A computational method based on a rapidly convergent form of the unlinked cluster expansion is presented, &&G's CoupIed-pair approximation (CPA) is derived in a basis of partially non-orthogonal orbitals which transform each palr function to diagonal form; this produces a simple (non-variational) set of equations from which may be extracted the energy and coefficients of a wavefunction constructed from the Hartree-Fock function, all double excitations and all unlinked clusters of these. The relationship of the CPA to simpler treatments is developed using the results of Hurley, and numerical results of a simple illustrative study of the BH3 molecule are given. Recent caIculations [ 1,2] have demdnstrated that the most rapid convergence of the CI expansion is obtained when this is expressed in the diagonal form: Here the wavefunction ~DE includes all double excitations from pairs of spin-orbi*Ws (IJ) occtipied in some determinant 'EO. This determinant will generally be the Hartree-Fock function, but might be, say, a Brueckner determinant. Each set of orbitals c1! E {I$} occurs only in correlating a particular pair of spin-orbitals -they are "natural spin orbit&" of the particular two-electron subsystem (II)_ Spin-orbit& LY in the set correIating (1.i) are orthogonal within that set and orthogonal to all of the occupied spin-orbit& in qO_ However, correlating spin-orbit& from pair fU) are not necessarily orthogonal to correlating spin-orbitais from any other pair. Of course, the exact diagonal&&ion of each pair expansion in \k,, requires prior knowledge of the form of this function and is thus of limited value. This is generally circumvented by the use of approximate natural expansions for each pair -"pseudo-natural-orbit&" of Meyer [l] or "pair-natural-orbit& of Ahlrichs and coworkers [2]. These approximtite natural orbitals are obtained from an (exact or approximate) calculation within the independent-pair approximation (WA), and to define this source unambiguously we prefer the term independent-pair narural (spin) orbital (I.PN(S)O) in the following analysis. The treatment below hoids'exactly if the expansions are exactly diagonal. Previous work by Meyer [l] suggests that if IPNO's are employed instead of the optimum functions (and no;l-diagonal substitutions are neglected), the error involved is less than 1% of the corre-* To whom correspondence should be addressed.