Unrestricted HF

In the HF scheme, the first origin of the correlation between electrons of antiparallel spins comes from the restriction that they are forced to occupy the same orbital (RHF scheme) and thus some of the same location in space. A simple way of taking into account the basic effects of the electronic correlation is to release the constraint of double occupation (UHF scheme = Unrestricted HF) and so use Different Orbitals for Different Spins (DODS scheme which is the European way of calling UHF). In this methodology, electrons with antiparallel spins are not found to doubly occupy the same orbital so that, in principle, they are not forced to coexist in the same spatial region as is the case in usual RHF wave functions. 

The method described previously is known as the unrestricted HF (UHF) method. It appears to be a logical choice. However, it does have one significant drawback the total single-determinantal wavefunction ... 

Another property calculated in the XH3—Y series is the X—Y bond dissociation energy. For this purpose the MP2 optimized geometries for the XH3 and Y (doublet spin) radicals were obtained using the unrestricted HF (UHF) method. For comparison to experiment, the electronic energy differences for the reaction... 

To distinguish between closed-shell and open-shell configurations (and determinants), one may generally include a prefix to specify whether the starting HF wavefunction is of restricted closed-shell (R), restricted open-shell (RO), or unrestricted (U) form. (The restricted forms are total S2 spin eigenfunctions, but the unrestricted form need not be.) Thus, the abbreviations RHF, ROHF, and UHF refer to the spin-restricted closed-shell, spin-restricted open-shell, and unrestricted HF methods, respectively. 

The most general version of Hartree-Fock (HF) theory, in which each electron is permitted to have its own spin and spatial wave function, is called unrestricted HF (UHF). Remarkably, when a UHF calculation is performed on most molecules which have an equal number of alpha and beta electrons, the spatial parts of the alpha and beta electrons are identical in pairs. Thus the picture that two electrons occupy the same MO with opposite spins comes naturally from this theory. A significant simplification in the solution of the Fock equations ensues if one imposes this natural outcome as a restriction. The form of HF theory where electrons are forced to occupied MOs in pairs is called restricted HF (RHF), and the resulting wave function is of the RHF type. A cal-... 

One way to deal with unpaired electrons is to use the unrestricted HF (UHF). Whereas regular ab initio calculations restrict the one-electron spatial orbitals to be identical for a- and (3-spin electrons (so-called restricted HF, RHF), in UHF the orbitals are allowed to be different in the SCF processes. Usually, the difference in the spatial orbitals for a and (3 electrons is only slight. Unfortunately, when applied to a radical, UHF stumbles in a pitfall (97). It is called spin contamination. Unrestricted HF wave functions cannot be trusted to correspond to pure spin states such as a doublet for radicals or a singlet or triplet for diradicals. Theoretically speaking, the UHF wave function may not be an eigenfunction of the spin operators. 

The favoring of a diradical mechanism here seems to be an artifact of semiem-pirical methods (Chapter 6) and unrestricted HF methods (Sections 5.2.3.5 and 5.2.3.6.5) see reference 11 in [41]. A DFT (Chapter 7) study also strongly supported the concerted mechanism [42],... 

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. 

Hartree-Fock (theory) (FIF) (restricted HF, RFIF for closed shells unrestricted HF, UHF for open shells)... 

Formally, Koopmans theorem applies to eigenfunctions of the Fock operator of a closed shell restricted HF wavefunction or an open shell unrestricted HF wavefunction... 

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 common way to treat free radicals is with the unrestricted HF method or UHF method. In this method, we employ separate spatial orbitals forthe a and the )3 electrons, giving two sets of MOs, one for a and one for electrons. Less commonly, free radicals are treated by the restricted open-shell HF or ROHF method, in which electrons occupy MO s in pairs as in the RHF method, except forthe unpaired electron(s). The theoretical treatment of open-shell species is discussed in [1,10, l(k, /)], in particular, compare the performance of the UHF and ROHF methods. 

The use of coupled-cluster (CC) wave functions within EOM theory for excitation energies, IPs and EAs has been developed [34,35] upon slightly different lines than outlined in Section 17.2. The CC wave function ansatz for Q,N) is written as usual in terms of an exponential operator acting on a single-determinant (e.g. unrestricted HF) reference function lO >... 

In the limit R-> 00, K12 becomes i(( 0 < < ), which is finite, but both fii (81 = /ill + 11) and 82 (82 = /122 + 2Ji2 12) go to the same limit (i.e., (H) -h i(00 0< )). Thus 81 - 82 tends to zero and E becomes infinite. The simplest way out of this difficulty is to improve the unrestricted HF description by means of perturbation theory. Computationally, this involves simply using the unrestricted spin orbitals and orbital energies in the formulas for the second and higher order energies written as sums over spin rather than spatial orbitals (e.g., Eqs. (6.72) and (6.75)). The potential energy curves for minimal basis H2, obtained in this way, are shown in Fig. 6.2. 

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