Spin contamination

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

For systems with unpaired electrons, it is not possible to use the RHF method as is. Often, an unrestricted SCF calculation (UHF) is performed. In an unrestricted calculation, there are two complete sets of orbitals one for the alpha electrons and one for the beta electrons. These two sets of orbitals use the same set of basis functions but different molecular orbital coefficients. 

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. 

As a check for the presence of spin contamination, most ah initio programs will print out the expectation value of the total spin (A . If there is no spin contamination, this should equal. v(.v + 1), where s equals times the number of unpaired electrons. One rule of thumb, which was derived from experience with 

Unrestricted calculations often incorporate a spin annihilation step, which removes a large percentage of the spin contamination from the wave function. This helps minimize spin contamination but does not completely prevent it. The final value of (,S y is always the best check on the amount of spin contamination present. In the Gaussian program, the option iop(5/14=2) tells the program to use the annihilated wave function to produce the population analysis. 

---

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. 

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. 

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. 

Another approach is to run an unrestricted calculation and then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). This gives a correction to the energy but does not affect the wave function. Spin projection nearly always improves ah initio results, but may seriously harm the accuracy of DFT results. 

A spin projected result does not give the energy obtained by using a restricted open-shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state, rather than the spin-projected state. In cases of very-high-spin contamination, the spin projection may fail, resulting in an increase in spin contamination. 

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. 

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more significant, use restricted open-shell methods. If all else fails, use highly correlated methods. 

Nearly every technical difficulty known is routinely encountered in transition metal calculations. Calculations on open-shell compounds encounter problems due to spin contamination and experience more problems with SCF convergence. For the heavier transition metals, relativistic effects are significant. Many transition metals compounds require correlation even to obtain results that are qualitatively correct. Compounds with low-lying excited states are difficult to converge and require additional work to ensure that the desired states are being computed. Metals also present additional problems in parameterizing semi-empirical and molecular mechanics methods. 

Many transition metal systems are open-shell systems. Due to the presence of low-energy excited states, it is very common to experience problems with spin contamination of unrestricted wave functions. Quite often, spin projection and annihilation techniques are not sufficient to correct the large amount of spin contamination. Because of this, restricted open-shell calculations are more reliable than unrestricted calculations for metal system. Spin contamination is discussed in Chapter 27. 

A few of the methods available are applicable to inorganic compounds. These include the PM3/TM method. However, the program is most useful for modeling organic compounds due to a lack of technical features often needed to contend with spin contamination, convergence failure, and so forth. 

As a final note, be aware that Hartree-Fock calculations performed with small basis sets are many times more prone to finding unstable SCF solutions than are larger calculations. Sometimes this is a result of spin contamination in other cases, the neglect of electron correlation is at the root. The same molecular system may or may not lead to an instability when it is modeled with a larger basis set or a more accurate method such as Density Functional Theory. Nevertheless, wavefunctions should still be checked for stability with the SCF=Stable option. ... 

Additional empirical corrections 1 and 2-electron higher-order corrections (size-consistent), spin contamination 2-electron higher-order correction (size-consistent), spin contamination, core correlation for sodium... 

The two sets of coefficients result in two sets of Fock matrices (and their associated density matrices), and ultimately to a solution producing two sets of orbitals. These separate orbitals produce proper dissociation to separate atoms, correct delocalized orbitals for resonant systems, and other attributes characteristic of open shell systems. However, the eigenfunctions are not pure spin states, but contain some amount of spin contamination from higher states (for example, doublets are contaminated to some degree by functions corresponding to quartets and higher states). 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

Another way of viewing spin contamination is to write the UHF wave function as a linear combination of pure R(0)HF determinants, e.g. for a singlet state. 

The amount of spin contamination is given by the expectation value of die operator, (S ). The theoretical value for a pure spin state is S S + 1), i.e. 0 for a singlet (Sz = 0), 0.75 for a doublet (S = 1/2), 2.00 for a triplet (S = 1) etc. A UHF singlet wave function will contain some amounts of triplet, quintet etc. states, increasing the (S ) value from its theoretical value of zero for a pure spin state. Similarly, a UHF doublet wave function will contain some amounts of quartet, sextet etc. states. Usually the contribution from the next higher spin state from the desired is... 

By including electron correlation in the wave function the UHF method introduces more biradical character into the wave function than RHF. The spin contamination part is also purely biradical in nature, i.e. a UHF treatment in general will overestimate the biradical character. Most singlet states are well described by a closed-shell wave function near the equilibrium geometry, and in those cases it is not possible to generate a UHF solution which has a lower energy than the RHF. There are systems, however, for which this does not hold. An example is the ozone molecule, where two types of resonance structure can be drawn. Figure 4.8. 

From the above it should be clear that UHF wave functions which are spin contaminated (more than a few percent deviation of (S ) from the theoretical value of S S + 1)) have disadvantages. For closed-shell systems an RHF procedure is therefore normally preferred. For open-shell systems, however, the UHF method has been heavily used. It is possible to use an ROHF type wave function for open-shell systems, but this leads to computational procedures which are somewhat more complicated than for the UHF case when electron correlation is introduced. 

---