Spin-unrestricted molecular orbital

Conversely, an unrestricted Uartree-Fock description implies that there are two different sets of spatial molecular orbitals those molecular orbitals, /j , occupied by electrons of spin up (alpha... 

The unrestricted approach defines two different sets of spatial molecular orbitals—those that hold electrons with spin up ... 

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

Due to the spin polarization effect, the magnetic orbitals can be difficult to identify from a spin-unrestricted calculation. Since the total energy of a Kohn—Sham determinant is invariant under unitary transformations between the spin-up orbitals among each other and spin-down orbitals among each other, one can arrange each spin-up orbital to overlap at most with each spin-down orbital on the basis of the corresponding orbital transformation (COT) (88—90). Then, the molecular orbitals (MOs) are ordered into pairs of maximum similarity between spin-up and spin-down orbitals and can be separated into three groups (i) the MOs with spatial overlap close to one (doubly occupied MOs),... 

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. 

The spin orbitals can be separated into a spatial part, if/ (orbital or molecular orbital) and a spin eigenfunction, a (or for the opposite spin). In restricted Hartee-Fock theory (RHF), the spatial part is independent on the spin state, in contrast to unrestricted Hartree-Fock (UHF) where it is spin dependent. Consequently, the RHF spin orbitals can be written as %i= whereas in UHF the corresponding relation is Xi= even number of electrons (closed shell system), but can easily be extended to treat also UHF1. 

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. 

The equations require to be modified for open-shell systems, in which some orbitals are doubly occupied and some singly (this is called spin-restricted Hartree-Fock theory). A further extension to the theory involves electrons of a and /3 spin being assigned to different molecular orbitals, type equations are described as unrestricted Hartree-Fock [31]. 

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)... 

Fig. 4.27. Molecular-orbital energy-level diagram for the (FeO,) " cluster calculated using the MS-SCF-Jfa method. The highest-energy occupied orbital is the 2 2g i containing one electron. Also shown are the energies of Fe and O atomic orbitals. Spin-up (f ) and spin-down ( i ) molecular orbitals are shown in this spin-unrestricted calculation on a regular octahedral (O ) cluster at an iron-oxygen distance of 2.17 A (after Tossell et al., 1974 reproduced with the publisher s permission).

Charge transfer between the constituent atoms is thought to be important for the stability of crystal structures. In our previous work [1] the transferred charge in the case of ARNi and AlNia was estimated from Auger parameter measurements. In the present study, performing non relativistic spin-restricted and spin-unrestricted DVY molecular orbital calculations of model cluster we obtain more detailed information on the particular orbitals involved in the charge transfer processes. 

The molecular and electronic structures of cyclic disulfide cation radicals of 1,2-dithietane 6 and 1,2-dithiete 7, and radical cations of 1,2-dithiolane 2 (2a-c represent stable conformations determined in terms of the symmetry restriction of Cs, Cz, and Czv), with emphasis on the nature of a two-center three-electron (Zc-ie) sulfur-sulfur bond have been examined by ab initio molecular orbital (MO) calculations <1997JMT(418)171>. Unrestricted Hartree-Fock (UHF)/ MIDI-4(d) computations showed that this bond in organodisulfide radical cation 2 is shorter in comparison to 1,2-dithiolane 2 and possesses partial Jt-bond character (structure A), as previously implied by electron spin resonance (ESR) spectroscopy <1982JA2318>, which correlates best with the form as the most favorable conformation of the cation radical 2. Contrary to the repulsive S-S interaction in the parent 1,2-dithiolane arising from the lone pairs of electrons, the hemi-7t-bond formed by one-electron oxidation should stabilize the five-membered ring of 2, or, for example, a similar cation radical of LA 3 which is involved in diverse biochemical reactions. 

Several researchers have recently devoted considerable effort to the derivation and efficient implementation of techniques based on spin-restricted reference determinants that reduce the computational discrepancy between closed- and open-shell systems. " This emphasis on spin-restricted techniques has resulted in part from a bias toward reference wavefunctions that maintain the spin symmetry of the exact wavefunction (such as the ROHF determinant), but also because of the possible efficiency advantages of spin-restricted methods over unrestricted techniques. Thus, since the component molecular orbitals are constrained to have identical spatial parts for each spin function, it should be possible to construct the correlated wavefunction in a manner that takes advantage of this symmetry. 

The one-electron Kohn-Sham equations were solved using the Vosko-Wilk-Nusair (VWN) functional [27] to obtain the local potential. Gradient correlations for the exchange (Becke fimctional) [28] and correlation (Perdew functional) [29] energy terms were included self-consistently. ADF represents molecular orbitals as linear combinations of Slater-type atomic orbitals. The double- basis set was employed and all calculations were spin unrestricted. Integration accuracies of 10 -10 and 10 were used during the single-point and vibrational frequency calculations, respectively. The cluster size chosen for Ag or any bimetallic was... 

For the development below, we will assume a closed-shell situation, with all electrons paired in molecular orbitals. In such a case OfSj = 1. In very many cases, however, an unrestricted Hartree-Fock (UHF) scheme is utilized for ground state properties. This theory is reasonably accurate for those cases in which each open-shell orbital has an electron of the same spin, i.e., the case that an open-shell has maximum multiplicity. In the UHF scheme Eq. [4] does not hold. Two Fock equations result, one for a and one for 3 spin molecular orbitals. In cases in which excited state properties are required, Eq. [4] is forced to hold in order to yield spectroscopic states, of known multiplicity. OfSJ can then become quite complex, and affects the form of the Fock operators that follow. ... 

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). 

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 ... 

The sum includes all mixing coefficients belonging to atomic orbitals molecular orbitals are numbered with i. We have also assumed, for simplicity, the "spin-restricted" case, meaning that there are identical molecular orbitals for "spin-up" and "spin-down" electrons (see Section 2.9), and these MOs have occupation numbers /, with /, = 0, 1, or 2 the "spin-unrestricted" case would simply have twice as many sums for both spin directions and spin orbitals. Although the formula looks quite powerful, nothing really new has entered if compared with the case of H2 it represents the general electron partitioning of a molecule in terms of Mulliken s recipe. 

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