| Hardness is a fundamental physical quantity describing macro physical property of materials, and is often considered in practical applications. In applications, hard and superhard materials are widely used and demanded. Hence, accurate values of hardness are of significant for materials'application. As the improving of the technology of measurement and experiment, people can get the value of hardness more easily and precise. However, many materials are only predicted and have not been synthesized, or there is no synthesized bulk samples for these materials and their hardness cannot be measured through experiment methods. For this kind of materials, people try to get some relevant physical quantities through stimulation and calculation, and then use these quantities (and, often, with other empirical parameters) to calculate the value of hardness. On the other hand, the hardness, as a macroscopic physical property of materials, has no accurate corresponding relationship to microscopic physical properties, so it cannot be deduced strictly and calculated through rigorous first-principle methods. The constructions of hardness calculating formulas are always based on the proposers'understanding to physics concepts and consideration to physical picture.In recent years, many based-on first-principle empirical and semi-empirical formulas were proposed,usually their calculation results can be consistent with experimental values pretty well in some scales. However, at the same time, there are some problems and not clarified points in these methods. In this thesis, we discussed and tried to correct the proposed by Prof. Faming Gao et al in 2006 method of hardness calculation, which calculate the hardness of materials by using the qualities of bond length and Mulliken overlap population, both of which are got by performing first-principle calculation. Gao's formula are as follows: (?)whereμis used as indices to indicate the kind of bonds, Hvμis the Vickers Hardness ofμ-type bond, Pμis the Mulliken overlap population of theμ-type bond, andνbμis the volume of a bond of typeμ. Gao chose the calculated hardness of the weakest bond as Hv, the hardness of materials.Our discussion mainly include the content as follows::Firstly,we pointed out and analyzed the rationality of this method. It is widely accepted that the absolute magnitude of the atomic charges yielded by population analysis have little physical meaning, since they display a high degree of sensitivity to the atomic basis set with which they were calculated. However, consideration of their relative values can yield useful information, provided a consistent basis set is used for their calculation. Therefore, calculations which include the values of Mulliken Overlap Population can be reasonable, provided we can find parameters, through which the relative quantity can be connected with the absolute physical quantity. The parameter in Gao's formula is the scale factor A (in this context, A is corresponding to the basis set of wave functions used by CASTEP).How to get the value of A: we used the experimental measured hardness value of diamond, the calculated Mulliken Overlap Population Pμ, and the calculated bond volume as known quantities, putting these values into Gao's formula, and then calculated out the undetermined coefficient A. According to the analysis in the above passage, the value of A would not change as the change of the kind of materials (only provided the basis set did not change).Secondly,we point out the existing problems of Gao's formula:(1) Calculate the hardness of materials only by using the Mulliken Overlap Population value of the weakest bond may not be proper;(2) There are size effects of simulation cell on the Hardness.Then we did the corrections as follows: About problem (1) we think that crystals are framework structures constructed by bonds and atoms, and for this kind of structure, until now, there are no sufficient evidence can support the opinion that the hardness of materials are consistent with the calculate Hvμof the weakest bond. We cannot accept this opinion only because such limited data provided by Gao in his paper. Therefore, the hardness Hv of complex crystals should be calculated by a geometric average of all bonds of different types.About problem (2) we found that the size and shape sensibility of hardness values calculated through Gao's formula are just because in this formula we used the Mulliken Overlap Population values which are sensibility to the size and shape of the simulation cells. And the reasons why the of Mulliken Overlap Population values have sensibility to the size and shape of the simulation cells are as follow: the wave functions of electrons, which should have been extended unlimited in space, are actually transmitted back into the simulation cell because of the Born-von Karman boundary conditions. The transmitted back electrons wave functions overlap with the original electrons wave functions and affect the calculation of Mulliken Overlap Population values. Therefore, the smaller the stimulation cells are the larger proportion the boundary and near boundary atoms occupy, and the larger of the deviation to real Mulliken Overlap Population values are. So we get the conclusion that using the Mulliken Overlap Population to measure the bond strength is proper, provided that the simulation cells are large enough so that the portion of transmitted back wave functions are smaller enough and can be ignored when calculating the Mulliken Overlap Population. For verifying this conclusion,we did the calculation of enlarging cells for many kinds of crystal materials and the result was consistent with the our discussion above. For example, for materials of zinc blende structure,their Mulliken Overlap Populations converge to certain values when the number of atoms in the simulation cells is larger than 80.According to our discussion above and calculating results,we did the corrections to Gao's formula as follows:First,we calculated the hardness Hv by a geometric average of all bonds of different types, then we got formula below. (?)Second, we get, in some precision, converging to certain values Mulliken Overlap Population through enlarging the simulation cells, and put these converged Mulliken Overlap Population values back to theabove formula. Finally, we figure out the new A, namely Arevised. After the two-step correction above, the final form of the formula is: (?)whereμis used as indices to indicate the kind of bonds, n is the sum of bond types in simulation cell. (B.3) could be more widely used and provides more accurate hardness for a material.
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