N}_{1,1}$ jumps another steady value 1. These two jumps result from the two level crossing as seen clearly from the figure inserted. The second level crossing also leads to a small dip in the curve of ${\cal N}_{1,1}$. The negativity ${\cal
N}_{1/2,1/2}$ displays a jump to a steady value of 1/2 near $J_2=1/2$. It is evident that the entanglement between NNN spins is enhanced by increasing NNN interactions. The competition between NN and NNN interactions leads to rich behaviors of quantum entanglement. Another observation is that there is a range of $J_2$, at which negativities ${\cal N}_{1/2,1}$ and ${\cal
N}_{1/2,1/2}$ are zero, and only ${\cal N}_{1,1}$ is not zero. This indicates that the NNN interaction must be strong enough to build up the entanglement of two spin halves.

*Entanglement versus $J_2$ and $T$*: As temperature increases the entanglement will decrease due to the mixing of less entangled excited states to the thermal state. It is obvious that there exists a threshold temperature after which the negativity is zero. In the frustrated system, there exists the parameter $J_2$, and with its increase, the negativity ${\cal N}_{1/2,1}$ will decrease to zero, while ${\cal N}_{1,1}$ and ${\cal N}_{1/2,1/2}$ increase from zero to their 