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I am trying to solve TDSE for a hydrogen atom in the b-spline basis set. $$i\dfrac{\partial}{\partial t}\Psi(t)=[H_{0}+D(t)]\Psi(t)$$ with initial condition $\Psi(t=-\infty)=\Psi_{g}$, where $\Psi_{g}$ is the initial field free atomic stationary state. The field free hamitonian $H_{0}$ and $D(t)$ represents the laser-atom interaction. $$D(t)=-E(t).r$$ $$\Psi(r,t)=\sum_{l=0}^{l_{max}}\sum_{i=1}^{N}c_{i}^{l}(t)\dfrac{B_{i}}{r}Y_{l}^{0}(\theta,\phi)$$ $$i \bar{S}.\dot{c}(t)=(\bar{H}_{0}+\bar{D})(t).c(t)$$ $\bar{S}$ and $\bar{H}_{0}$ are diagonal block matrices of $S$ and $H$ which are overlap and hamiltonian matrices. $$\bigg(\bar{S}+i(\bar{H}+\bar{D}(t))\dfrac{\delta t}{2}\bigg).c(t+\delta t)=\bigg(\bar{S}-i(\bar{H}+\bar{D}(t))\dfrac{\delta t}{2}\bigg).c(t)$$ the right side of this equation is known but the left side has an unknown part $c(t+\delta t)$ which we desire to achive by solving the $A(T).c(t+\delta t)=b$ acording to the article $c(t=-\infty)=c_{g}$ here is a question: how can I set $c_{g}$?

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