Restricted-unrestricted approach

The restricted-unrestricted approach (RU) in [155] has been applied to calculate the isotropic hyperfine coupling constants of a sequence of organic radicals and transition metal compounds. In the case of organic compounds, both spin-restricted and unrestricted approaches could accurately describe the isotropic hyperfine coupling constants which matched the accuracy achieved by coupled cluster methods. The situation is different for transition metal compounds for which the overall quality of the RU results is slightly better than the corresponding unrestricted results (see Table 13), independently on the exchange-correlation functional used in calculations. 

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. 

The ESR hyperfine coupling is determined by triplet perturbations. Thus, in principle one should use an unrestricted wave function to describe the reference state. However, it is also possible to use a spin-restricted wave function (Fernandez et al. 1992) and take into account the triplet nature of the perturbation in the definition of the response. Within such a (e.g., SCF or MCSCF) restricted-unrestricted approach, first-order properties are given as the sum of the usual expectation value term and a response correction that takes into account the change of the wave function induced by the perturbation (of the type (0 H° 0)). This restricted-unrestricted approach has also been extended to restricted Kohn-Sham density functional theory (Rinkevicius et al. 2004). 

Rinkevicius, Z., Telyatnyk, L., Vahtras, O., Agren, H. (2004). Density functional theory for hyperfine coupling constants with the restricted-unrestricted approach. Journal of Chemical Physics, 121, 7614. 

The philosophy of the restricted-unrestricted (RU) approach is a physically motivated compromise between the restricted and unrestricted methods to optimize the wave function with a spin-restricted approach and to account for perturbations with an unrestricted approach. That is, a ground state constructed from a and (3 spin-orbitals with common orbital parts is used with satisfies the variational condition... 

Most implementations of TDDFT to an open-shell system use an spin-unrestricted approach, because orbital-energy differences concerned with partially occupied orbitals are generally too small in a spin-restricted approach, and the orbital-energy difference in DFT is the leading term in the electron-excitation energy. 

Even though the majority of calculations of hyperfine coupling constants are performed using unrestricted approaches, the importance of spin contamination and its effect on the hyperfine coupling constants remains somewhat unclear, in particular in DFT calculations. Whereas restricted and unrestricted approaches give more or less identical results for simple radicals, the situation is less clear in the case of transition metal compounds, and a few examples of the differences between restricted and unrestricted approaches are collected in O Table 11-15, based on the results reported by Rinkevicius et al. (2004). We note that the differences between restricted and unrestricted approaches in general are small for the same functional, but that these differences become significant when spin contamination can be a problem, for instance for TiFs and MnOs. 

A detailed description for the CCSD(T)/CBS approach is found elsewhere [26, 59-61]. Briefly, the single-point electronic energies are calculated by using the restricted/ unrestricted coupled-cluster R/UCCSD(T) formalism [62-64] and extrapolated to the complete basis set limit (CBS) based on the correlation-consistent aug-cc-pVnZ n = D, T, and Q) basis sets [65, 66]. The CCSD(T) energies are then extrapolated to the CBS limit energies using expression (1) [67] ... 

Basis. The rotational mobility of a small ligand is relatively unrestricted in solution (anisotropy approaches 0). The mobility is restricted when the ligand binds to a large immobilized molecule such as the receptor. In a T-format fluorometer, the parallel and perpendicular components of the emission can be examined simultaneously. While precautions must be exercised in working with turbid suspensions, it is nonetheless practical to make continuous measurements of binding and dissociation. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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. 

Calculated geometries for a small number of diatomic and small polyatomic free radicals are compared with experimental structures in Table 5-18. These have been drawn from a somewhat larger collection provided in Appendix A5 (Tables A5-50 to A5-57). Except for triplet oxygen, all radicals possess a single unpaired electron (they are doublets). The usual set of theoretical models has been examined. All calculations involve use of the unrestricted open-shell SCF approach, where electrons of different spin occupy different orbitals, as opposed to the restricted open-shell SCF approach, where paired electrons are confined to the same orbital (see Chapter 2 for more detailed discussion). 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

In this section, we briefly discuss some of the electronic structure methods which have been used in the calculations of the PE functions which are discussed in the following sections. There are variety of ab initio electronic structure methods which can be used for the calculation of the PE surface of the electronic ground state. Most widely used are Hartree-Fock (HF) based methods. In this approach, the electronic wavefunction of a closed-shell system is described by a determinant composed of restricted one-electron spin orbitals. The unrestricted HF (UHF) method can handle also open-shell electronic systems. The limitation of HF based methods is that they do not account for electron correlation effects. For the electronic ground state of closed-shell systems, electron correlation effects can be accounted for relatively easily by second-order Mpller-Plesset perturbation theory (MP2). In modern implementations of MP2, linear scaling with the size of the system has been achieved. It is thus possible to treat quite large molecules and clusters at this level of theory. 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

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... 

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