Kernels

From Eq. (2) one sees that the projected energy (4) is given by the Fourier transforms of the overlap and energy kernels, which are defined by

$$I(\bm{R}) = \langle\Phi\vert\Phi(\bm{R})\rangle ,$$ (5)

$$H(\bm{R}) = \langle\Phi\vert\hat{H}\vert\Phi(\bm{R})\rangle ,$$ (6)

respectively. Therefore, properties of the kernels must be discussed first. Moreover, all results below depend only on the kernels; hence these results apply automatically to the EDF approaches, where very often does not start with the Hamiltonian but the diagonal energy density is extended to the energy kernel [8].

In Figs. 1 and 2 are shown, respectively, logarithms of the overlap kernels, $\ln(I(\bm{R}))$, and reduced energy kernels, $h(\bm{R})=H(\bm{R})/I(\bm{R})$, calculated in 9 doubly-magic spherical nuclei from $^{4}$He to $^{208}$Pb. Calculations were performed for the SLy4 [9] parametrization of the EDF, by using the code HFODD [10] (v2.40g) and the harmonic-oscillator (HO) basis of up to $N_0=14$ shells.

Figure 1: (Color online) Logarithms of the overlap kernels, $\ln(I(\bm{R}))$, calculated in 9 doubly-magic spherical nuclei (dots). Thick lines represent parabolic fits to the results, determined for $\vert\bm{R}\vert\leq3$fm.

Figure 2: (Color online) Same as in Fig. 1 but for the reduced energy kernels, $h(\bm{R})=H(\bm{R})/I(\bm{R})$. To guide the eye, thin lines connect calculated values (dots).

It can be seen that in the case of the translational symmetry, the so-called Gaussian Overlap Approximation (GOA) [1], given by

$$I(\bm{R}) = \exp(-{\textstyle{\frac{1}{2}}}a\bm{R}^2) ,$$ (7)

$$h(\bm{R}) = h_0-{\textstyle{\frac{1}{2}}}h_2\bm{R}^2 ,$$ (8)

is excellent. Sudden deviations of $h(\bm{R})$ from the parabolic dependence on $\bm{R}$, which can be seen in Fig. 2 around $\vert\bm{R}\vert=4$fm, are due to the finiteness of the HO basis used in the calculations. Nevertheless, values of $h_2$ can be very precisely determined in the parabolic region. This was confirmed by repeating the calculations for $N_0=20$ HO shells, whereby the above deviations appear around $\vert\bm{R}\vert=4.5$fm and the values of $h_2$ stay exactly the same.