Basis extension error

A very important source of error is the basis set superposition error if the basis used for molecule A is inadequate, the virtual orbitals of molecule B may be able to improve the description of A in a way which has nothing to do with the interaction, and this leads to a spurious stabilization. It is conventional to correct for it by means of the functional counterpoise method [22] in which reference calculations are performed for each molecule in the presence of the basis functions, but not the electrons or nuclei, of the other. This procedure overcorrects for the effect, since it makes available to molecule A the occupied space of molecule B as well as the virtual space. It is possible to carry out a reference calculation for A in which the occupied orbitals of B are projected out of the basis[23], but although this gives better results than the normal procedure it is probably too cumbersome for routine use. Note that the function counterpoise method demands a separate reference calculation for each of the interacting molecules at every relative position, since the basis extension error varies with the position of the orbitals of the other molecule, and the procedure is therefore very time-consuming. 

The only satisfactory solution at present appears to be the use of basis sets which are large enough to describe both molecules accurately. This is unfortunately often impracticable, since such a large basis can only be used for small molecules. Even in such cases difficulties remain, since electron correlation must be taken into account if dispersion effects are to be described. It is becoming clear that basis extension error is a much more serious problem in correlated calculations than in SCF ones[24], both because it is larger at a given level of basis and because it is difficult to correct for it in a consistent way[25]. A recent accurate calculation of the interaction between pairs of Be atoms[263 required s, p, d and f basis functions on each atom for a reasonably accurate result omission of the f functions led to a 40 error in the well depth. 

In Section III.E, EOM ionization potentials and electron affinities are compared with accurate configuration interaction (Cl) results for a number of atomic and molecular systems. The same one-electron basis sets are utilized in the EOM and Cl calculations, allowing for the separation of basis set errors from errors caused by approximations made in the solution of the EOM equation. EOM results are reported for various approximations including those for the extensive EOM theory developed in Section II. Section III.F presents results of excitation energy calculations for helium and beryllium to address a number of remaining difficult questions concerning the EOM method. 

In developing BWCCS.D theory, extensivity and size consistency have been a primary concern. In Table 1 a size-consistency test for the F2 molecule [131] is presented. The entries show that on applying the a posteriori extensivity correction the size-consistency error has been practically eliminated and that its absolute value does not increase with size of basis set. In other tests on CH2, SiH2 and twisted ethylene molecules [15] the extensivity error was smaller than 1 kcal/mol. This accuracy is suflScient for... 

Perturbation theory offers some advantages in this respect. It is also subject to basis superposition error, but the contributions in which such errors may occur can be identified, and it is possible to make an estimate of the error and to correct for it to some extent. The charge-transfer energy is subject to basis superposition error, but it is possible to estimate the contribution of this error to the result. The extension correlation and double charge transfer terms are wholly due to basis superposition effects, to lowest order in overlap at least[l8], and can be discarded. The charge-transfer correlation and dispersion terms, on the other hand, can have no basis superposition error at all because they can only arise when occupied orbitals of both molecules are present[193. 

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. 

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. 

The previous sections have presented an extensive description of some of the central concepts from the cognitive modeling perspective. These topics have been dealt with in some depth because they provide a comprehensive basis for the reduction of human error in the CPI. 

On the basis of extensive past analysis the iron content of an homogeneous ore sample is accepted to be 19.85%. An analysis of seven samples gave an average of 20.06% Fe. Within what probability levels is a determinate error indicated in the analysis if the absolute standard deviation of the results is 0.14% ... 

These single reference-based methods are limited to cases where the reference function can be written as a single determinant. This is most often not the case and it is then necessary to use a multiconfigurational approach. Multrreference Cl can possibly be used, but this method is only approximately size extensive, which may lead to large errors unless an extended reference space is used. For example, Osanai et al. [8] obtained for the excitation energy in Mn 2.24 eV with the QCISD(T) method while SDCI with cluster corrections gave 2.64 eV. Extended basis sets were used. The experimental value is 2.15 eV. 

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