I decide to ask my question (https://physics.stackexchange.com/questions/755354/how-to-find-optimal-parameters-in-gaussian-basis-set) in MATHEMATICS part of StackExchange too, because I think the question relate to some math approach.
I try to replicate some results of this article - https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.2220930140
The article here: https://drive.google.com/file/d/1q74hAn0UAdNd8DtPkoCsdYPkr61-xhr2/view?usp=sharing
I would like to get energies of ground state for Hydrogen atom in magnetic field. But I can't understand, how to find $\alpha_i$ and $\beta_i$ sets for Gaussian basis functions.
Brief description of the problem:
Hamiltonian of Hydrogen atom in magnetic field ($\vec{B}||z$) has the following form (formula (2) in the article): $$H=-\Delta-\frac{2}{r}+\gamma l_z+\frac{1}{4}(\gamma^2 \rho^2)$$ where $\quad r^2=\rho^2+z^2 \quad \gamma=\frac{\mu B}{R} \quad \mu=\frac{e \hbar}{2 m c} \quad R=\frac{mc^4}{2\epsilon_0^2\hbar^2}$
The authors use for the Hamiltonian matrix numerical diagonaliztion Gausian basis set (formula (4) in the article): $$\psi_i=\rho^{|m|}e^{im\phi}z^qe^{-\alpha_i r^2-\beta_i z^2}$$ where $m=0,\pm1,\pm2,\pm3...$ is the quantum number describing the $z $ component of the angular momentum, $q = 0,1$
the $\alpha_i$ and $\beta_i$ are coefficients of geometric progressions (formulas (8a) and (8b) in the article): $\alpha_i=\alpha_{i-1} \left( \frac{\alpha_N}{\alpha_1} \right)^{1/(N-1)}$, $\beta_i=\beta_{i-1} \left( \frac{\beta_M}{\beta_1} \right)^{1/(M-1)}$ . This is accomplished by specifying four quantities - the largest and smallest exponents from the $\alpha$ and $\beta$ sets: $\alpha_1$, $\alpha_N$, $ \beta_1$, $ \beta_M$
Let's consider the lowest energy ($m=0, q=0$), then the basis functions will take the form: $$\psi_i=e^{-\alpha_i r^2-\beta_i z^2}$$
But I really can't understand how need to find these parameters: $\alpha_1$, $\alpha_N$, $ \beta_1$, $ \beta_M$ ?
The authors write the following: "For given quantum numbers $m$ and $q$, the parameters $\alpha_1$, $\alpha_N$, $\beta_1$, and $\beta_M$ were varied in a four dimensional optimization search which minimized the sum of the eigen-values of interest"
Could you please explain to me what the authors mean?
How to find the $\alpha_1$, $\alpha_N$, $ \beta_1$, $ \beta_M$ parameters?