*From*: "Moshe Olshansky" <moshe_o |-at-| VNET.IBM.COM>*Subject*: combining basis sets - an addition*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: 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 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ --- Bureaucracy is a challenge to be conquered with a righteous attitude, a tolerance for stupidity, and a bulldozer when necessary -- Peter's Law 15. * Per-Ola Norrby * The Royal Danish School of Pharmacy, Dept. of Med. Chem. * Universitetsparken 2, DK 2100 Copenhagen, Denmark * tel. +45-35376777-506, +45-35370850 fax +45-35372209 * Internet: peon |-at-| medchem.dfh.dk, peo |-at-| compchem.dfh.dk ========================================================================= 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