  Predicting the $B_s^0-\bar{B}_s^0$ width difference $\Delta\Gamma_s$ relies
on the heavy quark expansion and on hadronic matrix elements of $\Delta B=2$
operators. We present the first lattice QCD results for matrix elements of the
dimension-7 operators $R_{2,3}$ and linear combinations $\tilde{R}_{2,3}$ using
nonrelativistic QCD for the bottom quark and a highly improved staggered quark
(HISQ) action for the strange quark. Computations use MILC ensembles of gauge
field configuations with $2+1+1$ flavors of sea quarks with the HISQ
discretization, including lattices with physically light up/down quark masses.
We discuss features unique to calculating matrix elements of these operators
and analyze uncertainties from series truncation, discretization, and quark
mass dependence. Finally we report the first Standard Model determination of
$\Delta\Gamma_s$ using lattice QCD results for all hadronic matrix elements
through $\mathcal{O}(1/m_b)$. The main result of our calculations yields the
$1/m_b$ contribution $\Delta \Gamma_{1/m_b} = -0.022(10)~\mathrm{ps}^{-1}$.
Adding this to the leading order contribution, the Standard Model prediction is
$\Delta \Gamma_s = 0.092(14)~\mathrm{ps}^{-1}$.
