HFB Diagonalization in Configurational Space

We use the same basis wave functions to expand upper and lower components of the quasiparticle states, i.e.,

$$\begin{array}{rcl} U_k({\bf r},\sigma,\tau)&=&\displaystyle\... ...m_\alpha V_{k\alpha}\Phi_\alpha({\bf r},\sigma),\\ \end{array}$$ (36)

where $\Phi_\alpha({\bf r},\sigma)$ are the HO or THO basis states. Note that the same basis $\Phi_\alpha({\bf r},\sigma)$ is used for protons and neutrons.

Inserting expression (36) into the HFB equation (16) and using the orthogonality of the basis states, we find that the expansion coefficients have to be eigenvectors of the HFB Hamiltonian matrix

$$\left( \begin{array}{cc} h^{(q_k)}-\lambda^{(q_k)} & \tilde{... ...\left( \begin{array}{l} U_{k} \\ V_{k} \end{array}\right) \;,$$ (37)

where the quasiparticle energies $E_{k}$, the chemical potential $\lambda^{(q_k)}$, and the matrices

$$h^{(q)}_{\alpha\beta}=\langle\Phi_\alpha\vert h_q\vert\Phi_\... ...beta}=\langle\Phi_\alpha\vert\tilde{h}_q\vert\Phi_\beta\rangle$$ (38)

are defined for a given proton ($q_k$=$+1/2$) or neutron ($q_k$=$-1/2$) block.

Proton and neutron blocks are decoupled and can be diagonalized separately. Furthermore, in the case of axially deformed nuclei considered here, $\Omega_k$=$\Lambda_k$+$\Sigma_k$ is a good quantum number and, therefore, matrices $h^{(q)}_{\alpha\beta}$ and $\tilde{h}^{(q)}_{\alpha\beta}$ are block diagonal, each block being characterized by a given value of $\Omega$. Moreover, for the case of conserved parity considered here, $\pi$= $(-1)^{n_z+\Lambda}$ is also a good quantum number, and each of the $\Omega_k$ blocks falls into two sub-blocks characterized by the values of $\pi$=$\pm1$. Finally, due to the time-reversal symmetry, the Hamiltonian matrices need to be constructed for positive values of $\Omega_k$ only.