Projected energies

Within the GOA, the Fourier transforms above can be analytically calculated, and the projected energy of Eq. (4) reads [1]

$$E^{\mbox{\scriptsize {GOA}}}_\Phi(\bm{P}) = h_0-E_{\mbox{\sc... ...{ZPM}}}+\frac{\hbar^2\bm{P}^2}{2M_{\mbox{\scriptsize {PY}}}} ,$$ (9)

where the so-called zero-point-motion correction and Peierls-Yoccoz (PY) mass are given by

$$E_{\mbox{\scriptsize {ZPM}}} = 3h_2/2a ,$$ (10)

$$M_{\mbox{\scriptsize {PY}}} = \hbar^2a^2/h_2 .$$ (11)

For completeness, in Figs. 3 and 4, I show the mass dependence of the GOA parameters $a$, $h_2$, $E_{\mbox{\scriptsize {ZPM}}}$, and $M_{\mbox{\scriptsize {PY}}}$, along with the fits of the power-law dependencies. One clearly sees that the PY mass is not equal to the total mass $mA$ of the nucleus. However, as discussed below, it is the PY mass, and not the physical mass $mA$, which is important for the translational-symmetry restoration.

Figure 3: (Color online) Dots show parameters $a$ (bottom) and $h_2$ (top) that are calculated by the parabolic fits to the logarithms of the overlap kernels (Fig. 1) and reduced energy kernels (Fig. 2). Lines give estimates of the power-law dependence.

Figure 4: (Color online) The GOA zero-point-motion corrections $E_{\mbox{\scriptsize{ZPM}}}$ and PY masses $M_{\mbox{\scriptsize{PY}}}$ (inset) calculated for the values of parameters $a$ and $h_2$ that are shown in Fig. 3.

The average energy of the system at rest, $E_\Phi(\bm{0})$, depends on the Slater determinant $\vert\Phi\rangle$, and in what follows we are interested in minimizing this energy with respect to $\vert\Phi\rangle$, that is, in performing the VAP or variation after symmetry restoration. To this end, in this study I follow the seminal idea by Harry Lipkin [6], who realized that the VAP calculations can be very easily performed by flattening the function $E_\Phi(\bm{P})$.

Indeed, from Eqs. (3) and (4) one obtains the so-called sum-rule property,

$$\int {\rm {d}}^3 \bm{P} E_\Phi(\bm{P}){\langle\Phi\vert\bm{P}\rangle}= \langle\Phi\vert\hat{H}\vert\Phi\rangle,$$ (12)

which tells us that the MF energy $\langle\Phi\vert\hat{H}\vert\Phi\rangle$ is equal to the average of the projected energies $E_\Phi(\bm{P})$ weighted by amplitudes $\langle\Phi\vert\bm{P}\rangle$. Therefore, minimization of $\langle\Phi\vert\hat{H}\vert\Phi\rangle$, that is, the standard MF method, corresponds to an entangled minimization of both the projected energies $E_\Phi(\bm{P})$ and amplitudes ${\langle\Phi\vert\bm{P}\rangle}$, whereas the VAP method pertains to the minimization of the energy $E_\Phi(\bm{0})$ only, and, of course, disregards the amplitudes ${\langle\Phi\vert\bm{P}\rangle}$ completely.

Note that the GOA amplitudes ${\langle\Phi\vert\bm{P}\rangle}$ are all positive,

$${\langle\Phi\vert\bm{P}\rangle}=\left(\frac{2\pi{a}}{\hbar^2... ...2} \exp\left(-{\textstyle{\frac{\hbar^2}{2a}}}\bm{P}^2\right),$$ (13)

and hence the MF minimization is bound to underestimate the momentum spread of the Slater determinant. It is then obvious that unless the center-of-mass and internal degrees of freedom are exactly separated, like is the case for the closed-shell HO [11,12,13] or for the coupled-cluster states [14], the MF and VAP Slater determinants can be different.