Dear Hussein,

there are several methods. The simplest way (as it is directly available in VASP), is using the stress-strain relationship, as described in the VASP manual (this describes the finite difference method, linear response is also possible):

http://cms.mpi.univie.ac.at/vasp/vasp/IBRION_5_IBRION_6.html

Another version of this approach would be to use the universal independent coupling strain (ULIC) method - which in principle needs fewer strained states, however you need to create strained configurations and determine Cij yourself. It is describe here:

R., Zhu J., Ye H. Calculations of single-crystal elastic constants made simple. Comput. Phys. Commun. 2010;181(3):671–675

A different approach would be to calculate it via the strain energy (again you need to create strained configurations). See e.g.

David Holec, Martin Friák, Jörg Neugebauer, and Paul H. Mayrhofer

Phys. Rev. B 85, 064101

Best,

Holger

Thank you so much for your answer, but what I need is to know or how I can  properly build  the INCAR file resulting in getting Cij  in VASP.. Please, which exact parameters do I need in INCAR file to get the correct Cij ones ?

Please check the above link in the manual where it is described (essentially it is setting IBRION = 6 and ISIF=3):

http://cms.mpi.univie.ac.at/vasp/vasp/IBRION_5_IBRION_6.html

Be aware, that before doing that you have to relax your structure (atomic positions and lattice parameter).

Dear Holger,  I checked the link and I have already set IBRION =6 ,ISIF=3 , of course ,after optimizing the structure. Below is the content of my INCAR file and please , look at it to correct me if I am wrong

Start parameter fo this run

ISTART=1

EDIFFG=1E-5

EDIFF=1E-2

Electronic Relaxation

ENCUT=500.0

LREAL=A

ALGO=FAST

MAXMIX=80

PREC=Medium

ISPIN=1

AMIN=0.01

Ionic Relaxation

IBRION=6

POTIM = 0.015

NSW=1

ISIF=3

NBANDS=408

LASPH = .TRUE

DOS realted values

ISMEAR=2

SIGMA = 0.3

EMAX=15

EMIN=-20

LORBIT=11

NDOS=3500

PREC=Medium

ISPIN=2

ICHARG=1

IALGO=48

LPLANE=T

The values depend of course on the system you calculate so it is hard to judge.

A few things that I can say from quickly looking at it are:

-Certainly you should not define parameters more than once (you have e.g. set ISPIN =1 and ISPIN=2).

-The sigma value is rather big - for metals ISMEAR=2 and SIGMA=0.2 is usually recommended.

-The LASPH is a rather special option - are you sure you need it.

-The DOS values like EMAX, EMIN, NDOS are not really necessary. For an exact DOS calculation you would rather do a separate run with ISMEAR=-5.

-PREC you might go for Accurate (at least to compare and check if it is necessary)

-EDIFF is usually set smaller than EDIFFG. EDIFF ~10-5 or 10-6 is reasonable.