Hartree-Fock scheme, unrestricted

Unrestricted Hartree-Fock Scheme. Exchange Polarization. 313... 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

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... 

This means that one has to be extremely careful in making physical interpretations of the results of the unrestricted Hartree-Fock scheme, even if one has selected the pure spin component desired. In many cases, it is probably safer to carry out an additional variation of the orbitals for the specific spin component under consideration, i.e., to go over to the extended Hartree-Fock scheme. In the unrestricted scheme, one has obtained mathematical simplicity at the price of some physical confusion—in the extended scheme, the physical simplicity is restored, but the corresponding Hartree-Fock equations are now more complicated to solve. We probably have to accept these mathematical complications, since it is ultimately the physics of the system we are interested in. 

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 ... 

This diagram is written in the sense of the "restricted Hartree-Fock scheme 18>. In the "unrestricted Hartree-Fock 19> sense each orbital of radical B is singly" occupied and LU is higher and HO is lower than the restricted Hartree-Fock SO, respectively (cf. Chap. 1)... 

One way to overcome this difficulty is to use different orbitals for different spins (DODS model). This technique introduced, into the Hartree-Fock scheme, gave rise to the unrestricted Hartree-Fock model (UHF), the wave-function being written as an open shell single Slater determinant[2) ... 

Scheme 83 and crystallographically characterized. With 2.942(2) A the Ti-Ti distance in the molecule is fairly short, and ab initio unrestricted Hartree-Fock calculations suggested the presence of a Ti-Ti bonding interaction. 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

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 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). 

In the later part of the 1950 s, It was evident that it was necessary to distlngush the new approach dealing with different orbitals for a-spln and P-spln from the previous approach starting out from symmetry restrictions the latter was called the Restricted Hartree-Fock (RHF) scheme, whereas the new approach was called the Unrestricted Hartree-Fock (UHF) scheme. For some time there was a certain amount of competition between the two schemes. In the late 1950 s, it was further shown that the RHF-scheme for closed-shell systems was completely se[f-consistent not only for atoms but also for molecules and solids [16.17] and that, if one started by imposing a symmetry requirement on the original Slater determinant, this assumption would be self-consistent, i.e. the final determinant would have the same symmetry property. Since symmetry properties are of such fundamental importance in quantum theory, one would hence anticipate that the RHF-scheme would... 

Another popular approach to the correlation problem is the use of perturbation theory. Fq can be taken as an unperturbed wave function associated with a particular partitioning of the Hamiltonian perturbed energies and wave functions can then be obtained formally by repeatedly applying the perturbation operator to Probably the commonest partitioning is the M ller-Plesset scheme, which is used where Fq is the closed-shell or (unrestricted) open-shell Hartree-Fock determinant. Clearly, the perturbation energies have no upper bound properties but, like the CC results, they are size-consistent. 

For the development below, we will assume a closed-shell situation, with all electrons paired in molecular orbitals. In such a case OfSj = 1. In very many cases, however, an unrestricted Hartree-Fock (UHF) scheme is utilized for ground state properties. This theory is reasonably accurate for those cases in which each open-shell orbital has an electron of the same spin, i.e., the case that an open-shell has maximum multiplicity. In the UHF scheme Eq. [4] does not hold. Two Fock equations result, one for a and one for 3 spin molecular orbitals. In cases in which excited state properties are required, Eq. [4] is forced to hold in order to yield spectroscopic states, of known multiplicity. OfSJ can then become quite complex, and affects the form of the Fock operators that follow. ... 

A combined approach starting from an unrestricted Hartree-Fock (UHF) scheme in L5-coupling in combination with radial four-component calculations was performed by Desclaux and co-workers [78]. These authors investigated the hyperfine structure of Ga and Br allowing diflFerent radial parts for all the spin orbitals. These functions were then jj recoupled corresponding to the electronic configurations... 

The methods mentioned above for the calculation of the first derivative of the energy have been extended to provide analytic evaluations of the second derivative. The necessary theory was first considered some time ago, but practical implementations of these schemes can be regarded as starting with those of Pople et al. in 1979. This important paper demonstrated how the complete second derivative of the molecular energy with respect to geometric parameters could be obtained in a single calculation. This paper used the unrestricted Hartree-Fock (UHF) method, but the methodology has since... 

The perturbation series can be truncated to various orders and one indicates the accuracy of MP methods applied within the Restricted Hartee-Fock (RHF) scheme by referring to the highest-orderterm allowed in the energy expansion. Thus a truncation to second-order corresponds to an MP2 approach, to third-order to an MP3 approach and so forth [27]. MP theory may also be used in the spin-Unrestricted Hartree-Fock (UHF) model. In this case, second- and third-order approximations of MP theory are indicated as UMP2andUMP3. 

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. ... 

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