Trotter approximation

Using the Trotter approximation for the exponential of a sum of noncommuting operators allows separation of the kinetic and potential contributions to the propagation in each matrix element in the above expression, so that... 

Path integral Monte Carlo simulations were performed [175] for the system with Hamiltonian (Eq. (25)) for uj = ujq/J = A (where / = 1) with N = 256 particles and a Trotter dimension P = 64 chosen to achieve good computer performance. It turned out that only data with noise of less than 0.1% led to statistically reliable results, which were only possible to obtain with about 10 MC steps. The whole study took approximately 5000 CPU hours on a CRAY YMP. 

For large P, (3/P is small and it is possible to find a good short-time approximation to the Green function p. This is usually done by employing the Trotter product formula for the exponentials of the noncommuting operators K and V... 

In order to get a first impression on the performance of the QC Liouville approach, it is instructive to start with a simple one-mode spin-boson model, that is. Model IVa [205]. In what follows, the QCL calculations used the first-order Trotter scheme (61) with a time step 8r = 0.05 fs. If not noted otherwise, we have employed the momentum-jump approximation (59) and the initial number of random walkers employed was N = 50 000. 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

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]. 

The restricted step method of the type discussed below were originally proposed by Levenberg and Marquardt [37,38] and extended to minimization algorithms by Goldfeld, Quandt and Trotter [39]. Recently Simons [13] discussed the restricted step method with respect to molecular energy hypersurfaces. The basic idea again is that the energy hypersurface E(x) can reasonably be approximated, at least locally, by the quadratic function... 

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

For both bosonic systems and fermionic systems in the fixed-node approximation, G has only nonnegative elements. This is essential for the Monte Carlo methods discussed here. A problem specific to quantum mechanical systems is that G is known only asymptotically for short times, so that the finite-time Green function has to be constructed by the application of the generalized Trotter formula [6,7], G(r) = limm 00 G(z/m)m, where the position variables of G have been suppressed. 

The Trotter formula proves that this approximation does converge to the right result in the f oo limit and is given by ... 

In this section, we will show how the thermal density matrix is used in PIMC to compute quantum viiial coefficients. Consider the Hamiltonian of a monatomic molecule like helium with mass m (Eq. 7). Using the primitive approximation (Eq. 4), Trotter formula (Eq. 5), and following the procedure outlined in Ref. [9], we can obtain the kinetic-energy operator matrix elements as ... 

Symplectic integrators may be constructed in several ways. First, we may look within standard classes of methods such as the family of Runge-Kutta schemes to see if there are choices of coefficients which make the methods automatically conserve the symplectic 2-form. A second, more direct approach is based on splitting. The idea of splitting methods, often referred to in the literature as Lie-Trotter methods, is that we divide the Hamiltonian into parts, and determine the flow maps (or, in some cases, approximate flow maps) for the parts, then compose the maps to define numerical methods for the whole system. 

For this decomposition, a short-time approximation to the evolution operator can be generated via the Trotter theorem [16] as... 

To be more precise, this time propagation scheme consists of two ingredients (i) application of the Trotter product formula to the time propagation operator, and (ii) an efficient use of the Fast Fourier transform (FFT). Recall that the increase of time to perform FFT scales to NlogN, where N is the number of grid points [219]. A time evolution operator for a short time interval At, which is composed of two noncommutable operators A and B, can be approximately represented with the TVotter product (decomposition) formifla as... 

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