Excitation operators coupled-cluster

Coupled-cluster with single and double excitation cluster operators Coupled-cluster with single, double, and triple excitation cluster operators... 

To compare the excitation-based coupled-cluster model with the configuration-based Cl model, we expand the exponential operator in (13.2.3) and collect terms to the same order in the excitation level ... 

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 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 ... 

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. 

In section c above, two different ways of writing the wave-function have been described by (4.12) and (4.13), which in principle can be extended to the exact wave-function. There is a third alternative which has many advantages and is connected with one of the presently dominating ways of solving the Schiodinger equation, and this is the coupled cluster wave-function. To define this wave-function it is convenient to define certain excitation operators. A general n-tuple excitation operator T is defined as... 

An interesting point of the coupled cluster method concerns the treatment of quadruple excitations. If the CCD method is considered, in which only the T2 operator is retained in the exponent, the amplitudes for these excitations are given as products of amplitudes for double excitations according to the term Ij. In fact is a sum of the 18 products of type (with phase factors) which can be formed... 

Another recently created correlated method is the no(virtual)pair DF (DCB) coupled-cluster technique of E. Eliav, U. Kaldor and I. Ishikawa [31,32,44]. It is based on the DCB Hamiltonian Equation 3. Correlation effects are taken into account by action of the excitation operator... 

The exponential operator T creates excitations from 4>o according to T = l + 72 + 73 + , where the subscript indicates the excitation level (single, double, triple, etc.). This excitation level can be truncated. If excitations up to Tn (where N is the number of electrons) were included, vPcc would become equivalent to the full configuration interaction wave function. One does not normally approach this limit, but higher excitations are included at lower levels of coupled-cluster calculations, so that convergence towards the full Cl limit is faster than for MP calculations. 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

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