Clustering and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math>-capture reaction rate from <i>ab initio</i> symmetry-adapted descriptions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mi>Ne</mml:mi><mml:mprescripts/><mml:none/><mml:mn>20</mml:mn></mml:mmultiscripts></mml:math>

We introduce a new framework for studying clustering and for calculating $\ensuremath{\alpha}$ partial widths using ab initio wave functions. We demonstrate the formalism for $^{20}\mathrm{Ne}$, by calculating the overlap between the $^{16}\mathrm{O}+\ensuremath{\alpha}$ cluster configuration and states in $^{20}\mathrm{Ne}$ computed in the $abinitio$ symmetry-adapted no-core shell model. We present spectroscopic amplitudes and spectroscopic factors, and compare those to no-core symplectic shell-model results in larger model spaces, to gain insight into the underlying physics that drives $\ensuremath{\alpha}$ clustering. Specifically, we report on the $\ensuremath{\alpha}$ partial width of the lowest ${1}^{\ensuremath{-}}$ resonance in $^{20}\mathrm{Ne}$, which is found to be in good agreement with experiment. We also present first no-core shell-model estimates for asymptotic normalization coefficients for the ground state, as well as for the first excited ${4}^{+}$ state in $^{20}\mathrm{Ne}$ that lies in a close proximity to the $\ensuremath{\alpha}{+}^{16}\mathrm{O}$ threshold. This outcome highlights the importance of correlations for developing cluster structures and for describing $\ensuremath{\alpha}$ widths. The widths can then be used to calculate $\ensuremath{\alpha}$-capture reaction rates for narrow resonances of interest to astrophysics. We explore the reaction rate for the $\ensuremath{\alpha}$-capture reaction $^{16}\mathrm{O}{(\ensuremath{\alpha},\ensuremath{\gamma})}^{20}\mathrm{Ne}$ at astrophysically relevant temperatures and determine its impact on simulated x-ray burst abundances.