Spin-restricted reference wavefunction

Several researchers have recently devoted considerable effort to the derivation and efficient implementation of techniques based on spin-restricted reference determinants that reduce the computational discrepancy between closed- and open-shell systems. " This emphasis on spin-restricted techniques has resulted in part from a bias toward reference wavefunctions that maintain the spin symmetry of the exact wavefunction (such as the ROHF determinant), but also because of the possible efficiency advantages of spin-restricted methods over unrestricted techniques. Thus, since the component molecular orbitals are constrained to have identical spatial parts for each spin function, it should be possible to construct the correlated wavefunction in a manner that takes advantage of this symmetry. 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

The choice of as the zeroth-order Hamiltonian requires the use of either a spin-restricted (closed-shell) Hartree-Fock (RHF) or spin-unrestricted Hartree-Fock (UHF) determinant as the zeroth-order (reference) wavefunction. Since spin-restricted open-shell Hartree-Fock (ROHF) reference functions are not eigenfunctions of the spin-orbital P, other partitionings are required (Refs. 127-134). 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

The term Restricted Hartree-Fock (RHF) is applied to those cases in which all the possible spin pairing in a system is allowed for by having electrons of both and p spin occupy the same space orbital. If this restriction is relaxed in writing out the determinantal wavefunction, the method of calculation is referred to as the Unrestricted Hartree-Fock (UHF) method. Unless Otherwise stipulated, the calculations referred to in this chapter are of the RHF variety. 

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],... 

These two issues recently initiated interest in the development of alternative open-shell CC schemes. " " A first suggestion was to start from a restricted open-shell HF (ROHF) instead of a UHF reference function. Since the ROHF wavefunction is already a spin eigenfunction, it can be shown that energies obtained from a spin-orbital based CC treatment correspond to so-called spin-projected energies ... 

The main restriction of the methods discussed in the main part of this chapter is the assumption of a single dominant configuration. This requirement is not met for systems with strong non-dynamical correlation effects such as biradicals (e.g. in bond-breaking situations) or transition metal compounds. In this case the reference function must be a multireference expansion. The fixed-amplitude approximation lends itself very well to a multireference extension of the formalism, as demonstrated first by Ten-no who devised an F12-based internally contracted geminal correction to the multireference MP2 (MRMP2) method. A closely related approach was considered by Torheyden and Valeev, who proposed a generalized perturbative correction to arbitrary wavefunctions and which they applied to a MRCI wavefunction. A spin-free formulation was reported recently.The approach was also used by Booth et al. to get a basis set limit estimate for their full Cl quantum Monte Carlo method, which can also be seen as an approach to tackle systems with multireference character. 

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