Víctor Luaña (y) & Alberto Otero-de-la-Roza ( y...

Electronic structure Crystal QE EOS Pressure LatDyn Elastic Electronic structure textbooks

Electronic structure of solids: quantum espresso

Víctor Luaña (†) & Alberto Otero-de-la-Roza (‡)(†) Departamento de Química Física y Analítica, Universidad de Oviedo

(‡) Universidad de California at Merced

European school on Theoretical Solid State ChemistryZCAM, Zaragoza, May 13–17, 2013

V. Luaña & A. Otero-de-la-Roza () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2013 1 / 86


Content

1 Electronic structure calculations

2 Crystalline structures

3 Quantum Espresso: pw+ps GNU code

4 Equation of state

5 Phase equlibrium and pressure

6 Lattice dynamics

7 Elastic constants and mechanical stability



Hartree-Fock and Kohn-Sham equations

Hartree-Fock (closed shell)Orbital approximation to solve the non-relativistic electronic stationary states:

HeΨ({xi}; {Rα}) = EΨ ⇒ Fψλ = ελψλ (1){−

1

2∇2

r +N∑α

Zα

riα+

∫R3

ρ(r′dr′)

|r− r′|+VX

}ψλ(x; {Rα}) = ελ({Rα})ψλ(x; {Rα}) (2)

where

VXψλ = −occ.∑

i

∫R3

ψ∗i (r′)ψλ(r′)

|r− r′|dr′ψi(r) (3)

is the Hartree (exact) exchange acting over orbital ψλ.

SCF: both, HF and KS equations must be solved iteratively until convergence (self-consistency).

Kohn-ShamOrbital ansatz to solve the density functional non-relativistic electronic ground state:

HeΨ0({xi}; {Rα}) = EΨ0 ⇒ Fψλ = ελψλ (4){−

1

2∇2

r +N∑α

Zα

riα+

∫R3

ρ(r′)dr′

|r− r′|+Vxc

}ψλ(x; {Rα}) = ελ({Rα})ψλ (5)

where Vxc is the unknown exchange and correlation functional.



Quantum Chemistry and Solid State formalisms I

Quantum ChemistryObjective: wavefunctions and total energies of any stationary state.Born-Oppenheimer: separate nuclear and electronic motions to simplify.Nonrelativistic: use classical kinetic energy, separate spin and orbital parts, ...Orbitals: way to produce antisymmetric many electron wavefunctions, via Slater determinants.HF orbitals fulfill Koopmans and Brillouin theorems.HF closed or open shell: the main method to produce orbitals: optimize the orbitals byminimizing the energy of an electronic configuration under the condition that the spinorbitalsremain orthonormal.Other methods to produce orbitals: UHF (unrestricted HF), GVB (generalized ValenceBond), MCSCF (Multiconfigurational SCF), ...Basis sets (molecular systems): primitive gaussians (GTO), usually combined to form con-tracted GTOs. Pople family: STO-3G, 3-21G, 6-311++G**, ... Dunning family: cc-PVDZ,cc-PVTZ, cc-PVQZ, aug-cc-PVDZ, ... CBS: Complete Basis Set (extrapolation).techniques: HF assumes an average interaction between electrons. The difference to theexact solution is the correlation problem. Correlation techniques: Möller-Plesset (Many BodyPerturbation Theory: MP2, MP3, ...), Configuration Interaction (CIS, CISD, CISDT, ...), CoupledCluster (CCSD, CCSDT, ...), ... Full CI.

lim→CBS,→FCI

calculation = exact



Quantum Chemistry and Solid State formalisms II

Size scaling (N: spinorbitals, electrons)

method lin.† HF, KS HF, KS‡ KS∗ MP2 MP3 CCSD CCSD(T)Q FCIO N N3 N4 N5 N6 N7 N10 N!

† Special linear versions of HF and KS methods.‡ KS with LDA, LSDA, or GGA xc functionals.∗ KS with hybrid functionals.

Solid State (Density Functional)Objective: electron density and total energy of the ground state.

Born-Oppenheimer:Nonrelativistic formalism: most solid state codes include relativistic corrections.

Different types of basis set formalisms: (1) planewaves and pseudopotentials (pw+ps); (2)GTO’s; (3) local orbitals and ps.; (4) FPLAPW (Full Potential Linear Augmented PlaneWaves);...

xc functionals: Metaphorical classification (Jacob’s Ladder): (1) LDA/LSDA [Ex: VWN91]; (2)GGA [Ex: PBE, PW91]; (3) meta-GGA [Ex: TPSS]; (4) hybrid [Ex: B3LYP]; (5) double hybrid[Ex: XYG3]; (¿?) the unknown exact functional.



Electronic structure solid state codes

code basis webpage price sourceabinit pw+ps www.abinit.org GNU yesCPMD pw+ps www.cpmd.org 0 yescrystal09 CGTO www.crystal.unito.it 1000 Eur. noelk FPLAPW elk.sourceforge.org GNU yesgpaw! PAW wiki.fysik.dtu.dk/gpaw GNU yesQE (pwscf) pw+ps www.quantum-espresso.org GNU yessiesta Loc+ps www.icmab.es/siesta 0 yesvasp pw+ps cms.mpi.univie.ac.at/marsweb yes yeswien2k FPLAPW www.wien2k.at yes yes

Others: adf, castep (free for UK academics), dacapo (free), fhi98md (free), fleur (free), g09,octopus (free), ...Visualizing codes: xcrysden (free), mercury, ...


www.abinit.org

www.cpmd.org

www.crystal.unito.it

elk.sourceforge.org

wiki.fysik.dtu.dk/gpaw

www.quantum-espresso.org

www.icmab.es/siesta

cms.mpi.univie.ac.at/marsweb

www.wien2k.at


KS equations on a crystal (quantum espresso version)

{−∇2

r +∑α

Vαsf + VH(r) + Vxc(r)}ψnk(r) = εn(k)ψnk(r) (6)

−∇2r : kinetic energy, Rydberg units are used throught.

Vαsf : non-local pseudopotential. Either norm-conserving, ultrasoft, or PAW (projected aug-mented wave) pseudopotentials can be used.

VH(r) =∂EH[ρ]

∂ρ=

∫R3

ρ(r′dr′

|r− r′|: Coulomb potential or Hartree term.

Vxc(r) =∂Exc[ρ]

∂ρ: exchange and correlation potential.

Bloch states: ψnk(r) = eikrunk(r), where ψnk(r) is periodical over reciprocal space and unk(r) isperiodical over real space.

ρ(r) =∑

nk

fnk |ψnk|2 =∑

nk

fnk |unk|2 , with fi ∈ {0, 1} (a typical insulator) or fi = 1/(1 + e−εi/kT)

(Mermin functional, used on metals).

k ∈ BZ1. Running special points and directions in iBZ1 (irreducible first Brillouin zone) is thebasis for band diagrams. Total properties (energy, DOS, etc) requires integration of BZ1.



KS calculations on a basis

Let φ~q+~G(~r) be a basis function φ~q(~r) at the reciprocal cell ~G. The basis is used to build the KS

orbitals:ψn~q(~r) =

∑n~q

cn~q(~G)φ~q+~G(~r). (7)

By minimizing the energy with cn~q(~G) as the variational parameters the KS equations transforminto

∀~G∑~G′

(H~q+~G,~q+~G′ − εn~qS

~q+~G,~q+~G′

)cn~q(~G) = ~0, (8)

that must be solved for every ~q ∈ BZ1. In the above equation:

H~q+~G,~q+~G′ =

⟨φ~q+~G(~r)

∣∣∣∣−12∇2 + V

∣∣∣∣φ~q+~G′ (~r)⟩ , S~q+~G,~q+~G′ =

⟨φ~q+~G

∣∣∣φ~q+~G′ ⟩ , (9)

and the KS equation takes the form of a generalized eigen equation HC = SCE that must besolvedfor every ~q.By sampling ~q-points along special directions on the BZ1 we get the band diagrams: εn~q.The total energy (E), Density of States (DOS, g(E)), and the Fermi surface in the case of a metal(εF(~q)), are produced by integration for all ~q ∈ BZ1. Mokhorst-Pack special points methods ispopular for doing this integration.



Planewaves (3D) and the movement of a free electron I

The hamiltonian is h = p2/2m, ~p = −i~~∇, and the solutions are planewaves (PW) again:

|~k〉 = ψ~k(~r ) = V−1/2ei~k·~r, ε~k =~2k2

2m, ~r,~k ∈ R3. (10)

with V being the volume of the normalization box. Some properties

The PWs are eigenfunctions of the momentum operator: ~p |~k〉 = ~~k |~k〉.The particle velocity is proportional to ~k, the wavevector : ~v = ~~k/m.The |~k〉 and |−~k〉 PWs are degenerated.The wavelength of a PW is λ = 2π/k.

To enforce periodic boundary conditions in a general way we define the parallelepipedic cell a˜

and

an arbitrary primitive translation~t = a˜

n, n ∈ Z3. For any~r ∈ R3:

ψ~k(~r +~t ) = ψ~k(~r ) =⇒ ei~k·~t = 1. (11)

If ~k is a vector in the reciprocal cell, ~k = a˜?k = 2πa

˜?h, the periodicity condition is

1 = ei~k·~t = ei2π(hTn) =⇒ h1nx + h2ny + h3nz ∈ Z for all n ∈ Z3. (12)



Planewaves (3D) and the movement of a free electron II

The wavevector of the periodic PWs is then

~k = 2π(h1~a ? + h2~b ? + h3~c ?) = 2πa˜

?h = a˜

?k, (h1, h2, h3) ∈ Z3. (13)

This discrete mesh of allowed ~k-points is distributed uniformly in the reciprocal space. Each ~k pointcan be associated with a small parallelepiped of volume v~k = (2π)3V? = (2π)3/V, where V is thevolume of the main cell. It is implicitely assumed that a primitive cell is used to describe thereciprocal space.The periodic PWs form an orthonormal set

〈~k|~k ′〉 =1V

∫V

ei(~k−~k ′)~rd~r = δ~k,~k ′ , (14)

where the integral is done on a unit cell. The set is also complete, which means that any 3Dfunction can be expanded as

f (~r ) =∑~k

f~kei~k·~r ⇐⇒ f~k =

∫V

f (~r )e−i~k·~rd~r. (15)



Brillouin zones I

The first Brillouin zone (BZ-1) is the common name for the Wigner-Seitz primitive cell of thereciprocal or ~k-space lattice. Whereas primitive and centered cells are both used for the directlattice, centering is avoiding in the manipulation of the ~k lattice.An alternative definition for BZ-1 is the set of points in ~k space that can be reached from the origin(~k = ~0) without crossing any Bragg plane. A Bragg plane for any two points in the lattice being theplane which is perpendicular to the line between the two points and passes through the bisector ofthat line.The concept of BZ-1 can be generalized. The second BZ (BZ-2) is the set of points that can bereached from the first zone by crossing only one Bragg plane. BZ-(n+1) is formed is the set ofpoints not in {BZ-1, BZ-2, ... BZ-(n−1)} that can be reached from BZ-n by crossing only one Braggplane. Alternatively, the n Brillouin zone can be reached from the origin by crossing n−1 Braggplanes, but not fewer.The construction of the BZ is illustrated in the next slides for a simple square lattice. An importantpoint to check is that every Brillouin zone has the same volume, namely the volume of a primitivereciprocal lattice. In addition, any BZ can be mapped back to the first zone by using just primitivetranslations, i.e. all the BZ’s are equivalent by translation symetry.



nn

2nd nn

3rd nn

4th nn



Al BZs are equivalent due to the translational part of the crystal space group. Furthermore, therotational symmetry determines the equivalence between different positions within a Brillouinzone. The special symmetry points receive particular names. Although there are different namingconventions, Γ is typically used to designate the origin of the reciprocal cell (~k = ~0).Let’s examine the BZ for the cubic P, I and F Bravais lattices, directly from the KVEC plots of theexcellent Bilbao Crystallographic Server. The plot and the list of special ~k positions is available forall the space groups.

Fm3m

L

zk z

Γ

k x

ky

W

VΣS 1

3X

Λ

∆

KM

Q

1X

U

XS

Im3m

k

k

x

y

zk z

HGNΣΓ ∆

DΛF

P

F1

RN1

3H

F2

Pm3m

zk z

Γ

Λ

k y∆

Z

k x

X

M

Σ

R

T

S

3X


http://www.cryst.ehu.es

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-kv-list?gnum=225




Band diagrams and electronic density of states

InN: a direct gap semiconductor



Crystal geometry

a

c

bα

γβ

Cell parameters: (a, b, c, α, β, γ).

Cell vectors matrix: h = (a, b, c)T .

Cell volume: V = a · (b× c) = b · (c× a) = c · (a× b).

Crystallographic coordinates: ri = hT xi = xia + yib + zic.

{x, y, x} ∈ [0, 1) (main cell).

Metric tensor: G = hT h.

Scalar product: ~x ·~y = xT Gy.



CaF2, fluorite type structure (C1)

Cubic, Fm3m (225), a = 5.4626 Å, Z = 4.Ca 4a (0, 0, 0)F 8c (1/4, 1/4, 1/4)

From the International Tables ofCrystalography:

Wyckoff Sim. Equiv. positions4a m3m (0, 0, 0)8c 43m (1/4, 1/4, 1/4), (1/4, 1/4, 3/4)

Centering vectors(0, 0, 0), (1/2, 1/2, 0), (1/2, 0, 1/2), (0, 1/2, 1/2)

Simple tasks: Cell volume? Density? Neigh-bor distances? Atoms in the main cell? Cellplot?



Some simple crystal structures

Grouped by structure symbols. (alternatives: http://cst-www.nrl.navy.mil/lattice/)

A1 Cu, Cubic, a = 3.609 Å, Fm3m, Z = 4.fcc Cu (4a) (0, 0, 0).A2 Li, Cubic, a = 3.46 Å, Im3m, Z = 2.bcc Li (2a) (0, 0, 0).A3 Be, Hexag., a = 2.2860, c = 3.5843 Å, (hcp ideal: c/a ≈ 1.63) P63/mmc, Z = 2.

Be (2c) (1/3, 2/3, 1/4).A4 C (diamante), Cubic, a = 3.5667 Å, Fd3m, Z = 8.

C (8a) (1/8, 1/8, 1/8).A9 C (grafito), Hexag., a = 2.456, c = 6.696 Å, P63/mmc, Z = 4.

C (2b) (0, 0, 1/4); C (2c) (1/3, 2/3, 1/4).

B1 NaCl, Cubic, a = 5.6402 Å, Fm3m, Z = 4.Na (4a) (0, 0, 0); Cl (4b) (1/2, 1/2, 1/2).

B2 CsCl, Cubic, a = 4.123 Å, Fm3m, Z = 1.Cs (1a) (0, 0, 0); Cl (1b) (1/2, 1/2, 1/2).

B3 β-ZnS (blenda), Cubic, a = 5.4060 Å, F43m, Z = 4.Zn (4a) (0, 0, 0); S (4c) (1/4, 1/4, 1/4).

B4 ZnO (zincita, wurtzita), Hexag., a = 3.2495, c = 5.2069 Å, P63mc, Z = 1.Zn (2b) (1/3, 2/3, z ≈ 0); O (2b) (1/3, 2/3, z ≈ 0.345).


http://cst-www.nrl.navy.mil/lattice/struk/index.html

http://cst-www.nrl.navy.mil/lattice/

http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=225



http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wp-list?gnum=227&gor=2







A1: Cu A2: Li A3: Be A4: diamante A9: grafito

B1: NaCl B2: CsCl B3: ZnS (blenda) B4: ZnO (wurtzita)

Plots made with tessel (http://web.uniovi.es/qcg/tessel/tessel.html) and POVRay(http://www.povray.org).


http://web.uniovi.es/qcg/tessel/tessel.html

http://web.uniovi.es/qcg/tessel/tessel.html

http://www.povray.org

http://www.povray.org


Crystal geometry

Molecular crystals are usually repre-sented including all the molecules thathave at least one atom in the main celland showing the crystal cell.

urea



The crystallographic information file (cif)

# Crystallography Open Database (COD),# http://www.crystallography.net/#data_1011280_chemical_name_systematic ’Calcium chloride’_chemical_name_mineral ’Hydrophilite’_chemical_compound_source ’synthetic’_chemical_formula_structural ’Ca Cl2’_chemical_formula_sum ’Ca Cl2’_publ_section_title;Die Kristallstruktur von Calciumchlorid, Ca Cl2;loop__publ_author_name’van Bever, A K’’Nieuwenkamp, W’

_journal_name_full;Zeitschrift fuer Kristallographie, Kristallgeometrie,Kristallphysik, Kristallchemie (-144,1977);_journal_coden_ASTM ZEKGAX_journal_volume 90_journal_year 1935_journal_page_first 374_journal_page_last 376_cell_length_a 6.24_cell_length_b 6.43_cell_length_c 4.2_cell_angle_alpha 90_cell_angle_beta 90_cell_angle_gamma 90_cell_volume 168.5

_cell_formula_units_Z 2_exptl_crystal_density_meas 2.22_symmetry_space_group_name_H-M ’P n n m’_symmetry_Int_Tables_number 58_symmetry_cell_setting orthorhombicloop__symmetry_equiv_pos_as_xyz’x,y,z’’-x,-y,z’’1/2+x,1/2-y,1/2-z’’1/2-x,1/2+y,1/2-z’’-x,-y,-z’’x,y,-z’’1/2-x,1/2+y,1/2+z’’1/2+x,1/2-y,1/2+z’

loop__atom_type_symbol_atom_type_oxidation_numberCa2+ 2.000Cl1- -1.000

loop__atom_site_label_atom_site_type_symbol_atom_site_symmetry_multiplicity_atom_site_Wyckoff_symbol_atom_site_fract_x_atom_site_fract_y_atom_site_fract_z_atom_site_occupancy_atom_site_attached_hydrogens_atom_site_calc_flagCa1 Ca2+ 2 a 0. 0. 0. 1. 0 dCl1 Cl1- 4 g 0.275(8) 0.325(8) 0. 1. 0 d

_cod_database_code 1011280



The protein databank file (PDB)

HEADER DE NOVO PROTEIN 19-AUG-03 1Q7OTITLE DETERMINATION OF F-MLF-OH PEPTIDE STRUCTURETITLE 2WITH SOLID- STATE MAGIC-ANGLE SPINNING NMRTITLE 3SPECTROSCOPY...DBREF 1Q7O A 1 3 PDB 1Q7O 1Q7O 1 3SEQRES 1 A 3 FME LEU MTY...CRYST1 1.000 1.000 1.000 90.00 90.00 90.00 P 1 1ORIGX1 1.000000 0.000000 0.000000 0.00000ORIGX2 0.000000 1.000000 0.000000 0.00000ORIGX3 0.000000 0.000000 1.000000 0.00000SCALE1 1.000000 0.000000 0.000000 0.00000SCALE2 0.000000 1.000000 0.000000 0.00000SCALE3 0.000000 0.000000 1.000000 0.00000MODEL 1HETATM 1 N FME A 1 -2.621 -3.439 0.640 1.00 0.00 NHETATM 2 CN FME A 1 -2.629 -4.738 0.349 1.00 0.00 CHETATM 3 O1 FME A 1 -1.977 -5.224 -0.578 1.00 0.00 O...HETATM 19 HE3 FME A 1 -3.705 -0.383 -4.831 1.00 0.00 HATOM 20 N LEU A 2 -0.242 -0.725 0.342 1.00 0.00 NATOM 21 CA LEU A 2 0.288 0.451 0.987 1.00 0.00 CATOM 22 C LEU A 2 -0.272 1.663 0.264 1.00 0.00 C...ATOM 38 HD23 LEU A 2 1.007 -1.229 3.026 1.00 0.00 HHETATM 39 N MTY A 3 -0.268 1.582 -1.067 1.00 0.00 NHETATM 40 CA MTY A 3 -0.778 2.668 -1.896 1.00 0.00 C...HETATM 58 HZ MTY A 3 5.192 2.389 -3.177 1.00 0.00 HTER 59 MTY A 3ENDMDL...CONECT 1 2 4 11CONECT 2 1 3CONECT 3 2...END



Crystal Data Bases I

Cambridge Structural Database (CSD) (http://www.ccdc.cam.ac.uk/products/csd/): The main source of crystal

structures for organic and organometallic compounds. Commercial, access by subscription. 596810 structures as of 2012-01-01.

Inorganic Crystal Structure Database (ICSD) (http://www.fiz-karlsruhe.de/icsd.html?&L=0): Natural and

synthetic inorganic compounds. Commercial, access by subscription. 142179 entries (nov 2011). See a fully functional subset athttp://icsd.ill.fr/icsd/ (3592 samples).

CRYSTMET (http://www.tothcanada.com/databases.htm): Metals, including alloys, intermetallics and minerals.Commercial, access by subscription. 139058 entries (Dec 1, 2010).

American Mineralogist Crystal Structure Database (AMCSD) (http://rruff.geo.arizona.edu/AMS/amcsd.php):

Minerals. Free access (financed by NSF). 3156 different minerals (many more entries, as a mineral may appear at differentpressures and temperatures).

Reciprocal Net (http://www.reciprocalnet.org/): Small but well choosen set of quite common molecules and materials.

Data from CSD and other sources. Free access (financed by NSF).

Protein Data Bank (PDB) (http://www.rcsb.org/pdb/): Structures of large biological molecules, including proteins and

nucleic acids. Free access (international support). 78992 structures (Jan 31, 2012). Indispensable portal for anyone working onbiomolecules.Crystallography Open Database (COD) (http://www.crystallography.net/): Voluntary effort to provide an open

alternative to CSD and ICSD. Absolutely free access. 158240 entries (feb 2, 2012) and growing fast. A sister PCOD databasespecializes on theoretically predicted structures (>1000000 in Nov 2009).

Structural Classification of Proteins (SCOP) (http://scop.mrc-lmb.cam.ac.uk/scop/): Further analysis of the

proteins contained in PDB: folds, superfamilies, evolutionary relationship, etc. Free access.

Nucleic Acids Data Bank (NADB) (http://ndbserver.rutgers.edu/): Similar to PDB, but specialized on

oligonucleotides. Free access (international support). 3745 structures (2008-01-17).


http://www.ccdc.cam.ac.uk/products/csd/

http://www.ccdc.cam.ac.uk/products/csd/

http://www.fiz-karlsruhe.de/icsd.html?&L=0

http://www.fiz-karlsruhe.de/icsd.html?&L=0

http://icsd.ill.fr/icsd/

http://www.tothcanada.com/databases.htm

http://www.tothcanada.com/databases.htm

http://rruff.geo.arizona.edu/AMS/amcsd.php

http://rruff.geo.arizona.edu/AMS/amcsd.php

http://www.reciprocalnet.org/

http://www.reciprocalnet.org/

http://www.rcsb.org/pdb/

http://www.rcsb.org/pdb/

http://www.crystallography.net/

http://www.crystallography.net/

http://scop.mrc-lmb.cam.ac.uk/scop/

http://scop.mrc-lmb.cam.ac.uk/scop/

http://ndbserver.rutgers.edu/

http://ndbserver.rutgers.edu/


Crystal Data Bases II

Mineralogy Database (http://webmineral.com/): Good database on minerals and gems maintained by commercial

dealers. Free access.Crystal Lattice Structures (http://cst-www.nrl.navy.mil/lattice/): Very good description of the common crystal

lattice structures of the elements and simple compounds. Free access.

Powder diffraction file (PDF) (http://www.icdd.com/): Largest DB on single phase powder diffraction pattern. Widely used

to identify compounds based on their fingerprint spectra. Commercial.

Database of Macromolecular Movements (http://molmovdb.mbb.yale.edu/molmovdb/): Analysis and prediction of thedynamical behaviour of macromolecules. Movies, morphings, etc. Free access.Do you know any other good structures database? Please, e-mail me the address and details(mailto:[email protected])


http://webmineral.com/

http://webmineral.com/



http://www.icdd.com/

http://www.icdd.com/

http://molmovdb.mbb.yale.edu/molmovdb/

http://molmovdb.mbb.yale.edu/molmovdb/

mailto:[email protected]

mailto:[email protected]


Quantum Espresso: pw+ps GNU code

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

PWSCF, CP, PHONON, FPMD, Wannier, ...Suite of codes for DFT electronic structure calculations and materials modeling. Based on the useof plane waves and pseudopotentials (both, norm-conserving and ultrasoft).

http://www.quantum-espresso.org/

Installation from source is simple:

1 cd2 mkdir -p src/qe/3 cd src/qe/4 wget http://qe-forge.org/frs/download.php/211/espresso-5.0.tar.gz5 cd espresso-5.06 ./configure # f i n d s compi lers , l i b r a r i e s , and creates "make . sys "7 ./make all # creates executables i n ~/ src / qe / espresso−5.0/ b in /8 cat << EOF >> ~/.bashrc9 export QE_HOME=~/pkgs/qe/espresso-5.0/

10 export PATH=${PATH}:${QE_HOME}bin:11 EOF12 source ~/.bashrc




Using QE in zcam2013:

1 . /apps/zcam2013/environ.zcam2013 # Create the environment2 pw input_file # Run quantum espresso3 jmol.sh # Run jmol

QE webpage (Documentation, pseudopotentials, ....)http://www.quantum-espresso.org/

Using QE in zcam:Check the existence of the QE_HOME variable. Edit your bash configuration file otherwise:

1 cat << EOF >> ~/.bashrc2 export QE_HOME=/apps/zcam2012/espresso-5.0/3 export PWGUI=/apps/zcam2012/PWgui-5.0/4 export PATH=$PATH:${QE_HOME}/bin5 export PATH=$PATH:${QE_HOME}/PWgui-5.0/6 export XCRYSDEN_TOPDIR=/apps/zcam2012/xcrysden/7 export XCRYSDEN_SCRATCH=${HOME}/scratch/8 export PATH="$PATH:$XCRYSDEN_TOPDIR:$XCRYSDEN_TOPDIR/scripts:$XCRYSDEN_TOPDIR/util"9 EOF




Get run examples from /apps/zcam2012/practicas-qe/The file qe_si.tgz contains inputs, pseudopotentials, and bash scripts to perform thecalculations:

1 # Unpack the tgz f i l e i n a d i r e c t o r y . . .2 mkdir si3 cd si4 cp -p /apps/zcam2012/practicas_qe/qe_si.tgz5 tar zxvf qe_si.tgz6 # . . . t h i s creates a number o f d i r e c t o r i e s7 #8 # bands dos energy ps pseudo rho tmp9 #

10 # Each d i r e c t o r y conta ins a ∗∗∗ r u n i t . sh∗∗∗ s c r i p t f i l e . Check and11 # modify them to s u i t your purposes .12 #13 # Not ice the d i r e c t o r i e s :14 # ( pseudo ) I t conta ins the pseudopoten t ia ls t h a t w i l l be used i n the15 # t u t o r i a l .16 # ( tmp ) I t w l l be used as scra tch d i r e c t o r y . Clean t h i s d i r e c t o r y once17 # you f i n i s h complete ly your work .



pwSCF: Input scheme (pw.x < input > output) I

The input file for pwSCF is structured into mandatory and optional "&NAMELISTS" and"INPUT_CARDS". Each namelist contains a number of control variables, most of them withappropriate default values, but a few are essential and must always be specified. The complete setof variables is described in the "INPUT_PW.txt" file.

&CONTROL: declare addresses and names for pseudopotentials, and other datafiles, select theamount of I/O (input/output), etc. Important variables: calculation ( task to be performed:’scf’, ’nscf’, ’bands’, ’relax’, ’md’, ’vc-relax’, ’vc-md’).

&SYSTEM: describe the system to be calculated. Enforced variables: ibrav (0–14, Bravais-lattice index). celldm (cell parameters required by the crystalline system); nat (number ofatoms in the unit cell); ntyp (number of atomic types); ecutwfc (kinetic energy cutoff).

&ELECTRONS: control the SCF process and the algorithm to be used. The most importantvariable is diagonalization, the main options being Davidson iterative (’david’) andconjugate gradient like (’cg’).

&IONS: needed when atoms move (structural relaxation and molecular dynamics runs) andignored otherwise.

&CELL: needed when the cell moves (structural relaxation and variable sell MD runs) andignored otherwise.

&PHONON: prepare the input for a PHONON task.



Each input card, on the other hand, corresponds to some vector or matrix.

ATOMIC_SPECIES:<atom> <mass> <pseudopotential> for every different atomic species.

ATOMIC_POSITIONS:<atom> <x> <y> <z> for every different atomic position.

K_POINTS:<number-of-k-points><x> <y> <z> <multiplicity>The k-points can correspond to the integration of iBZ1 or to special points used for depictingband structure.

CELL_PARAMETERS:

OCCUPATIONS:

CLIMBING_IMAGES: special data for the NEB (nudge elastic band) algorithm.

CONSTRAINTS:



pwSCF: Control of the input structure

There are several methods to introduce the crystal structure in pwscf. They are controlled byibrav, celldm, nat, and the CELL_PARAMETERS input card.

ibrav 0 1 2 3 4 5cell free cP (sc) cF (fcc) cI (bcc) hP hR(1)celldm (1) a (1) a (1) a (1) a (1) a (1) a

(3) c/a (4) C = cosαv1 a(1, 0, 0) a(−1, 0, 1)/2 a(1, 1, 1)/2 a(1, 0, 0) a(tx,−ty, tz)v2 a(0, 1, 0) a(0, 1, 1)/2 a(−1, 1, 1)/2 a(−1/2,

√3/2, 0) a(0, 2ty, tz)

v3 a(0, 0, 1) a(−1, 1, 0)/2 a(−1,−1, 1)/2 a(0, 0, c/a) a(−tx,−ty, tz)

• Cell: shows the crystalline system ([c]ubic, [t]etragonal, [o]rthorhombic, [h]exagonal orrhombohedric with hexagonal axes, [m]onoclinic, or [a]=triclininc) and the (P,I,F,A,B,C,R) centering.• hR: the threefold axis is either (1) c, or (2) the < 111 > direction.• hR: tx =

√(1− C)/2; ty =

√(1− C)/6; tz =

√(1 + 2C)/3; u = tz− 2

√2ty; v = tz +

√2ty.

The v1..v3 vectors, except for the a scale, form the R matrix that can be declared in theCELL_PARAMETERS input card.



ibrav -5 6 7 8 9 10cell hR(2) tP (st) tI (bct) oP (so) oC (bco) oF (fco)celldm (1) a (1) a (1) a (1) a (1) a (1) a

(4) C = cosα (3) c/a (3) c/a (2) b/a (2) b/a (2) b/a(3) c/a (3) c/a (3) c/a

v1 a(u, v, v)/√

3 (a, 0, 0) a(1,−1, c/a)/2 (a, 0, 0) (a, b, 0)/2 (a, 0, c)/2v2 a(v, u, v)/

√3 (0, a, 0) a(1, 1, c/a)/2 (0, b, 0) (−a, b, 0)/2 (a, b, 0)/2

v3 a(v, v, u)/√

3 (0, 0, c) a(−1,−1, c/a)/2 (0, 0, c) (0, 0, c) (0, b, c)/2

ibrav 11 12 -12 13 14cell oI (bco) mP(c) (sm-c) mP(b) (sm-b) mC aP (tric)celldm (1) a (1) a (1) a (1) a (1) a

(2) b/a (2) b/a (2) b/a (2) b/a (2) b/a(3) c/a (3) c/a (3) c/a (3) c/a (3) c/a

(4) cos γ (5) cosβ (4) cos γ (4) cosα(5) cosβ(6) cos γ

v1 (a, b, c)/2 (a, 0, 0) (a, 0, 0) (a, 0,−c)/2 (a, 0, 0)v2 (−a, b, c)/2 (b cos γ, b sin γ, 0) (0, b, 0) b(cos γ, sin γ, 0) b(cos γ, sin γ, 0)v3 (−a,−b, c)/2 (0, 0, c) (a sinβ, 0, c cosβ) (a, 0, c)/2 c(cosβ,H1,H2)

H1 = (cosα− cosβ cos γ)/ sin γ; H2 = (1 + 2 cosα cosβ cos γ − cos2 α− cos2 β − cos2 γ)1/2/ sin γ.



pwSCF: Task 1, energy at a single geometry I

Remember to copy the tutorial files and install them in your account:

1 # Unpack the tgz f i l e i n a d i r e c t o r y . . .2 mkdir si3 cd si4 cp -p /apps/zcam2012/practicas_qe/qe-si.tgz5 tar zxvf qe-si.tgz6 # . . . t h i s creates a number o f d i r e c t o r i e s7 #8 # bands dos energy ps pseudo rho tmp9 #

10 # Each d i r e c t o r y conta ins a ∗∗∗ r u n i t . sh∗∗∗ s c r i p t f i l e .11 # Ca lcu la t i on2 :12 # ( 1 ) t o t a l energy o f Si dimond phase at the exper imenta l geometry13 cd energy14 pw.x < si.scf.in > si.scf.out15 # Run the c a l c u l a t i o n severa l t imes exp lo r i ng the e f f e c t o f ecu t f c and the16 # s ize o f the Monkhorst−Pack g r i d . Check ∗∗ r u n i t 2 . sh∗∗ and ∗∗ r u n i t 3 . sh∗∗ .

Other calculations are in the directories: (opt1) geometry optimization, (bands) band diagram, (dos) electronicdensity of states, (eos1) fit the E(V) data to an analytical equation of state, (ph-opt) equilibrium geometry forseveral phases of Si, (ph-eos) get the E(V) data for several phases, (cij) elastic constants, (phon) phononcalculation, ....



Next file is a commented version of the si.scf.in input (remove the comments before doing the calculation).

1 &control2 calculation=’scf’, # type of run3 restart_mode=’from_scratch’, # start from an empty state4 title=’si’, #5 prefix=’si’, # keep restart files with this prefix6 pseudo_dir=’../pseudo/’ # look pseudopotentials in this dir7 outdir=’../tmp/’ # keep restart file in this dir8 /9 &system

10 ibrav=2, # Bravais cell, fcc in this case11 celldm(1)=10.2625, # Cell length (bohr) for the cube12 ntyp=1, # Only one type of atoms in the cell13 nat=2, # ... but two different positions14 ecutwfc=40.0, # Kinetic energy cutoff for the pw’s (Ry)15 ecutrho=720.0, # Cutoff for the electron density16 /17 &electrons18 conv_thr=1.0d-8 # SCF Energy convergence requested19 diagonalization=’cg’20 mixing_mode=’plain’21 mixing_beta=0.722 /23 ATOMIC_SPECIES24 Si 28.086 Si.pz-vbc.UPF25 ATOMIC_POSITIONS crystal26 Si -0.125 -0.125 -0.12527 Si 0.125 0.125 0.12528 K_POINTS automatic # Monkhorst-Pack grid29 5 5 5 1 1 1



Next is a shortened version of the pwscf output:

1 Program PWSCF v.4.3.2 starts on 11Apr2012 at 20:56:362 [......]34 G-vector sticks info5 --------------------6 sticks: dense smooth PW G-vecs: dense smooth PW7 Sum 2587 859 253 87843 16889 274189 bravais-lattice index = 2

10 lattice parameter (alat) = 10.2625 a.u.11 unit-cell volume = 270.2088 (a.u.)^312 number of atoms/cell = 213 number of atomic types = 114 number of electrons = 8.0015 number of Kohn-Sham states= 416 kinetic-energy cutoff = 60.0000 Ry17 charge density cutoff = 720.0000 Ry18 convergence threshold = 1.0E-0819 mixing beta = 0.700020 number of iterations used = 8 plain mixing21 Exchange-correlation = PBE ( 1 4 3 4 0)22 EXX-fraction = 0.002324 celldm(1)= 10.262500 celldm(2)= 0.000000 celldm(3)= 0.00000025 celldm(4)= 0.000000 celldm(5)= 0.000000 celldm(6)= 0.0000002627 crystal axes: (cart. coord. in units of alat)28 a(1) = ( -0.500000 0.000000 0.500000 )29 a(2) = ( 0.000000 0.500000 0.500000 )30 a(3) = ( -0.500000 0.500000 0.000000 )3132 reciprocal axes: (cart. coord. in units 2 pi/alat)33 b(1) = ( -1.000000 -1.000000 1.000000 )34 b(2) = ( 1.000000 1.000000 1.000000 )35 b(3) = ( -1.000000 1.000000 -1.000000 )



3637 PseudoPot. # 1 for Si read from file:38 ../pseudo/si.UPF39 MD5 check sum: a98eb339d92070d1daa3d6fcb438757140 Pseudo is Norm-conserving + core correction, Zval = 4.041 Generated using "atomic" code by A. Dal Corso (Quantum ESPRESSO distribution)42 Using radial grid of 1141 points, 2 beta functions with:43 l(1) = 044 l(2) = 14546 atomic species valence mass pseudopotential47 Si 4.00 28.08550 Si( 1.00)4849 48 Sym.Ops. (with inversion)5051 Cartesian axes5253 site n. atom positions (alat units)54 1 Si tau( 1) = ( 0.1250000 -0.1250000 -0.1250000 )55 2 Si tau( 2) = ( -0.1250000 0.1250000 0.1250000 )5657 number of k points= 1058 cart. coord. in units 2pi/alat59 k( 1) = ( -0.1250000 0.1250000 0.1250000), wk = 0.062500060 k( 2) = ( -0.3750000 0.3750000 -0.1250000), wk = 0.187500061 [......]62 k( 10) = ( 0.3750000 -0.3750000 1.1250000), wk = 0.18750006364 Dense grid: 87843 G-vectors FFT dimensions: ( 64, 64, 64)65 Smooth grid: 16889 G-vectors FFT dimensions: ( 36, 36, 36)6667 [......]6869 Self-consistent Calculation7071 [......]7273 total energy = -19.37231058 Ry



74 Harris-Foulkes estimate = -19.37231059 Ry75 estimated scf accuracy < 0.00000003 Ry7677 iteration # 6 ecut= 60.00 Ry beta=0.7078 Davidson diagonalization with overlap79 ethr = 3.45E-10, avg # of iterations = 2.68081 total cpu time spent up to now is 4.0 secs8283 End of self-consistent calculation8485 k =-0.1250 0.1250 0.1250 ( 2112 PWs) bands (ev):86 -5.5624 4.6045 5.9192 5.919287 [......]88 k = 0.3750-0.3750 1.1250 ( 2125 PWs) bands (ev):89 -2.8414 -0.4395 2.2179 4.32169091 ! total energy = -19.37231059 Ry92 Harris-Foulkes estimate = -19.37231059 Ry93 estimated scf accuracy < 1.2E-09 Ry9495 The total energy is the sum of the following terms:9697 one-electron contribution = 5.02037625 Ry98 hartree contribution = 1.09634466 Ry99 xc contribution = -8.69219473 Ry

100 ewald contribution = -16.79683678 Ry101102 convergence has been achieved in 6 iterations103104 Writing output data file si.save105 PWSCF : 3.99s CPU 4.08s WALL



Convergence of results

The two main variables affecting the convergence of the total energy in a pw+ps calculation are:(Left) the density of the BZ1 grid for integration (a Monkhorst-Pakt grid with n× n× n points with,s = 1, or without, s = 0, displacement from the origin; (Right) the kinetic cutoff energy for planewaves.

Si (A4) calculations with ’ecutwfc=60.0’.

n s Energy (Ry) cpu (s) s Energy (Ry) cpu (s)1 0 −18.12095095 2.75 1 −19.18542081 2.572 0 −19.18494284 2.80 1 −19.36052907 2.783 0 −19.32709942 2.60 1 −19.37116943 3.444 0 −19.35868324 3.66 1 −19.37231059 4.265 0 −19.36776428 3.93 1 −19.37247459 5.826 0 −19.37074242 4.87 1 −19.37250268 7.097 0 −19.37181213 5.80 1 −19.37250820 10.208 0 −19.37222133 7.18 1 −19.37250926 12.449 0 −19.37238614 7.92 1 −19.37250943 16.69

10 0 −19.37245518 9.95 1 −19.37250956 20.57



Si (A4) calculations with a 9× 9× 9(1,1,1) MP grid.

ecutfc Energy (Ry) cpu (s) ecutfc Energy (Ry) cpu (s)4 −19.07313086 4.40 28 −19.36845785 9.005 −19.14720944 4.53 32 −19.37045466 10.016 −19.19658829 4.58 36 −19.37162519 10.157 −19.23655922 4.94 40 −19.37219482 11.508 −19.26848609 4.94 44 −19.37237424 12.409 −19.29432113 4.96 48 −19.37240346 15.72

10 −19.31454844 4.97 52 −19.37241367 16.3111 −19.33030565 4.97 56 −19.37244934 16.0012 −19.34187693 5.08 60 −19.37250943 16.5913 −19.35024821 5.08 64 −19.37257347 21.4814 −19.35610643 5.20 68 −19.37262311 21.6016 −19.36202935 5.44 72 −19.37265408 21.9018 −19.36395765 5.45 76 −19.37266977 29.5320 −19.36484785 7.46 80 −19.37267589 30.0124 −19.36646425 7.66 100 −19.37268760 35.79



pwSCF: Task 2, band diagram for a single geometry I

Band structure is produced by solving the KS equations for a collection of k-vector points that follow a path alongthe border of the iBZ1. Like in the previous task, a calculation=’scf’ determines the energy and computesintegrals and orbitals for further tasks (this data can be already stored in the ../tmp/si/ files). There are twoprograms to produce a band diagram: (1) plotband.x produces a ps and a xmgr (input to xmgrace); (2)band_plot.x creates si_bands.dat to be used in gnuplot. The runit.sh script describes the wholeprocess:

1 cd ../bands2 pw.x < si.scf.in > si.scf.out3 pw.x < si.bands.in > si.bands.out4 bands.x << EOF > si.bands.pp.out5 &inputpp6 outdir=’../tmp/’, prefix = ’si’, filband = ’si_bands.dat’,7 /8 EOF9 plotband.x << EOF > si.plotband.out

10 si_bands.dat11 -6.0 1012 si_bands.xmgr13 si_bands.ps14 6.25515 1.0 6.25516 EOF17 band_plot.x << EOF18 819 6.020 si_bands.dat21 si_bands.out22 EOF

L

zk z

Γ

k x

ky

W

VΣ

3X

Λ

∆

KM

Q

1X

U

XS

Σ1

©bilbao crystallographic serverhttp://www.cryst.ehu.es

The Bilbao crystallographic server (http://www.cryst.ehu.es/cryst/get_kvec.html) provides illustra-tions, names and coordinates for the special points, lines, and faces of the iBZ1 for every crystal space group.The xcrysden code can also be used to create the list of k-points.


http://www.cryst.ehu.es/cryst/get_kvec.html


According to the current calculation, Si (A4 or diamond phase, a = 10.2625 Å) is a semiconductor,with an indirect bandgap of 0.6 eV (expt. value: 1.12 eV at 300 K). Notice that a good calculation ofthe band gap in a semiconductor is still one of the big unsolved problems of ab initio solid structurecalculations.

-15

-10

-5

0

5

10

L Γ X M Γ

ε (e

V)

k-points



pwSCF: Task 3, Density of States (DOS)

After the initial calculation=’scf’ task, than can bealready stored in the ../tmp/si/ files, obtaining andplotting the DOS is done by means of a non-consistentcalculation (calculation=’nscf’). Technically, theprocess requires integrating the orbital energies in theBZ1, using a special point quadrature (Blochl method,occupations=’tetrahedra’). The Fermi energy isdisplaced to zero "by hand".

1 cd ../dos2 pw.x < si.scf.in > si.scf.out3 pw.x < si.dos.in > si.dos.out4 dos.x << EOF > si.dos.pp.out5 &inputpp6 prefix=’si’, outdir=’../tmp/’,7 emin=-9.0, emax=16.0,8 /9 EOF

0.0

0.5

1.0

1.5

2.0

2.5

-15 -10 -5 0 5 10

DO

S

ε (eV)V. Luaña & A. Otero-de-la-Roza () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2013 44 / 86


pwSCF: Task 4, Electron density (ED)

The pp.x code of quantum espresso can be used to determine many different properties in a line,surface, or volume. Some of the properties available include: charge (ED), density of states(DOS), predicted STM images (scan tunnel microscope), electron localization function (ELF), etc.Next we can see the calculation of the electron density in a gausian cube format, and thetopological Bader analysis using the critic2 code [1].

1 cd ../rho2 pw.x < si.scf.in > si.scf.out3 pp.x < si.pp.in > si.pp.out4 critic2g si.incritic si.outcritic

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16

y

x

n

n



pwSCF: Task 5, Geometry optimization (1D)We can determine the E(V) curve of A4 phase by scanning the crystal energy vs. alat:

1 # ! / b in / bash # Scan energy vs . a l a t2 n=9; s=1; ecut=60; alat0=7.0; alat1=12.0; na=51;3 adelta=‘awk ’BEGIN{printf"%.6f",(’$alat1’-’$alat0’)/(’$na’-1)}’‘4 cat << __EOF__ > si-EvsV.dat5 # volume bohr ^36 # energy ry7 # z 18 __EOF__9 for ((i=1; i<=$na; i++)); do

10 alat=‘awk ’BEGIN{printf"%.6f",’$alat0’+(’$i’-1)*’$adelta’}’‘11 pw.x << EOF > pp.out12 &control13 calculation=’scf’, restart_mode=’from_scratch’, title=’si’,14 prefix=’si’, pseudo_dir=’../pseudo/’, outdir=’../tmp/’15 /16 &system17 ibrav=2, celldm(1)=${alat}, nat=2, ntyp=1, ecutwfc=${ecut}, ecutrho=720.0,18 /19 &electrons20 conv_thr = 1d-821 /22 ATOMIC_SPECIES23 Si 28.0855 si.UPF24 ATOMIC_POSITIONS crystal25 Si -0.125 -0.125 -0.12526 Si 0.125 0.125 0.12527 K_POINTS automatic28 ${n} ${n} ${n} ${s} ${s} ${s}29 EOF30 vol=‘awk ’/unit-cell volume/{print $4}’ pp.out‘31 ene=‘awk ’/\!/{print $5}’ pp.out‘32 awk ’BEGIN{printf"%16.9f %16.9f\n",’$vol’,’$ene’}’ >> si-EvsV.dat33 done



-9.70

-9.65

-9.60

-9.55

-9.50

-9.45

150 200 250 300 350 400 450

E (

Rydberg

)

V (bohr3)



Pressure-volume equation of state (EOS)

The asturfit package [2] can be used to fit the E(V) curve and determine the equilibrium propertiesand the pressure dependence of the mechanical properties of the crystal. These calculations aretypically based on the static model, in which the entropy terms and the vibrational zero pointenergy are ignored. Under such conditions

p = −(∂A∂V

)NT≈ −

dEdV, G ≈ H ≈ E + pV, BT = −V

(∂p∂V

)NT≈ V

d2EdV2

. (16)

The asturfit package [2] is an octave library of fitting routines but also implements three standalonetasks: (1) the recommended fitting procedure (asturfit, technically a linear fitting to an averageof Birch-Murnaghan EOS of different degrees); (2) fit many different EOS to a E(V) dataset(allfits); and (3) analysis of possible noise and anomalies in the dataset (checknoise). Atypical run is

1 asturfit << EOF > fit.out2 # volume bohr ^3 # Volume and3 # energy ry # energy are given4 # z 1 # per atom5 85.750000000 -17.0674337306 89.477700000 -17.2788037007 [......]8 EOF

V0 (bohr3) E0 (Rydberg) B0 (GPa) B′0 B′′0 (GPa−1)<BM> 274.1441(49) −9.686337(1) 88.6899(99) 4.2847(23) −0.077179(26)exp. 270.26 97.6 [3]



Comparison of crystal structures

We want to examine the relative stability of different crystal structures of Si. There are several ways of defining astructure in pwscf, involving: (1) the Bravais cell (ibrav), (2) some cell dimensions (celldm), (3) the atomiccoordinates (see ATOMIC_POSITIONS), and (4) the primitive cell vectors (see CELL_PARAMETERS).

name diam. (A4) graphite (A9) fcc (A1) bcc (A2) sc (Ah) βSn (A5)syst cubic hexag cubic cubic cubic tetragsg Fd3m P63/mmc Fm3m Im3m Pm3m I41/amdsgn 227 (2) 194 225 229 221 141 (2)cell a a, c a a a a, c

Wyck (8a)(1/8, 1/8, 1/8) (2b)(0, 0, 1/4) (4a)(0, 0, 0) (2a)(0, 0, 0) (1a)(0, 0, 0) (4a)(0, 3/4, 1/8)(2c)(1/3, 2/3, 1/4)

ibrav 2 0 2 3 1 7nat 2 4 1 1 1 2pos ±(1/8, 1/8, 1/8) (0, 0, 3/4) (0, 0, 0) (0, 0, 0) (0, 0, 0) (0, 3/4, 1/8)

(0, 0, 1/4) ±(1/2, 3/4, 3/8)(1/3, 2/3, 3/4)(2/3, 1/3, 1/4)

Eo (Ry) −9.686340 −9.663764 −9.520039 −9.499227 −9.600225 −9.647577Vo (au) 137.055 122.390 133.823 137.242 107.809 106.759ao (au) 10.3102 7.6407 8.1195 8.1881 4.7594 9.4521co (au) — 9.6831 — — — 4.7200do (au) 4.4651 4.4113 5.7414 5.6282 4.7594 4.7200



Yin and Cohen 26 (1982) 5668.



A typical 1D case:

1 # ! / b in / bash2 n=10; s=1; ecut=60;3 alat0=6.0; alat1=9.5; na=36; # a0=8.1195 bohr4 adelta=‘awk ’BEGIN{printf"%.6f",(’$alat1’-’$alat0’)/(’$na’-1)}’‘5 ###na= ‘awk ’BEGIN{ p r i n t f "%.6 f " , ( ’ $a la t1 ’− ’ $a lat0 ’ ) / ’ $adel ta ’ + 1 ) } ’ ‘67 datafile=’si-fcc-EV.dat’8 cat << EOF > ${datafile}9 # volume bohr ^3

10 # energy ry11 # z 112 EOF1314 for ((i=1; i<=$na; i++)); do15 alat=‘awk ’BEGIN{printf"%.6f",’$alat0’+(’$i’-1)*’$adelta’}’‘16 pw.x << EOF > pp.out17 &control18 restart_mode=’from_scratch’, title=’si_fcc_p’, prefix=’si_fcc_p’,19 pseudo_dir=’../pseudo/’, outdir=’../tmp/’,20 calculation=’relax’,21 etot_conv_thr=1e-6, forc_conv_thr=1e-5,22 /23 &system24 ibrav=2, celldm(1)=${alat}, nat=1,25 ntyp=1, ecutwfc=${ecut}, ecutrho=720.0,26 /27 &electrons28 conv_thr = 1d-829 /30 &ions31 /32 &cell33 /34 ATOMIC_SPECIES35 Si 28.0855 si.UPF



36 ATOMIC_POSITIONS crystal37 Si 0.0 0.0 0.038 K_POINTS automatic39 ${n} ${n} ${n} ${s} ${s} ${s}40 EOF4142 vol=‘awk ’/unit-cell volume/{print $4}’ pp.out‘43 ene=‘awk ’/\!/{print $5}’ pp.out‘44 awk ’BEGIN{printf"%16.9f %16.9f\n",’$vol’,’$ene’}’ >> ${datafile}45 done

# volume bohr ^3# energy ry# z 1

54.000000000 -9.31551628056.745200000 -9.348834600[....]



This is a possible input for the graphite phase:

1 &control2 restart_mode=’from_scratch’, title=’si_graph’, prefix=’si_graph’,3 pseudo_dir=’../pseudo/’, outdir=’../tmp/’,4 calculation=’vc-relax’, etot_conv_thr=1e-6, forc_conv_thr=1e-5,5 /6 &system7 ibrav=0, celldm(1)=1.00, nat=4, ntyp=1, ecutwfc=80.0, ecutrho=800.0,8 london=.true., occupations=’smearing’, smearing=’cold’, degauss=0.03,9 /

10 &electrons11 conv_thr = 1d-8,12 /13 &ions14 /15 &cell16 /17 ATOMIC_SPECIES18 Si 28.0855 si.UPF19 ATOMIC_POSITIONS crystal20 Si 0 0 0.7521 Si 0 0 0.2522 Si 0.333333333 0.666666667 0.7523 Si 0.666666667 0.333333333 0.2524 K_POINTS automatic25 16 16 8 1 1 126 CELL_PARAMETERS hexagonal27 7.732204062 -0.000000000 -0.00000000028 -3.866102031 6.696285145 -0.00000000029 -0.000000000 -0.000000000 9.608769495

The structure is layered, with the layers bonded by van der Waals forces, so we have used Grimme’s adaptation(london). This phase is metallic, so we have used the smearing option for the band occupation. We couldhave used ibrav=4, celldm(1)=a and celldm(3)=c/a. We have preferred, however, using the genericibrav=0 with a simple celldm(1)=1.0 scaling and providing explicitely the CELL_PARAMETERS block.



Surface exploration in a 2D problem:

β-Sn (A5) phase

6.00 6.50 7.00 7.50 8.00 8.50

a (bohr)

3.60

3.80

4.00

4.20

4.40

4.60

4.80

5.00

c (b

ohr)

-9.56

-9.55

-9.54

-9.53

-9.52

-9.51

-9.50

E (

ryd

berg

)

β-Sn (A5) phase

6.00 6.50 7.00 7.50 8.00 8.50

a (bohr)

3.60

3.80

4.00

4.20

4.40

4.60

4.80

5.00

c (b

ohr)



β-Sn (A5) phase

150 200 250 300 350 400

V (au)

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

c/a

-9.56

-9.55

-9.54

-9.53

-9.52

-9.51

-9.50

E (

ryd

berg

)

β-Sn (A5) phase

150 200 250 300 350 400

V (au)

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

c/a



We can add a pressure (in kbar) to the ’vc-relax’ and ’vc-md’ tasks:

1 # ! / b in / bash2 #3 # A5 or beta−Sn phase4 # Sn : a0=11.0199 , c0=6.0120 bohr c / a=0.54565 # Si ( p =0) : a0=7.3549 , c0=4.4903 bohr c / a=0.610567 n=10; s=1; ecut=60;8 p0=0; p1=2000; pdelta=40;9 np=‘awk ’BEGIN{printf "%d", (’$p1’-(’$p0’))/(’$pdelta’)+1}’‘

1011 datafile=’si-A5-EV.dat’1213 cat << EOF > ${datafile}14 # volume bohr ^315 # energy ry16 # z 217 EOF1819 alat=11.0; ca_ratio=0.61;2021 for ((i=1; i<=$np; i++)); do22 pp=‘awk ’BEGIN{printf"%.6f",’$p0’+(’$i’-1)*’$pdelta’}’‘23 echo "pressure: $pp kbar"24 pw.x << EOF > pp.out25 &control26 restart_mode=’from_scratch’, title=’si_A5’, prefix=’si_A5’,27 pseudo_dir=’../pseudo/’, outdir=’../tmp/’,28 tstress=.true., tprnfor=.true.,29 calculation=’vc-relax’,30 etot_conv_thr=1e-6, forc_conv_thr=1e-5,31 /32 &system33 ibrav=7, celldm(1)=${alat}, celldm(3)=${ca_ratio}, nat=1,34 ntyp=1, ecutwfc=${ecut}, ecutrho=720.0,35 /



36 &electrons37 conv_thr = 1d-838 /39 &ions40 /41 &cell42 press=${pp}, press_conv_thr=0.5, cell_factor=2.0,43 /44 ATOMIC_SPECIES45 Si 28.0855 si.UPF46 ATOMIC_POSITIONS crystal47 Si 0.0 0.0 0.048 K_POINTS automatic49 ${n} ${n} ${n} ${s} ${s} ${s}50 EOF5152 vol=‘awk ’/ unit-cell volume/{v=$4} /new unit-cell volume/{v=$5} END{print v}’ pp.out‘53 ene=‘awk ’/\!/{E=$5} END{print E}’ pp.out‘54 awk ’BEGIN{printf "%16.9f%17.9f%19.2f\n",’$vol’,’$ene’,’$pp’}’ >> ${datafile}5556 if ((i==1)); then57 vol0=$vol;58 alat0=$alat;59 else60 alat=‘awk ’BEGIN{printf ’$alat’*(’$vol’/’$vol0’)**(1.0/3.0)}’‘61 fi62 done



Statistical thermodynamics of vibrations I

Let the crystal be formed by N = rM atoms vibrating in 3D, where r is the number of atoms in each of the M unitcells. All the 3N degrees of freedom of movement correspond to vibrations, as the formally infinite crystal has notranslations nor rotations. Asuming harmonicity and independence of modes, each one of the 3N vibrations isassociated to a normal mode of frequency ωi = 2πνi = 2π

√ki/µi, where ki and µi are the effective force

constant and mass, respectively. The vibrational state of the i-th mode is described by a quantum numbervi ∈ {0, 1, 2, 3, ...} and has an energy εi = (vi + 1/2)~ωi. The vibrational state and energy of the whole crystalis described by

|v〉 = |v1, v2, ..., v3N〉 and Ev =3N∑

i=1

(vi + 1/2)~ωi. (17)

No restrictions exist on the collective vibrational quantum numbers, as vibrations behave as bosons with nullspin. In fact, the ground state corresponds to v = 0, all vibrational normal modes inactive, and the vibrationalenergy for this ground state is E0 =

∑3Ni ~ωi/2.

Using a classical, Boltzmann formulation for the canonical ensamble, the vibrational partition function is

Q(3N, V, T) =∑

ve−Ev/kT

=

∞∑v1=0

...

∞∑v3N=0

exp

{−

3N∑i=1

(vi + 1/2)~ωi/kT

}

=

3N∏i=1

∞∑vi=0

exp{−

(vi + 1/2)~ωi

kT

}=

3N∏i=1

e−~ωi/2kT∞∑

vi=0

(e−~ωi/kT

)vi

=3N∏

i=1

e−~ωi/2kT

1− e−~ωi/kT, (18)



Statistical thermodynamics of vibrations II

where k is the Boltzmann constant, T the absolute temperature, and we have used the summation formulaS = 1/(1− r) for an infinite series, 1 + r + r2 + r3 + ..., of a convergent r = exp(−~ω/kT) < 1 geometricprogression.The thermodynamic potential directly related to the canonical partition function is the Helmholtz function:

A(3N, V, T) = −kT ln Q(3N, V, T) =

3N∑i=1

[~ωi

2+ kT ln

(1− e−~ωi/kT

)], (19)

and all the thermodynamic properties can be derived from here. For instance:

E = −(∂ ln Q∂β

)N,V

, p = −(∂A∂V

)N,T

, S = k ln Q + E/T, ... (20)

where β = 1/kT.In a macroscopic crystal 3N is huge (of the order on the Avogadro number), and the summation over normalmodes can be converted to an integration:

A(3N, V, T) =

∫ ∞0

dωg(ω)

[~ωi

2+ kT ln

(1− e−~ωi/kT

)], (21)

where g(ω) is the vibrational density of states (DOS), normalized as∫ ∞0

g(ω)dω = 3N. (22)

The DOS represents the degeneracy of normal modes. Knowledge of DOS is all that is required to determinethe vibrational contribution to the thermodinamical properties of the crystal under the harmonic approach.



Einstein, Debye and QH Debye models I

The Einstein model [Ann. Physik 22 (1907) 180]Used to explain the main aspects of the heat capacity of solids. All crystal atoms vibrate with the samefrequency:

g(ω)dω = 3Nδ(ω − ωE)dω ⇒ Cv = 3Nk(~ωE

2kT

)2 1sinh2(~ωE/2kT)

(23)

, Increases from Cv(T → 0) = 0 to the Dulong-Petit limit Cv(T →∞) = 3Nk. / Decays exponentially asT → 0. / No explanation for thermal expansion.

The Debye model [Ann. Physik 39 (1912) 789]The solid is modelled as an elastic continuous medium:

g(ω)dω =

{9Nω3

Dω2dω if ω ≤ ωD,

0 if ω > ωD,⇒ Cv = 9Nk

(TθD

)3 ∫ θD/T

0

x4exdx(ex − 1)2

(24)

where ωD represents the highest frequency that could be propagated by the crystal, and θD = ~ωD/k is theDebye temperature. , Cv(T → 0) ≈ (T/θD)3 and Cv(T →∞) = 3Nk. / No explanation for thermalexpansion.



Quasiharmonic Debye model [Blanco et al. CPC 158 (2004) 57]The Debye model is used, but the Debye frequency is related to the static bulk modulus, which is different fordifferent volumes:

B(V) = V∂2AdV2≈ V

∂2EdV2

⇒ θD(V) =~k

(6π2V1/2r)1/3f (σ)

√Bm, (25)

where m is the molecular mass per formula unit, σ is the Poisson ratio, and

f (σ) =

{3/[

2(

2(1 + 2σ)

3(1− 2σ)

)3/2

+

(1 + σ

3(1− σ)

)3/2 ]}1/3

. (26)

In an isotropic solid σ = λ/2(λ+ µ), where λ and µ are the coefficients of Lamé. For a cubic solid:c11 ≈ λ+ µ, c12 ≈ λ, and c44 ≈ µ, so σ ≈ c12/(c11 + c12) ≈ c12/2(c12 + c44) ≈ 1/4.The vibrational contribution to the free energy is then

Avib(T, V) = rMkT[

9θ8T

+ 3 ln(1−e−θ/T)− D3(θ/T)

], D3(x) =

3x3

∫ x

0

t3dtet − 1

. (27)

, Thermal behaviour is inferred from E(V) data. , Correct limits for Cv when T → 0 and T →∞. , Thermalexpansion is explained for normal materials at low T. / No intrinsic anharmonicity. / Fusion?



Lattice dynamics: 1D, one mass system

a

b b bf j − 1 j j + 1

b b b

ξj−1 ξj ξj+1

m

Potential energy: U = U0 +12

N∑j=2

f (ξj − ξj−1)2

Kinetic energy: T =12

N∑j=1

mξ2j

Equations of motion (EOM):

mξj = −∂U∂ξj

= f (ξj+1 − 2ξj + ξj−1)

Born-von Karman (BvK) boundary conditions:ξj(t) = ξj+N(t)

Solution ansatz: ξj(t) = Aei(ωt+jka). Using ξj(t) in theEOM: −mω2 = f (eika − 2 + e−ika) = ... = −4f sin2 ka

2

Dispersion equation: ω =

√4fm

∣∣∣∣sinka2

∣∣∣∣

0 BZ1 π/a−π/a

ωmax

k

ωω



Example: 1D, two masses system I

Hamiltonian:

a

b bb bf 2j − 2 2j − 1 2j 2j + 1

b bb bξ2j−2 ξ2j−1 ξ2j ξ2j+1

m M

H = T + U =12

N∑j=1

(mξ22j + Mξ2

2j−1) + U0 +12

N∑j=1

f [(ξ2j − ξ2j−1)2

+ (ξ2j+1 − ξ2j)2]

Equations of motion (EOM):mξ2j = f (ξ2j+1+ξ2j−1−2ξ2j), Mξ2j+1 = f (ξ2j+2+ξ2j−2ξ2j+1).

Periodic conditions: ξj+2N(t) = ξj(t).

Solution ansatz: ξ2j(t) = Aei(ωt+2jka) and ξ2j+1(t) = Bei(ωt+(2j+1)ka).

Dynamical equation:[

2f − mω2 −2f cos ka−2f cos ka 2f − Mω2

] [AB

]= 0, k ∈ [−π/2a, π/2a]

Eigenvalues: ω4 −2fµω

2+

4f 2 sin2 kamM

= 0 ⇒ ω2± =

fµ

1±

√1−

4nM sin2 ka(m + M)2

where µ = mM/(m + M) is the reduced mass of the A and B particles.



Example: 1D, two masses system II

0 BZ1 π/2a−π/2a

√

2f/µ

√

2f/m

√

2f/M

k

ω

acoustic

optical

The acoustic branch, ω−(k), corresponds to the inphase motion of the A and B particles. In the opticalbranch, ω+(k), however, both move in oposition.

In a general 3D crystal there are 3 acoustic branches,that represent the elastic behaviour in the neighborboodof the Γ point (q = (0, 0, 0)). There are also 3r − 3optical branches, where r is the number of atoms in theprimitive unit cell.Density of States:

g(ω)dω = 2g(k)dk =2g(k)dω(dω/dk)

van Hove singularities: Critical points where dω/dk =0.



Lattice dynamics: general 3D crystal

Dynamical equations (eigensystem) To be solved for q ∈ BZ1:∑κ′β

(Dκα,κ′β(q)− ω2mqδκ,κ′δα,β)γmq(κ′β) = 0. (28)

κ, κ′: crystal primitive unit cells; α, β: atoms in a cell; q: reciprocal space vector; m: branch index.Dynamical matrix: Obtained using Density Functional Perturbation Theory (DFPT) or finitedifferences:

Dκα,κ′β(q) =1

√MκMκ′

cells∑b

∂2E∂u0κα∂ub

κ′β

eiqRb(29)

E: energy of a unit cell (electronic and ionic); Rb: origin of the b cell; uaκα(t): instantaneous

displacement (alpha: x, y, z) from the equilibrium position of the κ atom of cell a.Eigenvector displacement:

uaκα(t) =

1√

Mκγmq(κα)ei(qRa−ωmqt). (30)



P. Giannozzi et al., Phys. Rev. B 43 (1991) 7231.V. Luaña & A. Otero-de-la-Roza () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2013 69 / 86


P. Giannozzi et al., Phys. Rev. B 43 (1991) 7231.V. Luaña & A. Otero-de-la-Roza () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2013 70 / 86


Phonon calculation I

This is the sequence of steps required to perform a phonon calculation:

1 # Remove QE comment l i n e s2 pw.x < si.scf.in > si.scf.out # E l e c t r o n i c ground s ta te34 # Ca l cu l a t i on o f the fo rce constants on a q g r i d5 ph.x << EOF > si.ph.out6 phonon calculation # t i t l e l i n e ( f i l e : " s i . ph . i n " )7 &inputph8 outdir="../tmp/", # scra tch d i r e c t o r y9 prefix="si", # same name used on the unperturbed system

10 tr2_ph = 1d-14, # th resho ld f o r s e l f cons is tency11 ldisp = .true., # i f t r ue c a l c u l a t e phonons i n a g r i d o f q−po in t s12 nq1=4, nq2=4, nq3=4, # Monkhorst−Pack g r i d o f q−po in t s13 amass(1)=28.0855, # masses f o r the ntyp types o f atoms14 fildyn=’si.dyn’, # f i l e to w r i t e the dynamical mat r i x ( def : matdyn )15 /16 EOF1718 # Diagonal ize the dynamical mat r i x and w r i t e eigenvalues i n xcrysden format19 dynmat.x << EOF > si.dynmat.out20 &input # ( F i l e : " s i . dynmat . i n " ) . See PH/ dynmat . f9021 fildyn=’si.dyn1’, # dynamical mat r i x f i l e ( def : " matdyn " )22 asr=’simple’, # Type of acous t i c sum r u l e imposed23 amass(1)=28.0855, # Masses f o r the ntyp types o f atoms24 q(1)=0.0,q(2)=0.0,q(3)=0.0, # Ca lcu la te LO modes along q ( Def : q = (0 ,0 ,0 ) )25 /26 EOF2728 # Ca l cu l a t i on o f the r e a l space IFCs ( I n te ra tom ic Force Const )29 q2r.x << EOF > si.q2r.out30 &input # ( F i l e : " s i . q2r . i n " ) . See PH/ q2r . f9031 fildyn=’si.dyn’, # Inpu t f i l e name . Also on phonon (PH) i npu t .32 # Look f o r the " s i . dym { 0 . . 8 } " f i l e s



Phonon calculation II

33 zasr=’simple’, # Acoust ic sum r u l e used f o r the Born e f f e c t i v e charges34 flfrc=’si.fc’, # Output f i l e f o r the IFC i n r e a l space35 /36 EOF3738 # Ca l cu l a t i on o f the phonon DOS or d i spe rs ion curves39 matdyn.x << EOF > si.matdyn.out40 &input # ( F i l e : " s i . matdyn . i n " ) . See PH/ matdyn . f9041 asr=’simple’, # Acoust ic Sum Rules ( def : " no " ) .42 dos=.true., # " yes " ( c a l c u l a t e DOS) ; " no " ( do d ispe rs ion bands )43 amass(1)=28.0855, # Masses of the ntyp types o f atoms44 flfrc=’si.fc’, # q2r f i l e con ta in ing fo rce constants ( requ i red )45 fldos=’si.phdos’, # output f i l e f o r the phonon DOS ( def : " matdyn . dos " )46 # Uni ts : s ta tes versus omega i n cm^{−1}47 nk1=50,nk2=50,nk3=50, # uni form q−po in t g r i d f o r DOS c a l c u l a t i o n ( i f dos )48 /49 EOF



0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 100 200 300 400 500 600

φ-D

OS

ω (cm-1)

Si (p=0, diamond phase)



Elastic equation of state

��

��

��

��

��

� ��

� ��

� ��

��

� ��

For small deformations of the crystal cell around an equilibrium config-uration:

2φ = 2E − E0

V0=∑ij,kl

cijklεijεkl =∑ij,kl

sijklτijτkl. (31)

Using symmetric deformation components:

εij =12

[∂δxi

∂xj+∂δxj

∂xi

], δx = x− x0, x ∈ {a, b, c}. (32)

Strain-stress linear relationships:

τij =∂φ

∂εij, τij =

∑kl

cijklεkl, εij =∑

kl

sijklτkl. (33)

Voigt notation:τ11 τ22 τ33 τ23 τ31 τ12 ε11 ε22 ε33 2ε23 2ε31 2ε12τ1 τ2 τ3 τ4 τ5 τ6 ε1 ε2 ε3 ε4 ε5 ε6

Elastic constants and compliances:

cij =

(∂τi

∂εj

)ε′,0

, sij =

(∂εi

∂τj

)τ ′,0

, c = s−1. (34)



Deformations of the cubic cell

We need to calculate (E, V) for appropriate finite deformations of the unit cell: R→ R′ = R(1 + ε). In the caseof cubic systems:

R′ =

a 0 00 a 00 0 a

1 + ε1 2ε6 2ε52ε6 1 + ε2 2ε42ε5 2ε4 1 + ε3

(35)

V = V0{1 + ε1 + ε2 + ε3 + ε1ε2 + ε2ε3 + ε3ε1 + ε1ε2ε3 (36)

−4ε24 − 4ε2

5 − 4ε26 − 4ε1ε

24 − 4ε2ε

25 − 4ε3ε

26 + 16ε4ε5ε6}

Φ(ε1)⇒ c11 Φ(ε1, ε2)⇒ c12, c11 Φ(ε4)⇒ c44



Elastic moduli and the crystal systems

See Bhagavantam [4] or Nye [5]

Cubic Hexagonal Orthorhombicc11 c12 c12 0 0 0

c11 c12 0 0 0c11 0 0 0

c44 0 0c44 0

c44

c11 c12 c13 0 0 0c11 c13 0 0 0

c33 0 0 0c44 0 0

c44 0(c11−c12)/2

c11 c12 c13 0 0 0c22 c23 0 0 0

c33 0 0 0c44 0 0

c55 0c66

For a isotropic (cubic) medium c44 = (c11−c12)/2 (Cauchy relationship).

Trigonal (3, 3) Trigonal (3m, 32, 3m) Tetrag. (4, 4, 4|m)c11 c12 c13 c14 c15 0

c11 c13 −c14 −c15 0c33 0 0 0

c44 0 −c15c44 c14

c‡66

c11 c12 c13 c14 0 0c11 c13 −c14 0 0

c33 0 0 0c44 0 0

c44 c14

c‡66

c11 c12 c13 0 0 c16c11 c13 0 0 −c16

c33 0 0 0c44 0 0

c44 0c66

c‡66 = (c11−c12)/2.

Tetrag. (4mm, 42m, 422, 4|mmm) Monoclinic Triclinicc11 c12 c13 0 0 0

c11 c13 0 0 0c33 0 0 0

c44 0 0c44 0

c66

c11 c12 c13 0 0 c16c22 c23 0 0 c26

c33 0 0 c36c44 c45 0

c55 0c66

c11 c12 c13 c14 c15 c16c22 c23 c24 c25 c26

c33 c34 c35 c36c44 c45 c46

c55 c56c66



Catti’s method I

In the method pioneered by M. Catti [6, 7] deformations involve simultaneous activation of several components ofε such that the crystal symmetry is maximally conserved.For instance, the deformation εT[η, 0, 0, 0, 0, 0] in a cubic crystal has an elastic energy 2φ = εTcε = c11η

2 and itis clear that c11 can be obtained as (∂2φ/∂η2). This particular deformation transforms the unit cell from a cubeto a square-based prism of dimensions [a(1+η), a, a, 90, 90, 90], thus lowering the symmetry from cubic totetragonal. The symmetry reduction can have the added consequence that some of the atoms within the cell gaindegrees of freedom and their position within the cell should be reoptimized for each new value of η. Failing totake into account this inner strain can produce a significant impact on the calculated value of the elastic moduli.Example: Cubic CaF2 [fluorite; Fm3m (Num. 225); Ca (4a) (0, 0, 0); F (8c) (1/4, 1/4, 1/4)].

Strain Space G. Inner strain 2φ Cellη[1, 1, 0, 0, 0, 0] 4|mmm No 2(c11+c12)η

2 [a(1+η), a(1+η), a, 90, 90, 90]

η[1, 1,−2, 0, 0, 0] 4|mmm No 6(c11−c12)η2 [a(1+η), a(1+η), a(1−2η), 90, 90, 90]

η[1, 1, 1, 0, 0, 0] Fm3m No 3(c11+2c12)η2 [a(1+η), a(1+η), a(1+η), 90, 90, 90]

η[0, 0, 0, 1, 1, 1] R3m F (x, x, x) 3c44η2 [a, a, a, α, α, α], 2η = cos(90−α)

x ≈ 1/4

Ca and F remain on a symmetry fixed position for the first three proposed deformations, and F has only onedegree of freedom in the fourth one. [η, η, η, 0, 0, 0] is a breathing dilatation of the cubic unit cell and,consequently, the elastic energy is proportional to the bulk modulus. The c44 modulus could also be obtainedthrough a [0, 0, 0, 0, 0, η] deformation, but this would reduce the cell to moclinic symmetry, instead ofrhombohedral as the proposed [0, 0, 0, η, η, η].Designing a set of sensible deformations can be a creative task with a large influence on the computationaleffort. The best routes will elude creating inner strain and lowering the unit cell symmetry as much as possible.The bulk modulus, B, is related to the elastic moduli. In a cubic system, for instance, B = (c11 + 2c12)/3.The importance of the elastic characterization. Bulk modulus and elastic moduli are very importantproperties of the crystal beyond their significant role on establishing the stability of the crystal phase and helping



Catti’s method II

to predict the possible deformation of the unit cell shape in a potential phase transition. Elastic data is required incrystallography to calculate the thermal diffuse scattering correction to the Bragg diffraction intensities. Theinterpretation of the observed seismic waves, that constitute our best probe on Earth innards, depend on theassumed elastic properties of the solid phases present in the earth mantle. Material scientist are engaged in along quest for superhard materials that maintain their hardness on a high pressure, high temperature, or highradiation working ambient.



Elastic constants of SiNext scripts are designed to calculate the elastic constants of the diamond phase of Si. There are two mainscripts: gen.m, that prepares the input to pwscf for all the calculations, and extract.m, that analyzes the pwscfoutputs and produces the final results. Both scripts are written in octave, but extract.m calls the shell andcombines the use of grep, tail, and awk. The whole calculation is done as

1 # ! / b in / bash −f2 gen.m3 for i in {1..11}_{1..3}/ ; do4 cd $i5 echo $i6 pw.x < ${i%/}.scf.in > ${i%/}.scf.out7 cd ..8 done9 extract.m

The elastic constants are defined in terms of the conventional cubic cell, so we have performed all ourcalculations in this setting rather than using the rhombohedral primitive cell as in previous tasks.The calculation is based on three combined deformations: [111000], [001000], and [000111]. Several values ofthe deformations are performed on each case. Each point involves calculating the stress tensor(tstress=.true.) and relaxing the position of the ions (calculation=’relax’ and &ions/ namelist). Thetheoretical result reproduces well the experiments and describes a clearly stable phase.

[GPa] c11 c12 c44 B ε range pointscalc. 154.4 56.8 75.9 89.3 ±0.013 4calc. 153.9 57.4 75.6 89.6 ±0.050 11expt. 165,6 63.9 79.5 97.8

We can observe that a particular least squares fitting is performed to determine the cij from the strees (τ ) andstrain (ε): pinv(strain)*stress, where prinv() is the Moore-Penrose pseudo inverse. As the argument toprinv() is not a square matrix, this function returns the least squares coefficients.



1 # ! / usr / b in / octave −q2 # c i j determined by s t r a i n−s t ress r e l a t i o n s , l e a s t squares f i t .3 # The deformat ions correspond to :4 # ( 1 ) [100000] , ( 2 ) [000111] , ( 3 ) [111000]56 format long78 # the ca l cu la ted e q u i l i b r i u m geometry ( p r i m i t i v e c e l l )9 r_prim = [

10 -0.502258063 0.000000000 0.50225806311 0.000000000 0.502258063 0.50225806312 -0.502258063 0.502258063 0.00000000013 ] * 10.26250000;1415 # c a l c u l a t e the e q u i l i b r i u m convent iona l c e l l l eng th16 a_eq = (abs(det(r_prim))*4)^(1/3);17 r_eq = eye(3) * a_eq;1819 # atomic p o s i t i o n s i n the convent iona l c e l l20 x_eq = [21 0.125 0.125 0.12522 0.125 0.625 0.62523 0.625 0.125 0.62524 0.625 0.625 0.12525 0.875 0.875 0.87526 0.875 0.375 0.37527 0.375 0.875 0.37528 0.375 0.375 0.87529 ];3031 # the f i n i t e deformat ions , s t r a i n range32 #erange = [−0.013 −0.007 0.007 0 . 0 1 3 ] ;33 erange = [-0.05 -0.04 -0.03 -0.02 -0.01 0.001 0.01 0.02 0.03 0.04 0.05];3435 for i = 1:length(erange)36 for j = 1:3



37 s = sprintf("%d_%d",i,j);38 system(sprintf("mkdir -p %s",s));39 system(sprintf("cp -f si.UPF %s",s));4041 # def ine the s t r a i n tensor42 e = zeros(1,6);43 if (j == 1)44 e(1) = erange(i);45 elseif (j == 2)46 e(4) = e(5) = e(6) = erange(i);47 elseif (j == 3)48 e(1) = e(2) = e(3) = erange(i);49 endif5051 # pack the 6−vec to r i n the s t r a i n mat r i x52 emat = zeros(3,3);53 for k = 1:354 emat(k,k) = 1 + e(k);55 endfor56 emat(2,3) = emat(3,2) = e(4)/2;57 emat(1,3) = emat(3,1) = e(5)/2;58 emat(1,2) = emat(2,1) = e(6)/2;5960 # obta in the deformed l a t t i c e vec to rs61 r = r_eq * emat;6263 # w r i t e the i npu t64 fid = fopen(sprintf("%s/%s.scf.in",s,s),"w");65 fprintf(fid,"&control\n");66 fprintf(fid," title=’%s’,\n",s);67 fprintf(fid," prefix=’%s’,\n",s);68 fprintf(fid," pseudo_dir=’.’,\n");69 fprintf(fid," tstress=.true.,\n",s);70 fprintf(fid," calculation=’relax’,\n",s);71 fprintf(fid,"/\n");72 fprintf(fid,"&system\n");73 fprintf(fid," ibrav=0,\n");



74 fprintf(fid," celldm(1)=1.0,\n");75 fprintf(fid," nat=8,\n");76 fprintf(fid," ntyp=1,\n");77 fprintf(fid," ecutwfc=60.0,\n");78 fprintf(fid," ecutrho=720.0,\n");79 fprintf(fid,"/\n");80 fprintf(fid,"&electrons\n");81 fprintf(fid," conv_thr = 1d-8\n");82 fprintf(fid,"/\n");83 fprintf(fid,"&ions\n");84 fprintf(fid,"/\n");85 fprintf(fid,"ATOMIC_SPECIES\n");86 fprintf(fid," Si 28.0855 si.UPF\n");87 fprintf(fid,"\n");88 fprintf(fid,"ATOMIC_POSITIONS crystal\n");89 for k = 1:size(x_eq,1)90 fprintf(fid,"Si %.10f %.10f %.10f\n",x_eq(k,:));91 endfor92 fprintf(fid,"\n");93 fprintf(fid,"K_POINTS automatic\n");94 fprintf(fid,"4 4 4 1 1 1\n");95 fprintf(fid,"\n");96 fprintf(fid,"CELL_PARAMETERS cubic\n");97 for k = 1:398 fprintf(fid,"%.10f %.10f %.10f\n",r(k,:));99 endfor

100 fclose(fid);101 endfor102 endfor



1 # ! / usr / b in / octave −q2 format long34 # the ca l cu la ted e q u i l i b r i u m geometry ( p r i m i t i v e c e l l )5 r_prim = [6 -0.502258063 0.000000000 0.5022580637 0.000000000 0.502258063 0.5022580638 -0.502258063 0.502258063 0.0000000009 ] * 10.26250000;

10 # c a l c u l a t e the e q u i l i b r i u m convent iona l c e l l l eng th11 a_eq = (abs(det(r_prim))*4)^(1/3);12 r_eq = eye(3) * a_eq;13 # the f i n i t e deformat ions14 #erange = [−0.013 −0.007 0.007 0 . 0 1 3 ] ;15 erange = [-0.05 -0.04 -0.03 -0.02 -0.01 0.001 0.01 0.02 0.03 0.04 0.05];1617 nj = 3;18 tmat = smat = zeros(length(erange)*nj,6);19 n = 0;20 for i = 1:length(erange)21 for j = 1:nj22 n += 1;23 s = sprintf("%d_%d",i,j);24 ss = sprintf("%s/%s.scf.out", s, s);2526 ## def ine the s t r a i n tensor27 e = zeros(1,6);28 if (j == 1)29 e(1) = erange(i);30 elseif (j == 2)31 e(4) = e(5) = e(6) = erange(i);32 elseif (j == 3)33 e(1) = e(2) = e(3) = erange(i);34 endif3536 # pack the 6−vec to r i n the s t r a i n mat r i x



37 emat = zeros(3,3);38 for k = 1:339 emat(k,k) = 1 + e(k);40 endfor41 emat(2,3) = emat(3,2) = e(4)/2;42 emat(1,3) = emat(3,1) = e(5)/2;43 emat(1,2) = emat(2,1) = e(6)/2;4445 # obta in the deformed l a t t i c e vec to rs46 r = r_eq * emat;4748 # get the s t ress tensor49 order = sprintf("grep ’total stress’ %s -A 3 | tail -n 3 | awk ’{print $4, $5, $6}’",ss);50 [stat,out] = system(order);51 t = -str2num(out) / 10;52 tmat(n,1:3) = diag(t); tmat(n,4) = t(2,3);53 tmat(n,5) = t(1,3); tmat(n,6) = t(1,2);54 smat(n,:) = e;55 endfor56 endfor5758 # Ca lcu la te the C i j mat r i x59 cij = pinv(smat) * tmat;60 cij(2,2) = cij(3,3) = cij(1,1);61 cij(2,1) = cij(3,1) = cij(3,2) = cij(2,3) = cij(1,2) = cij(1,3);62 cij(4,4) = cij(5,5) = cij(6,6) = sum(cij(4,4:6));63 cij(4,5) = cij(4,6) = cij(5,6) = cij(5,4) = cij(6,4) = cij(6,5) = 0;64 cij(1:3,4:6) = cij(4:6,1:3) = 0;65 printf("C11 (GPa) = %.2f\n",cij(1,1));66 printf("C12 (GPa) = %.2f\n",cij(1,2));67 printf("C44 (GPa) = %.2f\n",cij(4,4));6869 # check the s t a b i l i t y cond i t i ons70 printf("Eigenvalues: %.2f %.2f %.2f %.2f %.2f %.2f\n",eig(cij));



Electronic structure textbooks

R. M. Martin, "Electronic Structure: Basic theory and practical methods" (Cam-bridge, 2004).L. N. Kantorovich, "Quantum Theory of the Solid State: An Introduction" (Kluwer,2004).E. Kaxiras, "Atomic and Electronic Structure of Solids" (Cambridge, 2003).M. Marder, "Condensed Matter Physics" (Wiley-Interscience, 2000).

[1] A. Otero-de-la Roza, V. Luaña, CRITIC2: topological analysis of scalar fields on crystals, Comput. Phys. Commun.(to be sub-mitted).

[2] A. Otero-de-la Roza, V. Luaña, GIBBS2: A new version of the quasi-harmonic model code. I. Robust treatment of the static data,Comput. Phys. Commun. 182 (2011) 1708–1720.

[3] M. A. Hopcroft, W. D. Nix, T. W. Kenny, What is the young’s modulus of silicon?, IEEE J. Microelectromechanical Systems 19(2010) 229–238.

[4] S. Bhagavantam, Crystal symmetry and physical properties, Academic, New York, 1966.

[5] J. F. Nye, Physical Properties of Crystals, Oxford UP, Oxford, UK, 1985, republication of the 1957 classic.

[6] M. Catti, Calculation of elastic constants by the method of crystal static deformation, Acta Cryst. A 41 (1985) 494–500.

[7] M. Catti, Crystal elasticity and inner strain: A computational model, Acta Cryst. A 45 (1989) 20–25.


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