Spin-unrestricted formalism

In this chapter, we later consider spin-polarized systems. One avenue of approach is to apply the spin unrestricted formalism, where SOs have different spatial orbitals for different spins. However, this procedure can introduce important spin contamination effects through the last term of Eq. (27) since the overlap matrix (5. These effects can be avoided by the use of spin-restricted theory. In this case only a single set of orbitals is used for a and / spins. 

In the Xa scattered wave approximation, the exchange potential for spin-up electrons may be different from that for spin-down electrons. In particular, when unpaired electrons are present, the exchange potentials, and hence the spin-up and spin-down orbitals and their energy levels, are different. Thus, MO calculations are performed using a spin-unrestricted formalism so that separate orbital energy levels are given for spin-up (a) and spin-down (p) electrons. 

We will consider the general case of a gradient-corrected functional, using a spin-unrestricted formalism. The XC energy is formally given by... 

Figure 4.1 Important features of ground- and excited-state PESs for ethylene photodynamics and demonstration of the inadequacy of TDDFT and CIS methods for this problem, (a) Sq and PESs for ethylene in the pyramidalization and torsion coordinates (defined in the inset) that dominate the photodynamics. This surface was calculated using multireference perturbation theory — CAS(2/2) PT2. The global minimum on Si occurs at twisted and pyramidalized geometries. (b-d) A quantitative comparison of the Si PES obtained with CAS(2/2) PT2, TDDFT/B3LYP, and CIS, respectively. All calculations use the 6-3IG basis set. The TDDFT and CIS calculations are performed in a spin-unrestricted formalism. Contour values are given in eV, and in all cases the energies are referenced to the Sq equilibrium geometry at the corresponding level of theory. Only the multireference calculation captures the Si minimum correctly.

In general, density functional methods in conjunction with the unrestricted formalism could satisfactorily reproduce the characteristics of the spin distribution and the... 

The restricted-unrestricted approach not only improves results from the unrestricted approach, but also allows to rigorously describe the effect of spin polarization for tlie hyperfine coupling constants as well as to provide ways to analyze the behavior of spin polarization (response term in RU approach, see fheory part) in problematic cases. The RU approach therefore provides a higher degree of control over the calculation and its analysis compared to the unrestricted formalism. It can consequently be recommended for investigations of hyperfine coupling constants in various molecular systems. 

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). 

Let us say a little more about these methods. The HF methods are divided into spin-restricted HF (RHF) and spin-unrestricted HF (UHF) methods. Closed-shell systems are almost always calculated using RHF. In this procedure, one set of molecular orbitals is calculated and pairs of electrons are entered to the lowest-energy orbitals. If the molecule has an odd number of electrons, one orbital will be singly occupied and the species is a radical (spin = 0.5, expectation value of the spin-squared operator =0.75). Inmost cases, however, radicals are calculated using the UHF formalism. UHF calculations determine two sets of molecular orbitals, one for each type of spin named alpha and beta. These MO sets are similar but not identical. A radical, for instance, has one more a than P electron. The UHF procedure is more flexible than RHF because the paired a and P orbitals, which correspond to doubly occupied MOs in the RHF formalism, need not be identical. So UHF allows for spin polarization but, on the other hand, spin-contamination occurs (i.e., states of higher spin are mixed into the wave function). 

Simple open shell cases may also be treated via this kind of perturbation theory. The high spin case with one electron outside a closed shell is of course easy when an unrestricted formalism is used. Dyall also worked out equations for the restricted HE formalism and the more complicated case of two electrons in two Kramers pairs outside a closed shell [32]. Also in this method the crucial step remains the efficient formation of two-electron integrals in the molecular spinor basis. 

Modern implementations of the MP2-F12 method combine the CABS approximation ]20] with robust density fitting techniques [21, 22] and local approaches ]23]. The coefficients are usually constrained at the values predetermined from the cusp conditions, as one half for singlet pairs and one quarter for triplet pairs in the spin-adapted formalism [24, 25]. The MP2-F12 methods have been extended to treat open-shell systems with unrestricted [26, 27, 28], restricted [29, 30] and multireference [29] formalisms. 

We shall now follow the unrestricted Hartree-Fock (UHF) formalism to obtain a restricted high-spin open-shell functions as proposed in [34], [35]. In order to eliminate spin contamination in the UHF function f i, the following spin purity constraint is imposed on the spatial orbitals ... 

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