Unrestricted spin orbitals

Section 2.S discusses electron spin and spin operators in many-electron systems and contains a description of restricted and unrestricted spin orbitals and spin-adapted configurations. Spin-adapted configurations, unlike many single determinants derived from restricted spin orbitals, are correct eigenfunctions of the total electron spin operator. Singlet, doublet, and triplet spin-adapted configurations as well as unrestricted wave functions, which are not eigenfunctions of the total electron spin operator, are described. 

Unrestricted determinants are formed from unrestricted spin orbitals. Unrestricted spin orbitals have different spatial orbitals for different spins. Given a set of K orthonormal spatial orbitals, ... 

As shown in Exercise 2.1, the 2K unrestricted spin orbitals form an orthonormal set, in spite of the fact that the a and fi spatial orbitals are not orthogonal. 

So fiir in this chapter we have discussed the Hartree-Fock equations from a formal point of view in terms of a general set of spin orbitals xj. We are now in a position to consider the actual calculation of Hartree-Fock wave functions, and we must be more specific about the form of the spin orbitals. In the last chapter we briefly discussed two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a (spin up) and jS (spin down) spin functions and unrestricted spin orbitals, which have different spatial functions for a and P spins. Later in this chapter we will discuss the unrestricted Hartree-Fock formalism and unrestricted calculations. In this section we are concerned with procedures for calculating restricted Hartree-Fock wave functions and, specifically, we consider here... 

Open-Shell Hartree-Fock Unrestricted Spin Orbitals... 

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. 

There are two types of spin orbitals restricted spin orbitals, which are constrained to have the same spatial function for a and spin functions and unrestricted spin orbitals, which have different spatial functions for a and spins. A restricted set of spin orbitals has the form X ( ) = whereas an unrestricted set has the form ... 

In addition, there are two methods that involve spin projection operators. The spin extended Hartree-Fock (EHF) method starts with a single determinant of unrestricted spin orbitals and applies a spin projection operator before the molecular orbital coefficients are calculated in an SCF procedure. The half projected Hartree-Fock (HPHF) method uses a two determi-nantal wavefunction composed of spin-unrestricted orbitals to represent singlet and = 0 triplet states for systems with an equal number of a and electrons. The second determinant i.s... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

The existing SCF procedures are of two types in restricted methods, the MO s, except for the hipest (singly) occupied MO, are filled by two electrons with antiparallel spin, while in unrestricted methods, the variation procedure is performed with individual spin orbitals. In the latter, a total wave function is not an eigenvalue of the spin operator S, which is disadvantageous in many applications because of a necessary annihilation of higher multiplets by the projection operator. Since in practical applications the unrestricted methods have not proved to be remarkably superior, we shall call our attention in this review mainly to the restricted methods. 

Figure 1 shows the one-electron energies of iron(II) porphine in both spin-restricted and spin-unrestricted (different orbitals for different spins) form. The geometry used has a planar porphine with an Fe-N distance of 2.01 A and no axial ligands. This calculation assumes a high-spin quintet state with the configuration (dxy) ... 

Much of the development of the previous chapter pertains to the use of a single Slater determinant trial wavefunction. As presented, it relates to what has been called the unrestricted Hartree-Fock (UHF) theory in which each spin-orbital (ftj has its own orbital energy 8i and LCAO-MO coefficients Cv,i there may be different Cv,i for a spin-orbitals than for (3 spin-orbitals. Such a wavefunction suffers from the spin contamination difficulty detailed earlier. 

The version of HF theory we have been studying is called unrestricted Hartree-Fock (UHF) theory. It is appropriate to all molecules, regardless of the number of electrons and the distribution of electron spins (which specify the electronic state of the molecule). The spin must be taken into account when the exchange integrals are being evaluated since if the two spin orbitals involved in this integral did not have the same spin function, a or ft, the integral value is zero by virtue of the orthonormality of the electron spin functions... 

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. 

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. 

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. 

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). 

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)... 

In this method the ionization energy is calculated from the weighted mean of the one-electron energies of the ground state and of a transition state in which 2/3 of an electron is removed from the spin orbital considered. The transition state calculations should be carried out through spin unrestricted calculations. 

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). 

As a result of the Pauli correlation, the effective Coulomb interaction between a 2s electron of spin a, say, and a Isa electron will be different from the interaction with a IsyS electron. Therefore, the Is spin-orbitals will have slightly different spatial parts. This is considered in the so-called unrestricted Hartree—Fock calculations. 

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