Elastic constants calculation - WIEN 2k

IR ELAST+WIEN2kA Package for calculating elastic tensors of hexagonalPhases by using second-order derivative with Wien2k PackageUser’s guide, Hex-elastic_13.2 (Release 27.08.2013)Morteza JamalGhods City-Tehran-Iran1

MANDATORY CONDITIONS:In any publication in the scientific literature please reference the program as follows:M. Jamal, Hex-elastic, http://www.wien2k.at/reg_user/unsupported/cubic-elast/ (2012).ACKNOWLEDGMENTI gratefully appreciate B.Z. Yanchitsky for fruitful discussions , P. Blaha and S. JalaliAsadabadi for suggestions, and Carol Phillips for editing.For suggestions or bug reports please contact the author by email:m_jamal57@yahoo.com2

1- IntroductionHex-elastic is a Package for finding elastic constants of hexagonal symmetries withWien2k. This Package calculates elastic constants by second-order derivative ( E”(δ) ) ofPolynomial fit ( E=E(δ) ) of Energy vs. strains (δ) at zero strain (δ =0). This called energyapproach [1].2- Background theory 2 (energy approach)Elastic constants are defined by means of a Taylor expansion of the total energyfor the system, with respect to a small strain ( ) of the lattice. In this Package weconsider the hexagonal crystal structure, which is spanned by three vectors, , and .The Bravais lattice vectors are normally written in a matrix form, i.e.:The distortion of the lattice ( ) is expressed by multiplying with asymmetric ( ) distortion matrix i.e. ( ), which iswritten as,And in Voigt notation ( It is often convenient t o change to the Voigt notation inorder to reduce the number of indices. The Voigt notation replaces 1,2, 3, (and ) 4, (and ) 5, (and ) 6 )3

We express the energy of the strained system by means of a Taylor expansion inthe distortion parameters,The linear terms vanish if the strain causes no changes in the volume of thecrystal. Otherwise, are related to the strain on the crystal and are elasticconstants and is the volume of unstrained hexagonal system and we use it toevaluate the elastic constants.There are five independent elastic constants for a hexagonal symmetry, calledC 11 , C 12 , C 13, C 33 , and C 55 [3]. Since we have five independent elastic constants,we need five different strains to determine these elastic constants. The fivedistortions used in the hex-elastic Package are described below [2,4]. The firstdistortion is written as:and it changes the size of the basal plane, while keeping the z axis constant. Thesymmetry of the strained lattice is therefore still hexagonal and the energy for thisdistortion can be obtained as:The second type of distortion is a volume conserved distortion and lead to Orthorhombicsymmetry and written as:4

and the energy for this distortion can be obtained as:The third strain we have used is given byThis strain changes C lattice parameter and keep the symmetry of the strained latticehexagonal and the energy for this distortion can be obtained byThe fourth elastic constant, C55, is determined by means of a deformation of the lattice,which produces an object with low symmetry. The deformation is written as:And it leads to triclinic symmetry and the energy for this deformation can be written asFinally, the last strain we have used is volume conserved and keeps the symmetry of thestrained lattice hexagonal and can be written as:And the energy for this strain is given by5

AndIn practice, to calculate elastic constants we fit to a polynomial of degree M andM changes from 2 till N-1 , N is number of data, and then elastic constants is computedby using second order derivative ( E”(δ) ) of Polynomial fit ( E=E( V, δ) ) of Energy vs.strains (δ) at zero strain (δ =0). In this situation we are able to check the sensitivity of ourresults (elastic constants) to the order of fit however final results are for a polynomial ofdegree 2 ( M=2) because we are in the regime of Hooke’s law.6

3- File structure and program flowThe following table describes input and output files for each program of the Hex-elasticPackage.Program needs generatesH_set_elast_lapw case.struct init.structruncommand1runcommand2pwdnamecommand_init_lapwH_command_run_lapwH_setup11m12, H_setupc33H_setupc1112, H_setupczzH_setupc55getcalljobHmakestructHinit.structpwdnameruncommand1runcommand2init.struct.stypHEX.jobauto_init_lapwruncommand1runcommand2.Vper.stypHEX.jobnumber.strainStypX_Y.structvol.optimizeH_modifyjob_lapwH_calljob_lapw HEX.job VstVeneH_fitdivELC number.strain ELCorder.fitvol.optimize ELC.output.stypELC.fitVstVeneH_ana_elastc_lapwVstVenecase.outputeosELC.fitvol.optimizeELC.psH_ana_elast_lapw ELC.output ELC-matrixcase.output_elasticH_InverseELC ELC-matrix INVELC-matrixMassRhoinit.struct.rhovol.optimizeH_ana_elastorder_lapw ELCorder.fit output-ordersgroupcheck_lapw StypX_Y.struct case.structcommand_intso_lapw.infSOcommand_initu_lapw.infLDAUauto_initso_lapw .infSO case.insoauto_initu_lapw .infLDAU case.inorbcase.indm/cBold font is OPTIONALItalic bold font means it is the user’s choice7

3-1- Short description for input and output filescase.structinit.structpwdnameruncommand1/2Is a Wien2k standard struct file.Is a copy of the case.struct file.Contains the name of the present work directory.Contains the run commands for running. It looks similar to:run_lapw –ec 0.0001 –p –in1new 2auto_init_lapwA C-shell program which automatically runs the initialization.It looks similar to:#!/bin/csh -fset RM = notif ( $RM == 'not' ) theninit_lapw -vxc 13 -ecut -6 -mix 0.2 -numk 3000 -belseinit_lapw -red 0 -vxc 13 -ecut -6 -mix 0.2 -numk 3000 -bendif……….Vper Defines the percent of changes for different strains..stypDefines the type of strain.HEX.jobA C-shell program which calculates the energy for each strainby using the Wien2k Package. It looks similar to:#!/bin/csh -f#STRAIN TYPE IS 1#Modify this script according to your needsunalias rmset co = 1set nameset bjset file = `pwd`set file = $file:tif (-e VstVene ) thenset i=`/bin/ls VstVene* |wc `echo " saving pervious VstVene to VstVene_$i[1]"cp VstVene VstVene_$i[1]rm VstVeneendif## to reuse previous scf runs (without a new scf run) set answscf=y# and use the same "savename".# When you make modifications (RKmax, k-mesh, XC-potentials) choose# answscf=no and a new savename (eg. "_pbe_rk8_1000k").set answscf=yset savename=8

if (-e cscl.clmsum && \! -z cscl.clmsum) thenx dstart -superendifif (-e cscl.clmup && \! -z cscl.clmup ) thenx dstart -super -upx dstart -super -dnendifforeach i ( \)echo "*******************************"echo $iset name=`echo "$name $i"`echo "*******************************"………Styp1_-3.0 \Styp1_-2.0 \Styp1_-1.0 \Styp1__0.0 \Styp1__1.0 \Styp1__2.0 \Styp1__3.0 \StypX_Y.structVstVenenumber.straincase.outputeosA Wien2k struct file for each value of changes and for eachstrain type where X and Y denote type of strain and value ofchanges, respectively.The main information file, contains values of changes (strains)and energies for each type of strain, for the calculation of theelastic constant.Contains the number of strains.A Wien2k output of equation of states(EOS).For finding the best values of elastic constants, findEOS and then copy the case.outputeos file in the "case"directory within the c11+c12, c11-c12 , c55, c33, and czz,directories. Otherwise, it sets the optimized volume from theoriginal struct file i.e. case.structvol.optimize Contains the optimized volume.ELCorder.fit Contains the elastic constants for different values of order offit.ELC.output Contains the elastic constants for order of fit =2ELC.fitContains the data to plot a curve of energy vs value of changes(strains) for each strain type.ELC-matrixDefines the elastic constant matrix for each symmetry.case.output_elastic Contains the final elastic constant values.INVELC-matrix Defines the inverse of elastic constant matrix.9

.rho.infSO.infLDAUContains density of mass and atomic volume.Contains information for making the case.inso file for runningspin-orbit coupling .Contains information for making the case.inorb andcase.indm/c files for LDA+U calculations.3-2- Flow and short description for programsThe Hex-elastic Package consists of several FORTRAN and SHELL SCRIPTS which aredescribed below. A flowchart of the program is shown in the following diagram. H_set_elast_lapw :Makes an elast-constant directory in the present work directory ( PWD ) and c11+c12,c11-c12 , c33, c55, and czz directories in the elast-constant directory. TheH_set_elast_lapw program also copies information of the "PWD" into the c11+c12, c11-c12 , c33, c55, and czz directories and calls "command_init_lapw",H_command_run_lapw, H_setupc1112, H_setupc11m12, H_setupc33, H_setupc55, andH_setupczz programs.. command_init_lapw :Gets information for making "auto_init_lapw". H_command_run_lapw :Gets the run commands for making “HEX.job”. H_setupcX (X=1112, 11m12, 55, ….) :Gets the type of strain and calls the “getcalljobH” program. getcalljobH:Calls “makestructH” program and makes the “HEX.job” file. makestructH :Makes the “StypX_Y.struct ” files where X and Y stand for the type of strain andvalue of changes, respectively and the vol.optimize file. H_modifyjob_lapw :Edits the job files according to the user’s needs. H_calljob_lapw :Calls the “HEX.job” files for running. H_ana_elast_lapw :10

Calls the H_ana_elastc_lapw program for calculating elastic constants then calculatesthe Voigt, Reuss, and Hill bulk, shear, and the Young modulus as well as the Poissonratio. After that it calls the “H_InverseELC” and “MassRho” programs and calculatessound velocity and Debye temperature then makes two output files in the elastconstantdirectory with the name case.output_elastic and the INVELC-matrix whichis the elastic compliance constants generated by inverting the elastic constant matrix.At the end it calls “H_ana_elastorder_lapw” program. H_ana_elastc_lapw :Calls the “H_fitdivELC” program with appropriate libraries for calculating C11-C12,C11+C12, C55, and …. H_InverseELC :Makes the Elastic compliance constants generated by inverting the elastic constantmatrix . MassRho :Finds density of mass and atomic volume. H_ana_elastorder_lapw :Checks the sensitivity of the elastic constants to the order of fit. sgroupcheck_lapw :Finds the best value of tol in the sgroup [5] program and copies case.struct_sgroup ascase.struct.11

H_set_elast_lapwcommand_init_lapwgeneratesauto_init_lapwH_command_run_lapwgeneratescommandrun1/2H_setupcXX=1112, 11m12, ….getcalljobHgeneratesHEX.jobmakestructHgeneratesStypX_Y.structsgroupcheck_lapwHEX.jobgeneratesVstVeneauto_init_lapwH_ana_elastorder_lapwgeneratesELCorder.fitH_ana_elast_lapwH_ana_elastc_lapwH_InverseELCMassRhoELASTIC CONSTANTS IS READYgeneratescase.output_elastic and INVELC-matrixProgram flow in Tetra-elastic12H_fitdivELC calls LibrariesgeneratesC 11 -C 12 , C 11 +C 12 , ….Dash arrow means user must run

4 – Elastic constants calculation1. Create a struct file and validate it by running "sgroupcheck_lapw".2. If Spin-Orbit calculations are required run "command_initso_lapw.3. If LDA+U calculations are required run "command_initu_lapw" and then"auto_initu_lapw".4. Run "H_set_elast_lapw”..5. Now you must adapt the job files according to your needs (you can run"H_modifyjob_lapw" in Terminal ).It is not necessary to do step 5 if you defined the “COMMAND RUN” commands instep 4.6. Now you must run the job files (you can run "H_calljob_lapw" ). It will take sometime.7. Run "H_ana_elast_lapw"..This package calculates elastic constants by second-order derivative ( E”(δ) ) ofPolynomial fit ( E=E(δ) ) of Energy vs. strains ( є) at zero strain (δ =0) so, you must usevalues of strain around zero and from the viewpoint of fit convergence, we usuallyexpect to see a minimum when we plot Energy vs. strain ( this Package plots it ). It isrecommended that the sensitivity of the results is checked to the order of fit. Thisprogram shows them.4-1– Notes about elastic constants calculationAfter using distortions for the calculation of C11-C12 and C55 the symmetry of thehexgonal compound changes and usually the number of atoms change. So when you run"command_initso_lapw" or "command_initu_lapw" , in the section name of an atom,type "all " ( for example: all Mn). With this command, you use SO orLDA+U calculations for example for all Mn atoms.When you want to rerun job files with modifications in (RKmax, k-mesh, XC-potentials )call the “command_init_lapw” and after that choose "answscf=no" in the “HEX.job”files and a new "savename" (eg. "_use_pbe_rk8").13

Optionally you can specify more cases by rerunning “H_setupcX” (X=1112, 11m12,55,… see section 5-3 ) . Specify also your ‘‘old’’ cases. The old results will then betaken automatically into account without recalculation (unless you modify job files i.e:set answscf=no ).For the calculation of the best values of elastic constants, please find EOS and then copycase.outputeos in the "case" directory within the c11+c12, c11-c12, c55, and …directories. Otherwise, it sets the optimized volume from the original struct file i.e.case.struct .4-2– One calculationTo calculate C 33 or C 55 the following steps should be performed for example forcalculation of C 55 :1. Make a directory for example c55.2. Make a "case" directory in c55 directory.3. Make a "case.struct" file in the "case" directory and name it "init.struct".Create a "pwdname" file and write in it "case." and save it.4. Run the command_init_lapw5. chmod +x auto_init_lapw6. For SO calculations, run the command_initso_lapw.7. For LDA+U calculations, run the command_initu_lapw and auto_initu_lapw.7-1. To avoid step 10, you can run “H_command_run_lapw” for setting the“COMMAND RUN” commands for making “HEX.job” .8. Run the H_setupc55 program.9. chmod +x HEX.job file.10. Modify the HEX.job file.11. Call HEX.job12. Call H_ana_elastc_lapw4-3– Run with more data pointsOptionally you can specify more data points, for the calculation of the elastic constants,by rerunning “H_setupcX” (X=1112, 11m12, 55,… ) . Specify also your ‘‘old’’ datapoints. The old results will then be taken automatically into account withoutrecalculation ( unless you modify the job files i.e: set answscf=no ). Please do thefollowing steps for this goal for example for c55.1. Cd to the “elast-constant” directory.2. Cd to the “c55” directory.3. cd the “case” directory.3-1) To avoid step 6, you can run “H_command_run_lapw” for setting the14

“COMMAND RUN” commands for making the “HEX.job” .4. Run the H_setupc55 program.5. If you want to rerun the job files with modifications in (RKmax, k-mesh, XCpotentials) call “command_init_lapw” and then choose "answscf=no" in“HEX.job” files and a new "savename" (eg. "_use_pbe_rk8").6. Modify the HEX.job file.7. Call HEX.job8. Call H_ana_elastc_lapw4-4– Elastic constants calculation for TiTi compound is a test case for elastic constants calculation. The Ti structure is describedin detail in the following:TiH LATTICE,NONEQUIV.ATOMS: 1 194 P63/mmcMODE OF CALC=RELA unit=bohr5.574694 5.574694 8.843922 90.000000 90.000000120.000000ATOM-1: X=0.33333333 Y=0.66666663 Z=0.75000000MULT= 2 ISPLIT= 4-1: X=0.66666667 Y=0.33333337 Z=0.25000000Ti1 NPT= 781 R0=0.00005000 RMT= 2.4000 Z: 22.0LOCAL ROT MATRIX: 1.0000000 0.0000000 0.00000000.0000000 1.0000000 0.00000000.0000000 0.0000000 1.0000000Select Xc = PBE-GGA, R_Kmax = 7, L_max = 8, and nkpoint = 5000Two dimensional search of Equation Of State for Ti by using 2Doptimize package.Equation of state: EOS2 (PRB52,8064) info 2a,b,c,d -3413.342702 15.961321 -415.6313341506.627412V0,B(GPa),BP,E0 234.6370 116.1081 3.7701Equation of state: Murnaghan info 2E=E0+[B*V/BP*(1/(BP-1)*(V0/V)**BP +1)-B*V0/(BP-1)]/14703.6Pressure=B/BP*((V0/V)**BP -1)V0,B(GPa),BP,E0 234.6314 115.7348 3.7979 -3415.259330vol energy de(EOS2) de(Murnaghan)Pressure(GPa)214.2200 -3415.251234 0.000003 0.000003 12.583226.1212 -3415.258025 -0.000014 -0.000015 4.590238.0223 -3415.259167 0.000023 0.000026 -1.616249.9234 -3415.255761 -0.000017 -0.000020 -6.497261.8245 -3415.248898 0.000005 0.000006 -10.380Sigma: 0.000014 0.00001615

Equation of state: Birch-Murnaghan info 2E = E0 + 9/16*(B/14703.6)*V0*[(eta**2-1)**3*BP + (eta**2-1)**2*(6-4*eta**2)]--> eta = (V0/V)**(1/3)Pressure = 3/2*B*(eta**7 - eta**5)*(1 + 3/4*(BP-4)*[eta**2 - 1])V0,B(GPa),BP,E0 234.6386 116.1166 3.7650 -3415.259333vol energy de(Birch-Murnaghan) Pressure(GPa)214.2200 -3415.251234 0.000003 12.545226.1212 -3415.258025 -0.000013 4.603238.0223 -3415.259167 0.000023 -1.618249.9234 -3415.255761 -0.000017 -6.506261.8245 -3415.248898 0.000005 -10.353Sigma: 0.000014After running job files, you will find the below values for elastic constants of Ti( do not forget to copy Ti.outputeos in "Ti" directory within c11+c12, c11-c12, c33, c55, andczz directories ).In the following examples you can find the percents that were used for strains.######################################## H_ana_elast_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ################################################################################### H_ana_elastc_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## using case.outputeos ## VstVene ## which have been created by ## HEX.job ###########################################-0.020000 -3415.258474-0.010000 -3415.2592480.000000 -3415.2591450.010000 -3415.2582040.020000 -3415.256548===============================================================Order of fit: 2 C11+C12 is: 255.0521 GPa, RMS: 0.231016E-04Order of fit: 3 C11+C12 is: 255.0521 GPa, RMS: 0.440928E-05Order of fit: 4 C11+C12 is: 264.5953 GPa, RMS: 0.352246E-12******************************************Polynomial fit for C11+C12 doneA RMS of 0.231016E-04 was achieved using a polynome of degree : 2At volume= 234.6314 bohr^3C11+C12 is: 0.017338 a.u or 255.0521 GPa******************************************16

Analyze done.....Do you want a hardcopy? (y/N)***************************************You can find data in ELC.output file.***************************************########################################### H_ana_elastc_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## using case.outputeos ## VstVene ## which have been created by ## HEX.job ###########################################-0.020000 -3415.259032-0.010000 -3415.2592540.000000 -3415.2591450.010000 -3415.2587510.020000 -3415.258066===============================================================Order of fit: 2 C33 is: 187.8748 GPa, RMS: 0.622939E-05Order of fit: 3 C33 is: 187.8748 GPa, RMS: 0.279662E-05Order of fit: 4 C33 is: 175.7690 GPa, RMS: 0.203369E-12******************************************Polynomial fit for C33 doneA RMS of 0.622939E-05 was achieved using a polynome of degree : 2At volume= 234.6314 bohr^3C33 is: 0.012771 a.u or 187.8748 GPa******************************************Analyze done.....Do you want a hardcopy? (y/N)***************************************You can find data in ELC.output file.***************************************########################################### H_ana_elastc_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## using case.outputeos ## VstVene ## which have been created by ## HEX.job ###########################################-0.020000 -3415.258881-0.010000 -3415.2590810.000000 -3415.2591450.010000 -3415.2591220.020000 -3415.258927===============================================================Order of fit: 2 Czz is: 353.0051 GPa, RMS: 0.899789E-05Order of fit: 3 Czz is: 353.0051 GPa, RMS: 0.735610E-05Order of fit: 4 Czz is: 209.7144 GPa, RMS: 0.287607E-12******************************************17

Polynomial fit for Czz doneA RMS of 0.899789E-05 was achieved using a polynome of degree : 2At volume= 234.6314 bohr^3Czz is: 0.023997 a.u or 353.0051 GPa******************************************Analyze done.....Do you want a hardcopy? (y/N)***************************************You can find data in ELC.output file.***************************************########################################### H_ana_elastc_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## using case.outputeos ## VstVene ## which have been created by ## HEX.job ###########################################-0.020000 -3415.258482-0.010000 -3415.2590130.000000 -3415.2591450.010000 -3415.2589620.020000 -3415.258437===============================================================Order of fit: 2 C11-C12 is: 108.6240 GPa, RMS: 0.990173E-05Order of fit: 3 C11-C12 is: 108.6240 GPa, RMS: 0.600696E-05Order of fit: 4 C11-C12 is: 95.6228 GPa, RMS: 0.909495E-12******************************************Polynomial fit for C11-C12 doneA RMS of 0.990173E-05 was achieved using a polynome of degree : 2At volume= 234.6314 bohr^3C11-C12 is: 0.007384 a.u or 108.6240 GPa******************************************Analyze done.....Do you want a hardcopy? (y/N)***************************************You can find data in ELC.output file.***************************************########################################### H_ana_elastc_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## using case.outputeos ## VstVene ## which have been created by ## HEX.job ###########################################-0.020000 -3415.273202-0.010000 -3415.2736520.000000 -3415.27374318

0.010000 -3415.2736520.020000 -3415.273202===============================================================Order of fit: 2 C55 is: 44.3437 GPa, RMS: 0.187853E-04Order of fit: 3 C55 is: 44.3437 GPa, RMS: 0.187853E-04Order of fit: 4 C55 is: 24.0148 GPa, RMS: 0.406738E-12******************************************Polynomial fit for C55 doneA RMS of 0.187853E-04 was achieved using a polynome of degree : 2At volume= 234.6314 bohr^3C55 is: 0.003014 a.u or 44.3437 GPa******************************************Analyze done.....Do you want a hardcopy? (y/N)***************************************You can find data in ELC.output file.***************************************Printing final Elastic constant At voulme= 234.6314 bohr^3 .=======================================================================C11+C12 = 255.0521 GPa C11-C12 = 108.6240 GPaC33 = 187.8748 GPaC55 = 44.3437 GPaCzz = C11+C12+2C33-4C13 = 353.0051 GPa=======================================================================LU decomposition successfulInverse SuccessfulYou can find Inverse Matrix in INVELC-matrix fileDone=======================================================================Atom name = TiAtomic Mass from Periodic table = 47.8670 (gr/mol)Atomic Mass from Periodic table = 7.9486*10^(-23) (gr)Volume in unit of cm^3 = 34.7687*10^(-24) (cm^3)Mass of Compound : 15.8973*10^(-23) (gr)Density of Compound : 4.5723 (gr/cm^3)=======================================================================C11 = 181.8380 GPaC12 = 73.2140 GPaC13 = 69.4491 GPaC33 = 187.8748 GPaC55 = 44.3437 GPa=======================================================================Prediction VOIGT Bulk modulus by using elastic constant valuesPrediction REUSS Bulk modulus by using elastic constant valuesPrediction HILL Bulk modulus by using elastic constant values= 108.419 (GPa)= 108.415 (GPa)= 108.417 (GPa)Prediction VOIGT Shear modulus by using elastic constant values = 51.228 (GPa)Prediction REUSS Shear modulus by using elastic constant values = 50.543 (GPa)Prediction HILL Shear modulus by using elastic constant values = 50.885 (GPa)Prediction VOIGT Young modulus by using elastic constant values = 132.772 (GPa)19

Prediction REUSS Young modulus by using elastic constant values = 131.235 (GPa)Prediction HILL Young modulus by using elastic constant values = 132.003 (GPa)Prediction VOIGT Poisson's coefficient by using elastic constant values = .295Prediction REUSS Poisson's coefficient by using elastic constant values = .298Prediction HILL Poisson's coefficient by using elastic constant values = .297================================================================================By using HILL dataTransverse elastic wave velocity = 3336.02 (m/s)Longitudinal elastic wave velocity = 6208.9 (m/s)The average wave velocity = 3724.88 (m/s)Debye Temperature = 428.093 (K)=====================================================================Press enter key to continue....############################################### H_ana_elastorder_lapw analyses Elastic ## constant ## C(2012) by Morteza Jamal ## ELCorder.fit files ###############################################CHECK THE SENSITIVITYOF YOUR RESULT TO THE ORDER OF FITPress enter key to continue....Order of fit for calculations were 4,4,4,4 and 4We select minimum value for ORDER OF FIT i.e. 4Press enter key to continue....######## ORDER OF FIT IS : 2 , At volume = 234.6314 (bohr^3) ########(c11-c12) = 108.624 (GPa)(c11+c12) = 255.052 (GPa)(c33) = 187.875 (GPa)(c55) = 44.344 (GPa)(czz=c11+c12+2c33-4c13) = 353.005 (GPa)________________________________________c11 = 181.838 (GPa)c12 = 73.214 (GPa)c13 = 69.449 (GPa)c33 = 187.875 (GPa)C55 = 44.344 (GPa)=======================================================================Prediction VOIGT Bulk modulus by using elastic constant valuesPrediction REUSS Bulk modulus by using elastic constant valuesPrediction HILL Bulk modulus by using elastic constant values= 108.419 (GPa)= 108.416 (GPa)= 108.417 (GPa)Prediction VOIGT Shear modulus by using elastic constant values = 51.229 (GPa)Prediction REUSS Shear modulus by using elastic constant values = 50.544 (GPa)Prediction HILL Shear modulus by using elastic constant values = 50.886 (GPa)20

Prediction VOIGT Young modulus by using elastic constant values = 132.774 (GPa)Prediction REUSS Young modulus by using elastic constant values = 131.237 (GPa)Prediction HILL Young modulus by using elastic constant values = 132.005 (GPa)Prediction VOIGT Poisson's coefficient by using elastic constant values = .295Prediction REUSS Poisson's coefficient by using elastic constant values = .298Prediction HILL Poisson's coefficient by using elastic constant values = .297=======================================================================######## ORDER OF FIT IS : 3 , At volume = 234.6314 (bohr^3) ########(c11-c12) = 108.624 (GPa)(c11+c12) = 255.052 (GPa)(c33) = 187.875 (GPa)(c55) = 44.344 (GPa)(czz=c11+c12+2c33-4c13) = 353.005 (GPa)________________________________________c11 = 181.838 (GPa)c12 = 73.214 (GPa)c13 = 69.449 (GPa)c33 = 187.875 (GPa)C55 = 44.344 (GPa)=======================================================================Prediction VOIGT Bulk modulus by using elastic constant valuesPrediction REUSS Bulk modulus by using elastic constant valuesPrediction HILL Bulk modulus by using elastic constant values= 108.419 (GPa)= 108.416 (GPa)= 108.417 (GPa)Prediction VOIGT Shear modulus by using elastic constant values = 51.229 (GPa)Prediction REUSS Shear modulus by using elastic constant values = 50.544 (GPa)Prediction HILL Shear modulus by using elastic constant values = 50.886 (GPa)Prediction VOIGT Young modulus by using elastic constant values = 132.774 (GPa)Prediction REUSS Young modulus by using elastic constant values = 131.237 (GPa)Prediction HILL Young modulus by using elastic constant values = 132.005 (GPa)Prediction VOIGT Poisson's coefficient by using elastic constant values = .295Prediction REUSS Poisson's coefficient by using elastic constant values = .298Prediction HILL Poisson's coefficient by using elastic constant values = .297=======================================================================######## ORDER OF FIT IS : 4 , At volume = 234.6314 (bohr^3) ########(c11-c12) = 95.623 (GPa)(c11+c12) = 264.595 (GPa)(c33) = 175.769 (GPa)(c55) = 24.015 (GPa)(czz=c11+c12+2c33-4c13) = 209.714 (GPa)________________________________________c11 = 180.109 (GPa)c12 = 84.486 (GPa)c13 = 101.604 (GPa)c33 = 175.769 (GPa)C55 = 24.015 (GPa)=======================================================================Prediction VOIGT Bulk modulus by using elastic constant valuesPrediction REUSS Bulk modulus by using elastic constant valuesPrediction HILL Bulk modulus by using elastic constant values= 123.486 (GPa)= 123.312 (GPa)= 123.399 (GPa)21

Prediction VOIGT Shear modulus by using elastic constant values = 35.720 (GPa)Prediction REUSS Shear modulus by using elastic constant values = 32.517 (GPa)Prediction HILL Shear modulus by using elastic constant values = 34.118 (GPa)Prediction VOIGT Young modulus by using elastic constant values = 97.736 (GPa)Prediction REUSS Young modulus by using elastic constant values = 89.669 (GPa)Prediction HILL Young modulus by using elastic constant values = 93.716 (GPa)Prediction VOIGT Poisson's coefficient by using elastic constant values = .368Prediction REUSS Poisson's coefficient by using elastic constant values = .378Prediction HILL Poisson's coefficient by using elastic constant values = .373=======================================================================You can find these data in the output-order file.C11 = 182 GPaC12 = 73 GPaC13 = 69 GPaC33 = 188 GPaC55 = 44 GPaOur calculation Other(TB) 6 Exp 6C11 182 171 160C12 73 58 90C13 69 46 60C33 188 203 181C55 44 64 4722

5– Installation of the Hex-elastic packageThe Hex-elastic package comes as a compressed tar file namely “hex-elastic-deriativemethod-2012-P.tar.gz”or “hex-elastic.tar.gz“. To install the package firstly copy the fileto a directory of your choice.Now, uncompress and expand it as: tar –zxvfhex-elastic.tar.gz cd hex-elastic23

Run buildHIRelast_lapwThis program helps you to create the "Makefile" and then compile hex-elastic. By default,the Makefile expects the lapack_lapw and blas_lapw to be in the location ../SRC_lib.This should be changed to the correct location by modifying the FOPT parameter asshown below.This Program helps you to define Fortran compiler, Fortran options, and Libraryoptions if you have installed WIEN2k. As you can see here this program definesFortran compiler, Fortran options, and Library options as automatically. Otherwiseyou can define compiler and linker options as well as the path of mkl librarydepending on the your selected system./home/MyLib/mkl/lib/em64t is the path of my mkl library.To make “Makefile” by the lapack_lapw and blas_lapw libraries in the location../SRC_lib and gfortran use the following options:Fortran compiler: gfortranFortran options: -ffree-formLibrary options (Lapack and BLAS): $(FOPT) –L/home/physicsprogram/SRC_lib–lpthread –static –llapack_lapw –lblas_lapwthe location ../SRC_lib should be changed to the correct location by modifying theFOPT parameter as shown above.PS: To install with -ffree-form, you should compile the lapack_lapw andblas_lapw libraries with -ffree-form options. Otherwise it might was caused error.24

If you view the OPTIONS file of the WIEN2k package you can use the FOPT,LDFLAGS, and R_LIBS of it for compiling.After defining the Fortran compiler, Fortran options, and Library options press Enter key.25

The Environment Variable ELASTH_PATH is then defined and added to the end of the.bashrc file. Thus you will be able to call hex-elastic’s programs for any location.26

If you view the .bashrc file you can seeNow, logout from your Linux system and then login.27

6– References[1] R. Stadler, W. Wolf, R. Podloucky, G. Kresse, J. Furthmller, J. Hafner, Phys. Rev. B54 (1996) 1729.[2] L. Fast, J. M. Wills, B. Johansson and O. Eriksson, Phys. Rev. B 51 (1995).[3] D.C. Wallace, Solid State Phys. 25 (1970) 301[4] http://cst-www.nrl.navy.mil/[5] B. Z. Yanchitsky, A. N. Timoshevskii, Determination of the space group and unit cellfor a periodic solid, Comp. Phys. Comm. 139 (2001) 235–242.[6] Michael J. Mehl and Dimitrios A. Papaconstantopoulos, Phys. Rev. B 54 (1996).28