Unrestricted Hartree-Fock theory

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%. 

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. 

An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68 69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 

Payne, P.W. (1979). Density functionals in unrestricted Hartree-Fock theory,... 

The equations require to be modified for open-shell systems, in which some orbitals are doubly occupied and some singly (this is called spin-restricted Hartree-Fock theory). A further extension to the theory involves electrons of a and /3 spin being assigned to different molecular orbitals, type equations are described as unrestricted Hartree-Fock [31]. 

From all this one must conclude that the determinantal and second-quantized formulations should be regarded as a poor man s group theory which, while convenient, hides the basic freeon dynamics. These fermion methods have the additional disadvantage that their antisymmetric fermion functions are not normally pure spin (freeon) states so that spin-projection may be required. A method for avoiding (approximately) spin projection is the employment of the variation principle to approximate the ground state e. g., unrestricted Hartree-Fock theory. Finally the use of the fermion formulations has lead to the spin paradigm as a replacement for the more fundamental freeon dynamics. 

Hartree-Fock (theory) (FIF) (restricted HF, RFIF for closed shells unrestricted HF, UHF for open shells)... 

Kama, S.P. (1996) Spin-unrestricted time-dependent Hartree-Fock theory, J. Chem. Phys., 104, 6590-6605. 

Potential energy curves for singlet and triplet A j, B, and B j states of COF j have been computed using ab initio projected-unrestricted Hartree-Fock theory with a contracted Gaussian type orbital basis set [273]. However, symmetry was strictly maintained for these excited states, so the poor agreement between the predicted and experimental band onsets (which was readily acknowledged by Brewer et al. [273]) comes as little surprise. 

Unbound state, 209-210 Unrestricted Hartree-Fock theory (UFIF), 23, 222-234 Urea, 304... 

Electronic structure calculations have been performed with the system of programs Gaussian94 (G94) [19]. Restricted Hartree-Fock theory (RHF) is used for closed shell systems, and Unrestricted Hartree-Fock theory (UHF) for open shell systems (radicals). The correlation energy corrections are introduced with Moller-Plesset perturbation theory up to second order, and results from spin projected calculations are used (PMP2 and PUMP2). 

The broken-symmetry approach was developed within the framework of unrestricted Hartree-Fock theory or spin-polarized density functional theory, i.e., Xa theory. " It is based on the assumption that the energy difference between the singlet and triplet states resulting from the interaction between two 5 = 1/2 spin centers is given by ... 

In this section, then, we first introduce a set of unrestricted spin orbitab to derive the spatial eigenvalue equations of unrestricted Hartree-Fock theory. We then introduce a basis set and generate the unrestricted Pople-Nesbet matrix equations, which are analogous to the restricted Roothaan equations. We then perform some sample calculations to illustrate solutions to the unrestricted equations. Finally, we discuss the dissociation problem and its unrestricted solution. 

To complete our discussion of unrestricted Hartree-Fock theory, we will use our minimal basis H2 model to investigate the description of bond dissociation by unrestricted wave functions. 

Fig. 4. Electron configuration diagrams highlighting the differences between restricted Hartree-Fock theory (RHF), restricted open-shell Hartree—Fock theory (ROHF), and unrestricted Hartree-Fock theory (UHF).

In another pap>er, Stashenko et al. studied the intramolecular Friedel-Crafts alkylation of urf/ju-aUyl-lV-benzylanihne, located all the stationary pwints on the PES with unrestricted Hartree-Fock theory, and computed energies using restricted MP2 theory [72]. However, this specific reaction was catalyzed with H (sulfuric acid) and not with a metal hahde. 

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