Baker-Campbell-Hausdorff

This approximation is a special case of the Baker-Campbell-Hausdorff lemma for additional discu.ssion and extensions to more general classes of methods. 

While each term in the resulting series has the form required by equation 12,29, the caveat is that as each commutator connects sites x and x that are a distance d apart, the terms of order n in Ai and Aj yield contributions to H x) that obey the third condition in equation 12.29 only for d = nd. In other words, we also need to require that the Baker-Campbell-Hausdorff series has a sufficient degree of convergence, t... 

In cases where the Hamiltonians (typically due to phase or amplitude switching in the rf fields) are discontinuously time-dependent, the average Hamiltonian may conveniently be set up using the semi-continuous Baker-Campbell-Hausdorff (scBCH) expansion [56] as... 

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

With the help of Eqs. (117) and (118) and the Baker-Campbell-Hausdorff relationship... 

The effective coupling tensor between two coupled spins in the toggling frame is only a good approximation of the effective coupling tensor in the (doubly) rotating frame if the higher order contributions in the Baker-Campbell-Hausdorff expansion [see Eq. (119)] can be neglected. This is the case if the term... 

T. S. Untidt and N. C. Nielsen, Closed solution to the Baker-Campbell-Hausdorff problem exact effective Hamiltonian theory for analysis of nuclear-magnetic-resonance experiments. Phys. Rev. E, 2003, 65, 021108-1-021108-17. 

A Baker-Campbell-Hausdorff expansion of the exponential time-evolution operator gives for the density (and similarly for other operators)... 

Reinserting this form of Uj t, — ) in the expectation value (63) and with the aid of the Baker-Campbell-Hausdorff expansion we arrive at the final expression... 

The last expression is the classical or canonical coherent state Iz). The Baker-Campbell-Hausdorff (BCH) formula, yields... 

Alternatively, we can multiply by e from the left and do the projection in the Baker-Campbell-Hausdorff manner [1]... 

This is the start of the celebrated Baker-Campbell-Hausdorff series [21] (often referred to as the BCH Lemma for short) which gives an explicit formula for the product of exponentials of non-commuting operators. 

The terms of the perturbation expansion for D can be computed using the Baker-Campbell-Hausdorff (BCH) expansion already introduced in Chap. 3. Recall that for linear operators X and Y, we can write the composition of their exponentials as... 

Note that when a symplectic integration method is used, we have, from the discussion in Chap. 3, a perturbed energy function and, moreover, from Theorem 3.1, the error in energy H is OQf), nonetheless the perturbations are large for a large stepsize h. In the Shadow Hybrid Monte-Carlo (SHMC) method [2, 3, 188], the accept-reject test is based on the modified Hamiltonian Hh (see Chap. 3), derived from the Baker-Campbell-Hausdorff expansion. SHMC can improve efficiency by decreasing the rejection rate. 

One direction in which improvements on the primitive propagator Eq. (19) can be found is to search for higher-order approximants, which allow one to obtain a given level of accuracy with a lower P. In this respect, an indispensable tool is the general Baker-Campbell-Hausdorff formula for two noncommutable operators A and B [66]... 

Splitting of the operators can be quite tricky. In the case of Eq. (6.113), the splitting is motivated by the observation that it is easy to implement exp(—iv(r)) in real space (at all points on the mesh), but it is difficult to take double derivatives at all points on the mesh. This difficulty can be overcome by using the fact that the derivatives can be implemented trivially in Fourier space. So, if somehow the Laplacian part could be split from w(r) part, then the progress can be made. This is the motivation for splitting the operator in Eq. (6.113) into derivative and nonderivative parts by Baker-Campbell-Hausdorff formula. For most practical purposes, a symmetric decomposition is carried out so that within an error of the order dt ... 

Using the Baker-Campbell-Hausdorff formula leads to the result that only terms corresponding to connected diagrams survive... 

To be more precise, the Baker-Campbell-Hausdorff lemma only guarantees that the direct term corresponds to connected diagrams. The coimectedness of diagrams associated with the coupling terms has to be proven a posteriori. 

Now we introduce curvy steps. An intuitive interpretation of what is done here is to expand the Taylor series of the exponential transformation to higher orders, such that the step directions are no longer straight lines, but instead they are curved. Invoking the Baker-(Campbell-)Hausdorff lemma (see, e.g.. Ref. 123), the unitary transformation of the density matrix can be written as... 

The first term of eq (8) above is manifestly extensive, while the connectivity property of the second term requires a careful treatment, since this involves a product of two matrix-elements and may not have terms with common orbital labels in the two factors. Using the Baker-Campbell-Hausdorff formula for the product of exponentials, the second term can be written as... 

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