Constricted Variational Density Functional Theory Approach to the Description of Excited States Constricted Variational Density Functional Theory Approach to the Description of Excited States

The aim of this chapter is to present constricted variational density functional theory (CV- DFT), a DFT-based method for calculating excited-state energies. This method involves constructing from the ground-state orbitals, a new set of ‘ occupied ’ excited-state orbitals. Consequently, a constraint is appliedtoensurethat exactlyoneelectron isfullytransferred from the occupied to the virtual space. This constraint also prevents a collapse to a lower state. With this set of orbitals, one obtains an electron density for the excited-state and therewith the CV-DFT excitation energy. This excitation energy can now be variationally optimized. With our successful applications to systems differing in the type of excitation, namely, charge-transfer, charge-transfer in disguise, and Rydberg excitations, as well as in size, we demonstrate the strengths of the CV-DFT method. Therewith, CV-DFT provides a valid alternative to calculate excited-state properties, especially in cases where TD-DFT has difficulties. Finally, our studies have shown that the difficulties arising in the TD-DFTexcitedstates are notnecessarilystemmingfromthefunctionalused,but fromthe application of these standard functionals in combination with the linear response theory.

The strengths and weaknesses of TD-DFT are well known and understood through extensive benchmark studies carried out over the years. The strengths explain its wide usage by delivering 'an excellent compromise between computational efficiency and accuracy' [70]. The weaknesses explain the ongoing fundamental studies searching for solutions in the cases where TD-DFT is found lacking. These include its deficiency in describing Rydberg transitions [71][72][73][74], chargetransfer (CT) transitions [75][76][77][78][79][80][81][82][83][84] and electronic transition with significant double contribution [42,43,[83][84][85][86][87]. TD-DFT is a formally exact theory; however, its practical application relies on the adiabatic formalism where use is made of the available ground-state exchange-correlation (XC) functionals [71,82,[88][89][90]. As a result, one can necessarily trace all the problems encountered in the application of TD-DFT to this approximation. The numerous research attempts to remedy the pitfalls in TD-DFT are classified as follows: In this chapter, we will explain the idea and theory of CV-DFT, before we have a look at different examples, where CV-DFT has been applied for transitions of Rydberg and chargetransfer type following the publications [99][100][101][102][103].

Theory
In this chapter, we review the theoretical framework of CV-DFT in a nutshell. We refer to the original publications [58][59][60][61][62]104] for a more in-depth description.

The CV-DFT scheme
Here, we only consider the excitation from the closed-shell ground state described with single where n occ is the number of occupied orbitals. CV-DFT starts from the ansatz which describes the excitation as an admixture of occupied ϕ i ; i ¼ 1; ⋯; n occ È É and virtual ϕ a ; a ¼ 1; ⋯; n vir È É ground-state orbitals [105]: where ϕ 0 i is the excited-state orbital and n vir the number of virtual orbitals. The transition matrix, U, only mixes between occupied and virtual orbitals (U ij ¼ U ab ¼ 0) and is skew symmetric (U ai ¼ ÀU ia ). In CV-DFT, we use the exponential expansion of U which leads to the unitary transformation Y: Thus, once the transition matrix, U, is determined, a new set of orbitals is obtained over the unitary transformation Due to the properties of the transition matrix, U, the 'occupied' excited-state orbitals can be written as The corresponding excited-state density becomes With these orbitals, some higher order contributions in U can arise in the density and therewith also in the energy, but we only keep up to the second order in U, as the contribution of higher order terms is negligible [60]. The second-order CV-DFT energy expression becomes where the two-electron integral is composed of a Coulomb and an exchange-correlation part: with The exchange-correlation integral is further decomposed into the local (KS) and nonlocal (HF): and where f r 1 ð Þ represents the regular energy kernel. We have shown that CV 2 ð Þ-DFT is equivalent to TD-DFT [59,106] within the Tamm-Dancoff approximation (TDA) [107].
In the infinite-order theory (CV(∞)-DFT), the new set of excited-state orbitals is obtained taking the sum in Eq. (3) to infinite order. These excited-state orbitals can be written in the convenient form of natural transition orbitals (NTO) [108]. For this, we decompose the transition matrix, U, into its singular values. Here, we also used a spin-adapted form for further description of the different spin states in the excited-state calculation: where Σ ii ¼ γ i and σ ∈ α; β È É depend on spin state (mixed and triplet states, respectively). This leads to the occupied and virtual NTOs as The resulting matrix W rotates ground-state KS orbitals as j runs over the occupied groundstate orbitals to give the corresponding ith 'occupied' NTO orbital (ϕ oα i ). For its virtual counterpart (ϕ vσ i ), V does the similar role as W with a running over the virtual ground-state orbitals. With these NTOs, we can rewrite Eq. (4) for the 'occupied' excited-state orbitals as where r α 0 and r β 0 are the ground-state density and Δr U, R indicates the excited-state density changes including relaxation effect. The F KS is the Kohn-Sham Fock operator.
Another idea is to restrict the transition matrix, U, in CV-DFT to the case of single NTO excitations, that is, Eq. (17) is approximated to include only one major excitation in the transition matrix. Three different forms of such restrictions on U were shown and discussed in the previous work [104], which referred to as SOR-R-CV(∞)-DFT, COL-RSCF-CV(∞)-DFT and SVD-RSCF-CV(∞)-DFT. Among the three methods, we have shown that SVD-RSCF-CV (∞)-DFT as rank 1 approximation is the most general form for such a single NTO excitation: where the v σσ 1 and w αα 1 are the vector of the largest singular value V σσ and W αα out of Eq. (17). The SVD-RSCF-CV(∞)-DFT was also shown to give the same excitation energies as ΔSCF-DFT within 0.1 eV [104].
As a roundup we list the current different versions of CV-DFT in Table 1.

Applications
In this section, we will show examples of excitations where different versions of CV-DFT have been applied successfully. These excitations are of Rydberg type or possess a dominant chargetransfer character; the work has been published in [99][100][101][102][103]. We would like to note that all CV-DFT-calculations presented here were carried out with developers versions of ADF [112,113] and we refer to the original publications for the technical details.

Rydberg excitations
It is well understood that the success of TD-DFT directly depends on how well the approximate exchange-correlation density functional used describes the potential ( b V KS XC ( r ! )). Further, it is evident that functionals based on the local density approximation (LDA) or the generalized gradient approximation (GGA) result in the potential,Ṽ KS XC ( r ! ), that is insufficiently stabilizing 'T' indicates that it is introduced theoretically. 'I' indicates that it is implemented into the code. when compared to b V KS XC ( r ! ) from an exact functional derived from high-level wave function theory [71][72][73][74]. This results in higher occupied orbital (ε i ) and virtual orbital (ε a ) energies . Additionally, the weakness ofṼ KS XC ( r ! ) becomes apparent at medium and large separations r from the polyatomic center of mass where it decays exponentially with r , while b V KS XC ( r ! ) decays as $ À1=r.
Excitation energies in TD-DFT are not necessarily affected by the instability ofṼ KS XC ( r ! ) for medium to large values of r in the valence region and in the density tail. This is primarily due to the dependence of the excitation energies in TD-DFT to the differenceε a Àε i . As can be noted, the large errors in the individual orbital energies might be canceled after the energy difference is calculated provided that the average potential experienced by ψ i and ψ a shows Resultantly, the success of TD-DFT for valence excitations is attributable to this phenomenon for transitions ψ i ! ψ a where the overlap S ia between the two densities r i and r a is large [92,114]. However, for cases such as Rydberg transitions [71][72][73][74] as well as charge-transfer excitations [77,79,82,90,96,97,115] where S ia is small, the error iñ It is a common practice in the case of small S ia to construct specialized potentials [71-74, 77, 79, 82, 90-98] in which the proper À1=r decay is enforced yielding acceptable results. The disadvantage here is that these parameterized potentials might yield inaccurate results for transitions in which S ia >> 0.
Since Rydberg transitions are characterized by a single orbital replacement ψ v ! ψ r , RSCF-CV(∞)-DFT will give results very similar to ΔSCF-DFT; this similarity in case of a single NTO transition has been demonstrated in [104]. Although for ΔSCF-DFT, states of the same symmetry as the ground-state almost always decompose to the ground-state, this weakness is absent in RSCF-CV(∞)-DFT. The RSCF-CV(∞)-DFT triplet and singlet transition energies for these single orbital replacement-type excitations are obtained as special case of Eqs. (26) and (27), with the singlet excitation energy given as 2ΔE M À ΔE T . In the analysis of Rydberg excitations based on RSCF-CV(∞)-DFT, the excitation energy is considered as a sum of the ionization potential (IP) of a neutral species, A, and the electron affinity (EA) of the resulting cation, A þ , after ionization: . Thus, errors in the excitation energies are due to error in the calculated EAs and IPs. Consequently, a method or 'specialized' XC functional that provide accurate EAs and IPs would in turn afford accurate Rydberg excitation energies [100]. Table 2 for comparison with the experimental data are the IPs (N 2 ) and EAs (N þ 2 ) calculated by RSCF-CV(∞)-DFT (or ΔSCF-DFT). The near-perfect agreement (RMSDs between 0.1 and 0.3 eV) with the experimental data is transferred to the excitation energies afforded by the RSCF-CV(∞)-DFT method. As noted previously by Verma and Bartlett for functionals used within TD-DFT [118][119][120] and the authors of the work discussed here [100]. A test set including 73 excitations (32 singlet, 41 triplet) from nine different species (N 2 , 5; CO, 7; CH 2 O, 8; C 2 H 2 , 8; H 2 O, 10; C 2 H 4 , 13; Be, 6; Mg, 6; Zn, 10) has been used. Broken down into the different species, the results are given in Table 3 in terms of mean absolute error (MAE) and root-mean-square deviation (RMSD).  [116]. f [117].

Shown in
Data represented in this table was first published in [100].  [100]. Table 3. Summary of RMSDs of Rydberg excitation energies a calculated with ΔSCF using an extended basis set b and five different functionals. Table 3 for RSCF-CV(∞)-DFT (or ΔSCF-DFT) are in general better than TD-DFT with the same functionals but at par with TD-DFT results with 'specialized' functionals [71][72][73][74].

The results in
With this benchmark the suitability of RSCF-CV(∞)-DFT without the need for sophisticated (or 'specialized') functionals for Rydberg excitations has been demonstrated. The origin of this good performance is attributable to the ability of RSCF-CV(∞)-DFT to afford good estimates of IPs and EAs for all functionals [100,121,122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the resultant excitation energies.

Charge-transfer excitations
In this subsection we will have a look at excitations with charge-transfer character.
It is well known that TD-DFT applied with standard local exchange and correlation functionals has difficulties for transitions with charge-transfer character between two spatially separated regions [82,91,109], a finding nicely explained by Drew, Weisman and Head-Gordon [114]. According to several authors, the reason lies in the exchange and correlation functional [79,82,91,123,124]. Indeed, a functional like CAM-B3LYP [125] includes a certain Hartree-Fock exchange and results in a clear improvement of TD-DFT excitation energies for transitions involving a charge-transfer character [79,124,126]. To further improve the asymptote of the exchange-correlation potential, long-range corrected hybrid scheme like the ones proposed in [76,95,98,127] and asymptotically corrected model potential scheme like in [128,129] have been designed. Of course modifying the functional is not the only approach, and it is not surprising also that other DFT-based approaches have been suggested, all having their own assets and drawbacks. Several of them have been applied for excitations involving chargetransfer character, for example, constrained orthogonality method (COM) [49,50], maximum overlap method (MOM) [56], constricted variational density functional theory (CV-DFT) [58] and its extensions [104,105], constrained density functional theory [130], self-consistent field DFT (ΔSCF-DFT) [131], orthogonality constrained DFT (OCDFT) [63], ensemble DFT [132,133] and subsystem DFT (FDE-ET) [134].
Ziegler et al. showed in [115] how the theoretical framework of CV-DFT is able to cope with excitations including a charge-transfer character and demonstrated this capability with different applications [102,109,121]. Here, we will have a look at examples out of three of these mentioned types.

C 2 H 4 XC 2 F 4 : long-range charge-transfer excitations
Ethylene tetrafluoroethylene, C 2 H 4 xC 2 F 4 , is a system well studied in literature [76,91,93,114,126,134]. It allows for the study of the dependence of excitation energies on the separation of the donor and acceptor and test for the expected À1=R behavior.
For the system C 2 H 4 xC 2 F 4 , two transitions are of particular interest, the excitations HOMO ! LUMO and HOMO-1 ! LUMO + 1, both resulting in an excited state of b 1 symmetry. With these transitions, a charge is transferred between the two molecules C 2 H 4 and C 2 F 4 . Although the concrete orbital localization is highly functional dependent, the orbitals HOMO-1, HOMO, LUMO and LUMO + 1 are from certain separation distance, dominantly located on one of the fragments, as visible in Figure 1 (see, e.g., [77,102]). It should be noted that for a classification to one of the aforementioned types, it is sufficient when the mentioned ground-state orbitals contribute the most, not necessarily uniquely.
The results obtained with CV-DFT and selected reference values for comparison are shown in Figure 2.   these excitations, similar energies are reported using the revised Hessian in [93]. In R-CV(∞)-DFT [61], relaxation of orbitals not directly participating is allowed (see Section 2.3), and it is of no surprise that excitation energies decrease. These results still correspond to a À1=R behavior (resulting in fitting coefficients c 1 for the values presented in Figure 2 of 1.1 and 0.9 E h a 0 ). For the HOMO ! LUMO transition, the values agree with those reported in [76] using LC-BLYP (MAD = 0.2 eV, RMSD = 0.2 eV). Thus, the extrapolated infinite separation value, ΔE R!∞ ¼ 12.7 eV, is close to the ΔE R!∞ ¼ 12.5 eV reported in [76].
Turning next to the triplet excitations for both CV(∞)-DFT and R-CV(∞)-DFT, similar findings are obtained. At longer distances, no spin interaction is expected; as envisioned the triplet excitation energies match values obtained for the corresponding singlet excitation. Excluding the HOMO ! LUMO triplet excitations with R < 6 Å, a nice À1=R behavior is obtained.
Until now all the applied methods have one thing in common: the transition matrix U has not been optimized. This means the character of the transition itself has not been changed.
With CV-DFT being a variational method, the transition matrix U can be optimized with the aim of minimizing the energy (see Section 2.3). In this case the RSCF-CV(∞)-DFT method [59][60][61] is applied, whose strength and merits have been demonstrated several times [100,104,121]. From Figure 2, it can clearly be seen that RSCF-CV(∞)-DFT minimizes the excitation energy at the expense of nearly distance-independent excitation energies and the loss of the À1=R long-range dependence. This energy gain stems from the optimization of the transition matrix U; a thorough explanation is given in [102]. In summary, the charge-transfer transitions, HOMO ! LUMO and HOMO-1 ! LUMO + 1, are dominated by single NTO transitions. Optimizing the transition matrix results in a mix of (mainly) two NTO transitions with (at least one) different participating fragments, meaning that the two charge-transfer excitations, clearly separated before, do mix now. This mixing of the two different excitations leads to a smaller destabilization and a larger stabilization, resulting in a clear reduction of the excitation energy [102]. An additional issue comes now from having a partial charge c A ∈ 0; 1 ð Þ located on fragment A and a partial charge 1 À c A on fragment B, even when these two fragments are further apart. Therefore, from a certain distance on this mixing should be suppressed. To block the optimization algorithm from mixing such unwanted excitations in RSCF-CV(∞)-DFT calculations, two different strategies have been proposed in [102]. But while working, they both depend highly on an arbitrarily chosen value for a threshold parameter. It remains to be seen, if a strategy without the need of such a parameter can be found for RSCF-CV(∞)-DFT.

Polyacenes: excitations with hidden charge-transfer character
The focus of this subchapter is on polyacenes, a system with an intramolecular charge-transferlike character, also referred to as charge-transfer in disguise [135]. The polyacenes are understood as a number n r of linearly fused benzene rings. Such linear polyacenes possess π ! π ⋆ excitations L a (or B 2u when the x-axis corresponds to the long molecular axis) and L b (or B 3u ) with distinct properties, described, for instance, in [136]. Additionally, these polyacenes have a singlettriplet gap for which a function of n r has been proposed. An extrapolation of this function gave rise to a discussion: if polyacenes with a certain size would have a triplet ground state [137][138][139][140][141][142][143]. Polyacenes and their derivatives have been used in a plethora of applications; an overview of some of these applications can be found in [144,145]. Thus, it is not surprising that polyacenes and their excitation energies have been studied extensively. While high-level calculations exist, see, for example, the work presented in [140,141,143], considering the size of larger polyacenes TD-DFT calculations is more common. But the latter ones applied with standard functionals have several difficulties. This is why different methods and strategies have been used, each one having its advantages, and we refer to [101] and references therein for more details. Before moving on to the results obtained with CV-DFT, it must be noted that the polyradical character in the ground-state builds up with increasing number of fused acenes, which was deduced by Ibeji et al. [143] and was confirmed by Plasser et al. [146]. This polyradical character gets bigger and for polyacenes larger than hexacene even big enough to lead to a 'breakdown of single reference approximation used to describe the ground-state of polyacenes in conventional DFT' [132]. Within CV-DFT we rely on a DFT ground-state description. The awareness of this limitation is the reason why only polyacenes as large as hexacene have been studied with CV-DFT.
We will now have a look at the singlet excitation energies. As these energies are not directly measurable, we will use the modified experimental values from Grimme and Parac [136] as reference, for simplicity referred to as experimental results.
As visible from Table 4 and Figure 3, CV(∞)-DFT with LDA results in vertical singlet excitation energies in a very good agreement with the experimental values [147], while for R-CV(∞)-DFT [101], the values deviate more from the experimental ones, although still in an acceptable agreement (a discussion of the difference is given in [101]). As can be seen from Table 4 and Figure 3, both versions of CV-DFT obtain a crossover between 1 1 B 2u and 1 1 B 3u for Anthracene onwards, which is in agreement with experimental findings.  [136]. b Out of [147]. c Out of [101].

Density Functional Calculations -Recent Progresses of Theory and Application
Next, take a look at the obtained triplet excitation energies for the studied polyacenes, shown in Table 5. The equivalency of CV 2 ð Þ-DFT and TD-DFT with the TDA stated in theory section (Section 2.2) is once again confirmed by the numbers in Table 5. It can also be seen that in the triplet case, the energies obtained with R-CV(∞)-DFT change only slightly in comparison with the values obtained with CV 2 ð Þ-DFT, on average by 0.05 eV (for comparison, singlet excitations have a MAD of 0.30 eV for 1 1 B 2u and 0.13 eV for 1 1 B 3u , values out of [101,147]). This surprisingly small difference is due to the nature of the excitation, and for a further discussion of the contributions, we refer to [101].
As previously pointed out in [104], R-CV(∞)-DFT results in triplet states of excitation energies being lower than the ones obtained by coupled cluster methods. Nevertheless, with a RMSD of 0.31 and 0.29 eV, respectively, when compared to the values given in [140 and 143], the results are in reasonable agreement (we note that coordinates were optimized slightly differently). The nature of the triplet excited states is in agreements with the findings of [148], namely, a 1 3 B 2u state for the first triplet excitation, T 1 ; for the second triplet excitation, T 2 ; 3 B 3u for Naphthalene; and 3 B 1g for Anthracene to Hexacene.  From Figure 4 one can see the singlet-triplet gap (ST) decreasing, resembling an exponential function. In order to estimate the ST gap for infinitely large polyacenes, giving an indication if there would be a ST crossover, several authors fitted the excitation energies to the function f n r ð Þ ¼ aexp Àbn r ð Þþc (see [140,142,143] [101]. For the 'adiabatic' or well-to-well excitations, results from different methods in literature are controversial about a possible ST gap crossover ( [140,142] versus [143,149,150]); for TD-DFT it even depends on the functional used [142]. Therefore, necessarily the findings presented here will agree with some findings, while disagree with others. It should be noted that these energies are very small, actually smaller than the estimated accuracy of the CV-DFT method, and with its error it must be regarded as giving only a tendency for no ST crossover. Two additional points of precaution which puts the value of the extrapolated results into question: (a) it has been shown in [142] how a small change of a single excitation energy can influence the obtained polymeric limit, and (b) one should have in mind the change of the ground-state character with the polyacene length and, thus, the number of fused acenes.
There are several adjustable parameters that can influence the excitation energies. These include the size of the basis set used, functionals used, geometry (optimized structures or experimental geometries), medium (since the complexes are anions), etc. Use was made of experimental structures which lead to higher excitation energies (0.1-0.3 eV) compared to optimized structures. Marginal influence of solvation was found for the three valence excitations; the calculated COSMO [161,162] excitation energies lower the energies by 0.01-0.02 eV [163,164]. Table 6 displays the RMSD between the first three experimental dipole-allowed transitions and the corresponding values calculated by RSCF-CV(∞)-DFT.
On average, the three functionals B3LYP, PBE0 with an intermediate fraction of HF exchange and LCBP86* have the lowest RMSD of 0.2 eV, whereas the local functionals (LDA, BP86, BPE) and BHLYP with the highest HF fraction and LCBP86 have a somewhat larger RMSD of 0.3 eV for both 3d and 4d + 5d averages. TD-DFT with the same functionals performs poorly for the 3d complexes but shows good agreement with experiment for the heavier tetraoxo complexes.  This shows a clear lack of consistency. However, RSCF-CV(∞)-DFT shows good and consistent performance for all complexes studied here.
Next, the excitation energies of the octahedral TM complexes [103] are presented. The analyses will be primarily focused on Cr(CO) 6 and [Fe(CN) 6 ] 4À where experimental excitation energies are available. The first system to be considered is Cr(CO) 6 ; the RSCF-CV(∞)-DFT results are displayed in Table 7. The results afforded by RSCF-CV(∞)-DFT are in good agreement with the experimental data even at the RSCF-CV(∞)-DFT/LDA level of theory. The RSCF-CV(∞)-DFT/ LDA results show performance identical to the TD-DFT/B3LYP [151] and better performance than TD-DFT with LDA and GGAs.
Considered next is the [Fe(CN) 6 ] 4À complex; the results are shown in Table 8. The RMSDs here were calculated with the lower limit of the experimental [153] excitation energies where ranges are applicable. There are some theoretical calculations carried with TD-DFT [151] and other DFT-based approaches [151] as well as some high-level ab initio methods [156]. Again, there are good performances even for the LDA and GGA functionals. The accurate excitation energies afforded by the RSCF-CV(∞)-DFT method when compared to the experimental data are as a result of, to some extent, fortuitous error cancelation.    A look now at the electronic density change that accompanies the electronic excitation. Figure 5 (a and b) shows the plot of the density changes associated with the electronic transitions in Cr (CO) 6 . The charge redistribution can be seen from the figure, where the density depletion (r occ ), the accumulation (r vir ) as well as the density change (Δr ex ¼ r vir À r occ ) occurs, resulting from the total change in density associated with the electronic transition. For the MLCT transition, the r occ (Figure 5a) is situated on the Cr metal center, the area or space spun by the density that is reminiscent of the d yz and the r vir is mostly situated on the equatorial CO ligands. The depletion density is in the yz plane, the accumulation density is situated on the CO ligand, and there is little interaction between them as can be seen from the difference (Figure 5b). The movement of density is from the metal center to the ligands corresponding to an intramolecular charge-transfer transition. It is clear from Figure 5c that this transition has a significant d ! d character. In the density plots that follow, there is a depletion in the density situated on the metal with some contribution from the CO in the xy-plane and accumulation of density largely on the central Cr metal along the yz-plane with some accumulation on the CO ligands in the same plane.
Displayed in Figure 6 are the density plots for [Fe(CN) 6 ] 4À . Similar features are seen here as seen for Cr(CO) 6 . The differences in the density plots representing the MC transition; there is more significant accumulation on the CN À ligands, and the density accumulation is in the same plane (xy-plane) as the depletion density (d x 2 Ày 2 -d xy ). As for the MLCT, the associated density movement is identical to that of Cr(CO) 6 (see Figure 6(b and c)).
The benchmark studies on the tetrahedral and octahedral TM complexes probed the ability of RSCF-CV(∞)-DFT to describe CT and hidden CT excitations. Use was made of the tetrahedral d 0 metal oxides as the first benchmark series since the tetra oxides have a long history as a challenging testing ground for new methods due to their complex electronic structure. In general there is either a comparable performance for RSCF-CV(∞)-DFT and TD-DFT in cases where TD-DFT shows good performances or RSCF-CV(∞)-DFT outperforms TD-DFT.
A trend that manifests itself at larger r for the TM complexes is the excitation energies which become more functional dependent and less accurate. Further, the accuracy of RSCF-CV(∞)-DFT for smaller r is not attributable to the ability of our method to afford accurate values of the IP of the complex and the EA of the cation formed alone; some error cancelation occurs when the IPs and EAs are combined to obtain the excitation energy. Finally, it is possible to plot the density change associated with the electronic transitions afforded by our method with regions of density depletion and accumulation supporting a qualitative classification of excitations as MLCT or MC.

Conclusion
In this chapter we presented the CV-DFT method and its different variants. While CV 2 ð Þ-DFT is consistent with (adiabatic) TD-DFT within the TDA approximation, CV-DFT allows to go to higher order. Indeed, its strength lies in going beyond linear response and therewith obtaining distance-dependent contributions to the excitation energy naturally. Additionally, the theory allows for the calculation of excitation energies for singlet and triplet states on the same footing. Of course as a variational method, CV-DFT relies on an accurate ground-state description. The theoretical framework allows us to apply special restrictions as done in [104] and therewith obtain a numerically stable method being numerically equivalent to ΔSCF-DFT.
How CV-DFT performs has been shown in Section 3 with the aid of selected examples of charge-transfer or Rydberg excitation type. With these examples, we could demonstrate how CV(∞)-DFT is able to reproduce the expected À1=R long-range behavior for charge-transfer excitations. When orbital relaxation is allowed, the excitation energies obtained by R-CV(∞)-DFT with LDA agree nicely with the findings of long-range corrected functionals. For short distance, the optimization of the transition matrix U is clearly beneficial [100,104,121]. But for medium-and long-range distances, a notion of care is to be taken as the optimization may lead to an unwanted mixing of transitions as shown in the case of C 2 H 4 xC 2 F 4 . Also, for excitations with hidden charge-transfer character, meaningful results are obtained with CV-DFT, for example, accurate results for the first singlet excitation energies of polyacenes [101,147] for polyacenes as large as hexacene. Not only is CV-DFT able to deliver meaningful results, even for the LDA functional, it has an incredible ability to provide a qualitative picture of the nature or type of excitation under consideration. This is seen in the case of the TM complexes, a complicated yet excellent test set for assessing the range of applicability of every newly developed method. In the case of Rydberg excitations, RSCF-CV(∞)-DFT produces meaningful results without the need for sophisticated (or 'specialized') functionals. This good performance is attributable to the ability of our method affords good estimates of IPs and EAs for all functionals [100,121,122]. Admittedly, fortuitous error cancelation in IPs and EAs obtained for both RSCF-CV(∞)-DFT and TD-DFT plays a role in the accuracy of the resultant excitation energies.