Embedding schemes

The addition of the ECPs to the cluster gives a better representation of the electrostatic potential hence of the electrostatic contribution to the surface bonding. What is still missing from this simplified approach is the polarization of the host crystal induced by an adsorbed species or by the presence of a defect. This effect can be particularly important for charged adsorbates or defects. The polarization, Epoi, induced by a charge on the surrounding lattice can be estimated by means of the classical Born formula [56]  

An alternative, more rigorous approach has been developed in recent years by Pisani and coworkers [65-67]. It is named perturbed cluster method and is based on the EMBED computer program [68]. With this approach, the properties of adsorbates at the surface of MgO have been studied at the HE and MP2 levels. The method relies on the knowledge of the one-electron Green function 

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One of the first cluster embedding schemes was put forth by Ellis and co-workers [172]. They were interested in studying transition metal impurities in NiAl alloys, so they considered a TMAl cluster embedded in a periodic self-consistent crystal field appropriate for bulk p -NiAl. The field was calculated via calculations, as was the cluster itself The idea was to provide a relatively inexpensive alternative to supercell DET calculations. 

Gutdeutsch U, Birkenheuer U, Kruger S and Rdsch N 1997 On cluster embedding schemes based on orbital space partitioning J. Chem. Phys. 106 6020... 

The variety of examples presented here can be seen as good evidence that simulations have become a valuable, partly indispensi-able tool for the study of chemicals in solution. The inclusion of ab initio QM procedures for the calculation of forces in every step of the simulation ensures the necessary accuracy of the simulation results to predict both structural and dynamical data, thus also providing a correct picture of the molecular and supermolecular species formed simultaneously in solution. The recently developed QMCF MD methodology has overcome several of the previous problems of MD simulations, mostly not only because of the possibility to renounce any kind of empirical or fitted solute-solvent potentials, but also because of an improved embedding scheme and the use of actual atom populations for the calculation of Coulombic forces. Besides its universality of application to various chemical compounds it also offers a straightforward way of further improvement and method-inherent quality control by the employment of correlated ab initio methods, although at a price which is not yet affordable with present computational facilities, but should become feasible within a few years. 

The description of the perturbation induced by the medium through a suitable external electric field (i.e. an electrostatic embedding scheme). 

The electrostatic embedding approach appears reasonable in cases of weak interactions, with negligible intermolecular charge transfer, provided that the interactions can be described as some average electric perturbation. By properly modifying the disposition of the point charges more realistic embedding schemes could also be introduced. 

An alternative solution to this problem is to apply the force obtained by Coulomb s law to both the QM and the MM counterparts, but then the embedding of MM charges has to be discarded as otherwise the Coulombic contributions for the QM atoms would be treated twice. On the other hand the improved description of the electron density makes the embedding technique a desirable feature. Therefore, an embedding scheme has to be sought which on the one hand allows the inclusion of the point charges to improve the quantum mechanical description, but on the other hand does not show any inconsistencies in the derivation of forces or with respect to momentum conservation. 

To derive an embedding scheme compatible with an application in a periodic environment the properties of the embedding technique and the QM calculations have to be recalled. Partial charges are included in the Hamiltonian as outlined in the... 

A Type A models are the simplest QM/MM models, using a simple mechanical embedding scheme. The QM/MM interactions are treated exactly as the same interactions would be in a purely classical MM calculation. 

In 51 the author presents a new procedure for constructing efficient embedded modified Runge-Kutta methods for the numerical solution of the Schrodinger equation. The methods of the embedded scheme have algebraic orders five and four. Applications of the new pair to the radial Schrodinger equation and to coupled Schrodinger equations show the efficiency of the approach. 

This constitutes a critical portion of the embedding scheme, leading to what may be called Potential Embedding. The prime in the summation denotes exclusion of the cluster atoms. The external densities are obtained by LDA atomic calculations. To avoid the spurious migration of the cluster electrons towards the external atoms, the attractive potential of the latter is truncated at a certain value, to simulate the Pauli exclusion principle. Typical truncation is at -0.2 hartrees relative to the Fermi energy Ep, with a range of 2.2 — 2.5 a.u. from each atomic nucleus. 

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