Restricted HF theory. An open-shell system

We now discuss the determination of optimum MOs for a state constrained to be of the form (6.1.1) in which n, orbitals are doubly occupied and the remaining it2 are singly occupied with spins parallel. In this context, the sets of doubly and singly occupied orbitals are usually referred to respectively as the closed shell and the open shell . Other states, in which the closed shell is fixed but there are several determinants corresponding to various orbital occupations and coupling schemes in the open shell, may sometimes be dealt with in the same way but first we consider the simple 1-determinant case. We also assume the wave- 

The MOs of the closed shell and the open shell may be collected in matrices T, and T2, with n, and 2 columns respectively, and we define density matrices for each shell, 

The energy expression that replaces (5.3.18) is then found to be (using A, B. and U, V. for closed- and open-shell orbitals respectively) 

for instance, the G term represents the coulomb-exchange effect of Ifae closed-shell electrons (density described by ViRi), while the G, term refers to the other open-shell electrons. 

The orthonormality conditions may be replaced by equivalent conditions on the density matrices these represent projections onto the subspaces spanned by doubly and singly occupied MOs and therefore satisfy the conditions (6.4.24), namely 

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