Open-shell species DODS

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with j3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Rp.strirted Open-shell Hartree-Fock (RQHF). For open-shell species a UHF treatment... 

Figures 2.1-2.3 and Table 2.1 emphasize the important differences between spin-orbitals that are considered equivalent in elementary treatments. This is particularly true for open-shell species, where the notion of pairing electrons of opposite spin in the same spatial orbital is generally unrealistic. Instead, one should visualize open-shell electronic distributions in terms of different orbitals for different spins (DODS), recognizing that distinct Coulomb and exchange forces will generally split paired electrons into spatially distinct spin-orbitals (see Sidebar 2.2). The DODS concept is automatically incorporated into open-shell NBO analysis, where analysis of a and p spin sets proceeds independently in separate output sections, with no presumed relationship between natural orbitals of the two spin sets. Only in rather exceptional cases (e.g., uncorrelated closed-shell singlet species in near-equilibrium geometry) will electrons be found to pair up in the restrictive manner envisioned in elementary textbooks. The DODS concept is generally a more satisfactory conceptual foundation on which to build an accurate and robust picture of closed- and open-shell electronic phenomena.

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