The Hartree-Fock equation (HF) for the hydrogen atom \(F_ i \varphi_ i (X) = ( - {1 \over 2} \Delta (x) + V(X)) \varphi_ i (X) = \varepsilon_ i \varphi_ i (X)\) is analyzed by means of the continuous wavelet transform to show how to improve the Gaussian approximation by an iterative process.
The one-dimensional space is treated, studying only the radial dependence of the wave function. Then the wavelet transform of the HF operator is performed and an iterative scheme is defined in position – momentum space. The first iterate is obtained analytically with a Gaussian function defined as the initial guess.
The problem of the singularity for \(x=0\) of the HF equation is bypassed. The improvement by the first iteration is assessed by comparison with the transform of the Slater function. The wavelet transforms are shown and analyzed.