Since the classic work of Roothaan [Rev. Mod. Phys. 32, 179 (1960)], the one-electron energies of a ROHF method are known as ambiguous quantities having no physical meaning. Together with this, it is often assumed in present-day computational studies that Koopmans' theorem is valid in a ROHF method. In this work we analyze the specific dependence of orbital energies on the choice of the basic equations in a ROHF method which are the Euler equations and different forms of the generalized Hartree-Fock equation. We first prove that the one-electron open-shell energies epsilon(m) derived by the Euler equations can be related to the respective ionization potentials I(m) via the modified Koopmans' formula I(m)= -epsilon(m)f(m) where f(m) is an occupation number. As compared to this, neither the closed-shell orbital energies nor the virtual ones derived by the Euler equations can be related to the respective ionization potentials and electron affinities via Koopmans' theorem. Based on this analysis, we derive the new (canonical) form for the Hamiltonian of the Hartree-Fock equation, the eigenvalues of which obey Koopmans' theorem for the whole energy spectrum. A discussion of new orbital energies is presented on the examples of a free N atom and an endohedral N@C(60) (I(h)). The vertical ionization potentials and electron affinities estimated via Koopmans' theorem are compared with the respective observed data and, for completeness, with the respective estimates derived via a DeltaSCF method. The agreement between observed data and their estimates via Koopmans' theorem is qualitative and, in general, appears to possess the same accuracy level as in the closed-shell SCF.

Recent discovery of magnesium isotope effect in the rate of enzymatic synthesis of adenosine triphosphate (ATP) offers a new insight into the mechanochemistry of enzymes as the molecular machines. The activity of phosphorylating enzymes (ATP-synthase, phosphocreatine, and phosphoglycerate kinases) in which Mg(2+) ion has a magnetic isotopic nucleus 25Mg was found to be 2-3 times higher than that of enzymes in which Mg(2+) ion has spinless, nonmagnetic isotopic nuclei 24Mg or 26Mg. This isotope effect demonstrates unambiguously that the ATP synthesis is a spin-dependent ion-radical process. The reaction schemes, suggested to explain the effect, imply a reversible electron transfer from the terminal phosphate anion of ADP to Mg(2+) ion as a first step, generating ion-radical pair with singlet and triplet spin states. The yields of ATP along the singlet and triplet channels are controlled by hyperfine coupling of unpaired electron in 25Mg+ ion with magnetic nucleus 25Mg. There is no difference in the ATP yield for enzymes with 24Mg and 26Mg; it gives evidence that in this reaction magnetic isotope effect (MIE) operates rather than classical, mass-dependent one. Similar effects have been also found for the pyruvate kinase. Magnetic field dependence of enzymatic phosphorylation is in agreement with suggested ion-radical mechanism.

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A new, ion-radical mechanism of enzymatic ATP synthesis was recently discovered by using magnesium isotopes. It functions at a high concentration of MgCl(2) and includes electron transfer from the Mg(H(2)O)(m)(2+)(ADP(3-)) complex (m = 0-4) to the Mg(H(2)O)(n)(2+) complex as a primary reaction of ATP synthesis in catalytic sites of ATP synthase and kinases. Here, the structures and electron transfer reaction energies of magnesium complexes related to ATP synthesis are calculated in terms of DFT. ADP is modeled by pyrophosphate anions, protonated (HP(2)O(7)H(2-), HP(2)O(7)CH(3)(2-)) and deprotonated (HP(2)O(7)(3-), CH(3)P(2)O(7)(3-)). The reaction generates an ion-radical pair, composed of Mg(H(2)O)(n)(+) ion and pyrophosphate anion-radical coordinated to Mg(2+) ion. The addition of the latter to the substrate P=O bond results in ATP formation. Populations of the singlet and triplet states and singlet-triplet spin conversion in the pair are controlled by hyperfine coupling of unpaired electrons with magnetic (25)Mg and (31)P nuclei and by Zeeman interaction. Due to these two interactions, the yield of ATP is a function of nuclear magnetic moment and magnetic field; both of these effects were experimentally detected. Electron transfer reaction does not depend on m but strongly depends on n. It is exoergic and energy allowed at 0 < or = n << infinity for the deprotonated pyrophosphate anions and at 0 < or = n < 4 for the protonated ones; for other values of n, the reaction is energy deficient and forbidden. The boundary between exoergic and endoergic regimes corresponds to the trigger magnitude n* (n* = 4 for protonated anions and 6 < n* << infinity for deprotonated ones). These results explain why ATP synthesis occurs only in special devices, molecular enzymatic machines, but not in water (n = infinity). Biomedical consequences of the ion-radical enzymatic ATP synthesis are also discussed.