  In recent years, the time-dependent variational principle (TDVP) method based
on the matrix product state (MPS) wave function formulation has shown its great
power in performing large-scale quantum dynamics simulations for realistic
chemical systems with strong electron-vibration interactions. In this work, we
propose a new stochastic adaptive single-site TDVP (SA-1TDVP) scheme to evolve
the bond-dimension adaptively, which can integrate the tra-ditional advantages
of both the high efficiency of single-site TDVP (1TDVP) variant and the high
accuracy of the two-site TDVP (2TDVP) variant. Based on the assumption that the
level statistics of entanglement Hamiltonians, which originate from the reduced
density matrices of the MPS method, follows a Poisson or Wigner distribution,
as generically predicted by random matrix theory, addi-tional random singular
values are generated to expand the bond-dimension automatically. Tests on
simulating the vibrationally-resolved quantum dynamics and absorption spectra
in the pyrazine molecule and perylene bisimide (PBI) J-aggregate trimer as well
as a spin-1/2 Heisenberg chain show that it can be automatic and as accurate as
2TDVP but reduce the computational time remarkably.
