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Non-analytic Spin-Density Functionals

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Density Functionals

Part of the book series: Topics in Current Chemistry ((TOPCURRCHEM,volume 365))

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Abstract

We examine the integer discontinuity (or derivative discontinuity) of the exact energy functionals of Kohn–Sham density-functional theory for the spin-polarized case. The integer discontinuity and its cause, the piecewise linearity of the energy in the grand canonical ensemble, are required to improve the predictive power of density-functional approximations to the exchange-correlation energy. We show how any spin-polarized DFA can be adapted to display the proper integer discontinuity. The formalism we present here can be used to improve functionals further within spin density-functional theory and fragment-based formulations of DFT.

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Notes

  1. 1.

    In this work by “subdensity-matrix” we mean a density matrix corresponding to a state with strictly an integer number of electrons.

  2. 2.

    The described ensemble is not a truly zero-temperature system because the spin-interactions are neglected.

  3. 3.

    Here we use Hund’s rules as a guide. Exceptions to these rules are known [39].

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Acknowledgements

We thankfully acknowledge support by the National Science Foundation CAREER program under Grant No. CHE-1149968. A.W. also acknowledges support from an Alfred P. Sloan Foundation Research Fellowship and a Camille-Dreyfus Teacher Scholar award.

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Authors and Affiliations

  1. Department of Chemistry, Purdue University, West Lafayette, IN, 47907, USA

    Martín A. Mosquera & Adam Wasserman

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  1. Martín A. Mosquera
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Correspondence to Adam Wasserman .

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Editors and Affiliations

  1. Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada

    Erin R. Johnson

Appendix. Calculation of Functional Derivatives

Appendix. Calculation of Functional Derivatives

The first term on the r.h.s. of (13) can be calculated using the chain rule:

$$ {\delta}_mE\left(\overline{N},\;\left[v\right]\right)={\left.\frac{\partial E}{\partial N}\right|}_u\kern0.24em {\left.{\delta}_mN+{\delta}_wE\left(\overline{N},\kern0.24em \left[v\right]\right)\right|}_N, $$

where | u and | N mean that these objects are fixed. The second term of the r.h.s. is interpreted as

$$ \begin{array}{l}\underset{\varepsilon \to 0}{ \lim}\frac{E\left(\overline{N},\left[u\left[{n}^{\prime}\right]\right]\right)-E\left(\overline{N},\left[u\left[n\right]\right]\right)}{\varepsilon }=\underset{\varepsilon \to 0}{ \lim}\frac{E\left(\overline{N},\left[v+\varepsilon \right]\right)-E\left(\overline{N},\left[v\right]\right)}{\varepsilon}\\ {}\kern13em ={\delta}_wE\left(\overline{N},\left[v\right]\right).\end{array} $$

The differential of N is trivial:

$$ \begin{array}{l}{\delta}_mN=\underset{\varepsilon \to 0}{ \lim}\frac{N\left[n+\varepsilon m\right]-N\left[m\right]}{\varepsilon}\\ {}\kern1.2em ={\displaystyle \int m=1}.\end{array} $$

When N is fixed, |ψ〉 is only determined by u, then

$$ \begin{array}{l}{\left.{\delta}_wE\right|}_N={\delta}_w{\left.\left\langle \psi \left|{\widehat{H}}_0\right|\psi \right\rangle \right|}_N+{\delta}_w{{\displaystyle \int \left.{\mathrm{d}}^3\mathbf{r}\kern0.24em n\left(\mathbf{r}\right)u\left(\mathbf{r}\right)\right|}}_N\\ {}\kern2.4em =\Big({\delta}_w\left\langle \psi \left|\right){\widehat{H}}_0\Big|\psi \right\rangle +\mathrm{c}.\mathrm{c}.+{\displaystyle \int {\mathrm{d}}^3\mathbf{r}}\kern0.24em n\left(\mathbf{r}\right)w\left(\mathbf{r}\right)+{\displaystyle \int {\mathrm{d}}^3\mathbf{r}}\kern0.24em u\left(\mathbf{r}\right){\delta}_wn\left(\mathbf{r}\right).\end{array} $$

Note that

$$ {\left.{\delta}_wn\left(\mathbf{r}\right)\right|}_N={\displaystyle \sum_{M\in \mathbb{N}}y\left(\overline{N}-M\right)\left[\right({\delta}_w{\left.\left\langle {\psi}_M\left|\right)\widehat{n}\left(\mathbf{r}\right)\Big|{\psi}_M\right\rangle \right|}_N+\mathrm{c}.\mathrm{c}.\Big]}. $$

This allows us to express δ w E| N as

$$ \begin{array}{l}{\delta}_wE\Big|{}_N={\displaystyle \sum_{M\in \mathbb{N}}y\left(\overline{N}-M\right)\left[\right({\delta}_w\left\langle \psi \left|{\widehat{H}}_0+{\displaystyle \int {\mathrm{d}}^3\mathbf{r}\;}\widehat{n}\left(\mathbf{r}\right)u\left(\mathbf{r}\right)\right|\psi \right\rangle +\mathrm{c}.\mathrm{c}.\Big]+{\displaystyle \int {\mathrm{d}}^3\mathbf{r}\kern0.24em n\left(\mathbf{r}\right)w\left(\mathbf{r}\right)}}\\ {}\kern2.5em ={\displaystyle \sum_{M\in \mathbb{N}}y\left(\overline{N}-M\right){E}_0\left(M,\kern0.24em \left[u\right]\right){\delta}_w\left(\left\langle {\psi}_M\Big|{\psi}_M\right\rangle \right)}+{\displaystyle \int {\mathrm{d}}^3\mathbf{r}}\kern0.24em n\left(\mathbf{r}\right)w\left(\mathbf{r}\right)\\ {}\kern2.4em ={\displaystyle \int {\mathrm{d}}^3\mathbf{r}}\kern0.24em n\left(\mathbf{r}\right)w\left(\mathbf{r}\right).\end{array} $$

Noting that δ m u[n] = w, we get (14).

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Mosquera, M.A., Wasserman, A. (2014). Non-analytic Spin-Density Functionals. In: Johnson, E. (eds) Density Functionals. Topics in Current Chemistry, vol 365. Springer, Cham. https://doi.org/10.1007/128_2014_619

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