Energy error

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

This equation relates the free energy difference between two systems to the individual perturbations x and the / and g distribution functions. The relationship (6.15) is important in both the characterization of free energy error and the development of improved free energy methods. 

Perhaps the most challenging part of analyzing free energy errors in FEP or NEW calculations is the characterization of finite sampling systematic error (bias). The perturbation distributions / and g enable us to carry out the analysis of both the finite sampling systematic error (bias) and the statistical error (variance). 

Fig. 6.5. Graphical illustration of the inaccuracy model and the relative free energy error in forward and reverse free energy calculations. A limit-perturbation Xf is adopted to (effectively) describe the sampling of the distribution the regions above x/ are assumed to be perfectly sampled while regions below it shaded area) are never sampled. We may also put a similar upper limit x f for the high-rr tail, where there is no sampling for regions above it. However, this region (in a forward calculation) makes almost zero contribution to the free energy calculation and its error. Thus for simplicity we do not apply such an upper limit here...

In order to usefully deal with truncations, a simple criterion is needed for assessing the energy error introduced by a truncation. In this context the concept of the normalization deficiency has proven to be effective. This quantity is defined as the difference between the sum of squares of the coefficients of the Ntot determinants in the untruncated wavefunction expansion and the corresponding sum that includes only the Ntr determinants selected by the truncation ... 

The normalization deficiency turns out to be useful for the extrapolation of truncated energies because it is found to be near-proportional to the energy error introduced by the truncations considered here. 

The effectiveness of the method is exhibited by Figure 2 in which the energy errors of truncated expansions are plotted versus the numbers of determinants in these expansions. For each of the four systems shown, one curve displays this relationship for the expansions generated by the just discussed a priori truncations, whereas the other curve is obtained a posteriori by starting with the full SDTQ calculation in the same orbital basis and, then, simply truncating the determinantal expansion based on the ordering established by the exact coefficients of the determinants. There is practically no difference in the number of determinants needed to achieve an accuracy of 1 mh. 

Figure 6 displays the HF and CISD correlation energy errors with respect to... 

Representability Defects of the 2-RDM, 2-HRDM, and 2-G and Energy Error Obtained When Solving the Regulated CSE with and Without the N-, S-Purification Procedure for Li2 and BeH2 Molecules... 

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