The calculation of the ground state energy of weakly bound van der waals trimers using the method of hyperspherical harmonics I. The Born—Oppenheimer and adiabatic approximations

A variational method for the calculation of vibrational energy levels of triatomic molecules using a Hamiltonian in hyperspherical coordinates

“…21t ap2 4p2 (15) This worked reasonably well for H/ , but was rather slow to converge for Na 3 + , as was to be expected due to the strong coupling between p and the angular coordinates discussed above. What is really required is a set of radial basis functions which are optimized with respect to an effective radial potential defined as the minimum of the potential with respect to the angular coordinates (16) In this way one ensures that the basis covers all the accessible regions of p-space and at the same time provides sufficiently fast oscillations where required.…”

“…It has been previous practice, 15,16 to consider a pure angular problem in the coordinates 9 and «I> for some fixed value(s) of the radial coordinate p. Conversely, this implies that the radial basis functions may be independently optimized before inclusion in the complete expansion set. At first we investigated a one-dimensional radial analysis at equilibrium values of the angular coordinates 9 and «1>, by considering only the one-dimensional Hamiltonian…”

“…The next stage in the radial analysis is to use the one-dimensional form of the Hamiltonian given in Eq. (15), with V replaced by V etr (p), and diagonalize the resulting matrix in the primary set of radial basis functions pertaining to Emax [p].…”

Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

“…(19) is still exact, if all terms in the expansion are kept, but this is prohibitively expensive [30]. Instead, we follow common practice and make a Born-Oppenheimer-like approximation [31][32][33][34]. Namely, we assert that ρ is a "slow" coordinate in the sense that we ignore the partial derivatives ∂Y {λ} /∂ρ in Eq.…”

“…The solution representing the BEC is a single wave function specified by the quantum numbers {λ}. We adopt this approximation in what follows; the derivative couplings between adiabatic functions can be reinstated by familiar means [31][32][33][34]. It is also worth noting that in the limit of infinite scattering length, the Born-Oppenheimer approximation again becomes exact [35].…”