Unified approach to spurious solutions introduced by time discretization Part II: BDF-like methods

It has been proved inter alia in part I of the present paper(Iserles et al., 1991) that irreducible multistep methods forordinary differential equations may possess period-2 solutionsas asymptotic states if and only if (–1)0, where the underlyingmethod is and $$\sigma \left(z\right):={\sum }_{k=0}^{m}{\sigma }_{\kappa}{z}^{k}$$. We provide an alternative proof of that statementand examine in detail properties of methods that obey (–1)=0.By using a variation of the original proof of the first Dahlquistbarrier (Henrici, 1962), we establish an attainable upper boundon the order of zero-stable multistep methods with the aforementionedfeature. Moreover, we modify the concept of backward differentiationformulae (BDF) to require that (–1)=0. A zero-stabilitybound on the ensuing methods is produced by extending the methodof proof in (Hairer & Wanner, 1983).     

The internal coordinate path Hamiltonian; application to methanol and malonaldehyde

The internal coordinate path Hamiltonian is introduced for the study of the vibrations of molecules which have one large amplitude motion. The Hamiltonian is represented in terms of a one path coordinate and 3N—7 normal coordinates. The variational method is used to solve the Schrödinger equation. The molecules studied are methanol and malonaldehyde. For methanol the internal coordinate is a dihedral angle, for malonaldehyde it is the difference in the distances between the migrating hydrogen and the neighbouring oxygen atoms. For methanol there is little coupling between the path and the normal coordinates and so no complications were encountered in the calculations which used harmonic surfaces generated by density functional and M?ller—Plesset theory. Fundamental frequencies were predicted. Malonaldehyde is a different story. There is substantial coupling between the path coordinate and several of the normal coordinates. This introduces many complications: an anharmonic surface is essential and large variational configuration interaction calculations are essential for convergence. Furthermore, because the Coriolis terms require the evaluation of derivatives of both the nuclear coordinates and the normal coordinate eigenvectors along the path, great care must be taken with these numerical procedures. B3LYP predicts too low a transition state which overemphasizes the large Coriolis terms near the transition state. This may be one of the reasons why our fundamental vibrations are in poor agreement with observation. It is most encouraging that the tunnelling splitting is 58 cm^?1 (obs. 21.56 cm^?1), obtained with our quartic density functional surface.     

Theoretical determination of the vibrational levels of NH+ 3 and its isotopomers

The vibrational levels associated with the electronic ground state X²A₂″ of NH⁺ ₃ have been determined up to 5000 cm^?1 by perturbation and variational calculations with full dimensionality of the molecule. For the variational part a new version of MULTIMODE was used which uses the ab initio electronic energy and its first derivative to define the potential energy function. These quantities were generated by the B97-1 density functional and RCCSD(T) approaches. For ND⁺ ₃, ND₂H⁺ and NDH⁺ ₂ the vibrational levels were calculated only by perturbation theory. The rotational constants for all the isotopomers were determined and the first transition dipole moments for NH⁺ ₃ and ND⁺ ₃ were plotted. A critical comparison of the perturbation and variational techniques suggests a possible further modification to the MULTIMODE algorithm for large systems.