Theory of the binding energy of an electron in the field of a linear electric quadrupole

A quantum-mechanical investigation has been carried out to determine the energy eigenvalues of an electron moving in the field of a finite electric quadrupole, having point charges -q,+2q, and -q arranged collinearly with each negative charge a distance scra from the positive charge. When the expectation value of the Hamiltonian is minimized in a variational calculation and then set equal to zero, the solution of the resulting equation gives a value of the minimum quadrupole moment required to assure the existence of a bound state. The values of the quadrupole moment Q studied here cover the range from zero to Q=(4×{}10⁴)ea₀², where a₀ is the effective Bohr radius. The minimum value of Q for binding is found to be 2.6ea₀².