Calculate the energy of a hydrogen atom with 2 Gaussians

Input parameters (in atomic units):

1. exponent α = 2. exponent β = ratio of coefficients γ =

= Eh

Change the 3 input parameters repeatedly to find the lowest energy of the hydrogen atom !

The expectation value for a unnormalized wave function ψ is given by E = <ψ/H/ψ>/<ψ/ψ>

In a linear ansatz with 2 basis functions: ψ = c1φ1 + c2φ2 we get:

E = (c121/H/φ1> + 2c1c21/H/φ2> + c222/H/φ2>) / (c1211> + 2c1c212> + c2222>)

Using the usual abbreviations the equation simplifies to:

E = (c12H11 + 2c1c2H12 + c22H22) / (c12S11 + 2c1c2S12 + c22S22)

Division of the nominator and of the denominator by c22 and abbreviating c1/c2 by γ, we obtain:

E = (γ2H11 + 2γH12 + H22) / (γ2S11 + 2γS12 + S22)

This is the equation used in the program. The basis functions applied are unnormalized Gaussians e-αr2 and e-βr2, respectively. The integrals are:

Hαβ = Sαβ ((3αβ)/(α + β) - 2((α + β)/π)1/2)

Sαβ = (π/(&alpha + β))3/2

Have a look to the source code of this web-page. There you find the program code in (JavaScript).

H. Huber, University of Basel, Dec. 99