Some of the exact conditions provided by the correlated orbital theory are employed to propose new non-empirical parameterizations for exchange-correlation functionals from Density Functional Theory (DFT). This reparameterization process is based on range-separated functionals with 100% exact exchange for long-range interelectronic interactions. The functionals developed here, CAM-QTP-02 and LC-QTP, show mitigated self-interaction error, correctly predict vertical ionization potentials as the negative of eigenvalues for occupied orbitals, and provide nice excitation energies, even for challenging charge-transfer excited states. Moreover, some improvements are observed for reaction barrier heights with respect to the other functionals belonging to the quantum theory project (QTP) family. Finally, the most important achievement of these new functionals is an excellent description of vertical electron affinities (EAs) of atoms and molecules as the negative of appropriate virtual orbital eigenvalues. In this case, the mean absolute deviations for EAs in molecules are smaller than 0.10 eV, showing that physical interpretation can indeed be ascribed to some unoccupied orbitals from DFT.

The Correlated Orbital Theory (COT)1 introduces clear interpretations for orbital eigenvalues (ε). Thus, the negative of occupied orbital eigenvalues should be ascribed to principal vertical ionization potentials (IPs) and these conditions must hold not only for the Highest Occupied Molecular Orbital (HOMO). In addition, the negative of some virtual (unoccupied) orbital eigenvalues as, for example, the Lowest Unoccupied Molecular Orbital (LUMO), should provide principal vertical electron affinities (EAs). The principal IP and EA meanings refer to the processes of removing one electron from any occupied orbital and adding one electron to an unoccupied orbital, respectively. At this point, it is important to emphasize that COT arguments are based on exact IPs and EAs, that is, the values including orbital relaxation and electron correlation effects. In this sense, COT offers an appealing quantitative molecular orbital theory for chemists since orbitals are no longer mere mathematical constructs from Hückel, Hartree-Fock (HF), or Kohn-Sham (KS) methods, but are now tied to observable quantities, rendering meaningful quantitative tools to rationalize several important phenomena such as reactivities, spectroscopy, chemical kinetics, and band structure in solids, among others.

In related work, it is pertinent to mention that the eigenvalues from ab initio Density Functional Theory (DFT)2 have also been investigated.3,4Ab initio DFT emphasizes the use of known orbital-dependent functionals to deal with exchange and correlation effects via the optimized effective potential (OEP) approach that turns these orbital-dependent functionals into multiplicative Vx and Vc potential energy operators, respectively. The exact exchange (EXX) functional is known as the HF form, subject to KS orbitals, and many have obtained an exchange potential using this OEP procedure. The use of orbital-dependent correlation functionals remains a challenge, but those from second-order perturbation theory (MBPT2) and linearized coupled-cluster theory have been obtained by the Bartlett group,2,5 and shown to be in agreement with the exact results from Umrigar et al.,6,7 who obtain their potentials in a different way by inverting the KS problem to ask what potential will generate a known accurate density from wavefunction theory. In other words, there is no functional in such studies. See Grabowski et al.8 for a systematic study of correlation potentials from a converging series of coupled-cluster calculations. In principle, one could use any orbital-dependent correlation functional from a multi-configurational self-consistent field (MCSCF) to full configuration interaction (CI) calculations in such OEP correlated studies. The relevance to this paper is that when KS-DFT is done as accurately as possible by using known orbital-dependent expressions, the eigenvalues are good approximations to the valence IPs, but an admixture of HF and OEP exchange (50%) is required to get the core IPs right, i.e., “generalized” KS.9 The quantum theory project (QTP) functionals will all be of this type with varying amounts of HF exchange as a function of the interelectronic distance.

The exact relations satisfied by COT also furnish a way to improve approximate electronic structure treatments. In this sense, the IP conditions from COT and from the IP theorem based on adiabatic Time-Dependent DFT (TD-DFT)10 were initially employed to attain CAM-QTP-00,11 a DFT method based on a reparameterization of CAM-B3LYP12 by using all the experimental values for the principal IPs of water (core and valence electrons). Later, the experimental valence IPs of water were again considered along with the results of other properties (electron excitation and atomization energies) to achieve CAM-QTP-01,13 which is also based on the same functional structure introduced by CAM-B3LYP.12 However, the exact electron affinity (EA) relations from COT were still not taken into account during the proposition of new exchange—correlation functionals. Moreover, both CAM-QTP-00 and CAM-QTP-01 can be considered as derived from empirical parameterizations once the experimental IPs of water were employed in the process.

Nowadays, there is strong evidence in the literature that the self-interaction error (SIE) is mitigated by satisfying the IP conditions mentioned before and vice versa.14,15 SIE is a recurrent problem in traditional DFT methods for one- and many-electron systems, which is particularly apparent in many studies like those for barriers in chemical reactions.16,17 COT tells us that −IPm = εm = m|ĥ+ĴK^+Σ^CC|m,1 where the self-energy taken from Coupled Cluster (CC) theory is frequency independent, unlike the traditional Dyson self-energy. Of course, this expression has no self-interaction of one- or many-particle form so to the degree that K^+Σ^CCVx + Vc, the effect should be ameliorated in KS. The primary condition for that is how well −IPm ≈ εm when εm corresponds to a KS orbital energy. As this is the condition imposed up to a threshold in all the QTP functionals, one would hope it would mitigate SIE.

Thus, SIE in KS-DFT, also known as delocalization or curvature error, can be assessed by doing calculations for fractional occupation numbers (ONs).16 Thus, while a straight line is expected for total electronic energies when the occupation number varies between integer numbers,18 it is well known in DFT that the delocalization error results in too low energies for the intermediate species with fractional occupations.16,17 These last authors also mention that functionals with some Coulomb-attenuated exchange seem to be the only ones that exhibit improvements in the treatment of systems with fractional occupation numbers. In this sense, it has been demonstrated that CAM-QTP-00 and CAM-QTP-01 provide promising results with respect to calculations done to investigate this error and that rCAM-B3LYP,17 which was developed to overcome the SIE in many-electron systems (N-SIE), accordingly gives satisfactory IPs from negative of occupied eigenvalues.15 On the other hand, although HF is manifestly free from self-interaction, it still suffers from the localization error and results in too high energies for fractional occupation number systems.16 The same occurs for a non-exact DFT method free from N-SIE such as optimized effective potential with exact exchange (EXX-OEP).16 

Historically, TD-DFT has faced serious difficulties dealing with Rydberg excitations, normally underestimating their transition energies.19 In addition, the correct description of charge-transfer (CT) excitation energies also constitutes a major challenge for popular DFT methods.19,20 However, both these problems are again linked to SIE and the usage of range-separated hybrid functionals seems propitious for overcoming such TD-DFT obstacles,19,20 which is true at least when the distances between charge donor and acceptor units remain large enough. In this case, CAM-QTP-00 and CAM-QTP-01 functionals are also shown to be successful in describing charge-transfer excitation energies.14 Moreover, CAM-QTP-01 presents an excellent performance in the treatment of excitations to Rydberg states.13 

Hence, the main objective of this study is to include a kind of EA condition from COT in a non-empirical parameterization process of exchange-correlation functionals from DFT. What this means is that consistency conditions in COT require that the LUMO eigenvalue of an atomic or molecular system (M) with N electrons, −EA(M(N)), must be equal to the HOMO eigenvalue of the related species containing one extra electron, −IP(M(N + 1)). This is done without the use of any experimental IP or EA value. We call this the HOMO-LUMO condition. A second difference from CAM-QTP-00 and CAM-QTP-01 is that the initial structure of the new functionals devised here is now based on CAM-BLYP.12 Therefore, we will pursue a minimal parameterization procedure starting from a simpler functional form. We also emphasize that the functionals devised here are universal in the sense that they may be applied to any atomic or molecular system. In the following, the new functionals are evaluated for localization/delocalization errors and their performance to predict geometries, vibrational frequencies, dipole moments, ionization potentials, excitation energies, reaction barrier heights, and electron affinities is comparatively assessed.

The DFT calculations have been carried out with the computational package NWChem 6.6.21 In this case, the unrestricted Kohn-Sham (KS) formalism is used for open-shell systems. On the other hand, the calculations based on the Coupled Cluster (CC) theory are done with ACES II.22 The basis sets23,24 are always employed in their spherical harmonic versions. We consider cc-pVDZ, cc-pVTZ, aug-cc-pVTZ, QZP, cc-pVQZ, aug-cc-pVQZ, aug-cc-pVQZ-PP and 6-311(3+,3+)G** basis sets along this work.25–42 For comparative reasons, the grid accuracy in DFT calculations is improved for obtaining barrier heights (xfine), as also done in Ref. 13. As expected, each of the transition state structures optimized here exhibit only one imaginary frequency. More details regarding the particular calculation procedures employed will be given along Sec. III since different approaches must be used for proper comparison with previous results from the literature.

The arguments from COT readily allow writing down the following relations for a molecule M during vertical processes:

IP(M)=εHOMO(M)=EA(M+)=εLUMO(M+)
(1)

and

EA(M)=εLUMO(M)=IP(M)=εHOMO(M),
(2)

where the orbital eigenvalues (ε) are clearly interpreted as orbital energies. This is a consequence of our COT model, but also follows in KS-DFT via the IP theorem derived from adiabatic TD-DFT.10 Accordingly, the generalization of Koopmans’ theorem for any one-electron self-consistent-field (SCF) equation, which relies on the combination between Janak’s theorem43 and the energy linearity theorem for fractional occupations,18 also assures that the same relations should be satisfied in terms of the HOMO and LUMO eigenvalues.44,45 The conditions illustrated in Eqs. (1) and (2) are accurately met within the coupled cluster theory with single and double (CCSD) excitations, as evidenced by the eigenvalues achieved with the equation-of-motion (EOM) formalism for IPs (IP-EOM-CCSD) and EAs (EA-EOM-CCSD) of some system pairs illustrated in Table I. The largest deviation found for these pairs with aug-cc-pVQZ basis sets was 0.15 eV, as obtained from the difference between the IP of F2 (0.29 eV) and the EA of F2 (0.14 eV). This remaining error is attributed here mainly to the truncation of CC excitations.

TABLE I.

Electron affinities (EAs) and ionization potentials (IPs) of system pairs as obtained in calculations with the aug-cc-pVQZ basis set (in eV).a

SpeciesEA-EOM-CCSDSpeciesIP-EOM-CCSD
B+ 8.23 8.31 
CN+ 14.17 CN 14.11 
CH+ 10.57 CH 10.67 
NO+ 9.70 NO 9.74 
LiH 0.29 LiH 0.30 
LiF 0.33 LiF 0.33 
F2 0.14 F2 0.29 
SpeciesEA-EOM-CCSDSpeciesIP-EOM-CCSD
B+ 8.23 8.31 
CN+ 14.17 CN 14.11 
CH+ 10.57 CH 10.67 
NO+ 9.70 NO 9.74 
LiH 0.29 LiH 0.30 
LiF 0.33 LiF 0.33 
F2 0.14 F2 0.29 
a

Only the valence electrons are included in the active space of these calculations, which used unrestricted wavefunctions as reference for open-shell cases. The molecular calculations are performed at the experimental geometries of the neutral systems.46 

Thus, we used versions of Eqs. (1) and (2) written for water and hydroxyl in the parameterization process, that is,

IP(H2O)=εHOMO(H2O)=EA(H2O+)=εLUMO(H2O+)
(3)

and

EA(OH)=εLUMO(OH)=IP(OH)=εHOMO(OH).
(4)

These two molecules are chosen because water has already been employed in the parameterization of CAM-QTP-00 and CAM-QTP-01 and hydroxyl is a closely related system with a noticeable electron affinity value. Thus, based on the previous equations, we can finally define the auxiliary quantities,

Δε1a=εHOMO(H2O)εLUMO(H2O+) andΔε1b=εLUMO(OH)εHOMO(OH),
(5)

which should be zero if COT conditions are exactly met.

Therefore, we reparametrized an exchange-correlation functional of DFT starting with the basic structure defining CAM-BLYP.12 This is a functional with long-range corrections, which relies on three parameters (α, β, and μ, using the same notation employed in the original reference) to regulate the fraction of exact exchange included along increasing electron–electron distances. In addition, the exact relation α + β = 1, which guarantees the correct limit for the exchange potential as interelectronic distances tend to infinity, is also used in the present study. Thus, we are left with the problem of optimizing two independent parameters (α and μ for instance). This process is done in such a way as to provide the smallest similar values for Δε1a and Δε1b as possible, trying to achieve the best well-balanced description of both relations (3) and (4). Therefore, we use the aug-cc-pVQZ basis sets and the experimental equilibrium geometries of water and hydroxyl46 to conduct this parameterization.

The procedure described before provided sets of combinations between the parameters α and μ that result in similar values for Δε1a and Δε1b. Of course, one peculiar choice is the functional starting with no exact exchange at short-range interelectronic interactions (α = 0.00, β = 1.00, and μ = 0.475), which will be named henceforth LC-QTP. In this case, Δε1a and Δε1b are around 0.75 eV. However, other molecules can be considered to select the best combination between α and μ among the ones previously found, particularly for transition metal applications, resulting in a functional more general as possible. Hence, we employ additional relations involving two transition metal compounds to this end, that is,

IP(CuH)=εHOMO(CuH)=EA(CuH+)=εLUMO(CuH+),
(6)
EA(CuH)=εLUMO(CuH)=IP(CuH)=εHOMO(CuH),
(7)
IP(ZnF)=εHOMO(ZnF)=EA(ZnF+)=εLUMO(ZnF+),
(8)

and

EA(ZnF)=εLUMO(ZnF)=IP(ZnF)=εHOMO(ZnF).
(9)

Again, these relations allow defining additional auxiliary values,

Δε2a=εHOMO(CuH)εLUMO(CuH+) andΔε2b=εLUMO(CuH)εHOMO(CuH),
(10)

along with

Δε3a=εHOMO(ZnF)εLUMO(ZnF+) andΔε3b=εLUMO(ZnF)εHOMO(ZnF).
(11)

Once more, the four quantities illustrated in relations (10) and (11) should be ideally null and we can take their sum as a parameter to choose between the previously found combinations of α and μ parameters. The aug-cc-pVQZ and aug-cc-pVQZ-PP (Cu and Zn) basis sets are employed and the calculations are done at the experimental geometries of CuH and ZnF.46 Hence, the functional that provides the minimum sum value (Δε2a + Δε2b + Δε3a + Δε3b = 0.80 eV) is named from now on CAM-QTP-02 (α = 0.28, β = 0.72, and μ = 0.335). In this case, Δε1a and Δε1b are around 0.71 eV. Table II summarizes the details of the Quantum Theory Project (QTP) functionals that are investigated in this study. We will return to the discussion regarding these quantities and the parameterization process in Sec. III C.

TABLE II.

Description of QTP functionals.a

NameαβμCorrelation functional
CAM-QTP-00b 0.54 0.37 0.29 0.20 VWN5 + 0.80 LYP 
CAM-QTP-01c 0.23 0.77 0.31 0.20 VWN5 + 0.80 LYP 
CAM-QTP-02 0.28 0.72 0.335 LYP 
LC-QTP 0.00 1.00 0.475 LYP 
NameαβμCorrelation functional
CAM-QTP-00b 0.54 0.37 0.29 0.20 VWN5 + 0.80 LYP 
CAM-QTP-01c 0.23 0.77 0.31 0.20 VWN5 + 0.80 LYP 
CAM-QTP-02 0.28 0.72 0.335 LYP 
LC-QTP 0.00 1.00 0.475 LYP 
a

The exchange functional used in the mixing with the Hartree-Fock exchange according to the α, β, and μ parameters is Becke88 (see Ref. 12 for the details regarding the parameter notation and expressions).

b

From Ref. 11.

c

From Ref. 13.

Initially, the results obtained for equilibrium geometries, harmonic vibrational frequencies, and dipole moments are discussed in this study and compared with available experimental data,46–49 as can be seen in Tables S1–S4 (supplementary material). The aug-cc-pVQZ basis set is selected to perform these calculations and the molecules considered are LiH, CH, NH, NH3, OH, H2O, HF, LiF, CN, HCN, CO, H2CO, CH3OH, N2H4, NO, H2O2, H2, Li2, N2, O2, and F2. Hence, the mean absolute deviation (MAD) and the maximum absolute deviation (MAX) values obtained with respect to experimental data are compared.

In general, the two new functionals introduced in this investigation, LC-QTP and CAM-QTP-02, provided results for these properties that are better than those from CAM-QTP-00 and worse than the ones from CAM-QTP-01. However, CAM-B3LYP is more accurate than all the QTP functionals in this comparison. Thus, the MAD (MAX) values given by CAM-QTP-02 are 0.017 (0.070) Å, 2.3 (7.4)°, 91 (254) cm−1, and 0.13 (0.37) D, respectively, for bond lengths, angles, vibrational frequencies, and dipole moments. In the same sequence, LC-QTP resulted in MAD (MAX) parameters equal to 0.017 (0.071) Å, 2.5 (7.5)°, 91 (249) cm−1, and 0.14 (0.41) D, while CAM-B3LYP provided 0.011 (0.046) Å, 2.0 (6.8)°, 48 (182) cm−1, and 0.08 (0.20) D. CAM-QTP-01 performed better than the other QTP functionals for these properties probably because it is the member of this family that maintains the closest resemblance to the original CAM-B3LYP in terms of the parameterization, with a noticeable difference seen only in the percentage of exact exchange included for long-range interelectronic interactions.

First, we will investigate the performance of the new functionals with respect to self-interaction error (SIE) in a one-electron molecular species, H2+. Thus, the potential energy curve (PEC) of this system obtained with the aug-cc-pVTZ basis set is presented in Fig. 1, along with other previous results calculated in Ref. 15 using the same basis set for HF, CAM-QTP-00, CAM-QTP-01, and rCAM-B3LYP methods. In this case, HF is able to provide the exact PEC and the correct dissociation limit. Nevertheless, traditional functionals such as BLYP and B3LYP give wrong PECs with respect to HF, especially at the dissociation limit.15 Thus, one can see that CAM-QTP-00 results in lower energies than HF near the equilibrium position. Next, the energies from rCAM-B3LYP and CAM-QTP-01 are nearly the same in this region, providing better values compared to HF than those from LC-QTP and CAM-QTP-02.

FIG. 1.

Energy of H2+ relative to the exact dissociation limit (−0.500 a.u.) against internuclear distances, as calculated with CAM-QTP-02 and LC-QTP by using aug-cc-pVTZ basis sets, together with previous results from Ref. 15.

FIG. 1.

Energy of H2+ relative to the exact dissociation limit (−0.500 a.u.) against internuclear distances, as calculated with CAM-QTP-02 and LC-QTP by using aug-cc-pVTZ basis sets, together with previous results from Ref. 15.

Close modal

On the other hand, turning our attention to the dissociation limit, the functional with the best performance in this region is now LC-QTP, followed by r-CAM-B3LYP, CAM-QTP-02, CAM-QTP-00, and CAM-QTP-01, in this order. Therefore, the errors with respect to HF results at 10.0 Å are −14.30, −13.70, −9.92, −9.79, and −8.30 kcal mol−1, respectively, for CAM-QTP-01, CAM-QTP-00, CAM-QTP-02, rCAM-B3LYP, and LC-QTP. Surprisingly, LC-QTP surpassed rCAM-B3LYP as the best functional at the dissociation limit of this one-electron system.

Next, CAM-QTP-02 and LC-QTP are evaluated for localization/delocalization errors in a many-electron atom, which are investigated by means of the deviations found for energy values of systems with a fractional number of electrons from an expected straight line that ideally would link a neutral species to its cation and anion, all in the ground state.17 Thus, these calculations are performed here for the fluorine atom using again the aug-cc-pVQZ basis set and the results obtained are displayed in Fig. 2.

FIG. 2.

Energy deviations obtained for fluorine with fractional occupations with respect to the ideal straight line linking the F atom to F+ and F against the number of electrons, as calculated with the aug-cc-pVQZ basis set.

FIG. 2.

Energy deviations obtained for fluorine with fractional occupations with respect to the ideal straight line linking the F atom to F+ and F against the number of electrons, as calculated with the aug-cc-pVQZ basis set.

Close modal

First, one can easily notice that CAM-B3LYP presents the worst results among the DFT methods compared, with much smaller energies than the ones expected for the fractional occupation number systems. CAM-QTP-01 shows an improvement over CAM-B3LYP but still seems to be affected by delocalization errors in a larger degree than the other functionals considered. LC-QTP is the next ranked functional in this trial. Interestingly, CAM-QTP-02 shows a behavior almost equal to that obtained with rCAM-B3LYP, which was developed to minimize N-SIE,17 and these two functionals are ranked here as the best ones for electron attachment to fluorine. From all these functionals, CAM-QTP-00 is the one with the smallest signs of delocalization errors for electron removal of fluorine. However, CAM-QTP-00 exhibits significantly larger energies than the ones expected for systems with fractional occupations during electron attachment to this atom.

Now we return to the discussion about the parameterization process. Figure 3 illustrates the energy results of CAM-QTP-02/aug-cc-pVQZ calculations with fractional occupation numbers (ONs) between the two pairs of systems used to parameterize the functionals presented here: (1) H2O+ and H2O (left) and (2) OH and OH (right). Accordingly, the experimental geometries of water and hydroxyl are considered in each case.46 As one can see, the points on the left tend to be arranged below the exact straight line linking water (ON = 0.0) and its cation (ON = −1.0). On the other hand, the points on the right show a trend to stay above the straight line between hydroxyl (ON = 0.0) and its anion (ON = 1.0). A simple numerical exercise shows that the slope of the curve obtained for CAM-QTP-02 in the closest region to the water cation is equal to −εLUMO(H2O+) while the slope determined near water provides −εHOMO(H2O). Accordingly, the same analysis resulted in slopes around −εLUMO(OH) and −εHOMO(OH), respectively, in the curve regions nearby hydroxyl and its anion. This is certainly in agreement with Janak’s theorem.43 Hence, by requiring the minimization of Δε1a and Δε1b to similar values during the parameterization, we are in fact assuring similar slopes at the beginning and at the end of each of both the curves seen in Fig. 3. Thus, the parameterization process proposed can also be seen as an alternative procedure to attain the closest behavior to the expected straight lines simultaneously for electron removal from water and electron attachment to hydroxyl.

FIG. 3.

Relative energy with respect to water (left) and hydroxyl (right) from CAM-QTP-02/aug-cc-pVQZ calculations for systems with fractional occupation numbers (ONs). In this case, ON = 0.0 represents H2O and OH while ON values of −1.0 and 1.0 are associated, respectively, with H2O+ and OH.

FIG. 3.

Relative energy with respect to water (left) and hydroxyl (right) from CAM-QTP-02/aug-cc-pVQZ calculations for systems with fractional occupation numbers (ONs). In this case, ON = 0.0 represents H2O and OH while ON values of −1.0 and 1.0 are associated, respectively, with H2O+ and OH.

Close modal

The performance of CAM-QTP-02 and LC-QTP in providing vertical IPs by means of the negative of eigenvalues attained for appropriate occupied orbitals (IPi = −εi,occ) is evaluated here by using the same prescription introduced in Ref. 13, that is, by means of calculations done with the aug-cc-pVTZ basis sets at the respective theoretical equilibrium geometries found with each DFT method. Moreover, the same set of reference vertical IPs is also considered for these comparisons, that is, the set constituted by 401 experimental values of valence IPs determined for 63 molecules, which was previously investigated by Chong and collaborators.50 The results encountered for MADs are presented in Table III together with previous data from Ref. 13.

TABLE III.

Mean absolute deviations (MADs) for vertical ionization potentials obtained from occupied orbital eigenvalues in calculations done with the aug-cc-pVTZ basis sets at the theoretical equilibrium geometries with respect to the experimental values (in eV).50 

CAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTP
MAD (all) 0.93 0.41 0.56 0.71 
MAD (HOMO) 0.32 0.24 0.23 0.39 
CAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTP
MAD (all) 0.93 0.41 0.56 0.71 
MAD (HOMO) 0.32 0.24 0.23 0.39 
a

From Ref. 13.

The values seen in Table III show that the MADs for vertical IPs provided by the new functionals for all the 401 reference values are between those from CAM-QTP-01 and CAM-QTP-00. This is quite impressive considering that CAM-QTP-01 was only inferior to M1151 (MAD = 0.35 eV) in the previous evaluation test conducted with 43 exchange-correlation functionals and also taking into account that CAM-QTP-00 was ranked in the eighth position at that time.13 Furthermore, as seen in Table III and in Fig. 4, the performance of CAM-QTP-02 in predicting the vertical IPs for HOMO in these 63 molecules is outstanding, surpassing CAM-QTP-01 as the most accurate functional in that aspect.13 By the way, although LC-QTP is less accurate than the other functionals in Table III for HOMO IPs, it is still superior to 39 functionals used in the previous test.13 These IPs for HOMO are presented in Table S5 (supplementary material). Hence, we can conclude that the new functionals maintain a high-level prediction capability for vertical IPs in terms of the respective eigenvalues and this finding is even more relevant as one considers that no experimental IP or EA values are used in their parameterization process, which is non-empirical.

FIG. 4.

Vertical ionization potentials as given by the negative of HOMO eigenvalues obtained in CAM-QTP-02/aug-cc-pVTZ calculations against experimental values (the dotted line represents the exact agreement).

FIG. 4.

Vertical ionization potentials as given by the negative of HOMO eigenvalues obtained in CAM-QTP-02/aug-cc-pVTZ calculations against experimental values (the dotted line represents the exact agreement).

Close modal

Next, the vertical excitation energies are evaluated in this work. Again, the same procedure employed in Ref. 13 is followed for comparative reasons. Thus, the geometries of a set constituted by 11 small organic compounds (ethylene, isobutene, trans-1,3-butadiene, H2CO, acetaldehyde, acetone, pyridine, pyrazine, pyrimidine, pyridazine, and s-tetrazine) are retrieved from the study conducted by Caricato et al.52 and the same basis set recommended by these authors, 6-311(3+,3+)G**, is used in these calculations within the TD-DFT approach. The reference experimental values for the vertical excitation energies of 69 states from these molecules (39 Rydberg and 30 valence states) are also given in Ref. 52.

Table S6 (supplementary material) shows the results found and Table IV illustrates the comparison of MADs provided by the new functionals investigated together with previous results found for CAM-QTP-00, CAM-QTP-01, and EOM-CCSD.13,52 First, one can see that CAM-QTP-01, CAM-QTP-02, and LC-QTP show a similar performance as compared to EOM-CCSD for valence states, with MADs between 0.46 and 0.49 eV for this subset. On the other hand, CAM-QTP-00 is not so successful for valence excitations, with a MAD of 0.71 eV. In addition, the best results of excitations to Rydberg states are achieved with EOM-CCSD, with a MAD of 0.11 eV. In the sequence, CAM-QTP-01, CAM-QTP-02, LC-QTP, and CAM-QTP-00 presented MADs of 0.17, 0.26, 0.42, and 0.52 eV for this subset of Rydberg states. The nice performance of CAM-QTP-01 for this property is expected, partly, as some of these excitation energies are employed in its parameterization process.13 However, by the comparison with CAM-QTP-00, we notice a clear improvement in the description of vertical excitation energies by the new functionals. In the case of excitation energies to Rydberg states for this group of molecules, CAM-QTP-02 is only inferior to three functionals in the set of nearly 60 different exchange-correlation functionals previously investigated.13 

TABLE IV.

Mean absolute deviations (MADs) for vertical excitation energies obtained in calculations with the 6-311(3+,3+)G** basis sets at the geometries presented in Ref. 52 with respect to the experimental values presented by the same authors (in eV).

CAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPEOM-CCSDb
MAD (all) 0.60 0.29 0.36 0.45 0.27 
MAD (valence) 0.71 0.46 0.49 0.48 0.47 
MAD (Rydberg) 0.52 0.17 0.26 0.42 0.11 
CAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPEOM-CCSDb
MAD (all) 0.60 0.29 0.36 0.45 0.27 
MAD (valence) 0.71 0.46 0.49 0.48 0.47 
MAD (Rydberg) 0.52 0.17 0.26 0.42 0.11 
a

From Ref. 13.

b

From Ref. 52.

Another investigation of excitation energies is also performed with the permanganate ion, MnO4. In this case, TD-DFT calculations are done at the experimental Mn-O distance of this Td species, 1.629 Å,53 with cc-pVDZ and aug-cc-pVTZ basis sets. Therefore, the excitation energies obtained are compared with experimental data in Table V.53 As one can see, there are slight differences, but the nominally best results are provided by CAM-QTP-02, followed by LC-QTP.

TABLE V.

Excitation energies in TD-DFT calculations at the experimental geometry of the permanganate ion, MnO4,a compared with experimental data (in eV).

Basis setStatesCAM-B3LYPCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPExpt.a
cc-pVDZ a1T2 2.68 2.86 2.66 2.68 2.59 2.27 
 b1T2 3.81 3.53 3.76 3.73 3.72 3.47 
 c1T2 4.23 3.93 4.14 4.09 4.08 3.99 
 d1T2 5.67 5.07 5.62 5.52 5.70 5.45 
 MAD 0.30 0.27 0.25 0.21 0.23  
aug-cc-pVTZ a1T2 2.77 2.98 2.77 2.79 2.70 2.27 
 b1T2 3.92 3.71 3.90 3.88 3.87 3.47 
 c1T2 4.30 4.04 4.21 4.16 4.16 3.99 
 d1T2 5.70 5.14 5.65 5.56 5.74 5.45 
 MAD 0.38 0.33 0.34 0.30 0.32  
Basis setStatesCAM-B3LYPCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPExpt.a
cc-pVDZ a1T2 2.68 2.86 2.66 2.68 2.59 2.27 
 b1T2 3.81 3.53 3.76 3.73 3.72 3.47 
 c1T2 4.23 3.93 4.14 4.09 4.08 3.99 
 d1T2 5.67 5.07 5.62 5.52 5.70 5.45 
 MAD 0.30 0.27 0.25 0.21 0.23  
aug-cc-pVTZ a1T2 2.77 2.98 2.77 2.79 2.70 2.27 
 b1T2 3.92 3.71 3.90 3.88 3.87 3.47 
 c1T2 4.30 4.04 4.21 4.16 4.16 3.99 
 d1T2 5.70 5.14 5.65 5.56 5.74 5.45 
 MAD 0.38 0.33 0.34 0.30 0.32  
a

From Ref. 53.

Another important property to judge the performance of new DFT functionals is the classical reaction barrier height (V). These quantities are considered following again the suggestions of Ref. 13 and for the same reactions. Thus, the aug-cc-pVTZ basis sets are chosen for this study and the geometries of each species are optimized. The reference values for two groups of reactions, hydrogen transfer (HT) and non-hydrogen transfer (NHT), are retrieved from the databases provided by Truhlar and co-workers.54,55 Hence, Tables VI and VII show the results found (without zero-point corrections) together with the respective values calculated by Jin and Bartlett.13 

TABLE VI.

Classical barrier heights (V) for forward (f) and reverse (r) hydrogen transfer reactions according to aug-cc-pVTZ calculations at the respective theoretical equilibrium geometries (in kcal mol−1).

CAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPReference 54 
ReactionsVfVrVfVrVfVrVfVrVfVrVfVr
Cl + H2 ↔ HCl + H 5.5 1.0 9.0 3.2 4.1 1.6 4.1 2.3 4.7 2.9 8.7 5.7 
OH + H2 ↔ H + H21.3 15.7 7.0 17.8 0.7 17.4 1.2 18.3 1.0 20.7 5.7 21.8 
CH3 + H2 ↔ H + CH4 8.9 11.2 11.4 13.4 8.0 12.4 8.1 13.2 8.7 14.2 12.1 15.3 
OH + CH4 ↔ CH3 + H23.3 15.3 10.4 19.3 3.2 15.5 4.2 16.1 3.6 17.8 6.7 19.6 
H + CH3OH ↔ CH2OH + H2 5.3 13.4 8.4 15.3 6.4 12.5 7.3 12.4 8.1 13.3 7.3 13.3 
H + H2 ↔ H2 + H 4.8 4.8 6.8 6.8 5.0 5.0 5.4 5.4 5.9 5.9 9.6 9.6 
OH + NH3 ↔ H2O + NH2 −0.1 9.5 9.4 16.8 0.8 10.2 2.3 11.4 1.9 12.1 3.2 12.7 
HCl + CH3 ↔ Cl + CH4 −0.4 6.3 2.0 9.8 −0.8 6.0 −0.5 6.5 0.3 7.5 1.7 7.9 
OH + C2H6 ↔ H2O + C2H5 0.7 17.2 7.8 20.4 0.7 17.5 1.7 17.9 1.2 19.8 3.4 19.9 
OH + CH3 ↔ O + CH4 5.4 8.5 9.2 16.8 5.2 8.9 5.8 10.1 7.0 8.7 8.1 13.7 
H + PH3 ↔ PH2 + H2 0.5 23.8 1.8 24.8 0.7 22.1 1.1 21.6 1.5 22.0 3.1 23.2 
H + HCl ↔ HCl + H 15.0 15.0 18.3 18.3 15.9 15.9 16.7 16.7 17.8 17.8 18.0 18.0 
H + H2S ↔ H2 + HS 0.9 16.6 2.5 18.7 1.2 14.9 1.7 14.6 2.2 15.2 3.5 17.3 
O + HCl ↔ OH + Cl 4.8 8.4 14.0 14.2 5.8 9.0 7.3 9.9 6.8 12.3 9.8 10.4 
CH4 + NH ↔ NH2 + CH3 18.6 7.3 24.0 10.8 18.6 7.0 19.4 7.5 19.5 8.6 22.4 8.0 
C2H6 + NH ↔ NH2 + C2H5 15.6 8.7 21.3 11.9 15.6 8.3 16.4 8.8 16.5 9.9 18.3 7.5 
C2H6 + NH2 ↔ NH3 + C2H5 9.9 16.9 15.2 20.4 9.5 16.9 10.2 17.4 10.4 18.8 10.4 17.4 
NH2 + CH4 ↔ CH3 + NH3 12.4 14.9 17.5 18.9 12.0 14.9 12.7 15.5 12.9 16.9 14.5 17.8 
Mean signed error −2.7 1.3 −2.7 −2.1 −1.4   
Mean absolute error 2.8 2.3 2.7 2.2 1.8   
Maximum absolute error 6.1 6.2 5.0 4.6 5.0   
CAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPReference 54 
ReactionsVfVrVfVrVfVrVfVrVfVrVfVr
Cl + H2 ↔ HCl + H 5.5 1.0 9.0 3.2 4.1 1.6 4.1 2.3 4.7 2.9 8.7 5.7 
OH + H2 ↔ H + H21.3 15.7 7.0 17.8 0.7 17.4 1.2 18.3 1.0 20.7 5.7 21.8 
CH3 + H2 ↔ H + CH4 8.9 11.2 11.4 13.4 8.0 12.4 8.1 13.2 8.7 14.2 12.1 15.3 
OH + CH4 ↔ CH3 + H23.3 15.3 10.4 19.3 3.2 15.5 4.2 16.1 3.6 17.8 6.7 19.6 
H + CH3OH ↔ CH2OH + H2 5.3 13.4 8.4 15.3 6.4 12.5 7.3 12.4 8.1 13.3 7.3 13.3 
H + H2 ↔ H2 + H 4.8 4.8 6.8 6.8 5.0 5.0 5.4 5.4 5.9 5.9 9.6 9.6 
OH + NH3 ↔ H2O + NH2 −0.1 9.5 9.4 16.8 0.8 10.2 2.3 11.4 1.9 12.1 3.2 12.7 
HCl + CH3 ↔ Cl + CH4 −0.4 6.3 2.0 9.8 −0.8 6.0 −0.5 6.5 0.3 7.5 1.7 7.9 
OH + C2H6 ↔ H2O + C2H5 0.7 17.2 7.8 20.4 0.7 17.5 1.7 17.9 1.2 19.8 3.4 19.9 
OH + CH3 ↔ O + CH4 5.4 8.5 9.2 16.8 5.2 8.9 5.8 10.1 7.0 8.7 8.1 13.7 
H + PH3 ↔ PH2 + H2 0.5 23.8 1.8 24.8 0.7 22.1 1.1 21.6 1.5 22.0 3.1 23.2 
H + HCl ↔ HCl + H 15.0 15.0 18.3 18.3 15.9 15.9 16.7 16.7 17.8 17.8 18.0 18.0 
H + H2S ↔ H2 + HS 0.9 16.6 2.5 18.7 1.2 14.9 1.7 14.6 2.2 15.2 3.5 17.3 
O + HCl ↔ OH + Cl 4.8 8.4 14.0 14.2 5.8 9.0 7.3 9.9 6.8 12.3 9.8 10.4 
CH4 + NH ↔ NH2 + CH3 18.6 7.3 24.0 10.8 18.6 7.0 19.4 7.5 19.5 8.6 22.4 8.0 
C2H6 + NH ↔ NH2 + C2H5 15.6 8.7 21.3 11.9 15.6 8.3 16.4 8.8 16.5 9.9 18.3 7.5 
C2H6 + NH2 ↔ NH3 + C2H5 9.9 16.9 15.2 20.4 9.5 16.9 10.2 17.4 10.4 18.8 10.4 17.4 
NH2 + CH4 ↔ CH3 + NH3 12.4 14.9 17.5 18.9 12.0 14.9 12.7 15.5 12.9 16.9 14.5 17.8 
Mean signed error −2.7 1.3 −2.7 −2.1 −1.4   
Mean absolute error 2.8 2.3 2.7 2.2 1.8   
Maximum absolute error 6.1 6.2 5.0 4.6 5.0   
a

Values without zero-point corrections provided by the authors of Ref. 13.

TABLE VII.

Classical barrier heights (V) for forward (f) and reverse (r) non-hydrogen transfer reactions (in kcal mol−1) according to aug-cc-pVTZ calculations at the respective theoretical equilibrium geometries.

CAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPReference 55 
ReactionsVfVrVfVrVfVrVfVrVfVrVfVr
H + NO2 ↔ OH + N2 13.07 77.60 16.94 94.33 13.69 78.65 14.51 80.95 15.27 78.95 18.14 83.22 
H + FH ↔ HF + H 33.19 33.19 40.35 40.35 34.68 34.68 35.89 35.89 36.82 36.82 42.18 42.18 
H + HCl ↔ HCl + H 15.03 15.03 18.35 18.35 15.94 15.94 16.65 16.65 17.83 17.83 18.00 18.00 
H + FCH3 ↔ HF + CH3 25.17 50.63 32.71 59.17 27.83 51.40 29.49 52.36 30.58 53.21 30.38 57.02 
H + F2 ↔ HF + F −0.33 102.24 1.91 118.74 0.07 103.21 0.54 105.33 0.85 101.32 2.27 106.18 
CH3 + FCl ↔ CH3F + Cl 3.27 57.83 9.15 68.61 5.23 61.28 6.19 63.22 7.60 64.40 7.43 60.17 
F + CH3F ↔ FCH3 + F −1.40 −1.40 1.82 1.82 −1.23 −1.23 −0.77 −0.77 0.09 0.09 −0.34 −0.34 
F⋯CH3F ↔ FCH3⋯F 12.13 12.13 16.09 16.09 13.48 13.48 14.26 14.26 14.77 14.77 13.38 13.38 
Cl + CH3Cl ↔ ClCH3 + Cl 2.35 2.35 4.62 4.62 3.82 3.82 4.25 4.25 6.63 6.63 3.10 3.10 
Cl⋯CH3Cl ↔ ClCH3⋯Cl 12.22 12.22 15.03 15.03 14.48 14.48 15.12 15.12 17.11 17.11 13.61 13.61 
F + CH3Cl ↔ FCH3 + Cl −13.44 20.59 −12.73 26.01 −13.17 21.89 −13.09 22.82 −10.59 22.86 −12.54 20.11 
F⋯CH3Cl ↔ FCH3⋯Cl 2.13 29.29 3.62 35.18 3.34 31.49 3.72 32.64 5.29 32.63 2.89 29.62 
OH + CH3F ↔ HOCH3 + F −3.14 17.71 0.16 21.96 −2.87 18.59 −2.39 19.30 −1.78 21.13 −2.78 17.33 
OH⋯CH3F ↔ HOCH3⋯F 10.09 48.17 14.20 53.03 11.62 50.98 12.44 52.12 12.86 53.19 10.96 47.20 
H + N2 ↔ HN2 9.38 12.24 12.06 14.96 9.87 13.10 10.58 13.36 11.35 13.22 14.69 10.72 
H + CO ↔ HCO 0.91 25.59 1.77 25.61 0.82 25.61 1.07 25.24 1.44 25.37 3.17 22.68 
H + C2H4 ↔ CH3CH2 0.48 43.71 0.55 47.48 0.22 44.85 0.36 45.17 0.83 45.64 1.72 41.75 
CH3 + C2H4 ↔ CH3CH2CH2 6.69 33.12 7.39 37.01 5.82 36.36 5.78 37.36 6.81 38.50 6.85 32.97 
HCN ↔ HNC 47.06 33.80 48.13 36.26 46.75 34.19 46.89 34.65 47.69 35.16 48.16 33.11 
Mean signed error −1.90 2.46 −0.79 −0.05 0.71   
Mean absolute error 2.38 3.02 2.22 2.15 2.54   
Maximum absolute error 8.99 12.56 7.50 6.29 5.99   
CAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02LC-QTPReference 55 
ReactionsVfVrVfVrVfVrVfVrVfVrVfVr
H + NO2 ↔ OH + N2 13.07 77.60 16.94 94.33 13.69 78.65 14.51 80.95 15.27 78.95 18.14 83.22 
H + FH ↔ HF + H 33.19 33.19 40.35 40.35 34.68 34.68 35.89 35.89 36.82 36.82 42.18 42.18 
H + HCl ↔ HCl + H 15.03 15.03 18.35 18.35 15.94 15.94 16.65 16.65 17.83 17.83 18.00 18.00 
H + FCH3 ↔ HF + CH3 25.17 50.63 32.71 59.17 27.83 51.40 29.49 52.36 30.58 53.21 30.38 57.02 
H + F2 ↔ HF + F −0.33 102.24 1.91 118.74 0.07 103.21 0.54 105.33 0.85 101.32 2.27 106.18 
CH3 + FCl ↔ CH3F + Cl 3.27 57.83 9.15 68.61 5.23 61.28 6.19 63.22 7.60 64.40 7.43 60.17 
F + CH3F ↔ FCH3 + F −1.40 −1.40 1.82 1.82 −1.23 −1.23 −0.77 −0.77 0.09 0.09 −0.34 −0.34 
F⋯CH3F ↔ FCH3⋯F 12.13 12.13 16.09 16.09 13.48 13.48 14.26 14.26 14.77 14.77 13.38 13.38 
Cl + CH3Cl ↔ ClCH3 + Cl 2.35 2.35 4.62 4.62 3.82 3.82 4.25 4.25 6.63 6.63 3.10 3.10 
Cl⋯CH3Cl ↔ ClCH3⋯Cl 12.22 12.22 15.03 15.03 14.48 14.48 15.12 15.12 17.11 17.11 13.61 13.61 
F + CH3Cl ↔ FCH3 + Cl −13.44 20.59 −12.73 26.01 −13.17 21.89 −13.09 22.82 −10.59 22.86 −12.54 20.11 
F⋯CH3Cl ↔ FCH3⋯Cl 2.13 29.29 3.62 35.18 3.34 31.49 3.72 32.64 5.29 32.63 2.89 29.62 
OH + CH3F ↔ HOCH3 + F −3.14 17.71 0.16 21.96 −2.87 18.59 −2.39 19.30 −1.78 21.13 −2.78 17.33 
OH⋯CH3F ↔ HOCH3⋯F 10.09 48.17 14.20 53.03 11.62 50.98 12.44 52.12 12.86 53.19 10.96 47.20 
H + N2 ↔ HN2 9.38 12.24 12.06 14.96 9.87 13.10 10.58 13.36 11.35 13.22 14.69 10.72 
H + CO ↔ HCO 0.91 25.59 1.77 25.61 0.82 25.61 1.07 25.24 1.44 25.37 3.17 22.68 
H + C2H4 ↔ CH3CH2 0.48 43.71 0.55 47.48 0.22 44.85 0.36 45.17 0.83 45.64 1.72 41.75 
CH3 + C2H4 ↔ CH3CH2CH2 6.69 33.12 7.39 37.01 5.82 36.36 5.78 37.36 6.81 38.50 6.85 32.97 
HCN ↔ HNC 47.06 33.80 48.13 36.26 46.75 34.19 46.89 34.65 47.69 35.16 48.16 33.11 
Mean signed error −1.90 2.46 −0.79 −0.05 0.71   
Mean absolute error 2.38 3.02 2.22 2.15 2.54   
Maximum absolute error 8.99 12.56 7.50 6.29 5.99   
a

Values without zero-point corrections provided by the authors of Ref. 13.

Table VI indicates small differences for HT reactions, but CAM-QTP-02 and LC-QTP provide the smallest MAD (between 1.8 and 2.2 kcal mol−1) and MAX (between 4.6 and 5.0 kcal mol−1) values. Moreover, CAM-QTP-00 (MAD = 2.3 kcal mol−1) and CAM-QTP-01 (MAD = 2.7 kcal mol−1) show only a slight improvement over CAM-B3LYP (MAD = 2.8 kcal mol−1). On average, CAM-QTP-00 is the only functional that overestimates HT barrier heights, while the other DFT methods tend to underestimate these energies. However, CAM-QTP-02 and LC-QTP present less underestimated barrier values than those from CAM-QTP-01 and CAM-B3LYP, as judged by the mean signed error.

Next, according to Table VII, the most accurate functional for NHT reactions is CAM-QTP-02 (MAD and MAX of 2.15 and 6.29 kcal mol−1, respectively). Now, LC-QTP (MAD = 2.54 kcal mol−1) seems only superior to CAM-QTP-00 (MAD = 3.02 kcal mol−1), while CAM-QTP-01 (MAD = 2.22 kcal mol−1) modestly outperforms CAM-B3LYP (MAD = 2.38 kcal mol−1). Moreover, CAM-QTP-00 and LC-QTP show an overestimation trend for these NHT barriers while CAM-B3LYP and CAM-QTP-01 underestimate such results on average. Finally, the mean signed error for CAM-QTP-02 (−0.05 kcal mol−1) is now very close to the optimal null value.

The data achieved in TD-DFT investigations of charge-transfer (CT) excitation energies are now discussed. First, Table VIII illustrates the results found for some complexes between aromatic (Ar) compounds and tetracyanoethylene (TCNE) with fixed inter-planar distances, which were also considered in Ref. 14. Accordingly, the geometries are taken from Ref. 20 and the cc-pVTZ basis sets are used in these calculations. The reference experimental values20,56 are also seen in the table. In this case, the best functional is CAM-QTP-01, with a MAD of 0.08 eV, followed by CAM-QTP-00 (MAD = 0.14 eV). The performance of CAM-QTP-02 is also excellent, with a MAD equals to 0.21 eV. In addition, LC-QTP provides the worst results among the QTP family, with a MAD of 0.44 eV. However, it is important to emphasize that traditional DFT methods are usually much inferior for charge transfer excitation energies. For example, CAM-B3LYP and B3LYP provide MADs of 0.72 and 1.59 eV, respectively, for this set.14 

TABLE VIII.

Excitation energies for aromatic(Ar)⋯tetracyanoethylene complexes with fixed inter-planar distances as calculated with TD-DFT and cc-pVTZ basis sets (in eV).a

ArCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPExpt.b
Benzene 3.79 3.69 3.85 4.08 3.59 
Toluene 3.46 3.35 3.50 3.73 3.36 
o-xylene 3.21 3.10 3.25 3.47 3.15 
Naphthalene 2.71 2.65 2.80 3.03 2.60 
 3.48 3.42 3.57 3.79 3.23 
MAD 0.14 0.08 0.21 0.44  
MAX 0.25 0.19 0.34 0.56  
ArCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPExpt.b
Benzene 3.79 3.69 3.85 4.08 3.59 
Toluene 3.46 3.35 3.50 3.73 3.36 
o-xylene 3.21 3.10 3.25 3.47 3.15 
Naphthalene 2.71 2.65 2.80 3.03 2.60 
 3.48 3.42 3.57 3.79 3.23 
MAD 0.14 0.08 0.21 0.44  
MAX 0.25 0.19 0.34 0.56  
a

The geometries were retrieved from Ref. 20.

b

From Refs. 20 and 56.

Another evaluation test that can be carried out for charge-transfer excitation energies is the study of their change along different donor-acceptor distances.14 Hence, we also carried out TD-DFT calculations for the complex between ethylene and tetrafluoroethylene using the same geometries provided by Zhao and Truhlar for the monomers57 while varying the inter-planar distance of this C2v complex (RDA). Once more, in agreement with Ref. 14, the cc-pVTZ basis set is used for this investigation. Figure 5 illustrates the results obtained with the new functionals together with previous data provided by the authors of Ref. 14 for the other methods considered. The reference results in this case are those from the similarity transformed equation-of-motion coupled cluster theory with single and double substitutions, STEOM-CCSD.58 Thus, one can see that the best agreement between TD-DFT and the reference values is now given by LC-QTP, followed by CAM-QTP-00, CAM-QTP-02, and CAM-QTP-01, in this order. Once more, functionals such as B3LYP and CAM-B3LYP are much worse in this case, largely underestimating the CT energies.14 

FIG. 5.

Charge-transfer excitation energies along the inverse of inter-planar distances (RDA) for the C2H4⋯C2F4 complex (C2v symmetry), as obtained in TD-DFT calculations with CAM-QTP-02 and LC-QTP, using cc-pVTZ basis sets, and previous results from Ref. 14. The point when 1/RDA = 0 refers to IP(C2F4)–EA(C2H4)19 and is obtained here as −εHOMO(C2F4) + εLUMO(C2H4). The corresponding linear regression lines are also displayed.

FIG. 5.

Charge-transfer excitation energies along the inverse of inter-planar distances (RDA) for the C2H4⋯C2F4 complex (C2v symmetry), as obtained in TD-DFT calculations with CAM-QTP-02 and LC-QTP, using cc-pVTZ basis sets, and previous results from Ref. 14. The point when 1/RDA = 0 refers to IP(C2F4)–EA(C2H4)19 and is obtained here as −εHOMO(C2F4) + εLUMO(C2H4). The corresponding linear regression lines are also displayed.

Close modal

The last property investigated here is the electron affinity (EA). Hence, we will analyze the accuracy of QTP functionals in providing vertical EAs by means of the negative of eigenvalues obtained for appropriate unoccupied orbitals (EAa = −εa,unocc). Thus, the aug-cc-pVQZ or aug-cc-pVQZ-PP (when available) basis sets (except by K and Ca atoms) are used for these calculations in atoms and in molecules at their respective theoretical equilibrium geometries. The reference values are determined in pairs of calculations done with the Coupled Cluster formalism including iterative single, double, and approximate triple excitations (CCSDT-3)59 based on restricted (closed or open shell) Hartree-Fock reference wavefunctions using aug-cc-pVQZ(-PP) basis sets for the neutral and anionic systems of interest, that is, EAref = E(neutral) − E(anion) = ΔE(CCSDT-3). In this case, the fixed experimental geometries of the neutral molecules46,47 are used for these reference EA determinations.

Table IX illustrates the results obtained for main group atoms between He and Kr. The available experimental EAs60 are also displayed for comparison. First, it is important to notice that the agreement between the reference values from ΔE(CCSDT-3) calculations and the experimental data is excellent, with a largest deviation of only 0.14 eV (nitrogen). Next, we will discuss the results obtained from LUMO eigenvalues and DFT methods. Thus, one can see that CAM-B3LYP provides the worst values for this property, with a MAD (MAX) of 1.02 (2.31) eV with respect to the reference data. In all cases, the calculations done with this functional result in larger electron affinities, that is, LUMO eigenvalues from CAM-B3LYP largely overestimate the EAs. The best DFT functional in this case is CAM-QTP-02, with a MAD (MAX) of only 0.16 (0.61) eV. Next, CAM-QTP-00 and LC-QTP show a similar performance, with MADs (MAXs) around 0.19-0.21 (0.74–0.81) eV. Finally, CAM-QTP-01 is inferior to the other functionals of the QTP family for atomic EAs, exhibiting a MAD (MAX) of 0.27 (0.86) eV.

TABLE IX.

Electron affinities of atoms as obtained from the negative of virtual eigenvalues in calculations done with aug-cc-pVQZ(-PP) basis sets with respect to the reference ΔE(CCSDT-3) results and available experimental values (in eV).

AtomsCAM-B3LYPCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPΔE(CCSDT-3)aExpt.b
He −1.95 −2.30 −2.40 −2.43 −2.45 −2.63  
Li 1.04 0.83 0.77 0.58 0.56 0.62 0.62 
Be 0.32 −0.16 −0.21 −0.26 −0.27 −0.27  
1.26 0.26 0.32 0.14 0.05 0.24 0.28 
2.66 1.16 1.51 1.21 1.14 1.24 1.26 
1.52 −0.04 0.34 0.06 0.07 −0.21 −0.07 
3.37 1.17 2.04 1.63 1.83 1.42 1.46 
5.71 2.89 4.26 3.73 4.21 3.40 3.40 
Ne −3.77 −4.54 −4.60 −4.67 −4.75 −5.28  
Na 1.12 0.82 0.75 0.59 0.56 0.54 0.55 
Mg 0.13 −0.17 −0.24 −0.27 −0.28 −0.21  
Al 0.96 0.34 0.23 0.13 0.06 0.43 0.43 
Si 2.14 1.26 1.21 1.04 0.92 1.40 1.39 
1.87 0.92 0.93 0.76 0.67 0.69 0.75 
3.35 2.11 2.26 2.03 1.90 2.03 2.08 
Cl 5.10 3.61 3.92 3.62 3.49 3.60 3.61 
Ar −1.85 −2.42 −2.52 −2.57 −2.64 −2.76  
Kc 0.98 0.68 0.59 0.45 0.43 0.40 0.50 
Cad 0.26 −0.12 −0.25 −0.31 −0.32 −0.06 0.02 
Gae 0.91 0.28 0.18 0.08 0.02 0.32 0.41 
Gee 2.10 1.24 1.18 1.03 0.91 1.35 1.23 
Ase 1.87 0.98 0.98 0.81 0.73 0.69 0.81 
See 3.23 2.12 2.20 2.00 1.87 2.00 2.02 
Bre 4.78 3.50 3.66 3.41 3.25 3.45 3.36 
Kre −1.03 −1.52 −1.60 −1.65 −1.69 −1.75  
MAD 1.02 0.19 0.27 0.16 0.21   
MAX 2.31 0.74 0.86 0.61 0.81   
AtomsCAM-B3LYPCAM-QTP-00CAM-QTP-01CAM-QTP-02LC-QTPΔE(CCSDT-3)aExpt.b
He −1.95 −2.30 −2.40 −2.43 −2.45 −2.63  
Li 1.04 0.83 0.77 0.58 0.56 0.62 0.62 
Be 0.32 −0.16 −0.21 −0.26 −0.27 −0.27  
1.26 0.26 0.32 0.14 0.05 0.24 0.28 
2.66 1.16 1.51 1.21 1.14 1.24 1.26 
1.52 −0.04 0.34 0.06 0.07 −0.21 −0.07 
3.37 1.17 2.04 1.63 1.83 1.42 1.46 
5.71 2.89 4.26 3.73 4.21 3.40 3.40 
Ne −3.77 −4.54 −4.60 −4.67 −4.75 −5.28  
Na 1.12 0.82 0.75 0.59 0.56 0.54 0.55 
Mg 0.13 −0.17 −0.24 −0.27 −0.28 −0.21  
Al 0.96 0.34 0.23 0.13 0.06 0.43 0.43 
Si 2.14 1.26 1.21 1.04 0.92 1.40 1.39 
1.87 0.92 0.93 0.76 0.67 0.69 0.75 
3.35 2.11 2.26 2.03 1.90 2.03 2.08 
Cl 5.10 3.61 3.92 3.62 3.49 3.60 3.61 
Ar −1.85 −2.42 −2.52 −2.57 −2.64 −2.76  
Kc 0.98 0.68 0.59 0.45 0.43 0.40 0.50 
Cad 0.26 −0.12 −0.25 −0.31 −0.32 −0.06 0.02 
Gae 0.91 0.28 0.18 0.08 0.02 0.32 0.41 
Gee 2.10 1.24 1.18 1.03 0.91 1.35 1.23 
Ase 1.87 0.98 0.98 0.81 0.73 0.69 0.81 
See 3.23 2.12 2.20 2.00 1.87 2.00 2.02 
Bre 4.78 3.50 3.66 3.41 3.25 3.45 3.36 
Kre −1.03 −1.52 −1.60 −1.65 −1.69 −1.75  
MAD 1.02 0.19 0.27 0.16 0.21   
MAX 2.31 0.74 0.86 0.61 0.81   
a

Only the valence s and p along with 3d electrons (when occupied) are included in the active space.

b

From Ref. 60.

c

Using the QZP basis set.

d

Using the cc-pVQZ basis set.

e

Calculations with effective core potentials (aug-cc-pVQZ-PP).

In the sequence, Table X shows the vertical electron affinities for the same molecules investigated in Sec. III B. As already mentioned, the results obtained for the DFT functionals are given by using the eigenvalues of appropriate unoccupied orbitals found in calculations done at the respective equilibrium geometries. The proper virtual orbital to be occupied according to symmetry arguments is always the LUMO for CAM-QTP-02 and LC-QTP, except by CO. In this molecule, the wavefunction symmetry found for the anion by CCSDT-3/aug-cc-pVQZ calculations requires the occupation of LUMO + 1 from DFT results, a σ molecular orbital that is almost degenerate with the pair of π orbitals ascribed to LUMO in this case. However, the other functionals presented additional inversions between LUMO and high-lying orbitals beyond that seen for CO and the choice is therefore made by using the eigenvalue corresponding to the lowest energy orbital of proper symmetry according to the reference results. Thus, CAM-B3LYP presented inversions of this kind for CO, H2CO, H2, and N2. CAM-QTP-00 showed such inversions for CO and H2, while CAM-QTP-01 also exhibited inverted results for N2 along with CO and H2.

TABLE X.

Vertical electron affinities of molecules as obtained from the negative of virtual eigenvalues in calculations done with aug-cc-pVQZ basis sets with respect to the reference ΔE(CCSDT-3) results and values achieved with the ΔSCF treatment using CAM-QTP-02 (in eV).

MoleculesCAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02aLC-QTPaΔE(CAM-QTP-02)aΔE(CCSDT-3)b
LiH 0.759 0.462 0.395 0.368 0.355 0.308 0.294 
CH 2.551 1.115 1.425 1.153 1.064 1.232 1.168 
NH 1.692 0.137 0.546 0.266 0.260 0.284 0.297 
NH3 −0.113 −0.436 −0.487 −0.522 −0.543 −0.581 −0.550 
OH 3.281 1.132 1.974 1.565 1.725 1.660 1.802 
H2−0.014 −0.394 −0.437 −0.477 −0.494 −0.552 −0.556 
HF 0.004 −0.406 −0.440 −0.481 −0.491 −0.570 −0.621 
LiF 0.801 0.509 0.455 0.435 0.424 0.365 0.333 
CN 5.472 4.140 4.383 4.132 4.041 4.220 3.834 
HCN −0.188 −0.390 −0.444 −0.461 −0.469 −0.477 −0.480 
CO −1.004 −1.306 −1.371 −1.402 −1.418 −1.391 −1.438 
H2CO −0.206 −0.444 −0.502 −0.524 −0.532 −0.558 −0.543 
CH3OH −0.164 −0.448 −0.495 −0.523 −0.535 −0.565 −0.535 
H2NNH2 −0.116 −0.436 −0.486 −0.522 −0.543 −0.568 −0.505 
NO 1.398 −0.280 0.137 −0.182 −0.219 −0.273 −0.465 
HOOH −0.112 −0.475 −0.517 −0.555 −0.571 −0.623 −0.620 
H2 −0.880 −1.115 −1.153 −1.175 −1.181 −1.215 −1.231 
Li2 0.665 0.293 0.188 0.136 0.119 0.237 0.336 
N2 −1.494 −1.730 −1.788 −1.812 −1.822 −1.862 −1.829 
O2 1.642 −0.294 0.309 −0.071 0.004 −0.177 −0.129 
F2 2.260 −0.400 0.680 0.171 0.419 −0.099 0.392 
MAD 0.823 0.193 0.177 0.096 0.076 0.085  
MAX 1.868 0.792 0.602 0.298 0.246 0.491  
MoleculesCAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02aLC-QTPaΔE(CAM-QTP-02)aΔE(CCSDT-3)b
LiH 0.759 0.462 0.395 0.368 0.355 0.308 0.294 
CH 2.551 1.115 1.425 1.153 1.064 1.232 1.168 
NH 1.692 0.137 0.546 0.266 0.260 0.284 0.297 
NH3 −0.113 −0.436 −0.487 −0.522 −0.543 −0.581 −0.550 
OH 3.281 1.132 1.974 1.565 1.725 1.660 1.802 
H2−0.014 −0.394 −0.437 −0.477 −0.494 −0.552 −0.556 
HF 0.004 −0.406 −0.440 −0.481 −0.491 −0.570 −0.621 
LiF 0.801 0.509 0.455 0.435 0.424 0.365 0.333 
CN 5.472 4.140 4.383 4.132 4.041 4.220 3.834 
HCN −0.188 −0.390 −0.444 −0.461 −0.469 −0.477 −0.480 
CO −1.004 −1.306 −1.371 −1.402 −1.418 −1.391 −1.438 
H2CO −0.206 −0.444 −0.502 −0.524 −0.532 −0.558 −0.543 
CH3OH −0.164 −0.448 −0.495 −0.523 −0.535 −0.565 −0.535 
H2NNH2 −0.116 −0.436 −0.486 −0.522 −0.543 −0.568 −0.505 
NO 1.398 −0.280 0.137 −0.182 −0.219 −0.273 −0.465 
HOOH −0.112 −0.475 −0.517 −0.555 −0.571 −0.623 −0.620 
H2 −0.880 −1.115 −1.153 −1.175 −1.181 −1.215 −1.231 
Li2 0.665 0.293 0.188 0.136 0.119 0.237 0.336 
N2 −1.494 −1.730 −1.788 −1.812 −1.822 −1.862 −1.829 
O2 1.642 −0.294 0.309 −0.071 0.004 −0.177 −0.129 
F2 2.260 −0.400 0.680 0.171 0.419 −0.099 0.392 
MAD 0.823 0.193 0.177 0.096 0.076 0.085  
MAX 1.868 0.792 0.602 0.298 0.246 0.491  
a

Calculations done at the respective theoretical equilibrium geometries of the neutral molecules.

b

Only the valence electrons are included in the active space of these calculations, which were performed at the experimental geometries of the neutral molecules.46,47

According to Table X and Fig. 6, the best functionals for electron affinity estimates directly from virtual eigenvalues in molecules are LC-QTP and CAM-QTP-02, with MADs (MAXs) around 0.08–0.10 (0.25–0.30) eV. Next, we notice that CAM-QTP-00 and CAM-QTP-01 are not so successful in this case, with MADs (MAXs) nearly two times larger, around 0.18–0.19 (0.60–0.79) eV. Finally, CAM-B3LYP is once more the worst functional for electron affinities, with a MAD (MAX) of 0.82 (1.87) eV. This last functional exhibits again a trend for overestimated EAs from virtual eigenvalues. The same conclusions are also drawn for a few investigated transition metal compounds, as seen in Table XI. For comparison, the vertical EA values provided by energy differences in CAM-QTP-02/aug-cc-pVQZ calculations for the neutral molecules and their anions, ΔE(CAM-QTP-02), are also presented in Table X. Thus, one can see that the accuracy of these ΔSCF results obtained with CAM-QTP-02 (MAD = 0.085 eV) is almost the same as those obtained from unoccupied orbitals in a single CAM-QTP-02/aug-cc-pVQZ calculation for the neutral system (MAD = 0.096 eV). Thus, there is no strong reason to use the more computationally demanding ΔSCF method for vertical EAs with the new DFT functionals developed here.

FIG. 6.

Vertical electron affinities as given by the negative of unoccupied orbital eigenvalues obtained in CAM-QTP-02 and LC-QTP calculations of 21 molecules with aug-cc-pVQZ basis sets against reference values (the dotted line represents the exact agreement).

FIG. 6.

Vertical electron affinities as given by the negative of unoccupied orbital eigenvalues obtained in CAM-QTP-02 and LC-QTP calculations of 21 molecules with aug-cc-pVQZ basis sets against reference values (the dotted line represents the exact agreement).

Close modal
TABLE XI.

Vertical electron affinities of molecules with transition metals as obtained from the negative of virtual eigenvalues in calculations done with aug-cc-pVQZ(-PP) basis sets with respect to the reference ΔE(CCSDT-3) results (in eV).

MoleculesCAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02aLC-QTPaΔE(CCSDT-3)b
CuH 1.212 0.546 0.489 0.399 0.361 0.418 
CuF 2.263 1.374 1.391 1.259 1.197 1.255 
ZnF 2.967 2.036 2.052 1.869 1.771 1.778 
AgF 2.424 1.563 1.555 1.426 1.360 1.320 
AgH 1.368 0.701 0.643 0.554 0.511 0.491 
ZnH 1.609 0.859 0.806 0.656 0.579 0.730 
CdH 1.727 0.981 0.929 0.784 0.710 0.894 
YF 0.666 0.201 0.078 0.013 −0.007 0.259 
MAD 0.886 0.154 0.145 0.089 0.098  
MAX 1.189 0.257 0.274 0.247 0.267  
MoleculesCAM-B3LYPaCAM-QTP-00aCAM-QTP-01aCAM-QTP-02aLC-QTPaΔE(CCSDT-3)b
CuH 1.212 0.546 0.489 0.399 0.361 0.418 
CuF 2.263 1.374 1.391 1.259 1.197 1.255 
ZnF 2.967 2.036 2.052 1.869 1.771 1.778 
AgF 2.424 1.563 1.555 1.426 1.360 1.320 
AgH 1.368 0.701 0.643 0.554 0.511 0.491 
ZnH 1.609 0.859 0.806 0.656 0.579 0.730 
CdH 1.727 0.981 0.929 0.784 0.710 0.894 
YF 0.666 0.201 0.078 0.013 −0.007 0.259 
MAD 0.886 0.154 0.145 0.089 0.098  
MAX 1.189 0.257 0.274 0.247 0.267  
a

Calculations done at the respective theoretical equilibrium geometries of the neutral molecules.

b

Only the valence electrons (considering also 3d electrons for Cu and Zn and 4d electrons for Y, Ag and Cd) are included in the active space of these calculations, which were performed at the experimental geometries of the neutral molecules.46 

The new functionals discussed in this work, CAM-QTP-02 and LC-QTP, are parameterized in a non-empirical way by using exact conditions from the correlated orbital theory. They provide geometries, vibrational frequencies, and dipole moments of similar accuracy as those given by other functionals belonging to the QTP family, CAM-QTP-00 and CAM-QTP-01. Additionally, CAM-QTP-02 and LC-QTP also show a nice performance in mitigating the self-interaction error, as seen for one electron (H2+) and many electron (F) systems.

The negative of the occupied eigenvalues obtained from these new functionals also provides excellent descriptions of vertical ionization potentials of valence electrons (one-particle spectrum), which is a characteristic shared by QTP functionals. In this case, CAM-QTP-02 is the most accurate functional already investigated for ionizations from the HOMO, with a MAD of only 0.23 eV. The deviations found for vertical electron excitation energies of small organic compounds provided by the two new functionals are between those from CAM-QTP-00 and CAM-QTP-01. Moreover, CAM-QTP-02 and LC-QTP are successful in predicting charge-transfer excitation energies, along with the prior QTP functionals.

The barrier heights of HT and NHT reactions are slightly improved with CAM-QTP-02 in comparison with results from previous functionals belonging to the QTP family. However, the most remarkable feature of CAM-QTP-02 and LC-QTP is an excellent description of vertical electron affinity values by using only the negative of unoccupied orbital eigenvalues, with MADs smaller than 0.10 eV for molecules.

See supplementary material for tables containing results for geometries, vibrational frequencies, dipole moments, vertical ionization potentials, and excitation energies.

The authors thank São Paulo Research Foundation (FAPESP), Grant No. 2016/18704-2, and U.S. Air Force Office of Scientific Research, Grant No. FA9550-14-1-0281, for financial support. The authors also acknowledge University of Florida Research Computing for providing computational resources. The authors thank Duminda S. Ranasinghe, Yifan Jin, Prakash Verma, Ajith Perera, and the other members of the Bartlett group for helpful discussions.

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