Unrestricted Hartree-Fock energy

Table 3 Unrestricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts, muon in atomic units (and MHz) after quartet spin state annihilation. The value of the total spin operator, < >, is also given after quartet spin state...

For open-shell systems, an extension of the Hartree-Fock method, called the unrestricted Hartree-Fock method, is sometimes used. For a closed-shell state, the lowest Hartree-Fock energy is generally obtained... 

You will need to decide whether or not to request Restricted (RHF) or Unrestricted (UHF) Hartree-Fock calculations. This question embodies a certain amount of controversy and there is no simple answer. The answer often depends simply on which you prefer or what set of scientific prejudices you have. Ask yourself whether you prefer orbital energy diagrams with one or two electrons per orbital. 

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. 

It should also be observed that there exists an approximation which is "intermediate between the unrestricted and the extended Hartree-Fock scheme. In starting from the former, the energy is increased by the mixing in of unappropriate spin states, and it can hence be essentially improved by selecting the component of the pure spin desired. It is clear that the energy obtained... 

Wood, J. H., and Pratt, G. W., Phys. Rev. 107, 995, "Wave functions and energy levels for Fe as found by the unrestricted Hartree-Fock method."... 

For the UHF calculations the undectet implies = 5 (septet implies 3z = 3) and the electrons are assigned as suggested by the ROHF calculation. For the 5 = 0 case the electrons are assigned to be AF coupled. The Projected Unrestricted Hartree-Fock (PUHF) energies correspond to the fully projected UHF values with 3 = 5. A brief overview of the PUHF method is given below, a more detailed description can be found in Reference 4. 

The wave funetion obtained eorresponds to the Unrestricted Hartree-Fock scheme and beeomes equivalent to the RHF ease if the orbitals (t>a and (()p are the same. In this UHF form, the UHF wave funetion obeys the Pauli prineiple but is not an eigenfunction of the total spin operator and is thus a mixture of different spin multiplicities. In the present two-eleetron case, an alternative form of the wave funetion which has the same total energy, which is a pure singlet state, but whieh is no longer antisymmetric as required by thePauli principle, is ... 

For most molecules studied, modest Hartree-Fock calculations yield remarkably accurate barriers that allow confident prediction of the lowest energy conformer in the S0 and D0 states. The simplest level of theory that predicts barriers in good agreement with experiment is HF/6-31G for the closed-shell S0 state (Hartree-Fock theory) and UHF/6-31G for the open-shell D0 state (unrestricted Hartree-Fock theory). The 6-31G basis set has double-zeta quality, with split valence plus d-type polarization on heavy atoms. This is quite modest by current standards. Nevertheless, such calculations reproduce experimental barrier heights within 10%. 

In diamond, Sahoo et al. (1983) investigated the hyperfine interaction using an unrestricted Hartree-Fock cluster method. The spin density of the muon was calculated as a function of its position in a potential well around the T site. Their value was within 10% of the experimental number. However, the energy profiles and spin densities calculated in this study were later shown to be cluster-size dependent (Estreicher et al., 1985). Estreicher et al., in their Hartree-Fock approach to the study of normal muonium in diamond (1986) and in Si (1987), found an enhancement of the spin density at the impurity over its vacuum value, in contradiction with experiment this overestimation was attributed to the neglect of correlation in the HF method. 

An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48], These frequencies are used to evaluate the zero-point energy Ezpe and thermal effects. 

The ordinary unrestricted Hartree-Fock (UHF) function is not written like either of these. It is not a pure spin state (doublet) as are these functions. The spin coupled VB (SCVB) function is lower in energy than the UHF in the same basis. 

Much of the development of the previous chapter pertains to the use of a single Slater determinant trial wavefunction. As presented, it relates to what has been called the unrestricted Hartree-Fock (UHF) theory in which each spin-orbital (ftj has its own orbital energy 8i and LCAO-MO coefficients Cv,i there may be different Cv,i for a spin-orbitals than for (3 spin-orbitals. Such a wavefunction suffers from the spin contamination difficulty detailed earlier. 

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