Gaussian overlap approximation

To compare cranking expressions with those obtained within the GOA, it is convenient to introduce the $\cal S$ matrices [23]:

$${\cal S}_{ij}^{(K)}=\sum_{\mu,\nu} \frac{\langle\mu \vert ... ...gle} {(\breve{E}_\mu+\breve{E}_\nu)^{K}} (\eta^+_{\mu\nu})^2.$$ (61)

It is immediately seen that for the mass tensor of Eq. (60) one has ${\cal M}^C = {\cal S}^{(3)}$. In the case of GOA, also assuming weak state dependence of pairing and neglecting the derivatives of $\lambda$ and $\Delta_{\mu\bar\mu}$, one obtains [24,23]

$${\cal M}^{\rm GOA}= {\cal S}^{(2)}\left[{\cal S}^{(1)}\right]^{-1}{\cal S}^{(2)}.$$ (62)

Evaluating the matrix elements of $h^i$ entering Eq. (61) perturbatively, one can express ${\cal S}$ explicitly through the matrix elements of the constraining field operators $\hat Q_i$:

$${\cal S}^{(K)}=\frac{1}{4}\left[{ M}^{(1)}\right]^{-1}{ M}^{(K)} \left[{ M}^{(1)}\right]^{-1},$$ (63)

where the energy-weighted moments ${M}^{(K)}$ are given by

$$M^{(K)}_{ij} = \sum_{\mu\nu} \frac{\langle\mu\vert\hat Q... ...gle} {(\breve{E}_\mu+\breve{E}_{\nu})^K} (\eta_{\mu\nu}^+)^2.$$ (64)