HO and THO Wave Functions

The solution of the HFB equation (16) is obtained by expanding the quasiparticle function (24) in a given complete set of basis wave functions that conserve axial symmetry and parity. The program HFBTHO (v1.66p) is able to do so for the two basis sets of wave functions: HO and THO.

The HO set consists of eigenfunctions of a single-particle Hamiltonian for an axially deformed harmonic oscillator potential. By using the standard oscillator constants:

$$\beta_z=\frac{1}{b_z}=\left(\frac{m\omega_z}{\hbar}\right)^{... ...frac{1}{b_\bot}=\left(\frac{m\omega_\bot}{\hbar}\right)^{1/2},$$ (27)

and auxiliary variables

$$\xi=z \beta_z,~~~\eta=r^2 \beta_\bot^2,$$ (28)

the HO eigenfunctions are written explicitly as

$$\Phi_\alpha({\bf r},\sigma)=\psi^\Lambda_{n_r}(r) \psi_{n_z}... ...rac{e^{\imath\Lambda\varphi}}{\sqrt{2\pi}}\chi_\Sigma(\sigma),$$ (29)

where

$$\begin{array}{rcl} \psi^\Lambda_{n_r}(r) &=&\beta_\bot \tild... ...i) = N_{n_z}\beta_z^{1/2}e^{-\xi^2/2} H_{n_z}(\xi). \end{array}$$ (30)

$H_{n_z}(\xi)$ and $L^\Lambda_{n_r}(\eta)$ denote the Hermite and associated Laguerre polynomials [31], respectively, and the normalization factors read

$$N_{n_z}=\left(\frac{1}{\sqrt{\pi}2^{n_z}n_z!}\right)^{1/2} ~... ...n_r}=\left(\frac{n_r!}{(n_r+\vert\Lambda\vert)!}\right)^{1/2}.$$ (31)

The set of quantum numbers $\alpha=\{n_r,n_z,\Lambda,\Sigma\}$ includes the numbers of nodes, $n_r$ and $n_z$, in the $r$ and $z$ directions, respectively, and the projections on the $z$ axis, $\Lambda$ and $\Sigma$, of the angular momentum operator and the spin.

The HO energy associated with the HO state (29) reads

$$\epsilon_\alpha=(2n_r+\vert\Lambda\vert+1)\hbar\omega_\bot+(n_z+{\textstyle{\frac{1}{2}}})\hbar\omega_z,$$ (32)

and the basis used by the code consists of $M_0$=$(N_{sh}$+$1)(N_{sh}$+$2)(N_{sh}$+$3)/6$ states having the lowest energies $\epsilon_\alpha$ for the given frequencies $\hbar\omega_\bot$ and $\hbar\omega_z$. In this way, for the spherical basis, i.e., for $\hbar\omega_\bot$=$\hbar\omega_z$, all HO shells with the numbers of quanta $N$=0...$N_{sh}$ are included in the basis. When the basis becomes deformed, $\hbar\omega_\bot$$\neq$$\hbar\omega_z$, the code selects the lowest-HO-energy basis states by checking the HO energies of all states up to 50 HO quanta. Note that in this case the maximum value of the quantum number $\Omega_k$, and the number of blocks in which the HFB equation is diagonalized, see Sec. 3.4, depend on the deformation of the basis.

The THO set of basis wave functions consists of transformed harmonic oscillator functions, which are generated by applying the local scale transformation (LST) [19,32,20] to the HO single-particle wave functions (29). In the axially deformed case, the LST acts only on the cylindrical coordinates $r$ and $z$, i.e.,

$$\begin{array}{llll} r & \longrightarrow & r^{\prime }\equiv ... ...prime }(r ,z) & = z \,\frac{f({\cal R})}{{\cal R}}, \end{array}$$ (33)

and the resulting THO wave functions read

$$\Phi_\alpha({\bf r},\sigma)=\sqrt{\frac{f^{2}({\cal R})}{ {\... ...rac{e^{\imath\Lambda\varphi}}{\sqrt{2\pi}}\chi_\Sigma(\sigma),$$ (34)

where

$${\cal R}=\sqrt{\frac{z^{2}}{b_{z}^{2}}+\frac{r^{2}}{b_\bot^{2}}},$$ (35)

and $f({\cal R})$ is a scalar LST function. In the code HFBTHO (v1.66p), function $f({\cal R})$ is chosen as in Ref. [23]. It transforms the incorrect Gaussian asymptotic behavior of deformed HO wave functions into the correct exponential form. Below, we keep the same notation $\Phi_\alpha({\bf r},\sigma)$ for both HO and THO wave functions, because expressions in which they enter are almost identical in both cases and are valid for both HO and THO variants.