Open access peer-reviewed chapter

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

Written By

Florian Senn, Issaka Seidu and Young Choon Park

Submitted: 05 April 2017 Reviewed: 12 September 2017 Published: 20 December 2017

DOI: 10.5772/intechopen.70932

From the Edited Volume

Density Functional Calculations - Recent Progresses of Theory and Application

Edited by Gang Yang

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Abstract

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 applied to ensure that exactly one electron is fully transferred 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-DFT excited states are not necessarily stemming from the functional used, but from the application of these standard functionals in combination with the linear response theory.

Keywords

  • CV-DFT
  • excited state
  • charge-transfer
  • Rydberg excitations
  • ZnBC-BC

1. Introduction

The behavior of atoms and polyatomic systems in the excited-state are of immense importance in the studies of several photophysical phenomena. Thus, the search for methods to study systems in their electronically excited state is the subject of ongoing research [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Resultantly, there are several methods to choose from within certain consideration such as system size, expected level of accuracy and nature of initial and final electronic state of the system under study. Therefore, some background knowledge is necessary for the accurate treatment of excited states with the available methods. These methods fall under different families, and the ab initio wave function family of methods includes multi-reference configuration interaction (MRCI) [14], multi-configurational self-consistent field (MCSCF) [15, 16], complete active space self-consistent field (CASSCF) [17], time-dependent Hartree-Fock (TD-HF) [18, 19, 20, 21], restricted active space self-consistent field (RASSCF) [22], RASPT2 [23], complete active space second-order perturbation theory (CASPT2) [24], equation-of-motion coupled cluster (EOMCC) [25], n-electron valence state perturbation theory (NEVPT) [26], spectroscopically oriented configuration interaction (SORCI) [27] and coupled cluster (CC) theory [28, 29].

However, the focus of this book is the Kohn-Sham density functional theory (DFT) [30] and methods based on it. In this chapter, our attention is on the calculation of excited states. Excited-state studies within DFT gained considerable attention owing to the increasing success of DFT in ground-state studies. Significant research effort toward the development of excited-state methods has resulted in a variety of approaches varying in both major and minor details, each method having its own advantages and disadvantages. The result of this endeavor includes self-consistent-field DFT (ΔSCF-DFT) [31, 32, 33] with extensions [34, 35, 36], time-dependent DFT [40, 41, 42, 43, 44], ensemble DFT [37, 38, 39, 45, 46, 47], constrained orthogonality method (COM) [48, 49, 50], restricted open-shell Kohn-Sham (ROKS) [47, 51, 52], constrained DFT (CDFT) [53], ‘taking orthogonality constraints into account’ (TOCIA) [54, 55], maximum overlap method (MOM) [56, 57], constricted variational density functional theory (CV-DFT) [58] and extensions [59, 60, 61, 62], orthogonality constrained DFT (OCDFT) [63] and guided SCF [64] among others. However, the most widely used by both expert and nonexpert is TD-DFT in the form of linear response adiabatic time-dependent DFT [40, 41, 65, 66, 67, 68, 69] (which we will refer to as TD-DFT) due to its successes.

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], charge-transfer (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:

  1. Finding the XC functionals with the correct short- and long-range behavior or going beyond the adiabatic approximation.

  2. Developing new DFT-based excited-state methods.

An often-encountered problem with the development of specialized functionals is that it usually performs very well for the purpose for which it was originally developed but unimaginably erratic for any other situation [71, 72, 73, 74, 77, 79, 82, 88, 91, 92, 93, 94, 95, 96, 97, 98].

Our contribution to this area of research is in the development of the constricted variational DFT (CV-DFT) [58, 59, 60, 61, 62], which combines the strengths of ΔSCF-DFT and TD-DFT methods without the need for ‘specialized’ functionals.

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 charge-transfer type following the publications [99, 100, 101, 102, 103].

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2. 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.

2.1. The CV-DFT scheme

Here, we only consider the excitation from the closed-shell ground state described with single Slater determinant, Ψ0=ϕ1ϕ2ϕiϕnocc, where nocc is the number of occupied orbitals. CV-DFT starts from the ansatz which describes the excitation as an admixture of occupied ϕii=1nocc and virtual ϕaa=1nvir ground-state orbitals [105]:

ϕi=anvirUaiϕaE1

where ϕi is the excited-state orbital and nvir the number of virtual orbitals. The transition matrix, U, only mixes between occupied and virtual orbitals (Uij=Uab=0) and is skew symmetric (Uai=Uia). In CV-DFT, we use the exponential expansion of U which leads to the unitary transformation Y:

Y=expU=k=0Ukk!.E2

Thus, once the transition matrix, U, is determined, a new set of orbitals is obtained over the unitary transformation

ϕoccϕvir=k=0Ukk!ϕoccϕvir.E3

Due to the properties of the transition matrix, U, the ‘occupied’ excited-state orbitals can be written as

ϕi'=jnoccYjiϕj+anvirYaiϕa.E4

The corresponding excited-state density becomes

ρ11=inoccϕ'i1ϕ'i1=inoccϕi1ϕi1+anvirinoccΔPaiϕi1ϕa1+ϕa1ϕi1+inoccjnoccΔPijϕi1ϕj1+anvirbnvirΔPabϕa1ϕb1E5

with the change in density matrix (ΔP). Later, one is given by

ΔPaj=inoccYaiYjiE6
ΔPjk=inoccYjiYkiδjkE7
ΔPab=inoccYaiYbi.E8

In CV-DFT, we apply the important condition that one electron is fully transferred from occupied into virtual spaces. This condition can be written as the following equation:

anvirΔPaa=1andinoccΔPii=1.E9

It should be noted that in CV-DFT we describe the excited state with a single Slater determinant. Thus, we obtain the mixed and triplet states. While this is uncritical for triplet excitations, for the singlet transition energy, we have to account for this by using the relation (which is also referred to as sum rule) [61]

ΔES=2ΔEMΔET.E10

2.2. CV(n)-DFT

The nth-order CV-DFT, CV(n)-DFT, is determined from the maximum order of U in the CV-DFT energy description. To understand how the order of the applied transition matrix, U, affects the excited-state energies, it is beneficial to discuss two extreme cases—second (n=2)-order and infinite (n=)-order CV-DFT.

The second-order CV-DFT (CV2-DFT) limits the U up to the second order in the Kohn-Sham energy description. For simplicity, the occupied excited-state orbitals in Eq. (4) are approximated to the second order in U:

ϕi'=ϕi+anvirUaiϕa12jnoccanvirUaiUajϕj+OU3.E11

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

EKSρ11=EKSρ0+aiUaiUaiεa0εi0+aiUaiUbjKai,bj+12aibjUaiUbjKai,jb+12aibjUaiUbjKai,jb+OU3E12

where the two-electron integral is composed of a Coulomb and an exchange-correlation part:

Kpq,st=Kpq,stC+Kpq,stXCE13

with

Kpq,stC=ψp1ψq11r12ψs2ψt2dν1dν2.E14

The exchange-correlation integral is further decomposed into the local (KS) and nonlocal (HF):

Kpq,stXCKS=ψpr1ψqr1fr1ψsr1ψtr1dr1E15

and

Kpq,stXCHF=ψp1ψq11r12ψs2ψt2dν1dν2E16

where fr1 represents the regular energy kernel. We have shown that CV2-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:

Uσα=VσσΣWααTE17

where Σii=γi and σαβ depend on spin state (mixed and triplet states, respectively). This leads to the occupied and virtual NTOs as

ϕioα=jnoccWααjiϕjαE18
ϕivσ=anvirVσσaiϕaσ.E19

The resulting matrix W rotates ground-state KS orbitals as j runs over the occupied ground-state orbitals to give the corresponding ith ‘occupied’ NTO orbital (ϕioα). For its virtual counterpart (ϕivσ), 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

ϕi'=cosγiϕioα+sinγiϕivσ.E20

Also, the condition of exciting exactly one electron (Eq. (9)) is now written as

inoccsinηγi2=1.E21

With the sum rule in Eq. (10), the excited-state CV()-DFT energy of the mixed state becomes

ΔEM=inoccsin2ηγiεivαεioα+12inoccjnoccsin2ηγisin2ηγjKioαioαjoαjoαKivαivαjvαjvα2Kioαioαjvαjvα+12inoccjnoccsinηγicosηγisinηγjcosηγjKioαivαjoαjvα+Kioαivαjvαjoα+2inoccjnoccsinηγisinηγisinηγjcosηγjKivαivαjoαjvαKioαioαjoαjvαE22

whereas the triplet exited-state energy has a simpler form:

ΔET=inoccsin2ηγiεivβεioα+12inoccjnoccsin2ηγisin2ηγjKioαioαjoαjoαKivβivβjvβjvβ2Kioαioαjvβjvβ.E23

The γ values out of Eq. (21) give information about the excitation character [60]. Keeping only the largest γ value in the excitation will give the most general form of single orbital replacement [104], which is used as the ΔSCF-DFT-like scheme within the RSCF-CV()-DFT formulation. This will be briefly mentioned in the next section.

2.3. SCF-CV()-DFT, R-CV()-DFT and RSCF-CV()-DFT

In Eqs. (22, 23), we obtain the excited-state energy of the mixed and triplet state. The transition matrix, U, is the same as one obtains within TD-DFT (and thus the TD-DFT excitation vector is implemented in CV-DFT). In SCF-CV()-DFT, U is optimized with the variational procedure [60]. For this step, we derived the gradient of the mixed and triplet excited state. The detailed procedures can be found in the [59, 60, 61, 104]. Further, also the orbitals which do not participate in the excitation can be changed after the excitation. We refer to this change as the relaxation of orbitals. This leads to R-CV()-DFT. To account for this orbital relaxation effect, we introduced R, which is orthogonal to U, and apply it on the orbitals from Eq. (4). Therewith, the ‘occupied’ and ‘virtual’ orbitals become

ψi1=ϕi1+cnvirRciϕc112cnvirknoccRciRckϕk1E24
ψa1=ϕa1knoccRakϕk112cnvirknoccRakRckϕc1.E25

It is possible to combine the approach of the variational optimization of the transition matrix and orbital relaxation, meaning the variational optimization of U and R, resulting in the most general form of CV-DFT (RSCF-CV()-DFT). The excitation energy expression of RSCF-CV()-DFT can be written for the mixed and triplet state, respectively:

ΔEM=EMU,Rρ0α+12ΔρMU,Rρ0β+12ΔρMU,REρ0αρ0β=FKSρ0α+12ΔρMU,Rρ0β+12ΔρMU,RΔρMU,Rdν1E26
ΔET=ETU,Rρ0α+12ΔρTU,Rρ0β+12ΔρTU,REρ0αρ0β=FKSρ0α+12ΔρTU,Rρ0β+12ΔρTU,RΔρTU,Rdν1E27

where ρ0α and ρ0β are the ground-state density and ΔρU,R indicates the excited-state density changes including relaxation effect. The FKS 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:

U=v1σσw1ααTE28

where the v1σσ and w1αα are the vector of the largest singular value Vσσ and Wαα 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.

Transition URelaxation RIntroduced
Order nOptimizationRestrictionsOrderOptimizationSingletsTriplets
CV(2)-DFTSecondNoNoN/ANoT [58], I [109]T [60], I [110]
CV(4)-DFTFourthNoNoN/ANoT [93], I [109]T [109], I [109]
CV()-DFTNoNoN/ANoT [59], I [111]T [60], I [60]
SCF-CV()-DFTYesNoN/ANoT [59], I [60]T [60], I [104]
RSCF-CV()-DFTYesNoSecondYesT [61], I [61]T [61], I [104]
SOR-R-CV()-DFTNoUai=δabδijSecondYesT [62]T [62], I [62]
COL-RSCF-CV()-DFTYesUai=δijSecondYesT [104]T [104], I [104]
SVD-RSCF-CV()-DFTYesγ1=1SecondYesT [104]T [104], I [104]

Table 1.

Variation of CV-DFT applied.

‘T’ indicates that it is introduced theoretically.

‘I’ indicates that it is implemented into the code.

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3. 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 charge-transfer 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.

3.1. 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 (V̂XCKS(r)). Further, it is evident that functionals based on the local density approximation (LDA) or the generalized gradient approximation (GGA) result in the potential, V˜XCKS(r), that is insufficiently stabilizing when compared to V̂XCKS(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 obtained with V˜XCKS(r) as opposed to those ((ε̂i),(ε̂a)) derived from V̂XCKS(r). Additionally, the weakness of V˜XCKS(r) becomes apparent at medium and large separations r from the polyatomic center of mass where it decays exponentially with r, while V̂XCKS(r) decays as 1/r.

Excitation energies in TD-DFT are not necessarily affected by the instability of V˜XCKS(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 difference ε˜aε˜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 ψi and ψa shows similar deviations from V̂XCKS(r). Resultantly, the success of TD-DFT for valence excitations is attributable to this phenomenon for transitions ψiψa where the overlap Sia between the two densities ρi and ρa 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 Sia is small, the error in V˜XCKS(r) gets more pronounced. It is a common practice in the case of small Sia to construct specialized potentials [71, 72, 73, 74, 77, 79, 82, 90, 91, 92, 93, 94, 95, 96, 97, 98] in which the proper 1/r decay is enforced yielding acceptable results. The disadvantage here is that these parameterized potentials might yield inaccurate results for transitions in which Sia >> 0.

Since Rydberg transitions are characterized by a single orbital replacement ψvψr, 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 2ΔEMΔET. 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+, after ionization: ΔEϕvϕr = EA(A+, ϕv, ϕr) + IP(A, ϕv). 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].

Shown in Table 2 for comparison with the experimental data are the IPs (N2) and EAs (N2+) 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 (N2, 5; CO, 7; CH2O, 8; C2H2, 8; H2O, 10; C2H4, 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).

Energy termLDABP86B3LYPLCBP86*cLCBP86dExpt.
IP(N2, σg)15.6315.5015.7415.9616.3815.58e,f
IP(N2, πu)17.4617.0716.8717.2217.1817.07f
EA(N2+, σg, 3sσg, S)−3.64−3.61−3.49−3.55−3.44−3.38g
EA(N2+, σg, 3pπu, S)−2.92−3.01−2.87−2.81−2.77−2.68g
EA(N2+, σg, 3pσu, S)−2.84−2.95−2.79−2.71−2.66−2.60g
EA(N2+, πu, 3sσg, S)−3.77−3.77−3.71−3.73−3.67−3.83g
EA(N2+, σg, 3sσg, T)−3.82−3.81−3.73−3.79−3.72−3.58h

Table 2.

IPa of N2 and EAa of N2+ calculated with ΔSCF using an extended basis setb and five different functionals.

Energies in eV.


[72].


Refers to LC functional combined with BP86 and ω = 0.40.


Represents LC functional combined with BP86 and ω = 0.75.


[116].


[117].


Evaluated as EA(A+, ϕv, ϕr, S) = ΔES(ϕvϕr) IP(A, ϕv).


Evaluated as EA(A+, ϕv, ϕr, T) = ΔET(ϕvϕr) IP(A, ϕv).


Data represented in this table was first published in [100].

SpeciesNo. of statesLDABP86B3LYPLCBP86*cLCBP86d
N250.270.340.050.230.62
CO70.220.430.130.120.37
CH2O80.210.280.120.200.34
C2H280.310.500.520.250.24
H2O100.270.170.140.210.24
C2H4130.150.200.28e0.280.29
Be60.450.600.470.310.23
Mg60.180.350.190.130.12
Zn100.180.250.270.340.46
RMSD0.240.320.240.230.32

Table 3.

Summary of RMSDs of Rydberg excitation energiesa calculated with ΔSCF using an extended basis setb and five different functionals.

Energies in eV.


[72].


Refers to LC functional combined with BP86 and ω = 0.40.


Represents LC functional combined with BP86 and ω = 0.75.


Comprising 12 states.


Data represented in this table was first published in [100].

The results in 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].

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.

3.2. 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 charge-transfer 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.

3.2.1. C2H4xC2F4: long-range charge-transfer excitations

Ethylene tetrafluoroethylene, C2H4xC2F4, 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 1/R behavior.

For the system C2H4xC2F4, two transitions are of particular interest, the excitations HOMO LUMO and HOMO-1 LUMO + 1, both resulting in an excited state of b1 symmetry. With these transitions, a charge is transferred between the two molecules C2H4 and C2F4. 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.

Figure 1.

C2H4xC2F4: Representation of ground-state KS orbitals (LDA) (R= 5.0 Å) (reprinted from [102], with the permission of AIP Publishing).

The results obtained with CV-DFT and selected reference values for comparison are shown in Figure 2.

Figure 2.

C2H4xC2F4 vertical excitation energies for singlets (circles) and triplets (triangles) using CV()-DFT (orange), R-CV()-DFT (red) and RSCF-CV()-DFT (dark red). The values for the revised hessian out of [96] (purple-filled circles), LC-BLYP out of [76] (black-filled circles) and SAC-CI out of [76] (gray-filled circles) are given as reference. The lines serve as a guide for the eyes, and when the excitation is not dominated by one of the charge-transfer excitations, we set its value to zero (and are therewith not visible in the figure). (reprinted from Senn F, Park YC. The Journal of Chemical Physics. 2016;145(24):244108-1 – 10). DOI: 10.1063/1.4972231. with the permission of AIP Publishing. Color specifications refer to the original figure).

First, consideration will be given to the singlet and triplet excitation results with different versions of CV-DFT, where the transition matrix, U, is not optimized, before turning to the most general form RSCF-CV()-DFT.

CV()-DFT results in a 1/R-like behavior, or when assuming a ΔER=c1/R+c0 function, fitting coefficients c1 for the results presented in Figure 2 of 1.1 and 0.9 Eha0 are obtained. For 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 1/R behavior (resulting in fitting coefficients c1 for the values presented in Figure 2 of 1.1 and 0.9 Eha0). 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, ΔER= 12.7 eV, is close to the ΔER= 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 1/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 1/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 cA01 located on fragment A and a partial charge 1cA 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.

3.2.2. Polyacenes: excitations with hidden charge-transfer character

The focus of this subchapter is on polyacenes, a system with an intramolecular charge-transfer-like character, also referred to as charge-transfer in disguise [135]. The polyacenes are understood as a number nr of linearly fused benzene rings. Such linear polyacenes possess ππ excitations La (or B2u when the x-axis corresponds to the long molecular axis) and Lb (or B3u) with distinct properties, described, for instance, in [136]. Additionally, these polyacenes have a singlet-triplet gap for which a function of nr 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 11B2u and 11B3u for Anthracene onwards, which is in agreement with experimental findings.

No. acene unitsExp.aCV()-DFTbR-CV()-DFTc
11 B2u11B3uΔEd11 B2u11B3uΔEd11B2u11B3uΔEd
24.664.130.534.734.390.344.584.420.16
33.603.64−0.043.683.73−0.053.463.75−0.29
42.883.39−0.512.913.32−0.412.693.33−0.63
52.373.12−0.752.353.03−0.682.153.04−0.89
62.022.87−0.851.932.82−0.891.742.83−1.09
MAD0.060.110.180.12
RMSD0.060.130.190.15

Table 4.

Vertical singlet excitation energies (in eV) for linear polyacenes.

Out of [136].


Out of [147].


Out of [101].


ΔE=ΔE11B2uΔE11B3u.


Figure 3.

Vertical singlet excitation energies for the states 11B2u (circles) and 11B3u (crosses) of linear polyacenes: R-CV()-DFT (maroon), ΔSCF-DFT (gray), CV-DFT (orange, out of [147]), [143] (dark blue), experimental values (black, out of [136]). The solid lines serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color specifications refer to the original figure).

Next, take a look at the obtained triplet excitation energies for the studied polyacenes, shown in Table 5. The equivalency of CV2-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 CV2-DFT, on average by 0.05 eV (for comparison, singlet excitations have a MAD of 0.30 eV for 11B2u and 0.13 eV for 11B3u, 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].

No. acene unitsVerticalAdiabatic
R-CV()-DFTaCV2-DFTaTDDFTbRef. [143]Ref. [140]R-CV()-DFTaExp.cRef. [143]
23.163.083.083.343.312.892.642.70
32.152.092.092.472.471.941.862.06
41.491.441.441.821.761.311.271.48
51.020.990.991.371.370.880.861.11
60.690.660.661.071.000.570.540.83
MADd0.050.050.310.280.090.19
RMSDd0.060.060.320.290.120.20

Table 5.

Vertical and adiabatic triplet excitation energies (in eV) for linear polyacenes.

Out of [101].


With LDA as functional.


Out of [143] and references therein.


To be understood as the deviation of the values obtained with R-CV()-DFT in comparison to the values of this column as reference values.


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 13B2u state for the first triplet excitation, T1; for the second triplet excitation, T2; 3B3u for Naphthalene; and 3B1g 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 fnr=aexpbnr+c (see [140, 142, 143]). With the results of R-CV()-DFT for the vertical transition, the limes of an infinitely long polyacene EvertSTnr=0.3±4.5 kcal mol−1 have been obtained and for the ‘adiabatic’ transition EadSTnr= (−1.6 ± 4.0) kcal mol−1 [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.

Figure 4.

Triplet excitation energies for the 13B2u states of linear polyacenes: R-CV()-DFT (maroon), ΔSCF-DFT (gray), SVD-R-CV-DFT (orange), [143] (dark blue), [140] (light blue), experimental values (black, out of [143] and references therein). The symbols are used to distinguish between vertical transitions (crosses) and adiabatic as well as ‘imitated adiabatic’ transitions (circles). The lines are the curves fitted to the function fnr=a expbnr+c and serve as guides for the eyes. (reprinted with permission from Senn F, Krykunov M. The Journal of Physical Chemistry. A. 2015;119(42):10575-10581. DOI: 10.1021/acs.jpca.5b07075. Copyright 2015 American Chemical Society. Color specifications refer to the original figure).

3.2.3. Charge-transfer excitations in transition metals

The complicated electronic structure of transition metal (TM) complexes [151] makes them ideal for testing the performance of newly developed methods. This section deals with charge-transfer (and hidden charge-transfer) excitations in these complexes, more precisely the 3d complexes MnO4, CrO42− and VO43−, as well as their 4d congeners RuO4, TcO4, MoO42− and 5d homologues OsO4, ReO4 and WO42− [99]. For these systems, the three lowest valence excitations involving transitions from 1t1, 2t2 to 2e and 3t2 are considered [99]. The comparison is made with available experimental data [152, 153] and high-level ab initio calculations [154, 155, 156, 157, 158, 159, 160]. 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.

ComplexLDABP86PBEB3LYPBHLYPPBE0LCBP86*LCBP86
MnO_40.410.320.330.150.620.190.240.37
CrO42−0.400.310.340.090.550.040.220.32
VO43_0.250.140.160.070.180.140.270.37
RuO40.320.280.280.210.440.220.190.31
TcO_40.100.130.130.250.130.290.270.17
MoO42_0.140.230.230.060.220.180.130.34
OsO40.530.510.500.270.390.310.210.26
ReO_ 40.360.430.430.140.250.160.140.16
WO42−0.430.510.510.140.110.070.110.16
Average 3df0.350.260.280.100.450.120.240.35
Average 4d + 5dg0.310.340.340.190.270.220.190.24
Total average 3dh0.330.310.320.160.340.180.210.28

Table 6.

RMSDs for tetraoxo excitation energies based on RSCF-CV()-DFT and a TZ2P basis seta, b, c, d, e.

Root-mean-square deviation.


The reference is the observed vertical excitation energies for the three first dipole-allowed transitions.


For MoO42_ and WO42− only, the first two experimental transitions are available.


Deviations are in eV.


No TDA was applied.


Average of the three 3d complexes.


Average of the six 4d and 5d complexes.


Average over all complexes.


Data represented in this table was first published in [99].

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]4− 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.

STATERASPT2bLDABP86PBEB3LYPPBE0BHLYPLCBP86*cLCBP86dExpt.e
Singlet
1T1g(MC)4.985.335.145.174.855.254.524.784.97
1T1u(MLCT)4.504.454.394.404.484.624.684.614.374.44
21T1u(MLCT)5.425.465.475.475.735.936.206.235.855.48
Triplet
13T1g(MC)4.28f4.724.884.914.544.624.154.184.49
13T2g(MC)4.64f4.634.594.604.394.454.174.454.59
RMSD0.060.020.040.030.180.340.540.540.27

Table 7.

Calculated excitation energiesa for Cr(CO)6 based on the RSCF-CV()-DFT method.

Energies in eV.


[154].


Represents LC functional combined with BP86 with ω = 0.75.


Refers to LC functional combined with BP86 using ω = 0.4.


[152].


[155].


Data represented in this table was first published in [103].

Considered next is the [Fe(CN)6]4− 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.

STATECASPT2bLDABP86PBEB3LYPPBE0BHLYPLCBP86*cLCBP86dExpt.e
Singlet
1T1g(MC)3.604.173.723.753.423.353.043.353.643.80–3.94
31T1u(MLCT)5.575.575.646.106.446.345.69–5.89
41T1u(MLCT)5.835.805.726.256.636.936.20
11T2g(MC)4.334.053.744.124.474.464.474.144.374.43–4.77
Triplet
13T1g(MC)2.673.603.393.412.982.902.492.562.932.94
RMSD0.20f0.420.410.330.250.440.40f0.29f0.44

Table 8.

Calculated excitation energiesa for [Fe(CN)6]4− based on the RSCF-CV()-DFT method.

Energies in eV.


[156].


Represents LC functional combined with BP86 with ω = 0.75.


Refers to LC functional combined with BP86 using ω = 0.4.


[153].


Calculated with three excitation energies.


Data represented in this table was first published in [103].

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 (ρocc), the accumulation (ρvir) as well as the density change (Δρex=ρvirρocc) occurs, resulting from the total change in density associated with the electronic transition. For the MLCT transition, the ρocc (Figure 5a) is situated on the Cr metal center, the area or space spun by the density that is reminiscent of the dyz and the ρvir 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 dd 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.

Figure 5.

Δρ associated with the CrCO6, red signifies depletion and green shows accumulation of density. (a) The density change associated with the 11T1u state. (b) Exemplifies the density redistribution associated with the 21T1u state. (c) Densities accompanying the calculated 11T1g state. (a) and (b) are MLCT-type transitions, and (c) is an example of MC-type transition. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT) Method. PhD dissertation. University of Calgary; 2016 for coloured pictures.

Displayed in Figure 6 are the density plots for [Fe(CN)6]4−. 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 ligands, and the density accumulation is in the same plane (xy-plane) as the depletion density (dx2y2dxy). As for the MLCT, the associated density movement is identical to that of Cr(CO)6 (see Figure 6(b and c)).

Figure 6.

Δρ associated with the Fe(CN)64. (a) the density change associated with the 11T1g state. (b) Exemplifies the density redistribution associated with the 31T1u state. (c) Densities accompanying the calculated 31T1u state. (a) Is an example of MC-type transition, and (b) and (c) are MLCT-type transitions. Red signifies loss and green shows gain in density. See Seidu I. Excited-State Studies with the Constricted Variational Density Functional Theory (CV-DFT) Method. PhD dissertation. University of Calgary; 2016 for coloured pictures.

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 d0 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.

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4. Conclusion

In this chapter we presented the CV-DFT method and its different variants. While CV2-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 1/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 C2H4xC2F4. 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.

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Acknowledgments

The authors would like to express their gratitude to the late Prof. Dr. Tom Ziegler for his unflinching support until his untimely passing away.

Further, the authors are grateful to Dr. Mykhaylo Krykunov for his helpful discussions.

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Written By

Florian Senn, Issaka Seidu and Young Choon Park

Submitted: 05 April 2017 Reviewed: 12 September 2017 Published: 20 December 2017