Connolly-Williams method

In the approach which has come to be known as the Connolly-Williams method (Connolly and Williams, 1983), a systematic inversion of the cluster expansion is effected with the result that the parameters in the effective Hamiltonian are determined explicitly. In this case, the number of energies in the database is the same as the number of undetermined parameters in the cluster expansion. Other strategies include the use of least-squares analysis (Lu et al. 1991) and linear programming methods (Garbulsky and Ceder, 1995) in order to obtain some best choice of parameters. For example, in the least-squares analyses, the number of energies determined in the database exceeds the number of undetermined parameters in the effective Hamiltonian, which are then determined by minimizing a cost function of the form... 

The basic analysis of the Connolly-Williams method may be spelled out as follows. If the effective Hamiltonian is characterized by M - - 1 distinct coupling constants Jq, J, ..., Jm, then we wish to perform M + 1 distinct total energy... 

Ab-initio studies of surface segregation in alloys are based on the Ising-type Hamiltonian, whose parameters are the effective cluster interactions (ECI). The ECIs for alloy surfaces can be determined by various methods, e.g., by the Connolly-Williams inversion scheme , or by the generalized perturbation method (GPM) . The GPM relies on the force theorem , according to which only the band term is mapped onto the Ising Hamiltonian in the bulk case. The case of macroscopically inhomogeneous systems, like disordered surfaces is more complex. The ECIs can be determined on two levels of sophistication ... 

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