coupled.

• Interaction is short-range.

Co-doping with Cu

C1-Cu43 C1-Cu53 C1-Cu59 C1-Cu19

C2-Cu89 C2-Cu67 C2-Cu23

D-C55 D-Cu19 D1-Cu41 E-Cu55

Codoping with Cu - results

-99.2C2-Cu23

-0.8E-Cu55

8.3-35.2C2-Cu67

14.9-2.1C2-Cu89

-22.8C1-Cu19

7.433.9C1-Cu59

-125.3-56.8C1-Cu53

-4.7C1-Cu43

-71.8D-Cu41

87.5-1.9D-Cu19

-44.2-65.9D-Cu55

FM-AFM (GGA+U)

(meV)

FM-AFM (GGA)

(meV)

Structures


• In most cases Cu induces a FM coupling

• Symmetric position of Cu (w.r.t Co atoms) induces FM coupling

• The splitting of the Cu-d band in AFM state

Cu-d band splitting

• No splitting for Cu-d in the AFM configuration for D-Cu55


• In most cases Cu induces a FM coupling

• Symmetric position of Cu (w.r.t Co atoms) induces FM coupling

• The splitting of the Cu-d band in AFM state

• Cu: Super-exchange mediator

• Cu and Co in different planes (perpendicular to the Wurtzite z axis)

induses FM coupling

• Possibly, growth of Cu and Co alternating impurity planes would lead to

enhanced magnetic properties.


• Structures with enhanced FM coupling:

• Structures with enhanced AFM coupling:

Co islands/dots in ZnO

• The 4-atom dot + the vacancy of the common Oxygen

X

60.6(1)

71.5(2)

71.0(1)

77.1(2)

80.1(1)

96.0(2)

0.00(1)

0.00(2)

E (meV)Structure


• The 4-atom dot + the vacancy of the common Oxygen

(1)Zn Lattice positions, (2)Relaxed Co atoms


• 3-4 atom dots prefer AFM state

• Far Oxygen vacancy: does not affect magnetic state

• Common Oxygen vacancy: FM state favorable

MgB2

2-D σσσσ-bonding hole pockets

3-D ππππ and ππππ∗∗∗∗ Fermi surfaces

σσσσσσσσ ππππ

ππππ

A

Tc= 39.5 K

ππππ∗∗∗∗σσσσ

Fermi Surface of MgB2

CollaboratorsCollaborators

• G. Kalosakas, K. Papagelis, K. Galiotis,

University of Patras

• Z. G. Fthenakis, Michigan State University

• A. N. Andriotis, IESL FORTH, Heraklio

• M. Menon, University of Kentucky

• E.K.U. Gross group, MPI Halle

Concluding remarksConcluding remarks

• Theoretical electronic structure methods can make real

predictions for material properties

• DFT is the workhorse of present day electronic

structure theory

• QE is a mature software applying DFT approximations

to periodic (and finite) systems able to calculate several

diverse properties like structural, electric, magnetic,

optical, superconducting properties.

Thank you !!!

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