Trotter factorizations

With this break up the reversible Trotter factorization of the propagator is... 

The system defined by the Liouvillian is called the reference system. Now applying the Trotter factorization to the propagator exp iLs + Fi arising from this subdivision gives the new propagator,[17]... 

The idea is now to replace the formal solution of the Liouville equation by the discretized version. The middle term gf the propagator in Eq. (51) can be further decomposed by an additional Trotter factorization to obtain... 

Application of the Trotter factorization for the exponential operator appearing in Eq. (6) leads to the expression... 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

As noted earlier, the fundamental equations of the QCL dynamics approach are exact for this model, however, in order to implement these equations in the approach detailed in section 2 the momentum jump approximation of Eq.(14) is made in addition to the Trotter factorization of Eq.(12). Both approximations become more accurate as the size of the time step 5 is reduced. Consequently, the results presented below primarily serve as tests of the validity and utility of the momentum-jump approximation. For a discussion of other simulation schemes for QCL dynamics see Ref. [21] in this volume. The linearized approximate propagator is not exact for the spin-boson model. However when used as a short time approximation for iteration as outlined in section 3 the approach can be made accurate with a sufficient number of iterations [37]. 

Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotter-factorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Parrinello simulations [20] and ab initio molecular dynamics algorithms [21]. 

Equation [146] is clearly time reversible, with the desirable side effect of being more accurate. We will now proceed with examining Eq. [144]. We can apply the Trotter factorization introduced in Eq. [146] to Eq. [144] to obtain... 

Equation [172] shows that the shear-rate-dependent term affects the update of only. As a result, the and Vy components in the bracketed term in Eq. [172] do not commute, and a Trotter factorization of this term must be performed ... 

The careful reader should have realized that we choose not to break up this operator with another Trotter factorization, as was done for the extended system case. In practice, one does not multiple-time-step the modified velocity Verlet algorithm because it will, in general, have a unit Jacobian. Thus, one would like the best representation of the operator that can be obtained in closed form. However, even in the case of a modified velocity Verlet operator that has a nonunit Jacobian, multiple-time-stepping this procedure can be costly because of the multiple force evaluations. Generally, if the integrator is stable without multiple-time-step procedures, avoid them. The solution to this first-order inhomogeneous differential equation is standard and can be found in texts on differential equations (see, e.g.. Ref. 53). 

As was done previously, we perform successive Trotter factorizations to Eq. [186] and obtain... 

We employed the following Trotter factorization to approximate the evolution operator to O(A)t ) ... 

The new reversible-RESPA [25] exploits the Trotter factorization [27] to make the algorithm reversible. Re-writing the equations of motion, for a system of N particles having coordinates q. .. qn and momentapi... p, to incorporate both the fast and slow components of the force 3uelds [28] ... 

Yoshida [398] gives an elegant method for creating a symplectic scheme of arbitrarily high order, using Trotter s results for compositions of linear operators [370], This work is related to methods suggested by Suzuki [354] in the context of Trotter factorization of quantum operators. Consider a scheme with order 2s (for > 1) and where the evolution of the system under the method is given by exp(/t /,), where... 

Invoking a Trotter factorization [75], we write the time evolution operator as... 

Here xj,k denotes the ikth discretization of coordinate xj and Xjfi=Xj,N- In practice, equation (12) is evaluated with N sufficiently large for the error introduced through the Trotter factorization of the short time propagators to be within the acceptable tolerance. 

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