Basis sets superposition error

1 Basis Set Superposition Error Since in practice, basis sets must be of some limited size far short of the HF limit, their incompleteness can lead to a spurious result known as basis set superposition error (BSSE). This is readily grasped in the context of the binding of two molecules, A and B, to form the complex AB. The binding energy is evaluated as 

The counterpoise method proposed by Boys and Bemardi attempts to remove some of the effect of BSSE. The counterpoise correction is defined as 

BSSE can, in principle, exist in any situation, including within a single molecule. There are two approaches toward removing this intramolecular BSSE. Asturiol et al. propose an extension of the standard counterpoise correction Divide the molecule into small fragments and apply the counterpoise correction to these fragments. For benzene, as an example, one can use C-H or (CH)2 fragments. 

Jensen s approach to remove intramolecular BSSE is to define the atomic counterpoise correction as 

Counterpoise correction should, in theory, be unnecessary for large basis 

Text books containing detailed basis set discnssions are 

Foresman, A. Frisch, Exploring Chemistry with Electronic Structure Methods Second Edition Gaussian, Pittsburgh (1996). 

Leach Molecular Modelling Principles and Applications Longman, Essex (1996). 

Levine, Quantum Chemistry Eourth Edition Prentice Hall, Englewood Cliffs (1991). W. J. Hehre, L. Radom, P. v. R. Schleyer, J. A. Pople, Ah Initio Molecular Orbital Theory John Wiley Sons, New York (1986). 

Evaluation of the ab initio interaction energy is substantially complicated by the so-called basis set superposition error (BSSE) [57]. This artifact arises from the final size of the basis set and is to be eliminated by the full counterpoise procedure (CP) [58]. The monomer energies are evaluated with the full basis set used for the dimer in the geometry of the complexes (designated E, E ). Then the equation (1) is modified as follows  

Elimination of BSSE in a trimer requires to carry out all calculations in the trimer-centered basis set. 

Since the CP correction provides an exact value of BSSE (it is not an estimate [57]) it is fully justified to make calculations even when this correction is large compared to the interaction energies. Such an effect has been noticed in correlated calculations of stacked clusters with medium sized diffuse basis sets. 

The corrections to be added to the B bond lengths to convert them to BC bond lengths are as foUows (A) C-H 0.0056 N-H 0.0079 C-N -0.0032 C-0 -0.0077, C-F -0.0063 C-Cl -0.0146 C-Br -0.0124 C=0 -0.0116. These values were not very accurately known in 1990 when this MM4 work was begun, but this is the way they have been fit. If one is to seriously read this book, it will be helpful to remember these three pieces of jargon, B, BC, and MM40. 

At relatively short distances where orbital overlap between the two molecules is significant, a special problan arises if one is using small (ordinary) basis sets. Because a molecule is being described with a too small basis set, it will tend to make excessive use of the other molecules empty orbitals as places into which to delocalize electrons. Of course, it should do this to some extent, but it does it to too great an extent when its own orbital description is incomplete. 

Here we must ask two questions with respect to any case that is under examination Is this effect sufficiently significant that we need to worry about it And, a second question (if the answer to the first question is affirmative) is How should we correct for this  

For an approximate overview, the stabilization energy due to basis set superposition error in the case of relatively small basis sets (6-31G ) will likely be of the order of 1 kcal/mol for small molecules containing three or four first-row atoms and increasingly larger for larger molecules. For larger basis sets, the problem tends to be reduced in importance, but it really doesn t go away for basis sets commonly used at present. 

There are many studies on the BSSE. For example, Simon et al. (1999) studied BSSE in systems of water molecules. Investigations of BSSE on hydrogen bonding were conducted by Simon et al. (1996) and Alagona and Ghio (1995). Fuentealba and Simon-Manso (1999) discuss BSSE in atomic clusters. 

Lately, density functional theory or DFT, which is based on the work of Hohenberg and Kohn (1964), has become popular because it is less expensive and handles both electron and exchange correlation satisfactorily. DFT is not one method, but a class of methods that calculate the total electronic energy as a functional of the electron density following the work of Kohn and Sham (1965). 

There can be no cross calculations between the methods, meaning one cannot, for example, take the difference of energies of a minimum calculated by HF and a transition state from a DFT method to obtain an activation energy. Doing so would produce bizarre results. As with the choice of basis sets, one needs to make a decision depending on the merits and appropriateness of the methods on the particular system in consideration. 

Johnson (1994) investigated the performance of different DFT methods. With materials important to the geosciences, Xantheas (1995) and Simon et al. (1999) have compared methods on water clusters, Harris et al. (1997) on iron hydrates, and Bacelo and Ishikawa (1998) on sodium hydrates. Gas phase acidities were investigated by Smith and Radom (1995). Recently, Bak et al. (2000) compared the accuracy in reaction enthalpies and atomization energies of different small systems using several methods and basis sets. 

MO-TST studies often include comparisons of reaction curves using several methods as well as basis sets (e.g., Xiao and Lasaga 1996). 

---

There are a number of other technical details associated with HF and other ah initio methods that are discussed in other chapters. Basis sets and basis set superposition error are discussed in more detail in Chapters 10 and 28. For open-shell systems, additional issues exist spin polarization, symmetry breaking, and spin contamination. These are discussed in Chapter 27. Size-consistency and size-extensivity are discussed in Chapter 26. 

There is also a local MP2 (LMP2) method. LMP2 calculations require less CPU time than MP2 calculations. LMP2 is also less susceptible to basis set superposition error. The price of these improvements is that about 98% of the MP2 energy correction is recovered by LMP2. 

It is a well-known fact that the Hartree-Fock model does not describe bond dissociation correctly. For example, the H2 molecule will dissociate to an H+ and an atom rather than two H atoms as the bond length is increased. Other methods will dissociate to the correct products however, the difference in energy between the molecule and its dissociated parts will not be correct. There are several different reasons for these problems size-consistency, size-extensivity, wave function construction, and basis set superposition error. 

Diffuse functions are those functions with small Gaussian exponents, thus describing the wave function far from the nucleus. It is common to add additional diffuse functions to a basis. The most frequent reason for doing this is to describe orbitals with a large spatial extent, such as the HOMO of an anion or Rydberg orbitals. Adding diffuse functions can also result in a greater tendency to develop basis set superposition error (BSSE), as described later in this chapter. 

BSSE (basis set superposition error) an error introduced when using an incomplete basis set... 

Simon, S., Duran, M., Dannenberg, J. J., 1999, Effect of Basis Set Superposition Error on the Water Dimer Surface Calculated at Hartree-Fock, Mpller-Plesset, and Density Functional Theory Levels , J. Phys. Chem. A, 103, 1640. 

BSSE Basis set superposition error DFT Density functional theory... 

A quantum theory of atoms in molecules insight on the effect of basis set superposition error removal... 

Saebe, S., Tong, W. and Pulay, P. Efficient elimination of basis set superposition errors by the local correlation method Accurate ab initio studies of the water dimer, J. Chem. Phys., 98, 2170-2175. 

Xantheas, S.S. (1996) On the importance ofthe fragment relaxation energy terms in the estimation of the basis set superposition error correction to the intermolecular interaction energy, J. Chem. Phys., 104, 8821-8824. 

---