Unrestricted Hartree-Fock model, spin

Here, occ means occupied and virt means virtual. In the restricted Hartree-Fock model, each orbital can be occupied by at most one a spin and one (i spin electron. That is the meaning of the (redundant) Alpha in the output. In the unrestricted Hartree-Fock model, the a spin electrons have a different spatial part to the spin electrons and the output consists of the HF-LCAO coefficients for both the a spin and the spin electrons. 

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. 

One way to overcome this difficulty is to use different orbitals for different spins (DODS model). This technique introduced, into the Hartree-Fock scheme, gave rise to the unrestricted Hartree-Fock model (UHF), the wave-function being written as an open shell single Slater determinant[2) ... 

The field of expansion coefficients is constrained to be the real numbers while retaining the possibility that each MO has an a and spin component. This is the Real General Unrestricted Hartree-Fock model (R)GUHF. The use of complex coefficients is sufficiently rare that this model is often called simply GUHF. 

The field of real numbers is retained for the coefficients, but now each MO is formed by linear combinations of basis functions having either an a spin factor or a. P spin factor. The most graphic name of this model is Different Orbitals for Different Spins (DODS). However, historically, this was the first Hartree-Fbck method to be used which had any of the common constraints removed and so has also come to be known as simply the Unrestricted Hartree Fock model (UHF). Obviously this name should really be used for CGUHF. 

Spin-density distributions for the radical anions of several compounds including dibenzothiophene have been calculated using the unrestricted Hartree-Fock method. Calculated values were of the same order as observed values but close agreement was not obtained. Hjrperfine splitting constants were also calculated using both Model A and Model B as defined earlier results favored Model B. ... 

Pople refers to a specific set of approximations as defining a theoretical model. Hence the ab initio or Hartree-Fock models employ the Born-Oppenheimer, LCAO and SCF approximations. If the system under study is a closed-shell system (even number of electrons, singlet state), the constraint that each spatial orbital should contain two electrons, one with a and one with P spin, is normally made. Such wavefunctions are known as restricted Hartree-Fock (RHF). Open-shell systems are better described by unrestricted Hartree-Fock (UHF) wavefunctions, where a and P electrons occupy different spatial orbitals. We have seen that Hartree-Fock (HF) models give rather unreliable energies. 

The Pariser-Parr-Pople Hamiltonian for the description of the 7i-electrons in trans-polyacetylene is reparametrized using ab initio Coupled Cluster Doubles calculations based on a Restricted Hartree Fock reference on trans-butadiene. To avoid the spin contaminations inherent in Unrestricted Hartree Fock (UHF) type calculations on polymethine chains in the doublet state the Annihilated Unrestricted Hartree Fock (AUHF) model is applied in our PPP calculations (tPA (CH) , polyenes H-(CH)2N-H, polymethines H-(CH)2N+1-H). In geometry optimizations on polymethine chains it is shown that in contrast to results from Hiickel type models the width of neutral solitons is strongly... 

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. 

For the sake of clarity we shall first discuss in Section 6.2.1 ground-state properties in the framework of the unrestricted Hartree-Fock approximation. In Section 6.2.2 the theory is extended to finite temperatures by using a functional integral formalism including spin fluctuations. Finally, in Section 6.2.3 we analyze the problem of electron correlations by exact diagonalization of the simpler single-band Hubbard model. 

Sulphur Heterocyclic Radicals.—Most studies in this field have used theoretical methods to derive the spin-density distribution and hyperfine splitting constants. In a few cases the unrestricted Hartree-Fock (UHF) formalism has been applied (PPP approximation). (The inclusion of d-orbitals slightly improved the spin-density results in the p-model. )... 

A common feature of the Hartree-Fock scheme and the two generalizations discussed in Section III.F is that all physical results depend only on the two space density matrices p+ and p, which implies that the physical and mathematical simplicity of the model is essentially preserved. The differences lie in the treatment of the total spin in the conventional scheme, the basic determinant is a pure spin function as a consequence of condition 11.61, in the unrestricted scheme, the same determinant is a rather undetermined mixture of different spin states, and, in the extended scheme, one considers only the component of the determinant which has the pure spin desired. 

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