Subject: Re: CCL:ADF: AE-Fit-functions (answer)

From: [email-masked] (Bert te Velde)
Date: Thu, 14 Aug 1997 10:07:24 +0100

>original question:
>
>>Has anybody developed fit-functions for the all-electron-basis
>>to be used in ADF? I ask this question because we would like to
>>do a geometry-optimization in an electric field by hand (coordinate
>>driving) In our opinion it is important to take into account the
>>inner shells too to explain a molecular electrostriction.
>>Are there other opinions?
>
>answer from Christoph Schwittek:
>
>>   Two comments on your question. First,  you can make the fit basis
>>yourself. Any combination of two basis functions contributes to the
>>density, and hence must be represented by the fit basis. For instance,
>>if your basis has two p-sets with coefficients alpha1 and alpha2,
>>then there is a d-type contribution in the density with coefficient
>>alpha1 + alpha2 that must be represented by the fit basis (in fact,
>>there are two more d-type contributions, with coefficients 2*alpha1
>>and 2*alpha2). We had in Calgary an old utility program with ADF
>>that would calculate from a given basis set all the possible combinations
>>for the fit and their respective overlap, so that one can choose a reasonabl=
>e
>>fit basis from that. The program is called "alphas" -- maybe you can
>>obtain it or its successor from Amsterdam. I have occasionally
>>created basis sets with smaller cores, and consequently had to
>>make a new fit basis each time.
>>   Second, as you know, ADF uses numerical integration throughout.
>>While this works well normally, I would be careful with all-electron
>>calculations of heavy atoms because the very steep density near
>>the nucleus might well strain the integration scheme over the limit,
>>at least this was my experience sometimes. Admittedly, I haven't
>>played with the detailed integration parameters here, and one can
>>probably do a lot by just increasing the integration accuracy around
>>the nuclei only.

1. about the numerical integration: ADF2.1 and later versions monitor the
actually used functions in a calculation to determine the integration grid
within the atomic core regions, so the fear for inadequate precision when
using steeper functions is not needed anymore. It might have been for
earlier versions when the program used fixed assumptions about what kind of
steep functions would occur for the different elements of the periodic
table.

2. the generation of all fitfunctions that would be needed according to the
products-of-basisfunctions principle can of course easily be generated by
any simple program. If there are people interested we may upgrade our own
variety of such a program a little so as to include it in the packet as one
of the utility programs.
This is not the end of the story, however, since fitsets generated in this
fashion are often close to linear dependency and one really has to remove
some functions to avoid numerical trouble. In addition, this basis-products
principle applies rigorously to single atoms but is not exact anymore when
considering products of basisfunctions centered on different atoms. The
limiting decaying behaviour is still correct but the true representation of
the involved charge-density has more terms that may play a role closer to
the nuclei and in the bonding region, and in particular components with
higher l-values. Therefore, we ourselves use the products-principle only
for a first set-up of fitsets.
The method we actually use when developing fitsets is to use the
products-of-basisfunctions only to determine the steepest and the most
diffuse fitfunction on each atom(type), for each l-value separately, up to
l=4, and then fill in the 'gap' between steep and diffuse in an
even-tempered way, using an overlap criterion for 'adjacent' fitfunctions
in the sequence to define some kind of 'density' of functions on the range:
the higher this density the more accurate the total set, but also the more
expensive the calculation, (and finally: too many functions will again
result in numerical stability problems due to linear dependency). S-sets
(l=0) are always chosen rather dense, and the numbers of used functions
decrease with increasing l,first because these are less important and
secondly because they imply a rapidly increasing computational effort (each
s-fitfunction has only one "m"-component, each p: 3 components, each d: 5,
etcetera)

3. We are currently looking into some re-investigation project on basis and
fit sets, including the development of fitsets for the all-electron basis
(in first instance  only for the lighter elements). The problem is not so
much to just create a bunch of fitfunctions and type them into the
database, but to do large numbers of test calculations to assess the
precision in practical terms: deviation of energy and other properties from
some limiting value.

Best regards

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Bert te Velde                  SCIENTIFIC COMPUTING & MODELLING
phone: +31-20-4447625          Chemistry Department, Vrije Universiteit
fax:   +31-20-4447643          De Boelelaan 1083
email: [email-masked]      1081 HV Amsterdam; The Netherlands
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