Unrestricted Wavefunctions

Unrestricted Wavefunctions.—Up to this point, little has been said about the form of the molecular orbitals used in the determinantal wavefunction. Since the choice 

In the restricted Hartree-Fock (RHF) method, two restrictions are placed on the molecular orbitals u in equation (11). The first is chat each ui transform according to one of the irreducible representations of the point group of the molecule. The second restriction is that the space functions f u come in identical pairs one with spin function a and the other with spin function /S. These are called, respectively, the symmetry and equivalence restrictions.190 

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 obvious advantage of this method is the retention of the single-determinant wavefunction. The drawback is that the resulting wavefunction is no longer an eigenfunction of the spin operator S 2 . There are cases where the eigenvalue equation 

In their work on the insertion of triplet oxygen to the hydrogen molecule, Bader and Gangi138 found it necessary to use the UHF method in order to investigate the changes of polarization of electron spins at various points on the potential energy surface. The spin density distribution function r(r) is defined as 

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For some cases, additional measures must be taken to force an unrestricted wavefunction to be used (for example, GuesseMur or Guess=Altar). 

A further alternative is to remove the higher-spin states from the unrestricted wavefunction by means of a spin-projection operator. Spin-projected energies are designated by a P prefix (e.g. PHF, PMP). 

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

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. 

On the other hand, DFT performs extraordinarily well. Spin contamination is a problem with DFT methods snch as B3LYP and so an unrestricted wavefunction must be used and spin correction appUed. Using this approach, Houk found 4 to lie 3.4 kcal mol" below 6 and 2.8 kcal mol below 7. Coupled with the computed KIEs, discussed next, the results of computational studies indicate that the Diels-Alder reaction proceeds by the concerted mechanism. 

Anytime a wavefunction may have diradical character, care must be taken to insure that the wavefunction is stable with respect to lifting the requirement of spin restriction. This can be simply tested by examining whether an unrestricted wavefunction is lower in energy. For the Cope rearrangement, the... 

Unrestricted wavefunction methods (see Ref. 3) lead to mixed spin states, but this is of no help in the present context. 

The values ( 1) reflect the biradical character in the silsesquioxanes (the lowest singlet and triplet energies strongly mixed in the unrestricted wavefunction), in agreement with the fairly small S-T energy separations. In fact, the S-T separations are smaller than in trimethylene (see Table 30.1). This indicates that these species, exemplified here by the various cube structures, behave as electronic isolators , as witnessed in the experimental investigations. At least this is strongly dictated for the 1.4-biradical structure-type. 

This is termed an unrestricted wavefunction, and it would have the form of Eq. 14.21. Here, the total number of orbitals—last of which is —equals the total number of electrons (n). 

In order to specify the proper electronic state, ozone calculations should be performed as unrestricted calculations, and the keyword Gue s=Mix should always be included. This keyword tells the program to mix the HOMO and LUMO within the wavefunction in an effort to destroy a-P and spatial symmetries, and it is often useful in producing a UHF wavefunction for a singlet system. Running a UHF GuesssMix Stable calculation confirms that the resulting wavefunction is stable, and it predicts the same energy (-224.34143 hartrees) as the previous Stable=Opt calculations. 

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. 

In addition, since the HPHF wavefunction exhibits a two-determinantal form, this model can be used to describe singlet excited states or triplet excited states in which the projection of the spin momentum Ms=0. The HPHF approximation appears thus as a simple method for the direct determination of excited states (with Afs=0)such as the usual Unrestricted Hartree Fock model does for determining triplet excited states with Ms = 1. 

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 ... 

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. 

Perturbative approximation methods are usually based on the Mpller-Plesset (MP) perturbation theory for correcting the HF wavefunction. Energetic corrections may be calculated to second (MP2), third (MP3), or higher order. As usual, the open- versus closed-shell character of the wavefunction can be specified by an appropriate prefix, such as ROMP2 or UMP2 for restricted open-shell or unrestricted MP2, respectively. 

Several of the methods referred to in this chapter use the URCCSD(T) procedure in which a spin-unrestricted CCSD(T) calculation is performed on a high-spin RHF reference wavefunction, as implemented in the MOLPRO program. H. J. Werner, P. J. Knowles, R. D. Amos, A. Bemhardsson, A. Beming, P. Celani, D. 

Here a is an adjustable parameter, usually determined by comparing Hartree-Fock and Xa atomic calculations. In the spin-unrestricted version, the spin-up and spin-down orbitals are distinct, so that in general the resulting wavefunction is not a spin eigenfunction. 

Pople-Nesbet Equations. The set of equations describing the best Unrestricted Single Determinant Wavefunction within the LCAO Approximation. These reduce the Roothaan-Hall Equations for Closed Shell (paired electron) systems. 

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

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