Hartree-Fock theory restricted open-shell

The equations require to be modified for open-shell systems, in which some orbitals are doubly occupied and some singly (this is called spin-restricted Hartree-Fock theory). A further extension to the theory involves electrons of a and /3 spin being assigned to different molecular orbitals, type equations are described as unrestricted Hartree-Fock [31]. 

Hartree-Fock (theory) (FIF) (restricted HF, RFIF for closed shells unrestricted HF, UHF for open shells)... 

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. 

Many-Body Perturbation Theory with a Restricted Open-Shell Hartree-Fock Reference. 

W. J. Lauderdale, J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, J. Chem. Phys., 97, 6606 (1992). Restricted Open-Shell Hartree-Fock Based Many-Body Perturbation Theory. Theory and Application of Energy and Gradient Calculations. 

Electronic structure calculations have been performed with the system of programs Gaussian94 (G94) [19]. Restricted Hartree-Fock theory (RHF) is used for closed shell systems, and Unrestricted Hartree-Fock theory (UHF) for open shell systems (radicals). The correlation energy corrections are introduced with Moller-Plesset perturbation theory up to second order, and results from spin projected calculations are used (PMP2 and PUMP2). 

We start with paper [1]. This work had put forward a first possible definition of the EUE density for an arbitrary wave function with any permitted spin value s > 0. As mentioned in the introduction, our main interest is the case of singlet states, and for them the EUE effects are really important and interesting. Indeed, for nonzero spin states (doublet-state radicals, triplet-state diradicals etc.), the manifestations of unpaired electrons can be described even within the restricted open-shell Hartree-Fock (ROHF) theory. The latter characterizes the unpaired spins by standard spin density matrices. In the singlet state, the spin density matrix disappears [2], and yet, electron correlation enforces electrons to be unpaired if physical and chemical circumstances require it (e.g., in bond breaking processes). 

Hurley, A. C., Introduction to the Electron Theory of Small Molecules Academic Press, New York, 1976. The restricted open-shell Hartree-Fock procedure, which we have not expanded upon is described. 

Fig. 4. Electron configuration diagrams highlighting the differences between restricted Hartree-Fock theory (RHF), restricted open-shell Hartree—Fock theory (ROHF), and unrestricted Hartree-Fock theory (UHF).

Lauderdale, W. J., Stanton, J. R, Gauss, J., D., W. J., 8c Bartlett, R. J. (1991). Many-body perturbation theory with a restricted open-shell Hartree-Fock reference. Chemical Physics Letters, 187, 21-28. 

As discussed in the present section and also in Sections 10.5 and 10.6, canonical Hartree-Fock theory offers many advantages. However, for open-shell RHF states, this approach is not always possible because of the constraints imposed by the spin- and space-symmetry adaptation of the wave fiinction. In such cases, the wave function must be calculated using the more general methods of Section 10.8 in the present section, we restrict our attention to closed-shell systems. In addition, the application of canonical Hartree-Fock theory is restricted to small and moderately large systems since it requires an amount of work that scales at least cubically with the size of the system for large systems, the methods of Section 10.7 must be used instead. 

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. 

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