Visualizing and characterizing excited states from time-dependent density functional theory

Time-dependent density functional theory (TD-DFT) is the most widely-used electronic structure method for excited states, due to a favorable combination of low cost and semi-quantitative accuracy in many contexts, even if there are well recognized limitations. This Perspective describes various ways in which excited states from TD-DFT calculations can be visualized and analyzed, both qualitatively and quantitatively. This includes not just orbitals and densities but also well-defined statistical measures of electron–hole separation and of Frenkel-type exciton delocalization. Emphasis is placed on mathematical connections between methods that have often been discussed separately. Particular attention is paid to charge-transfer diagnostics, which provide indicators of when TD-DFT may not be trustworthy due to its categorical failure to describe long-range electron transfer. Measures of exciton size and charge separation that are directly connected to the underlying transition density are recommended over more ad hoc metrics for quantifying charge-transfer character.


Introduction
2][3][4][5] For historical reasons, 6 that formulation is commonly known as ''time-dependent'' (TD-)DFT, 1,[5][6][7][8] despite the absence of time in its static, frequency-domain formulation.Semantics aside, the linear-response TD-DFT formalism has a pleasing familiarity for chemists, as it can be cast in the form of an eigenvalue problem in a space of singlysubstituted Slater determinants.This is analogous to the method of configuration interaction with single substitutions (CIS), 9 but incorporating dynamical electron correlation in the TD-DFT case.In favorable contexts, including the electronic spectroscopy of many medium-sized organic molecules, TD-DFT achieves a mean accuracy of B0.3 eV for vertical excitation energies, 1,[10][11][12][13][14] which is often sufficient for solution-phase spectroscopy.At the same time, TD-DFT's formal scaling and computational cost are comparable to ground-state DFT, 8,15 meaning that it is often the only ab initio method for excited states that can address large chemical systems.These considerations have made TD-DFT into the de facto workhorse of computational electronic spectroscopy.
The present work provides an overview of visualization and analysis methods for linear-response TD-DFT, going beyond molecular orbitals (MOs) and aiming to describe (and potentially quantify) how charge is rearranged upon electronic excitation.Both density-based and orbital-based visualization tools are considered, as are certain atomic partitions of the density change upon excitation, Dr(r) = r exc (r) À r 0 (r). (1.1) These can be used to characterize the nature of an excited state in both qualitative and quantitative terms.9][20][21][22] This obscures certain simplifications that are possible for CIS-and TD-DFTtype wave functions, for which the particle-hole picture is clear and explicit.The present work is limited to those particular ansa ¨tze, with an emphasis on connections between different visualization and analysis tools that exist in the literature.
The remainder of this work is organized as follows.Section 2 provides a brief introduction to the formalism of linearresponse TD-DFT and also introduces some visualization tools based on the density matrix, which are more incisive than simply plotting Dr(r) in real space.Orbital-based visualization tools, which remain the most popular means for qualitative characterization of an excited state, are introduced in Section 3. To quantify charge rearrangement upon excitation, it is useful to introduce an atomic partition of Dr(r) that can be made into a metric for CT, and can also assist in understanding states that are delocalized across more than one chromophore.These tools are introduced in Section 4. Section 5 introduces additional ways to quantify exciton delocalization that have a direct connection to the underlying Kohn-Sham wave function or transition density.Finally, the CT problem in TD-DFT is described in Section 6 along with a discussion of various metrics that can be used to indicate when (and for which excited states) this becomes an issue.

Theoretical background
We begin with a brief recapitulation of the linear-response TD-DFT formalism (Section 2.1), then introduce densities and density matrices for ground and excited states (Section 2.2).Attachment and detachment densities, 52 which are important tools for excited-state visualization, are introduced in Section 2.3.

Linear-response TD-DFT
Mathematical derivations of linear-response TD-DFT, starting from time-dependent response theory applied to the Kohn-Sham ground state, can be found elsewhere; [1][2][3][4][5] see ref. 1 for a pedagogical version.The linear-response formalism is what is most often implied by ''TD-DFT'', as it is (by far) the most common form.7][58][59][60][61][62][63] For computing excitation energies and most molecular electronic spectroscopy applications, however, the real-time method is much less efficient. 15][66][67][68][69] Starting from the ground-state solution of the Kohn-Sham eigenvalue problem, 70 F ˆcr = e r c r , (2.1)   the basic equation of linear response theory is This is a non-Hermitian eigenvalue problem for the excitation amplitudes x (n) = (x (n) ia ) and de-excitation amplitudes y (n) = (y (n) ia ), for the nth excited state whose vertical excitation energy is o n .Throughout this work, we use indices i, j,. . . to denote occupied MOs; a, b,. . . to indicate virtual (unoccupied) MOs; and r, s,. . . to denote arbitrary MOs.Spin indices are omitted here; see ref. 1 for a version of these equations that includes them.The matrices A and B in eqn (2.2) are Hessians with respect to orbital rotations. 1,71In the canonical MO basis that diagonalizes the Fock matrix F, their matrix elements are A ia;jb ¼ ðe a À e i Þd ij d ab þ @F ia @P jb (2.3a) and B ia;jb ¼ @F ia @P bj ; (2.3b) where P is the one-electron density matrix.Expressions for A and B in terms of electron repulsion integrals and the exchange-correlation (XC) kernel can be found elsewhere. 1,8,9astly, the quantities e a À e i in eqn (2.3a) are differences between virtual (e a ) and occupied (e i ) Kohn-Sham energy levels defined by the ground-state eigenvalue problem in eqn (2.1).
The difference e a À e i appears along the diagonal of A and constitutes a zeroth-order approximation to an electronic excitation energy, consistent with a zeroth-order picture in which an electronic transition consists in promotion of one electron from a single occupied MO into a single virtual MO, c ic a .
A TD-DFT calculation consists of solving eqn (2.2) for a certain number of excited states, each characterized by vectors x (n) and y (n) .These are subject to an unconventional normalization, consistent with the metric matrix in eqn (2.2). 4,[72][73][74] For brevity, we omit the state index n in eqn (2.4) and subsequent expressions.Amplitudes {x ia } and {y ia } parameterize the transition density matrix (TDM) for the excitation in question.As a position-space kernel, that object is 4,8,73 Tðr; It provides one possible visualization tool, usually in the form of the transition density, T(r) T(r,r).Often, eqn (2.2) is simplified by invoking the Tamm-Dancoff approximation (TDA), 3,75 in which the de-excitation amplitudes y ia are neglected.These amplitudes arise naturally in the equation-of-motion formalism for the one-particle density matrix, 4,74 yet in molecular TD-DFT calculations they are typically B100Â smaller than the largest x ia .(This may not always be the case for solids. 76,77) The matrix B is absent from the resulting TDA eigenvalue problem, which is simply Ax = ox. (2.6) For historical reasons, 78 the original eigenvalue problem in eqn (2.2) is sometimes called the random phase approximation (RPA), 9 in order to distinguish it from the simpler Hermitian eigenvalue problem in eqn (2.6).9][80][81] Where we need to make a distinction, we refer to eqn (2.2) as ''full'' TD-DFT and eqn (2.6) as TD-DFT/TDA.3][84][85] Triplet instabilities in the ground-state Kohn-Sham solution indicate that an unrestricted wave function would lower the energy with respect to the (unstable) closedshell solution, 86 and these instabilities manifest as negative excitation energies. 877][98] Beyond indicating an instability, solutions with negative excitation energies can lead to convergence failure in solving eqn (2.2), if the iterative eigensolver algorithm is predicated on the excitation energies being positive.Invoking the TDA decouples the stability problem from the excitation energy problem and is used in most calculations that are described here.
The TDA simplifies the structure of the transition density into a form where one can imagine a Kohn-Sham wave function 2,99 (determinant) whose form is analogous to the CIS ansatz, namely x ia jC a i i: (2.7) Here, |C a i i is a Slater determinant that differs from the ground state by a single substitution, c ic a .Given this form for |C exc i, the real-space kernel T(r,r 0 ) in eqn (2.5) can be connected to its more general definition in wave function theory, 17,73 which is Tðr;r 0 Þ ¼ N ð C Ã 0 ðr 0 ;r 2 ;...;r N ÞC exc ðr;r 2 ;...;r N Þdr 2 ÁÁÁdr N ; (2.8)   where C 0 (r 1 ,...,r N ) is the ground-state wave function.The definition of T(r,r 0 ) in eqn (2.8) is also valid for correlated wave functions. 18t has been argued that eigenvalue differences e a À e i should be good approximations to true excitation energies in exact Kohn-Sham theory, [99][100][101][102] albeit without spin coupling so there is no distinction between excitations to different spin multiplicities.To the extent that this remains true in approximate DFT, one might imagine that configuration mixing in eqn (2.7) occurs to a lesser extent in TD-DFT as compared to CIS, which is indeed observed to be the case. 103For example, Fig. 1a and b compare amplitudes x ia for S 0 -S 1 excitation of formaldehyde, computed using TD-DFT/TDA with the PBE functional and also with the Hartree-Fock functional, equivalent to the CIS method.The TD-PBE eigenvector consists almost exclusively of the 2b 2 -2b 1 amplitude whereas in a converged CIS calculation (including all virtual orbitals), this amplitude accounts for less than 20% of the norm of the transition density.(These calculations were performed using a realspace electronic structure code, 104 so there is no finite-basis approximation.CIS results with conventional Gaussian basis sets are shown in Fig. 1c).Truncating the virtual space leads to a more compact expansion and a larger 2b 2 -2b 1 coefficient in the CIS case, but this has a disastrous effect on the excitation energy (Fig. 1b).This is not a useful strategy.
That said, significant configuration mixing may be an unavoidable consequence of using hybrid functionals that contain some fraction of ''exact'' (Hartree-Fock) exchange.Virtual MOs in Hartree-Fock theory experience an N-electron potential rather than a (N À 1)-electron potential, 105 so the virtual levels e a are upshifted such that even the frontier virtual MOs are often unbound (e a 4 0).These unbound MOs are discretized continuum states, 106 and a large number of them will need to mix together in order to generate the localized wave function of a bound excited state.Inclusion of diffuse basis functions, which are often necessary to obtain converged excitation energies, 7,107 also generate significant configuration mixing as shown in Fig. 1c.
Configuration mixing muddies the picture of electron and hole, so it is desirable to have alternative ways of visualizing an excitation besides plotting a potentially large number of occupied and virtual MOs, corresponding to the significant amplitudes x ia .To that end, we next introduce excited-state electron densities that can be used to visualize an excitation in real space.

Densities and density matrices
Within TD-DFT, the density matrix for an excited state can be expressed as (2.9) Here, P 0 is the ground-state density matrix and is the unrelaxed difference density matrix.The quantity Z is the so-called ''Z-vector'' contribution that accounts for orbital relaxation in the excited state.This journal is © the Owner Societies 2024 The ''particle'' (or electron) and ''hole'' components of DP are available from the TD-DFT response vectors: 94,[108][109][110] Expressions for the matrix elements in the MO basis can be simplified to afford 109 Whereas DP is available from x and y alone, calculation of Z in eqn (2.9) requires solution of the coupled-perturbed equations that are associated with the TD-DFT excited-state gradient. 94,110The density matrix P exc that includes Z is known as the ''relaxed'' density matrix for the excited state, whereas is the unrelaxed density matrix.Examples illustrating the role of Z are deferred to Section 2.3.The quantities DP elec and DP hole can be conceptualized as separate densities for the excited electron and the hole that it leaves behind in the occupied space.More precisely, this is true of the real-space densities Dr elec (r) and Dr hole (r) that are encoded by these density matrices.Unlike the difference density, which exhibits both positive and negative regions in space, Dr elec (r) Z 0 everywhere in space, and Dr hole (r) r 0. Sometimes it is more informative to visualize these two quantities separately.It is therefore suggested that Dr elec (r) should be called the particle density and Dr hole (r) the hole density.(These terms are sometimes used differently, 18 but our usage is consistent with the idea of DP hole as the density matrix for the hole. 72) An example is depicted in Fig. 2, where the particle and hole densities can be visually superimposed by the reader to suggest the difference density Dr(r), which is also shown.The Z-vector contribution is omitted in this example, so these are unrelaxed densities.
The molecule in Fig. 2 is a polyfluorene oligomer with a single keto defect (fluorenone) in one of the terminal monomer units. 111It provides an example of how particle and hole densities are useful for interpreting excited states that are strongly mixed in the canonical MO basis, meaning there are numerous amplitudes x ia that are similar in magnitude.In this particular example, the frontier MOs are completely delocalized along the length of the oligomer, which is not atypical for p-conjugated chromophores.Nevertheless, it is obvious from the densities in Fig. 2 that the excited state in question is localized as a result of the defect.This is not obvious within the canonical MO basis, however, wherein the transition density consists of a roughly equal mixture of four different c ic a excitations, as shown in Fig. 3a.Localization arises from phase interference in a coherent superposition of four terms, but this is essentially impossible to discern by inspecting the relevant MOs alone.

Attachment and detachment densities
For CIS and TD-DFT calculations, the particle and hole densities defined in Section 2.2 coincide precisely with the attachment density and the detachment density, respectively, quantities that were originally defined in a manner that is not limited to single-excitation theories. 52This definition sheds additional light on the nature of DP elec and DP hole in TD-DFT.First, let us diagonalize a difference density matrix DP and express the result in the form where the nonzero blocks a and d are diagonal matrices that contain the positive and the negative eigenvalues of DP, respectively.Density matrices corresponding to the attachment and detachment densities are then defined as 16,17,52 and Note that DP attach is positive semidefinite and DP detach is negative semidefinite.This procedure could be followed for any difference density matrix, including the relaxed one from a TD-DFT calculation, or one that is obtained from a correlated wave function.In the special case that DP is the unrelaxed difference density matrix from a TD-DFT calculation [eqn (2.10)], it follows that DP attach DP elec and DP detach DP hole .Although this equivalence has been noted before, 1,17 it does not seem to be widely appreciated.It arises from a unique feature of single-excitation theories, namely, a direct correspondence between CI coefficients and matrix elements of the TDM. 18,73,74,112In the CIS case, for example, x ia = hC exc |a ˆ † a a ˆi|C 0 i.
Considering the specific case of DP in eqn (2.10), qualitative insight into the nature of an excited state can often be gleaned by analyzing its particle and hole components, DP elec and DP hole .It must be borne in mind, however, that electron/hole separation does not survive the contribution from orbital relaxation, i.e., from Z in eqn (2.9).Nonzero matrix elements Z ia = Z ai introduce occupied-virtual coupling, in contrast to the occupied-occupied and virtual-virtual terms that define the unrelaxed difference density [eqn (2.12)].However, one may construct the relaxed difference density, DP rlx = P exc À P 0 = DP + Z, (2.18)   and substitute this for DP in eqn (2.16).This defines attachment and detachment contributions to the relaxed density and recovers a particle/hole picture that includes orbital relaxation.
Orbital relaxation effects can be especially significant for states with CT character, as demonstrated in Fig. 4 for the case of a donor-acceptor complex consisting of naphthalene and tetracyanoquinone.Unrelaxed particle and hole densities (on the left in Fig. 4) suggest that the S 1 state of the complex has almost perfect CT character, with the excited electron localized on the acceptor (tetracyanoquinone) and the hole localized on the donor (naphthalene).A dipole moment change Dm = 14.9 D upon excitation underscores this CT character.However, the corresponding relaxed densities (on the right in Fig. 4) are both delocalized over both monomers.The change in dipole moment computed from the relaxed density is substantially reduced although still quite large: Dm = 10.9D. Note that the excitation energy is precisely the same regardless of which densities are used to visualize the transition, as is the ground-state dipole moment of 1.3 D, but the value of the excited-state dipole moment depends substantially on whether it is computed using the relaxed or unrelaxed density matrix for the excited state.
This example underscores the fact that the use of DP rlx rather than DP can have a significant effect on excited-state properties, 108,113,114 especially for states with a high degree of CT character. 108The relaxed dipole moment, which is the This journal is © the Owner Societies 2024 correct dipole moment for the excited state (according to linear response theory), is computed as m rlx x = tr(l x P exc ) for the x component, and its unrelaxed analog is m unrlx x = tr(l x P unrlx ).In Table 1, these two quantities are juxtaposed for the first excited state of formaldehyde ( 1 A 2 ), and for an excited state of pnitroaniline that is characterized by CT from its amino group to the nitro group.Even for the comparatively benign case of formaldehyde, use of the relaxed density alters the total dipole moment by more than 1 D for several different density functionals, bringing the value much closer to experiment.The contrast is more dramatic for p-nitroaniline, whose 1 CT state exhibits a large dipole moment (m E 13 D), 115,116 leading to significant orbital relaxation.For certain functionals, the unrelaxed density overestimates the excited-state dipole moment by more than a factor of two, although the effect decreases as the fraction of exact exchange is increased.(Relaxed and unrelaxed dipole moments differ by more than a factor of two for formaldehyde as well, 113,114 but the dipole moment is much smaller in that case.) Although relaxed densities are required for reliable and quantitative excited-state property calculations, there is much qualitative information to be gleaned from unrelaxed densities.For example, the CT nature of the donoracceptor transition in the naphthalene-tetracyanoquinone dimer (Fig. 4) comes through in both the relaxed and unrelaxed dipole moments, even if orbital relaxation serves to delocalize both particle and hole across both monomers.Other examples considered below will neglect the Z-vector contribution, which is adequate for a quick survey of the nature of the excited states.

Natural transition orbitals
Densities reveal how charge moves upon excitation but they sacrifice the phase (sign) information that is contained in the orbitals, which might be helpful for understanding the character of an excited state.If the number of significant amplitudes x ia is small, then the canonical Kohn-Sham MOs are a good way to visualize the state in question while retaining phase information, but this may be inconvenient if there are too many participating amplitudes, which is often the case with hybrid functionals and high-quality basis sets.
The quantity T(r,r 0 ) that is defined in eqn (2.5) and (2.8) does contain phase information, and can be plotted in threedimensional space by setting r = r 0 .However, the transition density T(r) T(r,r) cannot be directly interpreted in terms of the movement of charge.For example, consider the transition density for the fluorenone-terminated polyfluorene oligomer that was considered above (Fig. 2d).Although its nodal structure contains elements of the nodal structures of both Dr elec (r) and Dr hole (r), the transition density T(r) is clearly distinct from the difference density Dr(r).What can be gleaned from T(r) is the nature of the transition moment for the excitation in question, which must lie along the long axis of the molecule in Fig. 2d because the negative and positive lobes of T(r) approximately cancel along the short axis.Clearly, the result of the fluorenone defect is that this particular excited state is a property of the terminal monomer (fluorenone), not a property of the whole polymer.That fact is not obvious from the canonical MOs that participate in the transition, which are shown in Fig. 3a.The natural transition orbitals (NTOs), 120 which are plotted in Fig. 3b and introduced below, help to recover an electron/hole picture within a representation that preserves phase information.

Theory
Starting from DP elec and DP hole in eqn (2.12), phase information can be restored (in a manner that maximally preserves the Fig. 4 Relaxed and unrelaxed detachment densities (upper images, in orange) and attachment densities (lower images, in magenta) for the S 0 -S 1 transition of a donor-acceptor complex involving naphthalene and tetracyanoquinone.Calculations were performed at the TD-oB97X-D/6-31G* level within the TDA and all densities are plotted using 90% isoprobability contours.Unrelaxed densities (on the left), corresponding to density matrices in eqn (2.11), are localized on either the donor or the acceptor.Relaxed densities (on the right) are delocalized over both molecules, although the state maintains significant CT character as evidenced by the dipole moment change upon excitation, Dm. qualitative characteristics of these densities) by using their eigenvectors to define a change-of-basis for the canonical MOs.The transformation U o that diagonalizes DP hole defines a transformation of the occupied MOs that we express as The n occ Â n occ diagonal matrix K 2 contains the eigenvalues of DP hole .(It is the square of a diagonal matrix K that will be introduced in Section 3.2, where we will discover that the values l i have their own significance.)Eigenvalues of DP hole are nonnegative, which we indicate by writing them as l i 2 , and they are normalized such that In similar fashion, we introduce a matrix U v that diagonalizes DP elec , which defines a transformation of the virtual MOs.For singleexcitation wave functions, the matrices DP elec and DP hole have the same eigenvalues, up to a sign, 111,120 so K 2 is the same matrix in both eqn (3.1) and (3.2). (Extra zeros in the latter are needed to dimension the matrices consistently.) The matrix U o transforms the canonical occupied MOs into a set of ''hole'' orbitals that we will call {c hole i (r)}, while U v transforms the canonical virtual MOs into a corresponding set of ''particle'' (or ''electron'') orbitals {c elec i (r)}, where i = 1,. .., n occ in both cases; even for the virtual orbital transformation in eqn (3.2), there are only n occ nonzero eigenvalues.These transformed orbitals are the NTOs for the hole and for the excited electron, respectively.They are potent tools for qualitative analysis because they reduce the 2n occ n virt excitation amplitudes x ia and y ia into just n occ unique amplitudes, as discussed further in Section 3.2.For now, we simply note that the particle and hole densities are weighted averages of single-NTO probability densities: Examples of NTOs are provided in Sections 3.3 and 3.4.The term ''NTO'' was first suggested by Martin in 2003, 120 but this form of analysis was introduced much earlier by Luzanov and co-workers. 121,122It has since been rediscovered (and expressed in the notation used above) by others. 112,120,1235][126] Just as natural orbitals are eigenfunctions of P (even in the case of a correlated wave function), 124 with eigenvalues that are natural occupation numbers, the NTOs diagonalize the TDM.Within a single-excitation model, this is equivalent to diagonalizing the unrelaxed difference density matrix DP, although that equivalence is lost in the case of a correlated wave function.(In the latter case, one must distinguish between NTOs that diagonalize the TDM and the natural difference orbitals (NDOs) that diagonalize the difference density matrix. 18,19) Similarly, natural ionization orbitals diagonalize the difference density obtained upon electron removal. 127[130]

Interpretation
The transformations in eqn (3.1) and (3.2) fully define the NTOs in TD-DFT, but an equivalent and illustrative definition is possible.Keeping to the TDA case for simplicity, we consider x to be a rectangular matrix of dimension n occ Â n virt .Hole and particle NTOs are defined by separate unitary transformations of the occupied and virtual MOs (U o and U v , respectively), and an equivalent definition of these two transformations involves a singular value decomposition (SVD) of x: Here, K is the n occ Â n occ matrix of singular values l i , the same matrix that appears (as K 2 ) in eqn (3.1) and (3.2).According to eqn (3.4), the matrices U o and U v contain the left and right singular vectors of x, respectively, but they are identical to the eponymous transformations defined as eigenvectors of DP hole and DP elec .As compared to the how NTOs were introduced in Section 3.1, the construction in eqn (3.4) demonstrates more clearly why no more than n occ of the singular values are non-zero, and shows why the eigenvalues of DP hole and DP elec occur in pairs. 120From yet another point of view, eqn (3.4) is a special case of a corresponding orbitals transformation, [131][132][133] which selects a subset of virtual orbitals in one-to-one correspondence with the occupied orbitals.In this case, the NTOs are obtained from the corresponding orbitals transformation that diagonalizes the TDM.
If DP hole is dominated by a single NTO then so is DP elec , which is a consequence of the correspondence between amplitudes x ia and elements of the one-particle density matrix (Section 2.3).As a result, for single-excitation theories (only), the NTOs are equivalent to excited-state natural orbitals. 112For CIS-type wave functions, the eigenvalues in the natural orbital basis (i.e., the natural occupation numbers) can be specified in terms of the singular values of the transition amplitudes: 112 The values n r = 1 À l2 r represent the hole that is created, and n r = l 2 r correspond to the excited electron; this demonstrates why DP elec and DP hole have the same eigenvalues, up to a sign.Redundant orbitals (n r = 0) have been eliminated by the SVD in eqn (3.4).Although the direct connection between the excitation amplitudes, transition density, and one-electron density matrix for the excited state is a unique feature of the single-excitation ansatz, the concept of attachment and detachment densities as eigenfunctions of DP, separable based on the sign of the eigenvalues in eqn (2.16) and (2.17), is generalizable to wave functions of arbitrary complexity.The individual eigenfunctions of DP, which are the NDOs, 18 then generalize the concept of NTOs for many-body theories, without the need to introduce ''correlated NTOs''. 134otice also that the TDM is diagonal in the NTO basis: This constitutes another proof that the transformation to NTOs distills a given excitation into the smallest possible number of occupied/virtual orbital pairs.In a well-defined sense, the NTO basis is the best choice for conceptualizing excited states in terms of a one-electron promotion from an occupied MO into a virtual MO.The NTOs are state-specific, so this optimal basis changes from one excited state to the next.(State-averaged NTOs have been suggested as a compact basis for correlated wave function expansions. 18) In eqn (3.1) and (3.2), we have written the eigenvalues of DP elec and DP hole as l 2 r in order to emphasize the correspondence with probabilities x 2 ia in the canonical MO basis, whereas the singular values l r in eqn (3.4) are the amplitudes themselves, rotated into the NTO basis.
For chemists, there exists a temptation to designate the orbitals comprising the first NTO pair (having the largest singular values l i ) as ''HONTO'' and ''LUNTO'',  in analogy to the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO). (The tems ''HOTO'' and ''LUTO'' have also been used.111,158 ) This seems to be especially prevalent in the literature on thermally-activated delayed fluorescence (TADF) emitters.[144][145][146][147][148][149][150][151][152][153][154][155][156][157] As even some who use this terminology have acknowledged, 137 this usage is incorrect insofar as ''highest'' and ''lowest'' are typically used in the context of the aufbau principle, whereas orbital energies are undefined in the NTO basis because the Fock matrix is not diagonal.As such, it makes no sense to discuss the energies of NTOs, and this makes the ''HONTO'' and ''LUNTO'' terminology especially confusing when discussed alongside HOMO/LUMO energy gaps, as is often done in the TADF literature.In this author's view, the terms HONTO and LUNTO should be avoided, so that visual descriptions of NTOs are kept separate from arguments based on one-electron energy levels.In discussing the NTO pairs with the largest singular values, a more appropriate term is principal transition orbitals, or perhaps principal NTOs (pNTOs). The ful set of NTOs can be labeled pNTO, pNTO À 1, pNTO À 2,. .., in order of decreasing singular values l 1 4 l 2 4 l 3 4. ... That is the nomenclature that will be used here.

Examples
Eqn (3.3) demonstrates how the NTOs extract the most important contributions to the particle and hole densities, or in other words the most significant contributions to the unrelaxed attachment and detachment densities.In the case where there is only one significant singular value (l 2 1 E 1), then |c elec 1 (r)| 2 E Dr elec (r) and |c hole 1 (r)| 2 E Dr hole (r), which are unrelaxed attachment and detachment densities, respectively.This connection does not seem to be widely appreciated.
In some cases the use of densities rather than orbitals may be more convenient, especially when several singular values are significant.That situation is discussed in Section 3.4.On the other hand, the NTOs preserve phase information that is lost upon squaring the orbitals and this may be useful in some situations, e.g., to distinguish np* from pp* in cases of significant orbital mixing, or to reveal the pp* character in a case with complicated nodal structure, as in the example of Fig. 3b.For a very different example, we turn to X-ray spectroscopy at the titanium K-edge.Calculations on a Ti 16 O 32 H 2 cluster (Fig. 5) were performed at the PBE0/def2-ma-SVP level, 15 where the basis set is a minimally-augmented one (denoted ''ma''), 159 which is intended to describe any nascent band structure.The K-edge consists of transitions from Ti(1s) to valence virtual orbitals at almost 5000 eV, and to access core-level excitations these calculations invoke the core/valence separation approximation. 160That means omitting amplitudes x ia unless c i corresponds to a core orbital of interest, meaning Ti(1s) in the present example, while retaining the full virtual space.The principal particle NTO in this example exhibits essentially just one nonzero singular value (l 2 1 = 1.00) and is depicted in Fig. 5a, where its Ti(3d) character is evident along with some admixture of O(2p).The hole NTO is not shown but corresponds to the Ti(1s) on a nearby atom, meaning that the asymmetry of the cluster has localized this Ti(1s) -Ti(3d) pre-edge feature to one end of the cluster.
In the canonical MO basis, the same transition is completely scrambled and essentially uninterpretable.Two of the relevant canonical virtual orbitals are shown in Fig. 5b and c but there are 17 distinct amplitudes with |x ia | \ 0.1, the largest of which contributes only 19% of the norm of the transition eigenvector.Together, these 17 amplitudes contribute only 85% of the norm.Note that Fig. 5 indicates the fraction of |c(r)| 2 that is encapsulated within each depicted isosurface, which is necessary in order to make meaningful side-by-side comparisons of orbital localization or size.Isoprobability surfaces can be readily computed, 161 given volumetric data on a grid (e.g., in the format of a ''cube'' file), 106 and this functionality is available in some visualization programs. 162Orbitals should not be compared side-by-side unless one is plotting a consistent fraction of |c(r)| 2 , lest one give a false impression of relative size.
The example in Fig. 5 demonstrates that x may be characterized in some cases by just one singular value, with l 2 1 E 1.In such cases, the principal NTO pair can be used to distill the picture into one that involves just one occupied and one virtual orbital, without loss of information.Such is also the case for the S 0 -S 2 transition of the fluoronone-terminated polyfluorene that is shown in Fig. 3, for which l 2 1 = 0.96.The NTO basis presents a simple picture (Fig. 3b), but in the canonical MO basis there are four different amplitudes x ia that contribute significantly to the same transition (Fig. 3a).The latter are highly delocalized in the occupied space and form a localized hole only upon coherent superposition, whereas localization is immediately evident in the NTO basis.Higher-energy transitions of polyfluorenes do involve a larger number of significant NTO pairs, 136 which is not unusual.Since NTOs are the optimal particle/hole basis, the presence of more than one significant singular value l i is a signature of unresolvable multideterminant character in the excited-state wave function, which cannot be rotated away by unitary change of basis. 18,19,112The next section considers this in more detail.

Static correlation
The presence of more than one significant singular value (l i ) in the SVD of x can be driven by symmetry-induced orbital degeneracies.Such is the case for benzene, whose frontier MOs (Fig. 6) consist of a pair of degenerate e 1g orbitals (HOMO and HOMOÀ1) along with a pair of degenerate e 2u orbitals (LUMO and LUMO+1).In small basis sets, there is essentially no difference between the canonical MOs and the NTOs for the low-lying excited states and they can be used interchangeably in the following discussion.Four singly-excited determinants can be constructed from the aforementioned frontier MOs, and together these make up the first four singlet excited states of benzene.These states are: States S 1 and S 2 are optically dark in one-photon spectroscopy but S 3 is dipole-allowed and doubly degenerate. 163Calculations at the TD-B3LYP/6-31G* level conform to this simple fourorbital model with 499% fidelity.Although the TD-DFT results might at first seem complicated, in the sense that there is no excited state that is primarily HOMO(2e 1g ) -LUMO(1e 2u ) in nature, there is actually little else that these states could have been, given the symmetry of the system.Symmetry here is a parlor trick that makes the situation seem complicated by introducing unresolvable multiconfigurational character, wherein a minimum of four orbitals and two determinants is required to describe the low-lying excited states, even within a single-excitation theory such as CIS or TD-DFT.A more interesting example, which is not driven by symmetry, is the keto-defect polyfluorene oligomer whose S 0 -S 2 transition was considered in Fig. 2 and 3 and whose S 0 -S 3 transition is depicted in Fig. 7a.There is interest in these molecules for fabrication of organic light-emitting devices, [164][165][166][167][168][169] as this is one of the few classes of materials that can span the whole range of visible wavelengths at low operating voltage, 164 and with good emission properties for blue light. 165These properties arise from highly delocalized excited states of the p system that may exhibit large polarizabilities and hyperpolarizabilities, giving rise to nonlinear optical properties. 170n the present example, such states are accessed at higher excitation energies such as o = 4.5 eV for S 0 -S 3 .The oscillator strength for that delocalized transition ( f = 4.5) is about 25 times greater than that of the defect-localized S 0 -S 2 excitation.
For S 0 -S 3 , even the principal NTOs are delocalized over the length of the molecule (Fig. 7a), meaning that this is genuine delocalization and is not an artifact of the representation.The principal NTO pair accounts for only 67% of the transition density while a second NTO pair contributes another 20%.Irreducible mixing of more than one NTO pair is a signature of static correlation in the excited state. 18,19,112(Note that there is no contradiction with the use of a single-determinant formalism for the ground state, because the CIS wave function ansatz is multideterminantal.)From another point of view, the presence of more than one significant singular value in the TDM indicates that the natural orbitals of the ground state differ significantly from those in the excited state. 112A close examination of the NTOs in Fig. 7a reveals that c hole 1 (r) and c hole 2 (r) are out of phase with one another at the left end of the molecule, but evolve across its length such that they are in phase on the right terminus of the molecule.The same is true of c elec 1 (r) and c elec 2 (r), which suggests that the excited state in question can only properly be described using a minimum of two determinants.This would not be obvious from attachment/detachment density analysis.
Formal analysis suggests that static correlation, manifesting as more than one significant NTO pair, may occur in cases where a molecule consists of two or more weakly-interacting chromophores, 172 even if these chromophores are but different chemical moieties within the same molecule.An example is the molecule shown in Fig. 7b that consists of three identical ligands connected to a central 1,3,5-triazene moiety in a propeller motif, wherein each ligand may be considered a distinct chromophore.4][175][176][177] ) Canonical MOs for this molecule are not shown but are mostly delocalized over all three ligands, nevertheless the NTOs for the S 0 -S 1 transition are delocalized over just two of the three ligands (Fig. 7b).Although this could be inferred also from the particle and hole densities, what those densities cannot reveal is the role of static correlation: this excited state is evidently an irreducible combination of two particle/hole pairs.Another example with multiple chromophores within the same molecule is the poly[2-methoxy-5-(2ethylhexyloxy)-1,4-phenylene vinylene] (MEH-PPV) polymer that is shown in Fig. 7c. 171Here, breaks in the conjugation divide the polymer into several effective intramolecular chromophores, yet electronic coupling between them is sufficient to maintain coherence of the exciton across these gaps in conjugation.
The close connection between significant NTO pairs and static correlation suggests that the NTOs can be used to infer electron configurations, and in particular to detect changes in electron configuration across a potential energy surface.(In fact, a more descriptive name for the NTOs might be natural electron configurations. 178) Consider the electrocylic ring-opening of oxirane (C 2 H 4 O), a prototypical reaction for which the Woodward-Hoffmann rules were developed. 179,180Potential energy curves along the C-C-O bond angle of the ring are plotted in Fig. 8 and isosurface plots of the principal NTO pair are provided at various points, for transitions to S 2 and S 3 . 181The reaction involves a conical intersection between these two excited states, at an angle y CCO E 621, and the nonadiabatic transition that occurs there is accompanied by an abrupt switch in the qualitative nature of c elec 1 (r), as shown in Fig. 8a.On the other hand, c hole 1 (r) remains qualitatively consistent as the system passes through the intersection.By means of these NTOs, one may assign the diabatic character of either state: for y CCO o 621, the S 2 state is ns* and the S 3 state is n -3p z , whereas this character is reversed for y CCO 4 621.Away from any near-degeneracy between Born-Oppenheimer potential surfaces, no such abrupt change is seen in the nature of the dominant NTOs, as illustrated in Fig. 8b.

Atomic partitions
Orbitals and densities introduced above provide convenient tools to visualize excited states in real space.The present section describes tools that attempt to quantify charge rearrangement in Dr(r) by partitioning the density change into atomic contributions.

Mulliken analysis
Consider the 1 CT state of p-nitroaniline whose dipole moment change is listed in Table 1.Although the HOMO is nominally located on the amino group and the LUMO on the nitro group, both orbitals extend over a significant portion of this small molecule, thus the CT character of the state in question may not be obvious from the MOs.A Mulliken-style [182][183][184] partition of DP elec and DP hole might help to quantify the nature and extent of the charge rearrangement.In this approach, the charge that is transferred to atom A upon electronic excitation is defined as where S is the atomic orbital (AO) overlap matrix.Simultaneously, atom A may lose some charge if it contributes to the hole, which may be quantified as Other common atomic partitions of a density matrix can be applied equally well, to obtain Lo ¨wdin charges rather than Mulliken charges, 105 for example.These decompositions are subject to the same variability with respect to the choice of AO basis set that characterizes ground-state Mulliken or Lo ¨wdin atomic charges, and are intended only to aid qualitative understanding.Charges derived from the molecular electrostatic potential 185 are much more reliable when it comes to reproducing electrostatic properties of excited states, 24 such as dipole moments, while Hirshfeld charges have been recommended for tracking photochemical changes in electronic structure. 186

Charge-transfer numbers
8][189] Like the difference charges Dq elec A and Dq hole A , these quantities attempt to identify and quantify charge flow upon electronic excitation, based on atomic indices.For atoms or groups of atoms A and B, one might intuitively define an A -B charge transfer number according to 72,189 where is a transition amplitude expressed in the AO basis. 189(The quantity y mn is defined analogously and x mn is sometimes called a pseudo-density. 190,191) The idea behind eqn (4.3) is that squared amplitudes x 2 mn and y 2 mn are associated with probabilities for transfer of charge from m A A to n A B. However, the formula in eqn (4.3) accounts neither for the normalization condition in eqn (2.4), nor for the fact that the AOs are nonorthogonal.This may not be an issue when l A-B is used to analyze semi-empirical calculations, 188,[192][193][194][195][196][197][198] where the inherent minimal basis might be assumed to be orthonormal, but the same formula has been put forward for all-electron TD-DFT calculations in arbitrary basis sets. 72,189Normalization could be enforced in a straightforward fashion, 199 defining Failure to account for the AO overlap matrix, however, leads to significant discrepancies in CT numbers computed in small versus large basis sets. 199or this reason, an alternative definition due to Plasser et al. is preferable, 18,200 as it accounts for non-orthogonality of the AO basis functions.This definition starts from the normalization condition ð jTðr; r 0 Þj 2 drdr 0 ¼ 1: 4][205] Alternatively, one might exploit trace invariance to partition the summand in eqn (4.8) in the spirit of a symmetric (Lo ¨wdin) orthogonalization of DP. 105,[206][207][208] This means two factors of S 1/2 (DP)S 1/2 as opposed to the separate factors of (DP)S and S(DP) that appear in eqn (4.8). 27This leads to a definition to quantify the flow of charge from A to B, which amounts to a Lo ¨wdin-style partition of DP. 27 The quantity O A-B is a CT index in the spirit of l A-B but corrected to take proper account of the non-orthogonal AO basis set.A Mulliken-style partition has also been formulated, 18 in the spirit of eqn (4.8), however Lo ¨wdin populations are generally somewhat more stable and free of negative population artifacts. 27,184That said, the value of O A-B certainly depends on the choice of AO basis set, as does any Lo ¨wdin population analysis. 184he method based on eqn (4.9) has been called fragment transition density analysis, 199,203,209 because in the case of a correlated wave function one could imagine using the TDM in place of DP.For TD-DFT there is no distinction, although one could substitute DP rlx in place of DP, thereby using the relaxed density to understand charge flow.This is likely the better approach for analyzing CT states, for reasons discussed in Section 2.3.
The CT indices O A-B satisfy the normalization condition for single-excitation wave functions.(The normalization condition is more complicated for other types of wave functions. 18) An expression analogous to eqn (4.10) has been suggested for l A-B , 189 yet this claim seems suspicious for all-electron TD-DFT calculations in non-orthogonal basis sets.Several other concepts introduced by Luzanov et al. 72,189 in the context of the indices l A-B would seem to be rigorously valid only when the alternative definition O A-B is used instead.These include a gross excitation localization index (GLI), 72,189 where The quantity CT A is a measure of the charge that is shifted around in ways that involve atom A. It follows that X which suggests that GLI A provides an atomic or functional group partition of the excited electron.Along similar lines, it is possible to use the quantities O A-B to define the size of an exciton, although we postpone that discussion until Section 4.5.The CT character can be quantified using The net charge transferred from A to B is which has something of a history in TD-DFT.Its spectroscopy consists of dipole-allowed ultraviolet transitions to a pair of states known as 1 L a and 1 L b , [210][211][212] in a notation that comes from Platt's free-electron model. 2135][216][217][218][219][220] ) Setting aside detailed symmetry considerations, these two states have roughly perpendicular transition moments, along axes ''a'' and ''b''.The S 0 -1 L a transition is primarily a HOMO -LUMO excitation, with significant ionic character in PAHs, while S 0 -1 L b is a mixture of (HOMOÀ1) -LUMO and HOMO -(LUMO+1). 221TD-DFT calculations sometimes afford an unbalanced treatment of these two states, [220][221][222][223] which are quite close in energy in the case of DMABN. 224][247] Dual fluorescence represents an exception to Kasha's rule, [248][249][250][251] which states that emission typically occurs in a single band originating from the lowest excited state, insofar as radiationless internal conversion from higher-lying excited states is usually rapid and efficient.3][254][255][256][257][258][259][260] In this picture, the TICT state is stabilized in polar solvents, relative to the ''locally excited'' (LE) state, which has 1 pp* character, and is the origin of the longer-wavelength fluorescence band.][271][272] What is not in dispute is that the S 1 and S 2 states of DMABN exhibit different degrees of CT upon vertical excitation.In the gas phase, S 1 is the LE state and S 2 is the CT state, as evidenced by a dipole moment that is E6 D larger in S 2 than S 1 , even while the S 1 dipole moment is E3 D larger than that of S 0 . 224This interpretation is furthered by examining CT numbers and GLIs for both states, which are provided in Fig. 9 based on Luzanov's definition (l A-B ), normalized as percentages to sidestep issues with the normalization of eqn (4.3).These quantities suggest that the S 0 -S 1 transition is characterized by a single large CT number corresponding to electron transfer from the amino lone pair into the phenyl ring, yet the GLI analysis suggests that 73% of the excited electron is localized on the phenyl ring, consistent with the idea that S 1 is the pp* state.For the S 2 state, the CT numbers provide clear evidence of aminophenylcyano electron transfer, with smaller fractions of the excited electron localized on the amino and phenyl groups as compared to S 1 , and a larger fraction transferred to the cyano moiety.

Frenkel excitons and charge-resonance states
New forms of complexity emerge in systems having multiple electronic chromophores that are identical or near-identical and whose vertical excitation energies are degenerate or quasidegenerate.4][275][276] Consider the case of two identical monomers in a high-symmetry arrangement, such as a cofacial benzene dimer with D 6h symmetry, for which the pNTOs are illustrated in Fig. 10.
As discussed in Section 3.4, a minimum of four Slater determinants is required to describe the frontier excitations of the benzene monomer [eqn (3.7)], and the same is true for the dimer although the relevant pNTOs are delocalized over both monomers.
Collective excitations of electronically coupled chromophores can be conceptualized as linear combinations of basis states jC Ã 1 C 2 i and jC 1 C Ã 2 i in which one monomer or the other is excited.These are the Frenkel exciton (FE) states, as in the classic case of H-and J-aggregates of PAHs. 277In a highsymmetry system such as the benzene dimer, the mixing coefficients are equal: In lower-symmetry examples, the isolated-monomer excitations may not be exactly degenerate.Mixing may still occur but the coefficients need not be equal, so a more general expression is for mixing coefficients c 1 and c 2 .
When the chromophores are at close-contact (van der Waals) separation, there is also the possibility of intermolecular CT, which can be represented using basis states |C + 1 C À 2 i and or |C À 1 C + 2 i.For highly symmetric systems, these forward and backward CT states may be degenerate, leading to the formation charge-resonance (CR) states, which are characterized by equal amounts of forward and backward CT. 46,274,278 If the electron-transfer process is similar in energy to the S 0 -S 1 monomer excitation energy, then either CT excitons or else localized CT states may further mix with FE states.][281] Various scenarios are illustrated schematically in Fig. 11.The states depicted in Fig. 11 form an idealized basis to guide one's thinking about delocalization and excitation energy transfer in multichromophore systems.In the real world, these  This journal is © the Owner Societies 2024 basis states interact and mix, so the real picture may be more muddled.7][298][299][300] In the perylene diimide dimer, which is a common molecular scaffold for SF, 301,302 there has been much discussion of solvent-induced symmetry breaking that can convert CR states into localized CT states. 303,304Within a quantum chemistry calculation, even low-polarity dielectric boundary conditions (e = 3, as in organic thin films) can provide sufficient polarization to break the electronic symmetry and localize the CT states. 278n cases where mixing is significant, it can be challenging to develop a conceptual picture based on detailed calculations, even in a dimer system.Because each of the four wave functions |C FE AE i and |C CR AE i is delocalized over both chromophores, FE states cannot be distinguished from CR states on the basis of particle/ hole or attachment/detachment densities. 122The key to differentiating them is to recognize that the CT indices (O A-B or l A-B ) contain information about correlations between particle and hole that are averaged away in the densities Dr elec and Dr hole .This has been analyzed in terms of the cumulant of the two-electron density matrix, 305 but a more straightforward analysis is to use a 2 Â 2 matrix comprised of the quantities O A-B or l A-B , in which fragments A and B represent the two monomers. 200he matrix X comprised of the CT numbers O A-B is presented in Table 2 for the delocalized states |C FE AE i and |C CR AE i that appear Fig. 11, along with the four basis states that contribute to them.We use a slightly unusual indexing convention for X, which is for the 2 Â 2 case spanned by monomers 1 and 2, as in Table 2.This differs from the usual matrix indexing convention, and from the X matrices defined by Plasser and Lischka, 200 but is consistent with real-space plots of T(r hole ,r elec ) that appear in Section 5.In those plots, the particle and hole coordinates r elec and r hole have their coordinate origins in the lower left, per usual convention, so it makes sense to put the O 1-1 matrix element in the lower left of X, which presents a discretized version of the same information that is conveyed by T(r hole ,r elec ).
By means of the matrix X, the FE and CR states become easily distinguishable: matrix elements along the diagonal (O 2-1 and O 1-2 ) are associated with charge separation while the anti-diagonal direction (O 1-1 and O 2-2 ) is associated with charge-neutral excitations and thus FE states.Note that the full matrix X is necessary in order to make these distinctions and the GLI in eqn (4.11) is insufficient.The metric N CT [eqn (4.14)] differentiates charge-neutral excitations (both localized and delocalized) from charge-separated ones, and Dq 1-2 [eqn (4.15)] establishes the direction of charge flow.
This analysis is idealized in the sense that it assumes orthonormal basis functions, and is intended to demonstrate simply that the aforementioned metrics are capable of distinguishing between various types of delocalized states, thus providing information that Dr elec and Dr hole cannot.(In real systems, a given excitation may exhibit both FE and CT character, as discussed above.)These metrics rely on our ability to assign amplitudes x mn to atoms and are thus susceptible to the same basis-set sensitivity that can be problematic for Mulliken or Lo ¨wdin charge analysis.That said, this type of analysis has been used in real calculations to classify the excited states of pstacked dimers of naphthalene, 200 adenine, 200,283 and pyridine, 306 for example.

Participation ratio
Table 2 also introduces the participation ratio (PR) as a means to distinguish between localized and delocalized states.8][309][310][311][312] A generic definition is where p i is the probability of localization onto site i, for a system with n sites.In quantum mechanics, p i is usually the square of some coefficient that expresses the wave function as a linear combination of localized basis functions that are assignable to sites, say, The denominator in eqn (4.20) then involves the fourth power of the amplitudes a ki , We assume normalized coefficients henceforth, in which case the numerator in this expression equals unity, as in eqn (4.20).
If p i = 1/n, indicating equal probabilities for each site, then PR = n.In general, the PR may be interpreted as the number of sites over which the wave function delocalizes, and for that reason it has sometimes been called a collectivity index. 724][315][316][317][318] However, calling the quantity defined in eqn (4.20) a PR is consistent with the earliest examples in the literature, [307][308][309][310] and perhaps more importantly it means that the PR increases (rather than decreases) as additional monomers participate in the excitation.With eqn (4.20) taken to define the PR, then its inverse is meaning that IPR = 1/n if p i = 1/n.That is consistent with the idea of the inverse (reciprocal) of participation by n chromophores, and also appears to be standard notation in the literature on localization phenomena. 309,311,319,320However, both Mukamel and co-workers, [312][313][314] as well as Fleming and co-workers, [315][316][317][318][319][320][321][322][323] are inconsistent regarding whether eqn (4.20) defines the PR or the IPR.In view of the arguments above, PR should be defined by eqn (4.20) and its inverse, if needed, can be called 1/PR.For TD-DFT, one might define separate PRs for the electron and the hole: 200

PR elec ¼ X
(Note carefully the order of the summation indices and the fact that X is not symmetric.The quantities PR elec and PR hole are distinct.)Combining these two quantities affords a PR for the electron-hole pair: 200 Following appropriate coordinate transformations, each of these PRs involves a summation over x mn 4 , as in eqn (4.22).In the idealized case of the states presented in Fig. 11, one finds that the four localized states are characterized by PR e-h = 1 and are thus distinguishable from the four delocalized states, for which PR e-h = 2.This is indicated in Table 2.
The quantities PR elec and PR hole measure the size of the exciton in terms of the coordinates of the electron (r elec ) or the hole (r hole ), respectively.Their average, PR e-h , is thus a measure of exciton size along the extracule coordinate, 324 r elec + r hole .A complementary metric is the coherence length of the exciton (L coh ), 200,312 which measures exciton size in terms of the intracule coordinate, 324 r elec À r hole .The coherence length may be defined using CT indices according to 200 and a cartoon depiction for a conjugated polymer is provided in Fig. 12a.Note how off-diagonal elements of X (or DP) characterize coherences between atoms or fragments in the electronic excitation.Here, ''off-diagonal'' means O A-B where AaB; consult the model X matrices in Table 2 and beware of the unusual indexing convention introduced in eqn (4.19).A value L coh = 1 indicates that there is no contribution from A a B, which implies that the excitation is a superposition of localized excitations (a FE state), or that it is simply localized on a single site. 200he length scale over which a FE state is delocalized is measured in the extracule coordinate, and a sensible definition of a PR to describe this is 200

PR diag
This has been called a ''diagonal'' PR, 200 or sometimes a diagonal length scale, 312 as indicated by the notation.However, the extracule coordinate lies along the anti-diagonal direction in a real-space representation of T(r hole ,r elec ); see Fig. 12b.The indexing convention for X in eqn (4.19) reflects the fact that this matrix is essentially a coarse-graining of T(r hole ,r elec ).Within this convention, the length scale PR diag appears along the anti-diagonal direction of X, so that a heat map of the latter will resemble |T(r hole ,r elec )| 2 in Fig. 12b.
6][327][328] Excited states of a six-unit PPV polymer [(PV) 6 Ph] are considered in Table 3. 200 Although these calculations were performed using a many-body wave function method, they are characterized in Table 3 using the descriptors introduced above, which are equally valid for TD-DFT.Indices A and B in these (PV) 6 Ph examples correspond to PV monomer units and X is depicted as a grayscale heat map in Table 3.
For the lowest few singlet excited states, including the S 1 bright state, PR e-h ranges from 3.8-6.7 with PR e-h 4 5 in all but one case.This indicates near-complete delocalization of the exciton over the molecule.The S 1 , S 2 , and S 3 states are characterized by zero, one, and two nodes along the extracule coordinate, respectively, and could be interpreted as sequential states belonging to a single exciton band, with particle-in-a-box character along the exciton center-of-mass coordinate.The S 4 , S 5 , and S 6 states (again with zero, one, and two nodes, respectively) constitute a second exciton band.This is consistent with the idea that the intracule and extracule coordinates r elec AE r hole sometimes behave as separable quasiparticle coordinates in conjugated polymers. 171,329However, Plasser and Lischka question whether these should indeed be characterized as FE states, given their significant coherence lengths (e.g., L coh = 3.9 for S 1 and L coh = 5.0 for S 4 ). 200These values quantify the diagonal length scale in the X heat maps and can be interpreted as electron-hole separation, measured in units of PV monomers.The computed values suggest that the electron and hole are well separated, unlike the conventional FE picture of a tightly-bound electron and hole.
Evidence of static correlation in this PPV system can abe detected using the quantity where the l i are the singular values associated with each NTO pair.The quantity PR NTO is a participation ratio in the NTO basis; cf.eqn (4.20) and (4.22).For the six-unit polymer described in Table 3, the quantity PR NTO starts at a value of 1.6 for the S 1 state and increases monotonically as one moves up the excitation manifold, with PR NTO E 4 for states S 5 and S 6 .These two states are each characterized by an average of four significant particle/hole pairs, indicating significant static correlation.The PR makes an appearance in certain analytic theories of exciton transport in organic photovoltaic materials. 316,317,330or example, a simple analytic theory predicts that the effective Huang-Rys parameter (linear exciton-phonon coupling constant) should be S(n) = S(1)/PR in a polymer with n repeat units, where S(1) is the Huang-Rys parameter for the monomer unit. 316he PR is also used in studies of conjugated polymers to define an effective length scale for an exciton, which need not be the same as the conjugation length in the ground state.With this in mind, it is interesting to revisit the NTOs for MEH-PPV that are depicted in Fig. 7c, where three pairs of NTOs are needed to recover 81% of the transition density.These NTOs show that the excitation delocalizes around the bent portion of the molecule, suggesting that a purely geometric definition of broken conjugation is insufficient to understand exciton localization in this system, and similarly inadequate to define the effective size of the chromophore; other mechanisms such as dipole-dipole coupling and superexchange can drive delocalization even when geometric distortion leads to loss of conjugation. 171With an excited-state wave function in hand,  an effective chromophore size can be inferred by measuring the particle-hole separation for the exciton. 171This is the anti-diagonal coordinate in the X plots of Table 3, for a different PPV system.This analysis technique and other statistical measures of electronhole correlation are further described in the next section.

Exciton wave function
The concept of an ''exciton'' or bound particle/hole pair is ubiquitous in solid-state physics yet it can be difficult to connect that language to the MO-based concepts that are used in quantum chemistry. 20,331Any excited state in a singleexcitation theory consists of an excited electron and a hole in the occupied space.A connection between the two formlisms can be made by identifying virtual-occupied function pairs c a ðr elec Þc Ã i ðr hole Þ as a quasiparticle basis.The transition density kernel T(r,r 0 ) in eqn (2.5), written in the form T(r elec ,r hole ), is then identified as a ''wave function'' for the exciton.That said, the true excited-state wave function in a many-body formalism is C exc in eqn (2.8), which is used to construct T(r,r 0 ).Nevertheless, T(r elec ,r hole ) is often called a ''wave function'' in quasiparticle theories based on the two-particle Green's function and the Bethe-Salpeter equation. 332,333emantics aside, examination of T(r elec ,r hole ) reveals the separation and spatial correlations between particle and hole.In quantum chemistry, this form of analysis was pioneered by Mukamel, Tretiak, and Chernyak, 111,[312][313][314][334][335][336] and later used by others, [337][338][339][340][341] mostly in the context of organic photovoltaic materials and based upon semi-empirical CIS-type wave functions. These ieas were subsequently formalized by Plasser and co-workers, 18,19,22,200,342 who generalized them to wave functions of arbitrary complexity represented using nonorthogonal basis functions.Plasser and co-workers also applied these ideas to organic photovoltaics, 306,[343][344][345] albeit using TD-DFT and correlated wave functions rather than semi-empirical methods.

Electron-hole correlation
If T(r elec ,r hole ) is to serve as the exciton wave function then it might seem that |T(r elec ,r hole )| 2 should be the corresponding probability density, although this analogy breaks down when one realizes that the normalization condition in eqn (4.6) is not generally obeyed for correlated wave functions. 18,19(This fact has occasionally been used to quantify deviation from oneelectron character. 204,346,347) For single-excitation wave functions, eqn (4.6) is strictly valid and one may integrate over either r r elec or r 0 r hole to obtain separate one-particle densities for the electron and the hole. 18For TD-DFT, these quantities are the same as the particle and hole densities defined in Section 2.2.In terms of T(r elec ,r hole ), they are The CT matrix X can be used to demonstrate how various XC functionals afford qualitatively divergent behavior for excitonic states in multichromophore systems. 46TD-DFT excitation energies for charge-separated states are exquisitely sensitive to a functional's fraction of Hartree-Fock exchange (if any), 35 much more so than localized excitations such as pp* or np*. 31Adjusting the fraction of exact exchange has the effect of tuning charge-separated states in or out of resonance with FE states, which will affect whether (and to what extent) the FE states mix with either localized CT states or delocalized CR states.
As an example, Fig. 13 shows heat maps of X for singlet excited states in a (pentacene) 4 cluster, illustrating significant qualitative discrepancies between results obtained with different functionals.Using oB97X-V, 348 the states S 1 to S 4 are mostly FE states, as evident from the anti-diagonal character of the X heat maps; consult Table 2 and eqn (4.19) for a guide.For example, the S 1 and S 4 states primarily involve mixing basis states Optimally-tuned RSH functionals, 38,[43][44][45] including LRC-oPBE and a screened RSH approach (sRSH-oPBE) that affords correct asymptotic behavior within a low-dielectric crystal medium, 349 exhibit a much greater degree of charge separation.This separation resembles localized CT insofar as it is not symmetric about the antidiagonal of X (i.e., O A-B is very different from O B-A ).CAM-B3LYP presents an intermediate case where CT and FE character are both evident.
50,351 This analysis has sometimes been ported to all-electron TD-DFT calculations without regard for the fact that AO overlaps need to be considered. 352,353If those overlaps are ignored, or else if an This journal is © the Owner Societies 2024 orthogonalized minimal basis is employed, then there is little distinction between l A-B in eqn (4.3) and O A-B in eqn (4.9), if normalization is ignored for the purpose of inferring spatial correlations between particle and hole.Tretiak and co-workers use slightly modified CT indices, 312 namely for AaB (5.5)   in place of l A-B .When collected into a matrix n, these quantities measure electron-hole separation and overall exciton size in the same manner as the X matrix.Heat maps of n are depicted in Fig. 14 for a 20-unit PPV oligomer, 336 where indices A and B in x AB refer to PPV units.In these examples, the length PR diag in the anti-diagonal direction (extracule coordinate r elec + r hole ) signifies that the excitation is delocalized over essentially the entire oligomer, regardless of the XC functional that is employed.On the other hand, the coherence length (in the diagonal direction), which indicates charge separation, is rather sensitive to the fraction of Hartree-Fock exchange, as it was for (pentacene) 4 .For functionals with a large fraction of exact exchange, including Hartree-Fock theory itself, L coh approaches a limiting value of E2 monomer units, but for semilocal functionals such as BLYP and PBE the coherence length approaches the length of the entire polymer.]354 5.2 Quantifying exciton size A one-particle probability distribution that preserves certain types of electron-hole information is the electron-hole correlation function, 333 FðrÞ ¼ ð jTðr þ r hole ; r hole Þj 2 dr hole : (5.6) The function F(r) represents the probability of finding the centroids of the electron and the hole separated by a vector r.
The mean electron-hole distance can be sensibly defined as the expectation value of the vector between their barycenters,  R e-h = h8r elec À r hole 8i, ( which is computable by means of F(r): (Here, r = 8r8.)The CT character of the excitation in question can be estimated in terms of the fraction of an electron that is transferred (Q CT ), which can be defined as FðrÞdr: (5.9) The notation r A V molec indicates integration of the volume occupied by a single molecule in a crystal or other aggregate. 333][298][299][300] The SF process amounts to rapid spin-allowed conversion of a singlet excited state on one molecule into a pair of triplet excitations on two neighboring molecules, (5.10) 6][357] Following decoherence, SF ultimately results in two charge carriers (T 1 + T 1 ) for the price of a single photon.This photochemical two-for-one has the potential to overcome the thermodynamic limit on conversion efficiency for one-toone processes. 358,359However, there are basic mechanistic questions that are still being investigated, including the role of low-energy CT states, 46,[291][292][293][294][295] vibronic coherence, 356,[360][361][362][363][364][365][366] the nature of exciton/phonon couplings, 367 and whether the 1 (T 1 T 1 ) state may represent a trap rather than an intermediate. 368 the electron-hole correlation plots in Fig. 15, the origin corresponds to zero net separation between electron and hole (r elec = r hole ) but the plots do not indicate significant probability there.Rather, the regions of highest probability in the xy plane are those around (x = 0, y = AE1 nm), indicative of charge separation between nearest-neighbor molecules, although the extent of F(r) indicates delocalization over as many as three molecules. 292This leads to an exciton length 45 Å, as determined by eqn (5.8), with E50% CT character according to the definition in eqn (5.9). 333In contrast, plots in the xz plane of the crystal indicate no delocalization in the z direction, due to the large intermolecular spacing arising from bulky substituent groups.
The quantity T(r elec ,r hole ) can also be used to evaluate a variety of statistical properties of the joint electron/hole probability distribution. 20,22These measures are indicated schematically in Fig. 16 and include the root-mean-square (RMS) size of the electron and the hole, s elec = (hr elec Ár elec i À hr elec iÁhr elec i) 1/2 (5.11a) and the RMS value of the electron-hole separation, (5.12) The latter provides an alternative to R e-h in eqn (5.8), or L coh in eqn (4.26), as a way to characterize exciton size.All three quantities measure electron-hole separation but they are numerically distinct.These quantities play a central role in attempts to quantify the CT character of a given excited state, which will be explored in Section 6.
To examine these definitions a bit further, we define where hr elec i and hr hole i are the centroids of the attachment and detachment densities, respectively.Equivalently, these are the This journal is © the Owner Societies 2024 expectation values of the position operator, averaged over Dr elec (r) or Dr hole (r).For example, the x component of hr elec i is The quantity d + e-h in eqn (5.13) is another measure of exciton size.Its value depends on the choice of laboratory-fixed coordinate frame, but given a sensible choice for the coordinate origin it can be used to assess how the exciton migrates upon change in molecular geometry.The quantity d À e-h is a measure of the electron-hole separation (see Fig. 16a) but d À e-h = 0 for any centrosymmetric system. 22This means that d À e-h cannot detect charge separation in any system with inversion symmetry, which has important implications in solid-state systems.In a centrosymmetric (or near-symmetric) solid, the value of d À e-h may be zero or small, with a correspondingly small dipole moment change upon excitation, even for an exciton that is characterized by significant charge separation. 46That charge separation can be detected by examining the CT numbers O A-B (using heat maps of the X matrix, for example), but it would be useful to have a quantitative metric that might afford a length scale for charge separation.
In view of these remarks, d exc in eqn (5.12) seems to offer a more robust measure of electron-hole separation, as compared to d À e-h .The former satisfies mathematical bounds given by 22 and (5.15b) The physical interpretation of these bounds is that the RMS exciton size (d exc ) is limited by the center-to-center separation of electron and hole combined with the RMS size of each.
For MEH-PPV polymers (Fig. 7c), examination of d exc and d AE e-h leads to the conclusion that excitations in this system can be viewed as two independent quasiparticles in the intracule and extracule coordinates of the electron hole pair. 171As compared to geometric considerations, the RMS exciton size proves to be a better diagnostic for the effective size of the chromophore in a long, disordered polymer.That length scale (measured by d exc ) is sometimes longer than what might have been anticipated simply by counting conjugated bonds, due to electronic coupling between conjugatively distinct segments of the polymer.For the low-lying excited states of interest for optoelectronic applications, the value of d exc is effectively constant whereas d + e-h is observed to increase with excitation energy. 171ther statistical descriptors of an exciton include the covariance between the vectors r elec and r hole , defined as COV(r hole ,r elec ) = hr hole Ár elec i À hr hole iÁhr elec i. (5.16)This quantity is connected to the RMS exciton size via the relation 22,342 d 2 exc = (d À e-h ) 2 + s 2 exc + s 2 hole À 2COV(r hole ,r elec ).(5.17) The covariance can be used to compute Pearson's correlation coefficient (PCC) between the probability distributions for the electron and the hole, which is This quantity is defined such that À1 r PCC e-h r 1, ( with positive values indicating concerted motion of the two quasiparticles (Fig. 16d) and negative values indicating that they avoid each other dynamically (Fig. 16e). 22nalysis of correlations between the size of the electron and hole quasiparticles, as a function of conjugation length, suggests that the semilocal TD-DFT results for (PPV) 20 in Fig. 14 are consistent with quasiparticles avoiding one another, or in other words, more consistent with a CT state than with a bound exciton. 345Reducing the fraction of Hartree-Fock exchange is tantamount to eliminating electron-hole attraction, leading to an effectively repulsive interaction between the excited electron and the hole. 336As a result, TD-DFT using semilocal functionals contains no electron-hole interaction, and is inherently unable to describe bound excitons.This observation explains large errors for TD-DFT excitation energies in some conjugated p systems. 220,222,369,370In the (PPV) 20 example at least, the failure mode cannot be deduced from the MOs alone because the anti-diagonal length scales (PR diag ) are essentially identical for all functionals.Instead, realspace analysis of the transition density is required, visualizing electron-hole correlation. 18,345Diagnostics for charge transfer Results for models of crystalline tetracene (Fig. 13) and for conjugated polymers (Fig. 14) allude to systemic problems with the description of long-range CT in TD-DFT calculations.These problems are well documented, 1,9,28-36 but for completeness they are briefly recapitulated in Section 6.1.Nevertheless, for localized valence excitations TD-DFT affords a level of accuracy that is difficult to match with any other quantum chemistry method, except in very small molecules.[12][13][14]  The imbalance between TD-DFT's treatment of localized versus charge-separated excitations has spawned a small industry dedicated to providing diagnostic tools to determine which excitation energies may be problematic, as valence and CT excitations can mix in large systems.2]371 ) Historically speaking, the first such metric was introduced by Tozer and co-workers. 23That metric, which remains useful, is described in Section 6.2 followed by a more general discussion in Section 6.3 that introduces several other CT metrics and relates them back to statistical measures of electron-hole separation that were introduced in Section 5.2.These connections have seldom been made clear in the literature, and even more infrequently have they been discussed in terms of physically meaningful properties of the exciton.

CT problem in TD-DFT
Maitra 33 provides a fundamental discussion of how CT problems in TD-DFT arise from approximate XC functionals.A succinct and non-technical summary is that for any semilocal XC functional, including any generalized gradient approximation (GGA) that lacks Hartree-Fock exchange and even meta-GGAs that lack Hartree-Fock exchange, the interaction between the excited electron and the hole vanishes beyond the length scale at which electron and hole wave functions cease to overlap.This was mentioned above, in the context of understanding unbound excitons in conjugated polymers (Section 5.2), but it can be formulated and understood in a general way.
For well-separated donor and acceptor orbitals c i and c a , respectively, the TD-DFT orbital Hessian matrix elements reduce to A ia,jb E (e a À e i )d ij d ab À C HFX (ij|ab) (6.1a) where C HFX is the coefficient of Hartree-Fock exchange. 1For a semilocal functional (meaning C HFX = 0), eqn (6.1a) reduces to a block-diagonal form with diagonal matrix elements that are simply differences in Kohn-Sham energy levels, e a À e i .For a sufficiently large system, there is nothing to prevent spatially separated orbitals c i and c a from having an energy gap l = hc/ (e a À e i ) that happens to coincide with a visible or ultraviolet wavelength.These will manifest in semilocal TD-DFT as spurious excited states, 1,[28][29][30][31][32] which appear to move charge around at relatively low energies. 30These spurious CT excitation energies are much lower than one would estimate using a pointcharge formula for long-range CT, 9 which is Eqn (6.2) expresses the excitation energy o CT between wellseparated donor and acceptor moieties in terms of the ionization energy (IE) of the donor and the electron affinity (EA) of the acceptor, along with a Coulomb interaction of À1/(4pe 0 R) for an ion pair.What is missing in semilocal TD-DFT is the Coulomb penalty for separating charge, which is provided in hybrid TD-DFT by the Hartree-Fock exchange integral (ij|ab) in eqn (6.1a).
If the spurious CT states are relatively sparse in the excitation manifold then they will be optically dark, consistent with a vanishing transition moment between non-overlapping donor and acceptor orbitals (hc i |m x |c a i E 0).A solvated system, however, will engender a dense manifold of spurious CT states, some of whose energies will be (accidentally) near-resonant with genuine dipole-allowed transitions.This leads to intensity borrowing by the spurious CT states, with concomitant loss of intensity by the genuine bright state(s). 30Just as certain higherlying valence excitations predicted by MO theory may be absent from the spectrum, having been ''dissolved in the Rydberg sea'', 372,373 one may state that valence transitions in largescale TD-DFT can dissolve into a charge-transfer sea.Hybrid functionals with relatively small fractions of Hartree-Fock exchange, including B3LYP with C HFX = 0.20 and PBE0 with C HFX = 0.25, can still be susceptible to this problem, albeit less so than semilocal functionals. 30,31o diagnose and quantify anomalous CT in TD-DFT calculations, one may employ properties of the TDM to measure exciton size, delocalization, and charge-separated character, as discussed in Section 5. A schematic example is shown in Fig. 17 using ladder-type poly(p-phenylene) polymers. 338,374Here, semilocal TD-DFT calculations predict an exciton that is delocalized across the entire polymer, regardless of oligomer length, which is the same problem that was documented for (PPV) 20 in Fig. 14.Hybrid functionals significantly attenuate the charge separation but not the FE delocalization. 336 ''Optimally-tuned'' LRC functionals, wherein the range-separation parameter is adjusted to satisfy the IE theorem (IE = Àe HOMO ), are often used as a workaround for TD-DFT's CT problem.[37][38][39][43][44][45] The DMABN molecule provides an interesting case study.Fig. 18 characterizes the nature of its S 0 -S 1 and S 0 -S 2 transitions, accessing the 1 L a and 1 L b states that were introduced in Section 4.3. Accrding to the lore, one of these should be the LE(pp*) state and the other should exhibit nascent CT character that is enhanced upon twisting.At the planar ground-state geometry (on the left in Fig. 18), both transitions exhibit significant delocalization across the donor-p-acceptor framework, although the attachment density (representing the excited electron) is slightly enhanced on the cyano group in S 2 as compared to S 1 .Upon 901 twist of the amino group (on the right in Fig. 18), and in a polar dielectric medium, detachment densities for both transitions localize onto the amino lone pair.For the twisted geometry, S 1 is clearly the CT state and it is significantly stabilized by solvent polarization.In contrast, the excitation energy for the LE state is scarcely affected by the twist.
A point of historical debate was the fact that the PBE and B3LYP functionals both predict reasonably accurate excitation energies for the 1 L a and 1 L b states, for DMABN and other small donor-p-acceptor molecules. 23,375,376A resolution to this apparent paradox comes in the form of a metric for quantifying CT character, which will be introduced below and ultimately suggests that the extent of CT in the planar geometry of DMABN is not very large. 23Only in hindsight can this be inferred from the densities in Fig. 18, which do not suggest any dramatic difference between S 1 and S 2 at the ground-state geometry.(The difference is much more pronounced in the twisted geometry.)As such, DMABN serves as a cautionary tale warning that one must be careful with blanket statements that TD-DFT fails categorically for CT states, or at least one must be careful about what gets called a CT excitation.Energies for truly long-range CT will indeed be systematically (and catastrophically) underestimated by semilocal TD-DFT, but errors may be small if the donor and acceptor orbitals are not completely separated in space.In the planar geometry of ground-state DMABN, these orbitals are clearly not well-separated in space.A metric that might indicate this fact is introduced next.

Tozer's spatial proximity metric
A resolution to the DMABN paradox was provided by the very first CT metric to be introduced for TD-DFT calculations, by Tozer and co-workers. 23Their proposed metric is defined as where k ia = x ia + y ia (6.4)   and Note the absolute value signs in the integrand of O ia , which are necessary because occupied and virtual MOs are orthogonal, hc i |c a i = 0.For this reason, we resist using the term ''overlap'' to describe the spatial proximity of MOs.If we need a name for O ia , we will call it the ''spatial overlap'' of c i and c a .This and similar metrics are sometimes used to quantify the spatial proximity of HOMO and LUMO in donor-acceptor materials. 377n view of the normalization condition for x and y [eqn (2.4)], it is unclear why the definition of L does not involve both x + y and x À y.Perhaps it is in loose analogy to the expressions for the particle and hole density matrices [eqn (2.11)], which contain terms like (x + y) † (x + y) and (x + y)(x + y) † , although these expressions also contain (x À y) † (x À y) and (x À y)(x À y) † .(The latter terms have sometimes been erroneously omitted. 220) Whatever the reason, the definition of L in eqn (6.3) is used consistently in practice, 23,25,220,[378][379][380] yet the decision to abdicate proper normalization seems questionable and has implications for other CT metrics that are discussed in Section 6.3.Within the TDA there is no issue, since y = 0 and P ia x 2 ia ¼ 1, thus the denominator in eqn (6.3) has well-defined normalization.The implies that 0 r L r 1 within the TDA, but this need not be the case for full TD-DFT calculations that include the y ia amplitudes.
Fig. 18 Particle (attachment) and hole (detachment) densities for the S 0 -S 1 and S 0 -S 2 transitions in DMABN, in both its planar ground-state geometry (on the left) and with a 901 twist of the dimethylamino moiety (on the right).Opaque and wire mesh surfaces encapsulate 50% and 90% of each density, respectively.TD-DFT/TDA calculations were performed using LRC-oPBE/6-31G* with a dielectric constant of 37.5 (representing acetonitrile).
Tozer et al. find that 0.45 r L r 0.89 for localized valence excitations, whereas Rydberg excitations lie in the range 0.08 r L r 0.27. 23Examining excitation energy errors as a function of L, it becomes clear that there are approximate, functionaldependent thresholds below which TD-DFT results should not be trusted.Errors are large, for example, when L o 0.4 for B3LYP or when L o 0.3 for PBE, 23 and values of L correlate with excitation energy errors along flexible torsional coordinates that can lead to intramolecular CT in some conformations. 381esolution of the DMABN paradox comes in noting that its intramolecular CT excitation corresponds to L = 0.72 (TD-PBE) in the planar geometry, 23 which is not very CT-like.For the LRC-oPBE/6-31G* calculations that are shown in Fig. 18, the corresponding values are L(S 1 ) = 0.53 and L(S 2 ) = 0.67 in the planar geometry, indicating that the nominal CT state actually has somewhat larger spatial proximity between particle and hole.This is true in the twisted geometry as well, although values of L are much smaller and lie in the ''danger zone'': L(S 1 ) = 0.20 and L(S 2 ) = 0.22.Results at the TD-B3LYP/6-31G* level are similar.
While the L metric has proven successful as a diagnostic for TD-DFT errors, its numerical value does not provide much physical insight.Moreover, it may fail to detect problems when the excited state involves excitation from a relatively compact orbital into a much more delocalized orbital, 378 as the two MOs may share significant spatial proximity (in the sense of O ia ) yet the delocalized nature of the final state might still engender an erroneously low excitation energy.22]382 These can be rectified through the use of asymptotically-correct LRC functionals, 220,382 yet such states do not exhibit what might be understood as CT in intuitive chemical terms, involving donor and acceptor functional groups.Moreover, values of L do not portend any problems in such cases. 220Perhaps for these reasons, there has been significant effort devoted to identifying alternative CT metrics for use in TD-DFT.This is discussed in the next section.

Other CT metrics
Much of the work on alternative CT metrics for TD-DFT originates with Ciofini and co-workers, 24,229,239,354,[383][384][385][386][387][388] who introduce particle and hole densities but do not refer to them as such.Instead, these quantities are called r + (r) and r À (r) and are defined by regions of space where the excitation engenders either positive or negative change in the density, respectively: 354,385 Dr þ ðrÞ ¼ Dr elec (r) and Dr À (r) as Dr hole (r).To see this, recall that the attachment and detachment densities were defined by eigenvectors of DP corresponding to positive eigenvalues (DP attach ) or negative eigenvalues (DP detach ); see eqn (2.17).Since Dr elec (r) and Dr hole (r) have names that invoke both their physical meaning and their connection to the particle/hole formalism, we will use Dr elec (r) and Dr hole (r) in place of Dr + (r) and Dr À (r).Ciofini et al. 354 introduced what is now a widely-used measure of charge separation, which they call D CT and which is equal to the distance between the centroids of Dr elec (r) and Dr hole (r).However, D CT is simply d À e-h as defined in eqn (5.13).In more detail, where 8Á Á Á8 indicates the length of the vector that is defined by three different integrals, substituting x, y, or z for r in eqn (6.7) to define components of the vector, as in the definition of hx elec i in eqn (5.14).The metric D CT d À e-h is increasingly being used to analyze TD-DFT calculations, 24,229,239,354,[383][384][385][386][387][388][389] although most authors refer to it as D CT , ''Ciofini's CT metric'', or similar language that obscures its straightforward physical interpretation as the distance between barycenters of the particle and the hole. 20Although the physical interpretation has been noted elsewhere, 354,388 failure to introduce particle and hole densities per se obscures the conceptual origin of D CT and its connection to quantities such as the attachment and detachment densities.Calling this quantity d À e-h rather than D CT makes the physical meaning inherent in the nomenclature; the definition in eqn (5.13) is obvious and meaningful.More complicated generalizations of D CT have been suggested, 387 though it is not clear what advantages these may have as compared to straightforward moment analysis of the excitonic wave function, a `la eqn (5.2) and (5.12).As noted in Section 5.2, d À e-h 0 for any centrosymmetric system. 22To obtain a non-vanishing CT metric for systems with inversion symmetry, Ciofini et al. introduce alternative diagnostics that they call the ''t index'' 354,383 and the ''H index''. 229,239,354,383The latter quantity (H) is essentially (s elec + s hole )/2 but restricted to a one-dimensional donor-acceptor coordinate.It provides a measure of the exciton's spread, and the t-index is then defined as t = D CT À H.We suggest replacing t with an alternative measure of essentially the same information, the charge-displacement distance, 1 which we define as The quantity d CD represents the center-to-center distance of electron and hole, reduced by the average of the RMS size of either quasiparticle.It provides a physically meaningful way to combine electron-hole separation (d À e-h ) with something that measures the extent of the excitonic wave function, providing an intuitive way to convey the same information as the D CT and t indices.
A novel analysis tool introduced by Ciofini et al. 354 is the idea of Gaussian approximations to Dr elec (r) and Dr hole (r) that This journal is © the Owner Societies 2024 are based on the rigorous second moments of the transition density, i.e., the quantities s 2 elec and s 2 hole that are defined in eqn (5.11).These approximations provide a quantitative way to realize the cartoons in Fig. 16, which may be easier to conceptualize than the transition density itself because nodal structure is removed.Examples are depicted in Fig. 19 for a sequence of poly(p-phenyl)nitroaniline molecules.Due to the complex nodal structure along the conjugated backbone of these molecules (Fig. 19a), barycenters of Dr elec (r) and Dr hole (r) are more clearly evident in their Gaussian approximations (Fig. 19b).Whereas the particle and hole densities extend to the very edges of the molecule, the charge separation distance d À e-h is noticeably shorter and is indicated by the arrows in Fig. 19.This is even more clear in the examples of Fig. 20, where plots of the particle and hole densities appear to be considerably more delocalized than the quantitative measure afforded by d À e-h .The extent of spatial charge separation is therefore smaller than plots of Dr elec (r) and Dr hole (r) might lead one to imagine.Notably, the isocontour value that is used in this type of isosurface plot can be manipulated to make an orbital or density appear almost arbitrarily compact or diffuse.For that reason, the author recommends that such plots should always indicate the fraction of the density that is encapsulated within the isosurface.Only then can the size of two densities be compared side-by-side in a meaningful way. 161ong-distance charge separation is characterized by negligible overlap between particle and hole densities, meaning that the product Dr elec (r) Dr hole (r) E 0 everywhere in space.Based on that observtion, Etienne et al. 16,17,390 suggest a chargeoverlap metric f ¼ ð jDr elec ðrÞDr hole ðrÞj 1=2 dr: (6.9) We omit a normalizing denominator that is included in ref. 16, as it equals unity for TD-DFT calculations where both Dr elec (r) and |Dr hole (r)| integrate to exactly one electron.Roughly speaking, the integral in eqn (6.9) involves the blue region that is depicted schematically in Fig. 21a.The metric f is defined such that 0 r f r 1.If f = 0 then there is no spatial overlap between electron and hole, thus the excitation in question is entirely CT-like.An example that lies close to this limit is an end-to-end donor-acceptor electrontransfer excitation in a twisted push-pull chromophore whose particle and hole densities are plotted on the right side of Fig. 21b.The twisted geometry severs the conjugation of the p system, resulting in particle and hole densities that localize on opposite ends of the molecule and a small value of the chargeoverlap metric, f E 0.2.On the left in Fig. 21b is an excited state in a different molecule that lies near the opposite limit, where electron and hole are delocalized across the entire molecule and f E 0.8.
Other CT metrics have been proposed in the spirit of L in eqn (6.3) but attempting to find a diagnostic whose numerical value might be physically meaningful.One of these is a chargeseparation metric Dr, 25,26   R ia = hc i |r ˆ|c i i À hc a |r ˆ|c a i. (6.11) The quantity R ia is the displacement vector between the centroids of orbitals c i (r) and c a (r), thus Dr averages the charge displacement associated with each excitation c ic a , using weights k 2 ia .Although this seems like an intuitive and reasonable way to measure charge separation, the utility of Dr as a separate metric is questionable.Within the TDA, this quantity is which likely encodes similar information as compared to d À e-h .For full TD-DFT, Dr in eqn (6.10) employs the same curious choice of normalization that was used to define L, namely, use of k 2 jb in the denominator.Perhaps more damningly, Dr is not invariant to orbital rotations; its numerical value depends upon which MOs are used. 26It has been suggested to evaluate Dr in the NTO basis, as this affords good correlation between Dr and D CT , 26 but the need to make such a choice is a bothersome artifact of having sacrificed orbital invariance.For an alternative data set of intramolecular CT energies, 391 reasonable correlations are found between definitions of Dr based on either canonical MOs or NTOs, but those values also correlate well with d exc as shown in Fig. 22. 205 The quantities d exc and d À e-h , along with expectation values such as hr elec i and hr hole i, are invariant to unitary transformations of either the occupied MOs or the virtual MOs, just like excitation energies and other excited-state properties in TD-DFT.It is this invariance that provides the freedom to define NTOs, or to use localized MOs, [392][393][394][395][396] without affecting observables, because the aforementioned quantities are fundamental properties of the exciton, independent of representation.As such, these are less arbitrary ways to characterize the nature of a given excited state, as compared to a quantity such as Dr that depends upon the choice of representation.

Both d À
e-h and Dr vanish in centrosymmetric systems, 397 which is a significant limitation that is not shared by d CD .Alternatively, to obtain a non-vanishing metric in the presence of inversion symmetry, an effective ''electron displacement'' measure has been suggested, 397  The quantity s 2 r is the second moment of orbital c r .In a sense, Ds is conceptually similar to d exc in eqn (5.12), in the same way that Dr is comparable to d À e-h , with the important distinction that both Dr and Ds mangle the normalization when y is nonzero, and that the numerical values of both Dr and Ds depend upon the choice of representation.
Correcting the normalization by invoking the TDA, the electron displacement G = Dr + Ds is likely to contain similar information as d À e-h + d exc .As such, we suggest that d ˜CD = d À e-h + d exc (6.16)   is an alternative charge-displacement metric, constructed from well-defined properties of the exciton and independent of the choice of representation.It should be complementary (though not equivalent) to d CD in eqn (6.8).This analysis clarifies why numerical values of various charge-displacement metrics are found to be strongly correlated with one another. 26,205,247espite their shortcomings, the metrics Dr and G correlate well enough with the largest errors in TD-DFT excitation energies so that one may define a ''trust radius'' based on their values. 25,397For GGA functionals, and with G evaluated in the NTO representation, it is suggested that states with G r 1.8 Å  This journal is © the Owner Societies 2024 are ''safe'', in the sense that the excitation energy in question is is unlikely to be seriously affected by TD-DFT's underestimation of long-range CT energies. 397A trust radius G r 2.4 Å is suggested for global hybrid functionals with C HFX = 0.2-0.3,again with G evaluated using NTOs. 397For long-range excitations well beyond 2.0 Å, it is suggested that the use of either LRC functionals, or else global hybrids with C HFX Z 0.33, is mandatory. 25As an alternative, the close connection between G and d ˜CD suggests that the latter might also provide a reliability metric for TD-DFT excitation energies, while at the same time affording a physically interpretable (and representationinvariant) numerical value to quantify how charge moves (d À e-h ) and spreads (d exc ) upon excitation.The precise trust radius to use in conjunction with d ˜CD remains to be determined.
Lastly, a ''Mulliken-averaged configuration index'' (MAC) has been suggested for detecting spurious low-energy CT states. 388,398 , where o is the TD-DFT excitation energy, then the excited state in question is likely a ''ghost'' (spurious) CT state and should not be taken at face value. 398However, this metric should only be used for large values of d À e-h , because Mulliken's formula only makes sense for large donor-acceptor separation, whereas spurious CT states can appear at van der Waals contact distances. 30Given the crudeness of the approximation in eqn (6.17), it is also unclear how robust this metric will be.Proper statistical measures of electron-hole correlation seem preferable as means to define boundaries for trustworthiness in TD-DFT calculations.

Summary
TD-DFT is the workhorse method of computational electronic spectroscopy and is widely used by both computational and experimental chemists and materials scientists.Visualizing TD-DFT excitations in terms of NTOs, as a conceptually superior alternative to canonical MOs, has become standard practice but other visualization tools are also available and the connections amongst them are not always obvious to beginning users.The present work provides a theoretical foundation to understand how the NTOs relate to other common visualization tools including attachment and detachment densities, which are densities for the excited electron and the hole, respectively.Atomic or fragmentbased partitions of Dr(r), and quantitative measures of exciton size and electron-hole separation, have been introduced rigorously and demonstrated in numerous examples.Emphasis has been placed on understanding how various tools relate to one another, as previous literature has not always been clear in this regard.
CT numbers O A-B , which quantify electron flow from moiety A to B upon excitation, are a particularly useful atomic or functionalgroup partition.Arranged in the form of a matrix X, these quantities provide a simple visual representation of the transition density kernel T(r hole ,r elec ) that has sometimes been described as an ''exciton wave function''.Heat maps of the matrix X provide an easy way to distinguish localized versus delocalized excited states, or the presence of charge-separated character, even in centrosymmetric systems where symmetry prevents CT from manifesting as a change in dipole moment.Fragment-based analysis of T(r hole ,r elec ) can distinguish between delocalization caused by excitonic coupling, versus delocalization due to charge separation, possibilities that are not mutually exclusive but also not equivalent.In multichromophore systems, this analysis exposes qualitative differences in the low-energy states obtained using different XC functionals.
A variety of metrics have been discussed that are intended to quantify CT character in a given excitation, an important descriptor in view of TD-DFT's well-known tendency to underestimate long-range CT excitation energies, sometimes to the point of producing spurious low-lying states in large systems. 1 Some of these CT metrics have more desirable properties than others, such as correct normalization and invariance to unitary transformations of the MOs.The present work advocates for the use of direct measures of exciton size that correspond to well-defined expectation values, rather than ad hoc constructions.The former include the RMS electron-hole separation (d exc ), which is expressed in terms of the particle and hole densities Dr elec (r) and Dr hole (r).The mean separation between the centroids of those quantities (d À e-h ) can also be used, although it vanishes in centrosymmetric systems.In such cases, a charge-displacement metric (d CD or d ˜CD ) can be used instead.These quantities are directly interpretable and readily computable using third-party software, 162,209,399,400 based on formatted output from various electronic structure programs.The TheoDORE program is especially recommended, 209 as it implements measures of exciton size that are grounded in proper expectation values, as well as CT numbers O A-B that account for non-orthogonality of the AO basis functions.Much of this functionality exists in the Q-Chem program also, 401 without the need for third-party software.The author hopes that this Perspective will lead to better understanding and more erudite discussion of precisely what is being visualized or quantified when discussing the output of TD-DFT calculations.).This is consistent with the signs in eqn (2.12) that defines DP elec and DP hole , and with remarks made concerning the attachment and detachment densities in eqn (2.17).
In addition, eqn (2.13) for the normalization of the electron and hole densities holds only when the Tamm-Dancoff approximation (TDA) is invoked.A more general statement is that tr(DP elec ) = Àtr(DP hole ) always, consistent with the eigenvalues given above, whereas tr DP elec À Á TDA ¼¼ 1 holds only within the TDA.Note that tr DP elec À Á ¼ X ia x ia j j 2 þ y ia j j 2 according to eqn (2.12), whereas the normalization condition for linear response TD-DFT is P ia x ia j j 2 þ y ia j j 2 ¼ 1 [eqn (2.4)].
Therefore, tr(DP elec ) can exceed unity for full linear response TDDFT.Typically ||y|| B 10 À3 for small molecules, so the deviation from the TDA result is not large.The funding information in the Acknowledgments section of the original article was incomplete, the full funding information is shown here.
The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.

Fig. 1
Fig.1Bar graph of configuration mixing coefficients x ia for the 1 1 A 2 state of formaldehyde: (a) TD-PBE/TDA calculations using active spaces containing n virt virtual orbitals, as implemented in a real-space electronic structure code; (b) CIS calculations using the same active spaces; and (c) conventional CIS calculations in Gaussian basis sets.Calculated excitation energies provide a measure of convergence with respect to active space or basis set.Adapted from ref.103.
12b) These quantities are normalized such that tr(DP elec ) = 1 = Àtr(DP hole ).(2.13)Although we have not been explicit about spin indices, the spinorbital indices (i, a, etc.) could be limited to either a or b spin.By doing so, one could obtain a spin density matrix for either the particle (DP elec a À DP elec b ) or the hole (DP hole a À DP hole b ), whose real-space representation would reveal spin polarization for an open-shell system.

Fig. 3
Fig. 3 Transition density for the fluorenone-terminated polyfluorene oligomer that is also depicted in Fig. 2, viewed here in two different representations: (a) the canonical MO representation, with weights x 2 ia given as percentages, and (b) the NTO representation, with a single weight l 2 1 that is also given as a percentage.TD-DFT/TDA calculations were performed at the CAM-B3LYP/3-21G* level.Adapted from ref. 1; copyright 2023 Elsevier.

Fig. 9
Fig. 9 Analysis of intramolecular charge rearrangement for excitation to the S 1 and S 2 states of DMABN, using color-coded fragments representing the cyano, phenyl, and amino moieties.CT numbers l ˜A-B for selected atoms (given here as percentages) are shown in black, at the arrows, and gross excitation localization indices GLI A (also as percentages) are shown at the brackets.Calculations were performed at the TD-B3LYP/aug-cc-pVDZ level.Adapted from ref. 72; copyright 2010 John Wiley and Sons.

Fig. 10
Fig.10Principal NTO pairs for the lowest dipole-allowed FE state of the benzene dimer, in a cofacial D 6h arrangement.Calculations were performed at the TD-DFT/TDA level using CAM-B3LYP/6-31G* and these isosurfaces contain 80% of the orbital densities.

Fig. 11
Fig. 11 Different representations of FE (or excitonic resonance) excited states versus CT excited states, in a symmetric dimer whose ground-state wave function is denoted |C 1 C 2 i.Adapted from ref. 200; copyright 2012 American Chemical Society.

Fig. 12
Fig. 12 Guide to interpreting T(r hole ,r elec ) as a two-dimensional probability distribution, or X as a two-dimensional matrix, for a hypothetical excitation in a conjugated polymer.(a) Schematic illustrations of the coherence length L coh [eqn (4.26)], which measures electron-hole separation in the intracule coordinate r elec À r hole , along with the diagonal length PR diag that measures overall exciton size via the extracule coordinate r elec + r hole .The entire exciton should be construed as a superposition of electronhole pairs, each of which has a characteristic separation L coh , whereas the superposition extends over approximately PR diag distinct sites.(b) Schematic illustration of a two-dimensional probability distribution |T(r hole ,r elec )| 2 .Heat maps of X can be interpreted as two-dimensional plots with the same axes (r hole ,r elec ).In that case, distance is measured in units of atoms or functional groups (sites A in O A-B ), depending on how the molecule is partitioned.The overall size of the exciton is limited by the size of the molecule as an upper bound, and L coh is then limited by PR diag .This figure is based on a similar one in ref. 312.

Fig. 13
Fig. 13 Heat maps of the CT matrix X for the first four singlet excited states of the (pentacene) 4 model that is shown at the top.Matrix elements O A-B are obtained from a monomer-based partition of |T(r hole ,r elec )| 2 and results from four different XC functionals are shown.Darker blue color indicates larger values of O A-B whereas white indicates that O A-B E 0. Adapted from ref. 46; copyright 2020 American Chemical Society.

Fig. 14
Fig. 14 Heat maps of n for (PPV) 20 obtained from TD-DFT with various functionals.The n matrix is defined in eqn (5.4) and (5.5) and its interpretation is the same as that of X, representing |T(r hole ,r elec )| 2 but with the axes measured in units of PPV monomers.Equivalently, the heat map of n represents the probability of transferring charge from the site indicated on the horizontal axis to the site indicated on the vertical axis.Adapted from ref. 336; copyright 2007 American Institute of Physics.

Fig. 15
Fig. 15 Electron-hole correlation functions F(r) [eqn (5.6)] for the first four singlet excited states of a periodic crystal of 6,13-bis(triisopropylsilylethynyl) (TIPS) pentacene.These have been projected onto either the xy plane (upper panels) or the xz plane (lower panels).Corresponding cuts through the crystal structure are shown at the far right, with methyl groups removed from the TIPS side chains for clarity.Reprinted from ref. 292; copyright 2015 John Wiley & Sons.

Fig. 16
Fig. 16 Schematic depictions of statistical measures of electron-hole correlation, including (a) the average electron-hole separation, d À e-h ; (b) the RMS electron-hole separation, d exc ; (c) the RMS size of the electron, s elec ; and (d) and (e) Pearson's correlation coefficient for electron and hole, PCC e-h .Adapted from ref. 342; copyright 2018 American Chemical Society.

Fig. 17
Fig. 17 Cartoon depiction of exciton size versus conjugation length for ladder-type poly(p-phenylene).The quantity d exc [eqn (5.12)] is the RMS exciton size, which increases without bound when GGA functionals are used in TD-DFT.Reprinted from ref. 345; copyright 2017 American Chemical Society.
6b)There is no new information here, relative to what was discussed in Sections 2.2 and 2.3, because one may identify Dr + (r) as

Fig. 19 (
Fig.19(a) Electron and hole densities (in green and red, respectively) and (b) Gaussian approximations to these quantities, for a sequence of poly(pphenyl)nitroanilines, O 2 N-(C 6 H 4 ) n -NH 2 .Calculations were performed at the TD-PBE0/6-31+G* level using a solvent model.354Purple arrows connect centroids of the electron and hole densities.Adapted from ref.354; copyright 2011 American Chemical Society.

Fig. 20
Fig. 20 Particle densities Dr elec (r) (in red) and hole densities Dr hole (r) (in blue), for various push-pull chromophores that are indicated in the lower part of the figure.Each chromophore has the structure (CH 3 ) 2 N-p-NO 2 , where ''p'' indicates a large conjugated system.Examples include: (a) several oligomers of a,o-dimethylaminonitro-(p-phenylene vinylene) n ; (b) a tertiary amine of the form N(PhOCH 3 ) 2 (PhR), where Ph = phenyl and R is a pentathiophene side chain with a terminal nitro group; and finally, a,o-dimethylaminonitro-(p-thiophene) 5 with the central thiophene unit replaced by either (c) benzodifuranone or else (d) benzotriazole.Green arrows indicate the charge-separation distance, d À e-h .These arrows have been displaced away from the molecules for clarity but their endpoints coincide with the centroids of the particle and hole densities.Adapted from ref. 239; copyright 2012 American Chemical Society.

Fig. 21 (
Fig. 21 (a) Schematic view of the charge-separation metric f defined in eqn (6.9), using cartoon representations of the particle and hole densities made to resemble those in Fig. 19b.Roughly speaking, the integrand in eqn (6.9) is non-zero in the blue region of overlap between Dr elec (r) and Dr hole (r).(b) Examples of a localized excitation in the polyene O 2 N(CH) 10 NO 2 (on the left) and a CT excitation in the push-pull polymer O 2 N-(C 6 H 4 ) 5 -N(CH 3 ) 2 (on the right), with values of f indicated.Calculations in (b) were performed at the TD-PBE0/6-311++G(2d,p) level and the plots are adapted from ref.17.

Fig. 22
Fig. 22 Correlation between CT excitation energies (o CT ) and various CT metrics.The HCl molecule defies the overall trend and is indicated explicitly.Adapted from ref. 205.

2 )
This journal is © the Owner Societies 2024 Phys.Chem.Chem.Phys., 2024, 26, 3755-3794 | 3781 performed at the Ohio Supercomputer Center 402 using the Q-Chem program. 401My recent review 1 on visualization and characterization methods in time-dependent density functional theory (TD-DFT) contains a sign error in the definition of the natural transition orbitals.This affects eqn (3.1) and (3.2) and the accompanying text in the same paragraph.The corrected equations are These eigenvalues differ in sign from what was (erroneously) published in ref. 1.The text accompanying these equations should suggest that DP elec is positive semidefinite (eigenvalues l 2 i Z 0) whereas DP hole is negative semidefinite (eigenvalues Àl 2 i

Table 1 Excitation energies and excited-state dipole moments compared to experiment a
a Data are from ref. 113 except where indicated.b Minus signs indicate that the dipole moment changes direction upon excitation.c Ref. 117.d Ref. 118. e Ref. 119.f Ref. 115.
SP) mn into contributions m A A and n A B.A matrix DM, representing changes in bond orders, can be obtained by swapping SDP for SP in eqn (4.7): 27i et al.27provide examples illustrating the use of CT numbers O A-B in complicated cases of photochemical reactions involving transition metal complexes.Here, we consider a relatively simple example, 4-(dimethylamino)benzonitrile (DMABN),

Table 2
Descriptors for the excimer states of a symmetric dimer a
If we take |T(r elec ,r hole )| 2 seriously as a probability distribution for the exciton, it should afford the correlated probability of finding the hole at position r hole , given the presence of the excited electron at position r elec .A schematic view is provided in Fig.12b.According to eqn (5.3), this plot conveys the same qualitative information, in the same way, as does a heat-map plot of X, as in the examples of Table3, but it does so in real space whereas O A-B does so in atom or functional-group space.The coherence length L coh [eqn (4.26)] is a characteristic length scale for charge separation (intracule coordinate r elec À r hole ), whereas PR diag [eqn (4.27)] measures the total size of the excitation (extracule coordinate r elec + r hole ).
r hole Þj 2 dr hole (5.1a)Dr hole ðr hole Þ ¼ ð jTðr elec ; r hole Þj 2 dr defined asG = Dr + Ds.(6.13)This combines Dr from eqn (6.10) with A Koopmans-style approximation for long-range electron transfer from c i to c a , IE + EA E À(e i + e a ), (6.17) in conjunction with Mulliken's asymptotic formula for o CT [eqn (6.2)], suggests a definition This is a slightly modified version of the metric called M AC in ref. 398, replacing P jb x jb in the denominator with P jb x 2 jb , and substituting d À e-h for D CT .)The idea is that if o o o MAC ia x ia ðe i þ e a Þ P