Hamiltonian specified above in (\[eq:Hamiltonian\]), the NCSM solves the $A$-body Schrödinger equation using a matrix formulation $$\label{eq:Schrodinger}
    H_A \Psi_A(\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_A) = E \Psi_A(\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_A),$$ where the $A$-body wave function is given by a linear combination of Slater determinants $\phi_i$ $$\label{eq:Wavefunction}
    \Psi_A(\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_A) = \sum_{i=0}^k c_i \phi_i(\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_A),$$ and where $k$ is the number of many-body basis states, configurations, in the system. To obtain the exact A-body wave function one has to consider infinite number of configurations, $k = \infty$. However, in practice, the sum is limited to a finite number of configurations determined by $N_{\rm max}$. The Slater determinant $\phi_i$ is the antisymmetrized product of single particle wave functions $\phi_\alpha(\vec{r})$, where $\alpha$ stands for the quantum numbers of a single particle state. A common choice for the single particle wave functions is the HO basis functions. The matrix elements of the Hamiltonian in the many-body HO basis is given by $H_{ij} = \langle \phi_i|\hat{H}|\phi_j \rangle$. For these large and sparse Hamiltonian matrices, the Lanczos method is one possible choice to find the extreme eigenvalues [@PARLETT:SymmetricEigenvalueProblem:1998].

To be more specific, our limited many-body HO basis is characterized by two basis space parameters: $\hbar\Omega$ 