Abstract
We introduce neutral excitation densityfunctional theory (XDFT), a computationally light, generally applicable, firstprinciples technique for calculating neutral electronic excitations. The concept is to generalise constrained density functional theory to free it from any assumptions about the spatial confinement of electrons and holes, but to maintain all the advantages of a variational method. The task of calculating the lowest excited state of a given symmetry is thereby simplified to one of performing a simple, lowcost sequence of coupled DFT calculations. We demonstrate the efficacy of the method by calculating the lowest singleparticle singlet and triplet excitation energies in the wellknown Thiel molecular test set, with results which are in good agreement with linearresponse timedependent density functional theory (LRTDDFT). Furthermore, we show that XDFT can successfully capture twoelectron excitations, in principle, offering a flexible approach to target specific effects beyond stateoftheart adiabatickernel LRTDDFT. Overall the method makes optical gaps and electronhole binding energies readily accessible at a computational cost and scaling comparable to that of standard density functional theory. Owing to its multiple qualities beneficial to highthroughput studies where the optical gap is of particular interest; namely broad applicability, low computational demand, and ease of implementation and automation, XDFT presents as a viable candidate for research within materials discovery and informatics frameworks.
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Introduction
The firstprinciples calculation of the excitedstate energies of quantum systems holds crucial importance for the study of solar cells^{1}, organic light emitting diodes^{2}, and chromophores in biological systems^{3}, to name but a few. With some exceptions, densityfunctional theory (DFT), the primary ab initio workhorse for computing ground state properties^{4,5}, typically falls short on such tasks, although efforts are underway to extend the foundation of DFT to excited states^{6,7,8,9,10,11}. The most commonly used firstprinciples method for calculating excitation energies, at least of finite systems, is linearresponse timedependent density functional theory (LRTDDFT)^{12,13,14}. However, LRTDDFT has two significant limitations: its computational cost, which severely limits the size of the systems that it can be used to investigate^{15}, and its inability to treat double (twoelectron) or higherorder excitations within adiabatic approximations to the exchangecorrelation (XC) kernel^{16,17,18}. Such multiparticle excitations and indeed interconversions between excitation types are of considerable interest, not alone on fundamental grounds, but for example in the computational development of photovoltaic materials that intrinsically overcome the ShockleyQueisser limit.
Often, it is the lowest excitation energy of a given symmetry that is of principal interest, as this can be used to calculate the electronhole binding energy. In such cases, stateoftheart methods that rely on the coupling of all excitations, such as realtime TDDFT, are inefficient. These considerations strongly motivate the development of nonperturbative, variational methods based on timeindependent DFT for excited states. Ideally, the desired method should inherently capture response to all orders and preserve a favourable computational scaling, while avoiding slowlyconverging sums over virtual states.
Over the years, several such firstprinciples schemes have been developed for calculating neutral excitation energies, such as ensemble DFT^{19,20,21}, restricted openshell KohnSham DFT^{22,23,24,25}, constricted variational DFT^{26,27}, Δ selfconsistent field (ΔSCF) DFT^{28,29}, constrained DFT defined using virtual KohnSham states^{30}, and the maximum overlap method^{31,32}. These are underpinned by the existence of a variational DFT, with an equivalent noninteracting KohnSham (KS) state, for an individual excited state of interacting electrons^{6,7,28}. Each method, including the one introduced here which builds upon the foundation that others have provided, has its relative strengths and weaknesses in terms of computational cost and ease of both implementation and convergence. We refer the reader to ref. ^{33} for a recent review of TDDFT, and to ref. ^{26} for a foundational comparison between TDDFT and DFTbased variational approaches.
In this article, we introduce neutral excitation DFT (XDFT), an inexpensive, fully firstprinciples extension of constrained DFT (cDFT)^{34,35,36} for calculating neutral excitation energies in finite systems such as molecules and clusters. XDFT simulates an excitation in the KS system by reducing the electronic population of the ground state valence subspace by one electron, while keeping the total number of electrons unchanged. XDFT is quite unlike conventional cDFT in that no prior assumptions are made as to the spatial form of source or drain regions for charge constraint. It captures screening effects at all orders, unlike LRTDDFT, but retains a single Slater determinant of KohnSham orbitals and so it is readily portable to many standard DFT codes (this transpires to be drawback, as we will see, in the study of singlet excitations, motivating future developments). XDFT scales with the atom count, N, as per groundstate DFT, namely as \({\mathscr{O}}({N}^{3})\) when no quantum nearsightedness is exploited (we have implemented it within an \({\mathscr{O}}(N)\) code). This contrasts with methods like TDDFT, which typically scales as \({\mathscr{O}}({N}^{4})\)^{37} and the BetheSalpeter equation (BSE), which goes as \({\mathscr{O}}({N}^{6})\)^{38}. In addition, unlike LRTDDFT and BSE, which can be highly memory intensive when unoccupied states are treated explicitly, XDFT has a memory overhead comparable to that of standard DFT. Crucially, it avoids the calculation of unoccupied groundstate KS orbitals entirely, and we never calculate them here in practice. Therefore, in terms of computational efficiency, XDFT offers significant advantages over the aforementioned existing methods, as long as only the lowestenergy excited state of a given symmetry is of particular interest. XDFT offers ready compatibility with highthroughput frameworks^{39}, the study of largesystems, and variational KS methods beyond DFT.
Applying this technique, we calculate the lowest singlet and triplet single excitation energies of a representative set of organic molecules. Here we find a surprisingly good agreement with LRTDDFT values, notwithstanding the extreme simplicity of our method. We also move straightforwardly beyond singleparticle excitations, while still only using a small number of coupled DFT calculations. As we will see, XDFT can accurately reproduce the energies of excited states with a predominantly doubleexcitation character in the canonical testbed atom beryllium, which are inaccessible^{16,17,18} to adiabatic LRTDDFT.
In the next section, we describe the XDFT formalism. This is followed by a section that explores the connection between XDFT and exact theorems of excited state DFT and outlines the relevant approximations. The next section outlines certain important details of the calculation. Results, including single and double excitation energies and difference density are presented in the penultimate section. Finally, we present our conclusions.
The XDFT Formalism in Brief
A neutral excitation, within the quasiparticle picture, is the promotion of one or more electrons from occupied levels to empty ones, resulting in the creation of bound electronhole pairs and with consequent energy relaxation due to screening. To simulate this excited state, XDFT searches for the groundstate energy of the same system, but now subject to the extra condition of a given number of electrons with spin \(\sigma \), \({N}_{c}^{\sigma }\) being confined to a predefined subspace. This constraining condition may be written as
where ‘Tr’ denotes the trace, \({\hat{\rho }}^{\sigma }\) is the spindependent fermionic density operator and \(\hat{{\mathbb{P}}}\) is a projection operator onto the desired subspace. In conventional cDFT, the subspace spanned by \(\hat{{\mathbb{P}}}\) is a spatial region defined at the researcher’s discretion. If \(\hat{{\mathbb{P}}}\) spans two spatial regions with opposite weighting, for example, then one can enforce a chargeseparated density configuration for the simulation of chargetransfer excitations^{40,41,42,43,44}.
Here we come to the key development of XDFT. In order to access excitations beyond chargeseparated states of obvious spatial character, we free cDFT of all human assumptions by defining the subspace in terms of the groundstate KS eigenstates only. In doing so we retain the physical information encoded in the KS eigensystem, which is a key ingredient in LRTDDFT but discarded in normal cDFT. More technically speaking, in a neutral Nelectron system XDFT locates the energy of the lowest Melectron excited state (M may even be noninteger in principle) by confining N–M electrons within the valence KS subspace of the unconstrained DFT groundstate. This circumvents the need for any prior, empirical specification of subspaces and restores firstprinciples status. The projector that defines XDFT is
where \({\hat{\rho }}_{0}\) is the ground state density operator, \({\psi }_{i}\rangle \) is the i^{th} KS orbital and \({f}_{i}\) is its occupation number and the sum is over all KS levels. Note that, at zero temperature, \({f}_{i}\mathrm{=1(0)}\) for occupied (unoccupied) levels.
As in conventional cDFT, the ground state of the system subject to the “exciting” constraint in XDFT is found at the stationary point of the functional
where \({V}_{c}\) is a Lagrange multiplier. For a fixed \({V}_{c}\), the second term on the righthand side serves to modify the ground state potential by adding the term \({V}_{c}\hat{{\mathbb{P}}}\). One then minimises \(W[\hat{\rho },{V}_{c}]\) with respect to \(\hat{\rho }\), just as \(E[\hat{\rho }]\) is minimized in regular DFT. At the V_{c}dependent minima \(W[\hat{\rho },{V}_{c}]\) can be thought of as a function^{36}, W(V_{c}), of V_{c}. The maxima of W(V_{c}) occur at stable states of the constrained system^{45}, at which the value of W is the excitedstate total energy of interest. Once the energy of the first excited state, W, is determined by optimising V_{c}, then the lowest excitation (photon absorption) energy, \({E}^{\ast }\), can be evaluated as a total energy difference from the ground state DFT energy, \({E}_{0}\), namely as \({E}^{\ast }=W{E}_{0}\).
We note that excitations beyond the lowestenergy one of a given spin symmetry can be simulated in XDFT by employing multiple constraints. For example, if the KohnSham valence subspaces of the ground state and the first XDFT excited state are projected onto by \({\hat{{\mathbb{P}}}}_{0}\) and \({\hat{{\mathbb{P}}}}_{1}\), respectively, then the energy of the second excited state can be found by confining \(N1\) electrons within the subspace of \({\hat{{\mathbb{P}}}}_{0}\) using a Lagrange multiplier \({V}_{c}^{1}\) and, separately, confining \(N1\) electrons within the subspace of \({\hat{{\mathbb{P}}}}_{1}\) using a multiplier \({V}_{c}^{2}\). In general, the totalenergy of the I^{th} excited state system of a given spin symmetry will be found at the stationary point of
XDFT can be used to simulate combinations of charge and spin excitations. Given a closedshell ground state, the double (\(M=2\)) excitations and the triplet single excitation are straightforward to access with a single constraint. These both incur the cost of just two DFT calculations – the groundstate one and the constrained one. In both cases, the electronpromotion constraint can be applied to the sum of the density operators for each spin, and the triplet state can be selected by setting \({m}_{s}=1\). We have found that, for a common test set with different types of chemical bonds, XDFT allows us to directly access the lowestlying excitations using coupled groundstate calculations. Nonlinear response effects such as manyparticle excitations are treated on the same footing as linearresponse effects. In essence, while it has long been known that groundstate DFT contains the information needed to calculate neutral excitations, and relatively complex methods have been developed to explore it, we show here that constrained KohnSham DFT at a level easily implementable in any normal DFT code provides a variational approach to exploit this. We note in passing that, since the excited state KS wavefunctions are accessible through XDFT, one can potentially use them to calculate approximate oscillator strengths. This is an interesting topic for future investigations.
Formal Justification of the XDFT Method
XDFT is formally an orbitaldependent DFT, and its energy is separately invariant under arbitrary unitary transformations among the occupied KohnSham orbitals of the groundstate and of the constrained state. The XDFT constraint, which, for simulation of the first excited state, amounts to ejecting one electron from the valence subspace, encodes a welldefined manyparticle excitation of the noninteracting KohnSham system. Now, while this certainly does not imply a welldefined excitation of the interacting system, it may provide a good approximation in cases where the noninteracting and interacting wavefunctions are similar (modulo unitary transformations) in both the ground and excited states. This is expected to hold well for systems that do not exhibit strong static correlation, where furthermore we may expect a degree of cancellation of error when only looking at the differences of the ground and excitedstate total energies. To investigate the validity of the XDFT formalism without having to assume equivalence between the KohnSham and the interacting manyparticle wave function, in the following, we seek to establish a rigorous connection between the formally exact theorems of excited state DFT and the XDFT method. We start with a very brief discussion of these exact theorems.
Exact theorems for excited state DFT
Groundstate densityfunctional theory involves finding the groundstate (GS) density of a system of interacting electrons through the numberconserving minimisation of the Levy constrained search^{46} energy functional, \({E}_{{\rm{LL}}}[\rho ,{v}_{{\rm{ext}}}]\). Such minimisation is with respect to the density, \(\rho ({\bf{r}})\), and for a given external potential, \({v}_{{\rm{ext}}}({\bf{r}})\), where
Here, \(\hat{T}\) is the electron kinetic energy operator, \({\hat{V}}_{ee}\) is the electronelectron interaction operator, and \({F}_{{\rm{LL}}}[\rho ]\) is the minimum of \(\langle \hat{T}+{\hat{V}}_{ee}\rangle \) provided that the state \(\psi \rangle \) produces the density \(\rho ({\bf{r}})\). Now, for some external potential \({v}_{{\rm{ext}}}({\bf{r}})\), we may denote the k^{th} excited state, energy and density by \({\psi }_{k}\rangle \), \({E}_{k}\) and \({\rho }_{k}({\bf{r}})\), respectively. Furthermore, we may assume that, for the same or a different external potential \({v{\prime} }_{{\rm{ext}}}({\bf{r}})\), there exists a stationary state \(\psi {\prime} \rangle \) producing the same density \({\rho }_{k}({\bf{r}})\), but such that
Then, necessarily,
and consequently,
This leads us to an important result, shown by Perdew and Levy in ref. ^{6}, namely that a stationary excited state will correspond to a local density minimum of \({E}_{{\rm{LL}}}[\rho ,{v}_{{\rm{ext}}}]\) if and only if, for any external potential, there is no other stationary state that gives the same density and yields a lower value for \(\langle \hat{T}+{\hat{V}}_{ee}\rangle \).
In an alternate approach to this^{7,47}, the first excitedstate energy of a system subject to an external potential \({v}_{{\rm{ext}}}({\bf{r}})\) can be found by minimising, with respect to density \(\rho ({\bf{r}})\), the functional
Here, \({F}_{1}[\rho ,{v}_{{\rm{ext}}}]\) is a bifunctional defined as
where the minimisation of \(\langle \hat{T}+{\hat{V}}_{ee}\rangle \) is to be performed over states \(\psi \rangle \), which yield the density \(\rho ({\bf{r}})\) and are orthonormal to the ground state, \({\psi }_{0}\rangle \), of the external potential, \({v}_{{\rm{ext}}}({\bf{r}})\). Foundational justifications for numerous timeindependent excitedstate DFT schemes^{48,49} including \(\varDelta \)SCF^{28}, TOCIA^{50}, and OCDFT^{51} have been developed on the basis of this approach.
Remarkably, if the external potential is Coulombic, namely if, for a number of M nuclei, it has the form
where the α^{th} nucleus with charge \({Z}_{\alpha }\) is located at \({{\bf{R}}}_{\alpha }\), then two different stationary states in the presence of the same or different external potential cannot have the same density^{52}. In other words, the stationarystate density uniquely specifies the stationary state and the external potential. In this situation, as shown in ref. ^{52}, one can, in principle, omit the \({v}_{{\rm{ext}}}\) dependence of \({F}_{1}\) and find the first excited state density \({\rho }_{1}({\bf{r}})\) by minimising
XDFT may be placed in a formal context using this equation, as we now explain.
The XDFT approximation
Here we show how the XDFT method follows from Eq. (12) with the use of certain welldefined approximations. Let \({v}_{{\rm{KS}}}({\bf{r}})\) be the KS potential and \({\psi }_{0}^{{\rm{KS}}}\rangle \) be the KS ground state corresponding to an interacting system with an external potential \({v}_{{\rm{ext}}}({\bf{r}})\). Then, assuming representability where required, consider a noninteracting KS like system whose ground state density equals \({\rho }_{1}\), which is that of the first excited state of the interacting system. This system is subject to a local potential,
such that the exchangecorrelation potential \({v}_{{\rm{XC1}}}({\bf{r}})\) ensures that the correct density is recovered. Unfortunately, \({v}_{{\rm{XC1}}}({\bf{r}})\) is not known to us.
To facilitate the use of available approximations for exchangecorrelation, therefore, let us consider a different noninteracting KSlike system such that its lowestenergy stationary state \({\bar{\psi }}_{0}^{{\rm{KS}}}\rangle \) which satisfies the condition \(\langle {\bar{\psi }}_{0}^{{\rm{KS}}}{\psi }_{0}^{{\rm{KS}}}\rangle =0\), yields \({\rho }_{1}\). For such a noninteracting system, the XCpotential \({\bar{v}}_{{\rm{XC}}}({\bf{r}})\), which generates a KS potential \({\bar{v}}_{{\rm{KS}}}({\bf{r}})={v}_{{\rm{ext}}}({\bf{r}})+\int d{\bf{r}}{\boldsymbol{{\prime} }}\frac{\rho ({\bf{r}}{\boldsymbol{{\prime} }})}{{\bf{r}}{\bf{r}}{\boldsymbol{{\prime} }}}+{\bar{v}}_{{\rm{XC}}}({\bf{r}})\), contrasts with the standard groundstate XC potential, \({v}_{{\rm{XC}}}({\bf{r}})\), which is constructed in such way that the noninteracting groundstate density associated to \({v}_{{\rm{KS}}}({\bf{r}})={v}_{{\rm{ext}}}({\bf{r}})+\int d{\bf{r}}{\boldsymbol{{\prime} }}\frac{\rho ({\bf{r}}{\boldsymbol{{\prime} }})}{{\bf{r}}{\bf{r}}{\boldsymbol{{\prime} }}}+{v}_{{\rm{XC}}}({\bf{r}})\) coincides with the density of the interacting ground state for the external potential, \({v}_{{\rm{ext}}}({\bf{r}})\). We note, using Eq. (12), that \({\bar{v}}_{{\rm{XC}}}({\bf{r}})\) must be a unique functional of the density. We also note that, in contrast with the corresponding object in the treatment presented in ref. ^{53}, \({\bar{\psi }}_{0}^{{\rm{KS}}}\rangle \) is not generally the first excited state of the original, unconstrained KS system since \({\bar{\psi }}_{0}^{{\rm{KS}}}\rangle \) and \({\psi }_{0}^{{\rm{KS}}}\rangle \) are stationary states of KS Hamiltonians with different potentials \({\bar{v}}_{{\rm{KS}}}\) and \({v}_{{\rm{KS}}}\).
Using XDFT, we seek to obtain the noninteracting state \({\bar{\psi }}_{0}^{{\rm{KS}}}\rangle \) that is orthonormal to the noninteracting ground state \({\psi }_{0}^{{\rm{KS}}}\rangle \), by ejecting a single electron out of the valence subspace, i.e., by creating an electronhole pair in the KS system. Formally speaking, therefore, the XDFT method effectively amounts to making the central approximation
Variational collapse to the ground state density does not arise in XDFT, in spite of this approximation, since we only search for \({\bar{\psi }}_{0}^{{\rm{KS}}}\rangle \) that satisfy \(\langle {\bar{\psi }}_{0}^{{\rm{KS}}}{\psi }_{0}^{{\rm{KS}}}\rangle \mathrm{=0}\) and so the groundstate density generated \({\psi }_{0}^{{\rm{KS}}}\rangle \) is out of bounds. A similar approximation to Eq. (14), in terms of the exchangecorrelation energy functional, has been used in OCDFT^{51}. We refer the reader in particular to the very informative Table 3 of ref. ^{51}, where several properties of various timeindependent excitedstate DFT approaches are carefully compared. Using the same notation as is used in that Table, the properties of the XDFT method are tabulated in Table 1.
Proof that XDFT encodes the orthonormality of KS Slater determinants
In the following, we prove that the the density generated by XDFT belongs to a noninteracting state that is orthonormal to \({\psi }_{0}^{{\rm{KS}}}\rangle \). For an N electron system, let the excitedstate valence orbitals obtained with XDFT be \(\{{\psi }_{i}\rangle \}\) for \(i\mathrm{=1,}\ldots \mathrm{.,}N\). Then, since a unitary transformation of orbitals preserves the density, it is sufficient to prove that there is at least one unitary transformation of \(\{{\psi }_{i}\rangle \}\) that produces an “excited electron” orbital, i.e., an orbital that is orthonormal to all of the groundstate valence orbitals that generate \({\psi }_{0}^{{\rm{KS}}}\rangle \).
Let us define a candidate orbital \({\psi }^{\ast }\rangle \), with a view to describing the excitedstate component of the XDFT KS valence eigensystem, as
Then, noting that, for zerotemperature fermionic systems the density matrix is idempotent, and so
and that the set of orbitals \(\{{\psi }_{i}\rangle \}\) build a density operator \(\hat{\rho }\) obeying the XDFT condition \({\rm{Tr}}[\hat{\rho }{\hat{\rho }}_{0}]=N1\), by definition, we find that \({\psi }^{\ast }\rangle \) is not necessarily normalised, since
With the latter in hand, let us consider a unitary transformation \({\bf{U}}\) of the orbitals \(\{{\psi }_{i}\rangle \}\) such that
noting that unitarity imposes the requirement that
While the righthand side of Eq. (15) is not the linear expansion of \({\psi {\prime} }_{N}\rangle \) in terms of the set \(\{{\psi }_{i}\rangle \}\), we know that such a unique linear expansion exists, since \((\hat{{\mathbb{1}}}{\hat{\rho }}_{0})\) is a projection operator. Thus, in principle, we can uniquely specify a row of \({\bf{U}}\), \({U}_{iN}\) for all \(i=1,\ldots .,N\), such that \({\psi {\prime} }_{N}\rangle ={\psi }^{\ast }\rangle {C}^{\mathrm{1/2}}\), simultaneously satisfying the normalisation demand of unitary transformations. The problem of finding any suitable remaining \(\{{\psi {\prime} }_{i}\rangle \}\) boils down to constructing any \((N\times N)\) matrix \({\bf{U}}\) that satisfies the condition for unitarity
and whose Nth column is uniquely known. Avoiding double counting of equations for \(i\ne j\), Eq. (19) is a set of \(({N}^{2}+N\mathrm{2)/2}\) equations, omitting the equation for \(i=j=N\) since it contains known terms only. For the \(({N}^{2}N)\) unknowns, this is always solvable (for \(N > 2\), solvable with infinite solutions).
Now, recalling that \({\hat{\rho }}_{0}{\hat{\rho }}_{0}^{2}={\hat{\rho }}_{0}{\hat{\rho }}_{0}=\hat{0}\), we have
and therefore \({\psi {\prime} }_{N}\rangle \) is orthonormal to all of the groundstate valence orbitals. Hence, the Slater determinant constructed from the set of orbitals \(\{{\psi {\prime} }_{i}\rangle \}\) is necessarily orthonormal to the determinant \({\psi }_{0}^{{\rm{KS}}}\rangle \).
Methodological Details
Implementation and parameters
The linearscaling firstprinciples code ONETEP^{54}, within which we have implemented the XDFT formalism, variationally optimizes a minimal set of localized, nonorthonormal generalized Wannier Functions (NGWF), expanded in terms of psinc functions^{55,56}, to minimize the total energy. ONETEP is equipped with an automated conjugategradients method for optimizing the cDFT (or XDFT) Lagrange multiplier^{45,57,58}. We have used this, together with the PerdewBurkeErnzerhof (PBE) XC functional^{59} to calculate the lowest singlet excitation energies of the 28 closedshell organic molecules contained in the wellknown Thiel set^{60}. Our calculations are performed using scalar relativistic normconserving pseudopotentials, a planewave cutoff energy of 1500 eV and a radius of 14.0 a_{0} for the NGWFs. The MartynaTuckerman periodic boundary correction scheme^{61} was used with a parameter of 7.0 a_{0}. The constrained KS system was found to contain symmetryprotected partialfilling of the degenerate highest occupied state in certain molecules, and so we used finitetemperature ensemble DFT as implemented in ONETEP^{62} in all cases.
Multiplet sum method
Unlike the density and energy of the triplet first excited state, it is not straightforward to obtain the singlet counterparts with the XDFT method, since, given a closedshell ground state, the final state of a singlet single excitation can not be represented by a single Slater determinant (Sd). For the noninteracting KS system, any closedshell excited state corresponding to \([S=0,\,{m}_{s}=0]\) (with a noninteracting energy \({}^{S=0}{E}_{{m}_{s}=0}^{{\rm{K}}{\rm{S}}}\)) or openshell excited state \([S=1,\,{m}_{s}=0]\) (with \({}^{S=1}{E}_{{m}_{s}=0}^{{\rm{K}}{\rm{S}}}\)) can, fortunately, be expressed as a linear combination of the same pair of KohnSham Slater determinants (Sds), within a frozenorbital treatment. These two Sds, which are not eigenstates of \({\hat{S}}^{2}\), are then degenerate, with a noninteracting energy \({}^{Sd}{E}_{{m}_{s}=0}^{{\rm{K}}{\rm{S}}}\). Invoking the multiplet sum method^{33,63,64}, we can thereby express the noninteracting energy of a closedshell singlet state approximately as
In order to access one of these degenerate Sds that make up the singlet, in practice, we promote one electron by applying the XDFT constraint to one spin channel only, maintaining \({m}_{s}=0\). We note that, for a spinrestricted treatment, since the two Sds are degenerate, each of them has the same energy as the singlet and the triplet state and Eq. (21) is trivially satisfied. However, at this point, we make a final assumption that Eq. (21) may be used to approximately evaluate the energy of the interacting system. Keeping in mind that the three triplet states for \({m}_{s}=\,\mathrm{1,0,1}\) are degenerate, the energy of the singlet firstexcited state can then be approximated as
where, \({E}_{{\rm{Sd}}}\) and \({E}_{{\rm{t}}}\) are energies of interacting system obtained from XDFT calculations simulating single excitation while maintaining \({m}_{s}=0\) and \({m}_{s}=1\), respectively. The advantage of Eq. (22) is that it involves only the energies of two singleSd states that are available using XDFT. Each term on the right hand side of Eq. (22) derives from an interacting system that is obtained from an equivalent unrestricted KS system having the same density and spin density. In passing, we note that the Sd state is sometimes referred to in the literature as a contaminated singlet. A formalism using restricted (spinindependent) KS orbitals might offer an energy \({E}_{{\rm{Sd}}}\) that is more appropriate for use with Eq. 22.
Calculation flowchart
The XDFT flowchart involving coupled calculations for finding neutral gaps of finite systems is presented in Fig. 1. In practice, the task of determining the triplet and singlet single excitation energies boils down to calculating total energy differences.

1.
First, a standard DFT run is performed to calculate the closedshell groundstate energy \({E}_{0}\) and density operator \({\hat{\rho }}_{0}\).

2.
\({\hat{\rho }}_{0}\) is used to run an XDFT calculation confining \((N\mathrm{1)}\) electrons to the total valence subspace of the DFT run, with \({m}_{s}=1\) (i.e. fixing a spin moment of \(1\,{\mu }_{{\rm{B}}}\)). This gives the energy \({E}_{{\rm{t}}}\) of the lowest lying interacting triplet state.

3.
Finally, in order to obtain the energy \({E}_{{\rm{Sd}}}\), we run an XDFT calculation confining \((N\mathrm{/2)}1\) electrons to the spinup valence subspace of the DFT run while maintaining \({m}_{s}=0\). The singlet firstexcited state energy is then obtained from Eq. (22).

4.
Ultimately, the triplet and singlet neutral gaps are calculated, respectively, as
Results and Discussion
Excitation energy of Thiel set molecules
We have used XDFT to calculate the lowest singlet excitation energies of the 28 closedshell organic molecules contained in the wellknown Thiel set^{60}. In Fig. 2, we show a scatter plot of the singlet and triplet excitation energies calculated with XDFT against those obtained with LRTDDFT and the adiabatic PerdewBurkeErnzerhof (PBE) XC functional^{59} in ref. ^{65} (singlets) and ref. ^{66} (triplets). The LRTDDFT results are broadly in agreement with experimental values (see the supporting information in ref. ^{67}). The triplet energies, which do not rely on any additional approximation beyond XDFT such as multiplet sum, show almost perfect agreement with those calculated using groundstate DFT with \({m}_{s}\mathrm{=1}\). This may provide a practical way of validating the XDFT approximations when applied to a new system. The figure also demonstrates that, in spite of the multiplet sum approximation, XDFT yields singlet energies with a remarkably good accuracy, if adiabatic, semilocal LRTDDFT is taken as a reasonable benchmark. In terms of computational efficiency, we note that, for the representative molecule pBenzoquinone, an XDFT calculation run on 3 processors for the singlet excitation energy offers an approximate twofold reduction in computation time compared to its LRTDDFT counterpart. Unlike linearscaling LRTDDFT, XDFT does not require the prior optimization of a defined number of conductionband orbitals, which is a process that can demand some trialanderror before wellconverged results are obtained. It is to be noted that, as a result of the chargedelocalization error of semilocal XC functionals, XDFT, in its currentlyimplemented form, is applicable only to finite systems.
Charge difference density
The difference between the local part of the constrained density operator, \(\hat{\rho }\), and that of the ground state density operator, \({\hat{\rho }}_{0}\), can be viewed as an approximation to the difference density, from which transition dipole moments for example can be calculated. In Fig. 3 we show such plots for a representative molecule of the Thiel set^{60}, propanamide. Figure 3(a) shows an approximation to the difference density based on the groundstate KS orbitals, which neglects orbital relaxation and electronhole binding. Since it captures these effects, the singlet (b) and triplet (c) isosurfaces generated using XDFT (and, in the case of the singlet, the multiplet sum method^{33,64} applied to the total electron densities) reflect a greater degree of difference density localisation than (a). Due to Pauli exclusion, furthermore, the singlet (b) difference density attains a greater spatial localisation than the triplet one (c). Such conclusions are also supported by quantitative analysis of the charge densities.
Choice of XC functional
The accuracy of results obtained with any DFTbased method is necessarily dependent on the approximate XCfunctional used. We have found good agreement between XDFT and LRTDDFT results for the Thiel set, using the semilocal PBE functional for both methods. This choice of functional is motivated by the fact that, for many DFT and adiabatickernel LRTDDFT calculations, when compared to experimental results, PBE typically offers acceptable accuracy with relatively inexpensive calculations and is therefore a highly popular choice of functional.
However, nonlocal hybrid XCfunctionals containing a fraction of KS exactexchange typically improves the agreement with experimental results, at the expense of increasing the computational cost. This trend can be seen in Table 2, where we compare experimental singlet excitation energies with XDFT results evaluated using PBE and the B3LYP^{74} hybrid functional for six Thiel set molecules. These molecules are those for which the agreement between the XDFT(PBE) and experimental results is particularly poor (giving a Mean Absolute Error or MAE of 0.85 eV, whereas the MAE is 0.50 eV for the entire Thiel set). The MAE between XDFT and experiment for the six molecules reduces to 0.33 eV using XDFT(B3LYP). It is more probable that this improvement is primarily due to an improved description of the fundamental gap and electronhole binding, rather than an improvement in the performance of the XDFT or multiplet sum method approximations per se. It is to be noted that, although a hybrid functional improves the energies considerably, the discrepancies resulting from the multiplet sum approximation are present nonetheless. This can be seen by contrasting the MAE (w.r.t. experimental values) of the XDFT results (0.33 eV) with the LRTDDFT ones (0.08 eV). On the other hand, the fact that, for triplet excitations, the MAE of the XDFT energies is 0.08 eV indicates that the error in the singlet energies arises from the multiplet sum approximation and not from the XDFT approximation per se. It seems feasible that this agreement with experiment may be improved further within the XDFT framework, potentially using rangeseparated hybrid functionals, implicit dielectric screening, zeropoint phonon corrections, and a more elaborate treatment of spin contamination that circumvents the multiplet sum method.
Simulation of a double excitation
Finally, we explore the ability of XDFT to calculate energies of excitations with strong double (twoelectron) character, which are inaccessible by constuction to adiabatickernel LRTDDFT^{16,17,18}. The XDFT method is nonlinear, unlike LRTDDFT, in the sense that its Hartree and XC potentials are calculated selfconsistently with the density in the excited state, i.e., not just corrected to first order using the interaction kernel \({\hat{f}}_{{\rm{Hxc}}}\). Thus, XDFT is not limited to single excitations. As a proof of principle, we focus here on one particular excitation of a known, strong double (i.e. twoelectron) character, nothing that a more comprehensive study of excitations of more mixed singledouble character using XDFT would be necessary to fully establish the range of applicability of the method to such effects. We note, in passing, that XDFT is not restricted to exciting integer numbers of electrons, M, particularly when coupled with ensemble DFT. In the benchmark case of atomic beryllium, the first double excitation promotes two electrons from the 2s to the 2p orbitals^{75}. For the Be atom, 1s electrons are described by a pseudopotential, rendering the multiplet sum method unnecessary. Consequently, using the ground state valence density operator \({\hat{\rho }}_{0}\) to confine zero electrons to the total valence subspace from a standard groundstate DFT calculation, we can directly access the energies of the lowest lying doubly excited singlet \(({}^{0}E_{0}^{\mathrm{(2)}})\) and triplet \(({}^{1}E_{1}^{\mathrm{(2)}})\) states with two separate XDFT calculations with \({m}_{s}=0\) and \({m}_{s}=1\) respectively.
In Fig. 4 we plot the single and double excitation energies of Be calculated with semilocal and hybrid XCfunctionals. The singlet and triplet double excitation energies were obtained as \({}^{0}\,{E}^{\mathrm{(2)}\ast }={}^{0}\,{E}_{0}^{\mathrm{(2)}}{E}_{0}\) and \({}^{1}\,{E}^{\mathrm{(2)}\ast }={}^{1}\,{E}_{1}^{\mathrm{(2)}}{E}_{0}\), respectively. Our results agree well with those calculated with ensemble DFT in ref. ^{20}, for all four excitation types. The singlet singleelectron PBE excitation energy is also in very close agreement with our own LRTDDFT(PBE) result, indicating that the multiplet sum method is accurately applicable to this system. We note, however, that while our singlet double excitation energies agree surprisingly well with experimental values, this is much less the case for our triplet double energies. Experimentally, the singlet \(2{s}^{2}\to 2{p}^{2}\) excitation is slightly lower in energy than the triplet one, and this has been explained as resulting from a mixing of the singlet double with higher singlet single excitations^{82}. Our results would support the opposite conclusion about the mechanism behind this anomalous ordering (singlet below triplet), however, since it is the triplet state which is poorly described by the singledeterminant theory. In principle, XDFT is capable of accessing excitations of noninteger electron character (e.g, mixed single and double excitations) with the aid, e.g., of ensemble DFT^{62,83}, and this is a promising avenue for future investigation.
Conclusion
In summary, we introduce the XDFT method for calculating the excitedstate energies of finite systems by means of a small number of coupled DFT calculations. The XDFT method, which, with certain approximations, can be arrived upon from the exact theorems of excited state DFT, generalizes constrained DFT, in essence, by removing the necessity for users to predefine the targeted subspaces. Unlike standard implementations of LRTDDFT or BSE, no reference is made to unoccupied orbitals. XDFT closely reproduces the LRTDDFT values for triplet and also, with the help of an additional approximation, in most cases the singlet excitation energies of the Thiel molecular test set. XDFT(B3LYP) offers significantly improved singlet energies with respect to experiment, over XDFT(PBE). Interestingly, however, XDFT can access the energies of double excitations, in principle, effectively circumventing the requirement for nonadiabaticity in LRTDDFT. We demonstrate this in a successful application to the wellknown beryllium test case as a proof of principle.
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Acknowledgements
This work is supported by Science Foundation Ireland (SFI) through The Advanced Materials and Bioengineering Research Centre (AMBER, grant 12/RC/2278_P2), and of the European Regional Development Fund (ERDF). We acknowledge G. Teobaldi and N. D. M. Hine for their implementation of cDFT in ONETEP. We acknowledge the DJEI/DES/SFI/HEA Irish Centre for HighEnd Computing (ICHEC) and Trinity Centre for High Performance Computing (TCHPC) for the provision of computational resources. We also acknowledge Trinity Research IT for the maintenance of the Boyle cluster on which further calculations were performed.
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S.R., S.S. and D.D.O.R. designed the experiment. D.D.O.R. developed the method, software modifications, and figures. S.R. ran and analysed all calculations, developed the formal justification, and drafted the manuscript. S.R., S.S. and D.D.O.R. finalized the manuscript.
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Roychoudhury, S., Sanvito, S. & O’Regan, D.D. Neutral excitation densityfunctional theory: an efficient and variational firstprinciples method for simulating neutral excitations in molecules. Sci Rep 10, 8947 (2020). https://doi.org/10.1038/s41598020652094
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DOI: https://doi.org/10.1038/s41598020652094