Finite-size scaling for critical conditions for stable quadrupole-bound anions

We present finite-size scaling calculations of the critical parameters for binding an electron to a finite linear quadrupole field. This approach gives very accurate results for the critical parameters by using a systematic expansion in a finite basis set. The model Hamiltonian consists of a charge Q located at the origin of the coordinates and k charges -Q/k located at distances R(i), i=1, em leader,k. After proper scaling of distances and energies, the rescaled Hamiltonian depends only on one free parameter q=QR. Two different linear charge configurations with q>0 and q<0 are studied using basis sets in both spherical and prolate spheroidal coordinates. For the case with q>0, the finite size scaling calculations give an extrapolated critical value of q(c)=1.469 70+/-0.000 05 a.u. by using a basis set with prolate spheroidal coordinates. For the quadrupole case with q<0, we obtained an extrapolated critical value of mid R:q(c)mid R:=3.982 51+/-0.000 01 a.u. for stable quadrupole bound anions. The corresponding critical exponent for the ground state energy alpha=1.9964+/-0.0005, with E approximately (q-q(c))(alpha).