I hope the following construction will give you what you really need. If not, you'll have to explain why.
Take any nice locally finite covering $\mathbb R\subset\cup_j U_j$ and take any smooth partition of unity $1=\sum_j\psi_j^2$ subordinated to this covering. Take any smooth positive function $F$ on $\mathbb R$. Choose the numbers $n_j>0$ so that $\sum_j n_j(\psi_j^2+(\psi_j')^2)\le F/2$. Define the smooth function $\theta$ by
$$
(\theta^{\,\prime})^2\sum_j n_j\psi_j^2=F-\sum_j n_j(\psi_j^2+(\psi_j')^2)
$$
Finally, associate with each $U_j$ two functions $\varphi_{j,0}=\psi_j\cos\theta$ and $\varphi_{j,1}=\psi_j\sin\theta$ and enjoy the identities
$$
\sum_{j,k}\varphi_{j,k}^2=1, \quad \sum_{j,k}n_j(\varphi_{j,k}^2+(\varphi_{j,k}')^2)=F\,.
$$