# TDExponentialMethod

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#### TDExponentialMethod

Section Time-Dependent::Propagation
Type integer
Default taylor

Method used to numerically calculate the exponential of the Hamiltonian, a core part of the full algorithm used to approximate the evolution operator, specified through the variable TDPropagator. In the case of using the Magnus method, described below, the action of the exponential of the Magnus operator is also calculated through the algorithm specified by this variable.

Options:

• lanczos:
Allows for larger time-steps. However, the larger the time-step, the longer the computational time per time-step. In certain cases, if the time-step is too large, the code will emit a warning whenever it considers that the evolution may not be properly proceeding -- the Lanczos process did not converge. The method consists in a Krylov subspace approximation of the action of the exponential (see M. Hochbruck and C. Lubich, SIAM J. Numer. Anal. 34, 1911 (1997) for details). Two more variables control the performance of the method: the maximum dimension of this subspace (controlled by variable TDExpOrder), and the stopping criterion (controlled by variable TDLanczosTol). The smaller the stopping criterion, the more precisely the exponential is calculated, but also the larger the dimension of the Arnoldi subspace. If the maximum dimension allowed by TDExpOrder is not enough to meet the criterion, the above-mentioned warning is emitted.
• taylor:
This method amounts to a straightforward application of the definition of the exponential of an operator, in terms of its Taylor expansion.

$\exp_{\rm STD} (-i\delta t H) = \sum_{i=0}^{k} {(-i\delta t)^i\over{i!}} H^i.$

The order k is determined by variable TDExpOrder. Some numerical considerations from Jeff Giansiracusa and George F. Bertsch suggest the 4th order as especially suitable and stable.

• chebyshev:
In principle, the Chebyshev expansion of the exponential represents it more accurately than the canonical or standard expansion. As in the latter case, TDExpOrder determines the order of the expansion.

There exists a closed analytic form for the coefficients of the exponential in terms of Chebyshev polynomials:

$\exp_{\rm CHEB} \left( -i\delta t H \right) = \sum_{k=0}^{\infty} (2-\delta_{k0})(-i)^{k}J_k(\delta t) T_k(H),$

where $J_k$ are the Bessel functions of the first kind, and H has to be previously scaled to $[-1,1]$. See H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81, 3967 (1984); R. Kosloff, Annu. Rev. Phys. Chem. 45, 145 (1994); C. W. Clenshaw, MTAC 9, 118 (1955).

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