Spin-unrestricted Hartree-Fock

Choose LHH(spin Unrestricted Hartree-Fock) or RHF (spin Restricted Ilartree-Fock) calculations according to your molecular system. HyperChem supports UHF for both open-sh el I and closed-shell calcii lation s an d RHF for cUised-shell calculation s on ly, Th e closed-shell LHFcalculation may be useful for studyin g dissociation of m olectilar system s. ROHF(spin Restricted Open-shell Hartree-Fock) is not supported in the current version of HyperChem (for ah initio calculations). 

An initial equilibrium structure is obtained at the Hartree-Fock (HF) level with the 6-31G(d) basis [47]. Spin-restricted (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. The HF/6-31G(d) equilibrium structure is used to calculate harmonic frequencies, which are then scaled by a factor of 0.8929 to take account of known deficiencies at this level [48], These frequencies are used to evaluate the zero-point energy Ezpe and thermal effects. 

The simplest approximate wavefunction for an open-shell molecule is the spin-unrestricted Hartree-Fock function... 

An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68 69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 

When the second of the equivalence restrictions is removed, a single determinant wavefunction of lower energy is usually obtained. In fact, it is possible for a wave-function obtained in this way, a so-called unrestricted Hartree-Fock (UHF) wavefunction191 (perhaps more properly called a spin-unrestricted Hartree-Fock wavefunction) to go beyond the Hartree-Fock approximation and thus include some of the correlation energy. Lowdin192 describes this as a method for introducing a Coulomb hole to supplement the Fermi hole already accounted for in the RHF wavefunction. 

Molecular orbitals from linear combination of atom orbitals Spin-restricted Hartree-Fock. Wave function constructed from antisymmetrized product of doubly occupied spin orbitals (UHF, spin-unrestricted Hartree-Fock calculations are used for excited states and radicals)... 

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. 

There are a number of slightly more approximate methods for determining the electron affinity (EA) based on the restricted Hartree-Fock (RHF) and spin-unrestricted Hartree-Fock (UHF) methods. For closed shell anions, molecules which dissociate to... 

There have been few first principles calculations for ions and no form factor calculations. Brown and Burton s spin-unrestricted Hartree-Fock SCFMO calculations (93) for K2NaCrFegave /< = -2.19%, f =2.SS% and/<,-/ = -4.74%, in quite good agreement with the experimental estimates (Table 4), but a somewhat small value of fa — fn —6.5%) was estimated for MnFe"" (cf. Table 5). On the other hand the Xa multiple scattering calculation of Larsson and Connolly (94) indicates that the contribution to /<, — f from Cg spin polarization is dominant... 

The choice of as the zeroth-order Hamiltonian requires the use of either a spin-restricted (closed-shell) Hartree-Fock (RHF) or spin-unrestricted Hartree-Fock (UHF) determinant as the zeroth-order (reference) wavefunction. Since spin-restricted open-shell Hartree-Fock (ROHF) reference functions are not eigenfunctions of the spin-orbital P, other partitionings are required (Refs. 127-134). 

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). 

The SCF orbitals of open-shell systems may be obtained by the (spin) unrestricted Hartree-Fock (UHF) open-shell approach of Pople and Nes-bet (1954). In this method, the orbitals associated with an a spin function are different from those associated with a j8 spin function. So, for a doublet ground state In + 1 electrons), the wave function is written as ... 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

In a similar fashion, correction factors for the nitrogen-oxygen and oxygen-oxygen bond types are developed from a comparison of the theoretical results with various experimental values of N-O-H containing molecular species. In this case the bond additivity corrections depend both on the bond distance as well as the bond type. An additional correction factor for spin contamination in the spin-unrestricted Hartree-Fock wavefunction is included. 

The wavefunction W(X,b) will be given as a single Slater determinant made from spin-unrestricted Hartree-Fock (UHF) molecular orbital functions which are eigenfunctions of the perturbed Fock matrices... 

The perturbation series can be truncated to various orders and one indicates the accuracy of MP methods applied within the Restricted Hartee-Fock (RHF) scheme by referring to the highest-orderterm allowed in the energy expansion. Thus a truncation to second-order corresponds to an MP2 approach, to third-order to an MP3 approach and so forth [27]. MP theory may also be used in the spin-Unrestricted Hartree-Fock (UHF) model. In this case, second- and third-order approximations of MP theory are indicated as UMP2andUMP3. 

Technically, it is possible to define methods in which other restrictions than those on spin are lifted, but the most common unrestricted method is the spin-unrestricted method, and we follow the common usage of UHF for the spin-unrestricted Hartree-Fock method. 

There is a possibility for more than one solution of the Hartree-Fock equations if different electronic states come close on a potential energy surface. Within the spin-restricted Hartree-Fock (RHF) method, singlet and triplet instabilities are distinguished, The former involves the existence of another solution with lower energy and an electron distribution of lower symmetry, normally indicating that the initially assumed geometry is incorrect. Triplet instability involves rejection of the condition of double occupancy of molecular orbitals and a spin-unrestricted Hartree-Fock (UHF) method treatment is mandatory. The triplet instability is a necessary, but insufficient, condition for a biradical character of a ground state. 

In this section, we consider spin-unrestricted MPPT, taking as our unperturbed state the spin-unrestricted Hartree-Fock wave function and as our zero-order Hamiltonian the Fock operator. A spin-restricted treatment suitable for closed-shell states is given in Section 14.4, following the discussion of CCPT in Section 14.3. 

P. S. Bagus, B. Liu, and H. F. Schaefer III, Study of the contact-term contribution to the hyperfine structure obtained from spin-unrestricted Hartree-Fock wavefunctions, Phys. Rev. A 2 555 (1970),... 

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