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When is the Fukui function not normalized? : the danger of inconsistent energy interpolation models in density functional theory

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Abstract
When one defines the energy of a molecule with a noninteger number of electrons by interpolation of the energy values for integer-charged states, the interpolated electron density, Fukui function, and higher-order derivatives of the density are generally not normalized correctly. The necessary and sufficient condition for consistent energy interpolation models is that the corresponding interpolated electron density is correctly normalized to the number of electrons. A necessary, but not sufficient, condition for correct normalization is that the energy interpolant be a linear function of the reference energies. Consistent with this general rule, polynomial interpolation models and, in particular, the quadratic E vs N model popularized by Parr and Pearson, do give normalized densities and density derivatives. Interestingly, an interpolation model based on the square root of the electron number also satisfies the normalization constraints. We also derive consistent least-norm interpolation models. In contrast to these models, the popular rational and exponential forms for E vs N do not give normalized electron densities and density derivatives.
Keywords
Physical and Theoretical Chemistry, Computer Science Applications, LINEAR-RESPONSE KERNEL, FRACTIONAL PARTICLE NUMBER, DEGENERATE GROUND-STATES, CHEMICAL-REACTIVITY, ELECTROACCEPTING POWERS, ELECTROPHILICITY INDEX, DERIVATIVE DISCONTINUITIES, ELECTRODONATING POWERS, ABSOLUTE HARDNESS, ELECTRON-DENSITY

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MLA
Heidar Zadeh, Farnaz, et al. “When Is the Fukui Function Not Normalized? : The Danger of Inconsistent Energy Interpolation Models in Density Functional Theory.” JOURNAL OF CHEMICAL THEORY AND COMPUTATION, vol. 12, no. 12, 2016, pp. 5777–87, doi:10.1021/acs.jctc.6b00494.
APA
Heidar Zadeh, F., Miranda-Quintana, R. A., Verstraelen, T., Bultinck, P., & Ayers, P. W. (2016). When is the Fukui function not normalized? : the danger of inconsistent energy interpolation models in density functional theory. JOURNAL OF CHEMICAL THEORY AND COMPUTATION, 12(12), 5777–5787. https://doi.org/10.1021/acs.jctc.6b00494
Chicago author-date
Heidar Zadeh, Farnaz, Ramón Alain Miranda-Quintana, Toon Verstraelen, Patrick Bultinck, and Paul W Ayers. 2016. “When Is the Fukui Function Not Normalized? : The Danger of Inconsistent Energy Interpolation Models in Density Functional Theory.” JOURNAL OF CHEMICAL THEORY AND COMPUTATION 12 (12): 5777–87. https://doi.org/10.1021/acs.jctc.6b00494.
Chicago author-date (all authors)
Heidar Zadeh, Farnaz, Ramón Alain Miranda-Quintana, Toon Verstraelen, Patrick Bultinck, and Paul W Ayers. 2016. “When Is the Fukui Function Not Normalized? : The Danger of Inconsistent Energy Interpolation Models in Density Functional Theory.” JOURNAL OF CHEMICAL THEORY AND COMPUTATION 12 (12): 5777–5787. doi:10.1021/acs.jctc.6b00494.
Vancouver
1.
Heidar Zadeh F, Miranda-Quintana RA, Verstraelen T, Bultinck P, Ayers PW. When is the Fukui function not normalized? : the danger of inconsistent energy interpolation models in density functional theory. JOURNAL OF CHEMICAL THEORY AND COMPUTATION. 2016;12(12):5777–87.
IEEE
[1]
F. Heidar Zadeh, R. A. Miranda-Quintana, T. Verstraelen, P. Bultinck, and P. W. Ayers, “When is the Fukui function not normalized? : the danger of inconsistent energy interpolation models in density functional theory,” JOURNAL OF CHEMICAL THEORY AND COMPUTATION, vol. 12, no. 12, pp. 5777–5787, 2016.
@article{8501652,
  abstract     = {{When one defines the energy of a molecule with a noninteger number of electrons by interpolation of the energy values for integer-charged states, the interpolated electron density, Fukui function, and higher-order derivatives of the density are generally not normalized correctly. The necessary and sufficient condition for consistent energy interpolation models is that the corresponding interpolated electron density is correctly normalized to the number of electrons. A necessary, but not sufficient, condition for correct normalization is that the energy interpolant be a linear function of the reference energies. Consistent with this general rule, polynomial interpolation models and, in particular, the quadratic E vs N model popularized by Parr and Pearson, do give normalized densities and density derivatives. Interestingly, an interpolation model based on the square root of the electron number also satisfies the normalization constraints. We also derive consistent least-norm interpolation models. In contrast to these models, the popular rational and exponential forms for E vs N do not give normalized electron densities and density derivatives.}},
  author       = {{Heidar Zadeh, Farnaz and Miranda-Quintana, Ramón Alain and Verstraelen, Toon and Bultinck, Patrick and Ayers, Paul W}},
  issn         = {{1549-9618}},
  journal      = {{JOURNAL OF CHEMICAL THEORY AND COMPUTATION}},
  keywords     = {{Physical and Theoretical Chemistry,Computer Science Applications,LINEAR-RESPONSE KERNEL,FRACTIONAL PARTICLE NUMBER,DEGENERATE GROUND-STATES,CHEMICAL-REACTIVITY,ELECTROACCEPTING POWERS,ELECTROPHILICITY INDEX,DERIVATIVE DISCONTINUITIES,ELECTRODONATING POWERS,ABSOLUTE HARDNESS,ELECTRON-DENSITY}},
  language     = {{eng}},
  number       = {{12}},
  pages        = {{5777--5787}},
  title        = {{When is the Fukui function not normalized? : the danger of inconsistent energy interpolation models in density functional theory}},
  url          = {{http://doi.org/10.1021/acs.jctc.6b00494}},
  volume       = {{12}},
  year         = {{2016}},
}

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