Similarly, attempts have been made to make low-cost methods more accurate, for example by mixing wavefunction theory and density functional theory (DFT). On the other end of the spectrum, several groups have sacrificed some accuracy in order to develop very low cost methods; particularly popular are the semi-empirical DFT extended tight-binding methods. At their root, however, all of these methods rely on a basis set used to expand the unknown wavefunction (or electronic density) in terms of simple, known functions. The usual choice for molecular applications, for reasons of efficiency when calculating integrals, is the Gaussian-type orbital. These are expansions of the single-electron molecular orbitals in terms of atomic Gaussians of varying exponent and angular momentum. While there have been various investigations into, for example, midbond functions, or 'floating' stype orbitals, the usual procedure is to have a basis set associated with each atom type, and duplicate this set on each atomic centre. An assumption is then made that, by increasing the number of such functions, we approach a limit -the 'complete basis set' (CBS) limit -whereby the wavefunction has been represented exactly in the relevant Hilbert space, to within numerical limits. Whether this assumption is valid is so far unproven for most methods, although there is empirical evidence to support it. Unfortunately, basis sets designed in this way approach completeness very slowly. In particular, correlation energy tends to need many high-angular momentum functions, 22 greatly increasing the cost of the method. There are two principle reasons for this: Gaussian basis sets are incapable of accurately representing the wavefunction in the neighbourhood of a nucleus (the Coulomb hole); the exponents and positions of the basis functions are fixed and only adaptable via linear combination. The former problem can be addressed by either using different function types with the correct behaviour at the nucleus (such as Slater-type orbitals), or by adding geminals to the wavefunction that are asymptotically linear in the electron separation (so-called F12 methods). The necessity to fix the positions and exponents, however, comes primarily from a need for calculations to be computationally facile. In an attempt to alleviate this, the exponents for a given atom are carefully optimized on a method-by-method basis, according to criteria that depend on the desired use. Most commonly, they are energy-optimized, but there are situations where optimizing to other properties -such as the density or some 'completeness profile' -might be more useful. Moreover, any method using density-fitting approximations will need (possibly several) auxiliary basis sets, which also need to be optimized to minimize the fitting error. A typical F12 calculation, for example, might need up to four separate basis sets for each atom type.