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In Frank Jensen's Introduction to Computational Chemistry ($\mathrm{2^{nd}}$ edition), on page 99, it says:

For an ROHF wave function, it is not possible to choose a unitary transformation that makes the matrix of Lagrange multipliers in eq. (3.41) diagonal, and orbital energies from an ROHF (Restricted Open-shell Hartree–Fock) wave function are consequently not uniquely defined and cannot be equated to ionization potentials by a Koopmans-type argument.

Here ROHF eq. (3.41) is the non canonical Hartree-Fock equation:

$$F_i\phi_i=\sum\lambda_{ij}\phi_j$$

But in ATTILA SZABO, NEIL S. OSTLUND's Modern Quantum Chemestry, they prove that the Hartree-Fock equation can always be put into the canonical form:

$$F_i\phi_i=\lambda_{i}\phi_i$$

What am I misunderstanding here?

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    $\begingroup$ I'm guessing that "restricted open-shell" part is important. $\endgroup$ – Buck Thorn Mar 5 '19 at 10:56
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    $\begingroup$ ATTILA SZABO, NEIL S. OSTLUND showed that there is a basis where the hartree-fock equation has the form $F_i\phi_i=\lambda_{i}\phi_i$ that is, it does not work for every basis , only for a kind special of basis , so there is no issue in the two statements $\endgroup$ – amilton moreira Mar 5 '19 at 12:50
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The treatment found in Szabo-Ostlund, although perhaps it is not explicitly stated, is appropriate only for non-degenerate spin-singlet electronic states, which in the Hartree-Fock method correspond to closed-shell systems. For open-shell systems one may use the restricted open-shell Hartree-Fock method (ROHF), which is somewhat more complicated. In particular, in the ROHF method six arbitrary constants have to be introduced. Many choices for these parameters have been suggested. The total ROHF energy, as well as the overall ROHF wave function and therefore the expectation value of any operator, are independent of these six parameters, so that all these ROHF sub-variants are largely equivalent. However, the six parameters affect the Hartree-Fock orbitals and their orbital energies. Also, approximate post-HF methods such as MP2 or CISD will depend on the particular choice of those six parameters.

Knowles et al suggested in 1991 a particular choice leading to called "semi-canonical orbitals" whose orbital energies do obey Koopmans theorem and which are also a good choice for MP2 calculations.

I suggest the following paper for an introduction to the problem (the paper by Knowles metioned above is ref. 18): T. Tsuchimochi and G. E. Scuseria, ROHF Theory Made Simple, J. Chem. Phys. 133, 141102 (2010) https://arxiv.org/abs/1008.1607

The paper above shows that one can obtain semi-canonical ROHF orbitals by a minor modification of the unrestricted Hartree-Fock method and that this approach is numerically very stable.

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