Abstract: The trial function we used contains two linear parameters and is required to satisfy the cusp condition and Virial theorem. This function is accurate enough for calculating chemical properties and is used as the pair correlation function of the atoms.Writing the trial function φ_Гγ=ψ_Гγ⁽⁰⁾·F, where Г is the representation of the symmetry group of the system, γ labels the row of the representation and ψ_Гγ⁽⁰⁾ is the eigenfunction of , we showed the correlation function Fbelongs to the identity representation.We also showed the correlation function Fmay be choosen to be the same for both the ground state and the excited states.We proved that for any atomic and molecular system, the equation exists  where E(_r→) is the local energy eigenvalue of the trial function. This result is stated in a theorem, the average of local energy gradlant along any radial line is zero.For the calculation of the energy expectation value of the helium-like ions, the pair correlation function may be taken as:φ={[1+α(γ₁²+γ₂²)](1+(1/2)γ₁₂)+bγ₁₂[(γ₁+γ₂)γ₁₂+(1/2)(γ₁-γ₂)²]}e^{-z(γ₁+γ₂)}