Stephen R. Langhoff

Abinitio evaluation of the fine structure and radiative lifetime of the 3A2(n→π*) state of formaldehyde

The spin–orbit contribution to the zero‐field splitting (ZFS) in the CH2O 3 A 2(n→π*) state is evaluated using the full Breit–Pauli Hamiltonian. All calculations are carried out at the planar ground state geometry using a double‐zeta plus polarization basis of contracted Gaussian‐lobe functions augmented with diffuse s and p functions. Configuration–interaction wavefunctions, constructed using the 3 A 2 canonical orbitals, are used to describe the 3 A 2 state and all states coupling to it via the spin–orbit Hamiltonian. The excitation energies and oscillator strengths obtained from these wavefunctions are in good agreement with other theoretical calculations and with experiment. Of the 12 states considered in the second‐order perturbation theorytreatment of the spin–orbit interaction, the 1 A 1 ground and the nearby 3 A 1(π→π*) states were the most important. Rydberg states were observed to have very small spin–orbit matrix elements and consequently to have little effect on the ZFS. The spin–orbit contributions to the ZFS parameters D and E were −0.224 and 0.009 cm−1, respectively, which when added to the spin–spin contribution obtained in an earlier paper [S. R. Langhoff, S. T. Elbert, E. R. Davidson, Int. J. Quantum Chem. 7, 999 (1973)] give total values of D=0.314 cm−1 and E=0.04 cm−1. These results are larger than the best experimental results of D=0.141 cm−1 and E=0.02 cm−1, determined by Birss e t a l. [F. W. Birss, R. Y. Dong, and D. A. Ramsay, Chem. Phys. Lett. 18, 11 (1973)] from a rotational analysis of the 0+←0 bands of the 3 A 2←1 A 1 transition. An extensive calculation was also undertaken to assess the degree of convergence in the second‐order perturbation theorytreatment. The contribution of the lowest 100 singlet and triplet states of A 1, B 1, and B 2 symmetry were considered where each state was described by a 100‐term CI wavefunction. This calculation gives a spin–orbit contribution to D of −0.221 cm−1 essentially identical to the previous result providing evidence that the second‐order treatment has indeed converged. The radiative lifetimes of the three sublevels of the triplet state were determined using the same representations for the manifold of electronic states. In the high temperature limit, the radiative lifetime was determined to be between 0.02 and 0.06 sec, somewhat longer than the estimated experimental value of 0.01 sec. The mutual perturbation of the 1 A 1 ground and 3 A 2 states and the perturbation of the 3 A 2 state by the 1 A 1(π→π*) state were determined to be most important in determining the lifetime. These results ensure that the emitted light is polarized primarily along the carbon–oxygen bond in agreement with experiment. A critical examination of the quantitative validity of the numerical results is presented to assess the reliability of the theoretically determined lifetimes.

Ab initio evaluation of the fine structure of the oxygen molecule

The electron spin dipole‐dipole and spin‐orbit contributions to the zero‐field splitting in the O2 3Σg− ground state are calculated using the correct microscopic Hamiltonians. The calculations are carried out using minimum, double‐zeta, and double‐zeta‐plus polarization basis sets constructed from contracted Gaussian‐lobe functions. Configuration‐interaction wavefunctions are used for the ground state and all excited states that couple to the ground state via the spin‐orbit Hamiltonian. Values of 1.790 and 1.928 cm−1 are obtained for the splitting parameter λ for the double‐zeta and double‐zeta‐plus polarization bases, respectively, in good agreement with the accurate experimental value of 1.981 cm−1. The spin‐orbit contribution which comes primarily from the low‐lying excited 1Σg+ state accounts for about 2/3 of the total splitting. It is observed that only states which are single excitations from the ground state have a significant spin‐orbit contribution and that the over‐all effect of electron correla...

Variational calculations of vibrational properties of ozone

A variational method is used to obtain vibrational–rotational properties for ozone from an experimental quartic force field. Band positions, average structures, matrix elements for calculating infrared intensities, and effective rotational constants are reported for 16O3 and its 18O isotopic species. Also, the degree to which the vibrational energies and properties are converged is investigated as a function of the basis set parameters and basis set size, and of the method of obtaining the Hamiltonian matrix elements. A convenient procedure for assigning the vibrational states is developed for variational vibrational wavefunction expansions.

Ab initio study of the zero‐field splitting parameters of 3B1u benzene

The electron spin dipole–dipole contribution to the zero‐field splitting of benzene in its lowest triplet state (3B1u in D6h symmetry) is determined theoretically by ab initio methods. Two hexagonal conformations are considered, distinguished by having carbon–carbon bond lengths appropriate to the 1A1g ground state (1.395 A) and to the 3B1u state (1.427 A). In addition, two distorted forms of D2h symmetry are treated, one having a compressed or ’’quinoidal’’ structure and the other having an elongated structure. All calculations are carried out with a double‐zeta basis of contracted Gaussian‐lobe functions. The correct microscopic spin‐dipole Hamiltonian is used and all integrals are evaluated accurately. The hexagonal conformation with 1.427 A bond lengths gives results in best agreement with experiment. A large configuration‐interaction wavefunction leads to D=0.1676 cm−1, to be compared with D=0.1580 cm−1 obtained from the electron resonance spectra of benzene in a C6D6 host crystal. Both the quinoid a...

Ab initio study of perturbations between the X 1Σ+g and b 3Σ−g states of the C2 molecule

We report ab initio calculations of the energies and wavefunctions for the X 1Σ+g and b 3Σ−g states of C2 at several levels of accuracy. The best calculations, using a double‐zeta basis of Gaussian‐lobe functions and including dσ and dπ functions and extensive CI, give good agreement with experimental potential curves. These wavefunctions are used to compute spin–orbit coupling between the two states. The results agree well with the matrix elements obtained from the perturbations observed by Ballik and Ramsay. Some implications of our results for the theory of intersystem crossing are discussed.

Photoabsorption in formaldehyde

Theoretical investigations employing configuration‐interaction calculations and recently devised moment‐theory techniques are reported of the vertical electronic dipole excitation and ionization spectra in molecular formaldehyde. A double‐zeta basis of contracted Gaussian‐lobe functions, supplemented with appropriate polarization, diffuse, and bond functions, is employed in the construction of Fock spectra in C2v symmetry for X 1A1 and (n→π*)3A2 states near the ground‐state equilibrium geometry. The 50 occupied and virtual Fock orbitals obtained in each case are used in configuration‐interaction calculations of 200‐term eigenvectors of appropriate symmetry for each of the principle‐axis polarization directions, and for the lowest‐lying molecular ionic states. The ionization energies, discrete vertical transition frequencies and oscillator strengths, and associated approximate configurational assignments obtained are in general accord with experimental determinations and with the results of previously repo...

Spin‐orbit contribution to the zero‐field splitting in CH2

The spin‐orbit contribution to the zero‐field splitting in methylene is calculated at bond angles of 135° and 180° using the correct microscopic Hamiltonian. The two lowest states of 1A1 symmetry as well as the lowest states of 1,3A2 and 1,3B2 symmetry are considered in the second‐order perturbation treatment. Configuration‐interaction wavefunctions obtained using a double‐zeta‐plus polarization quality basis of contracted Gaussian‐lobe functions are used to describe each state. Spin‐orbit contributions to the splitting parameter, D, of 0.023 and 0.0245 cm−1 are obtained at 135° and 180°, respectively. Configuration interaction is found to reduce the spin‐orbit contribution at 135° by about 50%. Combining these results with the results for the spin‐spin dipole contribution to D gives D=0.807 cm−1 and E=0.049 cm−1 at 135° which is slightly higher than the current experimental value of D=0.76±0.02 cm−1. The increase in the value of the parameter D from 0.807 cm−1 at 135° to 0.934 cm−1 at 180° is observed to...

Average and directional Compton profiles for the N2, O2, and CH2O molecules. I. Effect of electron correlation

The average Compton profile, and the Compton profiles along directions parallel to the bonds and perpendicular to the plane of each of the N2, O2, and CH2O molecules, were calculated at two levels of approximation. The uncorrelated charge densities were obtained from wavefunctions calculated at the self‐consistent‐field (SCF) level and correlation was introduced by means of configuration interaction (CI). The CI wavefunctions were constructed by accepting all single excitations plus a selected set of double excitations from an initial reference list of configuration accouting for 40%–50% of the correlation energy for each of the molecules. Double‐zeta and double‐zeta‐plus‐polarization quality bases constructed from Gaussian‐lobe functions were used to generate these wavefunctions. The effect of electron–electron correlation on the charge density and its anisotropies was studied by a comparison of the various profiles at the two levels. It is found that this interaction, similar to its effect in atomic sys...

Molecular Fine Structure

Weak interactions in the complete many-body Hamiltonian are an important part of modern chemical theory and experiment.(1) As we have seen in this volume, these terms are usually neglected in the construction of electronic wave functions because they account for relatively small amounts of energy, typically on the order of cm-1. Nevertheless, they can be measured to very high accuracy, such as in EPR experiments that probe the coupling of the electron spins with themselves and with the angular momenta of their orbital motion.(2,3) These particular interactions, which are the subject of this chapter, are so small that they act as perturbations to split the nonrelativistic electronic states into a “fine structure” of levels. Since the spacings between these spin multiplets are often very sensitive to the details of the charge distribution, they provide a test of the zero-order wave functions that are used to calculate them.* We are, therefore, dealing with weak forces that have conspicious, measurable, and calculable effects.

Average and directional Compton profiles for the O2 and CH2O molecules. II. Anisotropy in the momentum charge distribution

The average Compton profiles and the profiles along various directions are obtained for the occupied canonical orbitals of the ground state of the oxygen and formaldehyde molecules. It is demonstrated that the oscillatory nature of the profiles calculated along the bond direction in a homonuclear diatomic molecule is strictly a consequence of molecular symmetry. The information extracted from these profiles give insight into the nature of the molecular orbitals comparable to that obtained from the electron density plots in coordinate space for formaldehyde. The profiles for individual orbitals are then used to explain the anisotropy in the total momentum distribution of the molecules which occurs upon chemical bonding.