Quantum Espresso

Electronic structure of solids: quantum espresso

Víctor Luaña (†) & Alberto Otero-de-la-Roza (‡) & Daniel Menéndez-Crespo (†) (†) Departamento de Química Física y Analítica, Universidad de Oviedo (‡) National Institute of NanoTechnology, Edmonton, Alberta, Canada

European school on Theoretical Solid State Chemistry ZCAM, Zaragoza, May 12–16, 2014

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 1 / 104 Electronic structure Crystal

Part I

crystallography and electronic structure calculations (th-1b)

Hartree-Fock and Kohn-Sham equations

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

HˆeΨ({xi}; {Rα}) = EΨ ⇒ Fˆψλ = λψλ (1) N Z Z ρ(r0dr0) n 1 2 X α ˆ o − ∇r + + +VX ψλ(x; {Rα}) = λ({Rα})ψλ(x; {Rα}) (2) 2 r 3 |r − r0| α iα R where occ. Z ∗ 0 0 X ψi (r )ψλ(r ) 0 VˆX ψλ = − dr ψi(r) (3) 3 |r − r0| i R

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

SCF: both, HF and KS equations must be solved iteratively until convergence (self-consistency). Kohn-Sham Orbital ansatz to solve the density functional non-relativistic electronic ground state:

HˆeΨ0({xi}; {Rα}) = EΨ0 ⇒ Fˆψλ = λψλ (4) N Z Z ρ(r0)dr0 n 1 2 X α ˆ o − ∇r + + +Vxc ψλ(x; {Rα}) = λ({Rα})ψλ (5) 2 r 3 |r − r0| α iα R

where Vˆxc is the unknown exchange and correlation functional.

Quantum Chemistry vs Solid State formalismsI

The Quantum Chemistry approach

Objective: 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 determi- nants. HF orbitals fulfill Koopmans and Brillouin theorems. HF closed or open shell (main method to produce orbitals): minimize the energy of an electronic configuration under orthonormality of the spinorbitals. Other methods to produce orbitals: UHF (unrestricted HF), GVB (generalized Valence Bond), 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 the exact solution is the correlation problem. Correlation techniques: Möller-Plesset (Many Body Perturbation Theory: MP2, MP3, ...), Configuration Interaction (CIS, CISD, CISDT, ...), Cou- pled Cluster (CCSD, CCSDT, ...), ... Full CI. lim calculation = exact →CBS,→FCI

Size scaling (N: spinorbitals, electrons) method lin.† HF, KS HF, KS‡ KS∗ MP2 MP3 CCSD CCSD(T)Q FCI O NN3 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 source abinit pw+ps www.abinit.org GNU yes CPMD pw+ps www.cpmd.org 0 yes crystal09 CGTO www.crystal.unito.it 1000 Eur. no elk FPLAPW elk.sourceforge.org GNU yes gpaw! PAW wiki.fysik.dtu.dk/gpaw GNU yes QE (pwscf) pw+ps www.quantum-espresso.org GNU yes siesta Loc+ps www.icmab.es/siesta 0 yes vasp pw+ps cms.mpi.univie.ac.at/marsweb yes yes wien2k 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, ...

KS equations on a crystal (quantum espresso version)

n 2 X α o −∇r + Vsf + VH(r)+ Vxc(r) ψnk(r) = n(k)ψnk(r) (6) α

2 −∇r : kinetic energy, Rydberg units are used throught. α Vsf : non-local pseudopotential. Either norm-conserving, ultrasoft, or PAW (pro- jected augmented wave) pseudopotentials can be used. Z 0 0 ∂EH[ρ] ρ(r dr VH(r)= = 0 : Coulomb potential or Hartree term. ∂ρ 3 |r − r | R ∂E [ρ] V (r)= xc : exchange and correlation potential. xc ∂ρ ikr Bloch states: ψnk(r)= e unk(r), where ψnk(r) is periodical over reciprocal space and unk(r) is periodical over real space. X 2 X 2 ρ(r) = fnk |ψnk| = fnk |unk| , with fi ∈ {0, 1} (a typical insulator) or fi = nk nk 1/(1 + e−i/kT ) (Mermin functional, used on metals). k ∈ BZ1. Running special points and directions in iBZ1 (irreducible ﬁrst Brillouin zone) is the basis 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: X ~ ψn~q(~r) = cn~q(G)φ~q+~G(~r). (7) n~q

By minimizing the energy with cn~q(G~ ) as the variational parameters the KS equations transform into X ~ ~ ∀~G H~q+~G,~q+~G0 − n~qS~q+~G,~q+~G0 cn~q(G) = 0, (8) ~G0 that must be solved for every ~q ∈ BZ1. In the above equation: 1 2 D E H ~ ~ 0 = φ ~ (~r) − ∇ + V φ ~ 0 (~r) , S ~ ~ 0 = φ ~ φ ~ 0 , (9) ~q+G,~q+G ~q+G 2 ~q+G ~q+G,~q+G ~q+G ~q+G and the KS equation takes the form of a generalized eigen equation HC = SCE that must besolved for every ~q. By sampling ~q-points along special directions on the BZ1 we get the band diagrams: n~q. The total energy (E), Density of States (DOS, g(E)), and the Fermi surface in the case of a metal (F(~q)), are produced by integration for all ~q ∈ BZ1. Monkhorst-Pack special points methods is popular for doing this integration.

Planewaves (3D) and the movement of a free electronI

2 ˆ The hamiltonian is ˆh = ˆp /2m, ~p = −i~∇~ , and the solutions are planewaves (PW) again: 2 2 ~ k |~ki = ψ (~r ) = V−1/2eik·~r, = ~ , ~r,~k ∈ 3. (10) ~k ~k 2m R with V being the volume of the normalization box. Some properties ˆ The PWs are eigenfunctions of the momentum operator: ~p |~ki = ~~k |~ki. The particle velocity is proportional to ~k, the wavevector : ~v = ~~k/m. The |~ki and |−~ki 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 3 3 cell a and an arbitrary primitive translation~t = a n, n ∈ Z . For any ~r ∈ R : ˜ ˜ ~ i~k·~t ψ~k(~r + t ) = ψ~k(~r ) =⇒ e = 1. (11)

If ~k is a vector in the reciprocal cell, ~k = a?k = 2πa?h, the periodicity condition is ˜ ˜ i~k·~t i2π(hTn) 3 1 = e = e =⇒ h1nx + h2ny + h3nz ∈ Z for all n ∈ Z . (12)

Planewaves (3D) and the movement of a free electronII

The wavevector of the periodic PWs is then ~ ? ~ ? ? ? ? 3 k = 2π(h1~a + h2b + h3~c ) = 2πa h = a k, (h1, h2, h3) ∈ Z . (13) ˜ ˜

This discrete mesh of allowed ~k-points is distributed uniformly in the reciprocal space. Each ~k point can be associated with a small parallelepiped of volume 3 ? 3 v~k = (2π) V = (2π) /V, where V is the volume of the main cell. It is implicitely assumed that a primitive cell is used to describe the reciprocal space. The periodic PWs form an orthonormal set Z ~ ~ 0 1 i(~k−~k 0)~r hk|k i = e d~r = δ~k,~k 0 , (14) V V where the integral is done on a unit cell. The set is also complete, which means that any 3D function can be expanded as Z X i~k·~r −i~k·~r f (~r ) = f~ke ⇐⇒ f~k = f (~r )e d~r. (15) V ~k

Brillouin zonesI

The first Brillouin zone (BZ-1) is the common name for the Wigner-Seitz primitive cell of the reciprocal or ~k-space lattice. Whereas primitive and centered cells are both used for the direct lattice, 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 the plane which is perpendicular to the line between the two points and passes through the bisector of that line. The concept of BZ-1 can be generalized. The second BZ (BZ-2) is the set of points that can be reached from the first zone by crossing only one Bragg plane. BZ-(n+1) is formed is the set of points not in {BZ-1, BZ-2, ... BZ-(n−1)} that can be reached from BZ-n by crossing only one Bragg plane. Alternatively, the n Brillouin zone can be reached from the origin by crossing n−1 Bragg planes, but not fewer. The construction of the BZ is illustrated in the next slides for a simple square lattice. An important point to check is that every Brillouin zone has the same volume, namely the volume of a primitive reciprocal lattice. In addition, any BZ can be mapped back to the first zone by using just primitive translations, i.e. all the BZ’s are equivalent by translation symetry.

4th nn

2nd nn

3rd nn VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 13 / 104 Electronic structure Crystal

4th nn

2nd nn

3rd nn VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 14 / 104 Electronic structure Crystal

Al BZs are equivalent due to the translational part of the crystal space group. Furthermore, the rotational symmetry determines the equivalence between different positions within a Brillouin zone. The special symmetry points receive particular names. Although there are different naming conventions, 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 the excellent Bilbao Crystallographic Server. The plot and the list of special ~k positions is available for all the space groups. ¯ Fm¯3m Im3m Pm¯3m

k z k k z z H3 X N1 3 R X3 R F2 F1

L S P Λ T Λ U F Q Λ D S Γ ∆ Γ ∆ Γ ∆ X k y X X 1 k k x K V y Σ M Σ N G H k y k x S 1 M Σ Z W k x

Band diagrams and electronic density of states

InN: a direct gap semiconductor

Crystal geometry

α b β γ

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). T Crystallographic coordinates: ri = h 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, Fm¯3m (225), a = 5.4626 Å, Z = 4. Ca 4a (0, 0, 0) F 8c (1/4, 1/4, 1/4)

From the International Tables of Crystalography: Wyckoff Sim. Equiv. positions 4a m¯3m (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? Neighbor distances? Atoms in the main cell? Cell plot?

Some simple crystal structuresI Grouped by structure symbols. (More structures in http://som.web.cmu.edu/StructuresAppendix2.pdf)

A1 Cu, Cubic, a = 3.609 Å, Fm¯3m, Z = 4. fcc Cu (4a) (0, 0, 0). A2 Li, Cubic, a = 3.46 Å, Im¯3m, Z = 2. bcc Li (2a) (0, 0, 0). p A3 Be, Hexag., a = 2.2860, c = 3.5843 Å, (hcp ideal: c/a = 8/3 ≈ 1.63) P63/mmc, Z = 2. Be (2c) (1/3, 2/3, 1/4). A4 C (diamond), Cubic, a = 3.5667 Å, Fd3m, Z = 8. C (8a) (1/8, 1/8, 1/8).

A9 C (graphite), 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).

A1: Cu A2: Li A3: Be A4: diamond A9: graphite

Some simple crystal structuresII

B1 NaCl, Cubic, a = 5.6402 Å, Fm¯3m, Z = 4. Na (4a) (0, 0, 0); Cl (4b) (1/2, 1/2, 1/2). B2 CsCl, Cubic, a = 4.123 Å, Fm¯3m, Z = 1. Cs (1a) (0, 0, 0); Cl (1b) (1/2, 1/2, 1/2). B3 β-ZnS (blend), Cubic, a = 5.4060 Å, F¯43m, Z = 4. Zn (4a) (0, 0, 0); S (4c) (1/4, 1/4, 1/4).

B4 ZnO (zincite, wurtzite), 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).

B1: NaCl B2: CsCl B3: ZnS (blend) B4: ZnO (wurtzite) Plots made with tessel (http://azufre.quimica.uniovi.es/software.html) and POVRay (http://www.povray.org).

Crystal geometry

Molecular crystals are usually represented including all the molecules that have at least one atom in the main cell and showing the crystal cell.

The crystallographic information ﬁle (cif)

# Crystallography Open Database (COD), _cell_formula_units_Z 2 # http://www.crystallography.net/ _exptl_crystal_density_meas 2.22 # _symmetry_space_group_name_H-M ’P n n m’ data_1011280 _symmetry_Int_Tables_number 58 _chemical_name_systematic ’Calcium chloride’ _symmetry_cell_setting orthorhombic _chemical_name_mineral ’Hydrophilite’ loop_ _chemical_compound_source ’synthetic’ _symmetry_equiv_pos_as_xyz _chemical_formula_structural ’Ca Cl2’ ’x,y,z’ _chemical_formula_sum ’Ca Cl2’ ’-x,-y,z’ _publ_section_title ’1/2+x,1/2-y,1/2-z’ ; ’1/2-x,1/2+y,1/2-z’ Die Kristallstruktur von Calciumchlorid, Ca Cl2 ’-x,-y,-z’ ; ’x,y,-z’ loop_ ’1/2-x,1/2+y,1/2+z’ _publ_author_name ’1/2+x,1/2-y,1/2+z’ ’van Bever, A K’ loop_ ’Nieuwenkamp, W’ _atom_type_symbol _journal_name_full _atom_type_oxidation_number ; Ca2+ 2.000 Zeitschrift fuer Kristallographie, Kristallgeometrie, Cl1- -1.000 Kristallphysik, Kristallchemie (-144,1977) loop_ ; _atom_site_label _journal_coden_ASTM ZEKGAX _atom_site_type_symbol _journal_volume 90 _atom_site_symmetry_multiplicity _journal_year 1935 _atom_site_Wyckoff_symbol _journal_page_first 374 _atom_site_fract_x _journal_page_last 376 _atom_site_fract_y _cell_length_a 6.24 _atom_site_fract_z _cell_length_b 6.43 _atom_site_occupancy _cell_length_c 4.2 _atom_site_attached_hydrogens _cell_angle_alpha 90 _atom_site_calc_flag _cell_angle_beta 90 Ca1 Ca2+ 2 a 0. 0. 0. 1. 0 d _cell_angle_gamma 90 Cl1 Cl1- 4 g 0.275(8) 0.325(8) 0. 1. 0 d _cell_volume 168.5 _cod_database_code 1011280

The protein databank ﬁle (PDB)

HEADER DE NOVO PROTEIN 19-AUG-03 1Q7O TITLE DETERMINATION OF F-MLF-OH PEPTIDE STRUCTURE TITLE 2WITH SOLID- STATE MAGIC-ANGLE SPINNING NMR TITLE 3SPECTROSCOPY ... DBREF 1Q7O A 1 3 PDB 1Q7O 1Q7O 1 3 SEQRES 1 A 3 FME LEU MTY ... CRYST1 1.000 1.000 1.000 90.00 90.00 90.00 P 1 1 ORIGX1 1.000000 0.000000 0.000000 0.00000 ORIGX2 0.000000 1.000000 0.000000 0.00000 ORIGX3 0.000000 0.000000 1.000000 0.00000 SCALE1 1.000000 0.000000 0.000000 0.00000 SCALE2 0.000000 1.000000 0.000000 0.00000 SCALE3 0.000000 0.000000 1.000000 0.00000 MODEL 1 HETATM 1 N FME A 1 -2.621 -3.439 0.640 1.00 0.00 N HETATM 2 CN FME A 1 -2.629 -4.738 0.349 1.00 0.00 C HETATM 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 H ATOM 20 N LEU A 2 -0.242 -0.725 0.342 1.00 0.00 N ATOM 21 CA LEU A 2 0.288 0.451 0.987 1.00 0.00 C ATOM 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 H HETATM 39 N MTY A 3 -0.268 1.582 -1.067 1.00 0.00 N HETATM 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 H TER 59 MTY A 3 ENDMDL ... CONECT 1 2 4 11 CONECT 2 1 3 CONECT 3 2 ... END

Crystal Data BasesI

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 at http://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 different pressures 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 on biomolecules. 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 database specializes 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.

Crystal Data BasesII

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). 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 the dynamical 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])

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 26 / 104 Part II

Examples of Quantum Espresso calculations (2014)

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 27 / 104 Quantum Espresso: pw+ps GNU code opEnSourcePackage forResearch inElectronicStructure,Simulation, and Optimization PWSCF, CP, PHONON, FPMD, Wannier, ... Suite of codes for DFT electronic structure calculations and materials modeling. Based on the use of plane waves and pseudopotentials (both, norm-conserving and ultrasoft). http://www.quantum-espresso.org/ Installation from source is simple:

1 cd 2 mkdir -p src/qe/ 3 cd src/qe/ 4 wget http://qe-forge.org/frs/download.php/211/espresso-5.0.tar.gz 5 cd espresso-5.0 6 ./configure # finds compilers, libraries, and creates"make.sys" 7 ./make all # creates executables in ~/src/qe/espresso −5.0/bin/ 8 cat << EOF >> ~/.bashrc 9 export QE_HOME=~/pkgs/qe/espresso-5.0/ 10 export PATH=${PATH}:${QE_HOME}bin: 11 EOF 12 source ~/.bashrc

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 28 / 104 Using QE in zcam2014:

1 . /home/apps/zcam2014/environ.zcam2014 # Create the environment 2 pw < input_file > output # Run quantum espresso

See in /.bashrc how the environ.zcam2014 is automatically run and the pw is really an alias to the real executable code. QE webpage (Documentation, pseudopotentials, ....) http://www.quantum-espresso.org/

In the VCL directory you will see a set of input and output examples. For instance:

1 $ cd 2 $ ls VLC 3 $ mkdir mydir 4 $ cp -r VLC/* mydir 5 $ cd mydir/si

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 29 / 104 pwSCF: Input scheme (pw.x < input > output)I

Input structured into mandatory and optional "&NAMELISTS" and "INPUT_CARDS". Each namelist contains a number of control variables, most of them with appropriate default values, but a few are essential and must always be speciﬁed. Full description: "INPUT_PW.txt" &CONTROL: declare addresses and names for pseudopotentials, and other dataﬁles, select the amount 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 of atoms 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 important variable is diagonalization, the main options being Davidson iterative (’david’) and conjugate gradient like (’cg’). &IONS: needed when atoms move (structural relaxation and molecular dynamics runs) and ignored otherwise.

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 30 / 104 pwSCF: Input scheme (pw.x < input > output)II

&CELL: needed when the cell moves (structural relaxation and variable sell MD runs) and ignored otherwise. &PHONON: prepare input for a PHONON task.

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 31 / 104 Each input card, on the other hand, corresponds to some vector or matrix. ATOMIC_SPECIES: for every different atomic species. ATOMIC_POSITIONS: for every different atomic position. K_POINTS: The k-points can correspond to the integration of iBZ1 or to special points used for depicting band structure. CELL_PARAMETERS: OCCUPATIONS: CLIMBING_IMAGES: special data for the NEB (nudge elastic band) algorithm. CONSTRAINTS:

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 32 / 104 pwSCF: Control of the input structure

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

ibrav 0 1 2 3 4 5 cell 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 = cos α

v1 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 or rhombohedric 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 = p(1 − C)/2; ty = p(1 − C)/6; tz = p(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 the CELL_PARAMETERS input card.

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 33 / 104 ibrav -5 6 7 8 9 10 cell 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 = cos α (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)/2 v2 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 14 cell 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) cos β (4) cos γ (4) cos α (5) cos β (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 sin β, 0, c cos β)(a, 0, c)/2 c(cos β, H1, H2) H1 = (cos α − cos β cos γ)/ sin γ; 2 2 2 1/2 H2 = (VLC1 + & AOR2 cos & DMCα cos () β cos γ −Electroniccos α structure− cos of solids:β − quantumcos espressoγ) / sin γ. ZCAM, Zaragoza 2014 34 / 104 Finding information in the Quantum Espresso output

The QE output is huge, like it happens with most computational science codes growth in long periods of time. Using tools like grep or scripting tools (awk, perl, python, ...) is imprescindible. In this table you will ﬁnd a most useful information for it:

keyword pos variable unit lattice parameter 5 a bohr ! 5 E Rydberg unit-cell volume 4 a bohr3

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 35 / 104 Electronic structure textbooks Bib

Part V

Electronic structure textbooks

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

ReferencesI

[1] A. D. Becke, E. R. Johnson, Exchange-hole dipole moment and the dispersion interaction, J. Chem. Phys 122 (2005) 154104.

[2] A. Otero-de-la Roza, E. R. Johnson, van der waals ineractions in solids using the exchange-hole dipole moment, J. Chem. Phys 136 (2012) 174109.

[3] A. Otero-de-la Roza, E. R. Johnson, A benchmark for non-covalent interactions in solids, J. Chem. Phys 137 (2012) 054103.

[4] A. Otero-de-la Roza, E. R. Johnson, V. Luaña, Critic2: a program for real-space analysis of quantum chemical interactions in solids, Comput. Phys. Commun. 182 (2013) 2232–2248.

[5] A. Otero-de-la Roza, V. Luaña, GIBBS2: A new version of the quasi-harmonic model code. I. Robust treat- ment of the static data, Comput. Phys. Commun. 182 (2011) 1708–1720, source code distributed by the CPC program library: URL:http://cpc.cs.qub.ac.uk/summaries/AEIY_v1_0.html.

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

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

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

[9] M. Catti, Calculation of elastic constants by the method of crystal static deformation, Acta Cryst. A 41 (1985) 494–500. [10] M. Catti, Crystal elasticity and inner strain: A computational model, Acta Cryst. A 45 (1989) 20–25.

VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza 2014 104 / 104