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Gaussian basis set in DFT calculation 
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#1
Oct2110, 05:46 PM

P: 45

I'm curious to know why chemists like to use Gaussian basis set in case of an abinitio (ex.DFT) calculation. I understand that the molecules that are of interest to chemists are nonperiodic and hence plane wave basis is not useful, but can't they use other real space basis like a grid? What makes Gaussian orbital so special? Also mathematically does the Gaussian orbitals of different atoms combined together form a complete basis set?



#2
Oct2210, 01:48 AM

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P: 3,593

Mainly because you can do all the integrals analytically.



#3
Oct2210, 12:48 PM

P: 426

DrDu said the main reason. There are lots of integrals.
Additionally, the contracted gauss type orbitals are also very efficient basis sets for representing occupied molecular orbitals. Nothing more efficient is known. A few dozen (or less) CGTO basis functions per atom are sufficient to get relative energies (say, of two molecular conformations) converged to below 1 kJ/mol, and you can easily put in more basis functions to approach that basis limit as close as you wish. Note that already 1kJ/mol is far more accurate than the intrinsic accuracy of DFT methods. Since you only need so few basis functions, you can also use dense linear algebra routines for most of the iterative solution procedures. But of course people regularly come up with shiny new FEM or realspace grid based HF/DFT programs. Only to realize that GTO based programs are much, much more efficient after all. In principle the GTOs form a complete basis (even GTOs on a *single* atom would do that!), but in practice you only really want to represent the space occupied molecular orbitals are likely to span. They are of course not good for representing continuum states etc. 


#4
Oct2210, 03:54 PM

P: 45

Gaussian basis set in DFT calculation
Hi DrDu. I'm still not clear. In DFT the PDE in nonlinear. Now what is the integration you are talking about? Recasting the PDE in functional form? If so can we solve it analytically for a functional of that sort? I can understand your argument for computed say Hartree potential from the poisson eq reformulation. But for DFT?
Hi cgk, thank you very much for the information. I now understand why chemists love GTO. However are you very sure about completeness? I agree with you that the Slater Orbitals/GTO of a single atom is complete to capture the wave function of that electron. For DFT we introduce GTO of different atoms. I'm not sure if the argument is true in this case. Can you refer me to some books/papers where they have proved that Slater Orbitals/GTO is a complete basis for an N electron wave function 


#5
Oct2210, 04:39 PM

P: 426

The integrals are integrals over the Coulomb interaction and over external potentials, kinetic energy operators and so on. They occur in the secondquantized expression for the Hamiltonian:
[tex]H = \sum_{rs} \langle rt+vs\rangle c^r c_s \sum_{rstu} \langle rs1/r_{12}tu\rangle c^r c^s c_u c_t[/tex] where the t/v are kinetic energy/external potential, and the cs are creation/destruction operators (regarding the empty vacuum). The rstu here are the orthogonal molecular orbitals, which need to be expressed in terms of some basis set. Especially the Coulomb integrals can be very numerous, and therefore basis sets are required in which these can be evaluated quickly. For finite elements of any kind this is not the case. Iirc there was a proof of the completeness of the GTO basis sets in "Molecular electronic structure theory" by Helgaker and others. It is also otherwise a very good textbook on that kind of stuff, if you are interested in that topic. 


#6
Oct2310, 07:16 PM

Sci Advisor
P: 1,866

This is a good question (perhaps better than you realize). You really have to view this as two questions, namely 1) Why GTOs are used in wave function methods and 2) Why GTOs are used in DFT methods.
For 1) DrDu's answer is right. It greatly simplifies the integrals. (See for instance Appendix A of Szabo and Ostlund, where they derive the analytical expressions for all the HartreeFock integrals of an stype orbital) Showing that they form a complete set is fairly simple. Handwavingly, you could just remember that the spherical harmonics (which are used for the angular part) form a complete set, and also recall that the eigenfunctions of the harmonic oscillator are gaussians and form a complete set. (Or arrive at the same by solving the 3d QHO) Of course, in reality a basis set is truncated and does not form a complete set, but there do exist complete basis sets (CBSes), such as G2, and also other methods of interpolating to the CBS limit. But it's not a major issue; the larger 'ordinary' basis sets are usually quite enough to bring (relative) basis set errors down within the error of the method. Anyway, so your basis is typically not orthonormal, and so you have to enforce orthonormality through some orthogonalization procedure. (e.g. canonical, symmetric, GramSchmidt) I'm not sure what cgk means with "nothing more efficient is known". I agree nothing more computationally efficient is known, but they're certainly not more efficient 'mathematically', i.e in terms of number of functions required for a given accuracy. For instance, the SlaterType functions which gaussians replaced, were more accurate in that respect. (e.g. for a HF calc on Helium, a single STO is the basisset limit!) It's just that the increased number of basis functions with gaussians was more than compensated for by the faster evaluation of the integrals. From the numberoffunctions standpoint, gaussians are a stupid choice. They're continuous at r=0, and they do not decay exponentially. So they fail to satisfy the few exact properties we know about the true wave function/density. This means it will always take a relatively large number of gaussians for a good approximation, especially due to the nuclear cusp. (since it always requires many continuous functions to correctly approximate a discontinuity) 2) Now, as for DFT, the situation is rather different. The function you are approximating (the density) is of course very similar to the true wave function that the basis sets were created to approximate, so in that respect they're a good choice. Since existing ab initio QC programs had HF/SCF methods implemented, as well as the basis sets, they had much of the code required to do DFT that way once it started getting popular, so to begin with it was just the most convenient choice. But rationale about integrals does not hold as well anymore. It's just as good and fine for the KohnSham one and twoelectron integrals as it is for HartreeFock. But you also need to evaluate the density functional, which means integrals of the form: [tex]\int f(\rho_\alpha, \rho_\beta) dV[/tex] and [tex]\int f(\rho_\alpha, \rho_\beta, \nabla\rho_\alpha, \nabla\rho_\beta) dV[/tex] For LSDA and GGAtype functionals, respectively, where the functional f often depends on [tex]\rho^{\frac{4}{3}}[/tex] and such. These integrals can't be calculated analytically at all, so all DFT codes (to the best of my knowledge) have to do some amount of numerical integration. At the same time you don't (always) have to perform the exchange integrals used in HartreeFock, which allows for different approaches to the Coulomb integral. So the rationale for using GTOs in wave function methods doesn't really hold for DFT. Which isn't to say GTOs don't work  they work very well, better than with wave function methods even, since DFT is more resistant to basis set errors. But the field is more open for trying other approaches. The immediately obvious idea would be to bring back STOs  which was done in the Amsterdam DFT program. Another is to go in the opposite direction and move to fully numerical, FEMtype basis sets, which has been done with e.g DMol. I haven't looked at any recent benchmarks, so I can't comment on the success of these approaches, but at the very least, they're competitive. Bearing in mind that molecular DFT is relatively young (becoming practical circa 1990, I'd say) and that a lot more research effort has been spent on GTO basis sets, I wouldn't categorically state that GTOs will remain the best choice for DFT. On the other hand, absent any remarkable new developments, I think they will remain the standard for wave function methods for the foreseeable future. 


#7
Oct2510, 03:03 AM

P: 426




#8
Oct2610, 02:43 AM

Sci Advisor
P: 1,866




#9
Oct2710, 02:05 AM

P: 426

The Gauss orbitals are, however, not actually constructed to form STOs. The only exceptions are the STO3G/STO6G basis sets and the elements hydrogen and helium, where the AOs actually are Slater functions. But if you use STO3G in an actual calculation of, say, reaction energies, you'll easily see that with this basis set it is quite possible get HF(!!) errors of 500% in the wrong direction. Real basis sets are designed to reproduce atomic AOs. This is most easily seen for the ccpVnZ basis sets. For the first and second row atoms these consist of one contracted function for each occupied shell of an atom (say, for N it would have one contracted 1s, 2s, and 1p function each) and a set of primitive polarization functions which are supposed describe the distortion of the orbitals due to the molecular surroundings. That means: if you actually do an ROHF calculation on N atom, you will get exactly the same HF number with the full ccpVTZ basis set and with the ccpVTZ stripped off all its functions except the three contracted ones (try it!). You will also get exactly the same number if you uncontract the basis set (i.e., have lots of primitive basis functions instead of the three contracted ones), because the contraction coefficients are actually determined as molecular AO coefficients of spherically averaged HartreeFock orbitals. Of course other basis sets are constructed in a somewhat different manner (e.g., ANORCC sets don't contain primitives, but are based on natural orbitals of atoms, cations/anions, and dimers; the def2 basis sets have exponents optimized on atomic HartreeFock and MP2 calculations and rather involved segmented contraction pattersn), but in the end all of them need to be able to represent the atomic parts of the orbitals. 


#10
Oct2710, 11:20 AM

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P: 1,866

Seems we got that misunderstanding straightened out at least. 


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