Benchmarking second order methods for the calculation of vertical electronic excitation energies: valence and Rydberg states in polycyclic aromatic hydrocarbons

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Benchmarking second order methods for the calculation of vertical electronic excitation energies: valence and Rydberg states in polycyclic aromatic hydrocarbons
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  Benchmarking Second Order Methods for the Calculation of Vertical Electronic ExcitationEnergies: Valence and Rydberg States in Polycyclic Aromatic Hydrocarbons † Heidi H. Falden, † Kasper R. Falster-Hansen, † Keld L. Bak, ‡ Sten Rettrup, † andStephan P. A. Sauer* ,†  Department of Chemistry, Uni V  ersity of Copenhagen, Uni V  ersitetsparken 5, DK-2100, Copenhagen Ø, Denmark, and Ingeniørhøjskolen i Århus, Dalgas A V  enue 2, 8000 Århus C, Denmark  Recei V  ed: April 22, 2009; Re V  ised Manuscript Recei V  ed: September 16, 2009 The performance of the six second order linear response methods RPA(D), SOPPA, SOPPA(CCSD), CIS(D),CC2, and CCSD, which include either noniterative or iterative doubles contributions, has been studied incalculations of vertical excitation energies. The benchmark set consisted of 39 valence and 76 Rydberg statesof benzene and five polycyclic aromatic hydrocarbons. As reference values we have used the results of thecorresponding calculations with the third order method CCSDR(3), which includes noniterative triplescontributions. In addition we have also carried out equivalent calculations at the level of the random phaseapproximation as well as with the configuration interaction singles and multireference configuration interactionsingles and doubles methods. Introduction Vertical electronic excitation energies can elegantly andconveniently be calculated with linear response or polarizationpropagator and equation of motion methods. 1 - 4 Several cor-related response theory methods have been developed over thelast 30 years based on multiconfigurational self-consistent field(MCSCF), 5,6 Møller - Plesset perturbation theory (MP), 7 - 17 andcoupled cluster (CC) wave functions 14 - 46 Among the most accurate methods are the CC basedmethods 45,47 - 49 including iterative triple or higher excitationssuch as, e.g., EOM-CCSDT 31,41,50 or CC3, 27,28,31 which is anapproximation to the latter, or several methods which includenoniterative triples corrections, 51 - 66 e.g., CCSDR(3). 55,56 Meth-ods including triple excitations are, however, very expensive,as they scale formally as  N  7 with the number of orbitals  N  . Theyare certainly not suitable yet for routine applications on mediumsize organic molecules.Second order linear response methods, where the singleelectron excitation contribution is evaluated through secondorder in the fluctuation potential and the two-electron contribu-tions are evaluated to lower order, are on the other handapplicable to much larger molecules. Several such approacheshave been presented, which differ mainly in how the contribu-tions of the two-electron excitations are treated. Although weare not dealing here with excitations which are dominated bytwo-electron excitations, the indirect contributions of these termsto the one-electron excitations are important and determinepartially the performance of the second order methods. Someapproaches are based on coupled cluster theory, such asCCSD, 18,21 - 24,30 where the double replacement dominatedexcitations are correct through first order and which scales as  N  6 , or an approximation to it called CC2, 26,29 where the doublereplacement dominated excitations are only correct throughzeroth order and which scales as  N  5 . With the implementationof analytical gradients and the fast resolution of the identityapproximation, 67 - 70 CC2 has become an important tool in thestudy of photochemical reactions.Alternatively methods based on MP directly have beenpresented such as the second order polarization propagatorapproximation (SOPPA), 7,8,10,12,13 where the double replacementdominated excitations are again only correct through zerothorder and thus scales also as  N  5 or SOPPA(CCSD) 14 - 16 whichstill has the same order of the excitations but employs coupledcluster singles and doubles amplitudes instead of the MPcorrelation coefficients. In the propagator calculation SOPPA-(CCSD) scales thus as  N  5 , but the generation of the CCSDamplitudes scales still as  N  6 . For linear response properties suchas frequency dependent polarizabilities, 16,17,71,72 oscillator strengthsum rules, 73 - 75 rotational g-factors, 76 - 79 and in particular indirectnuclear spin - spin coupling constants, 80 - 83 this leads to asignificant improvement over SOPPA, whereas for shieldings 84 a similar effect is not always observed. Excitation energies, onthe other hand, have not been studied yet with SOPPA(CCSD).More approximate and computationally less demanding aremethods which contain noniterative doubles corrections suchas CIS(D) 70,85 or RPA(D) 9,11 which can be derived as ap-proximations to CC2 or SOPPA by applying perturbation theoryto the response theory eigenvalue problem of CC2 and SOPPAusing CCS or RPA as the zeroth order solutions. This is thesame approach as it is employed in the derivation of CCSDR(3)from CC3.Based on a completely different idea are traditional config-uration interaction methods (CI), where excitation energies areobtained as the difference between two explicitly calculatedstates, i.e., eigenvalues of the CI matrix. The simplest approachbeyond taking orbital energy differences as qualitative ap-proximation for excitation energies is CIS. 86 Going beyond thisby inclusion of doubles leads to CISD, which however isnotouriously unbalanced in the calculation of excitation energiesas long as it is based on a single Hartree - Fock reference.Improvements require a multireference approach like MR-CISD, 87 - 89 which, however, becomes quickly too large as longas the whole space of virtual orbitals is included. The way out † Part of the “Walter Thiel Festschrift”.* To whom correspondence should be addressed. E-mail: sauer@kiku.dk. † University of Copenhagen. ‡ Ingeniørhøjskolen i Århus.  J. Phys. Chem. A  2009,  113,  11995–12012  11995 10.1021/jp9037123 CCC: $40.75  ©  2009 American Chemical SocietyPublished on Web 09/25/2009    D  o  w  n   l  o  a   d  e   d   b  y   C   O   P   E   N   H   A   G   E   N   U   N   I   V   L   I   B   R   A   R   Y  o  n   O  c   t  o   b  e  r   2   3 ,   2   0   0   9   |   h   t   t  p  :   /   /  p  u   b  s .  a  c  s .  o  r  g    P  u   b   l   i  c  a   t   i  o  n   D  a   t  e   (   W  e   b   )  :   S  e  p   t  e  m   b  e  r   2   5 ,   2   0   0   9   |   d  o   i  :   1   0 .   1   0   2   1   /   j  p   9   0   3   7   1   2   3  of this dilemma is to employ improved virtual orbitals 90 - 92 andafterward to truncate the virtual space.CC-based as well as MP-based second order linear responsemethods have been employed in calculation of excitation energiesforsmallbenchmarkmolecules 9,16,26,31,33,35,41 - 43,54 - 56,70,93 - 102 aswellon larger organic molecules. 8,10,11,102 - 104,104,105,105 - 112 FurthermoreThiel and co-workers 113 - 115 have recently investigated the perfor-mance of the CC-based methods, CC2, CCSD, CCSDR(3), andCC3, in comparison with each other as well as with CASPT2 andthree DFT approaches for a large set of organic chromophores.However, no systematic comparison of both MP and CC basedlinear response methods for larger molecules has to our knowledgebeen published so far.The goal of this study is therefore to answer the followingquestions:(1) Does CC2 reproduce on average the CCSDR(3) resultsmore closely than CCSD as was seen in the previous studiesby Thiel and co-workers? 113,115 (2) How do the MP based linear response methods performin the calculation of vertical excitation energies compared tocomparable CC based linear response methods using CCSDR(3)as reference?(3) How do the two SOPPA variants, RPA(D) andSOPPA(CCSD), perform compared to the “parent” methodSOPPA?(4) How does RPA(D) perform compared to CIS(D)?As benchmark set we have chosen 39 valence and 76 Rydbergstates singlet excitation energies in polycyclic aromatic hydro-carbons (PAH). PAHs constitute a large class of conjugated π  -electron systems that are key molecular species in manybranches of chemistry, such as interstellar, combustion, envi-ronmental, and materials science. 116 - 118 PAHs are detected, e.g.,in meteorites, in which they are strong candidates for the carriersof interstellar infrared absorption bands. 117 From this group wehave selected naphthalene, anthracene, phenanthrene, azulene,biphenylene, and the parent molecule benzene. The moleculesin this section were selected for their benchmark ability, sincethey have been extensively studied by chemists, physicists,environmental chemists, and so on. A wide range of bothexperimental data and computational results have been pub-lished. Previous calculations on the same systems includeSOPPA calculations on benzene, naphthalene, and anthracene 8,10 and CC2, CCSD, CCSDR(3), and CC3 calculations onbenzene 106,119 with identical basis sets and geometries as wellas CASSCF or CASPT2 studies, 120 - 124 a MRMP study 125 and arecent DFT/MRCI study. 126 Thus the current study presents alsothe first coupled cluster results including triples corrections forthe vertical excitation energies of naphthalene, anthracene,phenanthrene, azulene, and biphenylene. Details of the Calculations.  Vertical excitation energies werecalculated at the RPA, RPA(D), SOPPA, SOPPA(CCSD), CIS,CIS(D), CC2, CCSD, and CCSDR(3) levels of theory. Forbenzene and naphthalene we have also carried out MR-CISDcalculations with modified virtual orbitals. In all calculationsthe frozen core approximation was employed, which in ourprevious study 10 was shown to have no significant effect on thevertical excitation energies.All calculations were carried out with a local developmentversion of the DALTON 2.0 code, 127 which includes the atomicintegral direct implementation of the SOPPA, SOPPA(CCSD)and RPA(D) method, 9,10 and of the PEDICI program 128 - 131 which is interfaced to Dalton. Geometries.  With the exception of benzene we have em-ployed optimized geometries from the literature. They wereprimarily taken from the work of Martin et al., 116 who hadoptimized the geometries of naphthalene (cc-pVTZ), anthracene(cc-pVDZ), phenanthrene (cc-pVDZ), and azulene (cc-pVTZ)employing density functional theory (DFT) with the B3LYPfunctional and the basis set included in the parentheses. Thegeometry of biphenylene, taken from the work of Beck et al., 123 was also optimized at the DFT level with the B3LYP functionalbut with the 6-31G* basis set.Benzene stands out from the rest with an extensive amountof literature and is clearly a very common benchmark molecule.The commonly employed geometry is almost the experimentalgeometry 132 obtained in X-ray studies as reported by Stevenset al. 133 Roos and co-workers 120,122 employed this geometry intheir CASSCF and CASPT2 calculations and consequently alsoPacker et al. 8 in the first SOPPA calculation on benzene. Wecontinue with this practice. Azulene is in a subgroup of thepolycyclic aromatic hydrocarbons called nonalternant hydro-carbons due to the fact that it contains odd-membered rings. Itis isoelectric to naphthalene, and although naphthalene iscolorless and nonpolar, azulene is blue and has a large dipolemoment. An early report of the structure and electronic spectrumof azulene by Pariser 134 suggested that azulene belongs to the C  2 V   symmetry group. But as discussed by Hinchliffe andSoscu´n 135 the structure of azulene in the gas-phase is unknown.At different levels of theory two possibilities emerge: a  C  s  anda  C  2 V   structure with only a small difference between the twogeometries.  C  2 V   is more stable than  C  s  in the ground state.Murakami et al. 136 have calculated the in-plane asymmetric  b 2 normal mode of the 1  A 1  and 1  B 1  states and concluded that thebond equalized  C  2 V   structure yields a more stabilized structurethan the bond-alternating  C  s  structure in the ground state. Forthe calculations in this work the  C  2 V   geometry is therefore used.Phenanthrene like azulene has no center of symmetry. Possibleresonance structures of phenanthrene have been discussed byChakrabarti et al. 137 The classification of biphenylene has beenintensively discussed. 123,138 - 142 Depending on how the electronsare counted, either as two benzene molecules with 6 π   electronsor collectively as 12 π   electrons, which equals 4 n  with  n  )  3,biphenylene should be considered as an aromatic or as anantiaromatic molecule. In the work of Beck et al. 123 it was arguedthat the length of the bonds between the two ring systems islarge enough to ensure that there is no interaction between thetwo ring systems and that biphenylene can be regarded as anaromatic molecule. The experimentally observed ground stateenergy shows, that the bonds connecting the benzene moleculesare indeed unusually long as reported by Fawcett et al. 143 Thesituation is completely different, however, for the first excitedstates, where the four membered ring changes drastically bygoing from a rectangular structure in the ground state to a squarestructure, as shown by Elsaesser et al. 139 Basis Sets.  We have employed the same basis sets as in theprevious SOPPA studies on benzene, naphthalene, and anthra- Figure 1.  Structures of the studied PAHs: benzene, naphthalene,anthracene, phenanthrene (first row starting left), biphenylene, andazulene (second row starting left). 11996  J. Phys. Chem. A, Vol. 113, No. 43, 2009  Falden et al.    D  o  w  n   l  o  a   d  e   d   b  y   C   O   P   E   N   H   A   G   E   N   U   N   I   V   L   I   B   R   A   R   Y  o  n   O  c   t  o   b  e  r   2   3 ,   2   0   0   9   |   h   t   t  p  :   /   /  p  u   b  s .  a  c  s .  o  r  g    P  u   b   l   i  c  a   t   i  o  n   D  a   t  e   (   W  e   b   )  :   S  e  p   t  e  m   b  e  r   2   5 ,   2   0   0   9   |   d  o   i  :   1   0 .   1   0   2   1   /   j  p   9   0   3   7   1   2   3  cene. 8,10 They are of the atomic natural orbital (ANO) type andconsist of the C[4s3p1d]/H[2s1p], i.e., ANO1, basis set of Widmark et al. 144 However, in order to obtain a reasonabletreatment of the Rydberg excitations diffuse functions must beadded to these basis set. Due to the size of the Rydberg orbitalsit is suffice to put the diffuse functions at the center of mass(CM) of the molecule. The extra diffuse functions consist of 8even tempered s, p, and d sets of functions contracted to[1s1p1d] for benzene, 122 2 sets of uncontracted s, p, and dfunctions (2s2p2d) for naphthalene 121 and 3 sets of uncontracteds, p, and d functions (3s3p3d) for anthracene. 10 The extra diffusefunctions of naphthalene were also used in the calculations onazulene, and similarly the same diffuse functions were employedfor biphenylene and phenanthrene as for anthracene. Thecoefficients of the diffuse functions are listed in Table 1 wherethe contraction coefficients are given in parentheses. Additionof more diffuse functions was previously shown 29,106,119 tochange the results by less than 0.05 eV. Results and Discussion Vertical excitation energies have been calculated for benzene,naphthalene, anthracene, phenanthrene, azulene, and biphenyleneusing the RPA, RPA(D), SOPPA, SOPPA(CCSD), CIS, CIS(D),CC2, CCSD, and CCSDR(3) methods. In addition MR-CISDcalculations were carried out for benzene and naphthalene usingmodified virtual orbitals. We use the CCSDR(3) results asreference, with which we compare the results of the second ordermethods, because CCSDR(3) was shown to reproduce almostquantitatively the results of CC3 calculations not only for smallmolecules 55,56,101 but also for benzene, 106 naphthalene 115 and otherorganic chromophores 102,107,108,110,115 with only a few but predict-able exceptions. Experimental values 145 - 162 can be found forsome of the states included in our study. Benzene.  The calculated results for the lowest 12 singletexcitation energies of benzene are listed in Table 2. The statesare grouped according to their type of transitions, e.g. valenceor Rydberg. Within a group they are given in the order of increasing CCSDR(3) excitation energy. Only the  n  )  3Rydberg series, which converges to the first ionization potentialof 9.25 eV, was included in our study. The assignment of thetransitions follows that of Lorentzon et al. 122 They include twovalence states, 1  B 2 u  and 1  B 1 u , one mixed valence and Rydbergstate, 1  E  1 u , and 9 dominant Rydberg states. The weight of thesingle excitations, %  R 1 , in the CCSD excitation energycalculations is 95% for all states apart from the 1  B 2 u  state, whereit is only 91%.Considering the order of the calculated states first, we cansee that the iterative second order methods, SOPPA, SOP-PA(CCSD), CC2 and CCSD, as well as the noniterativemethods, RPA(D) and CIS(D), and MR-CISD reproduce theorder of the excited states as found in the CCSDR(3) calcula-tions. However, one should note, that we only consider thelowest excited state in every irreducible representation. CIS andRPA, on the other hand, have problems reproducing the orderof the 1  E  2 g  and 2  A 1 g  and of the 1  B 1 g  and 1  B 2 g  Rydberg states asgiven by CCSDR(3). Both pairs of transitions are predicted tobe energetically very close by all second-order methods as wellas by the reference CCSDR(3). Since CIS(D) and RPA(D) bothreproduce the correct order of states, we have here the firstexample for the important effect of the noniterative doublescorrections in CIS(D) and RPA(D). TABLE 1: Coefficients of the Diffuse Functions molecule s p dbenzene [1s1p1d] 0.024624 (0.4584) 0.042335 (0.0290) 0.060540 (0.0290)0.011253 ( - 2.0379) 0.019254 ( - 0.2025) 0.027446 ( - 0.2025)0.005858 (1.9778) 0.009988 ( - 0.2629) 0.014204 ( - 0.2629)0.003346 ( - 3.1952) 0.005689 ( - 0.4338) 0.008077 ( - 0.4338)0.002048 (3.7239) 0.003476 (0.0101) 0.004927 (0.0101)0.001324 ( - 3.1770) 0.002242 ( - 0.1587) 0.003175 ( - 0.1587)0.000893 (1.7028) 0.001511 (0.0831) 0.002137 (0.0831)0.000624 ( - 0.4214) 0.001055 ( - 0.0244) 0.001491 ( - 0.0244)naphthalene [2s2p2d] 0.009614 0.008778 0.006270azulene 0.003542 0.003234 0.002310anthracene [3s3p3d] 0.010000 0.010000 0.010000biphenylene 0.003300 0.003300 0.003300phenanthrene 0.001100 0.001100 0.001100 TABLE 2: Benzene: Vertical Singlet Excitation Energies (in eV) in Ascending Order of CCSDR(3) Energies state RPA RPA(D) SOPPA SOPPA(CCSD) CIS CIS(D) CC2 CCSD CCSDR(3) MR-CISDValence  ππ  *1  B 2 u  ( e 1 g f e 2 u ) 5.82 4.82 4.69 4.52 6.03 5.31 5.27 5.19 5.12 5.771  B 1 u  ( e 1 g f e 2 u ) 5.88 6.36 6.01 5.92 6.19 6.68 6.56 6.59 6.56 6.731  E  1 u  ( e 1 g f e 2 u  /3  p  z ) 7.16 6.87 7.03 6.94 7.18 7.06 7.01 7.16 7.15 7.34Rydberg  ππ  *1  E  2 g  ( e 1 g f (3 d   xz  /   yz )) 7.80 7.54 7.54 7.48 7.80 7.66 7.64 7.84 7.85 7.982  A 1 g  ( e 1 g f (3 d   xz  /   yz )) 7.77 7.55 7.55 7.49 7.77 7.67 7.65 7.85 7.86 7.991  A 2 g  ( e 1 g f (3 d   xz  /   yz )) 7.85 7.57 7.58 7.52 7.85 7.68 7.67 7.87 7.88 8.00Rydberg  πσ  *1  E  1 g  ( e 1 g f 3 s ) 6.55 6.20 6.17 6.12 6.55 6.38 6.33 6.47 6.45 6.681  A 2 u  ( e 1 g f (3  p  x   /   y )) 6.94 6.70 6.69 6.63 6.94 6.86 6.83 6.98 6.98 7.181  E  2 u  ( e 1 g f (3  p  x   /   y ) ) 7.11 6.76 6.75 6.69 7.12 6.91 6.88 7.05 7.04 7.251  A 1 u  ( e 1 g f (3  p  x   /   y )) 7.28 6.84 6.83 6.77 7.29 6.97 6.96 7.13 7.12 7.341  B 1 g  ( e 1 g f 3 d   xy ) 7.70 7.35 7.34 7.28 7.70 7.47 7.45 7.65 7.65 7.831  B 2 g  ( e 1 g f 3 d   xy ) 7.68 7.35 7.35 7.29 7.68 7.47 7.46 7.65 7.66 7.83 Calculation of Vertical Excitation Energies  J. Phys. Chem. A, Vol. 113, No. 43, 2009  11997    D  o  w  n   l  o  a   d  e   d   b  y   C   O   P   E   N   H   A   G   E   N   U   N   I   V   L   I   B   R   A   R   Y  o  n   O  c   t  o   b  e  r   2   3 ,   2   0   0   9   |   h   t   t  p  :   /   /  p  u   b  s .  a  c  s .  o  r  g    P  u   b   l   i  c  a   t   i  o  n   D  a   t  e   (   W  e   b   )  :   S  e  p   t  e  m   b  e  r   2   5 ,   2   0   0   9   |   d  o   i  :   1   0 .   1   0   2   1   /   j  p   9   0   3   7   1   2   3  Comparing now the results of the different methods with theCCSDR(3) results we find that the deviations of the RPA resultsfrom the CCSDR(3) reference values depend extremely on thestate under consideration. For the two valence states we findvery large deviations + 0.70 and - 0.68 eV, for 1  B 2 u  and 1  B 1 u ,and a very small one with only 0.01 eV for the mixed 1  E  1 u state, whereas the deviations for the Rydberg states range onlybetween  - 0.09 and  + 0.16 eV. Inclusion of the second orderand doubles corrections in RPA(D) changes this dramatically.The deviations from the CCSDR(3) reference values are thusin the range from - 0.20 to - 0.31 eV with the majority of energydifferences between  - 0.28 and  - 0.31 eV. All of the RPA(D)energies are lower than the CCSDR(3) values. The smallestdeviation is for the 1  B 1 u  transition, whereas the largest deviationsare found for the 1  E  2 g , 2  A 1 g , 1  A 2 g , and 1  B 2 g  states, i.e., for thetransitions to the 3d Rydberg orbitals. This implies that the twovalence states are in much better agreement with the referencevalues than at the RPA level but that the Rydberg states areactually in worse agreement. The SOPPA results are also alllower than the reference energies and the deviations are in therange from - 0.12 to - 0.55 eV. Most deviations are about - 0.30eV with three exceptions: the two valence transitions, 1  B 2 u  and1  B 1 u , with - 0.43 and - 0.55 eV and the 1  E  1 u  mixed transitionwith - 0.12 eV. The deviations for the Rydberg transitions varyonly between  - 0.28 and  - 0.31 eV. The deviations of theSOPPA(CCSD) results follow the SOPPA results with respectto which transitions are reproduced poorly. The range of deviations in the excitation energies is - 0.21 to - 0.64 eV againwith the same three exceptions, the valence transitions: the 1  B 2 u transition with  - 0.60 eV, the 1  B 1 u  transition with  - 0.64 eV,and the mixed 1  E  1 u  transition with - 0.21 eV. Taking these statesaside the remaining differences from the reference results forthe Rydberg transitions are in the range of  - 0.35 to - 0.37 eV.CIS, like RPA, does remarkably well for the Rydberg statesand quite badly for the valence states included in our study.However, whereas the CIS results for the Rydberg states areindistinguishable from the RPA results, the valence states arepredicted to be higher in energy. Similar to RPA(D), the additionof the second order and doubles corrections in CIS(D) leads toa much more uniform deviation from the CCSDR(3) results,although not as uniform as in the case of RPA(D). This impliesan improved agreement with the CCSDR(3) results for the twovalence states and a more uniform but in general worseagreement for the Rydberg transitions. On the other hand theCIS(D) results are all shifted to higher energies and thus in betteragreement with the reference values than the RPA(D) results.The CC2 results underestimate the reference results for theRydberg transitions by - 0.21 to - 0.12 eV which is in all casesslightly more than found in CIS(D). The situation is quitedifferent for the valence states: the result for the 1  B 1 u  state is inperfect agreement, whereas the 1  B 2 u  is overestimated by 0.15eV. Compared to SOPPA the CC2 results are all in betteragreement with the CCSDR(3) results, as was already pointedout previously. 106 The CCSD results are very close to thereference values, as should be expected since the CCSDR(3)method adds only a correction to the calculated CCSD energies.The valence states are higher in energy than CCSDR(3) valueswith deviations in the range of 0.01 - 0.07 eV. Again a slightlypoorer description of valence states is observed compared tothe description of the Rydberg states.The multireference space in the MR-CISD calculations isobtained by including all the configuration state functions(CSF’s) generated by all single excitations from the highestoccupied  e 1 g  orbitals into the lowest 11 optimized virtual orbitals.It leads to 23 reference CSF’s and a MR-CISD space formedby 28 935 182 spin adapted CSF’s after truncating the totalorbital space to 100 orbitals.The MR-CISD results are a clear improvement over the CISresults for the two valence states but are in worse agreementfor the Rydberg states than CIS. Compared with the othermethods including double excitations, we can see that we obtainsignificantly higher excitation energies in our MR-CISD cal-culations. This applies to the Rydberg states and in particularto the lowest excited valence state. Naphthalene.  The calculated results for 21 singlet excitedstates of naphthalene are listed in Table 3. The states are againgrouped according to their type of transitions, e.g., valence orRydberg and are sorted according to increasing CCSDR(3)excitation energy. The assignment of the transitions follows thatof Bak et al. 10 They include eight valence states, eleven Rydbergstates and two states, 2  B 1 g  and 4  B 2 u , which in earlier calculationsemploying smaller basis sets 121,8 were assigned as valencetransitions, but were later shown to exhibit significant Rydbergcharacter. 10 The weight of the single excitations, %  R 1 , in theCCSD excitation energy calculations is 95% for all Rydbergstates, whereas the valence states, and in particular the 1  B 3 u and 2  A g  states with only 91%, have smaller weights.Looking at the order of the states first again, we can see thatRPA and CIS are less able to reproduce the order of thetransitions as given by CCSDR(3) than in benzene. The twolowest states, 1  B 3 u  and 1  B 2 u , are interchanged, the energies of the 2  A g  and 3  B 1 g  states are predicted to be much too high andthe order of the two pairs of states 2  B 2 g  and 2  B 3 g  and 3  B 2 u  and3  B 3 u  are interchanged. Inclusion of the doubles contribution inRPA(D) and CIS(D) corrects some of these errors but not all.The two lowest valence states, e.g., have now the correct order,but the 2  A g , 3  A g , and 3  B 3 u  states are now too low in energy.Furthermore the 2  A u  and 2  B 1 u  transitions as well as the 2  B 2 u and 3  B 1 g  states and the 2  B 1 g  and 3  A g  transitions are now reversedby a few hundreds of an eV contrary to RPA and CIS. Theiterative second order methods show the correct ordering withone exception, the 2  A g  state relative to the 2  B 3 u  state, which atthe CCSDR(3) level is 0.19 eV higher in energy than 2  A g , buthas a lower energy at the SOPPA, SOPPA(CCSD) and eventhe CC2 level.Comparing now the results of the different methods with theCCSDR(3) results we find that like for benzene the deviationsof the RPA and CIS results for the valence states are partiallyvery large and depend strongly on the state. The RPA energiesare always smaller than the CIS energies and are in betteragreement with the CCSDR(3) results with one exception. Therange of deviations from the CCSDR(3) results is  - 0.26 to + 1.14 eV for RPA and 0.04 to 1.23 eV for CIS. The agreementfor the Rydberg transitions is better. The RPA and CIS resultsare identical and are scattered around the CCSDR(3) resultswith error bars of about  ( 0.2 eV, if we exclude the mixedvalence - Rydberg 2  B 1 g  state. This implies again that only theCCSD results are in better agreement with the CCSDR(3) resultsfor the Rydberg states. Adding the second order doublescorrections in CIS(D) reduces the deviations significantly forall valence states but the 1  B 2 u  state in CIS(D). RPA(D) on theother hand overestimates the corrections for three states, 1  B 1 g ,2  B 3 u , and 2  B 2 u , which are now too low in energy. Nevertheless,the spread of deviations for the valence states is very similar inboth methods,  - 0.80 to  + 0.03 eV for RPA(D) and  - 0.45 to + 0.33 eV for CIS(D), with the CIS(D) results just being shiftedto higher energies. For the Rydberg states we observe also againthat the doubles corrections, destroys the good agreement with 11998  J. Phys. Chem. A, Vol. 113, No. 43, 2009  Falden et al.    D  o  w  n   l  o  a   d  e   d   b  y   C   O   P   E   N   H   A   G   E   N   U   N   I   V   L   I   B   R   A   R   Y  o  n   O  c   t  o   b  e  r   2   3 ,   2   0   0   9   |   h   t   t  p  :   /   /  p  u   b  s .  a  c  s .  o  r  g    P  u   b   l   i  c  a   t   i  o  n   D  a   t  e   (   W  e   b   )  :   S  e  p   t  e  m   b  e  r   2   5 ,   2   0   0   9   |   d  o   i  :   1   0 .   1   0   2   1   /   j  p   9   0   3   7   1   2   3  the CCSDR(3) results by shifting the energies 0.2 to 0.4 eV(RPA(D)) or 0.1 to 0.3 eV (CIS(D)) toward lower energies. TheCIS(D) pure Rydberg excitation energies are thus all higher thanthe RPA(D) ones, but also smaller than the CCSDR(3) values.In SOPPA all valence states but the 3  B 1 g  and 4  A g  states arelower in energy than in RPA(D) and thus in worse agreementwith the CCSDR(3) reference values. However, the spread of deviations is slightly reduced to between - 0.67 and - 0.13 eV.Replacement of the Møller - Plesset correlation coefficients byCCSD amplitudes in SOPPA(CCSD) reduces the valenceexcitation energies even further and thus increases the deviationsfrom the CCSDR(3) results. The same trend is also observedfor the Rydberg states, where the SOPPA(CCSD) results areabout 0.1 eV lower than the SOPPA results, which are for themajority of the states also lower than the RPA(D) results.However, the span of deviations is also reduced in the seriesRPA(D), SOPPA and SOPPA(CCSD) to only 0.06 eV in thelatter method, meaning that SOPPA(CCSD) underestimates theCCSDR(3) results for the Rydberg states by 0.42  (  0.03 eV.The two coupled cluster methods, CC2 and CCSD, on theother hand, show the smallest deviations from the CCSDR(3)results for the valence states. CCSD always overestimates theCCDSR(3) results, whereas CC2 underestimates the majorityof the CCSDR(3) results. For 6 out of the 8 valence states CC2is in better agreement with the CCSDR(3) valence state valuesthan CCSD. For the Rydberg states the situation is reversed.The CCSD results are basically indistinguishable from theCCSDR(3) values, whereas the CC2 results are for the majorityof the Rydberg states in slightly worse agreement than theCIS(D) values.The multireference space in the MR-CISD calculations isobtained by including all the configuration state functions(CSF’s) generated by all single excitations from the highestoccupied orbitals, 1 b 2 g , 1 b 3 g , 2 b 1 u , and 1 a u , into a the lowest 12virtual orbitals. It leads to 49 reference CSF’s and a MR-CISDspace formed by 103 861 609 spin adapted CSF’s. As in thebenzene case the total orbital space is truncated to 100.For the MR-CISD results we observe quite a differentbehavior than for benzene. First of all the MR-CISD resultsare not always higher than the CCSDR(3) results. Second, theMR-CISD results are only for three of the valence states, 1  B 3 u ,2  B 3 u , and 2  B 2 u , a clear improvement over the CIS results. Allthe valence excitation energies are overestimated for the MR-CISD calculations compared with CCSDR(3). Generally, theRydberg states are are in good agreement with the CCSDR(3)results apart from the states 3  B 2 u , 3  B 3 u , and 2  B 1 g  where theexcitation energies are overestimated by 0.51, 0.30, and 0.51eV which is more than the CIS estimates. Azulene.  For azulene we have calculated excitation energiesto 4 valence and 16 Rydberg states. The results are given inTable 4. The weight of the single excitations, %  R 1 , in the CCSDexcitation energy calculations is ≈ 95% for all Rydberg states,whereas the valence states, and in particular the 1  B 2  and 2  B 2 states with only 91%, have smaller weights.The 5  A 1  and 6  A 1  Rydberg states are almost degenerate at theCCSDR(3) level, the latter is less than 0.01 eV higher in energy.All other methods, however, underestimate the energy of the6  A 1  Rydberg state slightly more than the energy of the 5  A 1 Rydberg state and predict therefore the wrong order of thesetwo Rydberg states. Experimentally these two states are alsovery close as reported by Foggi et al. 162 RPA and CIS get alsothe order of the 3  B 1  and 4  B 2  Rydberg states wrong, which iscorrected at the RPA(D) but not at the CIS(D) level. FurthermoreCIS(D) predicts the 4  A 1  Rydberg state to be lower than the 4  B 2 state contrary to all other methods.For the valence states we find again that the deviations of the RPA and CIS results from the CCSDR(3) reference resultsand from each other are are partially very large and dependstrongly on the state. For three out of the four states, the RPAresults are closer to the reference values than the CIS results.Adding the noniterative second order corrections in RPA(D)leads to large changes and improves the agreement withCCSDR(3) for two states. All RPA(D) excitation energies arealso smaller than the reference values. In the iterative MP TABLE 3: Naphthalene: Vertical Singlet Excitation Energies (in eV) in Ascending Order of CCSDR(3) Energies state RPA RPA(D) SOPPA SOPPA(CCSD) CIS CIS(D) CC2 CCSD CCSDR(3) MR-CISDValence  ππ  *1  B 3 u  (1 a u f 4 b 3 g ) 5.00 4.01 3.88 3.64 5.22 4.51 4.47 4.44 4.38 4.911  B 2 u  (1 a u f 4 b 2 g ) 4.75 4.94 4.34 4.17 5.08 5.23 4.89 5.13 5.01 5.471  B 1 g  (1 a u f 5 b 1 u ) 6.04 5.80 5.56 5.45 6.06 5.98 5.87 6.08 6.02 6.062  A g  (2 b 1 u f 5 b 1 u ) 7.22 6.11 5.65 5.48 7.31 6.41 6.18 6.21 6.08 6.73(2 b 1 u f 4 b 2 g )2  B 3 u  (1 a u f 4 b 3 g ) 6.50 5.85 5.65 5.38 7.02 6.38 6.15 6.43 6.27 6.66(2 b 1 u f 4 b 2 g )2  B 2 u  (2 b 1 u f 4 b 3 g ) 6.73 5.99 5.97 5.74 7.21 6.40 6.47 6.66 6.51 6.683  B 1 g  (1 b 3 g f 4 b 2 g ) 7.86 5.96 6.26 6.14 7.91 6.31 6.70 6.91 6.76 7.494  A g  (1 b 2 g f 4 b 2 g ) 7.78 7.40 7.43 7.29 7.78 7.48 7.50 7.71 7.56 8.46Rydberg  ππ  *1  B 2 g  (1 a u f 3  p  y ) 5.97 5.69 5.66 5.58 5.97 5.85 5.80 6.00 5.99 5.951  B 3 g  (1 a u f 3  p  x  ) 6.03 5.71 5.68 5.60 6.03 5.87 5.82 6.02 6.00 6.002  A u  (1 a u f 3 d   x  2 -  y 2 ) 6.53 6.27 6.24 6.15 6.53 6.39 6.35 6.58 6.57 6.512  B 1 u  (1 a u f 3 d   xy ) 6.58 6.26 6.28 6.19 6.58 6.38 6.38 6.62 6.61 6.662  B 2 g  (2 b 1 u f 3  p  x  ) 6.81 6.39 6.36 6.23 6.82 6.54 6.50 6.67 6.67 6.722  B 3 g  (2 b 1 u f 3  p  y ) 6.90 6.51 6.41 6.28 6.90 6.63 6.55 6.72 6.72 6.773  B 2 u  (1 a u f 3 d   xz ) 6.67 6.46 6.46 6.38 6.67 6.57 6.55 6.80 6.80 7.313  B 3 u  (1 a u f 3 d   yz ) 6.74 6.40 6.50 6.41 6.64 6.49 6.58 6.85 6.84 7.144  B 2 u  (1 a u f 4 d   xz ) 7.20 6.97 6.98 6.89 7.20 7.00 7.01 7.34 7.35 7.48Rydberg  πσ  *1  A u  (1 a u f 3 s ) 5.60 5.30 5.26 5.17 5.60 5.47 5.42 5.59 5.57 5.551  B 1 u  (2 b 1 u f 3 s ) 6.39 6.02 5.96 5.83 6.39 6.19 6.12 6.25 6.24 6.292  B 1 g  (1 a u f 3  p  z ) 6.56 6.37 5.97 5.86 6.75 6.66 6.25 6.58 6.39 6.903  A g  (2 b 1 u f 3  p  z ) 6.87 6.34 6.61 6.48 6.89 6.60 6.76 6.90 6.91 7.02 Calculation of Vertical Excitation Energies  J. Phys. Chem. A, Vol. 113, No. 43, 2009  11999    D  o  w  n   l  o  a   d  e   d   b  y   C   O   P   E   N   H   A   G   E   N   U   N   I   V   L   I   B   R   A   R   Y  o  n   O  c   t  o   b  e  r   2   3 ,   2   0   0   9   |   h   t   t  p  :   /   /  p  u   b  s .  a  c  s .  o  r  g    P  u   b   l   i  c  a   t   i  o  n   D  a   t  e   (   W  e   b   )  :   S  e  p   t  e  m   b  e  r   2   5 ,   2   0   0   9   |   d  o   i  :   1   0 .   1   0   2   1   /   j  p   9   0   3   7   1   2   3
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