The lowest order constrained variational (LOCV) method for the many-body problems and its applications

One always looks for a simplified technique and desirable formalism, to solve the Hamiltonian, and to find the wave function, energy, etc, of a many-body system. The lowest order constrained variational ($LOCV$) method is designed such that, to fulfill the above requirements. The $LOCV$ formalism is based on the first two, i.e., lowest order, terms of the cluster expansion theory with the $Jastrow$ correlation functions as its inputs. A constraint is imposed for the normalization of the total correlated two-body wave functions, which also forces the cluster expansion series to converge very rapidly. The variation of $Jastrow$ correlation functions subjected to the above normalization constraint, leads to the sets of Euler–Lagrange equations, which generates the required correlation functions. In order to satisfy the normalization constraint exactly, one has to define the long-range behaviors, for the two-body correlation functions, i.e., the Pauli function. The primary developments of $LOCV$ formalism, and some of its applications were reviewed in this journal by Max Irvine in 1981. Since then (1981–2022), the various extensions and applications of the $LOCV$ method are reported through the several published articles (nearly 180 items), which are the subjects of this review. (i) It is shown that the $LOCV$ results can be, as good as, the various more complicated and computer time-consuming techniques, such as the Fermi $hypernetted$ chain ($FHNC$), Monte Carlo ($MC$), G-matrix, etc, calculations. (ii) Moreover, the $LOCV$ method is further developed to deal with the more sophisticated interactions, such as the $AV18$, $UV14$, etc, nucleon–nucleon potentials, using the state-dependent correlation functions, and applicable to perform the finite temperature calculations. The extended $LOCV$ ($ELOCV$) method is also introduced for the state-independent media. (iii) Its convergence is tested through the calculation of three-body cluster series, with the state-dependent correlation functions, which confirm the old (1979) state-averaged predictions. Finally, its application to the $nucleonic$ and $\beta -$stable matter with and without the three-body force, the finite nuclei, the liquid helium 3, the neutron star, etc are performed and compared with the other many-body techniques. As we stated before, in this review, we definitely go through the most of above items.