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The elastic properties of the $B_1$-structured transition-metal nitrides andtheir carbide counterparts are studied using the {\it ab initio\} densityfunctional perturbation theory. The linear response results of elasticconstants are in excellent agreement with those obtained from numericalderivative methods, and are also consistent with measured data. We find thefollowing trends: (1) Bulk moduli $B$ and tetragonal shear moduli$G^{\prime}=(C_{11}-C_{12})/2$, increase and lattice constants $a_{0}$ decreaserightward or downward on the Periodic Table for the metal component or if C isreplaced by N; (2) The inequality $B > G^{\prime} > G > 0$ holds for$G=C_{44}$; (3) $G$ depends strongly on the number of valence electrons perunit cell ($Z_{V}$). From the fitted curve of $G$ as a function of $Z_{V}$, wecan predict that MoN is unstable in $B_{1}$ structure, and transition-metalcarbonitrides ($e.g.$ ZrC$_{x}$N$_{1-x}$) and di-transition-metal carbides($e.g.$ Hf$_{x}$Ta$_{1-x}$C) have maximum $G$ at $Z_{V} \approx 8.3$.Comment: 4 pages, 2 figures, submitted to PRL. 2 typos in ref. 15 were  correcte

Trends in elasticity and electronic structure of transition-metal nitrides and carbides from first principles