The accuracy and convergence of the Fock-space coupled cluster method discussed above depends on an appropriate partitioning of the function space into

P and

Q subspaces. Ideally, the

P space should include all functions which are important to the states considered, since the effective Hamiltonian is diagonalized in

P, whereas

Q-space contributions are included approximately. On the other hand, convergence of the coupled cluster iterations is enhanced by maximal separation and minimal interaction between

P and

Q. These two requirements are not always easy to reconcile. Relatively high

P functions have often strong interaction with or are energetically close to

Q states, making convergence slow or impossible. The offending functions are usually included in

P because of their significant contribution to the lower

P states, and we may not be particularly interested in the correlated states generated from them by the wave operator; however, the FSCC is an all-or-nothing method, and lack of convergence means that no states at all are obtained. The intermediate Hamiltonian coupled cluster method developed recently [

16] addresses this problem, making possible larger and more flexible

P spaces, thereby extending the scope of the coupled cluster method and increasing its precision.

The intermediate Hamiltonian method has been proposed by Malrieu [

17] in the framework of degenerate perturbation theory. The

P space is partitioned into the main

P_{m} and the intermediate

P_{i}, with the corresponding operators satisfying

Two sets of wave-like operators are defined and expanded in coupled-cluster normal-ordered exponential ansätze. Ω = 1 +

χ is a standard wave operator in

P_{m},

where |Ψ

_{m}〉 denotes an eigenstate of the Hamiltonian

H with the largest components in

P_{m}, and

R = 1 + ∆ is an operator in

P, satisfying

It should be noted that the last equation, and therefore all equations derived from it, applies when operating on

|Ψ

_{m}〉 but not necessarily on

|Ψ

_{i}〉. This feature distinguishes

R from a bona fide wave operator. The cluster equation for

S in the (

n) sector of the Fock space is [

16]

where

Q_{i} = 1 −

P_{i} =

Q +

P_{m}. No

P_{i}SP_{m} elements appear in the equation, so that

P_{i} acts as a buffer between

P_{m} and

Q, facilitating convergence and avoiding intruder states. Eq. (12) is valid provided

QSP_{m} ≃

QTP_{m}, which is rather easy to achieve and is checked in the calculation. After (12) is solved for

QSP_{m}, the equation for

QTP is solved,

E is an arbitrary constant, chosen to facilitate convergence. Tests have shown that

E may be changed within broad bounds (hundreds of hartrees) with minute effect (a few wave numbers) on calculated transition energies. The final step is the construction of the intermediate Hamiltonian

which gives upon diagonalization the correlated energies of |Ψ

_{m}〉,

The dimension of the

H_{I} matrix is that of

P; however, only the eigenvalues corresponding to |Ψ

_{m}〉 are required to satisfy (15). The other eigenvalues, which correspond to states |Ψ

_{i}〉 with the largest components in

P_{i}, may include larger errors.