CASPT2

Article pubs.acs.org/JCTC

Singlet La and Lb Bands for N‑Acenes (N = 2−7): A CASSCF/CASPT2 Study Fernanda Bettanin,† Luiz F. A. Ferraõ ,*,† Max Pinheiro, Jr.,† Adélia J. A. Aquino,‡,§ Hans Lischka,‡,§ Francisco B. C. Machado,*,† and Dana Nachtigallova*,∥,⊥ †

Departamento de Química, Instituto Tecnológico da Aeronáutica, São José dos Campos, 122228-900 São Paulo, Brazil Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409-1061, United States § School of Pharmaceutical Sciences and Technology, Tianjin University, Tianjin 300072, P.R. China ∥ Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo nam. 2, CZ-16610 Prague 6, Czech Republic ⊥ Regional Center of Advanced Technologies and Materials, Department of Physical Chemistry, Palacky University, Olomouc 77146, Czech Republic ‡

S Supporting Information *

ABSTRACT: In this work CASPT2 calculations of polyacenes (from naphthalene to heptacene) were performed to find a methodology suitable for calculations of the absorption spectra, in particular of the La (B2u state) and Lb (B3u state) bands, of more extended systems. The effect of the extension of the active space and of freezing σ orbitals was investigated. The MCSCF excitation energy of the B2u state is not sensitive to the size of the active space used. However, the CASPT2 results depend strongly on the amount of σ orbitals frozen reflecting the ionic character of the B2u state. On the other hand, the excitation energies of the B3u state are much more sensitive to the size of the active space used in the calculations reflecting its multiconfigurational character. We found a good agreement with experimental data for both bands by including 14 electrons in 14 π orbitals in the active space followed by the CASPT2(14,14) perturbation scheme in which both σ and π orbitals are included.

1. INTRODUCTION Due to the promising potential applications of materials which contain π-conjugated systems1−8 to be used as chemical sensors, organic semiconductors, in spintronics and nonlinear optics, a large number of studies on graphene and its model systems appeared recently in the literature.9−15 The electronic properties of polyacenes (PACs, Scheme 1) not only are very

the absorption spectra, are of A1g and B1g symmetries. According to previous theoretical results,18,29,31−33,37−39,43−46 two main conclusions can be drawn from these calculations: (1) the multireference character of the ground state electronic configuration increases with increasing number of fused benzene rings and (2) doubly excited configurations contribute to the wave functions of the low-lying excited states. For polyacenes larger than tetracene these configurations need to be included to correctly describe the ordering of the excited state of Ag and B3u symmetries observed experimentally.44 A multiconfigurational character of the wave function was found37,43 for the first excited state (B3u) of naphthalene, resulting from a linear combination of HOMO−1 → LUMO and HOMO → LUMO+1 configurations, while the wave function of the second excited state (B2u) corresponds almost exclusively to a HOMO → LUMO excitation. For acenes larger than naphthalene, the B2u and B3u states change their energy ordering. The A1g state is best described by the double excitation HOMO2 → LUMO2. The B1g state wave function has large contributions from the singly excited configurations

Scheme 1. Structure of Studied Polyacenes (n = 0−5)

attractive for material science,16−24 but also they can serve as a model system for studying the properties of ground and excited states of extended π systems23,25−42 by means of theoretical methods. PACs possess D2h symmetry with a totally symmetric (A1g) electronic ground state. The two low-lying excited states in the absorption spectra are short-axis polarized (La) and long-axis polarized (Lb) states of B2u and B3u symmetries, respectively.33 The two lowest excited states of gerade symmetry, not seen in © XXXX American Chemical Society

Received: March 21, 2017 Published: July 18, 2017 A

DOI: 10.1021/acs.jctc.7b00302 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation HOMO−2 → LUMO and HOMO → LUMO+2 for smaller members of the polyacenes group. With the increasing number of benzene rings, however, the contributions from double excitations become more important. For that reason, some of the methods frequently used for the calculations of excited states of extended systems which are not able to correctly describe the states with double-excitation character and to account for the multireference character of the wave function do not provide a reliable description of the excited states. Improvement of the description of the doubly excited configurations has been made using the extended ADC(2)-x approach.33 Still, the description of excited states of longer polyacenes remains problematic due to the increasing multireference character of the excited states. Due to the well-known problem of overstabilizations of charge-transfer states using conventional functionals,26,27 a somewhat problematic description of the charge separation in the large-size conjugated organic molecules can lead to an incorrect ordering of B3u and B2u states which are of covalent and ionic characters, respectively. Despite all the studies cited above, finding a methodology suitable for the description of the spectroscopic properties of even more extended molecules is still a great challenge for computational chemistry. Very recently, the ground and some electronic excited states of acenes up to 12 were analyzed based on the particle−particle random-phase approximation (ppRPA)44 giving new insight into the nature of these states. The goal of the current study is to propose a methodology suitable for characterization of optical properties of extended systems. In particular, this includes finding a method, capable of properly describing the states of different characters, which is, at the same time, computationally feasible for systems which include extended π-conjugated systems. Polyacenes ((PAC)N, where N is the number of the benzene rings) provide a suitable model for such studies due to the presence of two energetically close lying states of different characters, in particular the multireference covalent 11B3u (Lb band) and ionic 11B2u (La band) states. We present the discussion on the performance of both MCSCF and CASPT2 methods with respect to the character of the active space, including its size and with respect to the additional excitation constraints in both methods.

and au symmetries. Using this point group, the active space was constructed from one occupied and one virtual orbital of all four above listed symmetries, i.e. from eight electrons in eight π orbitals. With this size of the active space, the complete active space (CAS) was considered (CAS(8,8)). We have also investigated other modifications of the active space, in which the orbitals were chosen based on their occupation numbers obtained from MR-AQCC (Multireference Averaged Quadratic Coupled-Cluster)59 calculations for the ground state. In the case of active spaces larger than CAS (8,8), the extra orbitals, both occupied and virtual, were added with excitations restrictions (RAS) to reduce the computational cost. In particular, the excited state calculations of naphthalene and anthracene were performed using various CAS + RAS active space combinations to calibrate the methodology employed for larger polyacenes. To specify the reference CSFs in the CASPT2 calculations, the following notation is used throughout this paper: CASPT2(14,14)/CASSCF(8,8) + RAS (6,6)(s), for example, refers to the CASPT2 calculation using the reference set of the full CAS (14,14), where in the MCSCF step a combination of the CASSCF (8,8) and RASSCF(6,6) that includes only single excitations was used. The number of the configurations state functions generated in each one of these active spaces is shown in the Supporting Information (Table SI2). Different freezing schemes were adopted in both CASSCF and CASPT2 calculations to investigate the importance of σ orbitals correlating in the characterization of the excitation energies: freezing all the occupied σ orbitals (DOCC), only all 1s and 2s orbitals (2s), and only all the 1s orbitals (1s). Independently of the freezing scheme, all π orbitals were correlated in the CASPT2 calculations. The MCSCF frozen orbitals were obtained at the RHF level. The DFT optimizations were carried out using the Gaussian 09 program,60 and all the multiconfigurational calculations were carried out using the MOLPRO computational package.61,62

3. RESULTS The character of the 11Ag ground state wave function of polyacenes has been discussed previously.38,40−42,44,63 In agreement with the CASSCF results of Hajgato et al.,63 the square of the CI coefficient of the closed shell (HOMO2) configuration of naphthalene equates to 0.81 (81%). The next largest configuration corresponds to the double excitation (HOMO2 → LUMO2) configuration and contributes only about 2% to the ground state wave function. Therefore, there is a significant contribution to the ground state wave function (around 17%) from other configurations with individually small coefficients. Thus, although the wave function of the systems is dominated by a single reference, the system presents high electronic correlation. As a consequence, the total electron density can deviate from that corresponding to the closed shell configuration. As found by Yang et al.44 with the increasing size of polyacenes the contributions of the closed shell (HOMO2) configuration decrease, while the HOMO 2 → LUMO 2 configuration increases, respectively, with increasing of the acene size.44 Calculations using the density matrix renormalization group (DMRG),41,46 the two-electron reduced-densitymatrix (2-RDM),42 and MR-AQCC calculations38 show significantly larger open shell character. Also, the instability of the restricted DFT/B3LYP approach for acenes larger than pentacene40 points in this direction. The crucial step in the multiconfigurational description of the ground and lowest excited states, i.e. setting up the active space,

2. COMPUTATIONAL DETAILS The geometry of the (PAC)N=2−7 was optimized using density functional theory using the B3LYP functional47−49 and the 631G* basis set.50−53 The Cartesian coordinates of all studied molecules are presented in the Supporting Information (Table SI1). All the multiconfigurational calculations were carried out at these geometries using the 6-31G* basis set. In the first step, the state averaged MCSCF (Multi-Configurational SelfConsistent Field) method54,55 was used, with equal weights (i.e., in 1:1:1 ratio) for (X1Ag, (1)1B2u, and (1)1B3u) states, to incorporate the static correlation for the three singlet states in the wave function, generating the reference set of configuration state functions (CSF) and natural orbitals to be used in the dynamic correlation step. This was carried out using single state internally contracted CASPT256,57 without including level shifts. In some calculations with a small active space, a level shift was used in the zeroth-order Hamiltonian (of 0.2 au) combined with a correction on the second-order energy to avoid intruder states.58 The molecules were arranged in the xy plane with the long axis oriented along the x axis. Within D2h symmetry of the (PAC)N=2−7 the π orbitals present b1u, b2g, b3g, B


Article

Journal of Chemical Theory and Computation is challenging due to the growing number of π orbitals with increasing size of polyacenes. Since the use of the complete πactive space is computationally unfeasible for larger acenes, some criteria must be applied to choose a subset of orbitals included in the active space. One of the approaches is to select the active orbitals based on the occupation numbers of the natural orbitals. In our contribution AQCC calculations have been employed in such analyses. Within the polyacenes series, the complete π active space of anthracene, CAS(14,14) is already computationally demanding, although still feasible. A convenient way to reduce the computational cost is to divide this space into the complete (CAS) and restricted (RAS) active spaces. The former is constructed from one occupied and one virtual orbital of each symmetry within the π space, while the latter is selected based on the occupation numbers, respectively. As shown below, the results of the excitation energies of anthracene obtained with CAS(14,14) are well reproduced by the active space constructed as CAS(8,8) and RAS(6,6). Inspection of the natural orbital occupations for larger acenes (see Figure 1 and Figures SI1−SI3 for occupation numbers of

Naphthalene. The effect of the active space size and orbital freezing scheme has been tested for the calculated vertical excitation energies of the two lowest singlet excited states of naphthalene, namely 11B3u (Lb) and 11B2u (La). Results are given in Table 1 and Figures 2 and 3, together with previously reported experimental and theoretical data. The results for the Lb band (Figure 2 and Table 1) show that the excitation energies are sensitive to the size of the active space. With a smaller than full π-active space (ten electrons in ten orbitals) the results strongly depend on the freezing scheme used in the perturbative approach. Such dependence indicates that the active space in the reference wave function is not converged. The results become stable with respect to the freezing scheme once the full π-active space is used, regardless of the distribution of the electrons among the CAS and RAS spaces, as well as the level of excitation in the RAS space (i.e., single (S), single and double (SD), and single, double, triple, and quadruple (SDTQ) excitations). The resulting CASPT2 excitation energies are in the range of 3.91−4.25 eV, in good agreement with the estimated vertical excitation energies of 4.03−4.14 eV.39 Note that, using the full π-active space, the error between the calculated and estimated vertical excitation energy does not exceed 4% already at the MCSCF level. This behavior reflects the multiconfigurational character of the Lb state seen also from the inspection of the wave function character (see Table SI3). The two main configurations of the wave function of the Lb state are described by HOMO−1 → LUMO and HOMO → LUMO+1 excitations, with weights of 36% and 34%, respectively. An additional 20 and 26 other configurations with coefficients higher than 0.05 contribute to the wave function at the CAS(8,8) and CAS(10,10) levels, respectively. The results in Table 1 also show that with the full π-active space the excitation energies do not significantly depend on the freezing scheme used in the CASPT2 approach. In fact, good agreement with the experiment is reached already if all doubly occupied orbitals are kept frozen. Such independence reflects the covalent character of the Lb state. On the contrary the results for the La band (see Figure 3 and Table 1) show that the MCSCF excitation energies (gray bars) are not sensitive to the size of the active space. The different freezing schemes used in the perturbation treatment, however, greatly influence the resulting CASPT2 energies. As shown in Table 1, neglecting the σ-orbitals gives large errors in the CASPT2 excitation energies, even when the complete π-active space is used. In particular, allowing excitations from π orbitals only (DOCC freezing scheme) results in excitation energies which are overestimated by 1.3 eV compared to the vertical experimental value.40 On the contrary, freezing only 1s orbitals leads to excitations energies with errors less than 0.05 eV, independently of the reference MCSCF wave function. The freezing scheme, therefore, has a crucial effect on the results. The errors with respect to the vertical experimental value40 are about 28%, 9%, and 0.1% with DOCC, 2s, and 1s freezing schemes, respectively. The importance of the σ-orbitals in the perturbation scheme reflects the ionic character of the La state.64 The character of the wave function (Table SI3) is dominated by the HOMO → LUMO configuration with the weight of 78%, followed by the HOMO−1 → LUMO+1 configuration with the weight of 8%. The stability of the results with respect to the active space used in the calculations is in line with an almost single-reference character of the wave function. A balanced treatment of both excited states requires the use of the full π-active space and includes all but the σ (1s) orbitals

Figure 1. AQCC natural orbital occupation of π orbitals (occupied at the RHF level) included in the active space for the acenes series in the ground state (X 1Ag). The red line separates the orbitals included in the CAS (below the line) and RAS (above the line) active spaces. H and H-1 stand for HOMO and HOMO−1, respectively.

occupied and virtual orbitals, respectively) shows that the same scheme provides a sufficiently good and consistent description of the wave function of their ground and excited states. Also, the energies of the frontier orbitals obtained at the RHF level (see Figure SI4) provide the same trends. In particular, this active space ensures that the occupation numbers of the occupied and virtual orbitals lower than 1.9 and higher than 0.1, respectively, are included in the CAS. The results shown in Figures 1 and SI1−SI4 indicate that the incompleteness of the (14,14) active space should be less important with the increase of the acene size. The adequacy of the (14,14) active space is evident also from the occupation numbers of the ground states of even larger polyacenes with N = 8−10. Although the occupation numbers of the occupied and virtual CAS orbitals significantly deviate from 2 and 0, respectively, those of RAS orbitals are already very close to full occupancy or vacancy, respectively. C


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Journal of Chemical Theory and Computation

Table 1. Vertical Excitation Energies (eV) of the Two Lowest Excited States and the Energy Difference between the Excited States (La-Lb) of Naphthalene Calculated at the MCSCF and CASPT2a Levels Using Various Sizes of the Active Space Lb state method CAS(4,4) DOCC 2s 1s CAS(8,8) DOCC 2s 1s CAS(4,4) RAS(6,6)(S)b DOCC 2s 1s CAS(4,4) RAS(6,6)(SD)b DOCC 2s 1s CAS(4,4)RAS(6,6)(SDQT)b DOCC 2s 1s CAS(8,8) RAS(2,2)b DOCC 2s 1s CAS(10,10) DOCC 2s 1s ADC(2)-x33 DFT/MRCI31 experiment: ref 66d experiment: ref 67e vertical excitation energy39,f

MCSCF

La state CASPT2

MCSCF

La-Lb CASPT2

MCSCF

CASPT2

6.11 6.10 6.09

5.36 3.69 3.31

6.28 6.26 6.25

5.76 4.76 4.38

0.17 0.16 0.16

0.40 1.07 1.07

5.10 5.09 5.08

4.70 4.08 3.79

6.51 6.50 6.49

5.95 5.01 4.60

1.41 1.41 1.41

1.25 0.93 0.81

4.81 4.80 4.79

4.25 4.24 4.04

6.31 6.30 6.29

5.96 5.08 4.69

1.50 1.50 1.50

1.70 0.83 0.64

4.43 4.42 4.41

4.26 4.14 3.91

6.49 6.49 6.48

5.94 5.05 4.66

2.06 2.07 2.06

1.68 0.91 0.75

4.21 4.20 4.19

4.25 4.23 4.02

6.55 6.54 6.53

5.96 5.04 4.63

2.34 2.34 2.34

1.71 0.80 0.61

4.25 4.24 4.22

4.25 4.24 4.03

6.50 6.49 6.48

5.96 5.04 4.64

2.25 2.25 2.26

1.71 0.81 0.61

4.21 4.20 4.19 3.51 4.15 4.03 3.98 4.13

4.23 4.23 4.02 (3.99)c

6.55 6.55 6.54 4.47 4.66 4.38 4.34 4.66

5.95 5.03 4.63 (4.45)c

2.34 2.35 2.35 0.96 0.51 0.35 0.64 0.53

1.72 0.80 0.61 (0.46)c

a

Calculated with the 6-31G* basis set. bUsing CASPT2 with the configuration state functions of a CAS (10,10). cCalculated with the cc-pVDZ basis set. dAdiabatic excitation energies, in the gas phase. eAdiabatic excitation energies, in the neon matrix. fDerived from ref 66.

Figure 2. Vertical excitation energies of the 11B3u (Lb) band of naphthalene calculated using various approximations at the MCSCF/ CASPT2 level (see the text for notation) with the 6-31G* basis set. The vertical excitation energy estimated from the experiment is given in red color.

Figure 3. Vertical excitation energies of the 11B2u (La) band of naphthalene calculated using various approximations at the MCSCF/ CASPT2 level (see the text for notation) with the 6-31G* basis set. The vertical excitation energy estimated from the experiment is given in red color.

in the dynamical correlation (in our notation defined as CASPT2(1s)/CASSCF(10,10)). Using this scheme, the gap

between the excitation energies of the two excited states differs from the vertical experimental values by less than 0.1 eV. With a D


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Table 2. Vertical Excitation Energies (in eV) of the Two Lowest Excited States and the Energy Difference between the Excited States (La-Lb) of Anthracene Calculated at the CASPT2 and MCSCF Levels with Various Sizes of the Active Spacea 11B2u (La) state method

11B3u (Lb) state

La-Lb

MCSCF

CASPT2

MCSCF

CASPT2

MCSCF

CASPT2

CAS(8,8) CAS(4,4) RAS(6,6)(S)b CAS(8,8) RAS(2,2)(S)b CAS(8,8) RAS(6,6)(S)

5.11 4.97 4.96 5.23

4.81 4.48 4.23 3.93

5.37

CAS(14,14)

5.42

3.10 3.40 3.37 3.35 3.49c 3.35 3.43c 3.48 3.44e

0.30 0.49 0.73 1.30

CAS(8,8) RAS(6,6)(SD)

3.46 3.60 3.53 3.58 3.52c 3.51 3.51c 3.49 3.32e

0.36 0.20 0.16 0.23 0.03c 0.16 0.08c 0.01 −0.12e

ADC(2)-x33 DFT/MRCI31 experiment66,f experiment67,g estimated vertical excitation energy39,h

3.27 3.51 3.38 3.30 3.60

3.75 3.67 3.01 3.59 3.57 3.46 3.64

1.62 2.34 0.26 −0.08 −0.19 −0.04 −0.16

a Calculated with the 6-31G* basis set. bUsing CASPT2 with the configuration state functions of a CAS (10,10), see text for the notation. cUsing CASPT2 with the configuration state functions of a CAS (14,14). eCalculated with the cc-pVDZ basis set. fAdiabatic excitation energies, in the gas phase. gAdiabatic excitation energies. hDerived from ref 66.

more flexible cc-pVDZ65 basis set the energy gap of the two states drops to 0.46 eV, which is in excellent agreement with the experiment (0.52 eV).39 Anthracene. The results of vertical excitation energies of the 11B3u and 11B2u of anthracene (N = 3) together with those calculated using ADC(2)-x33 and DFT/MRCI31 and observed experimentally66 are presented in Table 2. The MCSCF and CASPT2 calculations were performed using (8,8), (10,10), and (14,14) active spaces; the latter corresponds to the full π-orbital active space. As in the case of naphthalene, for this active space, various distributions of the electrons between the CAS and RAS spaces were considered. Based on the results obtained for the La state of naphthalene, only 1s orbitals were frozen in the construction of the MCSCF wave function. In agreement with the results obtained for naphthalene, the distribution of electrons between CAS and RAS spaces does not affect the quality of the results when using the full π-active space. Restriction of the size of the active space from CAS(14,14) to CAS(8,8)RAS(6,6) results in only minor changes of the CASPT2 excitation energies. In particular, the CASPT2(14,14) excitation energies of La and Lb states do not differ by more than 0.3 and 0.6 eV, respectively. Importantly this agreement includes also the CASPT2 results which use the CAS(8,8)RAS(6,6) wave function with only single excitations allowed in the RAS space. The results reported in Table 2 indicate the same character of states as in the case of naphthalene. The excitation energies of the covalent Lb state depend more significantly on the size of the active space used for the construction of the MCSCF wave function, which is consistent with the character of the wave function (see Table SI3), and are not greatly sensitive to the inclusion of a higher level correlation treatment in the perturbative procedure. On the contrary, the excitation energies of the ionic La state are less sensitive to the active space used and crucially depend on the inclusion of the higher order correlation effects. According to the experimental studies performed which report the adiabatic excitation energies, the ordering of the two states of anthracene is reversed with respect to naphthalene placing the La state 0.19 eV66 and 0.16 eV67 below the Lb state,

respectively. The correction for the vertical excitation energies makes them almost degenerate with the difference39 of only 0.04 eV. This reordering is not reproduced by our results even with the full active π-space employed. At this level the (La-Lb) energy gap does not, however, exceed the value 0.1 eV, still providing a very good agreement with the experiment. The correct ordering of states is obtained with the more flexible ccpVDZ basis set (see Table 2). These calculations resulted in the energy difference of 0.12 eV, in excellent agreement with the experimental data.39,68,69 Polyacene Series. The vertical excitation energies of the series naphthalene to heptacene calculated at the MCSCF and CASPT2 levels together with experimental data are shown in Table 3 and Figure 4 for the 11B2u (La) state and in Table 4 and Figure 5 for the 11B3u (Lb) state, respectively. The selection of the active space and the orbital freezing was based on the previous results obtained for naphthalene and anthracene. In addition, the reliability of the description using more extended orbital freezing to describe trends in the excitation energies among the whole series of polyacenes was investigated. In these calculations freezing of 2s and doubly occupied orbitals was considered in the CASPT2(14,14)/CAS(8,8)RAS(6,6)(S) scheme. For the La band the calculations show a similar behavior to naphthalene and anthracene, i.e. freezing more than only 1s orbitals results in a large overestimation of the CASPT2 excitation energies even with a relatively large (14,14) active space. In particular, freezing of 2s orbitals and DOCC gives results which are overestimated by around 0.3 and 1.3 eV, respectively. A good agreement with experimental data39 is obtained only with the 1s orbitals freezing scheme, regardless of the partitioning of orbitals between the CAS and RAS spaces and of the excitation level used in the RAS space. In fact, with this freezing scheme the CASPT2 excitation energies agree even if the reference wave function is obtained using (8,8) and (10,10) active spaces. With these active spaces, however, the CASPT2 method gives excitation energies which are slightly underestimated with respect to the experimental results. The error increases with the increasing size of polyacene, but it does not make any significant deviation from the trends of La E


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Journal of Chemical Theory and Computation Table 3. Vertical Excitation Energies (in eV) of the 11B2u (La) Excited State of the Polyacene Series (Naphthalene (N = 2) to Heptacene (N = 7)) Calculated at the MCSCF (Normal Font) and CASPT2 (Italics)a Levels Using Various Sizes of the Active Spacei

excitation energies with respect to the size of the polyacene (see Figure 4). The CASPT2 results of the excitation energies of the Lb state obtained with the active space of 14 electrons distributed among 14 orbitals and 1s orbitals frozen in the MCSCF procedure, using either CAS(14,14) or CAS(8,8)+RAS(6,6) (see Table 4 and Figure 5), give excitation energies which are underestimated up to almost 0.4 eV with respect to the experiment.39 Note that calculations with both single and single−double excitations allowed in the RAS space give a similar picture. Closer agreement with the experimental data using the 2s freezing scheme in the MCSCF procedure is probably due to the error cancelation. The same trend is observed also for the active spaces with ten electrons in ten orbitals using CAS(4,4)+RAS(6,6) and CAS(8,8)+RAS(2,2) and 1s orbitals frozen. Note that with these active spaces the deviation from the vertical experimental results is larger compared to that found with the (14,14) active space. This behavior reflects the multiconfigurational character of this state discussed above. This is supported by the values of the weights (given by the square of CI coefficients) of state configurations given in Table SI3. The lack of the static correlation at the MCSCF treatment with increasing size of the acenes is evident from the trend of the MCSCF excitation energies (see Figure 5). This error is, however, compensated to a large extent at the CASPT2 level. Figure 6 compares the calculated energy gaps between the 11B2u (La) and 11B3u (Lb) states (La-Lb) among the polyacenes series, using various computational approaches, with experiment. The energy gap obtained with the DOCC orbital freezing scheme (red line in Figure 6) is consistently overestimated with respect to the experiment (pink triangles, Figure 6) which reflects neglecting of the dynamic correlation energy which has a larger effect on the values of ionic La state. Due to this overestimation the reversed ordering of the states is obtained for n = 6, much later than what is observed experimentally. The results which combine inclusion of both, extended σ-orbital correlation (1s orbital freezing) in the perturbation approach and a large active space (CAS(8,8)+RAS(6,6)) are shown by the blue line in Figure 6. These results give a similar tendency with respect to the increased system size as it is observed experimentally. In addition, the error of the calculated energy gap is lower than 0.4 eV. The transition moments of the excitations from the ground state (X 1Ag) to 11B3u and 11B2u states of polyacene series calculated at the MCSCF level using the CAS(14,14) active space are presented in Figure 7 (left). For the whole series studied, the values of the transition moments from the ground state to the 11B2u state are at least 3 times larger than those of 11B3u. Comparison of the transition moment values calculated with CAS (8,8) (Table SI4 in the Supporting Information) shows their stability with respect to the size of the active space. The oscillator strengths calculated using the MCSCF transition moments and using both MCSCF and CASPT2 excitation energies are given in Figure 7 (right). In agreement with the experimental observations,67,69,70 the oscillator strengths related to the La band are around 1 order of magnitude higher than those related to the Lb band. Larger differences in the oscillator strengths of the La band obtained using the MCSCF and CASPT2 energies reflect the ionic character of the 11B2u state and its sensitivity to the inclusion of corrections of higher correlation effects. As a result, for larger

N active space/freezing scheme CAS(8,8)/1s CAS(4,4) RAS(6,6) (S)/1sb CAS(8,8) RAS(2,2) (S)/1sb CAS(8,8) RAS(6,6) (S)/1s CAS(8,8) RAS(6,6) (S)/2sc CAS(8,8) RAS(6,6) (S)/DOCCc CAS(8,8) RAS(6,6) (SD)/1s CAS(14,14)/1s ADC(2)-x33 DFT/MRCI31 CC239 pp-RPA@R44 experiment

vertical excitation energy39,h

2

3

4

5

6

7

6.49 4.60 6.29 4.69 6.48 4.64 6.48 4.64 4.63c 5.04

5.11 3.46 4.97 3.60 4.96 3.53 5.23 3.58 3.52c 3.90

4.58 2.82 2.81 2.74 4.05 2.74 4.30 2.83 2.88c 3.19

3.71 2.26 3.79 2.36 3.93 2.28 3.76 2.29 2.30c 2.60

3.70 2.12 3.40 1.79 3.30 1.92 3.37 2.05 2.03c 2.30

3.08 1.71 2.80 1.69 3.54 1.67 3.37 1.69 1.64c 1.93

5.96

4.83

3.95

3.37

3.02

2.76

6.53

5.37

4.32

3.90

3.29

3.55

4.63 6.54 4.63 4.47 4.66 4.88 4.97 4.38d 4.34e

3.51 5.42 3.49 3.27 3.51 3.69 3.65 3.38d 3.30e

2.82 4.35 2.79 2.46 2.74 2.90 2.82 2.71d 2.60e

3.60

2.88

1.90 3.34 1.89 1.49 1.85 1.95 1.86 1.90d 1.80e 1.89f 2.02

1.61 3.64 1.66

4.66

2.22 3.95 2.20 1.90 2.22 2.35 2.26 2.23d 2.11e 2.21f,g 2.37

1.58

1.70f

a

In the case of naphthalene, all RAS(6,6) refers to RAS(2,2). bUsing CASPT2 with the configuration state functions of a CAS(10,10). c Using CASPT2 with the configuration state functions of a CAS(14,14), see the text for the notation. dReference 66. eReference 67. fReference 70. gReference 71. hDerived from ref 66 iThe 6-31G* basis set was used.

Figure 4. CASPT2 excitation energies of the 11B2u (La) band calculated for the polyacenes series using the CAS(8,8)RAS(6,6)(S) active space using a different orbital freezing scheme: * ref 39, ** ref 72.

F


Article


Table 4. Vertical Excitation Energies (in eV) of the 11B3u (Lb) Excited State of the Polyacene Series (Naphthalene (N = 2) to Heptacene (N = 7)) Calculated at the MCSCF (Normal Font) and CASPT2 (Italics)a Levels Using Various Sizes of the Active Spacei N active space/freezing scheme CAS(8,8)/1s CAS(8,8) RAS(6,6)(S)/1s

CAS(8,8) RAS(6,6)(S)/2sb CAS(8,8) RAS(6,6)(S)/DOCCb CAS(8,8) RAS(6,6)(SD)/1s CAS(14,14)/1s ADC(2)-x33 DFT/MRCI31 CC239 experiment

vertical excitation energy39,g

2

3

4

5

6

7

5.08 3.79 4.22 4.03 4.02b 4.24 4.25 4.19 4.02 4.19 4.02 3.51 4.15 4.46 4.03d 3.98e

4.81 3.10 3.93 3.35 3.49b 3.68 3.71 3.75 3.43 3.67 3.48 3.01 3.59 3.89 3.57d 3.46e

4.53 2.69 3.73 3.06 3.17b 3.35 3.41 3.63 3.12 3.57 3.15 2.67 3.22 3.52 3.32d 3.14e

4.51 2.95 4.07 2.61 2.58b 2.83 3.16 3.80 2.50 3.75 2.49 2.28 2.76−2.93 3.09 2.80d 2.81e

4.15 2.04 3.51 2.40 2.22b 2.44 2.67 3.39 2.19 3.29 2.40

4.13

3.64

3.39

4.28 2.42 3.92 2.88 2.89b 3.11 3.28 3.88 2.86 3.84 2.74 2.42 2.99 3.27 3.05d 2.88e 2.94f 3.12

2.87

a

In the case of naphthalene, all RAS(6,6) refers to RAS(2,2). bUsing CASPT2 with the configuration state functions of a CAS(14,14), see the text for the notation. dReference 66. eReference 67. fReference 70. gDerived from ref 66. iThe 6-31G* basis set was used.

Figure 5. Excitation energies of the 11B3u (Lb) band for the polyacenes series calculated at the CASPT2 and CASSCF levels employing the various sizes of the active spaces: * ref 39.

Figure 6. Energy gap between the excitation energies of La and Lb bands for the polyacene series calculated using different active spaces and an orbital freezing scheme: * ref 39.

acenes the oscillator strengths of this band are better reproduced using the CASPT2 energies.

extent of dynamic correlation. To obtain a balanced description of both states, an active space containing 14 electrons in 14 orbitals and only the 1s orbitals frozen proved to be adequate to provide a balanced description of the La and Lb states and very good agreement with experimental absorption energies. Similar results are obtained regardless of the partitioning of orbitals between the CAS and RAS spaces and regardless of the excitation level used in the RAS space. In summary, the performance of the multiconfigurational approach which covers both nondynamical and dynamic correlation with respect to the character of the excited states and the size of the systems was discussed. We show that the most important contributions to the excited state wave function are present using a reasonable size of the active space, and further correlation contributions are recovered at the CASPT2

4. CONCLUSION The vertical excitation energies for the La and Lb bands of the series naphthalene to heptacene were calculated at the MCSCF and CASPT2 levels employing various sizes of the active space and orbital freezing schemes with the goal to provide a reliable and efficient description of these states. These investigations show quite different requirements for the two states. In the valence bond formalism, the La state is of ionic character; it is sensitive to the freezing scheme of the σ-orbitals but does not require an extended active space. On the other hand, the covalent Lb state requires a large active space but is not sensitive to the freezing scheme within the σ-space, i.e. to the G


Article


Figure 7. [Left] Electronic transition moment for the La and Lb bands calculated at the CASSCF (14,14)/6-31G* level. [Right] Oscillator strength for La (circles) and Lb (squares) bands calculated with CASSCF (14,14) (full-filled) and CASPT2 (half-filled) energies. Experimental oscillator strength values: for naphthalene from * ref 69, for pentacene from ** ref 70, and for the series up to hexacene from *** ref 67.

■

level. We believe that the current study helps to provide a guide to further investigations of the excited states of extended πconjugated molecules. The applicability of this approach for such systems is currently under investigation.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.7b00302. Geometries, characterization of the ground and excited states wave functions, and transition dipole moments (PDF)

■

REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. ORCID

Adélia J. A. Aquino: 0000-0003-4891-6512 Hans Lischka: 0000-0002-5656-3975 Francisco B. C. Machado: 0000-0002-2064-3463 Dana Nachtigallova: 0000-0002-9588-8625 Funding

The authors acknowledge the research and fellowship support of the Fundaçaõ de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under process 2014/24155-6, to Coordenaçaõ de ́ Aperfeiçoamento de Pessoal de Nivel (CAPES), under processes CAPES/ITA No. 23038.005811/2014-89 and CAPES/PVE No. 8881.066022/2014-01, and to Conselho ́ Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq), Process Nos. 307052/2016-8 and 309051/2016-9. We also want to thank the FAPESP/Texas Tech University SPRINT program (project no. 2015/50018-9) for travel support. D.N. acknowledges the support of the Czech Science Fundation (project no. 16-16959S). This work was part of the Research Project RVO: 61388963 of the Institute of Organic Chemistry and Biochemistry Academy of Sciences of the Czech Republic. Notes

The authors declare no competing financial interest. H


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