Unrestricted Hartree-Fock approximation

Recent application of the TB method to transition metal clusters often made use of a convenient formulation in the language of second quantization.14 In this formalism, the TB Hamiltonian in the unrestricted Hartree-Fock approximation can be written as a sum of diagonal and nondiagonal terms15... 

As with the unrestricted Hartree-Fock approximation, the LSDA allows for different orbitals for different spin orientations. The LSDA gives a simplified treatment of exchange but also includes Coulomb correlations. 

We adopt the same chemisorption model as in our previous work [3], which within the unrestricted Hartree-Fock approximation involves a self-consistent calculation of the electronic charge on the adatom. The basis elements needed for the calculation are the... 

Our calculations for multi-electron atoms in magnetic fields are carried out under the assumption of an infinitely heavy nucleus in the (unrestricted) Hartree-Fock approximation. The solution is established in the cylindrical coordinate system (p, p, z) with the 2-axis oriented along the magnetic field. We prescribe to each electron a definite value of the magnetic quantum number mp. Each single-electron wave function depends on the variables p and (p, z)... 

We will study the solution to Eq.(2.26lb) first in the restricted and then in the unrestricted Hartree-Fock approximation. According to the restricted Hartree-Fock approximation ... 

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. 

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... 

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as H2, when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and p electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after J.A. Pople [6] ) for problems where the number of a andp electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction. 

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. 

Just as in the unrestricted Hartree-Fock variant, the Slater determinant constructed from the KS orbitals originating from a spin unrestricted exchange-correlation functional is not a spin eigenfunction. Frequently, the resulting (S2) expectation value is used as a probe for the quality of the UKS scheme, similar to what is usually done within UHF. However, we must be careful not to overstress the apparent parallelism between unrestricted Kohn-Sham and Hartree-Fock in the latter, the Slater determinant is in fact the approximate wave function used. The stronger its spin contamination, the more questionable it certainly gets. In... 

Although P3 procedures perform well for a variety of atomic and molecular species, caution is necessary when applying this method to open-shell reference states. Systems with broken symmetry in unrestricted Hartree-Fock orbitals should be avoided. Systems with high multireference character are unlikely to be described well by the P3 or any other diagonal approximation. In such cases, a renormalized elec-... 

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

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. 

In solving Eq. (2), an iterative process is used to adjust the until the best wavefunction is found [self-consistent field (SCF) theory]. For the open shell case where incompletely filled orbitals exist, spin-restricted Hartree-Fock (RHF) methods or unrestricted Hartree-Fock (UHF) methods may be used to calculate the energies.41 The extent of calculation, approximation, or neglect of the two-electron integral terms largely defines the computation method. 

Corrections for Improper HF Asymptotic Behaviour.—There are two techniques which may be used to obtain results at what is essentially the Hartree-Fock limit over the complete range of some dissociative co-ordinate in those cases where the single determinants] approximation goes to the incorrect asymptotic limit. These techniques are (i) to describe the system in terms of a linear combination of some minimal number of determinantal wavefunctions (as opposed to just one) 137 and (ii) to employ a single determinantal wavefunction to describe the system but to allow different spatial orbitals for electrons of different spins - the so-called unrestricted Hartree-Fock method. Both methods have been used to determine the potential surfaces for the reaction of an oxygen atom in its 3P and 1Z> states with a hydrogen molecule,138 and we illustrate them through a discussion of this work. 

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. 

From all this one must conclude that the determinantal and second-quantized formulations should be regarded as a poor man s group theory which, while convenient, hides the basic freeon dynamics. These fermion methods have the additional disadvantage that their antisymmetric fermion functions are not normally pure spin (freeon) states so that spin-projection may be required. A method for avoiding (approximately) spin projection is the employment of the variation principle to approximate the ground state e. g., unrestricted Hartree-Fock theory. Finally the use of the fermion formulations has lead to the spin paradigm as a replacement for the more fundamental freeon dynamics. 

The electronic structure methods are based primarily on two basic approximations (1) Born-Oppenheimer approximation that separates the nuclear motion from the electronic motion, and (2) Independent Particle approximation that allows one to describe the total electronic wavefunction in the form of one electron wavefunc-tions i.e. a Slater determinant [26], Together with electron spin, this is known as the Hartree-Fock (HF) approximation. The HF method can be of three types restricted Hartree-Fock (RHF), unrestricted Hartree-Fock (UHF) and restricted open Hartree-Fock (ROHF). In the RHF method, which is used for the singlet spin system, the same orbital spatial function is used for both electronic spins (a and (3). In the UHF method, electrons with a and (3 spins have different orbital spatial functions. However, this kind of wavefunction treatment yields an error known as spin contamination. In the case of ROHF method, for an open shell system paired electron spins have the same orbital spatial function. One of the shortcomings of the HF method is neglect of explicit electron correlation. Electron correlation is mainly caused by the instantaneous interaction between electrons which is not treated in an explicit way in the HF method. Therefore, several physical phenomena can not be explained using the HF method, for example, the dissociation of molecules. The deficiency of the HF method (RHF) at the dissociation limit of molecules can be partly overcome in the UHF method. However, for a satisfactory result, a method with electron correlation is necessary. 

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