Fock matrix open-shell systems

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. 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

The unrestricted L.C.A.O.—S.C.F. method reduces to the restricted method when a and electrons are assigned to spatially identical molecular orbitals. Thus under the INDO method the Hartree-Fock matrix elements for an open-shell system become... 

The coefficients c i are given by the solution of the corresponding SCF equation (equation 2). d s the diagonal matrix with the MO energies. For consideration of Cl we must distinguish between closed-shell systems and open-shell systems. For closed-shell systems the restricted Hartree-Fock (RHF) formalism is applied, whereas for open-shell systems one has the choice between the unrestricted Hartree-Fock (UHF) or the restricted open-shell formalism (ROHF). The Fock matrix elements were formulated on the CNDO, INDO and NDDO level by Sauer et al. ... 

The effective Hartree-Fock matrix equation for a many-shell system has been derived in Chapter 14 and used in several applications open shells and some MCSCF models. So far, it has been seen simply as the formally correct equation to generate SCF orbitals for these many-sheU structures without any interpretation. In particular, the fact that the effective Hartree-Fock matrix (the McWeenyan ) contains many arbitrary parameters has not been addressed, nor has the practical problem of the actual grounds for the choice of values for these parameters been systematised. In looking at this problem we must bear two points in mind ... 

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). 

We conclude that, for closed-shell and high-spin states, second-order optimizations can be carried out in the AO basis at a cost of n for each trial-vector transformation (10.8.8). For other open-shell CSF states, it is more difficult to simplify the construction of the Q matrix in order to carry out a second-order optimization in the AO basis. However, with the possible exception of the two-electron open-shell singlet state (10.1.7), Hartree-Fock wave functions for oth than high-spin states are of little interest except for systems of high spatial symmetry. In Exercise 10.7, an STO-3G Hartree-Fock wave function for HeH is calculated using Newton s method. 

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