Based on the idea of Schmidt orthogonalization, a fully diagonalized Chebyshev rational spectral method for solving second order differential equations on the whole line with asymptotic boundary conditions was proposed. Some Fourier-like Sobolev orthogonal basis functions were constructed and fully diagonal discrete algebraic equations were derived. Accordingly, both the exact solutions and the approximate solutions were represented as infinite and truncated Fourier series. The numerical results demonstrate the spectral accuracy. Compared with the existing algorithms, the results of numerical experiment indicate the cost of the computation with the present algorithm is less and the algorithm is easier to be implemented.