Restricted Hartree-Fock RHF

Quantum mechanics calculations use either of two forms of the wave function Restricted Hartree-Fock (RHF) or Unrestricted Hartree-Fock (UHF). Use the RHF wave function for singlet electronic states, such as the ground states of stable organic molecules. 

You can order the molecular orbitals that are a solution to equation (47) according to their energy. Electrons populate the orbitals, with the lowest energy orbitals first. Anormal, closed-shell, Restricted Hartree Fock (RHF) description has a maximum of two electrons in each molecular orbital, one with electron spin up and one with electron spin down, as shown ... 

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific ... 

This last Restricted Hartree-Fock (RHF) state, if allowed to go unrestricted, would probably result in the following UHF state ... 

So far there have not been any restrictions on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spinorbitals, eq. (3.20), being products of a spatial orbital times a spin function (a or /3). If there are no restrictions on the form of the spatial orbitals, the trial function is an Unrestricted Hartree-Fock (UHF) wave function. The term Different Orbitals for Different Spins (DODS) is also sometimes used. If the interest is in systems with an even number of electrons and a singlet type of wave function (a closed shell system), the restriction that each spatial orbital should have two electrons, one with a and one with /3 spin, is normally made. Such wave functions are known as Restricted Hartree-Fock (RHF). Open-shell systems may also be described by restricted type wave functions, where the spatial part of the doubly occupied orbitals is forced to be the same this is known as Restricted Open-shell Hartree-Fock (ROHF). For open-shell species a UHF treatment leads to well-defined orbital energies, which may be interpreted as ionization potentials. Section 3.4. For an ROHF wave function it is not possible to chose a unitary transformation which makes the matrix of Lagrange multipliers in eq. (3.40) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined, and cannot be equated to ionization potentials by a Koopman type argument. 

A theoretical study based on PM3 frontier molecular orbital (FMO) and potential energy surface (PES) analysis at the restricted Hartree-Fock (RHF)/6-31+G level was performed to examine the reaction of l-amino-2-ethoxycarbonyl-pyridinium mesitylenesulfonate and acrylonitrile in the presence of Hilnig s base leading to the formation of l,2-dihydropyrido[l,2-A]pyridazinium inner salt 17 <1999JOC9001>. The calculations indicated that both the [3+2] cycloaddition reaction and the ring expansion occurred in a concerted manner rather than through a stepwise mechanism via a zwitterionic intermediate 16 (Scheme 1). 

Such kind of calculations with a precise self-consistent account of crystal surrounding were performed by, at least, four scientific groups Baetzold [16], Das with coworkers [22,23], Winter etal. [28] and Ladik with coworkers [29]. Winter etal. [28] performed cluster calculation by restricted Hartree-Fock (RHF) method, so they did not take into account the electron correlation. The others groups used the umestricted Hartree-Fock (UHF) method for cluster calculations which allows to some extent the electron correlation. The strong covalent C-O bonding in planes and chains was revealed (in accordance with results obtained in Refs. [20,25,26]). For covalent systems,... 

Optimization at the restricted Hartree-Fock (RHF) and DFT (B3LYP) levels in the 6-31G basis set has also contributed to structural clarification of [l,2,5]thiadiazolo[3,4-f][l,2,5]thiadiazole 6 both approaches successfully predict that 6 ought to have planar equilibrium geometry. Direct a3 initio ipsocentric calculations of the total 7t-current density map for 6 permits classification as aromatic according to magnetic criteria. It exhibits a strong and uniform diatropic perimeter circulation, of comparable intensity to the parent monocycle 11 and similar to that of naphthalene <2004JA11202>. 

The second approach to treating nondynamical correlation has an air of the ostrich about it ignore the spin symmetry of the wave function and use unrestricted Haxtree-Fock (UHF) theory as the single configuration description [7]. Since the UHF wave function comprises one spin-orbital for each electron, a molecular UHF wave function should dissociate to atomic UHF wave functions, for example. This is certainly not the case for spin-restricted Hartree-Fock (RHF) molecules and atoms in general. And there is an attractive simplicity about UHF — no active orbitals to identify, and so forth. However, where nondynamical correlation would be important in an RHF-based treatment, the UHF method will suffer from severe spin-contamination, while where nondynamical correlation is not important the RHF solution may be lower in energy than any broken-symmetry UHF solution, so potential curves and surfaces may have steps or kinks where the spin symmetry is broken in the UHF treatment. 

In contrast to the semiempirical results, full optimization of the molecular geometry of 1-fluorosilatrane by restricted Hartree-Fock (RHF) calculations using the 3-21G, 3-21G and 6-31G basis sets did not find evidence for the existence of an endo minimum on the energy hypersurface193. Since the exo form was never found experimentally, the ab initio results support the view of the exclusive existence of silatranes in the endo form. As for the equilibrium Sk—N distance, the ab initio calculations using polarized basis sets overestimated its value (2.556 A) even more than the AMI and PM3 methods. 

Computational Details. Restricted Hartree-Fock (RHF) calculations were carried out using Gaussian 94 (45) and ACES II (46) on an IBM RISC/6000 computer. The gauge independent atomic orbitals (GIAO) method was used for the shielding calculations (47). All second-order many-body perturbation theory (MBPT2, also referred to as MP2) calculations were performed with ACES II (46). 

An initial equilibrium structure is obtained by geometry optimization at the Hartree-Fock (HF) level with the 6-31G(d) basis.68 69 Spin-restricted Hartree-Fock (RHF) theory is used for singlet states and spin-unrestricted Hartree-Fock theory (UHF) for others. 

---