How Does LCDFT Compare to SAC-CI for the Treatment of Valence

Article pubs.acs.org/JPCA

How Does LCDFT Compare to SAC-CI for the Treatment of Valence and Rydberg Excited States of Organic Compounds? Mojtaba Alipour* Department of Chemistry, College of Sciences, Shiraz University, Shiraz, Iran ABSTRACT: The computation of excitation energies for electronically excited states poses a challenge in quantum chemistry. In the present work, the performance of two related methodologies in this context, symmetry-adapted cluster-configuration interaction (SAC-CI) and time-dependent long-range corrected density functional theory (TDLCDFT), is compared in detail for the calculation of valence and Rydberg excitation energies against an experimental benchmark set comprising some organic compounds from different categories. Practically, the single- and double-linked excitation operators are considered in the SAC-CI wave functions. The considered LC density functionals include the combination forms of exchange and correlation functionals (BLYP, PBE, TPSS), pure functionals (tHCTH and B97-D), exchange-only functionals (HFS, HFB, and XAlpha), hybrid functionals (CAM-B3LYP, LC-ωPBE, ωB97, ωB97X), and dispersion-corrected hybrid functional ωB97X-D. Our results reveal that the SACCI gives the best performance for Rydberg excited states. However, the values of mean absolute deviation show that the applicability of some LC functionals is comparable to SAC-CI. For valence excited states, the functionals ωB97X-D, ωB97X, and LC-ωPBE outperform the other tested methods. Overall, the ωB97X-D functional is found to offer the best performance, and its validity compared with SAC-CI has also been verified by computing low-lying excited states of a few molecules as representative examples. Lastly, it is shown that not only is there a reasonable agreement between TDLCDFT and SAC-CI methods for the calculation of excitation energies but also the LC density functionals have quantitatively better overall performance for some excited states than the SAC-CI approach.

1. INTRODUCTION Treating the energetic position of excited states relative to the ground state as well as the geometrical and electronic properties of excited states is necessary for the explanation and interpretation of electronic spectra of molecular systems. Quantum-chemical calculations of excited states can provide useful information and do indeed contribute to the fundamental understanding of excited-state dynamics. Therefore, the validation of theoretical methods providing reliable predictions of the excited-state properties is an active field. In this respect, various ab initio wave function theory (WFT)based approaches such as symmetry-adapted cluster-configuration interaction (SAC-CI),1−4 coupled-cluster−linear response theory (CC-LRT),5 equation-of-motion−coupled cluster (EOM-CC),6 and so on have been proposed. However, the applicability of WFT-based methods is limited due to their high computational costs. Among all alternative computational schemes, time-dependent density functional theory (TDDFT)7,8 has become a popular tool for computing the signatures of electronically excited states and the properties directly related to the absorption and emission spectra of molecules, owing to its favorable balance between accuracy and efficiency. Conventional TDDFT is based on the linear response (LR) of the ground state to a time-dependent perturbation and may be called LR-TDDFT. The LR-TDDFT can be considered as a correlation-corrected version of linear-response time-dependent Hartree−Fock theory. Practical applications of LR-TDDFT usually employ the adiabatic approximation, by which the © 2014 American Chemical Society

exchange-correlation functional is independent of frequency, and it is usually taken to be the same as one of the exchangecorrelation approximations originally developed for the static ground state. In recent years, many articles have been published on the mathematical aspects, applicability, and benchmarking of TDDFT; see, for example, refs 9−25. However, TDDFT also has some limitations in its present formulations: (a) dependency of reliability of the results on the selected exchangecorrelation functional, (b) single determinant description of the ground-state wave function and consequently the inadequacy of TDDFT when multiple states are energetically close or when the excited state cannot be described in terms of single excitations, and (c) existence of a specific drawback for certain kinds of excitations such as those largely characterized by a charge transfer. Among several strategies that have been proposed to solve this problem, the most computationally tractable one is the long-range-corrected (LC) scheme.26,27 To date, perhaps the most successful approach in practice to include nonlocal exchange effects for finite systems is provided by LCDFT. This procedure was successfully applied to several problems such as nonlinear optical properties, charge-transfer excitations, Rydberg excitations, intermolecular interactions, and so on.28−35 However, in general, DFT does not provide as straightforward of an approach to systematically reliable results Received: January 9, 2014 Revised: February 10, 2014 Published: February 21, 2014 1741

dx.doi.org/10.1021/jp500236c | J. Phys. Chem. A 2014, 118, 1741−1747

The Journal of Physical Chemistry A

Article

⎛ 1 0 ⎞⎛ X ⎞ ⎛ A B ⎞⎛ X ⎞ ⎜ ⎟⎜ ⎟ = ε⎜ ⎟⎜ ⎟ ⎝ B A ⎠⎝ Y ⎠ ⎝ 0 − 1⎠⎝ Y ⎠

as the clear-cut hierarchy of WFT based methods. Therefore, it is necessary to test the accuracy of density functionals to determine the range of their applicability and identify areas for improvement. In the present investigation, as two successful theoretical methodologies in the context of excited state, the LC density functionals (from DFT) and SAC-CI method (from WFT) are compared for calculations of the valence and Rydberg excitations of an experimentally benchmark set of organic compounds. Indeed, in light of this work, we identify which LC density functionals are close to or dominate over SAC-CI accuracy for excitation energy computations. The rest of this paper is organized as follows. In the subsequent section, we present a terse description of the SACCI, LR-TDDFT, and LCDFT approaches. Then, a section is provided in which the details of our computational procedure are explained. In continuation, the results and discussion of the general trends of the benchmark calculations are presented. The last section concludes the paper.

The matrices A and B are defined as

∑ CISÎ I

∑ dK ΦK K

(7)

∫ ∫ ϕiα(r)ϕaα(r)⎢⎢ |r −1 r′| ⎣

+

⎤ δ 2E XC ⎥× δρα (r)δρβ (r′) ⎥⎦

ϕjβ (r′)ϕbβ (r′) dr dr′ (8)

in which EXC is the exchange-correlation energy, which, in turn, is defined in terms of exchange-correlation functional, and ρα and ρβ are electron spin densities. 2.3. LCDFT. In the LC scheme, the electron-repulsion operator 1/r12 is split into two parts by using the standard error function (erf)37,38 1 − erf(ωr12) erf(ωr12) 1 = + r12 r12 r12

(9)

with ω as a damping parameter. Accordingly, the total exchange term is partitioned into long-range (LR) and short-range (SR) contributions

(1)

Ex = ExLR + ExSR

(10)

The LR contribution to the exchange is computed as occupied occupied

(2)

ExLR = −

∑

∑

i

j

j i r12

(11)

where ϕ is the molecular orbital. The SR contribution to the exchange is a modified generalized gradient approximation (GGA) functional ExSR = −∑ σ

⎧ ⎪

∫ ρσ4/3kσ ⎨1 − ⎪

⎩

8 ⎡ 1/2 ⎛ 1 ⎞ aσ ⎢π erf⎜ ⎟ 3 ⎢⎣ ⎝ 2aσ ⎠

⎤⎫ ⎪ + 2aσ (bσ − cσ )⎥⎬ dr ⎥⎦⎪ ⎭

(12)

in which ρσ represents the density of electron with spin σ at point r and kσ is the enhancement factor. The explicit definitions of aσ, bσ, and cσ can be found in refs 26−28. In the Coulomb-attenuating method (CAM) of Yanai et al.,39 eq 9 is generalized by using two extra parameters α and β as follows

(3)

where dK is the coefficient of the function. To determine the SAC-CI coefficients dK in the SAC-CI wave function the variational principle can be applied ‐CI =0 e

Biaα , jbβ = K iaα , jbβ

⎡

The excitation operator ŜI is symmetry-adapted, which discriminates between the SAC and ordinary CC methods, and CI is the coefficient of the operator. Applying the variational principle, the SAC equations are obtained and iteratively solved to determine the energy and the coefficients. For excited states, by considering an excited function ΦK as ΦK ̂ = P̂ SK̂ |ΨSAC g > in which P is the operator that projects out the ground-state SAC wave function, the excited (e) state SAC-CI wave function can be described by a linear combination of the basis functions ‐ CI ΨSAC = e

(6)

K iaα , jbβ =

where Ψ0 is the reference determinant and Ŝ is the linear combination of the excitation operators ŜI

Ŝ =

Aiaα , jbβ = δijδabδαβ(εa − εi) + K iaα , jbβ

where the indices i, j and a, b label occupied and virtual orbitals, respectively, the indices α and β denote spin, εa and εi are orbital energies for Kohn−Sham orbitals ϕ a and ϕ i , respectively, and Kiaα,jbβ (as a coupling matrix)36 is given by

2. THEORETICAL FRAMEWORK As previously mentioned, the methods under study are SAC-CI and LCDFT in the time-dependent domain. Thus, it is worth outlining the underlying theories. However, we refer the interested readers to original references for further details. 2.1. SAC-CI. The SAC/SAC-CI method is a correlated electronic structure theory for the ground and excited states in various spin multiplicities.1−4 The SAC method belongs to the coupled-cluster (CC) theory, and in the case of a closed shell singlet state, the ground (g) state SAC wave function is written as ΨSAC = exp(S)̂ |Ψ0> g

(5)

(4)

1 − [α + β erf(ωr12)] α + β erf(ωr12) 1 = + r12 r12 r12

and then the excited states are obtained by single diagonalization. 2.2. LR-TDDFT. In the linear response formulation of the TDDFT equations for the calculation of excited states, solutions to the following eigenvalue problem yield the excitation energy, ε, and the corresponding transition vectors X and Y8

(13)

Here α and α + β define the exact exchange percentage at r12 = 0 and r12 = ∞, respectively, and the relations 0 ≤ α + β ≤ 1, 0 ≤ α ≤ 1, and 0 ≤ β ≤ 1 should be satisfied. In this case, the LR and SR contributions to the exchange are, respectively, as39−41 1742


The Journal of Physical Chemistry A occupied occupied

ExLR = αExHF − β

∑

∑

i

j

j i r12 (14)

where Ex

HF

molecule

is the Hartree−Fock (exact) exchange, and

ExSR = −

1 2

∑∫ σ

acetone

⎧ ⎪ 8 ⎡ 1/2 (1 ) ρσ4/3 kσ ⎨ − α − βaσ ⎢π ⎪ 3 ⎢⎣ ⎩

⎤⎫ ⎛ 1 ⎞ ⎪ erf⎜ dr ⎟ + 2aσ (bσ − cσ )⎥⎬ ⎪ ⎝ 2aσ ⎠ ⎦⎥⎭

butadiene

(15) cyclopentadiene

3. COMPUTATIONAL DETAILS Low-lying valence and Rydberg excite states of 10 molecules from various categories including alkenes, aldehydes, ketones, cycloalkenes, and heteroaromatic compounds have been computed. More specifically, the considered molecules are ethylene, trans-1,3-butadiene, formaldehyde, acetone, cyclopentadiene, furan, pyrrole, pyridine, pyrazine, and p-benzoquinone. All geometries were optimized at the MP2/6-311+G(d,p) level of theory. Moreover, calculations of excitation energies within SAC-CI and TDLCDFT approaches have been performed using the 6-311(2+,2+)G(d,p) basis set, which is similar to the standard 6-311++G(d,p) basis but contains a second set of diffuse functions on all atoms.42 We have used the angular momentum quantum numbers and exponential parameters for the additional diffuse subshells of atoms as follows: H: s, 0.00108434; C: sp, 0.0131928; N: sp, 0.0192470; and O: sp, 0.025451869.43 The use of the 6-311(2+,2+)G(d,p) basis set for the description of valence and Rydberg excitations has been advocated in previous investigations.43,44 For the SAC-CI calculations, we have employed the SAC-CI SD-R method in which the linked operators consist of the single (S) and double (D) excitation operators. In addition, the level-three accuracy (the energy thresholds (au) of λg = 1 × 10−6 and λe = 1.0 × 10−7) was adopted for the perturbation selection of linked operators. Various LC approximate density functionals are considered in the TDLCDFT calculations. LC scheme of Hirao and coworkers26 has been applied to three combination forms of exchange and correlation functionals BLYP,45−47 PBE,48,49 and TPSS;50 two pure functionals tHCTH51 and B97-D;52 and three exchange-only functionals HFS, HFB, and XAlpha.45,53−55 Moreover, LC hybrid functionals CAM-B3LYP,39 LC-ωPBE,56 ωB97,57 ωB97X,57 and dispersion-corrected ωB97X-D58 are also included in this study. All calculations were carried out using the Gaussian 09 suite of codes.59 We mention in passing that in the present work the excited states are examined in the gas phase. However, the solvent effect is important in such studies, and this issue has been considered in previous TDDFT treatments. In some other cases, it has been shown that the role of solvent is small. Therefore, it is common practice in the literature to compare calculated gas-phase vertical excitation energies to experimental data; see Table 1 in ref 25 and references therein. Moreover, while only gas-phase data are presented in this work and compared with SAC-CI gas-phase excitation energies reported in previous studies, we expect that reasonable agreements between LCDFT and SAC-CI results hold in solution as well. Nevertheless, the effect of solvent on the transition energies computed by LCDFT and SAC-CI is a problem worthy of

ethylene

formaldehyde

furan

p-benzoquinone

pyrazine

pyridine pyrrole

state A2 A2 B2 B2 Au Au Bg Bu A2 A2 B1 B2 B2 B1g B2g B1u B3u A2 A2 B2 B2 A2 A2 B1 B2 B2 Au B1g B3g B1u Au B2u B3u A2 B2 A2 B1

character valence Rydberg Rydberg Rydberg Rydberg Rydberg Rydberg valence Rydberg Rydberg Rydberg valence Rydberg Rydberg Rydberg valence Rydberg valence Rydberg Rydberg Rydberg Rydberg Rydberg Rydberg valence Rydberg valence valence valence valence valence valence valence valence valence Rydberg Rydberg

n → π* 3p 3s 3p 3p 3p 3s π → π* 3s 3p 3p π → π* 3p 3p 3p π → π* 3s n → π* 3p 3s 3p 3s 3p 3p π → π* 3p n → π* n → π* π → π* π → π* n → π* π → π* n → π* n → π* π → π* 3s 3p

ΔE (eV) 4.43 7.36 6.36 7.49 6.64 6.80 6.21 5.92 5.63 6.26 6.25 5.34 6.31 7.80 8.00 7.66 7.11 4.07 8.37 7.11 7.97 5.91 6.61 6.48 6.06 6.48 2.52 2.49 4.60 5.40 4.72 4.81 4.22 5.43 4.99 5.22 5.86

a

For more details on the excitation energies of the considered molecules, see refs 60−78.

investigation, which is not our focus in this work and can be explored elsewhere.

4. RESULTS AND DISCUSSION The benchmark set studied in this work including the type of molecules, excited states, and their characteristics and experimental excitation energies are collected in Table 1. For the purpose of the statistical evaluation of the methods performance for excited-state calculations, we used the mean signed deviation (MSD), mean absolute deviation (MAD), and maximum absolute deviation (MaxAD) with respect to experimental values. We first compare the results of SAC-CI and TDLCDFT for Rydberg and valence excited states separately. The benchmarked set can be divided into 22 Rydberg states and 15 valence states. (See Table 1.) Figure 1 illustrates the comparison of MADs of SAC-CI and LC density functionals in the calculations of Rydberg and valence excited states. For 1743



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Table 2. Statistical Analysis of the Deviations for the Computed Excitation Energies of the Benchmarked Set in This Worka

Figure 1. Comparison of SAC-CI and LC density functionals for valence and Rydberg excited-state calculations.

Rydberg excited states, the SAC-CI with MAD = 0.14 eV gives the best performance, followed by the LC exchange-only functionals HFS (MAD = 0.16 eV), XAlpha (MAD = 0.16 eV), and HFB (MAD = 0.19 eV) and the LC hybrid functionals ωB97X-D (MAD = 0.17 eV) and CAM-B3LYP (MAD = 0.21 eV). We see that the efficiency of some functionals is comparable to the SAC-CI method. Moreover, as can be observed from Figure 1, the LC-PBE (MAD = 0.73 eV) and LC-TPSS (MAD = 0.76 eV) functionals are nearly equivalent in performance for Rydberg excited states. The LC pure functionals tHCTH and B97-D with the MAD values of 0.98 and 0.86 eV, respectively, perform the worst among the considered methods for Rydberg excited states. However, it is important to note that because the LC density functionals include 100% Hartree−Fock exchange in the long-range limit, they have the correct asymptotic behavior and are supposed to yield good performance for Rydberg and charge-transfer excitations. For valence excited states, the functionals ωB97X-D (MAD = 0.25 eV), ωB97X (MAD = 0.25 eV), and LC-ωPBE (MAD = 0.26 eV) give the best performance in comparison with other methods. The next two functionals are the CAM-B3LYP and ωB97 with MAD values of 0.26 and 0.28 eV, respectively. Moreover, we see from Figure 1 that the values of MAD for LC functionals are even smaller than those of SAC-CI. Because LC density functional calculations are much less expensive than SAC-CI, it is very encouraging that these functionals outperform SAC-CI for the calculations of valence excited states. Lastly, a glance to Figure 1 is sufficient to conclude that the variations interval of MADs associated with valence excited states is less than that of Rydberg excitations. Up to this point, we have compared the performance of SAC-CI and various LC density functionals considering Rydberg and valence excited states. We now evaluate the overall performance of the considered methods. The relevant statistical measures of our investigation including MSD, MAD, MaxAD, and regression coefficient obtained using a standard least-squares linear fitting procedure (R2) are summarized in Table 2. Also, to make the key trends more visible, Figure 2 provides a graphical representation of the MADs and MaxADs in the computed values of excitation energies for the total number of excited states using SAC-CI and LC density functionals employed in this study. From Table 2, we find that the SAC-CI method and all of the considered LC density functionals, with the exception of CAM-B3LYP and ωB97X-D, overestimate the excitation energies and possess a positive MSD value. However, the MSD values by themselves cannot be used as estimators of methods performance because they

methods

MSDb

MADc

MaxADd

R2e

CAM-B3LYP LC-B97-D LC-BLYP LC-HFB LC-HFS LC-PBE LC-tHCTH LC-TPSS LC-ωPBE LC-XAlpha SAC-CI ωB97 ωB97X ωB97X-D

−0.086 0.63 0.37 0.03 0.02 0.55 0.71 0.58 0.36 0.08 0.24 0.39 0.28 −0.03

0.23 0.66 0.39 0.24 0.21 0.56 0.73 0.59 0.38 0.22 0.26 0.42 0.32 0.20

0.54 1.36 0.86 0.85 0.80 1.06 1.59 1.21 0.96 0.84 1.38 1.01 0.68 0.56

0.97 0.93 0.96 0.97 0.97 0.95 0.93 0.95 0.96 0.97 0.96 0.96 0.97 0.97

a

Unit: electronvolts. bMean signed deviation. cMean absolute deviation. dMaximum absolute deviation. eRegression coefficient obtained using a standard least-squares linear fitting procedure.

Figure 2. Graphical representations of the overall mean absolute deviation (MAD) and maximum absolute deviation (MaxAD) in the computed values of excitation energies for the considered molecules at the SAC-CI and LCDFT levels.

average over positive and negative values and do not offer insight into the magnitude of the errors. Therefore, other descriptors (MAD and MaxAD) are used to assess the absolute deviations (ADs) from the experimental data. Among all tested methods, the ωB97X-D functional gives the lowest MAD (0.20 eV) and offers the best performance. Interestingly, the next two methods are the LC exchange-only functionals HFS (MAD = 0.21 eV and MaxAD = 0.80 eV) and XAlpha (MAD = 0.22 eV and MaxAD = 0.84 eV), followed by CAM-B3LYP and LC-HFB. Observe, from Figure 2 and Table 2, that these functionals not only outperform other tested functionals but also dominate over SAC-CI approach, which is placed in sixth position of our ranking. Four functionals ωB97X, LC-ωPBE, LC-BLYP, and ωB97, and also two functionals, LC-PBE and LC-TPSS, offer near-equivalent overall performance. The LC-tHCTH (MAD = 0.73 eV and MaxAD = 1.59 eV) and LC-B97-D (MAD = 0.66 eV and MaxAD = 1.36 eV) give the poorest overall performance. The issue of ADs from the experiment aside, based on the consistency of theoretical estimates the regression coefficients reported in Table 2 show that the difference among R2 values is small for the considered LC functionals. However, LC-B97-D and LC-tHCTH functionals have the smallest regression coefficients. 1744



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1A2, 2A2, and 1B1. For the 2B2 state, the SAC-CI and ωB97X-D calculations provide similar excitation energies (6.37 and 6.24 eV, respectively), in excellent agreement with the experimental value of 6.31 eV. For furan, the SAC-CI and ωB97X-D methods offer near-equivalent treatments of Rydberg 1A2 state with the AD values of 0.12 and 0.10 eV, respectively. Moreover, the results of the two methods for the 2A2 state are comparable to experiment. However, although ωB97X-D predicts the excitation energy of 6.09 eV compared with experimental value of 6.06 eV for the valence π → π* excited state (1B2) of furan, the result of SAC-CI method (6.41 eV) deviated largely from the experiment. Related findings about such deviation for SACCI method even with various basis sets have also been pointed out in previous investigations.79,80 Finally, for the 1B1g state of ethylene, the SAC-CI performs better than the ωB97X-D functional. However, for the 1B3u state, functional ωB97X-D gives the AD value of 0.11 eV, lower than that of SAC-CI (AD = 0.25 eV).

Finally, let us compare in more detail the top ranked LC density functional in the current study (ωB97X-D) with SACCI for some low-lying excited states of a few molecules as illustrative examples. Table 3 lists the corresponding excitation Table 3. Excitation Energies for Some Low-Lying Excited States of Acetone, Butadiene, Cyclopentadiene, Furan, and Ethylene Computed Using SAC-CI and ωB97X-D Methods As Well As Corresponding Experimental Data ΔE (eV) molecule acetone

butadiene

cyclopentadiene

furan

ethylene

state

SAC-CI

ωB97X-D

experiment

2A2 1B2 2B2 1Au 2Au 1Bg 1A2 2A2 1B1 2B2 1A2 2A2 1B2 1B1g 1B3u

7.45 6.63 7.56 6.54 6.72 6.28 5.77 6.39 6.32 6.37 6.03 6.69 6.41 7.99 7.36

7.28 6.53 7.36 6.43 6.59 6.14 5.66 6.28 6.20 6.24 6.01 6.64 6.09 7.57 7.00

7.36 6.36 7.49 6.64 6.80 6.21 5.63 6.26 6.25 6.31 5.91 6.61 6.06 7.80 7.11

5. SUMMARY The performances of SAC-CI and TDLCDFT methodologies have been compared for the calculations of Rydberg and valence excited states of an experimental benchmark set consisting of some organic molecules from different categories with 37 total experimental excited state energies. The considered test set includes 15 valence states and 22 Rydberg states. It was found that the SAC-CI gives the best performance for Rydberg excited states. However, the values of MAD show that the applicability of some LC density functionals is comparable to SAC-CI. For valence excited states, the functionals ωB97X-D, ωB97X, and LC-ωPBE outperform the other tested methods. The ωB97X-D functional is shown to be superior for overall performance, and its applicability compared with SAC-CI was verified for some low-lying excited states of some molecules. Coming back to the main point posed in the current investigation about comparing between TDLCDFT and SAC-CI, we can conclude that the LCDFT through the framework of LR-TDDFT is able to predict excitation energies within an acceptable accuracy with respect to theSAC-CI method. Moreover, we observed that not only the TDLCDFT and SAC-CI calculations of the excitation energies show reasonable agreement but also the cost-effective LC density functionals dominate over SAC-CI for the prediction of some excited states.

energies of the excited states 2A2, 1B2, and 2B2 for acetone; 1Au, 2Au, and 1Bg for butadiene; 1A2, 2A2, 1B1, and 2B2 for cyclopentadiene; 1A2, 2A2, and 1B2 for furan; and finally 1B1g and 1B3u for ethylene. Moreover, shown in Figure 3 are the

■

Figure 3. Comparison between SAC-CI and ωB97X-D for some lowlying excited states of a few representative molecules. Acetone (a), butadiene (b), cyclopentadiene (c), furan (f), and ethylene (e).

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +98 711 6137160. Fax: +98 711 6460788.

ADs of the computed excitation energies using ωB97X-D and SAC-CI methods. The considered low-lying excited states of acetone are characterized as Rydberg excited states (Table 1). For A2 state of acetone, the corresponding excitation energy is computed to be 7.45 and 7.28 eV at the SAC-CI and ωB97X-D levels, respectively, in good agreement with the experiment value of 7.36 eV. The SAC-CI result for 1B2 state (AD = 0.27 eV) is deviated largely from experimental value, while for the 2B2 state, the SAC-CI performs better than ωB97X-D functional. For 1Au and 2Au Rydberg excited states of butadiene, the SAC-CI outperforms ωB97X-D. However, for the 1Bg state, the results of SAC-CI and ωB97X-D methods are in very good agreement with the experiment; the deviations do not exceed 0.07 eV. In the case of cyclopentadiene, the ωB97XD offers a better description for three Rydberg excited states

Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS The author thanks Shiraz University for computing resources. REFERENCES

(1) Nakatsuji, H.; Hirao, K. Cluster Expansion of the Wavefunction. Symmetry-Adapted-Cluster Expansion, Its Variational Determination, and Extension of Open-Shell Orbital Theory. J. Chem. Phys. 1978, 68, 2053−2065. (2) Nakatsuji, H. Cluster Expansion of the Wave Function. Excited States. Chem. Phys. Lett. 1978, 59, 362−364.

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