How to calculate

In order to calculate η(r) for a given chemical system either the wave function or the orbitals (= prerequisite quantity) must be available within some approximation (see section Quantum Mechanics). ELF does not depend on the kind of mathematical representation of the wave function or the orbitals, i.e. whether it is an LCAO or a plane wave expansion or something even more envolved, does not play a role for ELF as far as the first spacial derivative of the prerequisite quantity is continuous. This is an important property of ELF, which signifies that its underlying principle is a physical one, which does not depend on the arbitrary way how to mathematically describe the wave function. Of course it may very well depend on the quality of the physical description (level of approximation) of the system.

So, the general procedure to calculate η(r) for a given system consists of the following two subsequent steps:

1.) Calculation and storage of the prerequisite quantity for the system at the end of the SCF cycle. In principle this can be done with any quantum mechanical program system available.

2.) Calculation of η(r) from the prerequisite quantity. This may either be implemented directly in the quantum mechanical program system itself or it is accomplished by some interface program which is capable to correctly handle the output file(s) with the prerequisite quantity from step 1 and compute η(r) from it. Here we can distinguish between two principally different ways:

a) calculation of η(r) on a discrete grid and plugging the output on a file. This file can be processed in a subsequent step for purposes of graphical representation or topological analysis. It has to contain therefore at least the grid positions with associated η value. The graphical representation can be done with any suitable graphics program package on the market. However the data must be imported into the graphics package and this is done differently for different packages. So, either one has to stick to that/those graphics program package(s) for which the ELF output file was intended or one has to import it "by hand" into the preferred graphics package (for further information, see section Graphical Representation). Similar considerations apply for the topological analysis program package. As for this purpose the demands concerning precision are higher than for graphical representation the grid must be finer than for purely graphical purposes.

b) on-the-fly calculation of η(r) in order to be able to exactly find e.g. the position of critical points and separatrices.

In the following we give a table which lists to the best of our knowledge (!) all quantum mechanical first-principles program packages, from which η(r) can be computed in one or the other way.

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Quantum mechanical program package specification interface program name
ADF molecular systems, DFT DGrid
DeMon molecular systems, DFT  
GAMESS molecular systems, pHF  
GAUSSIAN molecular systems, pHF, DFT DGrid, TopMoD
MOLPRO molecular systems, pHF, DFT DGrid
MOLCAS molecular systems, pHF, DFT DGrid
TURBOMOLE molecular systems, pHF, DFT DGrid
CPMD molecular and crystalline systems, DFT, molecular dynamics implemented
CRYSTAL molecular and crystalline systems, HF, DFT TOPOND
SIESTA molecular and crystalline systems, DFT  
TB-LMTO-ASA crystalline systems, DFT implemented
VASP crystalline systems, DFT, molecular dynamics implemented
to be continued...

DFT: density functional theory
HF & pHF: Hartree-Fock and post-Hartree-Fock methods


An independant class of methods for the calculation of ELF for chemical systems can be clearly separated from the first-principles methods mentioned above due to the direct usage of an experimental requisite quantity. In all those cases published until now the prerequisite quantity used is the electron density ρ(r) (we shall not discuss the problems for a sufficiently accurate determination of ρ(r), which is outside the scope of this page). In order to calculate ELF from the electron density one must either

a) construct an approximate wave function (see, e.g. [KOHOUT1997], [GRIMWOOD2001]) and use the formula given by [BECKE1990], [SAVIN1992A], or

b) use a model functional for the (yet unknown) dependence of the kinetic energy density on the electron density t[ρ(r)] [TSIRELSON2002].

Clearly speaking, these methods should not be regarded as an experimental determination of ELF, as they use at best the same quantum mechanical models as the first-principles methods mentioned above. The only - but decisive difference - is that experimental information is additionally utilized. In electronic structure theory this is called a semiempirical calculation.



1.) Generally, ELF should be calculated from all electrons ("valence" and "core electrons") of the system. There exist cases, where the "ELF" calculated from the "valence electrons" ("valence ELF") is practically identical to the "all-electron ELF", which means that is represents a good approximation to the η(r) of the system. For situations where this is not the case, the "valence ELF" should be abandoned. Clearly speaking, there is no ELF for a subset of occupied orbitals, from which the energetically lower lying ones have been omitted. At best it can be a good approximation to ELF, but in general nobody knows what this quantity is.

2.) As long as the problem permits, use a single Slater determinant description. Beyond this approximation the definition of ELF is not unique. Similarly, for open-shell calculation of η(r) a specification of the formula actually used would be desirable.

3.) For graphical representation a mesh size of 0.1 bohr is very good, while for numerical detailed topological analysis or electron density integration (in the valence region), at least 0.05 bohr mesh size should be used.

4.) For the calculation of electronic basin populations the following hints can be given: a) determine the basin from the best possible approximation to the "all-electron ELF" (i.e. ELF, see point 1.). b) Use the total electron density for the integration. If the valence electron density is used, your result may be dependent on your definition, i.e. is not unique. Furthermore not all of your "valence electrons" are within the valence basin, some "valence electron density" is found in core basins meaning that this amount of electrons is missing in the valence region. This unpleasant feature is just an artefact of the arteficial division of the electron density into core and valence parts. If the total electron density is used for the electron density integration the correct number of valence electrons (given by the Aufbau principle for the isolated atoms, see section Shell Structure) can be expected, except for special bonding effects, as, e.g. the participation of inner shells on chemical bonding [KOHOUT2002]. Care must be taken for numerical integration of the total electron density using an equidistant grid. This kind of a grid can be made suitably fine (see point 3.) for an integration within the valence region. For core regions one very quickly gets arbitray numbers with this kind of a grid. This is an important issue if one wants to determine atomic charges from this procedure. For this isssue, as an approximation the "valence electron density" can be integrated within the basins attributed to one atom. The error made is dependent on the balance between corresponding "core electron density" of this atom leaking out into valence basins of other atoms and vice versa.

last update: 24.04.2006