With interaction between atomic nucleus in a hydrogen molecular ion,one-center expansion method and B-splines basis technology are used to build radial wave function and angular wave function of a hydrogen molecular ion.Energy levels E_b (m=0,-1,-2,-3,-4,-5) and equilibrium internuclear separations 2R are calculated in superstrong external parallel magnetic fields.Strength of magnetic field changes from 2.35×10⁶ T to 2.35×10⁸ T.Compared with results by adiabatic variational approximation,the precision increases to a range between 0.00015 Ry and 0.0488 Ry;Compared with results by variational approximation,ours increases to a range between 0.0029 Ry and 0.0257 Ry.It shows that the duplicate B-splines basis function expansion method can get better completeness of wave functions and higher calculation precisions.

The exciton wave function is built to calculate the exciton ground state evergy E_g in PbS quantum dots using special properties of B-splines.The quantum size effect of PbS quantum dots is investigated.Compared to other calculations,the E_g calculated shows better agreement with experimental results.The effects of effective mass and barrier height of potential on E_g are investigated.

The B splines method is used to calculate eigenvalues for one and two dimensional multi-well potential.Comparisons are made with calculations using Gaussian bases for one dimensional potential.The B splines bases is shown to be accurate,flexible,and efficient.