Structure of the exact wave function. III. Exponential ansatz

We continue to study exponential ansatz as a candidate of the structure of the exact wave function. We divide the Hamiltonian into $ND$ (number of divisions) parts and extend the concept of the coupled cluster (CC) theory such that the cluster operator is made of the divided Hamiltonian. This is called extended coupled cluster (ECC) including $ND$ variables (ECCND). It is shown that the S(simplest)ECC, including only one variable $(ND=1),$ is exact in the sense that it gives an explicit solution of the Schrödinger equation when its single variable is optimized by the variational or H-nijou method. This fact further implies that the ECCND wave function with $ND⩾2$ should also have a freedom of the exact wave function. Therefore, by applying either the variational equation or the H-nijou equation, ECCND would give the exact wave function. Though these two methods give different expressions, the difference between them should vanish for the exact wave function. This fact solves the noncommuting problem raised in Paper I [H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000)]. Further, ECCND may give more rapidly converging solution than SECC because of its non-linear character, ECCND may give the exact wave function at the sets of variables different from SECC. Thus, ECCND is exact not only for $ND=1,$ but also for $ND⩾2.$ The operator of the ECC, $exp(S),$ is an explicit expression of the wave operator that transforms a reference function into the exact wave function. The coupled cluster including general singles and doubles (CCGSD) proposed in Paper I is an important special case of the ECCND. We have summarized the method of solution for the SECC and ECCND truncated at order n. The performance of SECC and ECC2 is examined for a simple example of harmonic oscillator and the convergence to the exact wave function is confirmed for both cases. Quite a rapid convergence of ECC2 encourages an application of the ECCND to more general realistic cases.