Hartree restricted energy

Table 2 Restricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts. AE is the difference in energy between the carbon allotrope and its adduct. In all cases, except where indicated by f, only the six carbon atoms in the immediate vicinity of the muon have had there positions optimised, f means that a full geometry optimisation has been carried out. The type specifies the defect and for C70 is identified in Table 1. is the spin density at the muon in atomic units (and the hyperfine coupling constant in MHz). JMuon constrained to lie in equatorial plane. indicates geometry not fully optimized.

Calculated restricted Hartree-Fock energies and energy differences between the syn- and anti- conformers. Energies are given in Hartree (H) energy differences in milliHartree (mH)... 

Figure 3 Energy difference between the syn- and anti-confbrmer for a) restricted matrix Hartree-Fock energies b) total energies through second-order.

The next comparison to be made is that of OPM and Hartree-Fock results. In Sect. 2.2.2 it has been emphasized that the x-only OPM represents a restricted Hartree-Fock energy minimization One minimizes the same energy expression, but under the subsidiary condition of having a multiplicative exchange potential. How important is this subsidiary condition As Table 2.2 shows, the differences are rather small. For He the OPM energy is identical with the HF value, as in this case the HF equation can be trivially recast as a KS equation with the OPM exchange potential (2.52). Moreover, even for the heaviest elements the differences between OPM and HF energies are... 

The Hartree-Fock energy is not as low as the tme energy of the system. The mathematical reason for this is that our requirement that be a single determinant is restrictive and we can introduce additional mathematical flexibility by allowing i/r to contain many determinants. Such additional flexibility leads to further energy lowering. 

In this section, we shall ply the second-order Newton method to the optimization of the Hartree-Fock energy, considering both a method that works with the AO density matrix and is applicable to large systems, and a method that carries out rotations among the MOs and is applicable to general single-configuration states [16-18]. Our discussion is here restricted to minimizations - in the discussion of MCSCF theory in Chapter 12, we shall consider also the second-order localization of saddle points. 

You can order the molecular orbitals that arc a solution to etjtia-tion (47) accordin g to th eir en ergy, Klectron s popii late the orbitals, with the lowest energy orbitals first. normal, closed-shell, Restricted Hartree hock (RHK) description has a nia.xirnuin of Lw o electrons in each molecular orbital, one with electron spin up and one w ith electron spin down, as sliowm ... 

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 ... 

You will need to decide whether or not to request Restricted (RHF) or Unrestricted (UHF) Hartree-Fock calculations. This question embodies a certain amount of controversy and there is no simple answer. The answer often depends simply on which you prefer or what set of scientific prejudices you have. Ask yourself whether you prefer orbital energy diagrams with one or two electrons per orbital. 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

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. 

In the ordinary Hartree-Fock scheme, the total wave function is approximated by a single Slater determinant and, if the system possesses certain symmetry properties, they may impose rather severe restrictions on the occupied spin orbitals see, e.g., Eq. 11.61. These restrictions may be removed and the total energy correspondingly decreased, if instead we approximate the total wave function by means of the first term in the symmetry adapted set, i.e., by the projection of a single determinant. Since in both cases,... 

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