Open-shell cases

However, it is known that the open-shell hamiltonian itself is non-unique.34 In particular this means that the eigenvalues are dependent on the particular form of the hamiltonian for which the orbitals are canonical. Any attempt to use the extended form of the Phillips-Kleinman projection operator, which for open-shell cases can be written... 

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

As a final variant the SCF procedure may be solved by a Newton Raphson technique, a very important component of which comprises a partial or complete 4-index tramsformation of integrals at each cycle. As we show below, the integral transformation procedure is highly vectorisable. We feel that such a technique will perhaps prove profitable in slowly convergent close shell cases or complicated open shell cases. 

We can thus conclude that states of different spin multiplicity (singlets, doublets, triplets, quartets, etc.) of very diverse jr-electron systems (Kekule or non-Kekule, alternant or nonalternant, aromatic, nonaromatic or antiaromatic) can be satisfactorily described by the PPP-VB method with a severely truncated set of covalent or maximally covalent structures using the same simple OEAO basis set hi. In contrast, the MO description requires a different handling of closed and open shell cases and the amount of correlation recovered in states of different multiplicity may be rather unbalanced. 

Combining the LCAO approximation for the MOs with the HE method led Roothaan to develop a procedure to obtain the SCE solutions. We will discuss here only the simplest case where all MOs are doubly occupied with one electron that is spin up and one that is spin down, also known as a closed-shell wavefiinction. The open-shell case is a simple extension of these ideas. The procedure rests upon transforming the set of equations listed in Eq. (1.7) into matrix form... 

By minimizing the energy of d>, in Eq. (3.12), we obtain a set of coupled integro-differential equations, the Hartree-Fock equations, which may be expressed in the following form for closed-shell systems (for open-shell cases see Szabo and Ostlund, 1989) ... 

The simple closed-shell problem is generalised to open-shell cases by Roothaan (I960) and Roothaan and Bagus (1963). The present formalism gives an introduction to the generalisations. 

For open-shell and small (e.g. two-configuration) MCSCF wavefunctions, the construction of the AO density matrix according to Eq. (32) is computationally negligible compared to the evaluation of the integral derivatives. The open-shell case is thus computationally identical to the closed-shell case. 

Aspects of the relativistic theory of quantum electrodynamics are first reviewed in the context of the electronic structure theory of atoms and molecules. The finite basis set parametrization of this theory is then discussed, and the formulation of the Dirac-Hartree-Fock-Breit procedure presented with additional detail provided which is specific to the treatment of atoms or molecules. Issues concerned with the implementation of relativistic mean-field methods are outlined, including the computational strategies adopted in the BERTHA code. Extensions of the formalism are presented to include open-shell cases, and the accommodation of some electron correlation effects within the multi-configurational Dirac-Hartree-Fock approximation. We conclude with a survey of representative applications of the relativistic self-consistent field method to be found in the literature. 

Matrix theory for Dirac one-electron problems was set up in the last section, and we shall now generalize this, first for closed-shell atoms and then for the general open-shell case. We use the effective Hamiltonian of (95) as the starting... 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

More complicated open shell cases can be handled via the Configuration Interaction (Cl) method that allows a sophisticated choice of reference space. This makes the method applicable in cases where single reference methods fail due to the presence of low-lying excited states. This is often so in relativistic calculations where many applications concern complexes of heavy transition metals or f-elements. 

It has often been assumed that the MCSCF Fock matrix can be used as Hq for multiconfiguration root functions. This works for a few simple open-shell cases but is in general wrong. The MCSCF Fock matrix, which is used to find energy-optimized orbitals in MCSCF (such as CASSCF), is... 

Though used in some semiempirical applications by Paldus and Cizek [11] and one ab initio study [12] (see later), the CCD equations were not implemented into general purpose programs until 1978 by me and Purvis [5] and Pople et al. [13]. This general implementation included allowing for the open-shell case subject to an unrestricted Hartree-Fock reference function. 

All conventional coupled-cluster calculations were performed with the Mainz-Austin-Budapest version of the Aces II program [83] and with the MRCC program [89, 90]. Within all post Hartree-Fock calculations, both core and valence electrons were correlated (all-electron calculations). For the open-shell cases, the UHF reference wave function was used. The Hartree-Fock and GGSD(T) contributions to the IPs and EAs were computed within the correlation-consistent doubly augmented quintuple-d-aug-cc-pwCV5Z, basis... 

Theoretical Results. To solve the open-shell problem, various MO methods have been applied and, in some cases, compared with each other. Most commonly, the restricted Hartree-Fock formalism for the open-shell case (RHF) [15] and the unrestricted Hartree-Fock formalism before (UHF) [16] and after single spin annihilation (UHFASA) [17] have been used besides other methods, see [18 to 21] and footnote of Table 6, p. 234. 

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