Spin contamination open-shell molecules

Another way of constructing wave functions for open-shell molecules is the restricted open shell Hartree-Fock method (ROHF). In this method, the paired electrons share the same spatial orbital thus, there is no spin contamination. The ROHF technique is more difficult to implement than UHF and may require slightly more CPU time to execute. ROHF is primarily used for cases where spin contamination is large using UHF. 

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 fundamental difference between CN and Nj is simply that one molecule is centrosymmetric while the other is not. The lowest-energy UHF wave function in both cases suffers from unphysical spin localization, and it is illusory to believe that, 2 is the easier of the two molecules to calculate. The low spin-contamination solution to the UHF equations exists simply because of the molecular symmetry, while in CN the lower symmetry of the molecule allows the equations to converge to the lowest energy solution. This is a somewhat unappreciated difficulty in calculations on open-shell molecules. If one has appropriate elements of symmetry, then unphysical solutions can be avoided by enforcing the constraints on the wave function. Even if the constraints are not enforced, problems with the reference function are easily identifiable nonzero dipole moments along directions where the exact value must vanish by symmetry, unsymmetric spin densities, and so on. However, the issue is more diabolical in lower-symmetry species where localization does not break the framework molecular symmetry. In these cases, UHF and ROHF... 

In this paper we will refer specifically to the application of DFT methods to problems involving radicals. These open-shell molecules are more difficult to treat than closed shell ones at the conventional level due to the often inaccurate HF reference configuration. Conventional ab initio calculations of radicals is often complicated by spin contamination, a problem showed to be less severe in DFT calculations (24,37). Therefore, it is useful to apply this methodology to the determination of structure and reactivity of radicals. 

For open-shell molecules, spin contamination can be a problem, as mentioned earlier. DFT and coupled-cluster theory are resistant to spin contamination and may be helpful. One may also choose spin-restricted open-shell HF (ROHF) as a starting point, instead of UHF, unless dissociation behavior is important. ROHF has no spin contamination. 

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. 

The downside to the (spin)-unrestricted Hartree-Fock (UHF) method is that the unrestricted wavefunction usually will not be an eigenfunction of the operator. Since the Hamiltonian and operators commnte, the true wavefunction must be an eigenfunction of both of these operators. The UHF wavefunction is typically contaminated with higher spin states for singlet states, the most important contaminant is the triplet state. A procedure called spin projection can be used to remove much of this contamination. However, geometry optimization is difficult to perform with spin projection. Therefore, great care is needed when an unrestricted wavefunction is utilized, as it must be when the molecule of interest is inherently open shell, like in radicals. 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

Several empirical corrections are added to the resulting energies in the CBS methods to remove the systematic errors in the calculations. 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 Na atom to account for core-valence correlation effects. The CBS-4 and CBS-q methods also contain a one-electron correction term to improve the computed ionization energies and electron affinities. The mean absolute deviation for the 125 energies in the G2 test set is 2.0 kcal mol (CBS-4), 1.7 kcal mol (CBS-q), and 1.0 kcal mol (CBS-Q), respectively. These methods have not yet been tested for the 148 molecules in the enlarged G2 test set. 

It is well-known that such functions can suffer from large amounts of spin contamination and are not suited to obtaining any surfaces except those that are the lowest of a given S3niraietry. However the UHF function, unlike an RHF function, will usually allow a molecule to separate correctly into its fragments for all decomposition channels. In contrast multi-reference-function techniques that include all configurations required to achieve correct separation would be intractable for even most three- and four-atom molecules. To limit the uncertainty introduced in using a UHF function for open shells, we monitor the multiplicity in the calculations. For some cases, such as the A A" state of HNO in the present paper, it offers a caution on the interpretation of the results, while for other cases, such as the A HCO surface, no multiplicity problems are encountered. 

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