Use Of Linear Combination Research Articles

Extended bond orbital theory of piezoelectric and transverse charges

Extended bond orbitals are used to calculate the transverse charge and piezoelectric charge in cubic semiconductors. For both optical mode splitting and piezoelectricity, it gives improved agreement with experiment over the bond orbital model. It also clarifies the significance of the dimensionless quadrupole $\ensuremath{\Delta}{Q}_{\mathrm{xyz}}$. The success of this calculation gives additional support for the use of linear combination of atomic orbitals theory with universal parameters.

The correlation of solvent effects on physical and chemical properties

A comprehensive study of the existing empirical solvent parameters has been made covering spectroscopic, rate, and equilibrium data. Various multiparameter equations have been tested by use of linear combinations of existing empirical solvent parameters. The most generally successful treatment combines Dimroth's E values with dielectric and index of refraction functions. The results of these correlations are discussed with respect to the various solvent–solute interaction mechanisms.

Gaussian Basis SCF Calculations for OH−, H2O, NH3, and CH4

Extensive SCF LCAO MO calculations utilizing comparable Gaussian basis sets for OH−, H2O, NH3, and CH4 give SCF energies of −75.377, −76.034, −56.201, and −40.198 a.u., respectively. An ad hominem assumption of an error of 0.03 a.u. due to an incomplete basis set in each calculation leads to estimates of −75.41, −76.06, −56.23, and −40.23 a.u., respectively, for the Hartree—Fock limits, and of −0.39, −0.38, −0.33, and −0.28 a.u., respectively, for the correlation energies of these molecules. Details of the use of linear combinations of Gaussian functions as bases in molecular calculations are presented. The results show that considerable computation time can be saved with little sacrifice in accuracy for well-chosen combinations.

Verfahren zur programmgesteuerten Berechnung der Eigenwerte eindimensionaler Molekel‐Modelle

AbstractA method for calculating the eigenvalues of a particle moving in a one‐dimensional potential, given as V(ξ) = υ1 ξ + υ2ξ2 + υ3ξ3 +… is described. It is a variational method which makes use of linear combinations of HERMITE orthogonal functions \documentclass{article}\pagestyle{empty}\begin{document}$ \Phi (\xi) = \sum\limits_{n = 0}^{N - 1} {c_n u_n } (\xi) $\end{document}, the recursion properties of which allow for the calculation of the matrix elements Hnm and Snm in closed form without involving any further approximation. As the matrix H = (Hnm) is a band matrix, the corresponding eigenvalue problem can be solved by applying the LR‐transformation, which yields the eigenvalues in the order of their stability, so that the calculation may be stopped after the required number of the lowest levels has been calculated.

On the theory of beta-radioactivity I the use of linear combinations of invariants in the interaction Hamiltonian

The theory of β-radioactivity, based on the use of a linear combination of all five relativistic invariants, is developed for allowed transitions. The interaction Hamiltonian is taken as H β = GΣ k=1 5 C k J k , where the C k are coefficients that determine the combination of the invariants J k with 1 = scalar, 2 = vector, 3 = tensor, 4 = axial vector and 5 = pseudoscalar. The formula for the shape of β + and β −-spectra is calculated, taking into account the nuclear charge Z ( E energy of the electron; P and q are the momenta of the electron and the neutrino respectively): ▪ The probability for K-capture becomes for this interaction: ▪ The angular correlation between electron and neutrino is determined by (neglecting the influence of the nuclear charge): ▪ A general principle, viz. complete symmetry for the processes of β + and β −-emission, is proposed, which has as a consequence that only two types of combinations can exist: a) combinations of the invariants 1, 4 and 5, b) combinations of the invariants 2 and 3. We then obtain the result that in both cases the 1/ E-term in the β-spectrum drops out. Some considerations are given on the comparison of theory and experiment. Thus it is discussed that recoil-experiments with 6He are not yet sufficient to decide if the interaction is given by a “pure” invariant or by a linear combination. They should be completed by recoil-experiments with other nuclei e.g. 19Ne.