Half-projected Hartree-Fock Research Articles

Half‐Projected Hartree–Fock method: History and application to excited states of the same symmetry as the ground state

AbstractSpin projected wave functions are generalizations of the Hartree–Fock wave function. Among them, the Half‐Projected Hartree–Fock (HPHF) wave function is a nearly pure wave function of spin and recovers a small part of the spin correlation energy. This paper reviews the history of the HPHF theory, not only from the conceptual point of view but also providing a compilation of the publications of this method over the years until now. In addition, the extension of the HPHF method to the calculation of excited states with the same symmetry as the ground state will be discussed. The possible variational collapse during the calculation of a singlet excited state of the same symmetry as the ground state is avoided by orthogonalizing a pair of corresponding orbitals, one occupied and one virtual orbital, at every step of the variational process. As an example, the potential energy surfaces of the S0 ground and 1S1(n, π*) first excited state of the formic acid HCOOH are calculated.

Half-projected Hartree-Fock calculations for the spectroscopic parameters, frequencies, and intensities of the torsional mode of the lowest lying singlet excited states of hydrogen peroxide

Quantum chemical calculations have been performed in half-projected Hartree–Fock (HPHF) and CIS approaches for estimating the energy levels of the lowest two excited states of hydrogen peroxide. Geometry optimization was performed for the respective states at the equilibrium configuration with an elaborate quadrupole-zeta basis set augmented with polarization and diffuse functions. The model of the nonrigid rotor in an anharmonic potential was used for calculating the frequencies and intensities of the torsional mode, and the spectroscopic parameters were evaluated with a view to understand more closely the vibrational structure associated with the experimental ultraviolet (UV) spectrum of this molecule. While the spectroscopic parameters are new, reasonably good agreement is achieved for the frequencies of the torsional mode with the existing theoretical and experimental data for the 11B state. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 631–636, 1999

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.

A partially projected wave function for radicals

A partially projected wave function for odd electron systems with quantum number M=1/2, containing µu spin functions α and µ spin functions α, with fractional spin component αSz=1/2 and 3/2 are derived from the totally projected wave function. To obtain these wave functions new symmetry relations between Sanibel coefficients for the odd electron case have been found, as well as the relations between primitive spin functions and their spin permutations. The wave function for the doublet state is shown not to contain contamination of the quadruplet state, and the wave function for the quadruplet does not have contamination of the duplet. Both wave functions exhibit equal forms except in the signs of their summation terms. The number of primitive spin functions depends on the number of electrons (ns), it grows linearly as ns=(N+3)/2. It can be considered as a generalization of the half projected Hartree–Fock wave function to the odd electron case. The HPHF wave function is defined for even electron systems and consists of only two Slater determinants, it has been shown to introduce some correlation effects and it has been successfully applied to calculate the low-lying excited states of molecules. Therefore, this investigation is the first step to propose a method to calculate the excited states of radicals when other methods are impracticable.

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.

Deriving gradient formulae for SCF methods by using Brillouin-type theorems. Gradient in the HPHF method

It is discussed that the derivation of different SCF-type equations through the relevant (generalized) Brillouin theorems can be utilized to obtain the energy gradients of the method studied. This is illustrated for the case of the UHF method and used to obtain the gradient formula applicable in the framework of the half-projected Hartree—Fock method. The approach should also be applicable to MC-SCF theories.

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.

A spectroscopic study of ground state of lithium hydride with the HPHF model

In this work, matrix equations for obtaining the half-projected Hartree-Fock (HPHF) function from pseudo-eigenvalue equations in a non-orthogonal basis are given, similarly as in the unrestricted Hartree-Fock model. The potential energy curve for the ground state of the LiH molecule is calculated by using a 6-311(2d,2p) gaussian basis set. Finally, in order to test the HPHF model, some spectroscopic parameters are calculated. Relatively good agreement with experiment is found for the anharmonicity constant.

Half‐projected Hartree–Fock natural orbitals for defining CAS–SCF active spaces

AbstractWe have extended the range of systems to which the half‐projected Hartree–Fock (HPHF) method has been applied, including the triplet state of the wave function. In our implementation, DIIS overcomes the convergence difficulties reported in earlier studies. HPHF allows generation of a symmetry‐broken wave function in regions of the potential energy surface where the RHF wave function is triplet‐stable. The fractionally occupied natural orbitals (FONOS) of the HPHF wave function are good starting vectors for CAS–SCF calculations. A CAS–SCF in the space defined by the HPHF FONOS should be used instead of the unrestricted natural orbital CAS–SCF method in regions of triplet stability and for small active space problems. We draw extensive comparisons between the results of both the UNO–CAS and HPNO–CAS methods and those of full CAS–SCF calculations. © 1993 John Wiley & Sons, Inc.

The half‐projected Hartree–Fock model for computing thermally “forbidden” pericyclic reactions

AbstractThe possible use of a half‐projected Hartree–Fock (HPHF) wavefunction correlating a single pair of electrons is investigated in relation with the computation of potential energy hypersurfaces associated to pericyclic reactions of systems with an even number of electrons, in which the orbital configuration of the lowest‐energy singlet state is not conserved during the course of the reaction. The results of calculations, within the framework of the MINDO/3 Hamiltonian, for eleven stationary points on the potential energy hypersurfaces of three prototypical thermally “forbidden” pericyclic reactions indicate that, in general, the HPHF model provides a geometrical and energetical description that is equivalent to that obtained from a conventional two‐configuration self‐consistent‐field approach.

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 model for computing thermally “forbidden” pericyclic reactions and biradical processes. I. Formulation and results for some singlet biradical species

AbstractThe possible use of a half‐projected Hartree‐Fock (HPHF) wave function correlating a single pair of electrons, being defined as a linear combination of two unrestricted Slater determinants which are identical except that all the α and β spin functions are interchanged, is investigated in relation to the computation of the potential energy surfaces involving singlet biradical or biradical‐like species. The results of calculations, within the framework of the MINDO/3 semiempirical Hamiltonian, for a number of prototypical biradical molecules indicate that, in general, the HPHF model provides a geometrical and energetical description that is equivalent to that obtained from a conventional two‐configuration self‐consistent‐field wave function.

Half-projected Hartree–Fock calculations on several small molecules

The half-projected Hartree–Fock (HPHF) method is examined with respect to its ability to obtain molecular correlation, describe molecular potential energy surfaces, and provide a one particle basis for more elaborate treatments. The equivalence, aside from questions of efficiency, of two different HPHF algorithms is demonstrated. The results of calculations on H2O, C2, N2, and CH2 indicate that the performance of the HPHF method, with spin projection in appropriate cases, is roughly equivalent to limited MCSCF treatments. In particular, (1) a small but important fraction of the correlation energy, (2) qualitatively correct potential energy surfaces, and (3) good one particle orbital bases are obtained.

Half‐projected Hartree–Fock model for computing potential‐energy surfaces

AbstractThe possible uses of the half‐projected Hartree–Fock (HPHF) scheme for computing potential‐energy surfaces for both singlet and triplet states are investigated. For that purpose, a new procedure based on the simultaneous solution of two eigenvalue equations is presented for calculating the HPHF function. Applications are given in the cases of the singlet ground state and lowest triplet state of the lithium molecule. In both cases, the HPHF scheme is found to yield the correct general shape of the curves. The features of these curves are discussed. Finally, it is concluded that the HPHF model seems to be useful as a tool for determining potential‐energy surfaces, especially in the cases of large molecules where more sophisticated methods are unmanageable.

Stability conditions for the solutions of the half-projected Hartree-Fock scheme. The lithium-hydride ground state

The stability conditions for the solutions of a two-unrestricted-determinant function (the half-projected Hartree-Fock function) are deduced from the necessary requirements for the minimization of the energy functional. The theory is applied to the case of the LiH ground state, in order to investigate the various solutions encountered in the variational problem. It is found that the two lowest solutions are local minima. The number of these solutions is estimated and their significance discussed as a function of the nuclear separation.

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