Hamiltonian embedded-cluster

It is also possible to include in addition to the ionic part an additional external potential Vg t in the Hamiltonian in Eq. (11) in the self-consistent procedure. This could be the long-range static potential from a surrounding crystalline environment, as used in the embedded cluster method. Fig. 7. Clusters are here used to model a small section of an infinite solid or surface. 

To generalize the procedures of Sections 5.2 and 5.3.1 to the liquid phase, one can start from the full microscopic description of the system. The Hamiltonian for the whole system is partitioned into a gas-phase component, as given in Eq. (1), for a reactive embedded molecule or embedded cluster (note an embedded cluster is often called a supermolecule) in the absence of the solvent, and a solvent component that includes coupling between the solvent and reactive subsystem ... 

Visser O, Visscher L, Aerts P J C and Nieuwpoort W C 1992 Molecular open shell configuration interaction calculations using the Dirac-Coulomb Hamiltonian the f -manifold of an embedded cluster J. Chem. Phys. 96 2910-19... 

Summarising, the valence-only spin-orbit CGWB-AIMP embedded cluster Hamiltonian, which results from choosing the CGWB-AIMP for the isolated cluster, reads ... 

This is the Hamiltonian used in the calculations presented in Section 3. Simple changes in it, which are explicit in 2.1.2, would lead to a DKH-AIMP embedded cluster Hamiltonian. Embedding AIMPs have been obtained for a number of halide and oxide crystals [37]. AIMP embedded cluster calculations can readily be performed with the MOLCAS code [15]. 

Calculated spectroscopic constants of the Sf and 6d manifolds of Cs2ZrCl6 (PaCl6) corresponding to the CGWB-AIMP spin-orbit embedded cluster Hamiltonian. Available experimental data are shown in parentheses. ... 

Finally, algorithms have been developed which incorporate electron correlation effects explicitly in wave function based band theory for crystalline solids [16, 17]. These algorithms construct the many-electron Hamiltonian matrix for a periodic system by extracting the matrix elements from calculations on finite embedded clusters. In this way the incorporation of correlation effects leads to many-electron energy bands, not only associated with hole states and added-electron states but also with excited states. More recently, Pisani and co-workers [18] introduced a post-Hartree-Fock program based on periodic local second order Mpller-Plesset perturbation theory. 

The embedded-cluster results for water-adsorption energies are given in Table 11.13. This table demonstrates the dependence of the embedded-cluster results on the cluster size, basis set and Hamiltonian used. When the size of the cluster is increased, the energy difference between the associative and dissociative adsorption mechanisms decreases. 

The methods of choice must be adequate for manifolds of electronic states that are localized around a lanthanide ion in a solid host. The combination of a solid environment, a heavy element, and 4/, 5d, and other open-shells, demands the consideration of the effects of the solid host, the use of relativistic Hamiltonians up to spin-orbit coupling, the correct treatment of static and dynamic correlation, and handling large manifolds of quasi-degenerate excited states. We decided to use embedded-cluster wavefunction theory-based (EC-WFT) methods, with a two-component relativistic Hamiltonian to be used in two-steps, a multi-configurational variational treatment of static correlation, and a multireference second-order perturbation theory treatment of dynamic correlation. 

Since the pioneering cluster calculation on the KNiFs solid of Shulman and Sugano [6] there has been a wide variety of proposals of procedures to handle relatively localized electronic states of a solid with a molecule-like Hamiltonian that includes the relevant solid host effects, depending on the type of solids and on methodological flavors (Green s functions, wavefunctions, density functional, etc.). A recent summary of practical methods can be found in Huang and Carter [7]. Here we describe our choice of embedded-cluster method, particularly useful in ionic materials. 

Our choices of relativistic isolated cluster Hamiltonian allow us to write the embedded-cluster Hamiltonian of equation 9.1 as a sum of spin-free and spin-orbit coupling Hamiltonians,... 

With this embedding technique, one selects a representative cluster representing a solid, and embeds the cluster in a surrounding portion of the crystalline lattice, usually consisting of several coordination shells. The corresponding Hamiltonian reads... 

Clusters are studied in several forms. A study of the ionization energy and electron affinity of a metal cluster in the stabilized jellium model was recently performed by Sidl et al. [83]. A strictly variational procedure for cluster embedding, based on the extended subspace approach, has been presented by Gutdeutsch, Birkenheuer, and R6sch[84]. Initially used with the tight-binding model Hamiltonians, it has the potential to be extended to real Hamiltonians. 

Taking /SEk instead of Ek only amounts to choose the ground state as origin for the energy. Llusar et al showed by calculations of a MgO-embedded (Ni06) cluster, that the effective Hamiltonian technique applied on the spin-free part of the Hamiltonian, significantly improves the spin-orbit splitting. 

Electron spin resonance (ESR) has sometimes been used to characterize electronic and structural properties of transition-metal clusters embedded in frozen rare-gas matrices. Neglecting the spin-orbit coupling, the interaction between electrons and the nuclear magnetic moment of each atom in the cluster can be expressed by the simple Hamiltonian [116, 117] ... 

O. Visser, L. Visscher, P. J. C. Aerts, and W. C. Nieuwpoort, /. Chem. Phys., 96, 2910 (1992). Molecular Open Shell Configuration Interaction Calculations Using the Dirac-Coulomb Hamiltonian The f -Manifold of an Embedded Europate Anion (EuO ) Cluster. 

The methodology should allow a treatment of the embedding problem, that is to say, the study of a finite cluster of impurities embedded in a host lattice. Of particular importance in this context are the values of the Hamiltonian parameters that describe the coupling of the impurity cluster to the host medimn. 

The difficulties are multiplied when extra conditions of translational invariance are placed upon the self energy obtained through the treatment of a finite cluster in an effective medium [12]. Now, non-analytic behavior can arise. All objections raised above regarding the treatment of embedding remain valid in the present case. Finally, and quite importantly, one would need to average the elements of the self-energy with elements of the Hamiltonian of the system in an ill-defined procedure given that the Hermitian character of the latter is absent in the former. 

Here E is the so-called self-energy matrix and do is a matrix formed from the QP (correlation-corrected) one-electron energies. (For the derivation see Ladik. ) The Green s matrix can express the perturbation caused in the 0-th order Hamiltonian by taking into account correlation, the electron-hole interaction in excited (excitonic) states of extended systems, the perturbation effect of a cluster of impurities embedded in a periodic system, and so on. 

The isolated cluster Hamiltonian can be either the non-relativistic many-electron Hamiltonian or a suitable relativistic choice, both in their all-electron versions or in any effective core potential version. We will discuss later. The embedding potential acting on the cluster electrons reads ... 

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