2,304 research outputs found
Probabilistic performance estimators for computational chemistry methods: Systematic Improvement Probability and Ranking Probability Matrix. I. Theory
The comparison of benchmark error sets is an essential tool for the
evaluation of theories in computational chemistry. The standard ranking of
methods by their Mean Unsigned Error is unsatisfactory for several reasons
linked to the non-normality of the error distributions and the presence of
underlying trends. Complementary statistics have recently been proposed to
palliate such deficiencies, such as quantiles of the absolute errors
distribution or the mean prediction uncertainty. We introduce here a new score,
the systematic improvement probability (SIP), based on the direct system-wise
comparison of absolute errors. Independently of the chosen scoring rule, the
uncertainty of the statistics due to the incompleteness of the benchmark data
sets is also generally overlooked. However, this uncertainty is essential to
appreciate the robustness of rankings. In the present article, we develop two
indicators based on robust statistics to address this problem: P_{inv}, the
inversion probability between two values of a statistic, and \mathbf{P}_{r},
the ranking probability matrix. We demonstrate also the essential contribution
of the correlations between error sets in these scores comparisons
Probabilistic performance estimators for computational chemistry methods: the empirical cumulative distribution function of absolute errors
Benchmarking studies in computational chemistry use reference datasets to
assess the accuracy of a method through error statistics. The commonly used
error statistics, such as the mean signed and mean unsigned errors, do not
inform end-users on the expected amplitude of prediction errors attached to
these methods. We show that, the distributions of model errors being neither
normal nor zero-centered, these error statistics cannot be used to infer
prediction error probabilities. To overcome this limitation, we advocate for
the use of more informative statistics, based on the empirical cumulative
distribution function of unsigned errors, namely (1) the probability for a new
calculation to have an absolute error below a chosen threshold, and (2) the
maximal amplitude of errors one can expect with a chosen high confidence level.
Those statistics are also shown to be well suited for benchmarking and ranking
studies. Moreover, the standard error on all benchmarking statistics depends on
the size of the reference dataset. Systematic publication of these standard
errors would be very helpful to assess the statistical reliability of
benchmarking conclusions.Comment: Supplementary material: https://github.com/ppernot/ECDF
Range separation combined with the Overhauser model: Application to the H molecule along the dissociation curve
The combination of density-functional theory with other approaches to the
many-electron problem through the separation of the electron-electron
interaction into a short-range and a long-range contribution (range separation)
is a successful strategy, which is raising more and more interest in recent
years. We focus here on a range-separated method in which only the short-range
correlation energy needs to be approximated, and we model it within the
"extended Overhauser approach". We consider the paradigmatic case of the H
molecule along the dissociation curve, finding encouraging results. By means of
very accurate variational wavefunctions, we also study how the effective
electron-electron interaction appearing in the Overhauser model should be in
order to yield the exact correlation energy for standard Kohn-Sham density
functional theory.Comment: submitted to Int. J. Quantum Chem., special issue dedicated to Prof.
Hira
Study of the discontinuity of the exchange-correlation potential in an exactly soluble case
It was found by Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. {\bf 49},
1691 (1982)] and by Sham and Schl\"uter [Phys. Rev. Lett. {\bf 51}, 1884
(1983)] that the exact Kohn-Sham exchange-correlation potential of an open
system may jump discontinuosly as the particle number crosses an integer, with
important physical consequences. Recently, Sagvolden and Perdew [Phys. Rev. A
{\bf 77}, 012517 (2008)] have analyzed the discontinuity of the
exchange-correlation potential as the particle number crosses one, with an
illustration that uses a model density for the H ion. In this work, we
extend their analysis to the case in which the external potential is the simple
harmonic confinement, choosing spring-constant values for which the
two-electron hamiltonian has an analytic solution. This way, we can obtain the
exact, analytic, exchange and correlation potentials for particle number
fluctuating between zero and two, illustrating the discontinuity as the
particle number crosses one without introducing any model or approximation. We
also discuss exchange and correlation separately.Comment: Submitted to Int. J. Quantum Chem., special issue honoring Prof.
Mayer. New version, where an important error has been correcte
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