Restricted Hartree-Fock species

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... 

Compounds containing B—O Bonds.—Calculated bond lengths [r(H—B) = 2.1913 Bohr(1.1596 A) r(B—0) = 2.2284 Bohr(1.1792 A)] for the recently detected, linear HBO species were obtained from a restricted Hartree-Fock LCAO-MO-SCF study. ... 

G bases, except for the anions for which the MP2 energies were calculated in the former basis set only. Excitations from the core electrons were not included in the MP2 treatment. The restricted Hartree-Fock (RHF) method was used for the closed-shell molecules (parents and anions) and the unrestricted Hartree-Fock (UHF) method was applied to the spin doublet open-shell species (radicals and cations). Both methods are variants of the SCF approximation. A fuller description and explanation of the basis set and methods has been given previously8. 

Recently, an ab initio 6-31 G basis set calculation reaffirmed that phenol is significantly more stable than both 2,4- and 2,5-cyclohexadienone. Total electronic energies at 0 K and thermodynamic parameters were calculated for phenol, 2,4-cyclohexa-dienone, and 2,5-cyclohexadienone. Energy calculations were performed using the restricted Hartree— Fock (RHF) and second-order Moller—Plesset formalisms on the RHF optimized geometries. The study has shown the following (i) phenol is more stable than the two ketones by around 18 kcal/mol (ii) the ketone with the C-sp para to the carbonyl is slightly less stable than the one with the C-sp ortho to the carbonyl (iii) thermodynamic data calculated for the ketones versus the enol forms for the three species confirm the already known fact that phenol is more stable in the enolic form (Scheme 16). 

The common way to treat free radicals is with the unrestricted Hartree-Fock method or UHF method. In this method, we employ separate spatial orbitals for the oc and the jl electrons, giving two sets of MO s, one for oc and one for fj electrons. Less commonly, free radicals are treated by the restricted open-shell Hartree-Fock or ROHF method, in which electrons occupy MO s in pairs as in the RHF method, except for the unpaired electron(s). The theoretical treatment of open-shell species is discussed in various places in references [1] and in [12]. 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

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