Convergence rates for the coupling of FEM and collocation BEM

We prove convergence of the coupling of finite and boundaryelements where Galerkin's methd is used for finite elementsand collocation for boundary elements. We consider linear ellipticboundary value problems in two dimensions, in particular problemsin elasticity. The mesh width k of the boundary elements andthe mesh width h of the finite elements are required to satisfykßh with suitable ß. Asymptotic error estimatesin the energy norm and in the L²-norm are derived. Numericalexamples are included.     

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.     

31P and 13C chemical shielding tensors in the phosphoenolpyruvate moiety from rotary resonance recoupling 13C and 31P MAS and single crystal 31P NMR

A ³¹P and ¹³C NMR study of powder and single crystal samples of two phosphoenolpyruvate (PEP) compounds, the tris-ammonium salt monohydrate (NH₄)₃(PEP)·H₂O (1), and the mono-ammonium-salt (NH₄)(H₂PEP) (2) is presented. The P chemical shielding tensors in 1 are measured by ³¹P single crystal NMR on four minuscule samples and assigned without ambiguity by exploiting the orientation-dependent ³¹P-³¹p dipolar splittings of the resonance lines. The orientation of the ³¹P chemical shielding tensor is discussed in terms of the C_2v — and C₃-type distortions of the phosphate PO₄-coordination sphere. From ¹³C MAS NMR experiments with ³¹P rotary resonance recoupling on polycrystalline powder samples the orientations of the ³¹P chemical shielding tensors in 1 and 2 are obtained, for 1 in very good agreement with the ³¹P single crystal NMR results. Only some of the orientational parameters of the three ¹³C chemical shielding tensors in the PEP moiety of 1 could be derived from ¹³C MAS NMR experiments with ³¹P rotary resonance recoupling.     

The computation of Karplus equation coefficients and their components using self-consistent field and second-order polarization propagator methods

The Karplus equation has been investigated by ab initio computation of the spin-spin coupling constants for a series of rotated ethane geometries. The couplings have been calculated at the self-consistent field (SCF) level as well as using the second-order polarization propagator approximation (SOPPA) and the second-order polarization propagator approximation with coupled cluster singles and doubles amplitudes (SOPPA(CCSD)) and have been compared with results of previous calculations. The four principal components of the coupling constants rather than just the Fermi-contact have been calculated, and the common supposition that the Fermi-contact term is totally dominant has been confirmed. The derivatives of the orbital paramagnetic and orbital diamagnetic terms are significant but opposite in sign for the case of this rotation in ethane. It is found that the coefficients in the Karplus equation are largely overestimated at the SCF level, whereas the SOPPA(CCSD) results are in good agreement with coefficients derived from experimental coupling constant data or the results of multi-configurational self-consistent field (MCSCF) calculations. It is further observed that extending the Fourier series in the Karplus equation to include cos(3θ) and cos(4θ) terms neither significantly improves the quality of the fit nor significantly changes the values of the other coefficients. In order to simulate the Abraham and Pachler equation, calculations varying the nuclear charge on hydrogen have been performed. These will allow an abstract but flexible prediction of the effect of electronegative substituents.     

Light induced radical pair intermediates in photosynthetic reaction centres in contact with an observer spin label: spin dynamics and effects on transient EPR spectra

A novel strategy is discussed using site directed spin labelling to study the electron transfer process in photosynthetic reaction centres. An algorithm is presented for numerical simulations of the time resolved EPR spectra of radical pair states in the presence of an observer spin label. This algorithm accounts for spin dynamics, charge recombination and relaxation processes. It is shown that satisfactory agreement between experimental and simulated EPR spectra of the first stabilized radical pair state in photosystem I is achieved for various microwave frequencies. Transient EPR spectra for the radical pair state P^?+Q^?- in photosystem I were simulated for various distances and positions of the observer spin label with respect to the acceptor quinone molecule. It is shown that distances up to more than 20 Å give rise to observable changes in the transient EPR spectra. Both the additional spin-spin coupling between the quinone radical and the label and the polarization transfer processes contribute to the changes. Furthermore, the shape and intensity of the EPR spectrum of the spin label is altered by the coupling with the radical pair spins for distances up to 25 Å. Experiments on site directed spin labelled photosystem I are thus expected to provide valuable information on the dynamics of electron transfer in photosystem I.     

Two avenues to self-interaction correction within Kohn—Sham theory: unitary invariance is the shortcut

The most widely used density functionals for the exchange-correlation energy are inexact for one-electron systems. Their self-interaction errors can be severe in some applications. The problem is not only to correct the self-interaction error, but to do so in a way that will not violate size-consistency and will not go outside the standard Kohn-Sham density functional theory. The solution via the optimized effective potential (OEP) method will be discussed, first for the Perdew-Zunger self-interaction correction (whose performance for molecules is briefly summarized) and then for the more modern self-interaction corrections based upon unitarily invariant indicators of iso-orbital regions. For the latter approaches, the OEP construction is greatly simplified. The kinetic-energy-based iso-orbital indicator T^W σ(r)/Tσ(r) will be discussed and plotted, along with an alternative exchange-based indicator.     

Light induced radical pair intermediates in photosynthetic reaction centres in contact with an observer spin label: spin dynamics and effects on transient EPR spectra

(Molecular Physics, 2002, 100, 1311–1321)