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We introduce a new framework for studying clustering and for calculating alpha partial widths using ab initio wave functions. We demonstrate the formalism for $^{20}$Ne, by calculating the overlap between the $^{16}$O$+α$ cluster configuration and states in $^{20}$Ne computed in the ab initio 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 alpha-clustering. Specifically, we report on the alpha partial width of the lowest $1^-$ resonance in $^{20}$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 $^{20}$Ne that lies in a close proximity to the $^{16}$O$+α$ threshold. This outcome highlights the importance of correlations for developing cluster structures and for describing alpha widths. The widths can then be used to calculate alpha-capture reaction rates for narrow resonances of interest to astrophysics. We explore the reaction rate for the alpha-capture reaction $^{16}$O$(α,γ)^{20}$Ne at astrophysically relevant temperatures and determine its impact on simulated X-ray burst abundances.