footnote

Last modification: 1996/3/25
Inclusion of higher order terms of N and Z to the correction formula does not substantially decrease the r.m.s. error. With terms of degrees from zero to three (10 terms in total), the r.m.s. error is 107 keV. Addition of up to sixth-order terms (28 terms) leads to an r.m.s. error of 93 keV.

On the other hand, addition of only one more term

c6 ETD A1/3
decreases the r.m.s. error to 86 keV, where ETD is the space integral of
B5 rho Laplacian rho + B6 rhon Laplacian rhon + B6 rhop Laplacian rhop
(see Ref. [BFH85] for the definitions of B5 and B6).

Why does the above term work well ? It is because the principal part of the error of the total energy comes from the discrete approximations to derivatives with a ~ 1 fm. Such errors are included in different manners in the partial energies (the space integrals of any terms of the hamiltonian density functional). Therefore, we expect that linear combinations of such partial energies can efficiently cancel the errors in the approximations to derivatives. Since such formulae do not utilize the magic numbers of spherical shape, it is expected to work not only for spherical solutions but also for deformed ones.

Further addition of a term

c7 (Deltan + Deltap) A
reduces the error to 77 keV, where Deltan and Deltap are the pairing gaps.