Spin-restricted coupled-cluster theory

The spin-restricted coupled-cluster theory presented here is given in the spin-orbital basis. As such, there are no immediate computational savings relative to the spin-unrestricted theory. However, the spin equations (13.9.22) may be used to reduce the number of parameters in the projected equations (13.9.26) and thereby the computational cost of the evaluation. However, we do not here go into the practical details of such an approach to the calculation of spin-restricted coupled-cluster wave functions. 

Spin-restricted coupled-cluster theory does not provide a wave function that is an eigenfunction of the total spin, just as it does not provide an eigenfunction of the Hamiltonian. Rather, spin-restricted theory gives a wave function for which the spin equations are solved in the truncated projection manifold and the Schrodinger equation in the subspace of spin-adapted CSFs p). In a sense, spin-restricted coupled-cluster theory may thus be said to provide a balanced treatment of energy and spin. Neither the spin equations nor the energy equations are satisfied in the orthogonal complement p) to the projection manifold ... 

However, although spin-restricted coupled-cluster theory does not provide a spin-adapted wave function, it does afford a description where the expectation value of the spin operator is exact. 

In spin-restricted coupled-cluster theory, the projection manifold is partitioned into two spaces a space containing CSFs of the same spin symmetry as the reference state and a space containing the remaining states. As an illustration, consider a spin-restricted CCSD calculation for a doublet reference state... 

Open-shell coupled-cluster theory can be formulated in terms of the one-particle spinor or spin-orbital basis. However, spin-restricted or Kramers-restricted open-shell theories are complicated by the ambiguous role of the open-shell orbital or Kramers pair. We develop here the basic outline of open-shell Kramers-restricted coupled-cluster theory. 

M. Urban, P. Neogrady, and I. Hubac, Spin Adaptation in the Open-Shell Coupled-Cluster Theory with a Single Determinant Restricted Hartree-Fock Reference. In R. J. Bartlett (Ed.) Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, Vol. 3. (World Scientific, Singapore, 1997), pp. 275-306. 

T. D. Crawford, T.. Lee, and H. F. Schaefer,/. Chem. Phys., 107, 7943 (1997). A New Spin-Restricted Triple Excitation Correction for Coupled Cluster Theory. 

P. Neogrady, M. Urban, and I. Hubac, /. Chem. Phys., 100, 3706 (1994). Spin Adapted Restricted Hartree-Fock Reference Coupled-Cluster Theory for Open-Shell Systems. 

The Kramers-restricted form of the Hamiltonian that was used in Cl theory is not suitable for Coupled Cluster theory because it mixes excitation and deexcitation operators. One possibility is to define another set of excitation operators that keep the Kramers pairing and use these in the exponential parametrization of the wavefunction. This would automatically give Kramers-restricted CC equations upon rederivation of the energy and amplitude equations. A more pedestrian but simpler alternative is to start from the spin-orbital formulation and inspect the relations that follow from the Kramers relation of the two-electron integrals. This method does also readily give the relations between the Kramers symmetry-related amplitudes. We will briefly discuss the basic steps in this approach, a detailed description of a possible algorithm is given in reference [47],... 

For open-shell molecules, spin contamination can be a problem, as mentioned earlier. DFT and coupled-cluster theory are resistant to spin contamination and may be helpful. One may also choose spin-restricted open-shell HF (ROHF) as a starting point, instead of UHF, unless dissociation behavior is important. ROHF has no spin contamination. 

In the closed-shell theory of Section 13.7, we developed a spin-restricted theory in which both of these problems are solved. Thus, for closed-shell systems, we may calculate a singlet coupled-cluster wave function at a fraction of the cost of the corresponding spin-unrestricted wave function. Unfortunately, for open-shell systems, the problems are more complicated and it is no longer obvious how we should best satisfy the Schrbdinger and spin equations in coupled-cluster theory. 

Chapter 13 discusses coupled-cluster theory. Important concepts such as connected and disconnected clusters, the exponential ansatz, and size-extensivity are discussed the Unked and unlinked equations of coupled-clustCT theory are compared and the optimization of the wave function is described. Brueckner theory and orbital-optimized coupled-cluster theory are also discussed, as are the coupled-cluster variational Lagrangian and the equation-of-motion coupled-cluster model. A large section is devoted to the coupled-cluster singles-and-doubles (CCSD) model, whose working equations are derived in detail. A discussion of a spin-restricted open-shell formalism concludes the chapter. 

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. 

Let us examine the convergence of the coupled-cluster hierarchy towards the FCI wave function. In Table 5.11, we have — for the water molecule in the cc-pVDZ basis — listed the differences between the (spin- and space-restricted) CCSD, CCSDT and CCSDTQ energies and the FCI eneigy [3]. Comparing with the Cl energies in Table 5.9, we find that the coupled-cluster hierarchy converges faster and more uniformly towards the FCI limit. In particular, there is a significant reduction in the error at each level of truncation - not just at even-order levels as in Cl theory. 

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