Unrestricted Hartree-Fock computations

Tripos a molecular mechanics force field, also the name of a company that sells computational chemistry software TST (transition state theory) method for computing rate constants UHF (unrestricted Hartree-Fock)... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Figure 5-1. H2 potential curves computed within the restricted and unrestricted Hartree-Fock (RHF and UHF) and Kohn-Sham (RKS and UKS) formalisms. 

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. 

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

The molecular and electronic structures of cyclic disulfide cation radicals of 1,2-dithietane 6 and 1,2-dithiete 7, and radical cations of 1,2-dithiolane 2 (2a-c represent stable conformations determined in terms of the symmetry restriction of Cs, Cz, and Czv), with emphasis on the nature of a two-center three-electron (Zc-ie) sulfur-sulfur bond have been examined by ab initio molecular orbital (MO) calculations <1997JMT(418)171>. Unrestricted Hartree-Fock (UHF)/ MIDI-4(d) computations showed that this bond in organodisulfide radical cation 2 is shorter in comparison to 1,2-dithiolane 2 and possesses partial Jt-bond character (structure A), as previously implied by electron spin resonance (ESR) spectroscopy <1982JA2318>, which correlates best with the form as the most favorable conformation of the cation radical 2. Contrary to the repulsive S-S interaction in the parent 1,2-dithiolane arising from the lone pairs of electrons, the hemi-7t-bond formed by one-electron oxidation should stabilize the five-membered ring of 2, or, for example, a similar cation radical of LA 3 which is involved in diverse biochemical reactions. 

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. 

Another example is provided by the EA of the state of the Na atom to generate the Na anion. Because the state is open shell, one would have to employ the unrestricted Hartree-Fock method to evaluate its orbitals and orbital energies to use in an EOM or GF EA calculation. However, one could, alternatively, compute the EA of Na by evaluating the IP of Na . The advantage would be that the Na is closed shell, so one could employ restricted Hartree-Fock methods to compute the requisite orbitals and orbital energies. 

Kolos and Wolniewicz (130) published some very accurate absolute calculations on hydrogen. The SCF—M. O. method was used by Sahni, Sahwney and Hanley (131) to calculate coe, weXe, B , a , R and D of LiH. The unrestricted Hartree-Fock method as a tool for the computation of potential h 7persurfaces was investigated by Salotto and Burnelle (132). 

In this research, the coupled-cluster single and double substitutions method with a perturbative treatment of triple excitations CCSD(T) [16-18] was adopted based on unrestricted Hartree-Fock reference wave functions. For the CCSD(T) computations, the cc-pVDZ, cc-pVTZ, and cc-pVQZ quality basis sets were used, where cc-pVnZ is an abbreviation for the correlation-consistent polarized valence basis sets of Dunning et al. [19-22]. With the coupled-cluster methods used in this research, the core orbitals are frozen. That is, the Is-like molecular orbital is frozen for O, while the Is2s2p3s3p3d-like molecular orbitals are frozen for Br. 

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. 

Finally, it should be pointed out that the perturbation-theoretic methods [25] (MP2, etc.), used with such success for molecules near their equilibrium geometries, are much less satisfactory when used to compute a PES. Obviously, where several reference configurations are required these single-reference treatments cannot be expected to perform well, and it does not seem possible to overcome such problems by the use of unrestricted Hartree-Fock (UHF) methods to define a single reference CSF the UHF PES itself will often display discontinuities from spontaneous symmetry breaking, and this inevitably compromises the subsequent perturbation theory treatment. Recent efforts [26] to devise projected UHF-based schemes may overcome these problems, but this is simply another approach to generating a multireference wave function. 

More physically correct than ROHF, and much easier to implement computationally, is another scheme called unrestricted Hartree-Fock (UHF), wherein the manifold of occupied MOs is not subdivided into closed and open shells. Instead, a standard HF program is used to carry out parallel sets of HF calculations on two different sets of MOs, one containing only the a and the other only the p electrons. The resulting pairs of UHF MOs for a and p electrons, which are identical in an ROHF calculation, have similar nodal properties, but they differ from each other in spatial detail. The restriction, Inherent in the ROHF scheme, that paired electrons of opposite spin occupy identical MOs is thus removed in UHF calculations. ... 

Transition metal elements are a real test of the efficacy of quantum chemical theories. Primarily, this is because many (if not most) of these systems have unpaired electrons, i.e., are open shell. Open shell systems therefore require additional approximations beyond standard Hartree-Fock theory to make the calculations tractable computationally. These include the UHF (unrestricted Hartree-Fock) and configuration interaction (Cl) approaches. Both approaches are computationally costly and have complicating considerations that make their general application more difficult and less straightforward. Also, it is often difficult to determine which spin state is the correct or natural one, and properties can vary substantially with differences in spin state. 

A. W. Salotto and L. Burnelle, Investigations on the unrestricted Hartree-Fock method as a tool for computing potential energy surfaces, J. Chem. Phys. 52 2936 (1970). 

The ab initio method used to obtain the potential surface employs many-body perturbation theory (MBPT) relative to an unrestricted Hartree-Fock (UHF) reference function. Correlation is treated to fourth order in MBPT and includes contributions from all single, double, and quadruple excitations. A Gaussian basis of double zeta plus polarization functions (DZP) is used. This model is computationally efficient and has been shown to provide accurate force fields and geometries and reliable thermochemical data for a wide variety of molecular systems.It has also been used to calculate reaction paths for some triatomic systems. 

Numerous steps have been undertaken in order to overcome these shortcommings of the standard CCSD method. The simplest way to achieve a proper dissociation limit (size-consistency) is to employ the unrestricted Hartree-Fock (UHF) reference. This often works rather well, except that UHF solution(s) exist(s) only in a limited range of internuclear separations and, at the onset of the RHF triplet instability the computed energies display a nonanalytic behavior. Of course, in more general situations, the UHF solution may dissociate to a wrong limit [cf., e.g. Refs. 4J0)]. not to mention the multiplicity and often haphazard behavior of various broken-spin-symmetry solutions, spin contamination, etc 4), Thus, this approach is usually reserved for computation of dissociation energies rather than for the generation of accurate PESs. 

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