Determination of Total Energy Tight Binding Parameters from First Principles Calculations Using Adaptive Simulated Annea
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Determination of Total Energy Tight Binding Parameters from First Principles Calculations Using Adaptive Simulated Annealing Anders G. Froseth, Peter Derlet1 and Ragnvald Hoier Department of Physics, NTNU, Trondheim, NORWAY 1 Present address: Paul Scherrer Institute, Nano-Crystalline Materials Group, Villigen, SWITZERLAND
ABSTRACT Empirical Total Energy Tight Binding (TETB) has proven to be a fast and accurate method for calculating materials properties for various system, including bulk, surface and amorphous structures. The determination of the tight binding parameters from first-principles results is a multivariate, non-linear optimization problem with multiple local minima. Simulated annealing is an optimization method which is flexible and ”guaranteed” to find a global minimum, opposed to classical methods like non-linear least squares algorithms. As an example results are presented for a nonorthogonal s,p parameterization for Silicon based on the NRL tight binding formalism. INTRODUCTION In recent years the tight binding approach has become increasingly popular for use in atomic structure calculations. This is because the tight binding approach is much less expensive computationally than common ab initio whitout sacrificing too much in accuracy. The tight binding Hamiltonians are often found by fitting the Slater-Koster integrals to a bandstructure and energy database produced by ab initio calculations. This approach can be made very accurate. But fitting the parameters is often a very lengthy business. One often needs a very large parameter space with multiple local minima for the cost function. This especially becomes a problem when developing models for systems with more than one atom type. One way to proceed would be to implement a global optimization algorithm that can handle any set of complex constraints at a moderate cost of computer time.
TOTAL ENERGY TIGHT BINDING The empirical, total energy tight binding model developed by Cohen et al.[1] for the study of transition metals has proven successful for the description of broad range of elements [1,2,3]. One of the main features of this model is the absence of the common repulsive pair potential term in the total energy expression. Instead, the onsite terms of the Hamiltonian are made environment dependent. One therefore avoids the problem of fitting the bandstructure and total energy separately, as the total energy now is equivalent to the bandstructure energy.
S7.4.1
The secular equation for the general tight binding Hamiltonian can be written ( H − εS ) Ψ = 0 ,
(1)
where H is the Hamiltonian matrix, S is the overlap matrix, and Ψ is the eigenstate vector. Employing the Slater-Koster two-center approximation[4], each of these matrix elements can be written as a linear combination of Slater-Koster(SK) parameters. The SK parameters are given the form of a linear combination of Gaussians : Pγ ( R ) = (eγ + f γ R + g γ R 2 ) exp( − hγ2 R ) FC ( R ) ,
(2)
where γ=1..10 for an s,p,d atomic basis. R is the interatomic distance and FC is a cut-off fun
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