Basis-set convergence

As noted in Chapter 6, basis-set flexibility is key to accurately describing the molecular wave function. When methods for including electron correlation are included, this only becomes more true. One can appreciate this in an intuitive fashion from thinking of the correlated wave function as a linear combination of determinants, as expressed in Eq. (7.1). Since the excited determinants necessarily include occupation of orbitals that are virtual in the HF determinant, and since the HF determinant in some sense uses up the best 

The greater dependence on basis-set quality of correlated calculations compared to those of the HF variety has prompted many developers of basis sets to optimize contractions via some scheme that includes evaluating results from the former. For instance, the correlation consistent prefix of the cc-pVrcZ basis sets discussed in Chapter 6 highlights this feature. 

Note that the scaling behavior of methods more highly correlated than MP2 is expected in general to be less favorable than MP2. This derives from the greater sensitivity to basis set exhibited by determinants involving excitations beyond double, since still more virtual orbitals must be occupied. 

For the very small systems in Table 7.1, it is possible to approach the exact solution of the Schrodinger equation, but, as a rule, convergence of the correlation energy is depressingly slow. Mathematically, this derives from the poor ability of products of one-electron basis functions, which is what Slater determinants are, to describe the cusps in two-electron densities that characterize electronic structure. For the MP2 level of theory, Schwartz (1962) 

Note tliat compared to the R12 benchmark energies, the CCSD energies with individual basis sets are quite slowly convergent. Even with the staggeringly large cc-pV6Z basis set, the mean unsigned error over the seven molecules remains 4.2 mEi,. The extrapolated values 

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Abstract. This paper presents results from quantum molecular dynamics Simula tions applied to catalytic reactions, focusing on ethylene polymerization by metallocene catalysts. The entire reaction path could be monitored, showing the full molecular dynamics of the reaction. Detailed information on, e.g., the importance of the so-called agostic interaction could be obtained. Also presented are results of static simulations of the Car-Parrinello type, applied to orthorhombic crystalline polyethylene. These simulations for the first time led to a first principles value for the ultimate Young s modulus of a synthetic polymer with demonstrated basis set convergence, taking into account the full three-dimensional structure of the crystal. 

Secondly, the ultimate properties of polymers are of continuous interest. Ultimate properties are the properties of ideal, defect free, structures. So far, for polymer crystals the ultimate elastic modulus and the ultimate tensile strength have not been calculated at an appropriate level. In particular, convergence as a function of basis set size has not been demonstrated, and most calculations have been applied to a single isolated chain rather than a three-dimensional polymer crystal. Using the Car-Parrinello method, we have been able to achieve basis set convergence for the elastic modulus of a three-dimensional infinite polyethylene crystal. These results will also be fliscussed. 

Regarding mechanical properties of polymers, the efficiency of the Car-Parrinello approach has enabled us to evaluate the ultimate Young s modulus of orthorhombic polyethylene, and demonstrate basis set convergence for that property. 

Dargel, T. K., Hertwig, R. H., Koch, W., Horn, H., 1998, Towards an Accurate Gold Carbonyl Binding Energy in AuCO+ Basis Set Convergence and a Comparison between Density Functional and Conventional Methods , J. Chem. Phys., 108, 3876. 

Martin, J. M. L., El-Yazal, J., Francois, J.-P, 1995a, Basis Set Convergence and Performance of Density Functional Theory Including Exact Exchange Contributions for Geometries and Harmonic Frequencies , Mol. Phys., 86, 1437. 

Martin, J. M. L., 2000, Some Observations and Case Studies on Basis Set Convergence in Density Functional Theory in Density Functional Theory A Bridge between Chemistry and Physics, Geerlings, P., De Proft, F., Langenaeker, W. (eds.), VUB Press, Brussels. 

In the preceding section, we observed the slow basis-set convergence of the doubles contributions to the AE of CO. In the present section, we shall make an attempt at understanding the reasons for the slow convergence and to see if this insight can help us design better computational schemes. 

For systems devoid of nondynamical correlation effects, this is the largest individual contribution to the molecular binding energy. Its basis set convergence is relatively rapid, yet our discussion will be disproportionately long because a number of the dramatis personae that reappear in the remainder of the story need to be introduced here. 

Some representative results can be found in Table 2.2. For the G2-1 set of electron affinities, W1 theory has a mean absolute error of 0.016 eV [26]. Not unexpectedly - given the slow basis set convergence of electron affinities - the extra effort invested in W2 theory pays off with a further reduction of the mean absolute error to 0.012 eV. Accuracy appears to be limited principally by imperfections in the CCSD(T) method for the atoms B-F and Al-Cl, using even larger basis sets we achieve 0.009 eV at the CCSD(T) level, which decreases to 0.001 eV if approximate full Cl energies are used. 

For a subset of 27 G2-2 molecules with fairly small experimental uncertainties, W1 theory had MAD of 0.7 kcal/mol, compared to the average experimental uncertainty of 0.4 kcal/mol. Some systems exhibit deviations from experiment in excess of 1 kcal/mol in the cases of BF3 andCF4, very slow basis set convergence is responsible, and W2 calculations in fact remove nearly all remaining disagreement with experiment for the latter system. (The best available value for BF3 is itself a theoretical one, so a comparison would involve circular reasoning.) Other molecules (N02 and C1NO) suffer from severe multireference effects. 

The higher-order contributions to the correlation energy [such as CCSD(T)-MP2] are more than an order of magnitude smaller than their second-order counterparts. However, the basis set convergence to the CCSD(T)-R12 limit does not follow the simple linear behavior found for the second-order correlation energy. This is a consequence of the interference effect described in Eq. (2.2). The full Cl or CCSD(T) basis set truncation error is attenuated by the interference factor (Fig. 4.9). The CBS correction to the higher-order components of the correlation energy is thus the difference between the left-hand sides of Eqs. (2.2) and... 

Table 1. The basis-set convergence of the Cauchy moments S(k) [a.u.] for Ne calculated with CCSD model and the n-aug-cc-VXZ basis-set family (all electrons correlated)...

The relative energies of the three protonated species are well reproduced by all methods from the Gn family. This can largely be explained by (a) the fact that all these methods involved CCSD(T) or QCISD(T) steps (and apparently triple excitations are quite important here) (b) the relatively rapid basis set convergence noted above, which means that it is not really an issue that the CCSD(T) and QCISD(T) steps are carried out in relatively small basis sets. CBS-QB3 likewise reproduces the relative energetics quite well. 

This approximation can be justified from a perturbation theory viewpoint that assumes the smallness of ff and is analogous to the treatment of connected triples in CCSDT-1 [64]. The simplification in the equations allowed the CCSD(R12) and CCSD(T)(R12) methods to be implemented by a modest extension of the computational elements developed in the MP2-R12 implementations. Since they do not rely on the SA, they need an auxiliary basis set for the RI, but the rapid basis-set convergence can be obtained. 

Helgaker, T., Klopper, W., Koch, H., Noga, J. Basis-set convergence of correlated calculations on water. J. Chem. Phys. 1997, 106, 9639-46. 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

Table 7.1 Basis set convergence for HF and full CI energies of CO and O, respectively...

As for basis-set convergence, triple- calculations at the MP2 and CCSD levels are provided for comparison to die double- results. For this particular property, the results for p-benzyne are not terribly sensitive to improvements in the Ilexibility of the basis set. In the pyridynium ion case, die CCSD results are also not very sensitive, but a large effect is seen at the MP2 level. This has more to do with the instability of the perturbation expansion than any intrinsic difference between the isoelectronic aryiies. 

As for basis-set convergence, triple- calculations at the MP2 and CCSD levels are provided for comparison to the double- results. For this particular property, the results... 

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