Expectation value coupled cluster method

R. J. Bartlett and J. Noga, The expectation value coupled-cluster method and analytical energy derivatives. Chem. Phys. Lett. 150, 29-36 (1988). 

In contrast to variational methods, perturbation theory and coupled-cluster methods achieve their energies from a transition formula < I H I F > rather than from an expectation value... 

Some work has been reported on relativistic coupled cluster methods most notably by Kaldor, Ishikawa and their collaborators.232 These calculations are carried out within the no virtual pair approximation and are therefore analogous to the non-relativistic formulation. Perturbative analysis of the relativistic electronic structure problem demonstrated the importance of the negative energy branch of the spectrum in the calculation of energies and other expectation values. 

Care must be taken in using the expressions above for obtaining nonlinear optical properties, because the values obtained may not be the same as those obtained from Eq. [4]. The results will be equivalent only if the Hellmann-Feyn-man theorem is satisfied. For the case of the exact wavefunction or any fully variational approximation, the Hellmann-Feynman theorem equates derivatives of the energy to expectation values of derivatives of the Hamiltonian for a given parameter. If we consider the parameter to be the external electric field, F, then this gives dE/dP = dH/d ) = (p,). For nonvariational methods, such as perturbation theory or coupled cluster methods, additional terms must be considered. 

In non-variational approaches such as Moller-Plesset perturbation theory or coupled cluster methods the wavefunction is not at all variationally optimized. However, it is possible to choose ( o hi such a way that the Hellmann-Feynman theorem is fulfilled to a certain extent, while the transition expectation value in Eq. (9.88) still gives the energy. 

The last method used in this study is CCSD linear response theory [37]. The frequency-dependent polarizabilities are again identified from the time evolution of the corresponding moments. However, in CCSD response theory the moments are calculated as transition expectation values between the coupled cluster state l cc(O) and a dual state... 

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 additivity of E and the separability of the equations determining the Cj coefficients make the MPPT/MBPT energy size-extensive. This property can also be demonstrated for the Coupled-Cluster energy (see the references 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 (because their energy functional is not of the expectation-value form for which the upper bound property has been proven). 

The method described above is of general validity and can be applied to transition metal clusters of arbitrary shape, size, and nucleanty. It should be noted that in the specific case of a system comprising only two interacting exchanged coupled centers, our general treatment yields the same result as that of Bencini and Gatteschi (121), which was specifically formulated for dimers. In this case, the relation between the spin-projection coefficient and the on-site spin expectation value is simply given by... 

Certain Schrodinger equation based methods, such as coupled cluster theory, are not based on a variational principle. They fall outside schemes that use the energy expectation value as a optimization function for simulated annealing, although these methods could be implemented within a simulated annealing molecular dynamics scheme with alternative optimization function. 

Nevertheless, some effort has been extended toward developing coupled-cluster treatments that are either exactly or partially spin-adapted. Rigorously spin-adapted methods have been presented in the literature [214-216], but most applications have used either the partially spin-adapted variants [217,218] and the spin-restricted approach [219] in which the expectation value of (S2) is... 

Non-variational methods such as Mpller-Plesset perturbation theory (MP) or the coupled cluster (CC) method, where the energy can be expressed as a transition or asymmetric expectation value... 

In the case of the variational methods, SCF, MCSCF and CI, o0 = l o) and we have the normal expectation value. For the non-variational methods such as Mpller-Plesset perturbation or coupled cluster theory, the energy is calculated as a transition expectation value, where I FoO =... 

In Section 12.2 it will be discussed that this approach for the calculation of expectation values is called the unrelaxed method, because the conditions for the molecular orbital coefficients were not included as additional constraints in the coupled cluster Lagrangian given in Eq. (9.95) or Eq. (9.98). A coupled cluster Lagrangian including orbital relaxation takes the following form... 

Frequency-dependent response functions can only be computed within approximate electronic structure models that allow definition of the time-dependent expectation value. Hence, frequency-dependent response functions are not defined for approximate methods that provide an energy but no wave function. Such methods include MoUer-Plesset (MP) perturbation theory, multiconfigurational second-order perturbation theory (CASPT2), and coupled cluster singles and doubles with non-iterative perturbative triples [CCSD(T)]. As we shall see later, it is possible to derive static response functions for such methods. 

---