Assessing convergence

A more detailed decomposition of macromolecular dynamics that can be used not only for assessing convergence but also for other purposes is principal components analysis (PCA), sometimes also called essential dynamics (Wlodek et al. 1997). In PCA the positional covariance matrix C is calculated for a given trajectory after removal of rotational and translational motion, i.e., after best overlaying all structures. Given M snapshots of an N atom macromolecule, C is a 3N X 3A matrix with elements... 

Using all the above, Cizek presents the explicit, spin orbital and spin-adapted CC doubles equations (CCD) i.e., T = T2 (then called coupled-pair many-electron theory) in terms of one- and two-electron integrals over an orthogonal basis set. Assisted by Joe Paldus with some computations, he also reports some CCD results for N2, which though limited to only ti to Ug excitations, uses ab initio integrals in an Slater-type-orbital basis. He also does the full Cl calculation to assess convergence, a tool widely used in Cizek s and Paldus work and by most of us, today. He also reports results for the minimum-basis TT-electron approximation to benzene. 

The case of water is particularly convenient because the required high Ka states may be detected in the solar absorption spectrum. However, it is difficult to observe the necessary high vibrational angular momentum states in molecules, which can only be probed by dispersed fluorescence or stimulated emission techniques. On the other hand, it is now possible to perform converged variational calculations on accurate potential energy surfaces, from which one could hope to verify the quantum monodromy and assess the extent to which it is disturbed by perturbations with other modes. Examples of such computed monodromy are seen for H2O in Fig. 2 and LiCN in Fig. 12. 

Before commenting on the values that appear in Table I, the level of convergence of the number of clusters with time needs to be assessed. This question is addressed in Figure 4 (see color insert). The number of clusters with a 99%, 75%, and 50% joined weight in the unfolded state is plotted as a function of time in the upper, middle, and lower panels, respectively. Even for the longest simulation (i.e., 200 ns), it is not possible to assess whether the number of clusters with a 99% weight... 

Clark, L. A., Livesley, W. ]., Schroeder, M. L., Irish, S. L. (1996). Convergence of two systems for assessing specific traits of personality disorders. Psychological Assessment, 8, 294-303. 

In another study, nucleic acid dendrimers were synthesized via both convergent and divergent approaches, and their purity assessed using PAGE [23, 24], In Figure 10.6 is shown convergent synthetic route, which involves ... 

Figure 2. Rate of convergence of truncated SDTQ-CI expansions based on a priori a posteriori ordering and error assessment. Filled circles Truncations based on anticipated a priori estimates. Open circles Truncations determined a posteriori from the full wavefunctions.

Finally, to assess the convergence of the commutator expansion in the effective Hamiltonian as the bond is stretched, we computed a second-order energy using the L-CTSD amplitudes, denoted CASSCF/L-CTSD(2). Here the energy expression is evaluated as As seen from Tables VI... 

Lyman, E., Zuckerman, D.M. Ensemble based convergence assessment of biomolecular trajectories. Biophys. J. 2006, 91, 164—72. 

So how accurate are DFT calculations It is extremely important to recognize that despite the apparent simplicity of this question, it is not well posed. The notion of accuracy includes multiple ideas that need to be considered separately. In particular, it is useful to distinguish between physical accuracy and numerical accuracy. When discussing physical accuracy, we aim to understand how precise the predictions of a DFT calculation for a specific physical property are relative to the true value of that property as it would be measured in a (often hypothetical) perfect experimental measurement. In contrast, numerical accuracy assesses whether a calculation provides a well-converged numerical solution to the mathematical problem defined by the Kohn-Sham (KS) equations. If you perform DFT calculations, much of your day-to-day... 

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