  Numerically exact simulations of quantum reaction dynamics, including
non-adiabatic effects in excited electronic states, are essential to gain
fundamental insights into ultrafast chemical reactivity and rigorous
interpretations of molecular spectroscopy. Here, we introduce the tensor-train
split-operator KSL (TT-SOKSL) method for quantum simulations in tensor-train
(TT)/matrix product state (MPS) representations. TT-SOKSL propagates the
quantum state as a tensor train using the Trotter expansion of the
time-evolution operator, as in the tensor-train split-operator Fourier
transform (TT-SOFT) method. However, the exponential operators of the Trotter
expansion are applied using a rank adaptive TT-KSL scheme instead of using the
scaling and squaring approach as in TT-SOFT. We demonstrate the accuracy and
efficiency of TT-SOKSL as applied to simulations of the photoisomerization of
the retinal chromophore in rhodopsin, including non-adiabatic dynamics at a
conical intersection of potential energy surfaces. The quantum evolution is
described in full dimensionality by a time-dependent wavepacket evolving
according to a two-state 25-dimensional model Hamiltonian. We find that
TT-SOKSL converges faster than TT-SOFT with respect to the maximally allowed
memory requirement of the tensor-train representation and better preserves the
norm of the time-evolving state. When compared to the corresponding simulations
based on the TT-KSL method, TT-SOKSL has the advantage of avoiding the need of
constructing the matrix product state Laplacian by exploiting the linear
scaling of multidimensional tensor train Fourier transforms.
