Local Multipole Expansions in MD Simulations

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs) 

As proposed by Marzari and Vanderbilt [219], an intuitive solution to the problem of the non-uniqueness of the unitary transformed orbitals is to require that the total spread of the localized function should be minimal. The Marzari-Vanderbilt scheme is based on recent advances in the formulation of a theory of electronic polarization [220, 221]. By analyzing quantities such as changes in the spread (second moment) or the location of the center of charge of the MLWFs, it is possible to learn about the chemical nature of a given system. In particular the charge centers of the MLWFs are of interest, as they provide a classical correspondence to the location of an electron or electron pair. 

For the analysis of a supramolecular assembly it is most convenient to write the total dipole moment M to a good approximation as a sum of individual molecular dipoles nf, 

The expectation value r, of the position operator for a MLWF i is thus is often called a Wannier function s center (WFC). With this definition the electronic part of the supercell dipole moment reads 

We now discuss a recently developed method to derive atomic charges from WFCs [225]. This method is closely related to the D-RESP procedure of the Roethlisberger group [226]. We consider a molecule of M atoms with charges Za and atomic positions R. The electronic distribution of the molecule is described by n WFCs with charges — at positions r. has a value of one for the spin polarized case and a value of two for spin restricted calculations. The electrostatic potential of the molecule derived from the WFCs is defined as 

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