Spin-unrestricted Hartree-Fock UHF Method

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 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. 

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. 

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

The term Restricted Hartree-Fock (RHF) is applied to those cases in which all the possible spin pairing in a system is allowed for by having electrons of both and p spin occupy the same space orbital. If this restriction is relaxed in writing out the determinantal wavefunction, the method of calculation is referred to as the Unrestricted Hartree-Fock (UHF) method. Unless Otherwise stipulated, the calculations referred to in this chapter are of the RHF variety. 

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. 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

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 ... 

Symmetry and stability analysis. The semi-empirical unrestricted Hartree-Fock (UHF) method was used for symmetry and stability analysis of chemical reactions at early stage of our theoretical studies.1,2 The BS MOs for CT diradicals are also expanded in terms of composite donor and acceptor MOs to obtain the Mulliken CT theoretical explanations of their electronic structures. Instability in chemical bonds followed by the BS ab initio calculations is one of the useful approaches for elucidating electronic structures of active reaction intermediates and transition structures.2 The concept is also useful to characterize chemical reaction mechanisms in combination with the Woodward-Hoffman (WH) orbital symmetry criterion,3 as illustrated in Figure 1. According to the Woodward-Hoffmann rule,3 there are two types of organic reactions orbital-symmetry allowed and forbidden. On the other hand, the orbital instability condition is the other criterion for distinguishing between nonradical and diradical cases.2 The combination of the two criteria provides four different cases (i) allowed nonradical (AN), (ii) allowed radical (AR), (iii) forbidden nonradical (FN), and (iv) forbidden radical (FR). The charge and spin density populations obtained by the ab initio BS MO calculations are responsible for the above classifications as shown in Fig. 1. 

G bases, except for the anions for which the MP2 energies were calculated in the former basis set only. Excitations from the core electrons were not included in the MP2 treatment. The restricted Hartree-Fock (RHF) method was used for the closed-shell molecules (parents and anions) and the unrestricted Hartree-Fock (UHF) method was applied to the spin doublet open-shell species (radicals and cations). Both methods are variants of the SCF approximation. A fuller description and explanation of the basis set and methods has been given previously8. 

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 ... 

Thus far, the spin-independent forms of the various calculation methods have been described without clear notification, despite these forms being available only in the electronic state calculations for closed-shell molecules, in which electrons occupy molecular orbitals two by two. To take open-shell molecules containing unpaired electrons into consideration, spin orbitals should be explicitly considered. As a simple method for calculating the electronic states of open-shell molecules, Pople and Nesbet developed the unrestricted Hartree-Fock (UHF) method (Pople and Nesbet 1954), which independently deals with the spatial orbitals for a and P spins, in 1954. 

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. 

The situation is analogous to the unrestricted Hartree-Fock (UHF) method, cf. p. 408. This extension of the DFT is known as spin density functional theory (SOFT). 

The density matrix idempotency relations described here may be easily extended to the unrestricted Hartree-Fock (UHF) method when the orbitals for a and / spins are treated independently. 

Since for the open shells the total number of electrons with the jS spin is not equal to the number of electrons with the a spin, their electron surroundings must be different and the assumption of equivalence of the space functions describing the spin orbitals of the a- and j3-electrons will not be rigorous. In the more general approximation represented by the unrestricted Hartree-Fock (UHF) method, the electrons with the a- and jS-spins correspond to different orbitals (p , and cpf, cp, ...,( (Fig. 2.1), and the one-determinant... 

The method of different orbitals for different spins (DODS) is regarded as one possible way of accounting for a part of the correlation. However, in using one single Slater determinant built up from different spatial orbitals for the electrons with spin a and [the unrestricted Hartree-Fock (UHF) method], the difficulty arises that the many-elec-tron wave function will not be an eigenfunction of the total spin operator... 

The situation is more complicated if the restriction of double occupancy is dropped. Here we discuss the unrestricted Hartree-Fock (UHF) method in some detail. In this case one has different density matrices for the a and p electrons, P and pP, respectively. As a consequence, one has also different Fockians for the a and P electrons F and F. The expressions of their matrix elements can be derived from Eq. (10.55) by performing the integration over spin functions ... 

In the former, electrons are assigned to orbitals in pairs, the total spin is zero, so the multiplicity is 1. In this case, the restricted Hartree-Fock method (RHF) can be applied. For radicals with doublet or triplet states, the unrestricted Hartree-Fock (UHF) has to be applied. In this method, a and, 3 electrons (spin up and spin down) are assigned to different spatial orbitals, so there are two distinct sets I and FJf... 

Early determinations of RSE values employed unrestricted Hartree-Fock (UHF) theory in combination with 3-21G [9] or 4-31G [10] basis sets to evaluate the RSE according to Eq. 1. The appropriate consideration of correlation effects, the avoidance of spin contamination, and the treatment of thermochemical corrections have in detail been studied in the following, in particular by Bauschlicher [11], Coote [12-14], Morokuma [15-18], and Radom [19-25]. Highly accurate RSE and BDE results can be obtained with high level compound methods such as the G2 [26-30] and G3 [31-34] schemes (and variants thereof [11,15-18]), as well as extrapolation methods such as the CBS schemes [35,36], Wl, or W2 [37-39]. Generally, the accurate... 

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. 

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. 

---