The variation of the functional with respect to the linear parameters a$_{n}$ and tan δ leads to the following equations To obtain the last equation  the normalization relation of Eq. (9) has been used. From these two equations  Ψ$_{c}$ and the first order estimate of the phase shift (tan δ ) 1 st can be determined. It should be noted that the first equation implies < Ψ$_{c}$ | H − E | Ψ$_{t}$ > = 0. Furthermore  from the general relation for A in Eq. (4)  and using the second equation in Eq. (16)  the following integral relation results Replacing the two relations of Eq.(16) into the functional of Eq.(14)  a second order estimate of the phase shift is obtained Multiplying Eq. (18) by A one gets On the other hand  a first order estimate for the coefficient B can be obtained from the general relation in Eq. (4)  i.e.  Therefore  replacing Eq.(20) in Eq.(19)  a second order integral relation for B is obtained. The above results can be summarized as follow These equations extend the validity of the integral relations  given in Eq.(6) for the exact wave functions  to trial wave functions. To be noticed that F  ˜ G are solutions of the Schr¨ odinger equation in the asymptotic region  therefore ( H − E ) F → 0 and ( H − E ) ˜ G → 0 as the distance between the particles increases. As a consequence the decomposition of 7 B 2 nd = − m ℏ 2 < Ψ$_{t}$ | H − E | F > A = m ℏ 2 < Ψ$_{t}$ | H − E | ˜ G > [tan δ ] 2 nd = B 2 $^{nd}$/A . (21) m ℏ 2 [ < F | H − E | Ψ$_{t}$ > − < Ψ$_{t}$ | H − E | F > ] = B 1 st . (20) B 2 nd = B 1 st − m ℏ 2 < F | H − E | Ψ$_{t}$ > . (19) [tan δ ] 2 nd = (tan δ ) 1 st − m ℏ 2 < F | H − E | (1 /A )Ψ$_{t}$ > . (18) m ℏ 2 < Ψ$_{t}$ | H − E | ˜ G > = A . (17) < φ$_{n}$ | H − E | Ψ$_{t}$ > = 0 < ˜ G | H − E | Ψ$_{t}$ > = 0 . (16)