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I'm doing my thesis, and I have to solve a TDSE for molecular alignment - non-adiabatic and non-resonant laser induce alignment. And I really need your help to solve it. I tried split operator method to solve it but I was unable to solve it:

$$i\frac{\partial\Psi_{JM}(\theta,\phi,t)}{\partial t}=\left[BJ^2-\frac{E(t)^2}{2}(\alpha_\parallel\cos^2\theta+\alpha_\perp\sin^2\theta\right]\Psi_{JM}(\theta,\phi,t).$$

I tried split operator method like this:

$$\Psi(\theta,\phi,t+\Delta t)=\exp\left(-iH_o\frac{\Delta t}{2}\right)\exp\left(-iV\left(t+\frac{\Delta t}{2}\right)\Delta t\right)\exp\left(-iH_o\frac{\Delta t}{2}\right)\sum_{jm}c(t)\Psi_{jm},$$

but the sum of $c^2$ after a time step is not $1$, it could be $2$, $3$, $4$...

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Split-operator is going to work perfectly fine for non-resonant alignment simulations. The drifting normalization is probaby due to not small enough time-step or an implementation error. In rigid rotor alignment simulations the direct exponentiation of the molecule+field hamiltonian in spherical harmonics basis is also a possible way of solving the TDSE. There are a number of computer codes capable of doing this type of calculation. I suspect that in order for you to get a practical answer to your question, it should be more specific. Then I should be able to help.

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