Convergence behavior

Similarly, improvement in the accuracy of the nuclear dynamics would be fruitful. While in this review we have shown that, in the absence of any approximations beyond the use of a finite basis set, the multiple spawning treatment of the nuclear dynamics can border on numerically exact for model systems with up to 24 degrees of freedom, we certainly do not claim this for the ab initio applications presented here. In principle, we can carry out sequences of calculations with larger and larger nuclear basis sets in order to demonstrate that experimentally observable quantities have converged. In the context of AIMS, the cost of the electronic structure calculations precludes systematic studies of this convergence behavior for molecules with more than a few atoms. A similar situation obtains in time-independent quantum chemistry—the only reliable way to determine the accuracy of a particular calculation is to perform a sequence of... 

In variable-density flows, this relation is used to couple the PDF code to the FV code by replacing the mean density predicted by the FV code with p X. Because the convergence behavior of the FV code may be sensitive to errors in the estimated mean density, particle-number control is especially critical in variable-density Lagrangian PDF codes. 

HOC1,309,310 HArF,311 and C1HC1.71 Most of these calculations were carried out using either the complex-symmetric Lanczos algorithm or filter-diagonali-zation based on the damped Chebyshev recursion. The convergence behavior of these two algorithms is typically much less favorable than in Hermitian cases because the matrix is complex symmetric. 

In the original W1/W2 paper [1], we opted for the geometric formula in view of the observed geometric convergence behavior. In a subsequent validation study [26] on a much wider variety of systems, we however found the two-point formula to be much more reliable, and we have adopted it henceforth. 

For the smaller basis sets used in W1 theory, the regime where the leading Eoo + A/L3 term dominates convergence behavior has not yet been reached, and using the formula in its unmodified form leads to overestimated (in absolute value) CCSD limits. One unelegant solution would be the use of three-term extrapolations like Eoo + A/L3 + B/L4, but in light of the poor quality of the VDZ basis set this is a most unsatisfactory alternative. Another alternative is the use of a two-point extrapolation Eoo + A/LQ, in which a is a fixed empirical parameter. By minimizing the deviation from the W2 CCSD limit for the so-called W2-1 set of 28 molecules (vide infra), we determined a = 3.22, which is the value used in W1 theory and its variants. 

Moritz, G., Hess, B.A., Reiher, M. Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings. J. Chem. Phys. 2005,122(2), 024107. 

Turning to the basis set dependence of the harmonic frequencies of HF, we see that the uncorrected CD s first increase, and then decrease with increasing basis set quality. Although the frequencies are improved by applying the counterpoise correction, the counterpoise-corrected results show the same general convergence behavior as the uncorrected results. Note the fortuitous cancellation of errors in the... 

The convergence behavior of the uncorrected as well as the corrected 0) is improved substantially by including diffuse functions in the basis set. The counterpoise-corrected values are again slightly better behaved. 

As seen above, application of the counterpoise procedure or the addition of diffuse functions may result in Rg(nZ) or coAnZ) curves which are better behaved. As another example. Figure 9 shows the harmonic frequency of HCl, computed with the cc-pVnZ and aug-cc-pVnZ basis sets. As for HF, application of the counterpoise correction improves the convergence behavior of computed with the regular sets, although the convergence behavior of the corrected co, s is far from simple. Addition of the set of diffuse functions to the basis as well as use of the counterpoise correction results in a very smooth dependence of on basis set. 

As mentioned earlier in this section, the convergence behavior of computed properties generally becomes less exponential as the quantities become less related to energies. By way of illustration, neither the application of the counterpoise procedure nor the addition of diffuse functions to the basis set improves the convergence behavior of the computed anharmonicities co x of the HF molecule (see Figure 10). Even in this case, however, both the uncorrected and corrected curves appear to be converging to the same limits. 

The convergence behavior of the corrected and uncorrected intramolecular r (HF) is very similar to ArHF (see previous section). The monomer deformation energies are small for this system as well. At the CCSD(T)/aug-cc-pVQZ level, AU (rA)+AUf (fb) is just 0.03 kcal/mol. However, as we have seen in the previous section, for systems in which the monomers are strongly distorted from their uncomplexed geometries the deformation energy can be significantly larger [see also (Xantheas, 1996)]. 

For strongly bound diatomics like Nj, HF, and HCl, the counterpoise procedure can improve the convergence behavior of r and co, (although the counterpoise-corrected values are not always well represented by a simple exponential formula). For highly ionic molecules like HF and HCl, it was found that addition of diffuse functions to the basis set in addition to the counterpoise correction also improves the convergence of r and co. ... 

The following case study demonstrates the convergence behavior for the LH mechanism (50) with irreversible first stage (i.e. r i = 0). In this case the kinetic polynomial (51) always has (structurally unstable with respect to feasibility) zero root whereas three other roots could be found from the cubic equation... 

---