Spin contamination error

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

Because Jab calculated by Eq. (18.8) is a value that the spin contamination error is approximately eliminated, it should be equal to Jab value calculated by the approximately spin-projected LS energy (i p) as... 

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

The advantage of unrestricted calculations is that they can be performed very efficiently. The alpha and beta orbitals should be slightly different, an effect called spin polarization. The disadvantage is that the wave function is no longer an eigenfunction of the total spin <(5 >. Thus, some error may be introduced into the calculation. This error is called spin contamination and it can be considered as having too much spin polarization. 

When it has been shown that the errors introduced by spin contamination are unacceptable, restricted open-shell calculations are often the best way to obtain a reliable wave function. 

Because of cancellation of errors in reactions such as (4.5), reasonable results are often obtained, even at quite simple levels of theory. However, it has been found [21, 34] that larger errors may occur with unrestricted methods if there is a significant difference between the degrees of spin contamination for the two radicals in the reaction. 

It is often assumed that there will be substantial cancellation of errors associated with the calculation of stabilization energies via reactions such as (6.2). However, this is not always the case. In particular, it has recently been shown [21, 34] that stabilization energies calculated for the cyanomethyl and cyanovinyl radicals show large variation with level of theory. For these situations, methods such as UMP2 perform very poorly because errors associated with spin contamination in the reactant and product radicals are very different and do not cancel. 

The second and third terms on the right-hand side provide a measure of the convergence from lower to higher levels of theory, while the last term accounts for errors associated with spin contamination of the groimd state or UHF instability. 

Note that the error estimates obtained from Eq. 19 are, in the majority of cases, less than 2kcalmol This indicates that the computational method is converging well from the lowest level of perturbation theory through to the highest. The only exceptions to this are the divalent singlet radicals (compounds of the form XSnY), in which there is a nonzero correction for UHF instability. In these cases, the ad hoc error estimation method indicates a higher level of uncertainty because the presence of UHF instability is an indication that the MP4(SDTQ) level of theory is not fully adequate to describe the electronic ground state of the molecule (the same is true of a nonzero spin contamination correction). 

In principle, transition-metal clusters may best be treated with multi-determinant wave-function methods (139), but in practice due to their size often only DFT calculations are feasible and method-inherent errors have to be taken care of, e.g., the problem of spin contamination and the approximate nature of the exchange-correlation functionals available. 

G, respectively. ROHF theory is more accurate than UHF theory in this case, presumably owing to moderate spin contamination in the latter. Projecting out the spin contamination at the PUHF level reduces the error by almost one half, while going to second-order perturbation theory (which introduces electron correlation and also probably reduces the spin contamination compared to UHF) provides an improvement of about the same order. 

UHF Methods. A major drawback of closed-shell SCF orbitals is that whilst electrons of the same spin are kept apart by the Pauli principle, those of opposite spin are not accounted for properly. The repulsion between paired electrons in spin orbitals with the same spatial function is underestimated and this leads to the correlation error which multi-determinant methods seek to rectify. Some improvement could be obtained by using a wavefunction where electrons of different spins are placed in orbitals with different spatial parts. This is the basis of the UHF method,40 where two sets of singly occupied orbitals are constructed instead of the doubly occupied set. The drawback is of course that the UHF wavefunction is not a spin eigenfunction, and so does not represent a true spectroscopic state. There are two ways around the problem one can apply spin projection operators either before minimization or after. Both have their disadvantages, and the most common procedure is to apply a single spin annihilator after minimization,41 arguing that the most serious spin contaminant is the one of next higher multiplicity to the one of interest. 

Several empirical corrections are added to the resulting energies in the CBS methods to remove the systematic errors in the calculations (see Table 10). The CBS-Q method contains a two-electron correction term similar in spirit to the higher level correction used in G2 theory, a spin correction term to account for errors resulting from spin contamination in UHF wavefunctions for open-shell systems, and a correction to the sodium atom to account for core-valence correlation effects. The CBS-4 and CBS-q methods also contain a one-electron... 

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. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

For species involving open-shell ground states (for example, O2, NO2, and OH), one can base an MP calculation on the unrestricted SCF wave function (Section 15.3), giving calculations designated UMP2, UMP3, and so on. Unrestricted SCF wave functions are not eigenfunctions of S, and this spin contamination can sometimes produce serious errors in UMP-calculated quantities [K. Wolinski and P. Pulay, J. Chem. Phys., 90,3647 (1989)]. Alternatively, several versions of open-shell MP perturbation theory that are based on the ROHF wave function have been developed. It is not clear which of these ROHF MP methods is best [T. D. Crawford, H. F. Schaefer, and T. J. Lee, /. Chem. Phys.,m, 1060 (19%)]. 

Spin contamination is the main source of error in the evaluation of /, but deviations from the Ising model may also account for part of it. However, despite the large disagreement between the calculated and experimental values of J, the prediction of the relative stability of different magnetic phases is correct. Moreover, investigation of the same properties with the other systems previously mentioned always reproduced phase stabilities correctly and J values were calculated approximately within the same error bar. 

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