Restricted Hartree-Fock method, single

The Hartree-Fock description of the hydrogen molecule requires two spinorbitals, which are used to build the single-determinant two-electron wave function. In the Restricted Hartree-Fock method (RHF) these two spinorbitals are created from the same spatial... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

M. Urban, P. Neogrady, and I. Hubac, Spin Adaptation in the Open-Shell Coupled-Cluster Theory with a Single Determinant Restricted Hartree-Fock Reference. In R. J. Bartlett (Ed.) Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, Vol. 3. (World Scientific, Singapore, 1997), pp. 275-306. 

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. 

G.E. Scuseria, The open-shell restricted Hartree-Fock singles and doubles coupled-cluster method including triple excitations CCSD (T) Application to Ca+. Chem. Phys. Lett. 176, 27-35 (1991)... 

Each spin orbital is a product of a space function fa and a spin function a. or ft. In the closed-shell case the space function or molecular orbitals each appear twice, combined first with the a. spin function and then with the y spin function. For open-shell cases two approaches are possible. In the restricted Hartree-Fock (RHF) approach, as many electrons as possible are placed in molecular orbitals in the same fashion as in the closed-shell case and the remainder are associated with different molecular orbitals. We thus have both doubly occupied and singly occupied orbitals. The alternative approach, the unrestricted Hartree-Fock (UHF) method, uses different sets of molecular orbitals to combine with a and ft spin functions. The UHF function gives a better description of the wavefunction but is not an eigenfunction of the spin operator S.2 The three cases are illustrated by the examples below. 

G. E. Scuseria, Chem. Phys. Lett., 176,27 (1991). The Open-Shell Restricted Hartree-Fock Singles and Doubles Coupled-Cluster Method Including Triple Excitations CCSD(T) Application to C3. 

There is, however, a method which has not been fully exploited yet, in particular for extended systems, namely the single-determinant method without any restrictions on the spin orbitals. Thanks to very important work by Fukutome and collaborators we now have a clear picture of the various possible forms of such spin orbitals, which includes, in addition to the well-known doubly filled restricted Hartree-Fock (RHF) orbitals, alternant molecular orbitals and other forms of different orbitals for different spins, Overhauser s spin density waves, as well as others. It has been known for a long time that a sufficiently general single determinant can... 

In Table 6.3, the values of De for RfCU are compared with those obtained within various approximations using relativistic effective core potentials (RECP) Kramers-restricted Hartree-Fock (KRHF) (Han et al 1999), averaged RECP including second-order M0ller-Plesset perturbation theory (AREP-MP2) for the correlation part (Han et al. 1999), RECP coupled-cluster single double (triple) [CCSD(T)] excitations (Han et al. 1999), and a Dirac-Fock-Breit (DFB) method (Malli and Styszynski 1998). The AREP-MP2 calculation of De gives 20.4 eV, while the RECP-CCSD(T) method with correlation leads to 18.8 eV. Our value of De of 19.5 eV is just between these calculated values. 

The few attempts at describing excited states in transition metal complexes within the Restricted Hartree Fock (RHF) formalism were rapidly abandoned due to the computational difficulties (convergence of the low-lying states in the open-shell formalism) and theoretical deficiencies (inherent lack of electronic correlation, inconsistent treatment of states of different multiplicities and d shell occupations). The simplest and most straightforward method to deal with correlation energy errors is the Configuration Interaction (Cl) approach where the single determinant HF wave function is extended to a wave function composed of a linear combination of many de-... 

The Roothaan-Hall equations are not applicable to open-shell systems, which contain one or more unpaired electrons. Radicals are, by definition, open-shell systems as are some ground-state molecules such as NO and 02. Two approaches have been devised to treat open-shell systems. The first of these is spin-restricted Hartree-Fock (RHF) theory, which uses combinations of singly and doubly occupied molecular orbitals. The closed-shell approach that we have developed thus far is a special case of RHF theory. The doubly occupied orbitals use the same spatial functions for electrons of both a and spin. The orbital expansion Equation (2.144) is employed together with the variational method to derive the optimal values of the coefficients. The alternative approach is the spin-unrestricted Hartree-Fock (UHF) theory of Pople and Nesbet [Pople and Nesbet 1954], which uses two distinct sets of molecular orbitals one for electrons of a spin and the other for electrons of / spin. Two Fock matrices are involved, one for each type of spin, with elements as follows ... 

Hartree-Fock theory employs a single Slater determinant. In the restricted Hartree-Fock (RHF) method, one spatial function 4>i is multiplied by an a (representing spin up, spin quantum number ms = +j) or P (representing spin down, nis = — ) spin function with the properties... 

Theoretical Results. To solve the open-shell problem, various MO methods have been applied and, in some cases, compared with each other. Most commonly, the restricted Hartree-Fock formalism for the open-shell case (RHF) [15] and the unrestricted Hartree-Fock formalism before (UHF) [16] and after single spin annihilation (UHFASA) [17] have been used besides other methods, see [18 to 21] and footnote of Table 6, p. 234. 

The restricted open-shell Hartree-Fock (ROHF) and the unrestricted Hartree Fock Method ( UHF) approximations permit, however, open-shell systems to be described, while maintaining the simplicity of the single-determinant approximation. This is made at the stage of self-consistent electronic-structure calculations. Afterwards, the obtained spin-orbitals can be used to get the correct total spin many-determinant wavefunction and to calculate the corresponding electron energy. 

The unrestricted and restricted open-sheU Hartree-Fock Methods (UHF and ROHF) for crystals use a single-determinant wavefunction of type (4.40) introduced for molecules. The differences appearing are common with those examined for the RHF LCAO method use of Bloch functions for crystalline orbitals, the dependence of the Fock matrix elements on the lattice sums over the direct lattice and the Brillouin-zone summation in the density matrix calculation. The use of one-determinant approaches is the only possibility of the first-principles wavefunction-based calculations for crystals as the many-determinant wavefunction approach (used for molecules) is practically unrealizable for the periodic systems. The UHF LCAO method allowed calculation of the bulk properties of different transition-metal compounds (oxides, perovskites) the qrstems with open shells due to the transition-metal atom. We discuss the results of these calculations in Chap. 9. The point defects in crystals in many cases form the open-sheU systems and also are interesting objects for UHF LCAO calculations (see Chap. 10). 

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