Hartree-Fock theory projected

The second chapter introduces the student to orbitals proper and offers a simplified rationalization for why orbital interaction theory may be expected to work. It does so by means of a qualitative discussion of Hartree-Fock theory. A detailed derivation of Hartree-Fock theory making only the simplifying concession that all wave functions are real is provided in Appendix A. Some connection is made to the results of ab initio quantum chemical calculations. Postgraduate students can benefit from carrying out a project based on such calculations on a system related to their own research interests. A few exercises are provided to direct the student. For the purpose of undergraduate instruction, this chapter and Appendix A may be skipped, and the essential arguments and conclusions are provided to the students in a single lecture as the introduction to Chapter 3. 

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. 

Potential energy curves for singlet and triplet A j, B, and B j states of COF j have been computed using ab initio projected-unrestricted Hartree-Fock theory with a contracted Gaussian type orbital basis set [273]. However, symmetry was strictly maintained for these excited states, so the poor agreement between the predicted and experimental band onsets (which was readily acknowledged by Brewer et al. [273]) comes as little surprise. 

Electronic structure calculations have been performed with the system of programs Gaussian94 (G94) [19]. Restricted Hartree-Fock theory (RHF) is used for closed shell systems, and Unrestricted Hartree-Fock theory (UHF) for open shell systems (radicals). The correlation energy corrections are introduced with Moller-Plesset perturbation theory up to second order, and results from spin projected calculations are used (PMP2 and PUMP2). 

It is worthwhile to mention that the projected oscillator orbitals bear some similarities to the trial perturbed wave function in the variation-perturbational technique associated with the names of Kirkwood [30], Pople and Schofield [31] (KPS), to calculate multipole molecular polarizabilities. The closely related Karplus-Kolker [32, 33] (KK) method and its variants [34, 35] use a similar Ansatz for the perturbed orbitals. In these latter methods, which were formulated originally as simplified perturbed Hartree-Fock theories, the first-order perturbed... 

Huzinaga S, McWilliams D, Cantu AA. Projection Operators in Hartree-Fock Theory. Adv Quantum Chem. 1973 7 187. 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

It has also been shown that for a series of benzene rings in paracyclophanes" and in benzenoid hydrocarbons," a very good correlation exists between the geometry-based index HOMA and the Hartree—Fock energy (HF) at the 6-3IG level of theory. It is clear that in all these cases the selection process favors a projection from a multidimensional space to a lower... 

Finally, it should be pointed out that the perturbation-theoretic methods [25] (MP2, etc.), used with such success for molecules near their equilibrium geometries, are much less satisfactory when used to compute a PES. Obviously, where several reference configurations are required these single-reference treatments cannot be expected to perform well, and it does not seem possible to overcome such problems by the use of unrestricted Hartree-Fock (UHF) methods to define a single reference CSF the UHF PES itself will often display discontinuities from spontaneous symmetry breaking, and this inevitably compromises the subsequent perturbation theory treatment. Recent efforts [26] to devise projected UHF-based schemes may overcome these problems, but this is simply another approach to generating a multireference wave function. 

Abstract Some previous results of the present author are combined in order to develop a Hermitian version of the Chemical Hamiltonian Approach. In this framework the second quantized Bom-Oppenheimer Hamiltonian is decomposed into one- and two-center components, if some finite basis corrections are omitted. (No changes are introduced into the one- and two-center integrals, while projective expansions are used for the three- and four-center ones, which become exact only in the limit of complete basis sets.) The total molecular energy calculated with this Hamiltonian can then presented as a sum of the intraatomic and diatomic energy terms which were introduced in our previous chemical energy component analysis scheme. The corresponding modified Hartree-Fock-Roothaan equations are also derived they do not contain any three- and four-center integrals, while the non-empirical character of the theory is conserved. This scheme may be useful also as a layer in approaches like ONIOM. 

At the post-Hartree-Fock level. Wind and Heully have projected out the BSSE effect in Cl theory. In calculations on the interaction of a hydrogen molecule and a helium atom, the authors found that the results were similar to those of the counterpoise method. Muguet and Robinson more recently proposed a localized approach allowing not only energy calculations, but various molecular properties to be calculated with none of the BSSE complications,... 

---