Subject: combining basis sets - an addition

From: "Moshe Olshansky" <[email-masked]>
Date: Tue, 5 Dec 95 14:02:34 IST

Dear netters!
About a week ago I asked the following question:

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

P.S. and now I have an additional question:
     I am a mathematician,  not a chemist,  so let me look at the
     basis sets purely mathematically.  If one has a complete (and
     hence necessarily infinite) basis set,  he/she gets a limit
     of Hartree-Fock model.  Otherwise (with limited basis set) one
     gets some approximation to this limit and the more complete the
     basis set is the better is the approximation.  Now assume one
     uses a certain "standard" basis set and gets some result (from
     Hartree-Fock model).  And now we add ANY additional function to
     this basis set.  This does not make the basis set less complete
     and so it should lead to at least as good (or even better) an
     approximation as the original basis did (it is also possible
     to get the original solution by taking that additional function
     with zero coefficient for every electron).
     Is there anything wrong with this statement?

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I want to thank all those who replied.  As I understand,  adding
just any function will indeed lower the model energy but the
problem is that for unbalanced basis set the wave function
with lower energy does necessarily better describe the chemical
properties.
Below are the responses I received.

Moshe Olshansky
e-mail: [email-masked]

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Date: Tue, 28 Nov 95 10:08:10 -0600
From: [email-masked] (Steven Bachrach)

Yes, the addition of an extra function MUST result is an energy equal
or lower than obtained with the smaller basis set.

Having a balanced basis set is only important in two circumstances
(1) when you are far from the HF limit (i.e. using a small basis set)
and need to get as much of the "correct"soultion as possible
(2) when you are trying to do some type of density decomposition in
terms of orbital occupancy.

Realize that one could describe the LiF molecule with simply a huge
number of basis functions centered on just Li. The solution would
be identical to one obtained with a large number of orbitals centered
on both atoms.

Steve

Steven Bachrach				
Department of Chemistry
Northern Illinois University
DeKalb, Il 60115			Phone: (815)753-6863
[email-masked]			Fax:   (815)753-4802

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Date: Tue, 28 Nov 1995 13:08:57 -0330 (NST)
From: Uli Salzner <[email-masked]>
Subject: basis stes

Dear Moshe,
it is  correct that by adding any function to a basis set you get a
mathematically improved representation of the wavefunction. The total energy
will be lower. The danger is that you might introduce physical or chemical
errors by lowering the energy unevenly for the different atoms. The
electrons will then flow to the regions where their energy is lower. Try to
calculate ethane with an STO-3G basis set on one carbon atom and 6-311+G* on the
other carbon and check the charges of both carbons. Mathematically the
wavefunction is certainly more complete than with STO-3G on both carbons but the
calculation will not reproduce the chemical fact that both carbons are
identical.

In a geometry optimization not only the distribution of the electrons but also
the position of the atoms can be influenced by an unbalance basis set. The reason
is that if you use a small basis set, the energy of the molecule gets lower when
the atoms are closer together and can use the functions of the neighbor, thus
improving their basis sets. This can be due to insufficient basis sets on all
atoms or on only one.

If you want to see the effect of using different basis sets on a chemical
problem, you may look at the results we got for the bond angle of CaF2 with
different basis sets: U. Salzner and P. v. R. Schleyer, Chem. Phys. Lett. 1990,
172, p461. Fortunately the problem is normally less severe. CaF2 is an extrem
case because the potential energy surface is v e r y flat.

Bye,
Uli

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From: [email-masked]
Date: Tue, 28 Nov 1995 12:08:52 -0500 (EST)

It depends what you mean by 'better'.  What you describe will lower the
energy, since that is the function you are variationally minimizing.
There is no guarantee that any other function, like the dipole moment for
example, will converge 'smoothly' as you add basis functions.  Furthermore,
the HF approximation is not necessarily very accurate, so approaching that
limit might not be what you intuitively expect as an improvement.

--
 John Gunn ([email-masked]) | "The world will not be free until
 Departement de Chimie / CERCA        | the last king is strangled with
 Universite de Montreal               | the entrails of the last priest."
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Date: Tue, 28 Nov 1995 12:20:15 -0500
From: [email-masked] (Ryszard Czerminski X 217)

Dear Moshe,

I am not chemist either (I am physicist) so let me say
what is my understanding of the concept of "balanced basis set".

Your statement that adding "ANY additional function"
will make results at least as good is definitely true
in the sense that it will produce at least as low total
energy as wave function without it.

On the other hand this is not always the value you are
after. Sometimes you are interested in charge distribution
(dipole moments etc...), sometimes in energy differences
(when studying molecular complexes).

In such cases adding any arbitrary function might make
your results worse not better, because these values
are not covered by variational principle. This is why,
to some extend, generating "well balanced basis set" is
sort of "alchemical art" to me.

As I understand it, this is the question of compromise
between resources (CPU time, memory, etc....) and quality
of results for values not necessarily obtained from
variational principle (in usual formulation it covers
only expected value of the hamiltonian i.e. total
energy of the system).

With unlimited computational resources the whole
idea of well balanced basis sets would be moot.

Best regards,

Ryszard Czerminski
+--------------------------------------------------------------+
| Biosym/Molecular Simulations   | phone : (617)229-8875 x 217 |
| 16 New England Executive Park, | fax   : (617)229-9899       |
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Date: Tue, 28 Nov 1995 14:03:26 -0500 (EST)
From: "E. Lewars" <[email-masked]>

Hello,   This is a comment on your query as to whether increasing the size
of a basis set should always improve the calculated results.  In practice
it does not always give better results (see Warren Hehre, "Practical
Methods for Electronic Structure Calculations", Wavefunction Inc., 1995).
As a mathematician, you realize we are trying to span an infinite-dimentional
vector space with a finite basis set, and it seems that as this set gets
bigger the results should get better.  However, there is no guarantee that
the approach to perfection is smoothly asymptotic; it may oscillate.  And
in fact a bigger basis set *can* lead to worse results.
Best Wishes
Errol
E. Lewars
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Date: Tue, 28 Nov 1995 11:49:18 -0800
From: Rene Fournier <[email-masked]>

   Hello ;

   What you wrote is correct in a sense, but one has to be careful about
what is meant by "good" or "better".  By virtue of the "variational principle",
adding ANY basis function will lower the total Hartree-Fock energy (the energy
required to pull to infinity all electrons while keeping the nuclei fixed)
and it will get closer to the true total energy.  In that sense, results are
better.  However we are always interested in energy DIFFERENCES, NOT the total
energy.  Say you underestimate the bond energy of a diatomic AB with a certain
basis set and when you add certain functions the energies of A, B, and AB all
go down, by 0.1 eV, 0.2 eV, and 0.25 eV respectively.  You have better total
energies for each of A, B and AB but the dissociation energy is smaller by
0.05 eV and worst than the original one.  I think this situation is common
with small or medium size basis sets, and not only for dissociation energies
but for all properties related to energy differences: ionization potentials,
electron affinities, excitation energies, barriers to reaction, harmonic
vibrational frequencies, energy differences between conformers, equilibrium
geometries.
   If one judges the quality of results with respect to experiment
the issue becomes even more cloudy.  If a limit Hartree-Fock calculation
overestimates a bond length by 0.10 Angstrom, then using a certain grossly
incomplete basis set might bring the Hartree-Fock calculation in perfect
agreement with experiment but in error by 0.10 Angstrom from the complete
basis set result.  Is that good or bad ?   It is good in the context of an
empirical approach that works systematically, but it is bad if one has an
"ab initio" approach.  For example, scaled harmonic frequencies calculated
by Hartree-Fock with some small basis are "good" from an empirical point
of view.
   My overall impression from the quantum chemistry literature is this.
When using a small or medium basis set, adding basis functions can worsen
energy differences almost as likely as they can improve it, unless one uses
"chemical intuition" to choose precisely what basis functions to add.  When
using very large basis sets, I think that adding more basis functions almost
always improves all results (or leaves them unchanged).

      Here is a humoristic illustration of this.  This graph pretends to
show the error on a typical property measured relative to experiment as a
function of the level of theory and the "3 zero-error regions" where quantum
chemists try to work: "Pauling's", "HF/6-31G" (or today we might say
"BLYP//6-31G" ?!), and the "really good calculation".  A similar graph
may apply also if the x-axis was labeled "basis set size" and the graph
referred only to Hartree-Fock calculations with error measured relative to
limit Hartree-Fock.
( Note: I made this graph from memory from a similar one I saw in a
        lecture by P. O. Lowdin; my apologies for possible inacurracies
        or misrepresentation.  I think it was a very good graph! )

    ^
    |
    |
    |
    x
    x
 E  |x
 r  |x
 r  |x
 o  |x
 r  | x
    | x
    | x
    |  x                                         Really, REALLY tough
    |  x                           x  x          /   fully ab initio
    |  x     Pauling's level    x        x      /      calculation
    |   x     of theory       x              x /
    |   x    /              x                   x
    |   x   /             x                  |--->  x
    |    x /             x                   |--->      x
   0|   |x  |       |   x |                  |                x
----|---|-x-|-------|--x--|------------------|-----------------------x--->
    |0  | x |       | x   |                  |      Level of theory;
    |      x         x   \                          Computational effort
    |       x       x     \
    |        x     x       \
    |          x x           Hartree-Fock 6-31G
    |                         level of theory

    Sincerely,
               Rene Fournier.
 |-------------------------------|-----------------------------|
 | Rene Fournier                 |  [email-masked]  |
 | Department of Physics         |  [email-masked]   |
 | University of Nevada          |  phone : (702) 895 1706     |
 | Las Vegas, NV 89154-4002 USA  |  FAX   : (702) 895 0804     |
 |-------------------------------|-----------------------------|
=========================================================================
Georg Schreckenbach                      Tel: (Canada)-403-220 8204
Department of Chemistry                  FAX: (Canada)-403-289 9488
University of Calgary                    Email: [email-masked]
2500 University Drive N.W.,  Calgary,  Alberta,  Canada,  T2N 1N4
==============================================================================
From: "Victor M. Rosas Garcia" <[email-masked]>
Date: Tue, 28 Nov 1995 23:02:46 -0500

I'm a chemist, not a mathematician, but I like this kind of problems so, here I
go:

I'd say, yes, there is something wrong with the statement.  First I want to
point out what I consider is a small contradiction in your argument, first you
say:

"let me look at the basis sets purely mathematically"

which I understand as dismissing any considerations of "physical meaning". Then
you say:

"If one has a complete (and hence necessarily infinite) basis set,  he/she gets
a limit of Hartree-Fock model."

Now we are considering the basis sets within the frame of a physical model (the
Hartree-Fock approximation) and therefore we are not considering them purely
mathematically.

Having said that, my reasoning is as follows:
Inasmuch as the basis set complies to certain requirements of the physical
model (e.g. a functional form that will "imitate" Slater Type Orbitals), the
addition of ANY function (which in the general case does not comply to those
requirements) will affect negatively the result of the calculation.  I mean, as
far as the Hartree-Fock model is concerned.

just my $0.02

Victor

--
-----------------------------------------------------------------------
Victor M. Rosas Garcia                   * "How can we contrive to be
[email-masked]                *  at once astonished at the
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-------------------------------------------------------------------------
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Date: Wed, 29 Nov 1995 11:10:00 +0100
From: [email-masked] (Per-Ola Norrby)

        It of course depends on what you mean with "better".  The SCF will
minimize the energy, adding any new function should give an energy closer
to the HF limit.  However, most of the time this is not really interesting.
When you want energies, you usually want relative energies, and then it's
quite important that you make the same approximations, that is, calculate
at a constant level of theory, so that systematic errors cancel.  Also, as
you said in the part of the message I deleted, adding functions in an
unbalanced way will definitely affect the charge distribution, probably not
making it "better" :-)  Specific questions can sometimes be answered by
including functions that are not atom-centered, but then you get the
problem of findning a completely reproducable way of doing that for any
system.

        Per-Ola Norrby

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Date: Wed, 29 Nov 1995 13:25:18 +0100 (NFT)
From: [email-masked]

Dear Moshe,

concerning your question about adding any basis-function to an existing
basis-set, I think you are right in principle, that this function
doesn't make the basis worse, i.e., the energie becomes 'better' in the
sense, that it reaches the exact solution. BUT, often the energy is not
the quantity a chemist is interested in. Especially, if one takes a
look at the charges at a certain atom in a system, one uses the so-called
Mulliken-analysis (as you may know), to get this information.
Unfortunatly, this quantity isn't even an obsevable, so one cannot get it
be taking the expectation-value of an hermitian operator. One takes the
difference between the nuclear charge at the atom and the sum of the
diagonal-elements of P*S, which belong to this atom. Now, if you have an
unbalanced basis-set, you will get charges which are far from reality. In
the worst case, think of a complete basis, where all the functions are
centered on a single atom. If you do now a Mulliken-analysis, you will
find no electrons on the other atoms, though the solution of the
HF-equations is exact.
So, take care of your basis-set, and choose it well for the chemical
problem you have to solve.

Markus Oppel
Chair for theoretical chemistry
University of Erlangen-Nuernberg
Germany
[email-masked]