Unrestricted Hartree-Fock UHF

Quantum mechanics calculations use either of two forms of the wave function Restricted Hartree-Fock (RHF) or Unrestricted Hartree-Fock (UHF). Use the RHF wave function for singlet electronic states, such as the ground states of stable organic molecules. 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

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. 

In the former, electrons are assigned to orbitals in pairs, the total spin is zero, so the multiplicity is 1. In this case, the restricted Hartree-Fock method (RHF) can be applied. For radicals with doublet or triplet states, the unrestricted Hartree-Fock (UHF) has to be applied. In this method, a and, 3 electrons (spin up and spin down) are assigned to different spatial orbitals, so there are two distinct sets I and FJf... 

Minimal END has also been applied to a model system for intramolecular electron transfer. The small triatomic system LiHLi is bent C2V structure. But the linear structure presents an unrestricted Hartree-Fock (UHF) broken symmetry solution with the two charge localized structures... 

Early determinations of RSE values employed unrestricted Hartree-Fock (UHF) theory in combination with 3-21G [9] or 4-31G [10] basis sets to evaluate the RSE according to Eq. 1. The appropriate consideration of correlation effects, the avoidance of spin contamination, and the treatment of thermochemical corrections have in detail been studied in the following, in particular by Bauschlicher [11], Coote [12-14], Morokuma [15-18], and Radom [19-25]. Highly accurate RSE and BDE results can be obtained with high level compound methods such as the G2 [26-30] and G3 [31-34] schemes (and variants thereof [11,15-18]), as well as extrapolation methods such as the CBS schemes [35,36], Wl, or W2 [37-39]. Generally, the accurate... 

A first quantitative assessment of the effect by ab initio calculations was presented by Schleyer s group (Crans et al. 1980). They used unrestricted Hartree-Fock (UHF) calculations at the fully geometry optimized, split valence, 4-3IG basis-set level to investigate the interaction of a ic-donor and a Jt-acceptor both bonded to a CH radical centre. The existence of a... 

Such kind of calculations with a precise self-consistent account of crystal surrounding were performed by, at least, four scientific groups Baetzold [16], Das with coworkers [22,23], Winter et al. [28] and Ladik with coworkers [29]. Winter et al. [28] performed cluster calculation by restricted Hartree-Fock (RHF) method, so they did not take into account the electron correlation. The others groups used the unrestricted Hartree-Fock (UHF) method for cluster calculations which allows to some extent the electron correlation. The strong covalent C-0 bonding in planes and chains was revealed (in accordance with results obtained in Refs. [20,25,26]). For covalent systems,... 

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. 

We shall now follow the unrestricted Hartree-Fock (UHF) formalism to obtain a restricted high-spin open-shell functions as proposed in [34], [35]. In order to eliminate spin contamination in the UHF function f i, the following spin purity constraint is imposed on the spatial orbitals ... 

The version of HF theory we have been studying is called unrestricted Hartree-Fock (UHF) theory. It is appropriate to all molecules, regardless of the number of electrons and the distribution of electron spins (which specify the electronic state of the molecule). The spin must be taken into account when the exchange integrals are being evaluated since if the two spin orbitals involved in this integral did not have the same spin function, a or ft, the integral value is zero by virtue of the orthonormality of the electron spin functions... 

Finally we describe several methods that combine molecule-dependent empirical parameters with a moderate level ab initio molecular orbital method. The BAC-MP4 method of Melius and coworkers115-118 combines a computationally inexpensive molecular orbital method with a bond additivity correction. This procedure uses a set of accurate experimental data to obtain a correction for bonds of different types that is then used to adjust calculated thermochemical data such as enthalpies of formation. Quite accurate results can be obtained if suitable reference molecules are available and if the errors in the calculation are systematic. The computational methodology is based on an MP4/6-31G(d,p)//HF/6-/31G(d) calculation. A pairwise additive empirical bond correction is derived for different bonds from fitting to experimental enthalpies of formation or in some cases to high quality ab initio computations. In addition, for open-shell molecules an additional correction is needed to compensate for spin contamination of the wavefunction from higher spin states in the unrestricted Hartree-Fock (UHF) method. 

Calculations of the electronic structure of the phosphorus molecule were carried out by ab initio MO LCAO Hartree-Fock-Roothaan SCF method in the restricted (RHF), restricted-open (ROHF) or unrestricted Hartree-Fock (UHF)... 

Molecular parameters of seven radical 1,3,2-dithiazolyls 1, 3, 4, 8-11 were systematically investigated by ab initio and DFT methods <2001PCA7615>. Statistical analysis of the differences between the calculated and experimental values for radical 8 indicates that both methods overestimate most distances nevertheless, the unrestricted Hartree-Fock (UHF) performs better than the UB3LY1 method for compatible basis sets. The mean deviation of distances decreases from 1.0pm for 3-21G(d) to 0.2pm for 6-31 G(2df) basis sets in the UHF method. The mean difference in the calculated and experimental interatomic angles shows low basis set sensitivity. 

The spin orbitals can be separated into a spatial part, if/ (orbital or molecular orbital) and a spin eigenfunction, a (or for the opposite spin). In restricted Hartee-Fock theory (RHF), the spatial part is independent on the spin state, in contrast to unrestricted Hartree-Fock (UHF) where it is spin dependent. Consequently, the RHF spin orbitals can be written as %i= whereas in UHF the corresponding relation is Xi= even number of electrons (closed shell system), but can easily be extended to treat also UHF1. 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

Fig. 3 Unrestricted Hartree-Fock (UHF) optimized structures (in A) of Qoion of D5d> D3d) and D2h symmetry. The bond distances are shown only for the fragment of hatched carbon atoms. Italics are center-carbon distances. Atom numbering denotes symmetry-unique carbons (reproduced from [20])...

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