Electronic structure of solids: quantum espresso

Transcription

1 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 / 74

2 Part I Examples of Quantum Espresso calculations (2014) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

3 Task 1.1: CdSe (wurtzite phase) Simple case: ionic system, cubic geometry, 1 cell parameter. First, we must find the 0 K, 0 GPa geometry. 1 &control 2 title= cdse, prefix= cdse, pseudo_dir=../peuso/, calculation= vc-r 3 / 4 &system 5 ibrav=0, celldm(1)=1.0, nat= 4, ntyp= 2, 6 ecutwfc=50.0, ecutrho=500.0, 7 / 8 &electrons 9 conv_thr = 1d-8, 10 / 11 &ions 12 / 13 &cell 14 / 15 ATOMIC_SPECIES 16 se Se.pbe-n-rrkjus_psl.0.2.UPF 17 cd Cd.pbe-dn-rrkjus_psl.0.2.UPF ATOMIC_POSITIONS crystal 20 cd cd se se K_POINTS automatic VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

4 Task 1.2: phonon structure (ph.x) Create cdse.ph.in: 1 &inputph 2 tr2_ph = 1d-14, 3 ldisp =.true., epsil =.true., nq1=4, nq2=4, nq3=4, 4 amass(1)= , amass(2)= , 5 prefix= cdse, fildyn= cdse.dyn, 6 reduce_io=.true., 7 / The phonon structure is a big task, compared to the cpu invested in pw.x Many usefult results can be obtained cheaper using the E(V) points and the gibbs technology[1, 2, 3] The most relevant parts of the ph.x input are 1 ldisd: allows you to calculate the phonons in a grid of 1BZ points. Hereinafter: q-points, 2 nqx: grid size 3 nqx: calculates for an insulating dielectric (this is used to calculate the non-analytic contribution to the symmetry points other than Γ so, if you are a non-metal, put this always) 4 prefix: It MUST be the same used for the pw.x previous run obtaining phonons in the Γ point is far more cheaper and enough in many cases (Raman spectra, for instance). In that case, dynmat.x analyzes the relevant information from the ph.x output. Check the ( ) Si examples for more information. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

5 Task 2.1: The A4 phase of C Diamond geometry is simple: everything is fixed by symmetry except the cubic cell parameter. Doing calculations at a single point we must concentrate on the convergence in the number of k-points. Furthermore, it is an electrical insulator. 1 &control 2 calculation= scf, restart_mode= from_scratch, 3 title= si, prefix= si, 4 pseudo_dir=../pseudo/, outdir=../tmp/ 5 / 6 &system 7 ibrav=2, celldm(1)= , nat=2, ntyp=1, 8 ecutwfc=60.0, ecutrho=300.0, 9 / 10 &electrons 11 conv_thr = 1d-8 12 / 13 ATOMIC_SPECIES 14 C C.UPF ATOMIC_POSITIONS crystal 17 C C K_POINTS automatic Interested in transport properties? Follow Dr. Jaime Ferrer, also in Oviedo. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

6 Task 2.2: the graphite phase of C I We have examined ionic and covalent bonds. In graphite the layers are bound by weak van der Waals forces. Fortunately, quantum espresso includes appropriate density functionals to deal with the situation. Let us examine the XDM model, designed by Becke and Johnson[4] and implemented in QE by Johnson and Otero-de-la-Roza[5, 6]. 1 &control 2 calculation= vc-relax, restart_mode= from_scratch, 3 title= graphite, prefix= graphite, 4 pseudo_dir=../pseudo/, outdir=../tmp/, 5 / 6 &system 7 ibrav= 4, celldm(1)= , celldm(3)=2.7264, nat=4, ntyp=1, 8 occupations= fixed, 9 smearing= methfessel-paxton, degauss=0.02, 10 xdm=.true., xdm_a1=0.4073, xdm_a2=2.4150, 11 ecutwfc=30.0, ecutrho=180.0, 12 / 13 &electrons 14 conv_thr=1.0d-8, 15 / 16 &cell 17 press_conv_thr=0.5d0, press=0.d0, cell_dynamics= bfgs 18 / 19 ATOMIC_SPECIES 20 C C.pbe-rrkjus.UPF VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

7 Task 2.2: the graphite phase of C II ATOMIC_POSITIONS {alat} 23 C C C C K_POINTS automatic Notice the weird atomic coordinates. The alat keyword corresponds to the coordinates/celldm(1) ratio. Next form is easier to follow: 1 ATOMIC_POSITIONS crystal 2 c c c c Using { 1 /3, 2 /3, 1 /4} would be even better, but that is still not possible in quantum espresso. Further notes: VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

8 Task 2.2: the graphite phase of C III Namelists simplify hugely programming the interface. However, they produce very obscure bugs. Be careful! Remember: namelists are read in order: &control, &system, &electrons, &ions, &cell. Any other order causes an error. The ecut s are too small for a research calculation. Use ecutrho 4 ecutwf on Troulier-Martins pseudopotentials, or ecutrho 6 8 ecutwf on ultrasoft and PAW ones. There are many control variables and most have been avoided to reduce space: every one has a default value. The XDM parameters depend on the xc functional, included in the UPF (pseudopotential) file. This a1 and a2 values correspond to a BLYP xc. Consult Otero-dela-Roza and Johnson s works for other cases[5, 6] and check ucmerced.edu/wiki/xdm#quantum_espresso In Dr. Johnson s website (gatsby.ucmerced.edu) you will find many wonders. Pay attention to the materials of the NCIplot workshop celebrated in Paris in Lessons are still available and useful. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

9 Task 3.1: the ice-i phase I Let me end with a simple material, but the most awesome one. 1 &control 2 title= ice-1, prefix= ice-1, pseudo_dir=../pseudo/, 3 / 4 &system 5 ibrav=0, celldm(1)=1.0, nat=12, ntyp=2, 6 ecutwfc=30.0, ecutrho=300.0, 7 / 8 &electrons 9 conv_thr = 1d-8, 10 / 11 &ions 12 / 13 &cell 14 / 15 ATOMIC_SPECIES 16 h h.upf 17 o o.upf ATOMIC_POSITIONS crystal 20 o h h o h VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

10 Task 3.1: the ice-i phase II 25 h o h h o h h K_POINTS automatic CELL_PARAMETERS hexagonal VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

11 Homeworks 1 2 Create a nice E(V) grid for diamond and estimate its properties using asturfit. 3 Repeat examining phonons and using gibbs2 4 Examine the b/a ratio in graphite. Does the RPT ratio approach the ideal value? 5 You can extend the same questions to any element in the C period. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

12 Part II Examples of Quantum Espresso calculations ( ) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

13 pwscf: Task 1.1, Compute a single geometry of the A4 phase of Si I The A4 phase is very simple: everything is fixed by symmetry except the cubic cell parameter. Doing calculations at a single point we must only concentrate on the convergence in the number of k-points, and the system is an insulator. 1 &control 2 calculation= scf, 3 restart_mode= from_scratch, 4 title= si, 5 prefix= si, 6 pseudo_dir=../pseudo/, 7 outdir=../tmp/ 8 / 9 &system 10 ibrav=2, 11 celldm(1)= , 12 nat=2, 13 ntyp=1, 14 ecutwfc=60.0, 15 ecutrho=120.0, 16 / 17 &electrons 18 conv_thr = 1d-8 19 / 20 ATOMIC_SPECIES VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

14 pwscf: Task 1.1, Compute a single geometry of the A4 phase of Si II 21 Si si.upf 22 ATOMIC_POSITIONS crystal 23 Si Si K_POINTS automatic Try: 1 k-point convergence. 2 Different types of pseudopotentials (Vanderbilt, Troullier-Martins, norm-conserving,...). Each student should try a different type and exchange results and cpu time. 3 Effect of relativistic pseudos (inon rel, scalar, full rel,...) 4 convergence in ecutwfc (it has more impact in the cpu time) 5 convergence in ecutrho (it has more impact in the memory required) See the final output: less bleh. The cpu spent is at the end. Check final energy: grep! bleh.output. Check cpu time: grep CPU bleh,output. You can explore the output and prepare a little script using your favorite tool to extract the relevant results. In any real calculation remember to ensure convergence of your results to avoid later feel embarrased. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

15 pwscf: Task 2.1, Find the equlibrium geometry of the wurtzite phase of the CdSe crystal I 1 &control 2 title= crystal, 3 prefix= crystal, 4 pseudo_dir=., 5 calculation= vc-relax, 6 / 7 &system 8 ibrav=0, 9 celldm(1)=1.0, 10 nat= 4, 11 ntyp= 2, 12 ecutwfc=50.0, 13 ecutrho=500.0, 14 / 15 &electrons 16 conv_thr = 1d-8, 17 / 18 &ions 19 / 20 &cell 21 / 22 ATOMIC_SPECIES 23 se se.upf 24 cd cd.upf VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

16 pwscf: Task 2.1, Find the equlibrium geometry of the wurtzite phase of the CdSe crystal II ATOMIC_POSITIONS crystal 27 cd cd se se K_POINTS automatic CELL_PARAMETERS hexagonal Notice the vc-relax task to optimize the geometry. All the geometry of the cell: the cell lattice parameters and the free (internal) coordinates of the atoms. Alternative: use relax to only optimize the internal coordinates at fixed values of the cell lattice parameters. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

17 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 plane waves. Si (A4) calculations with ecutwfc=60.0. n s Energy (Ry) cpu (s) s Energy (Ry) cpu (s) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

18 Si (A4) calculations with a 9 9 9(1,1,1) MP grid. ecutfc Energy (Ry) cpu (s) ecutfc Energy (Ry) cpu (s) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

19 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 along the border of the ibz1. Like in the previous task, a calculation= scf determines the energy and computes integrals and orbitals for further tasks (this data can be already stored in the../tmp/si/ files). There are two programs 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 whole process: 1 cd../bands 2 pw.x < si.scf.in > si.scf.out 3 pw.x < si.bands.in > si.bands.out 4 bands.x << EOF > si.bands.pp.out 5 &inputpp 6 outdir=../tmp/, prefix = si, filband = si_bands.dat, 7 / 8 EOF 9 plotband.x << EOF > si.plotband.out 10 si_bands.dat si_bands.xmgr 13 si_bands.ps EOF 17 band_plot.x << EOF si_bands.dat 21 si_bands.out 22 EOF k x X 1 X 3 Γ k z bilbao crystallographic server Σ Λ L K Σ 1 Q M U S X V ky W VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

20 pwscf: Task 2, band diagram for a single geometry II The Bilbao crystallographic server ( provides illustrations, 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. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

21 According to the current calculation, Si (A4 or diamond phase, a = 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 of the band gap in a semiconductor is still one of the big unsolved problems of ab initio solid structure calculations ε (ev) L Γ X M Γ k-points VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

22 pwscf: Task 1.3, Density of States (DOS) After the initial calculation= scf task, than can be already stored in the../tmp/si/ files, obtaining and plotting the DOS is done by means of a non-consistent calculation (calculation= nscf ). Technically, the process requires integrating the orbital energies in the BZ1, using a special point quadrature (Blochl method, occupations= tetrahedra ). The Fermi energy is displaced to zero "by hand". 1 cd../dos 2 pw.x < si.scf.in > si.scf.out 3 pw.x < si.dos.in > si.dos.out 4 dos.x << EOF > si.dos.pp.out 5 &inputpp 6 prefix= si, outdir=../tmp/, 7 emin=-9.0, emax=16.0, 8 / 9 EOF DOS VLC & AOR & DMC () Electronic structure of solids: quantum espresso 5 10 ZCAM, Zaragoza / 74

23 pwscf: Task 1.4, 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 the topological Bader analysis using the critic2 code [7]. 1 cd../rho 2 pw.x < si.scf.in > si.scf.out 3 pp.x < si.pp.in > si.pp.out 4 critic2g si.incritic si.outcritic 5 n 4 y n x VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

24 pwscf: Task 1.5, Geometry optimization (1D) We can determine the E(V) curve of A4 phase by scanning the crystal energy vs. alat: 1 #! / bin / bash # Scan energy vs. a l a t 2 n=9; s=1; ecut=60; alat0=7.0; alat1=12.0; na=51; 3 adelta= awk BEGIN{printf"%.6f",( $alat1 - $alat0 )/( $na -1)} 4 cat << EOF > si-evsv.dat 5 # volume bohr ^3 6 # energy ry 7 # z 1 8 EOF 9 for ((i=1; i<=$na; i++)); do 10 alat= awk BEGIN{printf"%.6f", $alat0 +( $i -1)* $adelta } 11 pw.x << EOF > pp.out 12 &control 13 calculation= scf, restart_mode= from_scratch, title= si, 14 prefix= si, pseudo_dir=../pseudo/, outdir=../tmp/ 15 / 16 &system 17 ibrav=2, celldm(1)=${alat}, nat=2, ntyp=1, ecutwfc=${ecut}, ecutrho=720.0, 18 / 19 &electrons 20 conv_thr = 1d-8 21 / 22 ATOMIC_SPECIES 23 Si si.upf 24 ATOMIC_POSITIONS crystal 25 Si Si K_POINTS automatic 28 ${n} ${n} ${n} ${s} ${s} ${s} 29 EOF 30 vol= awk /unit-cell volume/{print $4} pp.out 31 ene= awk VLC & AOR /\!/{print & DMC () $5} Electronic pp.out structure of solids: quantum espresso ZCAM, Zaragoza / 74

25 E (Rydberg) V (bohr 3 ) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

26 Pressure-volume equation of state (EOS) The asturfit package [2] can be used to fit the E(V) curve and determine the equilibrium properties and the pressure dependence of the mechanical properties of the crystal. These calculations are typically based on the static model, in which the entropy terms and the vibrational zero point energy are ignored. Under such conditions p = ( A V ) NT de dv, G H E + pv, B T = V ( ) p V d2 E V NT dv 2. (1) The asturfit package [2] is an octave library of fitting routines but also implements three standalone tasks: (1) the recommended fitting procedure (asturfit, technically a linear fitting to an average of 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). A typical run is 1 asturfit << EOF > fit.out 2 # volume bohr ^3 # Volume and 3 # energy ry # energy are given 4 # z 1 # per atom [...] 8 EOF V 0 (bohr 3 ) E 0 (Rydberg) B 0 (GPa) B 0 B 0 (GPa 1 ) <BM> (49) (1) (99) (23) (26) exp [8] VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

27 Comparison of crystal structures We want to examine the relative stability of different crystal structures of Si. There are several ways of defining a structure in pwscf, involving: (1) the Bravais cell (ibrav), (2) some cell dimensions (celldm), (3) the atomic coordinates (see ATOMIC_POSITIONS), and (4) the primitive cell vectors (see CELL_PARAMETERS). name diam. (A4) graphite (A9) fcc (A1) bcc (A2) sc (A h ) βsn (A5) syst cubic hexag cubic cubic cubic tetrag sg Fd 3m P6 3 /mmc Fm 3m Im 3m Pm 3m I4 1 /amd sgn 227 (2) (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 nat pos ±(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) E o (Ry) V o (au) a o (au) c o (au) d o (au) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

28 Yin and Cohen 26 (1982) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

29 A typical 1D case: 1 #! / bin / bash 2 n=10; s=1; ecut=60; 3 alat0=6.0; alat1=9.5; na=36; # a0= bohr 4 adelta= awk BEGIN{printf"%.6f",( $alat1 - $alat0 )/( $na -1)} 5 ###na = awk BEGIN{ p r i n t f "%.6 f ", ( $alat1 $alat0 ) / $adelta + 1 ) } 6 7 datafile= si-fcc-ev.dat 8 cat << EOF > ${datafile} 9 # volume bohr ^3 10 # energy ry 11 # z 1 12 EOF for ((i=1; i<=$na; i++)); do 15 alat= awk BEGIN{printf"%.6f", $alat0 +( $i -1)* $adelta } 16 pw.x << EOF > pp.out 17 &control 18 restart_mode= from_scratch, title= si_fcc_p, prefix= si_fcc_p, 19 pseudo_dir=../pseudo/, outdir=../tmp/, 20 calculation= relax, 21 etot_conv_thr=1e-6, forc_conv_thr=1e-5, 22 / 23 &system 24 ibrav=2, celldm(1)=${alat}, nat=1, 25 ntyp=1, ecutwfc=${ecut}, ecutrho=720.0, 26 / 27 &electrons 28 conv_thr = 1d-8 29 / 30 &ions VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

30 31 / 32 &cell 33 / 34 ATOMIC_SPECIES 35 Si si.upf 36 ATOMIC_POSITIONS crystal 37 Si K_POINTS automatic 39 ${n} ${n} ${n} ${s} ${s} ${s} 40 EOF 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 [...] VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

31 This is a possible input for the graphite phase: 1 &control 2 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 &system 7 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 &electrons 11 conv_thr = 1d-8, 12 / 13 &ions 14 / 15 &cell 16 / 17 ATOMIC_SPECIES 18 Si si.upf 19 ATOMIC_POSITIONS crystal 20 Si Si Si Si K_POINTS automatic CELL_PARAMETERS hexagonal 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 could have used ibrav=4, celldm(1)=a and celldm(3)=c/a. We VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

32 Surface exploration in a 2D problem: c (bohr) β-sn (A5) phase E (rydberg) a (bohr) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

33 β-sn (A5) phase c/a V (au) E (rydberg) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

34 We can add a pressure (in kbar) to the vc-relax and vc-md tasks: 1 #! / bin / bash 2 # 3 # A5 or beta Sn phase 4 # Sn : a0= , c0= bohr c / a= # Si ( p =0): a0=7.3549, c0= bohr c / a= n=10; s=1; ecut=60; 8 p0=0; p1=2000; pdelta=40; 9 np= awk BEGIN{printf "%d", ( $p1 -( $p0 ))/( $pdelta )+1} datafile= si-a5-ev.dat cat << EOF > ${datafile} 14 # volume bohr ^3 15 # energy ry 16 # z 2 17 EOF alat=11.0; ca_ratio=0.61; for ((i=1; i<=$np; i++)); do 22 pp= awk BEGIN{printf"%.6f", $p0 +( $i -1)* $pdelta } 23 echo "pressure: $pp kbar" 24 pw.x << EOF > pp.out 25 &control 26 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, VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

35 31 / 32 &system 33 ibrav=7, celldm(1)=${alat}, celldm(3)=${ca_ratio}, nat=1, 34 ntyp=1, ecutwfc=${ecut}, ecutrho=720.0, 35 / 36 &electrons 37 conv_thr = 1d-8 38 / 39 &ions 40 / 41 &cell 42 press=${pp}, press_conv_thr=0.5, cell_factor=2.0, 43 / 44 ATOMIC_SPECIES 45 Si si.upf 46 ATOMIC_POSITIONS crystal 47 Si K_POINTS automatic 49 ${n} ${n} ${n} ${s} ${s} ${s} 50 EOF vol= awk / unit-cell volume/{v=$4} /new unit-cell volume/{v=$5} END{print v} pp 53 ene= awk /\!/{E=$5} END{print E} pp.out 54 awk BEGIN{printf "%16.9f%17.9f%19.2f\n", $vol, $ene, $pp } >> ${datafile} if ((i==1)); then 57 vol0=$vol; 58 alat0=$alat; 59 else fi alat= awk BEGIN{printf $alat *( $vol / $vol0 )**(1.0/3.0)} 62 done VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

36 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

37 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 unit cells. All the 3N degrees of freedom of movement correspond to vibrations, as the formally infinite crystal has no translations nor rotations. Asuming harmonicity and independence of modes, each one of the 3N vibrations is associated to a normal mode of frequency ω i = 2πν i = 2π k i /µ i, where k i and µ i are the effective force constant and mass, respectively. The vibrational state of the i-th mode is described by a quantum number v i {0, 1, 2, 3,...} and has an energy ɛ i = (v i + 1/2) ω i. The vibrational state and energy of the whole crystal is described by v = v 1, v 2,..., v 3N and E v = 3N (v i + 1/2) ω i. (2) No restrictions exist on the collective vibrational quantum numbers, as vibrations behave as bosons with null spin. In fact, the ground state corresponds to v = 0, all vibrational normal modes inactive, and the vibrational energy for this ground state is E 0 = 3N i ω i /2. i=1 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

38 Statistical thermodynamics of vibrations II Using a classical, Boltzmann formulation for the canonical ensamble, the vibrational partition function is Q(3N, V, T) = { } 3N e Ev/kT =... exp (v i + 1/2) ω i /kt v v 1 =0 v 3N =0 i=1 3N { = exp (v } i + 1/2) ω i 3N ( ) = e ω i/2kt e ω v i i/kt kt i=1 v i =0 i=1 v i =0 3N e ω i/2kt = 1 e ω i/kt, i=1 where k is the Boltzmann constant, T the absolute temperature, and we have used the summation formula S = 1/(1 r) for an infinite series, 1 + r + r 2 + r , of a convergent r = exp( ω/kt) < 1 geometric progression. The thermodynamic potential directly related to the canonical partition function is the Helmholtz function: 3N [ ωi ( )] A(3N, V, T) = kt ln Q(3N, V, T) = 2 + kt ln 1 e ω i/kt, (4) i=1 (3) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

39 Statistical thermodynamics of vibrations III and all the thermodynamic properties can be derived from here. For instance: ( ) ( ) ln Q A E =, p =, S = k ln Q + E/T,... (5) β N,V V N,T where β = 1/kT. In a macroscopic crystal 3N is huge (of the order on the Avogadro number), and the summation over normal modes can be converted to an integration: [ ωi ( )] A(3N, V, T) = dωg(ω) kt ln 1 e ω i/kt, (6) where g(ω) is the vibrational density of states (DOS), normalized as g(ω)dω = 3N. (7) 0 The DOS represents the degeneracy of normal modes. Knowledge of DOS is all that is required to determine the vibrational contribution to the thermodinamical properties of the crystal under the harmonic approach. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

40 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 same frequency: ( ) 2 ωe 1 g(ω)dω = 3Nδ(ω ω E )dω C v = 3Nk 2kT sinh 2 (8) ( ω E /2kT) Increases from C v(t 0) = 0 to the Dulong-Petit limit C v(t ) = 3Nk. Decays exponentially as T 0. No explanation for thermal expansion. The Debye model [Ann. Physik 39 (1912) 789] The solid is modelled as an elastic continuous medium: { 9N g(ω)dω = ωd 3 ω 2 ( ) dω if ω ω D, T 3 θd /T x 4 e x dx C v = 9Nk 0 if ω > ω D, θ D 0 (e x 1) 2 (9) where ω D represents the highest frequency that could be propagated by the crystal, and θ D = ω D /k is the Debye temperature. C v(t 0) (T/θ D ) 3 and C v(t ) = 3Nk. No explanation for thermal expansion. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

41 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 for different volumes: B(V) = V 2 A dv 2 V 2 E dv 2 θ D (V) = B k (6π2 V 1/2 r) 1/3 f (σ) m, (10) where m is the molecular mass per formula unit, σ is the Poisson ratio, and { /[ ( ) 3/2 ( ) 3/2 2(1 + 2σ) 1 + σ ] } 1/3 f (σ) = (11) 3(1 2σ) 3(1 σ) In an isotropic solid σ = λ/2(λ + µ), where λ and µ are the coefficients of Lamé. For a cubic solid: c 11 λ + µ, c 12 λ, and c 44 µ, so σ c 12 /(c 11 + c 12 ) c 12 /2(c 12 + c 44 ) 1/4. The vibrational contribution to the free energy is then A vib (T, V) = rmkt [ 9θ 8T + 3 ln(1 e θ/t ) D 3 (θ/t) ], D 3 (x) = 3 x x 3 0 t 3 dt e t 1. (12) Thermal behaviour is inferred from E(V) data. Correct limits for C v when T 0 and T. Thermal expansion is explained for normal materials at low T. No intrinsic anharmonicity. Fusion? VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

42 Lattice dynamics: 1D, one mass system a f j 1 j j +1 ξ j 1 ξ j ξ j+1 m Potential energy: U = U N f (ξ j ξ j 1 ) 2 2 j=2 Kinetic energy: T = 1 N m ξ j 2 2 j=1 Equations of motion (EOM): m ξ j = U ξ j = f (ξ j+1 2ξ j + ξ j 1 ) Born-von Karman (BvK) boundary conditions: ξ j (t) = ξ j+n (t) Solution ansatz: ξ j (t) = Ae i(ωt+jka). Using ξ j (t) in the EOM: mω 2 = f (e ika 2 + e ika ) =... = 4f sin 2 ka 2 4f Dispersion equation: ω = m sin ka 2 π /a ω ω max 0 BZ1 π/a k VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

43 Example: 1D, two masses system I a f 2j 2 2j 1 2j 2j +1 ξ 2j 2 ξ 2j 1 ξ 2j ξ 2j+1 Hamiltonian: m M H = T + U = 1 N (m ξ 2j 2 2 2j 1 2 ) + U N f [(ξ 2j ξ 2j 1 ) 2 + (ξ 2j+1 ξ 2j ) 2 ] 2 j=1 j=1 Equations of motion (EOM): m ξ 2j = f (ξ 2j+1 +ξ 2j 1 2ξ 2j ), M ξ 2j+1 = f (ξ 2j+2 +ξ 2j 2ξ 2j+1 ). Periodic conditions: ξ j+2n (t) = ξ j (t). Solution ansatz: ξ 2j (t) = Ae i(ωt+2jka) and ξ 2j+1 (t) = Be i(ωt+(2j+1)ka). [ ] [ 2f mω 2 2f cos ka A Dynamical equation: 2f cos ka 2f Mω 2 B Eigenvalues: ω 4 2f µ ω2 + 4f 2 sin 2 ka = 0 ω± 2 = f mm µ 1 ± ] = 0, k [ π/2a, π/2a] where µ = mm/(m + M) is the reduced mass of the A and B particles. 1 4nM sin2 ka (m + M) 2 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

44 Example: 1D, two masses system II ω 2f/µ 2f/m π /2a optical acoustic 0 BZ1 π/2a 2f/M The acoustic branch, ω (k), corresponds to the in phase motion of the A and B particles. In the optical branch, ω + (k), however, both move in oposition. k In a general 3D crystal there are 3 acoustic branches, that represent the elastic behaviour in the neighborbood of the Γ point (q = (0, 0, 0)). There are also 3r 3 optical branches, where r is the number of atoms in the primitive 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. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

45 Lattice dynamics: general 3D crystal Dynamical equations (eigensystem) To be solved for q BZ1: ( D κα,κ β(q) ωmqδ 2 κ,κ δ α,β )γ mq(κ β) = 0. (13) κ β 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 finite differences: D κα,κ β(q) = 1 MκM κ cells b 2 E e iqrb (14) u 0 κα u b κ β E: energy of a unit cell (electronic and ionic); R b : origin of the b cell; u a κα(t): instantaneous displacement (alpha: x, y, z) from the equilibrium position of the κ atom of cell a. Eigenvector displacement: u a κα(t) = 1 Mκ γ mq(κα)e i(qra ω mqt). (15) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

46 P. Giannozzi et al., Phys. Rev. B 43 (1991) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

47 P. Giannozzi et al., Phys. Rev. B 43 (1991) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

48 Phonon calculation I This is the sequence of steps required to perform a phonon calculation: 1 # Remove QE comment l i n e s 2 pw.x < si.scf.in > si.scf.out # E l e c t r o n i c ground s t a t e 3 4 # Calculation of the force constants on a q g r i d 5 ph.x << EOF > si.ph.out 6 phonon calculation # t i t l e l i n e ( f i l e : " s i. ph. i n " ) 7 &inputph 8 outdir="../tmp/", # scratch d i r e c t o r y 9 prefix="si", # same name used on the unperturbed system 10 tr2_ph = 1d-14, # threshold f o r s e l f consistency 11 ldisp =.true., # i f t r u e c a l c u l a t e phonons i n a g r i d of q p o i n t s 12 nq1=4, nq2=4, nq3=4, # Monkhorst Pack g r i d of q points 13 amass(1)= , # masses f o r the ntyp types of atoms 14 fildyn= si.dyn, # f i l e to w r i t e the dynamical matrix ( def : matdyn ) 15 / 16 EOF # Diagonalize the dynamical matrix and w r i t e eigenvalues in xcrysden format 19 dynmat.x << EOF > si.dynmat.out 20 &input # ( F i l e : " s i. dynmat. i n " ). See PH/ dynmat. f90 21 fildyn= si.dyn1, # dynamical matrix f i l e ( def : " matdyn " ) 22 asr= simple, # Type of acoustic sum r u l e imposed 23 amass(1)= , # Masses f o r the ntyp types of atoms 24 q(1)=0.0,q(2)=0.0,q(3)=0.0, # Calculate LO modes along q ( Def : q = ( 0, 0, 0 ) ) 25 / 26 EOF 27 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

49 Phonon calculation II 28 # Calculation of the r e a l space IFCs ( Interatomic Force Const ) 29 q2r.x << EOF > si.q2r.out 30 &input # ( F i l e : " s i. q2r. i n " ). See PH/ q2r. f90 31 fildyn= si.dyn, # Input f i l e name. Also on phonon (PH) input. 32 # Look f o r the " s i. dym { } " f i l e s 33 zasr= simple, # Acoustic sum r u l e used f o r the Born e f f e c t i v e charges 34 flfrc= si.fc, # Output f i l e f o r the IFC i n r e a l space 35 / 36 EOF # C a l c u l a t i o n of the phonon DOS or d i s p e r s i o n curves 39 matdyn.x << EOF > si.matdyn.out 40 &input # ( F i l e : " s i. matdyn. i n " ). See PH/ matdyn. f90 41 asr= simple, # Acoustic Sum Rules ( def : " no " ). 42 dos=.true., # " yes " ( c a l c u l a t e DOS) ; " no " ( do d i s p e r s i o n bands ) 43 amass(1)= , # Masses of the ntyp types of atoms 44 flfrc= si.fc, # q2r f i l e containing force constants ( required ) 45 fldos= si.phdos, # output f i l e f o r the phonon DOS ( def : " matdyn. dos " ) 46 # Units : states versus omega i n cm^{ 1} 47 nk1=50,nk2=50,nk3=50, # uniform q p o i n 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 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

50 φ-dos Si (p=0, diamond phase) ω (cm -1 ) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

51 Elastic equation of state For small deformations of the crystal cell around an equilibrium configuration: 2φ = 2 E E 0 V 0 = c ijkl ɛ ij ɛ kl = s ijkl τ ij τ kl. (16) ij,kl ij,kl Using symmetric deformation components: ɛ ij = 1 [ δxi + δx ] j, δx = x x 0, x {a, b, c}. (17) 2 x j x i Strain-stress linear relationships: τ ij = φ ɛ ij, τ ij = kl c ijkl ɛ kl, ɛ ij = kl s ijkl τ kl. (18) Voigt notation: τ 11 τ 22 τ 33 τ 23 τ 31 τ 12 ɛ 11 ɛ 22 ɛ 33 2ɛ 23 2ɛ 31 2ɛ 12 τ 1 τ 2 τ 3 τ 4 τ 5 τ 6 ɛ 1 ɛ 2 ɛ 3 ɛ 4 ɛ 5 ɛ 6 Elastic constants and compliances: ( ) ( ) τi ɛi c ij =, s ij =, ɛ j ɛ,0 τ j τ,0 c = s 1. (19) VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

52 Deformations of the cubic cell We need to calculate (E, V) for appropriate finite deformations of the unit cell: R R = R(1 + ɛ). In the case of cubic systems: R = a a ɛ 1 2ɛ 6 2ɛ 5 2ɛ ɛ 2 2ɛ 4 (20) 0 0 a 2ɛ 5 2ɛ ɛ 3 V = V 0 {1 + ɛ 1 + ɛ 2 + ɛ 3 + ɛ 1 ɛ 2 + ɛ 2 ɛ 3 + ɛ 3 ɛ 1 + ɛ 1 ɛ 2 ɛ 3 (21) 4ɛ 2 4 4ɛ2 5 4ɛ2 6 4ɛ 1ɛ 2 4 4ɛ 2ɛ 2 5 4ɛ 3ɛ ɛ 4ɛ 5 ɛ 6 } Φ(ɛ 1 ) c 11 Φ(ɛ 1, ɛ 2 ) c 12, c 11 Φ(ɛ 4 ) c 44 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

53 Elastic moduli and the crystal systems See Bhagavantam [9] or Nye [10] Cubic Hexagonal Orthorhombic c 11 c 12 c c 11 c 12 c c 11 c 12 c c 11 c c 11 c c 22 c c c c c c c c 44 0 c 44 c 44 0 (c 11 c 12 )/2 c 55 0 c 66 For a isotropic (cubic) medium c 44 = (c 11 c 12 )/2 (Cauchy relationship). Trigonal (3, 3) Trigonal (3m, 32, 3m) Tetrag. (4, 4, 4 m) c 11 c 12 c 13 c 14 c 15 0 c 11 c 12 c 13 c c 11 c 12 c c 16 c 11 c 13 c 14 c 15 0 c 11 c 13 c c 11 c c 16 c c c c 44 0 c 15 c c c 44 c 14 c 66 c 44 c 14 c 66 c 44 0 c 66 c 66 = (c 11 c 12 )/2. Tetrag. (4mm, 42m, 422, 4 mmm) Monoclinic Triclinic c 11 c 12 c c 11 c 12 c c 16 c 11 c 12 c 13 c 14 c 15 c 16 c 11 c c 22 c c 26 c 22 c 23 c 24 c 25 c 26 c c c 36 c 33 c 34 c 35 c 36 c c 44 c 45 0 c 44 c 45 c 46 c 44 0 c 55 0 c 55 c 56 VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

54 Catti s method I In the method pioneered by M. Catti [11, 12] deformations involve simultaneous activation of several components of ɛ such that the crystal symmetry is maximally conserved. For instance, the deformation ɛ T [η, 0, 0, 0, 0, 0] in a cubic crystal has an elastic energy 2φ = ɛ T cɛ = c 11 η 2 and it is clear that c 11 can be obtained as ( 2 φ/ η 2 ). This particular deformation transforms the unit cell from a cube to a square-based prism of dimensions [a(1+η), a, a, 90, 90, 90], thus lowering the symmetry from cubic to tetragonal. The symmetry reduction can have the added consequence that some of the atoms within the cell gain degrees of freedom and their position within the cell should be reoptimized for each new value of η. Failing to take into account this inner strain can produce a significant impact on the calculated value of the elastic moduli. Example: Cubic CaF 2 [fluorite; Fm 3m (Num. 225); Ca (4a) (0, 0, 0); F (8c) (1/4, 1/4, 1/4)]. Strain Space G. Inner strain 2φ Cell η[1, 1, 0, 0, 0, 0] 4 mmm No 2(c 11 +c 12 )η 2 [a(1+η), a(1+η), a, 90, 90, 90] η[1, 1, 2, 0, 0, 0] 4 mmm No 6(c 11 c 12 )η 2 [a(1+η), a(1+η), a(1 2η), 90, 90, 90] η[1, 1, 1, 0, 0, 0] Fm 3m No 3(c 11 +2c 12 )η 2 [a(1+η), a(1+η), a(1+η), 90, 90, 90] η[0, 0, 0, 1, 1, 1] R 3m F (x, x, x) 3c 44 η 2 [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 one degree 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 c 44 modulus could also be obtained through a [0, 0, 0, 0, 0, η] deformation, but this would reduce the cell to moclinic symmetry, instead of rhombohedral as the proposed [0, 0, 0, η, η, η]. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

55 Catti s method II Designing a set of sensible deformations can be a creative task with a large influence on the computational effort. 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 = (c c 12 )/3. The importance of the elastic characterization. Bulk modulus and elastic moduli are very important properties of the crystal beyond their significant role on establishing the stability of the crystal phase and helping to predict the possible deformation of the unit cell shape in a potential phase transition. Elastic data is required in crystallography to calculate the thermal diffuse scattering correction to the Bragg diffraction intensities. The interpretation of the observed seismic waves, that constitute our best probe on Earth innards, depend on the assumed elastic properties of the solid phases present in the earth mantle. Material scientist are engaged in a long quest for superhard materials that maintain their hardness on a high pressure, high temperature, or high radiation working ambient. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

56 Elastic constants of Si Next scripts are designed to calculate the elastic constants of the diamond phase of Si. There are two main scripts: gen.m, that prepares the input to pwscf for all the calculations, and extract.m, that analyzes the pwscf outputs and produces the final results. Both scripts are written in octave, but extract.m calls the shell and combines the use of grep, tail, and awk. The whole calculation is done as 1 #! / bin / bash f 2 gen.m 3 for i in {1..11}_{1..3}/ ; do 4 cd $i 5 echo $i 6 pw.x < ${i%/}.scf.in > ${i%/}.scf.out 7 cd.. 8 done 9 extract.m The elastic constants are defined in terms of the conventional cubic cell, so we have performed all our calculations 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 of the 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). The theoretical result reproduces well the experiments and describes a clearly stable phase. [GPa] c 11 c 12 c 44 B ɛ range points calc ± calc ± VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

57 1 #! / usr / bin / octave q 2 # c i j determined by s t r a i n s t r e s s r e l a t i o n s, l e a s t squares f i t. 3 # The deformations correspond to : 4 # ( 1 ) [100000], ( 2 ) [000111], ( 3 ) [111000] 5 6 format long 7 8 # the c a l c u l a t e d 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 = [ ] * ; 15 # c a l c u l a t e the e q u i l i b r i u m conventional c e l l length 16 a_eq = (abs(det(r_prim))*4)^(1/3); 17 r_eq = eye(3) * a_eq; # atomic p o s i t i o n s i n the conventional c e l l 20 x_eq = [ ]; # the f i n i t e deformations, s t r a i n range VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

58 32 #erange = [ ] ; 33 erange = [ ]; 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)); # define the s t r a i n tensor 42 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 endif # pack the 6 vector in the s t r a i n matrix 52 emat = zeros(3,3); 53 for k = 1:3 54 emat(k,k) = 1 + e(k); 55 endfor 56 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; # obtain the deformed l a t t i c e vectors r = r_eq * emat; 63 # w r i t e the i n p u t VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

59 64 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 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 endfor 92 fprintf(fid,"\n"); 93 fprintf(fid,"k_points automatic\n"); 94 fprintf(fid," \n"); 95 fprintf(fid,"\n"); VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

60 96 fprintf(fid,"cell_parameters cubic\n"); 97 for k = 1:3 98 fprintf(fid,"%.10f %.10f %.10f\n",r(k,:)); 99 endfor 100 fclose(fid); 101 endfor 102 endfor 1 #! / usr / bin / octave q 2 format long 3 4 # the c a l c u l a t e d 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 = [ ] * ; 10 # c a l c u l a t e the e q u i l i b r i u m conventional c e l l length 11 a_eq = (abs(det(r_prim))*4)^(1/3); 12 r_eq = eye(3) * a_eq; 13 # the f i n i t e deformations 14 #erange = [ ] ; 15 erange = [ ]; nj = 3; 18 tmat = smat = zeros(length(erange)*nj,6); 19 n = 0; 20 for i = 1:length(erange) 21 for j = 1:nj 22 n += 1; 23 s = sprintf("%d_%d",i,j); VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

61 24 ss = sprintf("%s/%s.scf.out", s, s); ## define the s t r a i n tensor 27 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 endif # pack the 6 vector in the s t r a i n matrix 37 emat = zeros(3,3); 38 for k = 1:3 39 emat(k,k) = 1 + e(k); 40 endfor 41 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; # obtain the deformed l a t t i c e vectors r = r_eq * emat; 48 # get the s t r e s s tensor 49 order = sprintf("grep total stress %s -A 3 tail -n 3 awk {print $4, $ 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 endfor VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

62 56 endfor # Calculate the C i j matrix 59 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)); # check the s t a b i l i t y c o n d i t i o n s 70 printf("eigenvalues: %.2f %.2f %.2f %.2f %.2f %.2f\n",eig(cij)); VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

63 qcg Bib Part III Oviedo s Quantum Chemmistry Group VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

64 qcg Bib Codes in the QCG group family since 1982 we have been doing and publishing scientific codes, mainly in fortran. we have developed a paradigm of making the codes easy to use and powerful the list is large and increasing: pi, pi7, optim, derive, navigator, pairpot, tessel, critic, promolden,... we are scientific programmers as well as quantum chemists (computational quantum physical chemists, if you like) escher is one of our last tools: a mixture of octave and python routines in which the authors, DMC, AOR and VLC, are currently working. The task for ESCHER is helping to prepare inputs for solid state codes, provide arithmetic calculations and facilitate the analysis of its output. I hope ESCHER will be ready for ZCAM-2015 and available, as always, from our webpage. Stay tuned. We love thermodynamics and symmetry, but our real passion is chemical bonding. Topological theories (aka QTAIM) of chemical bonding. There are other useful codes created by our best programmers, and some of them appear en this course: asturfit[2] and gibbs.[1, 3] The codes are maintained and you can learn about them in our web page (azufre.quimica.uniovi.es). VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

65 qcg Bib Main Programmers in the Oviedo s QCG group Oviedo s Quantum Chemistry group around VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74

66 qcg Bib This is a tribute to our friend and colleague Dr. Miguel Álvarez Blanco, whose untimely death left a deep hole in our hearts yet not in our brains. We miss him but he has not left us and never will. Dr. Miguel Álvarez Blanco ( ), Dr. Aurora Costales castro, Miguel y Jaime Álvarez Costales. VLC & AOR & DMC () Electronic structure of solids: quantum espresso ZCAM, Zaragoza / 74