Double Perturbation Theory for He-like Systems

The He‐like system is considered to be a spherical Harmonic oscillator with the electrons attracted to the nucleus by a Hooke's‐law force, under two perturbations: (1) electronic repulsion, $H^{\text{(1,0)}}$ , and (2) the difference between the harmonic oscillator potential and the actual nuclear Coulomb attraction, $H^{\text{(0,1)}}$ . For the ground state, the first‐order wavefunctions with respect to both perturbations are found in closed form. Exact expressions for both of the single perturbation energies through third order and the mixed perturbation energy ${\epsilon }^{\text{(1,1)}}$ are derived. Calculations have been carried out on H⁻, He, Li⁺, and Be²⁺ for different choices of $f$ , the Gaussian parameter of the zeroth‐order harmonic oscillator wavefunction ${\psi }^{\text{(0,0)}}$ . The choice ${\text{f\hspace{0.17em}=\hspace{0.17em}(4\hspace{0.17em}/\hspace{0.17em}3π}}^{\text{1/2}}\text{) [Z\hspace{0.17em}−\hspace{0.17em}(1\hspace{0.17em}/\hspace{0.17em}2√2)]}$ corresponding to the variational one‐term Gaussian orbital of a He‐like system gives the best total calculated energy, − 2.88210 a.u. for helium compared to the experimental and “exact” value of − 2.90372 a.u. From the study of the convergence of the perturbation energies, it is concluded that as far as correlation is concerned the first‐order perturbation with respect to $H^{\text{(1,0)}}$ is accurate enough. Higher order perturbations with respect to the one‐electron operator $H^{\text{(0,1)}}$ will be needed for highly accurate results. However, even at the first‐order level the Gaussian character of ${\psi }^{\text{(0,0)}}$ is improved tremendously by the perturbation $H^{\text{(0,1)}}$ . This is shown by the values of ${\epsilon }^{\text{(0,2)}}{\mathrm{\hspace{0.17em}+\hspace{0.17em}\epsilon }}^{\text{(0,3)}}$ , of the order of − 0.4 to − 0.6 a.u.