Restricted Hartree-Fock model

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. 

Among the many ways to go beyond the usual Restricted Hartree-Fock model in order to introduce some electronic correlation effects into the ground state of an electronic system, the Half-Projected Hartree-Fock scheme, (HPHF) proposed by Smeyers [1,2], has the merit of preserving a conceptual simplicity together with a relatively straigthforward determination. The wave-function is written as a DODS Slater determinant projected on the spin space with S quantum number even or odd. As a result, it takes the form of two DODS Slater determinants, in which all the spin functions are interchanged. The spinorbitals have complete flexibility, and should be determined from applying the variational principle to the projected determinant. 

Restricted Hartree-Fock method, 34 136 Retinal pigment models, 32 451-452 Reverse spillover, 34 2, 4, 11, 16, 31, 44, 46,... 

Such a procedure is referred to as restricted and individual models as restricted models, for example, restricted Hartree-Fock (or RHF) models. ... 

Comparative data for heavy-atom bond lengths and skeletal bond angles for molecules incorporating one or more third or fourth-row, main-group elements are provided in Appendix A5 Table A5-39 for Hartree-Fock models with STO-3G, 3-2IG and 6-3IG basis sets. Table A5-40 for the local density model, BP, BLYP, EDFl andB3LYP density functional models and the MP2 model, all with the 6-3IG basis set, and in Table A5-41 for MNDO, AMI and PM3 semi-empirical models. 6-31G, local density, density functional and MP2 calculations have been restricted to molecules with third-row elements only. Also, molecular mechanics models have been excluded from the comparison. A summary of errors in bond distances is provided in Table 5-8. 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

MLCT metal to ligand charge transfer LMCT ligand to metal charge transfer HOMO highest occupied molecular orbital AOM angular overlap model RHF restricted Hartree-Fock... 

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. 

To account for the magnetic hyperfine interaction in, e.g., the ground state of a lithium atom, it is necessary in the core model to go beyond the usual, restricted Hartree-Fock approximation, which assumes that the spatial parts of one-electron orbitals are independent of mB (the spin-... 

When the orbitals are determined in this manner, with the only restriction being the orthonormality constraint, equation (12), they yield the best possible antisymmetrized product wavefunction (i.e., a single determinant wavefunction) for the system in question since the resulting fP(x R) yields the lowest possible value for E(R). The heart of the Hartree-Fock model is to replace the detailed and accurate description of the repulsions between every pair of electrons in the system by the average field that each electron exerts on every other. This is a consequence of the one-electron or orbital basis of the model which leads to a product wavefunction. The probability, Pa(ri,r2)dridr2, that electron 1 be in some small volume element about ri when electron 2 is simultaneously in some small volume element about ra is, in a simple product-type wavefunction, given by the product of the two singleparticle probabilities,... 

Bubble Diagrams.—The bubble diagrams which are shown in Figures 13 and 14 are required when the reference function for a closed-shell system is not defined by the matrix Hartree-Fock model, when a restricted Hartree-Fock reference... 

Up to now most quantum mechanical studies of the ground and excited states of model heme complexes have focused primarily on diamagnetic systems (36), with less frequent treatment of heme systems with unpaired spins (37-42). With the inclusion of a restricted Hartree-Fock treatment (37, 38) within an INDO formalism parameterized for transition metals (39, 40, ), it is now possible to calculate the relative energies of different spin states of ferric heme complexes in an evenhanded fashion at a semiempirical level. 

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... 

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