USPEX allows the calculation of the hardness.
It involves the concept of hybrid global optimization, where global optimization with respect to the hardness is conducted in the space of local minima of the (free) energy.
The Lyakhov-Oganov model is based on the model of Li et al. (PRL 2009). To calculate hardness H, we used the formula for the Knoop hardness (in GPa):
where V is the volume of the unit cell and Nk is the number of bonds of the type k in the unit cell.
Xk and fk are the electron-holding energy of the bond k and its ionicity indicator,
which are defined as in the original work of Li et al. χik, χjk are the electronegativities of atoms i, j in bonds.
CNik, CNjk are the coordination numbers of atoms i and j. Lyakhov and Oganov (2011) have replaced these discrete and not always well-defined numbers by a continuous quantity specific for each bond in a coordination polyhedron: instead of CN, the ratio valence/bond_strength is used. The bond valence (s) is computed using the classical Brown's bond valence model.
Here the sum goes over all bonds k' in which atom i participates, and Δ is deviation from reference covalent bond length. This definition satifies exactly the sum rule,
The most defining feature of the Lyakhov-Oganov approach is in the use of multicolor graph theory - i.e. not only bond strengths, but also bond topology is used for computing the hardness. For molecular, chain and layered structures, 3-dimensionality is maintained by the weak bonds, which define the hardness of the crystal. Therefore, 2 types of bonds are now included in the hardness calculation:
(1) strong bonds, with s > goodBonds
(2) minimum set of additional weak bonds necessary to maintain a 3D-connected topology.
The details of the methodology are described in Phys. Rev. B. 84, 092103.
To use this feature, we need a few inputs:
(1) The structure file POSCAR (must contain element symbol line).
(2) Set the parameters of goodBond, valence, valence electrons. You could check the manual of USPEX to learn how to set them.
The output file contains the following information: Length : bond length. Delta : deviation from bond length defined by the reference for the univalent covalent radii. N_k : number of bonds of the type k in the unit cell. s_i/s_j : bond valence, see above equations for the details.
Make sure the output bonds are reasonable in order to get correct hardness value.