Open-shell Wave Functions Research Articles

Ab initio study on magnetic properties of 1,1′,5,5′-tetramethyl- 6,6′-dithioxo-3,3′-biverdazyl homo-biradical and 3-(2′6′-di-t-butyl-4′-phenoxyl)-1,5-dimethyl-6-thioxoverdazyl hetero-biradical

The ab initio method has been applied to investigate the magnetic properties of 1,1′,5,5′-tetramethyl-6,6′-dithioxo-3,3′-biverdazyl homo-biradical and 3-(2′,6′-di-t-butyl-4′-phenoxyl)-1,5-dimethyl-6-thioxoverdazyl hetero-biradical. The second-order Møller–Plesett perturbation method for the restricted open-shell wavefunctions and the method of Borden and Davidson are very useful to predict the magnetic properties of homo- and hetero-biradicals.

Open-shell Møller–Plesset perturbation theory based on the asymptotic method of obtaining SCF orbitals

Asymptotic method of obtaining restricted open-shell wavefunctions satisfying the correct variational conditions has been extended to excited states. Perturbation theory calculations based on this methodology have been carried out for the correlation energy in the NH 2 ( 2B 1) molecule and the vertical excitation energy in BeF (B 2∑ +→X 2∑ +). Our results for NH 2 are closely related to known RMP results. Unlike spin-restricted UHF methods however, the proposed scheme deals as do the OPT1, OPT2 and ZAPT theories with the identical spatial parts for the α and β orbitals, which has computational advantages. The results for the excitation energy are in agreement with experimental data.

Asymptotic method of building restricted open-shell wavefunctions satisfying the generalized Brillouin's theorem

Amended unrestricted Hartree-Fock (UHF) equations which determine an optimum set of molecular orbitals (MOs) for high-spin open-shell molecules have been obtained. The Slater determinant built from the MOs is an eigenfunction of S 2 and corresponds to a minimum of energy which is equivalent to that obtained by the Roothan method. Such a set satisfies the correct variational conditions unlike other spin-restricted UHF methods. An open-shell calculation based on the MOs has a computational cost similar to closed-shell calculations.

Second-order Møller-Plesset perturbation theory for molecular Dirac-Hartree-Fock wavefunctions. Theory for up to two open-shell electrons

Møller-Plesset perturbation theory is developed to second order for a selection of Kramers-restricted Dirac-Hartree-Fock closed-and open-shell reference wavefunctions. The open-shell wavefunctions considered are limited to those with no more than two electrons in open shells, but include the case of a two-configuration SCF reference. Denominator shifts are included in the style of the OPT2 method of Murray and Davidson, but with a modification which avoids the orbital dependence of the original formulation. An implementation which uses unordered integrals with labels is described, and application to the ground state of the O 2 molecule is presented.

Adatom-based (2 × 2 n) reconstruction on Ge(111)

Atomic cluster calculations are reported which model adatom generated (2 × 2) and c(2 × 8) reconstruction on the germanium (111) surface, for both the T4 and H3 adsorption sites. The clusters are four atomic layers deep and the positions of atoms in the first three atomic layers are found by minimising the total energy calculated with the MNDO quantum chemical Hamiltonian, according to the prevailing symmetry. Both closed and open shell wavefunctions are considered. We find at least four different electronic states with similar energies but different geometries. The lowest energy is found for the c(2 × 8)H3 configuration with a p-like dangling bond on the rest atom, while for the (2 × 2) structure an s-like dangling bond orbital on the adatom is most favourable.

A molecular cluster model of the V o. center in paratellurite

A RHF-SCF electronic structure calculation has been performed by means of the MELD program package for a cluster model of the V o .-center, a basic radiation-induced defect in paratellurite identified recently by ESR as an oxygen vacancy defect with a net positive charge. A “perfect” Te 2O 7H 6 cluster has been constructed with an effective core potential and an STO-3G valence basis set complemented by a single Gaussian d-function on Te. The charge distribution inside this cluster has been optimized by adjusting positions and nuclear charges of the peripheral “hydrogens”. A Mulliken analysis of a Te 2O 6H 6 defect cluster derived from the perfect one by removing an O − ion and relaxing both Te's showed that the unpaired spin is primarily concentrated on the Te nearest to the oxygen vacancy with appreciable spin densities on nearby atoms and a substantially modified distribution of the surplus charge. The open-shell wave function has also been employed to derive hyperfine constants, taking into account relativistic corrections in independently calculated values of |; ψ 5 s(0) | 2 and 〈 r −3〉 5 p for the Te atom, but neglecting core polarization. The calculated spin populations and hyperfine parameters account for all major features of the ESR spectra.

On the evaluation of matrix elements of symmetry operators in a determinantal basis

A general algebraic theorem is presented to calculate the matrix elements of symmetry operators in a basis of determinantal wavefunctions. It is also shown how the matrix elements of complementary open-shell wavefunctions are related to each other.

Relativistic corrections in a series of helium excited states

Relativistic corrections to order (υ/ c) 2 are applied to the helium excited states 2 1,3P, 3 1,3D,…, 8 1,3K. Simple correlated open-shell wavefunctions are employed and the Breit operators H 1 through H 5 treated as perturbations. Account is also taken of mass polarization and lowest-order one electron Lamb shift. The energies thus calculated agree with experiment to within 2.2 cm −1 or better.

Ligand Ordering Parameters

Lanthanide and actinide single-ligand crystal field parameters are shown to define similar spectrochemical series to the cubic parameter for 3d ions, providing direct evidence that the dominant electronic processes which cause crystal field splittings are the same for all open-shell ions. The contrast between the nephelauxetic series and the spectrochemical series and the existence of a strong empirical correlation between the former and ligand polarizability show that the nephelauxetic series cannot be determined by the admixture of ligand and open-shell wavefunctions as is often assumed. The influence of ligand polarizability on various spectroscopic parameters is then discussed.

Spin-Dependent Operators in the Occupation-Branching Number Representation

In two preceding papers, a general method was given of building up open-shell wavefunctions which are eigen-functions of the spin operators S2. This method leads to a second quantizationtype formalism, where degenerate multiplets are described in what we call the occupation-branching number representation. In the paper referred to as Paper II, equivalent monoelectronic and bielectronic operators were defined and a diagrammatic technique was developed. In the present paper, the equivalent spin-dependent operators in the occupation-branching number representation are built up, in the cases of monoelectronic and bielectronic operators. The spin-orbit and the spin-spin coupling are the most important operators of this kind. The associated diagrammatic representation is given.

Variational Approach to Perturbation Theory. I. Application to the Calculation of Atomic Polarizabilities

A variational approach to perturbation theory is derived which establishes the relationship between variational and conventional approaches that utilize expansions in terms of atomic or molecular eigenfunctions. This approach is based upon the observation that the variational approach to perturbation theory is formally equivalent to a variational method for constructing a special set of excited-state wavefunctions. A mathematical demonstration of this equivalence has been formulated. With the use of this new approach to perturbation theory, we have calculated both the static and dynamic polarizabilities of the two-electron helium isoelectronic series, H−, He, and Li+ and a four-electron system represented by the beryllium atom. Multiconfiguration open-shell wavefunctions were employed. These flexible wavefunctions permit a quantitative analysis of the near-degenerate s—p orbital mixing in the ground state of the beryllium atom. The inclusion of this orbital degeneracy effect results in a substantial reduction in the calculated polarizability.

Hyperfine Splitting of the Ground State of Lithium-like Ions

The hyperfine splitting of the 1s2 2s 2S states of the three-electron systems is calculated using the simplest form of open-shell wave functions. The result for lithium is in good agreement with the very accurate calculation of Nesbet and with experiment. The values for the isoelectronic sequence can be closely represented by an expansion in powers of Z-1 of the form = a03{0.5Z3 -1.472Z2+0.838Z+0.616} where f is the operator introduced by Nesbet.

Open-Shell Calculations of Hyperfine Splitting for the Ground State of Li-Like Ions

The hyperfine splitting of the ground state of lithium is calculated using analytic open-shell wavefunctions. The difference in the experimental and calculated value for the Li7 nucleus by this method is 36.468 Mc/sec as compared to a difference of 57.011 Mc/sec using Hartree-Fock wavefunctions. The nuclear-spin-independent contribution to the hyperfine splitting is computed for the Li-like ions He— to Ne7+ using both open- and closed-shell wavefunctions, and the results are discussed with reference to contributions from the orbital and core electrons.