Restricted and Unrestricted Hartree-Fock

Using the concepts from Chapter 16, the variational problem can be considered as a rotation of the coordinate system. In the original function space, the basis functions, the Fock operator depends on all the Mbasis functions, and the corresponding Fock matrix is non-diagonal. By performing a rotation of the coordinate system to the molecular orbitals, however, the matrix can be made diagonal, i.e. in this coordinate system the Fock operator only depends on Nocc functions. 

So far there has not been any restriction on the MOs used to build the determinantal trial wave function. The Slater determinant has been written in terms of spin-orbitals, eq. (3.20), being products of a spatial orbital and a spin function (a or P). 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 p 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, and 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 choose a unitary transformation that makes the matrix of Lagrange multipliers in eq. (3.41) diagonal, and orbital energies from an ROHF wave function are consequently not uniquely defined and cannot be equated to ionization potentials by a Koopmans-type argument. 

ELECTRONIC STRUCTURE METHODS INDEPENDENT-PARTICLE MODELS 

So far there have not been any restrictions on the MOs used to build the determinantal ki-AVP fimrlinii Thft. Slater He.tp.nninnnt hns beer written in terms nf spinnrbitals, eg 

---

Local spin density functional theory (LSDFT) is an extension of regular DFT in the same way that restricted and unrestricted Hartree-Fock extensions were developed to deal with systems containing unpaired electrons. In this theory both the electron density and the spin density are fundamental quantities with the net spin density being the difference between the density of up-spin and down-spin electrons ... 

Figure 5-1. H2 potential curves computed within the restricted and unrestricted Hartree-Fock (RHF and UHF) and Kohn-Sham (RKS and UKS) formalisms. 

Tel. 904-392-1597, fax. 904-392-8722, e-mail aces2 qtp.ufl.edu Ab initio molecular orbital code specializing in the evaluation of the correlation energy using many-body perturbation theory and coupled-cluster theory. Analytic gradients of the energy available at MBPT(2), MBPT(3), MBPT(4), and CC levels for restricted and unrestricted Hartree-Fock reference functions. MBPT(2) and CC gradients. Also available for ROHE reference functions. UNIX workstations. 

Restricted and Unrestricted Hartree-Fock 70 4.8.2 Unrestricted and Projected M0ller-Plesset Methods —m ... 

Figure 11 Potential energy curves for H2 generated by the GAUSS2 program using an STO-3G basis set, for both restricted and unrestricted Hartree—Fock.

In the cases other than [case A] and [case E, so called open-shdl SCF methods are employed. Theorbitalconcept becomes not quite certain. The methods are divided into classes which are "restricted" and "unrestricted" Hartree-Fock procedures. In the latter case the wave function obtained is no longer a spin ergenfunction. 

---