The Hausdorff Expansion

This expression is usually referred to simply as the Hausdorff expansion, and although it may not immediately appear to be a simplification of the coupled 

As shown explicitly in Refs. 80, 84, and 92, the creation and annihilation operators described earlier may be used to represent dynamical operators such as the electronic Hamiltonian  

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( j)p l j)r t s) is its antisymmetrized two-electron counterpart. Equation [53] contains general annihilation and creation operators (e.g., or ) that may act on orbitals in either occupied or virtual subspaces. The cluster operators, T , on the other hand, contain operators that are restricted to act in only one of these spaces (e.g., al, which may act only on the virtual orbitals). As pointed out earlier, the cluster operators therefore commute with one another, but not with the Hamiltonian, f . For example, consider the commutator of the pair of general second-quantized operators from the one-electron component of the Hamiltonian in Eq. [53] with the single-excitation pair found in the cluster operator, Tj  

The anticommutation relations of annihilation and creation operators given in Eqs. [19], [20], and [21] may be applied to the two terms on the right-hand side of this expression to give 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera- 

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The only nonzero terms in the Hausdorff expansion are those in which the Hamiltonian, has at least one contraction with every cluster operator, T , on its right. 

This is often referred to as the connected cluster form of the similarity-transformed Hamiltonian. This expression makes the truncation of the Haus-dorff expansion even clearer since the Hamiltonian contains at most four annihilation and creation operators (in n) can connect to at most four cluster operators at once. Therefore, the Hausdorff expansion must truncate at the quartic terms. 

The factor of Vs appearing in the first three equalities arises from product of the factor of V2 from the Hausdorff expansion and the Vi from the definition of (Eq. [104]). 

The coupled cluster energy, on the other hand, does not suffer from this lack of size extensivity for two reasons (1) the amplitude equations in Eq. [50] are independent of the coupled cluster energy and (2) the Hausdorff expansion of the similarity-transformed Hamiltonian in Eq. [106], for example, guarantees that the only nonzero terms are those in which the Hamiltonian is con-... 

The difference between H(t) and /70(/,) stems from the kinetic energy operator in the adiabatic Hamiltonian H0, which can be treated as a perturbation. Using the Cambell-Baker-Hausdorff expansion, to the first order we have... 

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... 

Using the truncated Hausdorff expansion, we may obtain analytic expressions for the commutators in Eq. [52] and insert these into the coupled cluster energy and amplitude equations (Eqs. [50] and [51], respectively). However, this is only the first step in obtaining expressions that may be efficiently implemented on the computer. We must next choose a truncation of T and then derive expressions containing only one- and two-electron integrals and cluster amplitudes. This is a formidable task to which we will return in later sections. 

This last point also has interesting consequences for the higher excitation amplitude equations such as that for. For example, one term that arises in the general Hausdorff expansion is V5i(l 2vTi)c. This term does not contribute to the T3 amplitude equation... 

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 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. 

The key entities are the similarity transformed PTE Hamiltonian H(O) = e f/(0)jye and the similarity transformed molecular electrostatic potential operators )7 = e Vjye. Both operators can be expressed as a terminated Backer-CampbeU-Hausdorff expansion. Specifically, H 0) terminates at the four-fold commutator, because it has at most two-particle interactions ... 

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,... 

Another approach to derive the evolution of the density operator is based on the so-called Hausdorff formula, which results from the power series expansion of (7(t) and (7 (t) in Eq. (185) ... 

The Baker-CampbeU-Hausdorff formula is a fundamental expansion in elementary Linear Algebra and Lie group theory (J. E. Campbell, Proc. London Math. Soc. 29, 14 (1898) H. F. Baker, Proc. London Math. Soc. 34, 347 (1902) F. Hausdorff, Ber. Verhandl. Saechs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 58, 19 (1906)). 

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