Density-based partitioning methods for ground-state molecular calculations.

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Density-Based Partitioning Methods for Ground-State Molecular Calculations Jonathan Nafziger† and Adam Wasserman*,‡ †

Department of Physics and ‡Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, United States ABSTRACT: With the growing complexity of systems that can be treated with modern electronic-structure methods, it is critical to develop accurate and efficient strategies to partition the systems into smaller, more tractable fragments. We review some of the various recent formalisms that have been proposed to achieve this goal using fragment (ground-state) electron densities as the main variables, with an emphasis on partition density-functional theory (PDFT), which the authors have been developing. To expose the subtle but important differences between alternative approaches and to highlight the challenges involved with density partitioning, we focus on the simplest possible systems where the various methods can be transparently compared. We provide benchmark PDFT calculations on homonuclear diatomic molecules and analyze the associated partition potentials. We derive a new exact condition determining the strength of the singularities of the partition potentials at the nuclei, establish the connection between charge-transfer and electronegativity equalization between fragments, test different ways of dealing with fractional fragment charges and spins, and finally outline a general strategy for overcoming delocalization and static-correlation errors in density-functional calculations.

1. ATOMS-IN-MOLECULES Chemistry relies heavily on the idea that atoms do not entirely lose their identity when they come together to form molecules. Atoms and functional groups within different molecules behave in similar ways. However, quantum mechanics provides no convenient demarcation to establish which electrons belong to which fragment within a molecular context. To build chemical models, we would very much like to know how we can expect atoms and functional groups within molecules to behave in different chemical contexts, and how the molecular environment changes and deforms them. To pursue this question, it is helpful to provide a definition of what is meant by an atom within a molecule. Density-functional theory (DFT) gives us one possibility for keeping track of such changes. For a given molecular system we can divide its total electronic density, n(r), into atomic or fragment densities, nα(r). The question is then how to define the density of each fragment such that the fragment densities sum to the total electronic density,

experiment can distinguish one choice as more correct than any other. In that paper they advocated for the use of the Hirshfeld or stockholder partitioning2 of atoms-in-molecules, but they pointed out that various other choices may be better suited in certain circumstances. The Hirshfeld or stockholder fragment densities are given by nα(r) =

(1)

There are obviously many sets of fragment densities that may satisfy this equation, but is there an optimal way to choose them? We certainly have some intuition about how we would like atoms-in-molecules to behave. Some features of isolated atomic densities should be shared. We would expect the fragments to be well localized spatially, and we would like them to decay monotonically with distance from the nuclei. These ideas suggest a definition that looks for similarities to the density of the isolated species. Many possibilities exist, and as pointed out by Parr, Ayers, and Nalewajski1 the choice is necessarily ambiguous because no © XXXX American Chemical Society

∑α nα0(r)

n(r) (2)

where nα0 (r) is the density of the isolated α-fragment. Nalewajski and Parr showed that this definition of atoms-inmolecules yields fragments that are most similar to the isolated atoms according to an information theory measurement called the Kulback Liebler information distance.3 Another way to achieve similarity with the isolated atomic species is to follow the suggestion by Parr et al.4,5 that atoms-inmolecules be defined as the set of fragment densities having equal chemical potentials and minimum promotion energies from their ground state. They defined the promotion energy as the difference in energy of the atomic fragments when they are in the molecule as compared to when they are isolated. Thus, the promotion energy is a measure of the extent to which the isolated atomic densities must be deformed to satisfy the condition that all of the fragment densities add up to the correct molecular density. This minimization of fragment energies is also used in the partition theory of Cohen and Wasserman,6 forming a basis for partition density-functional theory (PDFT). 7,8 Palke9 and Guse 10 both developed

∑ nα(r) = n(r) α

nα0(r)

Received: April 25, 2014 Revised: July 3, 2014

A

dx.doi.org/10.1021/jp504058s | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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algorithms and performed calculations to find the atoms-inmolecules defined by Parr et al. for the hydrogen molecule and its positive ion. Guse’s method has some similarities with the algorithms of PDFT (the details of which will be explained in sections 6−9). The Hirshfeld or stockholder definition of atoms-inmolecules has strong parrallels with the definition of Parr et al., because both are based on a minimization principle. Rather than minimizing the energy difference between isolated atoms and the atoms in their molecular environment, the Hirschfeld partitioning minimizes the information distance between the two. Also well-known are Bader’s atoms-in-molecules.11,12 These are based on the topology of the molecular density. The fragments are nonoverlapping pieces of the total density, divided by boundaries where the gradient of the density normal to the boundary surface is zero. Bader’s quantum theory of atoms-in-molecules (QTAIM) contrasts strongly with the proposed definition of both Hirschfeld and Parr et al. because Bader’s QTAIM fragments have sharp discontinuities and are nonoverlapping, whereas the other two fragment definitions lead to smooth fragment densities that may overlap one another. From a density-functional viewpoint, the fragments of Parr et al. are attractive because, as can be shown, each fragment density is v-representable. Zhang and Wasserman studied the transferability of these fragment definitions and found that the method based on energy minimization led to the best transferability, at least in simple model systems.13 On the other hand, Bader severely critiqued the promotionenergy definition of atoms-in-molecules. He argued that QTAIM fragments are based on firm foundations of quantum mechanics and Schwinger’s principle of stationary action, whereas the atoms of Parr et al. are based on an arbitrary definition for the minimum promotion energy.14,15 The energies of each fragment in Bader’s method add directly to the energy of the system, whereas with the atoms of Parr et al. only the densities are additive and the energy contains nonadditive contributions. In other words, Parr et al. use fragments that are fictitious systems, not actually tied to the reality of quantum mechanics. Nevertheless, DFT regularly makes use of another equally fictitious system with great success (KS electrons), so we see little merit in Bader’s argument.

and in the early 1990s several general methods were developed that can be seen as self-consistent versions of the Gordon−Kim model. Cortona,23 Wesołowski and Warshel24 and Boyer and Mehl25 all developed formalisms for fragment-based DFT calculations. The starting point is the same for all these methods. The total electronic energy, E, is understood as a functional of the set of fragment densities {nα(r)}. It is divided into fragment contributions as E[{nα}] = Vnuc[n] + E H[n] + E XC[n] +

∑ Ts[nα] α

+

Tsnad[{nα}]

(3)

In eq 3, Vnuc, is the electron−nuclear energy, EH is the Hartree energy, EXC is the exchange−correlation energy, Ts is the noninteracting kinetic energy, and Tnad is the nonadditive s noninteracting kinetic energy. This last term is simply the difference between the kinetic energy functional evaluated for the sum of fragment densities and the sum of energy functionals evaluated for each of the fragment densities: Tsnad[{nα}] = Ts[n] −

∑ Ts[nα] α

(4)

This rewriting of the total energy is of course still in principle exact as long as eq 1 is satisfied. However, in practical calculations, the nonadditive kinetic energy term may be evaluated using approximate functionals whereas fragment kinetic energies may be calculated using KS orbitals. This is precisely the method of Gordon and Kim, except that the fragment densities were not calculated self-consistently. Allowing the calculations to be done self-consistently was an important step forward. Cortona in 199123 performed calculations for a nonmagnetic crystal with fragment densities centered at each lattice point. The nonadditive kinetic energy was approximated using the Thomas−Fermi functional, and the fragments were approximated as spherically symmetric. Cortona used the method to perform calculations for NaCl and KCl, treating all the atoms in the crystal lattice as separate fragments, and finding the total energy by minimizing it with respect to each of the fragment densities. All of the fragments in Cortona’s calculation reach self-consistency and subsequent calculations in which all fragments are fully relaxed and self-consistent are commonly referred to as subsystem-DFT.26 In 1993 Wesołowski and Warshel24 developed frozen-density embedding theory (FDET). They considered initially a twofragment system where one fragment was a solute molecule and the other fragment was its solvent. Rather than self-consistently relaxing both fragment densities, the larger solvent fragment was frozen as an approximation and allowed to create an effective potential for the solute, which was found selfconsistently. FDET may also be made completely selfconsistent by using freeze−thaw cycles in which the fragment being held frozen is alternated. In this case subsystem-DFT and FDET become equivalent because they use the same underlying equations, the Kohn−Sham equations for constrained electron density (KSCED, see section 3 below). These methods have been used for studying many systems including the spectroscopy of complex systems and solvatochromatic shifts.27−30 For recent reviews of subsystem-DFT and FDET calculations there are several excellent reviews available.26,31,32 Govind et al. in 199833 developed a method to embed different levels of calculation into a DFT calculation. The

2. FRAGMENT-BASED DENSITY-FUNCTIONAL METHODS In addition to its use as an interpretive tool in chemistry, the division of the density into fragments has also been used as a practical tool in electronic structure calculations. Gordon and Kim in 197216 essentially used eq 1 to construct the molecular density of rare-gas dimers from isolated atomic Hartree−Fock densities. They made non-self-consistent calculations based on the assumption that the density of the dimer was not significantly changed from superimposed atomic densities. They then used density functionals based on the electron gas to estimate the relevant binding energy curves. Other workers added corrections to account for self-interaction errors17,18 and to include induction effects and dispersion forces.19 For an early review on the Gordon−Kim model see ref 20. Vela, Cedilla, and Gázquez21 made the connection between calculations of Gordon and Kim and the atoms-in-molecules of Parr et al. Self-consistent versions of the Gordon−Kim model were not far behind. Senatore and Subbaswamy were the first in 198622 B



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4. NONUNIQUENESS OF SOLUTIONS OF THE EXACT KSCED EQUATIONS If Tnad s and its functional derivatives are calculated exactly, these equations will exactly reproduce a given KS calculation performed on the whole system. We refer to this as an “exact solution to the KSCED equations”, but it should be noted that this does not mean that the results would exactly reproduce the fully interacting system, unless of course the exact XC functional is used. It is known that the exact solution to the KSCED equations is not unique. We show below that in fact any set of noninteracting v-representable densities, {nα}, satisfying eq 1, will necessarily be a solution to the KSCED equations. This indicates that with the exact functional for Tnad s , a self-consistent solution to eqs 7 and 8 yields different results dependent on the choice of initial fragment densities and the method of convergence. Interestingly, this nonuniqueness disappears when an approximation is employed for Tnad s . In such a case, sets of fragment densities summing to the same total density will not have the same total energy, and because the total energy is minimized, the KSCED equations will yield the set of fragment densities that minimizes the error due to the approximate nonadditive kinetic energy functional employed. It has been observed that local and semilocal approximations for the nonadditive kinetic energy lead to fragments that reduce their mutual overlap.48,49 Each approximation to Tnad results in a s unique set of fragment densities and fragment potentials that together satisfy the KSCED equations. Nonunique potentials corresponding to the exact Tnad are s illustrated in Figure 1, where we compare the embedding

method uses the ideas of Wesołowski and Warshel and Cortona, except that one of the subsystems may be treated with a more accurate wave function method. The wave function method is used to calculate the energy and density of a chosen subsystem in the presence of the embedding potential that is generated from DFT calculations on the other fragments. This allows specific parts of a larger DFT calculation to be treated with higher accuracy. This method is usually referred to as embedding-DFT. See the review of Huang and Carter for a more in depth account of this method.34 The self-consistent atomic deformation theory (SCAD) of Boyer, Mehl, and co-workers,25,35−39 which was also developed in 1993, can be viewed as a version of subsystem-DFT requiring that fragment densities be written as atomic densities. Although it is not the focus of the present article, there are also timedependent versions of fragment-based DFT.40−45

3. FRAGMENT SELF-CONSISTENCY To calculate fragment densities in the methods mentioned in the previous section, the total energy of eq 3 is minimized by allowing each fragment density to vary subject to the constraint that it is normalized to an integer number of electrons, which are then said to belong to that fragment. As we shall see, this leads to an embedding potential for each fragment. In FDET this potential is updated for one fragment at a time, leaving the other fragments frozen, whereas in subsystem-DFT all fragment embedding potentials are updated simultaneously. The minimization leads to an Euler equation for each fragment: 0=

μα =

δE[{nα}] − μα δnα(r)

(5)

δTs[nα] δTsnad[{nα}] + + vnuc[n](r) + vH[n](r) δnα(r) δnα(r) + vXC[n](r) (6)

where the potentials are the functional derivatives of the energy terms in eq 3. The lagrange multiplier, μα, enforces the constraint that fragment α has Nα electrons. It is also referred to as the chemical potential of fragment α. One then assumes that each fragment density has an auxiliary Kohn−Sham system of noninteracting electrons with identical densities.46 This leads directly to the fragment Kohn−Sham equations with constrained electron density (KSCED):23,24,47

{

1 − ∇2 + vαKSCED[nα](r) ϕi , α(r) = ϵi , αϕi , α(r) 2

}

Figure 1. Exact embedding potentials for the helium dimer. The two helium atoms are placed at x = ±3 and labeled as left (nucleus at x = −3) and right (nucleus at x = +3). The top and bottom figures show different methods for converging the calculation. In the top frame the embedding potentials are iterated simultaneously, and in the bottom frame freeze-and-thaw cycles are applied to reach self-consistency. For each of these sets of potentials the exact total energy and exact molecular density are recovered, although the fragment densities are different in each case.

(7)

where the KSCED potential, vKSCED , is given by α vαKSCED[nα](r) =

δTsnad[{nα}] + vnuc[n](r) + vH[n](r) δnα(r) + vXC[n](r)

potentials and densities obtained for the helium dimer using two different methods of convergence, the freeze-and-thaw cycle47 and simultaneous relaxation.23 In both cases the densities of the isolated fragments were used as the initial guess, but the fragment densities at convergence are clearly different for the two convergence methods. The total energies and densities from both are the same, despite ending with

(8)

The last three potential terms simply constitute the KS potential for the entire molecular system. These equations form the basis for both FDET and subsystem-DFT. The KSCED potential may be divided further into fragment and nonadditive terms as will be explored in section 5. C



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Note that for approximate Tnad s this equality does not hold, but for the exact case we can plug eq 11 into eq 8 to obtain

differing sets of fragment densities. This occurs whenever the nonadditive terms match exactly the corresponding fragment energy terms. In this case, the total energies will be identical for any set of fragment densities that add up to the same total density. Unique potentials from approximate Tnad can be seen in s Figure 2, where the simplest approximation for Tnad s , the

vαKSCED[nα](r) = μm − μα + vnα(r)

Because μm − μα is just a constant, the new KSCED potential for fragment α is equivalent to vnα(r), which is exactly the potential needed to produce the fragment density nα. Thus, provided Tnad is exact, we have shown that any arbitrary set of s noninteracting v-representable densities that satisf y ∑αnα(r) = n(r) is a solution to the KSCED equations. In this way we see that the nature of the KSCED equations with approximate nonadditive functionals differs qualitatively from that of exact nonadditive functionals; in the first case there is a unique solution, whereas in the second there are many. Nonuniqueness in the exact case is an important consideration because several methods have been developed to perform calculations with the exact KSCED equations using OEP-like techniques.31,52,58 Though computationally expensive, these techniques can be used as a benchmark to compare with calculations using approximate Tnad functionals. To this end, it s would be ideal to have a method known to provide a unique solution when the exact nonadditive functionals are used. In the next section we explore how to achieve unique solutions, even in the exact case.

Figure 2. Approximate embedding potentials for the He2 dimer using the Thomas−Fermi (TF) approximation of Tnad s . The two monomers are placed at x = ±3 and labeled as left (nucleus at x = −3) and right (nucleus at x = +3). The left two plots show the solution to the KSCED equations, which is unique for the TF approximation, but the potentials are not global, meaning there is a different embedding potential for each fragment. The right two plots show results from PDFT using the local-Q approximation, where there is a global partition potential shared by each fragment. In each case the log of the fragment densities are plotted above the corresponding embedding potentials. Both cases are compared with the exact partition potential (solid black) which is unique and global.

5. PATHS TO UNIQUENESS Uniqueness has been achieved from two directions. First, through the development of partition density-functional theory (PDFT)7,8 in 2010, based on the earlier partition theory,6 and second through a re-formulation of embedding-DFT by Huang, Pavone, and Carter in 2011.56,57 In both cases the key to ensuring uniqueness of the fragment densities is a global embedding potential. Although each fragment density in standard subsystem-DFT is independently varied to minimize the total energy, the requirement that some portion of each fragment’s effective potential is global (i.e., the same for all fragments) constrains the solution to be unique. This does not mean that each fragment KS system sits in the same potential (although this would clearly lead the unique set of fragment densities that are all identical), but instead, just the parts of each fragment’s effective potential that depends on other fragment’s densities are identical for all fragments. Cohen and Wasserman proved that there is a one-to-one mapping between this type of global one-body potential, shared by all fragments, and the corresponding set of fragment densities.59 Both PDFT and the method of Huang, Pavone, and Carter rely on this theorem to ensure the uniqueness of the fragment densities. The first step in both of these developments is to define the energy associated with each fragment. In other words, each piece of the energy in eq 3 must be divided into additive contributions from the fragments and nonadditive contributions from the interaction:

Thomas−Fermi approximation,50,51 is used. In this case both freeze-and-thaw and simultaneous relaxation yield the same fragment densities and fragment embedding potentials, whereas in the exact case (Figure 1) they gave different results. We plot the log of the density to show that the fragment densities have very small cusps at the location of the other nuclei. To show the nonuniqueness of the exact KSCED equations, we start by identifying the functional derivative of the nonadditive kinetic energy. If the fragment densities are noninteracting v-representable (nα is a ground-state density for Nα noninteracting electrons in the potential vnα), then we write down the Euler equation for a system of noninteracting electrons corresponding to density nα: δTs[nα] + vnα(r) = μα δnα(r)

(9)

Also, the molecular Euler equation with chemical potential μm reads δTs[n] + vnuc[n](r) + vH[n](r) + vXC[n](r) = μm δn(r)

(12)

E[{nα}] = (10)

∑ Eα[nα] + Eint[{nα}] α

These two expressions can be combined to find the functional derivative of the nonadditive kinetic energy:

(13)

This partitioning leads to a definition of fragment energies and a corresponding interaction energy. The interaction energy is defined as the quantity that makes eq 13 exact, but it can also be written as a sum of nonadditive functionals:

δTsnad[{nα}] = μm − μα + vnα(r) − vnuc[n](r) − vH[n](r) δnα(r) − vXC[n](r)

nad E int[{nα}] = Tsnad[{nα}] + E Hnad[{nα}] + E XC [{nα}] nad [{nα}] + Vnuc

(11) D

(14)



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To make this division, the external potential, vnuc(r), is divided into fragments vαnuc(r). This can be done arbitrarily as long as the fragment potentials sum to the total external potential. There are a few subtle differences between the PDFT and Huang−Pavone−Carter definition of fragment energies, as we will mention later. Huang, Pavone, and Carter continue the derivation along the same lines as subsystem-DFT, by minimizing the total energy with respect to fragment densities. The corresponding KSCED potentials are then divided into contributions from the local fragment and a term arising from the interactions between the fragments: α α vαKSCED[nα](r) = veff [nα](r) + vemb [{nα}](r)

The first potential,

vαeff,

(15)

depends just on the fragment density,

α α α α veff [nα](r) = vnuc (r) + vcoul [nα](r) + vXC [nα](r)

whereas the embedding potential, fragment densities: α vemb [{nα}](r) =

electrons and thus may potentially have a noninteger number of electrons. The correct way to describe an isolated system with fractional numbers of electrons is through ensembles, and thus we use them to handle the energies for the PDFT fragments. It should be noted that it is the minimization of the sum of fragment energies that provides the uniqueness discussed in the previous section, and not the ensemble treatment of fragment energies. The derivation of PDFT7 starts with the definition of the energy, Eα[nα], of each fragment. Each fragment energy is the energy of Nα electrons in the fragment potential, vαnuc(r). The sum of fragment energies is the energy of a fictitious system of noninteracting fragments constrained to have the same total density, n(r), as the interacting system of fragments. When a fragment has a noninteger number of electrons, its energy, Eα[nα], is defined through an ensemble of two systems, each containing an integer number of particles:

vαemb,

(16)

depends on all the other

Eα[nα] = ωαEvα[n p + 1] + (1 − ωα)Evα[n p ] α

(18)

Here, ωα is the noninteger portion of Nα and pα is the lower bounding integer of Nα. These fragment energies are defined as

β ∑ (vnuc (r) + vHβ[nβ ](r)) β≠α nad nad + vXC [{nα}](r) + v kin [{nα}](r)

α

Evα[n p ] = Ts[n p ] + E H[n p ] + E XC[n p ]

(17)

α

Huang, Pavone, and Carter then impose the additional constraint that this embedding potential must be the same for all fragments.57 In a later paper, they formulate the embedding problem as a potential functional of this global embedding potential.56 The globality of the embedding potential ensures that the fragments are unique through the theorem of Cohen and Wasserman.59 PDFT7,8,60 takes a slightly different route. Rather than directly minimizing the total energy, the sum of fragment energies (first term of eq 13), is minimized subject to the constraint that the sum of fragment densities is equal to the molecular density. This minimization of fragment energies guarantees the uniqueness of the fragment densities and the globality of the embedding potential, due to the same theorem.59 This slightly different route will still lead to the exact groundstate energy of the molecular system given the same assumptions made in the more general case of subsystemDFT. The KSCED equations will lead to fragments that sum to the exact molecular density as long as the molecular density is noninteracting pure state v-representable and it is decomposable into fragment densities that are noninteracting pure state v-representable.61 These requirements apply to PDFT as well, except that the fragment densities in PDFT need only be noninteracting ensemble v-representable. Additionally, both in the formulation of ref 56 and in PDFT there is the requirement that there exists a global embedding potential that is added to the external potential of all fragments. The proof of Cohen and Wasserman establishes the existence and uniqueness of this global potential.59

α

+

∫

α

α vnuc (r)

α

3

n p (r) d r

(19)

α

With this ensemble construction, the noninteracting kinetic energy, the hartree, and exchange−correlation energy functionals only ever act on densities with integral numbers of electrons. The densities npα(r) and npα+1(r) are defined as those that minimize the sum of fragment energies, Ef ≡

∑ Eα[nα] (20)

α

and simultaneously satisfy two constraints. First, the sum of fragment densities, nf(r) ≡ ∑αnα(r), is equal to the molecular density, where each fragment density is calculated from the ensemble as nα(r) = ωαn p + 1(r) + (1 − ωα)n p (r) α

α

(21)

Second, the total fragment density integrates to N, the total number of electrons in the system, N=

∫ nf (r) d3r = ∑ Nα α

(22)

Overall, this constrained minimization of Ef is transformed into an unconstrained optimization of G through the introduction of two lagrange multipliers: the partition potential, vp(r), and the molecular chemical potential, μm. G[{nα}] = Ef [{nα}] +

6. DERIVATION OF PDFT The fragments of PDFT are designed to connect smoothly to a system of dissociated fragments. To this end, the definition of PDFT fragments is meant to mimic the behavior of a physical system of isolated fragments. This is the motivation for minimizing the sum of fragment energies. To further mimic a physical system, the PDFT fragments are allowed to exchange

∫

∫ vp(r)(nf (r) − n(r)) d3r − μm

( n f (r) d3r − N )

(23)

The molecular chemical potential, μm, controls the total number of electrons in the system. The functional G[{nα}] must then be minimized with respect to every element of the fragment ensembles, yielding the Euler equations: E


The Journal of Physical Chemistry A 0=

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cancels it such that the total energy is unaffected.56 In contrast, the fragment energies of PDFT do not include energetic contributions from the corresponding partition potential, vp(r). The only contributions of the partition potential to the fragment energies are through the fact that the partition potential deforms the fragment densities so that they sum to the correct total density. As mentioned in the previous section, we can write the fragment densities as functionals of the partition potential, and therefore we can also write all of the energy components of eq 28 as functionals of the partition potential:

δG δn p (r) α

= (1 − ωα)

δEα [n p ] α

δn p (r)

+ (1 − ωα)vp(r) − (1 − ωα)μm

α

=

δEα [n p ] α

+ vp(r) − μm

δn p (r) α

=

δTs[n p ] α

+ vH[n p ](r) + vXC[n p ](r) + vα(r) + vp(r) − μm

δn p (r)

α

α

α

E[vp] = Ef [vp] + Ep[vp]

(24)

To solve eq 24, we use a KS system of pα electrons in the external potential vα(r) + vp(r) to obtain the density npα(r) =

Assuming that the density is ensemble v-representable, the Hohenberg−Kohn theorem62 indicates that if we find the minimum of the total energy with respect to vp(r), then the corresponding density nf(r) must be equal to the correct total density, n(r). Variation of the total energy with respect to the partition potential gives us a new Euler equation:

p

∑i a |ϕi (r)|2 , where the fragment orbitals are determined by the pa

pa

fragment KS equations:

{− 12 ∇ + v 2

eff α [n pα ](r)

}

+ vp(r) ϕi , p (r) = ϵi , αϕi , p (r) α

α

(25)

0=

and the fragment effective potential is given by vαeff [n p ](r) = vH[n p ](r) + vXC[n p ](r) + vα(r) α

α

α

(26)

=

Through these equations we see that each fragment density may be written as a functional of the partition potential and the occupation number Nα = pα + ωα, nα = nα[vp ,Nα](r)

δE δvp(r)

∑ α

δEp δEα + δvp(r) δvp(r)

(30)

Let us consider first the first term, which we expand by using the chain rule:

(27)

∑

This ensures that our fragment energies are minimized. Our next step is then to find how the partition potential and occupation numbers may be calculated such that both constraints are satisfied.

α

δEα = δvp(r)

⎛

∑ ∫ ⎜⎜(1 − ωα) ⎝

α

+ ωα

7. PARTITION POTENTIAL OPTIMIZATION We must choose the lagrange multiplier vp(r) such that it correctly enforces its constraint that the sum of fragment densities be equal to the molecular density. To do this, we write the total molecular energy (corresponding to nf(r)) as the sum of fragment energies, Ef, and the partition energy, Ep. The partition energy is defined exactly through this equation but may be approximated in practical calculations: E[{nα}] = Ef [{nα}] + Ep[{nα}]

(29)

δEvα

δn p (r′) α

δn p (r′) δvp(r) α

δn p + 1(r′) ⎞ α ⎟ d3r′ δn p + 1(r′) δvp(r) ⎟⎠ α δEvα

(31)

Because each of the npα(r) comes from the solution to eq 24 we can write eq 31 as

∑ α

δEα = δvp(r)

∫ (μm − vp(r′)) ⎛

∑ ⎜⎜(1 − ωα)

(28)

α

⎝

δn p (r′) α

δvp(r)

+ ωα

δn p + 1(r′) ⎞ α ⎟ d3r′ δvp(r) ⎟⎠ (32)

The partition energy is thus very close in definition to the interaction energy, Eint, of eq 13 used by Huang, Pavone, and Carter in ref 57. However, the notation should be kept distinct due to a few subtle but important differences. First, in some cases the Huang−Pavone−Carter method restricts the number of electrons in each fragment to be an integer so that the method may be integrated with wave function methods.57 In a later paper Huang and Carter allow for fragment occupations to be noninteger, but their noninteger densities are constructed by fractionally occupying the HOMO from a single KS calculation56 rather than by using the ensemble construction of PDFT. Any such small difference in how fragment energies are defined will lead to different fragment densities and energies. The other difference between Eint and Ep is that fragment energies corresponding to Eint contain the energy term, ∫ nα(r) vemb(r) d3r, and thus the interaction energy contains a corresponding term, −∫ n(r) vemb(r) d3r, which

Using our definition of nf(r), we find that the first term becomes ∫ (μ − vp(r′))(δnf(r′)/δvp(r)) d3r′. Because variation of the partition potential is independent of variation in the occupation numbers it will only induce norm-conserving changes in nf(r). As a result, the integral over the constant term μ vanishes, and therefore

∑ α

δEα =− δvp(r)

δn (r′)

∫ vp(r′) δvf (r) p

d3r′ (33)

Now we may consider the second term of eq 30. Cohen and Wasserman59 established that there is a one-to-one mapping between nf(r), the sum of fragment densities minimizing Ef[{nα}], and the partition potential, vp(r). This indicates that nf(r) is a functional of vp(r) allowing us to use the chain rule to expand the functional derivative of Ep: F


The Journal of Physical Chemistry A δEp δvp(r)

=

∫∫∑ ∑ α

x = 0,1

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placing Na electrons on the left (a) and Nb electrons on the right (b). In these cases we calculate a target molecular density and corresponding chemical potential ahead of time using a one-dimensional real-space grid. We then make a search over partition potentials and directly optimize eq 23. Our optimizer works by calculating values of eq 23 along with its the first and second-order functional derivatives with respect to the partition potential. With these derivatives the optimization of eq 23 is quite efficient and our optimizer typically converged within 6 to 10 iterations. The functional derivative of G with respect to the partition potential is the difference between the sum of the fragment densities and the target density:

δn p + x(r″) δn (r′) α f d3r′ d3r″ δn p + x(r″) δnf (r′) δvp(r) δEp

α

(34)

Combining eqs 33 and 34, we see that the only way that the Euler equation will always be satisfied is if vp(r) =

∫∑ ∑ α

x = 0,1

δEp

δn p + x(r′)

δn p + x(r′)

δn f (r)

α

α

d3r′ (35)

or vp(r) =

∫∑ ∑ α

vp, α , x(r′) Q α , x(r′,r) d3r′

x = 0,1

δG = n f (r) − n(r) δvp(r)

(36)

where the fragment-dependent partition potential and Qfunctions are given by vp, α , x(r) =

The second derivative is simply the response of the fragment densities to changes in the partition potential:

δEp δn p + x(r)

δn (r′) δ 2G = f δvp(r) δvp(r′) δvp(r)

(37)

α

and Q α , x(r′,r) =

α

(38)

The Q-functions must satisfy the sum-rule:

∑∑

8

Q α , x(r′,r) = δ(r′ − r)

x = 0,1

α

(39)

8. LOCAL-Q APPROXIMATION The Q-functions represent the way in which the fragment densities respond when a small variation, δnf, is made to the molecular density. Some of this variation in density will be distributed to each component of each fragment ensemble. These functions are not known as explicit density functionals and in practical calculations they may be approximated using the local-Q approximation:8 (r′,r) = Q plocal ,x α

n p , x(r) α

n f (r)

δ(r − r′)

(40)

However, it is important to note that when the exact Ep functional is used, either through inversion58 or through use of the exact kinetic energy density functional (known only in limiting cases), then the choice of Q-function approximation is irrelevant as long as the sum rule of eq 39 is satisfied. This is due to the interesting fact that in these cases, at convergence, the functional derivative of the partition energy with respect to any of the fragment densities is exactly the same. In cases where approximate nonadditive functionals are used to construct the partition energy functional then using the local-Q approximation may restrict the partition potential, leading to densities that are not necessarily optimal with respect to the total energy. A potential functional formalism such as the one described by Huang and Carter56 will avoid this issue and yield the optimal global partition potential. We use simple numerical models to check the validity of the local-Q approximation, eq 40. For example, use a double well potential with N = Na + Nb noninteracting electrons: v(x) = −Za cosh(x + R /2)2 − Zb cosh(x − R /2)2

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This response can be calculated as a sum of fragment responses, which in turn are calculated using the fragment orbitals and perturbation theory. For this model system we can easily find the numerically exact Q-functions to compare with the local-Q approximation. In the following, we chose the values of Za = Zb = 3 and Na = Nb = 3 and considered two values of the separation between the wells, R = 3, 10. We found the corresponding molecular density and the resulting fragment densities. Then, for small changes in the molecular density we reoptimized the fragment densities and occupations. The results of these variations yield the exact Q-functions. The exact Q-functions along with the corresponding local-Q approximation are displayed in Figure 3. We note

δn p + x(r′) δn f (r)

(42)

Figure 3. Numerically evaluated Q-functions, Qb(x′,x) = δnb(x′)/ δnf(x), for one-dimensional noninteracting electrons. (No ensembles are needed when fragments for noninteracting electrons are calculated.) The top row consists of calculations for the separation R = 3, and the bottom row is for separation R = 10. The left column shows the numerically exact Q-function; the right column is the localQ approximation from eq 40. Note that the local-Q approximation works well in both cases but actually improves at large separation.

that at both separations the local-Q approximation accounts for nearly all of the exact Q-function. For R = 10, the only nonlocal contribution resides close to the bond-midpoint where the fragment densities are both very small.

9. OCCUPATION NUMBER OPTIMIZATION The occupation numbers are also optimized to satisfy the PDFT requirement that the sum of fragment energies is

(41) G



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how this difference in chemical potential in the isolated fragments affects charge transfer between fragments in the molecule. The results, displayed in Figure 4, indicate that the fragment occupations are locked into integers until the absolute

minimized. In general, there are two possibilities that may occur when eq 23 is optimized with respect to the occupations numbers, {Nα}. The first possibility is that there will be a minimum of G at a noninteger value of one or more of the Nα. In this case we can find the stationarity condition by taking the derivative of G with respect to the noninteger part of the occupation number, ωα. 0=

∂G ∂ωα

= Eα[n p + 1] − Eα[n p ] + α

α

∫ vp(r)(n p +1 − n p ) − μm α

α

= μα − μm (44)

Here we have defined a chemical potential for each fragment in a molecule, and we see that because μm is a global quantity, all of these fragment chemical potentials must be equal.59 It is important to note that the chemical potential derived from PDFT fragment energies will be different from those defined by subsystem-DFT and FDET, where fragment energies may be defined differently than in PDFT. The notable difference is that in subsystem-DFT and FDET the electron−nuclear energy is not split into fragment and nonadditive pieces, and instead each fragment energy includes the interaction between the fragment density and all of the nuclei, rather than just a subset of nuclei assigned to a fragment. This difference in fragment energy has the effect that, in general, chemical potentials within subsystemDFT and FDET are equalized even at integer occupation numbers.54,63,64 Within PDFT it is also possible for there to be no minimum at noninteger values. In this case the occupation numbers will be locked into integer values and there will not necessarily be chemical potential equalization between the fragments. We will explore these two possibilities in the next section. The partition potential and occupation numbers may be simultaneously optimized to find the extrema of the functional of eq 23. The first derivative with respect to the occupation numbers is given in eq 44, and its second derivative is zero. The cross derivatives with respect to occupation numbers and the partition potential are identified as the fragment Fukui functions: ∂ δG = fα (r) = n p + 1(r) − n p (r) α α ∂ωα δvp(r)

Figure 4. Charge transfer between fragments as a function of the difference in chemical potential of the isolated fragments. The chemical potential difference is μ0b − μ0a. When this difference is above +1, charge transfers to fragment a, and when it is below −1, charge transfers in the opposite direction. Between about −1 and +1, no charge tranfer takes place due to a cusp in G with respect to particle number.

difference in chemical potential is greater than a value close to μ0b − μ0a = 1. In this region the infimum of G has a cusp at Na = 2 and Nb = 2, which means that there is no minimum in between integers and the fragment chemical potentials cannot equalize. In the two outer regions, charge is transferred between the fragments and the fragment chemical potentials are equalized. See ref 66 for further discussion of chemical potential equalization in PDFT. For any of these potentials we can also fix the occupation numbers to arbitrary values, provided Na + Nb = N. In Figure 5 we compare occupation numbers fixed at integers with the optimized PDFT occupation numbers. We considered the cases where Za = 5.5 and Za = 3.0 with two choices for fixed

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10. CHARGE TRANSFER BETWEEN FRAGMENTS In cases where the optimized occupation numbers are not integers, some charge has transferred from one fragment to another. The PDFT occupation numbers can then serve as a method of population analysis. We use our one-dimensional PDFT solver to study how electrons are transferred between fragments. Placing a total of Na + Nb = 4 electrons in the potential of eq 41, we vary the strength of the well in fragment a from Za = 1 to Za = 6 while fixing Zb = 3. This effectively tunes the electronegativity of fragment a. As we make the well deeper, some of the charge transfers from fragment b to fragment a. As a measure of the difference in electronegativity, we look at the differerence in the chemical potential of the isolated fragments. The negative of the chemical potential has been identified with the electronegativity.4,65 We can then observe

Figure 5. Fragment densities with various occupation numbers. The left-hand column has plots of the density, and the right-hand column displays the log of the densities. The first row is for occupation values Na = 3 and Nb = 1; the second row is for occupation values Na = Nb =2. The last row uses occupation numbers Na = 2.2469 and Nb = 1.7531, the optimized PDFT occupations. This illustrates that the optimized fragment occupations yield the most localized fragments. It also shows that when chemical potential equalization is reached, the fragments all have the same assymptotic behavior. H



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occupation numbers (Na = 3 and Nb = 1 as well as Na = Nb = 2). The optimized PDFT occupation numbers are Na = 2.2469 and Nb = 1.7531. As in previous studies of PDFT,67 regardless of the choice for occupation numbers, we were always able to to find a global partition potential satisfying the constraint nf(r) = n(r). When the PDFT occupation numbers are used, the chemical potentials of all fragments are identical, but when occupation numbers are fixed, the chemical potentials do not equalize. Because the PDFT chemical potentials control the exponential tails of the fragment densities, this equalization (or lack of) can be seen in the log of the density (plotted in the second column of Figure 5). In the first two rows of plots, which correspond to fixed occupation numbers, the fragment densities decay at different rates whereas the PDFT occupation numbers lead to densities that decay at exactly the same rate. This differs from fragment chemical potentials in subsystemDFT and FDET which, due to differences in definition for the fragment energy, do not control the exponential tails of the corresponding fragment densities.64 We also see that the PDFT occupation numbers lead to fragment densities that are much more localized, resulting in a smaller overlap between fragments. This has important implications for calculations using approximate nonadditive kinetic energy functionals, when it is preferable to have as little overlap as possible, as large errors are associated with regions of large overlap.68 By reducing the regions of overlap between fragments, we reduce the dependence on these approximate (noninteracting) kinetic energy functionals.

∂ [nother(r )] = −2Zvapnother(r) ∂r ̅ r=0 r = ra

Because the derivatives and spherical averaging are linear operations, we can combine eqs 46−48 to get a Zext [na(r) + nother(r)]

=

a Zvap = Zext

r = ra

r = ra

a + (Zext + Zvap)na(r)

(49)

r = ra

n f (r) − na(r) n f (r)

a = Zext r = ra

nother(r) n f (r)

r = ra

(50)

It can be shown that this singularity is contained entirely in the potential-energy contribution to the partition potential when the local-Q approximation is used. The external potential component of the partition energy is nad Vnuc =

∫ nf (r) vext(r) dr − ∑ ∫ nα(r) vα(r) dr α

=

∑ ∫ nα(r) vβ(r) dr (51)

α≠β

The functional derivative of this quantity with respect to a fragment density nα(r) is simply ∑β≠αvβ(r), so the contribution to the partition potential due to this portion of the partition energy is nad vnuc (r) =

∫ ∑ ∑ vβ(r′) Q α(r′,r) d3r′ α

β≠α

(52)

Plugging this result into the local-Q approximation of eq 40, we find nad vnuc (r) =

∑ ∑ vβ(r) nα(r)/nf (r) α

β≠α

(53)

If we consider the contribution from a single fragment potential that contains a singularity, we can see that the local-Q approximation satisfies eq 50. This effect can be seen in the partition potentials of Figure 10, most notably in the cases of H2+ and H2 where the fragments are relatively close together, and thus the ratio of nother(r) to nf(r) is larger.

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12. CONNECTION TO SUBSYSTEM-DFT In the formalism of Huang, Pavone, and Carter,57 the connection to subsystem-DFT is clear: an additional constraint forces the fragment embedding potentials to be identical. In PDFT, it is the Q-functions that allow us to connect to subsystem-DFT calculations. The fragment contributions to the partition potential, vp,α,x(r), given in eq 36, are closely related to the embedding potentials of subsystem-DFT or FDET. However, in PDFT calculations these fragment contributions to the partition potential are averaged together using the Qfunctions to obtain a global vp(r). In this way any code capable of performing subsystem-DFT calculations could use the localQ functions to average the embedding potentials for each fragment and obtain the unique partition potential for a PDFT calculation. Thus, the Q-functions provide a bridge between subsystem-DFT calculations and PDFT calculations. This averaging has interesting features that may help address some

Here, nf̅ (r) is the spherical average of the density a distance r away from ra. In PDFT the total density, nf(r), is the sum of the fragment density na(r) and all the other fragments, nother(r). The fragment a is the fragment whose potential va(r) contains the singularity from the external potential. Because the partition potential is global, both na(r) and nother(r) must also satisfy cusp conditions with their effective potentials at point ra. Fragment a contains the singularity in question plus any singularity from the partition potential: ∂ a [na(r )] = −2(Zext + Zvap)na(r) ∂r ̅ r=0 r = ra

Zvapnother (r)

which leads directly to

11. SINGULARITIES IN THE EXACT PARTITION POTENTIAL We now derive an exact condition that the partition potentials must satisfy. The presence of singularities in the external potential leads to cusps in the molecular density. These cusps in turn lead to singularities in the partition potential at the location of the singularities of the external potential. If we consider Coulomb potentials, −Z/r, then the strength, Zavp, of the singularity in the partition potential at the point ra is proportional to the ratio of fragment densities at the same point. To derive the condition, we start with Kato’s cusp condition69 for the molecular density due to a singularity in the external potential Zaext at the point ra: ∂ a = −2Zext [n f (r )] n f (r) ∂r ̅ r=0 r = ra

(48)

(47)

The other fragments live in an effective potential which, at the point ra, only contains singularities due to the partition potential: I



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are ones for which the embedding potential cannot be directly split into two pieces, one coming from the molecule and the other coming from the fragment. Jacob et al. proposed such an approximation that switches off the embedding potential near the frozen nuclei to fulfill the exact limit they found,70 and Fux et al. used this correction to improve the description of coordination bonds in FDET calculations.73 Lastra et al. also proposed an approximation based on switching, but instead focused on correcting just the kinetic contribution to the embedding potential in two-fragment cases where one fragment is negligable and the other fragment contains two electrons.72 An attractive feature of PDFT is that the Q-function averaging of eq 36, even within the local-Q approximation of eq 40, leads to fragment effective potentials that have the correct behavior at the location of nuclei in other fragments, according to the formula derived in section 11. Figure 7 compares partition and embedding potentials for the helium dimer. The solid line shows the potentials obtained

issues encountered with approximate subsystem-DFT embedding potentials, as indicated next. Examining eq 17, we see that the embedding potential for a given fragment has attractive coulomb singularities at the nuclei belonging to all other fragments. Jacob et al. showed in 2007 that the exact nonadditive kinetic energy potential cancels these contributions in the limit of large separation between fragments.70 However, approximate nonadditive kinetic energy potentials do not cancel these singularities and leave significant coulomb attraction near nuclei from other fragments. This is illustrated in Figure 6 for the helium dimer by decomposing the

Figure 6. Comparison of the components of an embedding potential in the region of the nonactive fragment. The total is in solid black, the potential energy component is in dashed green, the kinetic energy component is in dot-dashed blue, and the XC component is in dotted red. In the top figure the exact kinetic energy is used, and in the bottom the Thomas−Fermi kinetic energy is used. In the top figure the kinetic energy has a positive singularity, which cancels a large portion of the negative singularity from the external potential component, so overall the embedding potential is fairly flat. In the bottom figure the Thomas−Fermi kinetic energy does not cancel well with the negative singularity from the external potential component, and as a result, the embedding potential is not accurate in this region.

Figure 7. Comparison of exact vs Thomas−Fermi (TF) approximations for subsystem-DFT and PDFT for the He2 dimer. The two monomers are placed at x = ±3, and we refer to these fragments as left (nucleus at x = −3) and right (nucleus at x = +3). The solid black line is both the exact partition potential and an exact embedding potential for the left monomer (they are slightly different but indistinguishable here but may differ greatly, as in Figure 1). The two dotted lines represent calculations using the Thomas−Fermi kinetic energy functional. The dash-dotted line is the global TF partition potential, and the dashed line is the TF embedding potential for the left monomer.

embedding potential for one fragment into its kinetic, nuclear, Hartree, and exchange−correlation components. The figure shows the components of the embedding potential for the left fragment centered at x = −3 bohr, but it shows them near the right fragment, centered at x = +3 bohr. In the top frame, the components of the embedding potential are calculated exactly whereas in the bottom frame the Thomas−Fermi functional is used for the kinetic component. In the exact case, the large components of the embedding potential nearly cancel out, leaving an embedding potential that, in the case of He2, varies on the order of 10−2 hartree. On the other hand, the potential from the approximate kinetic energy functional cannot balance the other components in the region of the nonactive fragment, and as a result there is a relatively large singularity in the embedding potential at the location of the nonactive nuclei. In purely electrostatic embedding this can lead to orbitals from fragment α localizing at the nuclei from other fragments (known as the electron leak problem), but it has been shown that this does not happen in subsystem-DFT.71 In subsystemDFT, the nonadditive kinetic energy potential provides enough repulsion at these nuclei so that occupied orbitals do not localize. However, unoccupied orbitals could localize at those points, with implications for fragment-based linear response.43 Jacob et al.70 and Lastra et al.72 both proposed nondecomposable approximations to the embedding potential to correct this problem. These nondecomposable aproximations

with the exact Ts[n]. When the exact functional is used, the partition potential is virtually indistinguishable from the two embedding potentials of subsystem-DFT (however, the embedding potentials are not unique as discussed in section 4). Figure 7 makes it clear that, when the Thomas−Fermi functional is used, the partition potential and the embedding potentials differ significantly. The partition potential is global and, therefore, the same for both fragments, whereas the embedding potential is different for both (Figure 7 shows it for the left fragment, centered at x = −3 bohr).

13. NONADDITIVE KINETIC ENERGY Most fragment-based methods require calculation of the nonadditive kinetic energy, Tnad (eq 4), and its functional s derivative. Some methods avoid this by doing a molecular KS calculation ahead of time. Then the partition potential (or fragment embedding potentials in the case of nonglobal potentials) may simply be chosen so as to satisfy the density constraint. Alternatively, Manby et al.74 proposed a method based on projectors which ensures that orbitals from different subsystems are orthogonal to each other. In this case the kinetic J



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Table 1. Energies and Components of Ep, in Atomic Unitsa molecule H2+ H2+ H2 He2 Li2 Be2

(exact)

a

Ef

Ep

Vnad nn

Vnad ext

Tnad s

Enad H

Enad XC

2.0 2.0 1.446 6 5.122 4.522

−0.4470 −0.4250 −0.9119 −5.6689 −14.6750 −28.8749

−0.6556 −0.6589 −0.9173 −0.6668 −1.8068 −3.5770

0.5 0.5 0.6916 0.6667 1.7571 3.5383

−0.5704 −0.5654 −1.3476 −1.3334 −3.6781 −7.1230

−0.0852 −0.0818 −0.1521 0.000 04 0.0049 0.0604

−0.0421 −0.0457 0.5842 0.6668 1.8866 3.5338

0.0421 0.0341 −0.0019 −0.0001 −0.0201 −0.0001

a The LDA is used in all calculations (except in the first line corresponding to H2+). Calculations are performed at equilibrium geometries, except for He2.

14. BENCHMARK PDFT CALCULATIONS FOR HOMONUCLEAR DIATOMIC MOLECULES To obtain benchmark PDFT results free of basis-set errors, we use our own all-electron real-space code (CADMium, for “Chemical Atoms in Diatomic Molecules”). Following the work of many others79−83 we use prolate spheroidal coordinates to create a two-dimensional mesh. We then solve the Kohn−Sham equations on this mesh, treating the cylindrical problem analytically. We used the Libxc package to evaluate XCfunctionals.84 Table 1 displays energies for several diatomic molecules along with the sum of fragment energies, the partition energy, and its components. The partition energy is strictly electronic and does not include contributions from nonadditive nuclear− nuclear repulsion Vnad nn . However, it is interesting to note that the magnitude of the electronic partition energy is consistently close to, but not equal to, Vnad nn . The classical electrostatic terms: nad nad Vnad nn , Vnuc, and EH sum to approximately zero. Also, in Figures 8 and 9, the partition potentials for several diatomics are compared along with the density difference

energies of the subsystems are additive and the nonadditive portion vanishes. Although approximate functionals for Tnad (eq 4) are widely s available, they are generally not well-suited for partitions that cut across covalent bonds.48,75,76 However, a number of methods have been developed that use inversion techniques to calculate embedding potentials exactly.31,52−58 In our work, we use the method mentioned by Jacob et al.70 and implemented by both Fux et al.31 and Goodpaster et al.54 We first use a Wu−Yang inversion77 to find the potential corresponding to the sum of fragment densities, nf(r). This inversion yields the potential vs(r), and the corresponding chemical potential, μm. We can then use eq 11 to find the functional derivative with respect to the first term of eq 4. We can similarly use the potentials and chemical potentials corresponding to each np(r) and np+1(r) to obtain the functional derivative of the second term of eq 4. These potentials are available without the need for an inversion. Although the procedure is not particularly efficient we do obtain the numerically exact noninteracting kinetic energy for the molecular system. Alternatively, an explicit density functional may be used to evaluate Tnad and its functional s derivative, in which case the computational cost will be greatly reduced at the cost of some accuracy. An alternative to the above PDFT algorithm is used by Elliot et al. in 20107 and Nafziger et al. in 2011.58 The full derivation of the update formula can be found in Elliot et al., but we briefly describe how it works. At each step, k, the current guess for the partition potential is used to determine each fragment density. Then the sum of fragment densities is inverted to obtain its corresponding potential, v[nkf (r)](r). This potential is compared to the molecular potential (which is the potential that produces the correct molecular density) to update the partition potential according to vpk + 1(r) = vpk(r) + (v(r) − v[n fk ](r))

Figure 8. PDFT(LDA) calculations on H2+ and H2. The left column displays the difference between the left fragment density and the isolated atomic density, and the right column displays the corresponding partition potential. The density for the fragment in the molecule sits in the same potential but with the addition of the partition potential. The density for the fragment in the molecule sits in the same potential as the isolated fragment, but with the addition of the partition potential. In both plots, the partition potential is depressed in the bonding region (see also Figure 10), increasing the density in that region.

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We can see that this update will only have a fixed point when v(r) − v[nkf (r)](r) = 0. The Hohenberg−Kohn theorem ensures that this can only occur when the sum of fragment densities is equal to the molecular density. The inversion calculations used in these algorithms can be quite difficult. We used a real-space grid for the inversions of the KS potential, thus avoiding many of the problems associated with inversions using basis sets. Although inversions with basis sets are possible, greater care must be taken to ensure that the result is unambiguous.78

between one of the two fragments and the corresponding isolated fragment density. The isolated fragment and the fragment in the molecule share the same potential except for the partition potential which deforms the fragment in the molecule. In general, we can see that the partition potential K



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the partition potentials for H2+ and H2 are the largest, followed by Be2 and then Li2. He2 has the smallest partition potential. To analyze vp(r), it is useful to split it into components, in the same way as the energy. These components are compared in Figure 11. The Hartree and the external potential

Figure 9. PDFT(LDA) calculations on He2, Li2, and Be2. The left column displays the difference between the left fragment density and the isolated atomic density, and the right column displays the corresponding partition potential. In all three plots, the partition potential is depressed in the bonding region (see also Figure 10), increasing the density in that region. In the case of Be2, the partition potential squeezes each fragment and elongates it along the bond axis. In the case of Li2, the partition potential displays distinct plateau structures surrounding both nuclei. The edges of this plateau correspond to the transition between regions where the core orbital contributes the most density and regions where the HOMO contributes the most density.

Figure 11. Components of the partition potential for H2, He2, Li2, and Be2 using the LDA. The total partition potential is in solid black, the Hartree plus external potential component is in brown, the kinetic energy component is in blue, and the XC component is in green.

components are larger in magnitude than the other components, but they have opposite sign so their sum is plotted for better comparison. We may compare the relative sizes of the components in each partition potential. For H2, the Hartree plus external contribution plays a dominant role, whereas in the molecules with larger bond lengths the kinetic component plays an increasingly important role. In He2, the exchange−correlation and kinetic components play comparable roles, whereas the Hartree plus external potential term is relatively small. For the case of H2+ we compare results using LDA with exact PDFT. Figure 12 displays the components of the partition potential for H2+ using LDA. In the exact case, the Hartree term exactly cancels with the exchange term, leaving a total HXC contribution of zero. We can see that, though the LDA XC component is close to canceling the Hartree contribution, it does not completely succeed.

typically has an attractive well between the two fragments, pulling some of the fragment density into the bonding region. It also contains features closer to the nuclei, where it reshapes the spherical isolated fragments into fragments that fit into the molecule. Figure 10 displays these same partition potentials but only along the bond axis, to compare vp’s for the different molecules.

Figure 10. Plots of the partition potential along the bond axis for H2+, H2, He2, Li2, and Be2. The location of the nuclei can be identified from the singularity features in the partition potential. Eq 50 gives the strength of these singularity features.

The singularities in the partition potentials of H2+ and H2 are clearly visible and are also visible but smaller for Li2 and Be2. This agrees with eq 50, as the equilibrium distances are larger for Li2 and Be2, so the density from one fragment in those cases is very small at the location of the other nucleus. For the case of He2, the density of each fragment is so small at the location of the other nuclei that the corresponding singularity in the partition potential is not visible in this plot. We also note that

Figure 12. Left: components of the partition potential for H2+ using the LDA. Right: Hartree and XC components of the partition potential, along with their sum. In the exact case, the XC component precisely cancels with the hartree component, which is shown in solid black. However, the LDA XC component does not exactly cancel with the hartree component. L



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Other authors56 have favored an alternative option for treating fractional numbers of electrons, in which the density is constructed by simply fractionally occupying the HOMO:

Spin decomposition is tricky, as usual. Consider the case of Li2, shown in Figure 13. Each of the Li fragments has 3

n(r) =

∑ fi

α

|ϕi |2 α

(55)

iα

Here the f iα will be 1 when its eigenvalue is less than the chemical potential and 0 when its eigenvalue is greater than the chemical potential. However, when the HOMO eigenvalue is equal to the chemical potential, the corresponding occupation number can take on a value in between 0 and 1. We refer to this method for constructing densities with fractional numbers of electrons as fractional orbital occupation (FOO). We use our CADMium code to illustrate the differences between these two methods of handling fractional numbers of electrons. Table 2 displays results for a few diatomic molecules using both our ensemble method (ENS) and FOO. These calculations were performed using LDA exchange and correlation. Both calculations use exact expressions for the partition energy, in the sense that they exactly reproduce a molecular LDA KS-DFT calculation. Thus, the total energy is not affected by the choice of ENS vs FOO. However, this choice affects both the sum of fragment energies, Ef, and the partition energy, Ep, even when their sum is not affected. For the case of ensembles, the partition energy is given by

Figure 13. Left: one of the Li fragments within a Li2 molecule as compared to an isolated Li atom and the density for the Li2 molecule. Right: each Li fragment has 3 electrons and comes from an open shell calculation; however, the Li2 molecule is closed shell. This figure displays the ensemble spin-up component with one electron and the ensemble spin-up component with two electrons, which combine to give a fragment density corresponding to a fractional spin-up density. These are LDA results at the equilibrium bond length.

electrons. Because the ground state of the molecule is spinunpolarized, each of the fragment densities is calculated as an ensemble of two oppositely spin-polarized systems: One system has two spin-up electrons and one spin-down electron whereas the other has two spin-down electrons and one spin-up electron. These component densities are averaged together to form a closed-shell density. The up-spin densities of the two ensemble components are shown on a log-scale in the right panel of Figure 13. These spin densities are averaged together to obtain the ensemble spin density for the fragment, shown in the left panel, where it is also compared with an isolated Liatom density.

EpENS[{nα}] = E[n f ] −

∑ ((1 − ωα)Eα[n p ] α

α

+ ωαEα[n p + 1])

(56)

α

whereas with FOO, the partition energy is EpFOO[{nα}] = E[n f ] −

∑ (Eα[nα]) (57)

α

15. FRACTIONAL CHARGES AND SPINS Various fragment-based methods formulated for wave functionin-DFT embedding do not admit fractional numbers of electrons in the fragments. Although wave function methods could in principle be defined for fractional numbers of electrons through ensembles, this would require additional calculation using already expensive methods.56 In these cases, it is simpler to fix the fragments to have integer values. However, as was demonstrated in Figure 5, there are certainly cases where chemical intuition and practicality indicate that a fragment contains fractional numbers of electrons. As mentioned in the first section, it is difficult to assign physical meaning to fragments within a molecule. However, to the extent that it is possible, PDFT seeks to treat these fragments as physical systems. These fragments represent open systems, and therefore, in the case of noninteger occupations, the correct representation is achieved through ensembles.85

The key difference between the two options is that the npα always integrate to integer values, whereas the nα may integrate to noninteger values. In the ensemble case, this means that the fragment energy functionals are never evaluated with fragment densities with fractional charges and spins. This difference is important due to the well-known problems associated with approximate density functionals and fractional spins and charges. For a recent overview of these issues of XC functionals, see the work of Cohen, Mori-Sanchez, and Yang.86−89 The dependence of the energy on particle number is known for the exact case due to the work of Perdew, Parr, Levy, and Balduz using ensembles,85 and independently via arguments based on size-consistency and translational invariance.90 The errors incurred in by local and semilocal XC functionals are illustrated in Figure 14 for the simple case of a hydrogen atom with a fractional number of electrons. The errors that are

Table 2. Table of Energies PDFT Energies Comparing Ensemble (ENS) vs FOO Treatment of Fractional Charges and Spinsa

a

molecule

D

Eiso

Ef

Ep

Vnn

Etot

H2+ (ENS) H2+ (FOO) H2 (ENS) H2 (FOO) Li2 (ENS) Li2 (FOO)

2 2 1.446 1.446 5.122 5.122

−0.4787 −0.5787 −0.9574 −0.8913 −14.6867 −14.6693

−0.4250 −0.5438 −0.9119 −0.8293 −14.6750 −14.6553

−0.6589 −0.5401 −0.9173 −1 −1.8068 −1.8264

0.5 0.5 0.6916 0.6916 1.7571 1.7571

−0.5839 −0.5839 −1.1377 −1.1377 −14.7246 −14.7246

All calculations used the LDA. M



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for standard local or semilocal functionals to properly describe the delocalized electron of stretched H2+. On the other hand, many standard density-functional approximations do quite well for localized densities that integrate to an integer number of electrons. The use of ensembles within PDFT suggests a possible solution. The fragments of PDFT have always been found to be well localized, and if a fragment does contain a fractional number of electrons or spins, then it is calculated as an ensemble of two systems with integer numbers of electrons and spins, and therefore the fragment energy will have the correct linear behavior for fractional numbers. Thus, if we look at the behavior of the energy for a fragment with arbitrary fractional spin and fractional charge within a molecule for fixed partition potential, it will exhibit precisely the “flat-plane behavior” required by the exact functional.87 It will have linear dependence on fractional charge and spin even when the LDA is used. This flat-plane behavior remains even as the molecule is stretched, so in the case of stretched H2 and H2+ the sum of fragment energies will be very close to the correct molecular energy. The partition energy, on the other hand, might ruin the correct behavior, restoring the errors of LDA. Clearly, we “simply” need to correct the behavior of the partition energy to improve molecular dissociation within DFT. In this way, PDFT provides a different framework for addressing the issue. The goal is to develop improved functionals for the XC components of the partition energy rather than the XC functional itself. The partition energy is a functional of the fragment densities and thus contains information about how overlapped or dissociated the fragments in a system may be. Exact conditions for the partition energy can aid in this process of functional development. This idea is explored further in ref 93.

Figure 14. Energies of the hydrogen atom with fractional charge (left) and fractional spin (right). In both cases the exact is linear and is shown in solid black. PBE is shown by the dashed line, and the LDA is shown by the dot-dashed line.

due to approximate noninteracting kinetic energy functionals are illustrated in Figure 15. In these cases, the functionals perform best for densities closest to the integers and perform poorly for densities corresponding to a fractional number of electrons.

Figure 15. Kinetic energies of the hydrogen atom with fractional charge (left) and fractional spin (right). In both cases the exact is linear and is shown in black. Thomas−Fermi is shown by the dotdashed line, and the LC94 (PW91k) is shown by the dashed line. These energies were evaluated for the exact densities.

16. CONCLUDING REMARK Fifty years after the formulation of density-functional theory (DFT),62 the ground-state electron density continues to be the main actor on the quantum-chemistry stage. But new actors are coming into play: the fragment densities that minimize the sum of the fragment energies subject to eq 1. Whereas any set of noninteracting v-representable densities adding to the correct molecular density is a solution to the KSCED eqs 7 and 8, the path to uniqueness offered by PDFT and the Huang−Pavone− Carter embedding method is appealing from both formal and practical perspectives. The derivation of PDFT presented in sections 6−9, along with the detailed discussion comparing existing alternatives throughout this article, should make this clear. We want to highlight once again the importance of allowing fragments to have fractional charges and spins. As new exchange−correlation functionals appear and faster computational algorithms are developed, it is timely to investigate exact conditions satisfied by the partition energy of PDFT (or interaction energies of embedding-DFT), so that functionals of the sets of fragment densities can be developed to be used in conjunction with standard functionals. Exact conditions, such as the one derived in section 11, should be helpful in this regard. Partitioning a molecule is not like splitting a cake, so this is not a piece of cake.

Clearly, these errors need to be addressed in any fragmentbased method that allows for fractional charges and spins, but the inability of standard density functionals to correctly handle these densities also causes well-known issues in standard Kohn−Sham calculations with integer numbers of electrons. In particular, dissociation curves are strongly affected. The two paradigmatic systems exhibiting such errors are the hydrogen molecule and its ion in the case where their bonds are stretched. When a closed-shell molecule, such as H2, dissociates into open-shell fragments, unless symmetry is broken, there will be regions with fractional spins. The associated error caused by incorrect treatment of these fractional spins is known as staticcorrelation error. Similarly, in cases such as H2+, where a charge is shared between two distant nuclear centers, incorrect treatment of the fractional charges on each center leads to delocalization error. The question of how to best correct these shortcomings remains largely open in DFT, in spite of recent progress on the development of explicit density functionals that yield the correct behavior.91,92 Part of the difficulty in addressing this issue is that it is not enough for functionals to be able to treat a whole system with fractional numbers of electrons, but the functional must also be able to recognize when a system with an integer number of electrons contains regions with fractional numbers of electrons or spins (e.g., stretched H2+ and stretched H2). It is not possible

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AUTHOR INFORMATION

Corresponding Author

*A. Wasserman. E-mail: [email protected]. Notes

The authors declare no competing financial interest. N



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Biographies

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Jonathan Nafziger received a B.A. in Physics from Goshen College in 2008. He is now a Ph.D. candidate at Purdue University in the department of Physics and Astronomy where has been studying the fundamentals of density-functional theory with Prof. Adam Wasserman.

Adam Wasserman is an Assistant Professor of Chemistry at Purdue University. He obtained his B.Sc. degree from the National University of Colombia in Bogotá, and his Ph.D. degree from Rutgers University under the guidance of Kieron Burke. Before joining the faculty at Purdue, he was a postdoctoral fellow with Eric Heller at Harvard University. Adam is a 2012 NSF-CAREER recipient and a 2013 Alfred P. Sloan Research Fellow.

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ACKNOWLEDGMENTS We thank the anonymous reviewers for their helpful comments. This work was supported by the Office of Basics Energy Sciences, U.S. Department of Energy, under grant No. DEFG02−10ER16196. A.W. also acknowledges support from an Alfred P. Sloan Foundation Research Fellowship.

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