combining basis sets - an addition



 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: moshe_o |-at-| vnet.ibm.com
 =========================================================================
 Date: Tue, 28 Nov 95 10:08:10 -0600
 From: smb |-at-| smb.chem.niu.edu (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
 smb |-at-| smb.chem.niu.edu			Fax:   (815)753-4802
 =========================================================================
 Date: Tue, 28 Nov 1995 13:08:57 -0330 (NST)
 From: Uli Salzner <uli |-at-| smaug.physics.mun.ca>
 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
 =========================================================================
 From: gunnj |-at-| CERCA.UMontreal.CA
 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 (gunnj |-at-| cerca.umontreal.ca) | "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."
 =========================================================================
 Date: Tue, 28 Nov 1995 12:20:15 -0500
 From: ryszard |-at-| msi.com (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       |
 | Burlington, MA 01803-5297      | e-mail: ryszard |-at-| msi.com     |
 +--------------------------------------------------------------+
 =========================================================================
 Date: Tue, 28 Nov 1995 14:03:26 -0500 (EST)
 From: "E. Lewars" <elewars |-at-| alchemy.chem.utoronto.ca>
 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
 =========================================================================
 Date: Tue, 28 Nov 1995 11:49:18 -0800
 From: Rene Fournier <fournier |-at-| mail.physics.unlv.edu>
    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                 |  fournier |-at-| physics.unlv.edu  |
  | Department of Physics         |  fournie |-at-| ned1.sims.nrc.ca   |
  | 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: schrecke |-at-|
 zinc.chem.ucalgary.ca
 2500 University Drive N.W.,  Calgary,  Alberta,  Canada,  T2N 1N4
 ==============================================================================
 From: "Victor M. Rosas Garcia" <rosas |-at-|
 irisdav.chem.vt.edu>
 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
 rosas |-at-| irisdav.chem.vt.edu                *  at once astonished at the
 Virginia Tech doesn't necessarily share  *  world and yet at home in it?"
 the opinions you just read.	         *  G. K. Chesterton
 -------------------------------------------------------------------------
 =========================================================================
 Date: Wed, 29 Nov 1995 11:10:00 +0100
 From: peon |-at-| medchem.dfh.dk (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: oppel |-at-| pctc.chemie.uni-erlangen.de
 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
 oppel |-at-| pctc.chemie.uni-erlangen.de