Coupled cluster operator

The computational complexity of the coupled-cluster method truncated after a given excitation level m - for example, m = 2 for CCSD - may be discussed in terms of the number of amplitudes (Nam) in the coupled-cluster operator and the number of operations (Nop) required for optimization of the wavefunction. Considering K atoms, each with Nbas basis functions, we have the following scaling relations ... 

For example, the amplitudes if-- or d-- , in which orbitals (j), and (j) are localized on fragment X and orbitals (j)y and are localized on fragment Y, will be zero. Thus, the total coupled cluster exponential operator may be written as a product of independent coupled cluster operators for each fragment, namely, ... 

We use capital letters (as in (3.11)) for active spin-orbitals. Among those, the letters I, J. etc. are used for active spin-orbitals which are occupied in 0) and letters A, B, etc. are used for active spin-orbitals which are unoccupied in 0). Lowercase indices (i, j, k, 1,...) denote the occupied orbitals and (a, b, c, d,...) denote orbitals which are unoccupied in all the CAS determinants. These orbitals are inactive . Also we use italic-style letters, a, b, c, d,... and i,j, k, /,..., to label orbitals in the cluster amplitudes and indicate that the coupled-cluster operators corresponding to these amplitudes can involve both active and inactive orbitals. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples... 

The field- and time-dependent cluster operator is defined as T t, ) = nd HF) is the SCF wavefunction of the unperturbed molecule. By keeping the Hartree-Fock reference fixed in the presence of the external perturbation, a two step approach, which would introduce into the coupled cluster wavefunction an artificial pole structure form the response of the Hartree Fock orbitals, is circumvented. The quasienergy W and the time-dependent coupled cluster equations are determined by projecting the time-dependent Schrodinger equation onto the Hartree-Fock reference and onto the bra states (HF f[[exp(—T) ... 

A standard method of improving on the Hartree-Fock description is the coupled-cluster approach [12, 13]. In this approach, the wavefunction CC) is written as an exponential of a cluster operator T working on the Hartree-Fock state HF), generating a linear combination of all possible determinants that may be constructed in a given one-electron basis,... 

For our purposes, CC theory and its finite order MBPT approximations offer a convenient, compact description of the correlation problem and give rapid convergence to the basis set (i.e. full Cl) limit with different categories of correlation operators (see Fig. 15.1). The coupled-cluster wave function is... 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

To summarize the theory dynamic correlations are described by the unitary operator exp A acting on a suitable reference funchon, where A consists of excitation operators of the form (4). We employ a cumulant decomposition to evaluate all expressions in the energy and amphtude equations. Since we are applying the cumulant decomposition after the first commutator (the term linear in the amplimdes), we call this theory linearized canonical transformation theory, by analogy with the coupled-cluster usage of the term. The key features of the hnearized CT theory are summarized and compared with other theories in Table II. 

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