Size of the basis set

The size of the basis set is, however, only one criterion for judging the level of an ab-initio calculation. The situation is best illustrated by what has become known as a Pople diagram [27], as shown in Figure 7-24. 

The different levels of ab-initio theory are represented on two axes. The vertical one indicates the size of the basis set, which we have already discussed. However, the diagram shows that we can never reach the correct result (top right-hand... 

Figure 7-24. The Pople diagram . The vertical axis gives the size of the basis set and the horizontal axis the correlation treatment. The basis sets and methods given are chosen from the examples discussed in the text. Their positions on the axes (but not the order) are arbitrary.

The number of excited determinants thus grows factorially with the size of the basis set. Many of these excited determinants will of course have different spin multiplicity (triplet, quintet etc. states for a singlet HF determinant), and can therefore be left out in the calculation. Generating only the singlet CSFs, the number of configurations at each excitation level is shown in Table 4.1. 

The HF error depends only on the size of the basis set. The energy, however, behaves asymptotically as exp(—L),L being the highest angular momentum in the basis set, i.e. already, with a basis set of TZ(2df) (4s3p2dlf) quality the results are quite stable. Combined witii the fact that an HF calculation is the least expensive ab initio method, this means that tire HF error is not the limiting factor. 

It is usually observed that the CP correction for methods including electron correlation is larger and more sensitive to the size of the basis set, than that at the HE level. This is in line with the fact that the HE wave function converges much faster with respect to the size of the basis set tlian correlated wave functions. 

In practice a DFT calculation involves an effort similar to that required for an HF calculation. Furthermore, DFT methods are one-dimensional just as HF methods are increasing the size of the basis set allows a better and better description of the KS orbitals. Since the DFT energy depends directly on the electron density, it is expected that it has basis set requirements similar to those for HF methods, i.e. close to converged with a TZ(2df) type basis. 

This is a process which increases as the third power of the size of the basis set, and the optimization of the function is therefore an method. The Edmiston-... 

The calculated dipole moment is remarkably insensitive to the size of the basis set. Note that the SVWN value in this case is substantially better than BLYP and BPW91, i.e. this is a case where the theoretically poorer method provides better results than the more advanced gradient methods. Inclusion of exact exchange again improves the performance, and provides results very close to the experimental value, even with quite small basis sets. 

The two previous sections outline the main formal and applicative results obtained in our search for a theoretical framework where the number of variables which are explicitly taken into account would be as small as the observables allow. This framework should permit the use of different levels of approximation for the Hamiltonian operator and its orbital representation. That is, the size of the basis set and the kind of approximation used for the integrals should not be predetermined by the formalism. 

Thus, by following the hydrogenic model, we know not only the kind of angular symmetry but also the value n of the quantum number of the suitable polarization functions. In the case of a true hydrogenic atom these STO appear in a given linear combination. To limit the size of the basis set, one could use an unique polarization... 

For this and other problems, the spread in the results depends on the initial size of the basis set. For small basis sets, the final branching ratios depend on the initial conditions—position and momentum parameters that define each basis function. Because in the TDB only 10 (out of 30) initial conditions were chosen independently, it is instructive to compare its results to the ones obtained when only 10 independent basis functions are used to represent the initial wavefunction. The purpose of this comparison is to examine the dependence of the branching ratios on the initial conditions and not to demonstrate an improvement in the results. Such an improvement is expected because the... 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

Figure 21. The (So — S2) absorption spectrum of pyrazine for the reduced three-dimensional model using different spawning thresholds. Full line Exact quantum mechanical results. Dashed line Multiple spawning results for — 2.5, 5.0, 10, and 20. (All other computational details are as in Fig. 20.) As the spawning threshold is increased, the number of spawned basis functions decreases, the numerical effort decreases, and the accuracy of the result deteriorates (slowly). In this case, the final size of the basis set (at t — 0.5 ps) varies from 860 for 0 = 2.5 to 285 for 0 = 20.

HF frequencies are generally larger than the corresponding experimental data. The inclusion of electron correlation improves the results, but the theoretical frequencies are still higher than the experimental ones. Both the introduction of electron correlation and the size of the basis set seem to be important in order to obtain reliable results. 

There is a wide variety of ab initio techniques available for the study of radical thermochemistry, ranging from quite cheap and approximate methods to much more expensive and accurate approaches. The quality of results yielded by these procedures depends on the size of the basis set used and on the degree of electron correlation included. In practice, it is necessary to strike a balance between the required accuracy and the computational cost that can be afforded. 

The first kind of simplification exclusively concerns the size of the basis set used in the linear combination of one center orbitals. Variational principle is still fulfilled by this type of "ab initio SCF calculation, but the number of functions applied is not as large as necessary to come close to the H. F. limit of the total energy. Most calculations of medium-sized structures consisting for example of some hydrogens and a few second row atoms, are characterized by this deficiency. Although these calculations belong to the class of "ab initio" investigations of molecular structure, basis set effects were shown to be important 54> and unfortunately the number of artificial results due to a limited basis is not too small. 

Using the aug-cc-pVTZ-J basis set for fluorine, cc-pVTZ for carbon and cc-pVDZ for hydrogen (LDBS apTJ/pT/pD) it is possible to reproduce each contribution to the vicinal F-F coupling from the most complete basis set apTJ/apTJ/apTJ without considerable loss of accuracy (< 1 Hz) while reducing the size of the basis set by 62 functions or 28%. 

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