$^7$Be($p,\gamma$)$^8$B S-factor from {\it ab initio} wave functions: I. Overlap integral calculation*

Nuclear structure of $^7$Be and $^8$B is studied within the {\it ab initio} no-core shell model (NCSM). Starting from realistic inter-nucleon interactions, $^7$Be and $^8$B wave functions are obtained in basis spaces up to $10\hbar\Omega$ with the matrix dimensions exceeding $7\times 10^7$. These wave functions are then used to calculate cluster form factors or overlap integrals of the $^8$B ground state with the $^7$Be$+p$ as a dependence on the separation between the proton and the center-of-mass of $^7$Be. Due to the use of the harmonic oscillator basis, the overlap integrals have incorrect asymptotics. To fix this problem, we perform a least-square fit of Woods-Saxon potential solutions to the NCSM overlap integrals in the 0-4 fm range under the constraint that the experimental $^8$B binding energy with respect to the $^7$Be$+p$ is reproduced. The corrected overlap integrals can then be used for the $^7$Be($p,\gamma$)$^8$B S-factor calculation. Support from the LDRD contract No. 04-ERD-058 as well as partial support from the DOE grants SCW0498 and DE-FG02-04ER41338 is acknowledged.