Half-projected Hartree-Fock Method Research Articles

The multiconfiguration half-projected Hartree–Fock (MCHPHF) method and its application to low-lying singlet excited states of ketene, allene and benzene

Numerous applications of the half-projected Hartree–Fock (HPHF) method have demonstrated that this approach is appropriate for determining low-lying singlet excited states of molecules. The computational effort is comparable to that with a standard unrestricted Hartree–Fock (UHF) calculation. To circumvent the problems of molecules with degenerate or quasi-degenerate excited states, the HPHF method has been formulated by introducing a multiconfigurational ansatz for the linear combination of projected two-determinantal wave functions. Calculation of the first excited states of π-electron systems results in excitation energies that are close to experimental data.

The simplex method for geometry optimization in half-projected Hartree-Fock calculations of excited states

Theoretical investigations of excited states of molecules with the help of the half-projected Hartree-Fock method and their rotational spectra require the determination of molecular structures with high accuracy (in the order of cm −1). To fulfil this condition and to obtain the minima on the potential energy surface, the simplex algorithm has been chosen to perform the geometry optimization. A computer program has been developed to treat excited and ground states of molecular systems with many degrees of freedom and using a large number of basis functions. Formic acid has been chosen to illustrate how the method works. The method has further been applied to determine the optimized geometry of cyclobutanone in its first excited state in order to clarify the discussion on spectroscopic data. With the help of this method not only can one scan a large part of the potential energy hypersurface; additionally, one always obtains a conformation with the lowest energy in any degree of accuracy, even in the case of very shallow regions.

Multiple solutions and spatial symmetry properties of singlet and triplet states in the half-projected Hartree-Fock method

Symmetry properties of different solutions for singlet and triplet states of some simple molecules are investigated in the framework of the half-projected Hartree-Fock (HPFH) method. HPHF initial-guess functions were obtained by using mixtures with, or substitutions by, virtual orbitals of restricted Hartree-Fock (RHF) occupied orbitals. The relation existing between the solutions and the spatial symmetry of the initial guess has been established. The elaborated criteria have been tested for the lowest singlet and triplet states of CH 2, CF 2 and CHF.

The half-projected Hartree-Fock method: some new aspects

The half-projected Hartree-Fock (HPHF) theory set forth by Smeyers (An. Fis., 67 (1971) 17) is extended in order to obtain all the necessary expressions for the calculations of the first and second derivatives of the energy functional. This permits the implementation of well known direct optimization techniques in the framework of the HPHF method. We also propose a treatment of linear response properties based on a HPHF reference function and derive the equation defining the Tamm-Dankoff (monoexcited CI) approximation.

Hamiltonian and overlap matrix elements between half-projected Slater determinants

We obtain expressions for the Hamiltonian and overlap matrix elements between half-projected Slater determinants of the type “different orbitals for different spins” and differing by one or two orbitals. We consider the use of these expressions in the construction of direct orbital optimization schemes in the half-projected Hartree-Fock method (HPHF), in the study of the instability of the Hartree-Fock solution, in the study of the singly-excited configuration interaction and random phase methods and in the interpretation of the static response function in the HPHF method.

The half-projected Hartree-Fock method

A SCF method based on the solution of two eigenvalue problems, in the same manner as for the normal UHF procedure, is formulated for determining the half-projected Hartree-Fock (HPHF) function for singlet ground states of molecules, the HPHF function being defined as a linear combination of two Slater determinants containing only spin eigenfunctions with even quantum number. A computer program has been written and is described, and results are presented for two simple linear molecules. An important part of the correlation energy is obtained for these molecules.

The half-projected Hartree-Fock method

However, this function is not a spin eigenfunction and often the best orbitals, found by minimizing the total energy, are found to be RHF orbitals. To remedy these deficiencies of the UHF method, L6wdin [3] has suggested projecting the UHF function to give a spin eigenfunction of the desired multiplicity, the resulting function being known as the projected Hartree-Fock (PHF) function. Unfortunately, this method, which has been applied only to very small systems [4-8], becomes far too complex for larger systems due to the large numbers of Slater determinants involved in the PHF function. Another two-determinantal function has been suggested by Smeyers and Doreste-Suarez [8]. This function given by