The coupled-cluster energy

The additivity of E and the separability of the equations determining the Cj eoeffieients make the MPPT/MBPT energy size-extensive. This property ean also be demonstrated for the Coupled-Cluster energy (see the referenees given above in Chapter 19.1.4). However, size-extensive methods have at least one serious weakness their energies do not provide upper bounds to the true energies of the system (beeause their energy funetional is not of the expeetation-value form for whieh the upper bound property has been proven). 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

The computational problem, then, is determination of the cluster amplitudes t for aU of the operators included in tlie particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wave functions expressed as dctcnninants of the HF orbitals. This generates a set of coupled, nonlinear equations in the amplitudes which must be solved, usually by some iterative technique. With the amplitudes in hand, the coupled-cluster energy is computed as... 

P. Piecuch, J. Cizek, and J. Paldus, Int. ]. Quantum Chem., 42, 165 (1992). Behaviour of the Coupled Cluster Energy in the Strong Correlated Limit of the Cyclic Polyene Model. Comparison with the Exact Results. 

The determination of a coupled cluster wave function does not follow the conventional variational procedure but a non-variational procedure where the excitation amplitudes are determined by a projection technique. We have that the coupled cluster energy for a molecule in vacuum is given by... 

Presently, we are able to determine the coupled cluster energy based on the variational Lagrangian and expectation values for real operators... 

This property of the coupled cluster energy is commonly known as size consistency. ... 

This is the natural truncation of the coupled cluster energy equation an analogous phenomenon occurs for the amplitude equation (Eq. [45]). This truncation depends only on the form of f and not on that of T or on the number of electrons. Equation [49] is correct even if T is truncated to a particular excitation level. 

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. 

The ostensible impracticality of a variational coupled cluster theory raises an important question regarding the physical reality of the coupled cluster energy as computed using projective, asymmetric techniques. Quantum mechanics dictates that physical observables (such as the energy) are expectation... 

The two-electron component, however, produces four equivalent fully contracted terms, and therefore contributes to the coupled cluster energy ... 

This equation is not restricted to the CCSD approximation, however. Since higher excitation cluster operators such as T3 and T4 cannot produce fully contracted terms with the Hamiltonian, their contribution to the coupled cluster energy expression is zero. Therefore, Eq. [134] also holds for more complicated methods such as CCSDT and CCSDTQ. Higher excitation cluster operators can contribute to the energy indirectly, however, through the equations used to determine the amplitudes, and t-h, which are needed in the energy equation above. 

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 coupled-cluster electronic state is uniquely defined by the set of the cluster amplitudes and these amplitudes are used to obtain the coupled-cluster energy from Eq. (33). Due to the fact that the Ansatz of the coupled-cluster wave function has the exponential parametrization [Eq. (28)] the energy is size-extensive. This is an obvious advantage of the coupled-cluster formalism compared to some other techniques (e.g. configuration interaction). For a general discussion of coupled-cluster theory and the coupled-cluster equations see Refs. [5, 36]. 

One of the special cases of coupled-cluster theory is the singles-and-doubles (CCSD) model [37]. The cluster operator Eq. (29) is restricted to contain only the singles and doubles excitation operators. The importance of this model can be seen from the fact that, for any coupled-cluster wave function, the singles and doubles amplitudes are the only ones that contribute directly to the coupled-cluster energy. In the explicitly correlated CCSD model the conventional cluster operator containing the T and T2 operators is supplemented with an additional term that takes care of the explicit correlation (written with red font)... 

Similarly to the MP2 energy, Eq. (9.68), the coupled cluster energy is obtained as a transition expectation value by projecting against the Hartree-Fock wavefunction... 

In the CCSD model, for example, the excited projection manifold comprises the fiill set of all singly and doubly excited determinants, giving rise to one equation (13.2.19) for each connected amplitude. For the full coupled-cluster wave function, the number of equations is equal to the number of determinants and the solution of the projected equations recovers the FCI wave function. The nonlinear equations (13.2.19) must be solved iteratively, substituting in eac iteration the coupled-cluster energy as calculated from (13.2.18). 

As a result, only singles and doubles amplitudes contribute directly to the coupled-cluster energy -irrespective of the truncation level in the cluster operator. Of course, the higher-order excitations contribute indirectly since all amplitudes are coupled by the projected equations (13.2.23). 

Having examined the coupled-cluster energy and seen that it is no higher than quadratic in the cluster amplitudes, let us now turn our attention to the structure of the linked projected coupled-cluster equations (13.2.23) ... 

Inserting the Hamiltonian (13.2.42) into the expressions for the energy (13.2.22) and the nonlinear equations (13.2.23), we obtain the following expressions for the coupled-cluster energy and the amplitude equations ... 

For reasons of computational practicality and efficiency, molecular electronic coupled-cluster energies are determined using a nonvariational projection technique. The chief deficiency of this approach is not so much the loss of boundedness (since the coupled-cluster energy is nevertheless rather accurate), but the difficulties that it creates for the calculation of properties as the conditions for the Hellmann-Feynman theorem are not satisfied - even in the limit of a eomplete one-electron basis. Fortunately, as discussed in Section 4.2.8, this situation may be remedied by the construction of a variational Lagrangian [14]. In this formulation, the conditions of the Hellmann-Feynman theorem are fulfilled and molecular properties may be calculated by a proeedure that is essentially the same as for variational wave functions. The Lagrangian formulation of the energy is also related to a variational treatment of coupled-cluster theory applicable to excited states, as discussed in Section 13.6. 

The calculation of the zero-order multipliers and Lagrangian requires the solution of one set of linear equations. The resulting expression for the coupled-cluster energy is then variational with respect to the amplitudes as well as their multipliers. Note that the linear equations for the multipliers (13.5.6) are similar in structure to the Newton equations for the amplitudes (13.4.5), both containing the coupled-cluster Jacobian (13.4.4). 

---