Cluster operator single

Cluster operator (general, single, double,. .. excitations)... 

In this section, we derive basic equations for the monoexcited and biexcited cluster amplitudes at the CCSD level of approximation, i.e. with the cluster operators 7 being approximated by their singly and doubly excited cluster components... 

On the one hand, we can strive for a single cluster operator T, defining the valence universal wave operator U, U = exp(T), which will transform all the model space states ]< > ) into some linear combinations of the exact states jfl i), f = 1,2, , M, which in turn span the target space M, i.e.. 

CCSD Coupled cluster with single and double substitution operators... 

Similarly the CC amplitudes are determined by projecting the Schrodinger equation from the left against all excitations included in the cluster operator, for example, all single and double excitations from the HF reference function. If we denote this set of excited states by >, the cluster amplitude equations have the form... 

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

Using these so-called second-quantized operators, we may define the single-orbital cluster operator... 

Note that commutation of cluster operators holds only when the occupied and virtual orbital spaces are disjoint, as is the case in spin-orbital or spin-restricted closed-shell theories. For spin-restricted open-shell approaches, where singly occupied orbitals contribute terms to both the occupied and virtual orbital subspaces, the commutation relations of cluster operators are significantly more complicated. See Ref. 36 for a discussion of this issue. 

In this expression, hp = pW q) represeiits a matrix element of the one-electron component of the Hamiltonian, h, while (pqWrs) s ( lcontains 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 ... 

Next we consider the first quadratic term from Eq. [123], which involves two T1 cluster operators. The reference expectation value of the one-electron component, is zero because the single construction operator pair in... 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory ... 

Questions concerning possible modifications to the descriptions of excited determinants and cluster operators Tm due to the renormalization and/or the use of normal product operators must be addressed. Excited determinants with respect to the Fermi vacuum can be written straightforwardly using creation and annihilation operators a singly excited determinant is given by... 

Application of the time-independent Wick theorem to the single-excitation operator X% Xiy present in both the description of singly excited determinants with respect to the Fermi vacuum and in the cluster operator T, gives... 

---