Mass polarisation term

The first term in (1.12) represents the kinetic energy due to translation of the whole molecule through space this motion can be separated off rigorously in the absence of external fields. In the second term, /i is the reduced nuclear mass, M M2/(Mi + M2), and this term represents the kinetic energy of the nuclei. The third term describes the kinetic energy of the electrons and the last term is a correction term, known as the mass polarisation term. The transformation is described in detail in chapter 2 and appendix 2.1. An alternative expression equivalent to (1.12) is obtained by writing the momentum operators in terms of the Laplace operators,... 

The above terms represent the kinetic energy due to translation, the kinetic energy of the nuclei, the kinetic energy of the electrons, and finally a correction term, commonly known as the mass polarisation term. 

H[ is the operator of the relative kinetic energy of the nuclei, H is a correction to the kinetic energy of the electrons, is a mass polarisation correction and // denotes the reduced mass of the nuclei. The explicit expression for H R) in terms of elliptic coordinates is given in Ref(Kolos and Wolniewicz, 1964). In the BO approximation the term is neglected. 

The last two terms in (3.267) represent mass polarisation corrections to the spin-orbit and orbit orbit interactions respectively. From equations (2.73) and (3.250), the term Kspin.nuci is given by... 

The first term consists of both the kinetic energy of the electrons and the complete Coulomb energy, the second is the vibrational kinetic energy, the third is the rotational kinetic energy, the fourth is the mass polarisation energy involving the total linear momentum of the electrons P = Pi > the fifth and sixth are the parts of the electronic... 

This situation, despite the fact that reliability is increasing, is very undesirable. A considerable effort will be needed to revise the shape of the potential functions such that transferability is greatly enhanced and the number of atom types can be reduced. After all, there is only one type of carbon it has mass 12 and charge 6 and that is all that matters. What is obviously most needed is to incorporate essential many-body interactions in a proper way. In all present non-polarisable force fields many-body interactions are incorporated in an average way into pair-additive terms. In general, errors in one term are compensated by parameter adjustments in other terms, and the resulting force field is only valid for a limited range of environments. 

This result is remarkably simple as compared to the usual methods. For a spin-polarised potential V, Kraft, Oppeneer, Antonov and Eschrig (1995) used the elimination method and found the corrections as a sum of 9 terms, which is equivalent to our Eq.(ll). They notice that three terms of their sum have a known physical meaning (spin-orbit, Darwin and mass-velocity corrections), but the other terms have no special name . 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

In this, M is the molecular mass, p is density, is Avogadro s number, cq is the permittivity of free space, is the relative permittivity/dielectric constant and a is the molecular polarisability. For a full discussion of the dielectric behaviour of polymer-based materials, reference to the excellent works by Kremer (2003) and Jonscher (1983) is highly recommended. Nevertheless, simplistically, permittivity can be thought of in terms of the number and nature of the polarisable species present in the system, plus their dynamics. Since the dielectric response of a given moiety is affected by its environment, dielectric spectroscopy can provide local structural information. In practice, the relative permittivity of a material is a complex quantity ... 

Electrode-level models describe the performance of SOFC electrodes in detail. They take into account the distribution of species concentrations, electric potential, current, and even temperature in the electrode. Their purpose is to (i) interpret the performance (polarisation curve) of electrodes in terms of rate-limiting resistances such as kinetic (activation), mass transfer, and ohmic resistance and (ii) predict the local polarisation in full-scale cell and stack models. 

In a similar manner, the species mass balance equations, Eqs. (1), (2), may be coupled with the electrochemical rate at each point of the reaction zone (at or near the TPB). In the continuum-level modelling discussed in Section 11.2, the concentration polarisation of the electrode, riconc was related to a limiting current of the reactant, e.g., Eq. (9). A more fundamental and general expression for the concentration overpotential (the term overpotential denotes exclusively the local polarisation) at any point of the electrode reaction zone is the so-called Nernst equation for example... 

If the reaction kinetics of the electrode is assumed to be very rapid, mass transfer and ohmic resistance are the dominant resistances. Assuming a reaction zone that coincides with the electrode-electrolyte interface, the diffusion fluxes in stationary operation can be expressed simply in terms of bulk gas partial pressures and gas-phase diffusivities. This is illustrated schematically in Figure 11.8, which compares anode- and cathode-supported cell designs for the simple case of a H2/O2 fuel cell. The decrease in concentration polarisation at the cathode, rjcc- is obvious in the case of an anode-supported cell, while the model shows that concentration polarisation at the anode, tiac is relatively insensitive to anode thickness. The advantage of the mass transfer-based approach is that analytical expressions are obtained for the polarisation behaviour. These are rather simple if activation overpotential is excluded but may still become elaborate in the case of an internally reforming anode where a number of reactions (discussed in Section 11.3) may occur simultaneously within the pores of the anode. 

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