# Trying to solve TDSE for Hydrogen atom, but I cant determine initial condition

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 $$c_{g}$$?