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## Generalized Hirshfeld Partitioning with Oriented and Promoted Proatoms

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The project was supported by NSERC, Compute Canada, and the Canada Research Chairs

Corresponding authors: AYERS Paul W., Email: ayers@mcmaster.ca

In this study, we show how to generalize Hirshfeld partitioning methods to possibly include non-spherical proatom densities. While this generalization is numerically challenging (requiring global optimization of a large number of parameters), it is conceptually appealing because it allows the proatoms to be pre-polarized, or even promoted, to a state that more closely resembles the atom in a molecule. This method is based on first characterizing the convex set of proatom densities associated with the degenerate ground states of isolated atoms and atomic ions. The preferred orientation of the proatoms' densities are then obtained by minimizing the information–theoretic distance between the promolecular and molecular densities. If contributions from excited states (and not just degenerate ground states) are included in the convex set, this method can describe promoted atoms. While the procedure is intractable in general, if one includes only atomic states that have differing electron-numbers and/or spins, the variational principle becomes a simple convex optimization with a single unique solution.

Abstract

In this study, we show how to generalize Hirshfeld partitioning methods to possibly include non-spherical proatom densities. While this generalization is numerically challenging (requiring global optimization of a large number of parameters), it is conceptually appealing because it allows the proatoms to be pre-polarized, or even promoted, to a state that more closely resembles the atom in a molecule. This method is based on first characterizing the convex set of proatom densities associated with the degenerate ground states of isolated atoms and atomic ions. The preferred orientation of the proatoms' densities are then obtained by minimizing the information–theoretic distance between the promolecular and molecular densities. If contributions from excited states (and not just degenerate ground states) are included in the convex set, this method can describe promoted atoms. While the procedure is intractable in general, if one includes only atomic states that have differing electron-numbers and/or spins, the variational principle becomes a simple convex optimization with a single unique solution.

Keywords： Hirshfeld partitioning ; Stockholder atoms in molecules ; Nonspherical proatoms ; Information theory ; Degenerate ground states ; Promoted atomic reference states, Electron density ; Conceptual density functional theory

HEIDAR-ZADEH Farnaz, AYERS Paul W.. Generalized Hirshfeld Partitioning with Oriented and Promoted Proatoms. 物理化学学报[J], 2018, 34(5): 514-518 doi:10.3866/PKU.WHXB201710101

HEIDAR-ZADEH Farnaz, AYERS Paul W.. Generalized Hirshfeld Partitioning with Oriented and Promoted Proatoms. Acta Physico-Chimica Sinica[J], 2018, 34(5): 514-518 doi:10.3866/PKU.WHXB201710101

## 1 Motivation

Within the conceptual density functional theory (conceptual DFT) framework, chemical reactivity indicators are commonly defined as the derivatives of the ground-state energy and the grand potential with respect to the number of electrons, chemical potential and external potential 1-6. These global, local and nonlocal functions measure the sensitivity of a molecule to electron transfer and electrostatic interactions 7-14. However, to pinpoint the molecule's preferred reactive site in an electrophilic or a nucleophilic attack, it is common to partition the (non)local indictors among the constituent atoms and define the corresponding condensed reactivity indicators 15-27. As atoms-in-molecules are not uniquely defined within quantum mechanics, this requires selecting a method to decompose a molecule into atomic subsystems by distributing either the molecular orbitals 28-35 or the molecular electron density 36-41. Here, we are interested in generalizing the Hirshfeld partitioning approaches to utilize non-spherical reference proatom densities.

In 1977, Hirshfeld proposed an approach for identifying atoms in a molecule based on the stockholder perspective 40. Specifically, given the spherically-averaged densities of the isolated neutral reference atoms, ρA0(r), let the density of an atom in a molecule, ρA(r), be determined by the fraction of the molecular electron density, ρmol(r), that was "invested" by that reference atom, defined intuitively as

${w_A}\left( \mathit{\boldsymbol{r}} \right) = \frac{{\rho _A^0\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\rho _B^0\left( \mathit{\boldsymbol{r}} \right)} }}$

The density of the atom in a molecule (AIM) is then

${\rho _A}\left( \mathit{\boldsymbol{r}} \right) = {w_A}\left( \mathit{\boldsymbol{r}} \right){\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)$

and obviously the sum of the AIM densities is the molecular density, guaranteeing an exhaustive partitioning,

$\sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {{\rho _A}\left( \mathit{\boldsymbol{r}} \right)} = {\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)$

The reference-atom densities in Eq.(1) are commonly referred to as the proatom densities, and their sum is called the promolecular density,

$\rho _{{\rm{mol}}}^0\left( \mathit{\boldsymbol{r}} \right) = \sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {\rho _A^0\left( \mathit{\boldsymbol{r}} \right)}$

While Hirshfeld presented the partitioning in Eq.(1) as a heuristic, Nalewajski and Parr provided a mathematical interpretation for it 42-44. Their motivation was to define AIM to be as close as possible to isolated neutral atoms, and to maximize that closeness by minimizing the Kullback-Leibler divergence between the density of AIM and the reference atoms, subject to the constraint in Eq.(3). The result of the Nalewajski-Parr variational formulation,

$\underbrace {\min }_{\left\{ {{\rho _A}\left( \mathit{\boldsymbol{r}} \right)\left| {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right) = \sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {{\rho _A}\left( \mathit{\boldsymbol{r}} \right)} } \right.} \right\}}\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\int {{\rho _B}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _B}\left( \mathit{\boldsymbol{r}} \right)}}{{\rho _B^0\left( \mathit{\boldsymbol{r}} \right)}}} \right){\rm{d}}\mathit{\boldsymbol{r}}} }$

is the Hirshfeld AIM density, Eqs.(1)–(2). Once one realizes that the Hirshfeld partitioning tries to maximize the similarity between the AIM densities and those of the neutral reference atoms in an information-theoretic sense, it is not surprising that Hirshfeld partitioning tends to systematically underestimate the magnitude of AIM charges (relative to other approaches like electrostatic potential fitting and natural population analysis) 45.

To "improve" the Hirshfeld partitioning, one can use a different method to measure the dissimilarity between the AIM density and the proatom density or one can use a different definition for the proatom 46. Both techniques have been explored in the literature. There is a very broad class of dissimilarity measures that retain the partitioning in Eq.(1) 47-52, but the Kullback-Leibler divergence in particular has appealing mathematical properties 53, 54. It seems most effective, then, to change the definition of the proatom density. There are many attempts to do this in the literature, most (but not all 55) of which are restricted to the spherically-symmetric proatom densities 53, 56-67.

To understand why aspherical proatoms are not commonly employed, suppose that one had aspherical proatomic densities denoted by ρA; αA, βA, γA0(r), where (αA, βA, γA) are the Euler angles controlling the orientation of proatom A. The best orientation for the proatom density is obtained by minimizing the information-theoretic divergence in Eq.(5) with respect to Euler angles

$\underbrace {\min }_{\left\{ {{\alpha _A},{\beta _A},{\gamma _A}} \right\}_{A = 1}^{{N_{{\rm{atoms}}}}}}\underbrace {\min }_{\left\{ {{\rho _A}\left( \mathit{\boldsymbol{r}} \right)\left| {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right) = \sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {{\rho _A}\left( \mathit{\boldsymbol{r}} \right)} } \right.} \right\}}\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\int {{\rho _B}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _B}\left( \mathit{\boldsymbol{r}} \right)}}{{\rho _{B,{\alpha _B},{\beta _B},{\gamma _B}}^0\left( \mathit{\boldsymbol{r}} \right)}}} \right)} }$

or, equivalently 47, 48,

$\underbrace {\min }_{\left\{ {{\alpha _A},{\beta _A},{\gamma _A}} \right\}_{A = 1}^{{N_{{\rm{atoms}}}}}}\int {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\rho _{B,{\alpha _B},{\beta _B},{\gamma _B}}^0\left( \mathit{\boldsymbol{r}} \right)} }}} \right){\rm{d}}\mathit{\boldsymbol{r}}}$

Because of the objective function's complicated nonlinear dependence on the Euler angles, Eq.(7) is a difficult global optimization problem.

Moreover, it is not clear how one should choose the non-spherical densities. For example, for the sulfur atom, should one use a non-spherical density corresponding to the Cartesian p-orbitals, {pm}m=x, y, z or the ones defined by the (complex) spherical Harmonics, {Y1m}m=−1, 0, +1? The answer almost certainly depends on the molecule that one is considering: one would expect to prefer a different representation for MgS (perhaps the spherical harmonics) and SO2 (perhaps the Cartesian orbitals). In some molecules (perhaps SF6), the spherically averaged sulfur proatom density might be preferable.

In the next section, we will propose a method that avoids the problem of choosing an optimal orientation for the aspherical atomic densities. However the problem of global optimization, while perhaps somewhat reduced, will remain.

### 2.1 Oriented proatoms in degenerate ground states

For simplicity, we will start by restricting ourselves to neutral proatoms, as in traditional Hirshfeld partitioning. The neutral atoms that can have non-spherical ground-state electron densities always have degenerate ground states. Let {Ψg}g=1G denote an orthonormal basis for the G-dimensional manifold of ground-state wave-functions defined by

$\left| \mathit{\Psi } \right\rangle = \sum\limits_{g = 1}^G {{c_g}\left| {{\mathit{\Psi }_g}} \right\rangle }$

$1 = \sum\limits_{g = 1}^G {{{\left| {{c_g}} \right|}^2}}$

The electron densities associated with pure (as opposed to mixed) ground states of this system have the form

$\begin{array}{l}\rho \left( \mathit{\boldsymbol{r}} \right) = \left\langle {\sum\limits_{f = 1}^G {{c_f}{\mathit{\Psi }_f}\left| {\hat \rho \left( \mathit{\boldsymbol{r}} \right)} \right|} \sum\limits_{g = 1}^G {{c_g}{\mathit{\Psi }_g}} } \right\rangle \\\;\;\;\;\;\;\;\; = \sum\limits_{f = 1}^G {\sum\limits_{g = 1}^G {c_f^ * {c_g}{\rho _{fg}}\left( \mathit{\boldsymbol{r}} \right)} } \end{array}$

where in the second line we have introduced the transition density denoted by,

${\rho _{fg}}\left( \mathit{\boldsymbol{r}} \right) = \left\langle {{\mathit{\Psi }_f}\left| {\hat \rho \left( \mathit{\boldsymbol{r}} \right)} \right|{\mathit{\Psi }_g}} \right\rangle = \rho _{gf}^ * \left( \mathit{\boldsymbol{r}} \right)$

Here and in Eq.(10), the electron density operator is defined as

$\hat \rho \left( \mathit{\boldsymbol{r}} \right) = \sum\limits_{i = 1}^N {\delta \left( {\mathit{\boldsymbol{r}} - {\mathit{\boldsymbol{r}}_i}} \right)} = {\psi ^\dagger }\left( \mathit{\boldsymbol{r}} \right)\psi \left( \mathit{\boldsymbol{r}} \right)$

where in the second line we have, for convenience, introduced the field operators.

In analogy to Eq.(7), we can consider the density expression of Eq.(10) as the proatom density and minimize the information loss with respect to the wave-function coefficients of each atom,

$\underbrace {\min }_{\left\{ {{c_{g;A}}\left| {1 = \sum\limits_{g = 1}^{{C_A}} {{{\left| {{c_g};A} \right|}^2}} } \right.} \right\}_{g = 1;A = 1}^{{G_A};{N_{{\rm{atoms}}}}}}\int {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\sum\limits_{f = 1}^{{G_B}} {\sum\limits_{g = 1}^{{G_B}} {c_{f;B}^ * {c_{g;B}}{\rho _{fg;B}}\left( \mathit{\boldsymbol{r}} \right)} } } }}} \right)}$

Both the objective function and the constraints are nonlinear in the wave-function coefficients, so this is again a (difficult) global optimization problem. In addition, spherical proatom densities (which are not pure-state v-representable 68, 69) are excluded from Eq.(13).

Notice that an objective functions of the form

$D\left[ {\left\{ {{c_{k;A}}} \right\}_{A = 1;k = 1}^{{N_{{\rm{atoms}}}},{K_A}}} \right] \equiv \int {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {\sum\limits_{k = 1}^{{K_A}} {{c_{k;A}}{\rho _{k;A}}\left( \mathit{\boldsymbol{r}} \right)} } }}} \right){\rm{d}}\mathit{\boldsymbol{r}}}$

is convex because

$\frac{{\partial D}}{{\partial {c_{k;B}}\partial {c_{l;C}}}} = \int {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right){\rho _{k;B}}\left( \mathit{\boldsymbol{r}} \right){\rho _{l,C}}\left( \mathit{\boldsymbol{r}} \right)}}{{{{\left( {\sum\limits_{A = 1}^{{N_{{\rm{atoms}}}}} {\sum\limits_{k = 1}^{{K_A}} {{c_{k;A}}{\rho _{k;A}}\left( \mathit{\boldsymbol{r}} \right)} } } \right)}^2}}}{\rm{d}}\mathit{\boldsymbol{r}}}$

is positive definite. This motivates us to choose a new set of variables,

${d_{fg}} = c_f^ * {c_g}$

In terms of these new variables, the electron density can be written as

$\rho \left( {\mathit{\boldsymbol{r}};\left\{ {{d_{fg}}} \right\}_{f,g = 1}^{G,G}} \right)\sum\limits_{g = 1}^G {{d_{gg}}{\rho _{gg}}\left( \mathit{\boldsymbol{r}} \right)} + \sum\limits_{f = 1}^G {\sum\limits_{g = 1}^{f - 1} {\left( {{d_{fg}}{\rho _{fg}}\left( \mathit{\boldsymbol{r}} \right) + {d_{gf}}{\rho _{gf}}\left( \mathit{\boldsymbol{r}} \right)} \right)} }$

The new variables must satisfy the constraints,

$0 \le {d_{gg}} \le 1,\sum\limits_{g = 1}^G {{d_{gg}} = 1} ,{d_{fg}} = d_{gf}^ * ,{d_{fg}}{d_{gf}} = {d_{ff}}{d_{gg}}$

These constraints follow directly from the definition (16) and the normalization constraint on the wave-function, Eq.(9) 70, 71. The last constraint in Eq.(18) is especially important: without this constraint the electron density in Eq.(17) can become negative.

To relax the requirement that the electron density corresponds to a pure state, we can relax the final equality in Eq.(18) 70, 71,

${d_{fg}}{d_{gf}} \le {d_{ff}}{d_{gg}}$

We can now rewrite Eq.(13) as the optimization of a convex objective function subject to some constraints,

$\underbrace {\min }_{\left\{ {\begin{array}{*{20}{c}} {\left. \begin{gathered} \hfill \\ \hfill \\ {d_{gg;A}} \in \mathbb{R} \hfill \\ {d_{fg;A}} \in \mathbb{C} \hfill \\ \end{gathered} \right|}&\begin{gathered} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - {d_{gg;A}} \leqslant 0 \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{g = 1}^{{G_A}} {{d_{gg;A}} - 1} = 0 \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{d_{fg;A}} - d_{gf;A}^ * = 0 \hfill \\ {d_{fg;A}}{d_{gf;A}} - {d_{ff;A}}{d_{gg;A}} \leqslant 0 \hfill \\ \end{gathered} \end{array}} \right\}_{f,g = 1;A = 1}^{{G_A},{G_A};{N_{{\rm{atoms}}}}}}\int {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)\ln \left( {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\rho _B^0\left( {\mathit{\boldsymbol{r}};\left\{ {{d_{fg;B}}} \right\}_{f,g = 1}^{{G_B},{G_B}}} \right)} }}} \right){\rm{d}}\mathit{\boldsymbol{r}}}$

where $\rho _B^0\left( {{\bf{r}};\left\{ {{d_{fg;B}}} \right\}_{f,g = 1}^{{G_B},{G_B}}} \right)$ is given by Eq.(17). The optimization of a convex objective function with respect to linear equalities and convex inequalities has at most one minimum, and the problem in Eq.(20) is almost in the standard form of a convex optimization problem. However, the final inequality constraint,

$g\left( {{d_{fg}},{d_{gf}},{d_{ff}},{d_{gg}}} \right) \equiv {d_{fg}}{d_{gf}} - {d_{ff}}{d_{gg}} \le 0$

is not convex because the Hessian matrix,

$\left[ {\begin{array}{*{20}{c}}{\frac{{{\partial ^2}g}}{{\partial d_{fg}^2}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{fg}}\partial {d_{gf}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{fg}}\partial {d_{ff}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{fg}}\partial {d_{gg}}}}}\\{\frac{{{\partial ^2}g}}{{\partial {d_{gf}}\partial {d_{fg}}}}}&{\frac{{{\partial ^2}g}}{{\partial d_{gf}^2}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{gf}}\partial {d_{ff}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{gf}}\partial {d_{fg}}}}}\\{\frac{{{\partial ^2}g}}{{\partial {d_{ff}}\partial {d_{fg}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{ff}}\partial {d_{gf}}}}}&{\frac{{{\partial ^2}g}}{{\partial d_{ff}^2}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{ff}}\partial {d_{gg}}}}}\\{\frac{{{\partial ^2}g}}{{\partial {d_{gg}}\partial {d_{fg}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{gg}}\partial {d_{gf}}}}}&{\frac{{{\partial ^2}g}}{{\partial {d_{gg}}\partial {d_{ff}}}}}&{\frac{{{\partial ^2}g}}{{\partial d_{gg}^2}}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0&1&0&0\\1&0&0&0\\0&0&0&{ - 1}\\0&0&{ - 1}&0\end{array}} \right]$

has negative eigenvalues. The constrained optimization problem in Eq.(20), therefore, typically has many local minima. Nonetheless, solution of Eq.(20) allows one to choose the best proatom density among the set of ensemble-state-representable ground state proatom densities.

### 2.2 Promoted proatoms in quasi-degenerate ground states

To make contact with the valence-bond picture, it would be desirable to include promoted and ionized states of the reference proatoms. This is easily achieved using the previous approach because non-ground-state wave-functions can be included in Eq.(8),

$\left| \mathit{\Psi } \right\rangle = \sum\limits_{g = 1}^{{N_{{\rm{wf}}}}} {{c_g}\left| {{\mathit{\Psi }_g}} \right\rangle }$

Equation (20) still holds in this case, and a complete (but chemically unreasonable and computationally intractable) description would be obtained by including all possible wave-functions in Eq.(23). Sensible truncations are clearly required, perhaps by using similar strategies to those one uses to select dominant resonance structures in valence-bond approaches.

Notice that the optimization in Eq.(20) becomes tractable if all the transition densities are zero. i.e.,

$\forall f \ne g:{\rho _{fg}}\left( \mathit{\boldsymbol{r}} \right) = \left\langle {{\mathit{\Psi }_f}\left| {{\psi ^\dagger }\left( \mathit{\boldsymbol{r}} \right)\psi \left( \mathit{\boldsymbol{r}} \right)} \right|{\mathit{\Psi }_g}} \right\rangle = 0$

If the wave-functions are Slater determinants, this is true for double-excitations. For exact eigenfunctions of the atoms, this is true when $\left| {{\psi }_{f}} \right\rangle$ and $\left| {{\psi }_{g}} \right\rangle$ correspond to eigenstates of the number of electrons, the spin, or the spin-projection. (i.e., different eigenfunctions of $\hat{N}$, ${{\hat{S}}^{2}}$, or ${{\hat{S}}_{z}}$) If we restrict ourselves to those eigenstates, then we can rewrite Eq.(20) as a convex optimization problem with respect to linear constraints,

$\begin{gathered} \underbrace {\min }_{\left\{ {\begin{array}{*{20}{c}} {{d_{N,S,{M_S};A}} \in \mathbb{R}}&{\left| \begin{subarray}{l} \;\;\;\;\;\;\;\;\;\;\;\; - {d_{N,S,{M_S};A}} \leqslant 0 \\ \sum\limits_{N,S,{M_S}} {{d_{N,S,{M_S};A}} - 1} = 0 \end{subarray} \right.} \end{array}} \right\}_{f,g = 1;A = 1}^{{G_A},{G_A};{N_{{\rm{atoms}}}}}}\int {{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)} \hfill \\ \;\;\;\;\;\;\; \times \ln \left( {\frac{{{\rho _{{\rm{mol}}}}\left( \mathit{\boldsymbol{r}} \right)}}{{\sum\limits_{B = 1}^{{N_{{\rm{atoms}}}}} {\sum\limits_{N,S,{M_S}} {{d_{N,S,{M_S};B}}{\rho _{N,S,{M_S};B}}\left( \mathit{\boldsymbol{r}} \right)} } }}} \right){\rm{d}}r \hfill \\ \end{gathered}$

This formulation allows promoted proatoms, however because it does not allow one to adjust the orientation of the proatoms, the densities that are used in Eq.(25) should be spherically averaged. Nevertheless, Eq.(25) is a tractable convex optimization, which can be solved using the same iterative strategies employed by the minimal basis iterative stockholder analysis 53.

## 3 Summary

We have developed a variational procedure, Eq.(20), which determines the optimal ground state proatom density in Hirshfeld partitioning. This procedure includes the traditional spherically-averaged neutral proatoms as a special case, but allows oriented and polarized proatoms to be used when that would allow the molecular electron density to be more closely approximated by the promolecular density. Unfortunately, the variational procedure is a convex optimization that is constrained by a non-convex inequality constraint, and so it will generally have many local minima. This procedure, then, is impractical except for very small molecules.

By including contributions from the wave-functions of excited and ionized states in Eq.(20), one can consider promoted and oriented proatoms. This variational procedure is also intractable in general, except when all the states under consideration have vanishing transition densities, Eq.(24). This condition is achieved if one considers only wave-functions corresponding to different eigenvalues for $\hat{N}$, ${{\hat{S}}^{2}}$, or ${{\hat{S}}_{z}}$, which allows one to generalize the Hirshfeld method to the variational principle in Eq.(25). This extended variational Hirshfeld framework requires the optimization of a convex objective function subject to linear constraints, and thus has a unique solution. We believe that the proatoms used in Eq.(25) provide a very promising direction for further generalizing Hirshfeld partitioning.

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