J. Phys. Chem. A 2010, 114, 8591–8600
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A Multireference Coupled-Cluster Study of Electronic Excitations in Furan and Pyrrole† Xiangzhu Li* and Josef Paldus‡ Department of Applied Mathematics, UniVersity of Waterloo, Waterloo, Ontario, Canada N2L 3G1 ReceiVed: December 7, 2009; ReVised Manuscript ReceiVed: January 15, 2010
A multireference (MR), general-model-space (GMS), state-universal (SU), coupled-cluster (CC) method that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space (GMS SU CCSD) is employed to investigate a number of low-lying excited states of two five-membered, six π -electron aromatic ring molecules, namely, furan and pyrrole. An extended basis set that includes diffuse functions and molecule-centered Rydberg functions is employed. Computed vertical excitation energies are compared with experimental values as well as with other theoretical results, namely, those obtained by the equation-of-motion (EOM) CCSD method or, equivalently, the symmetry adapted cluster configuration interaction (SAC-CI) method, by the MR CI method, by MR perturbation theory, and by other methods whenever available. For excited states dominated by single-excitations the GMS SU CCSD energies represent an improvement relative to the most closely related EOM CCSD results by about 0.1 eV. I. Introduction The standard ab initio post-Hartree-Fock approaches that can handle excited states include the multireference (MR) configuration interaction (CI) method,1 MR perturbation theory (PT),2 and, within the framework of coupled-cluster (CC) formalism,3-6 the equation-of-motion (EOM) CC method,7 the linear response (LR) type CC methods,8 and the symmetry adapted cluster CI (SAC-CI) method.9,10 The EOM and LR CC, as well as the SAC-CI, approaches are conceptually identical, except for some computational approximations involving perturbation selection characterizing the latter approach. More recently, we have formulated a general-model-space (GMS) version11-14 of the multireference (MR), state-universal (SU), coupled-cluster (CC) method,15 which enables the use of arbitrary references and, hence, can be used to compute excited states, providing an alternative option for the treatment of excited states in a size-extensive fashion. In the EOM, LR CC, and SAC-CI approaches (for an overview and literature see, e.g., refs 3-7 and 10), one first generates the ground-state SR CC wave function and subsequently computes excitation energies relative to this state using a linear Ansatz for the excitation operator (e.g., as response properties in the LR CC theory). No wave functions are actually constructed for the excited states. These approaches perform well at the SD level unless the states are dominated by double excitations, in which case triple corrections are required.16 They also cannot be expected to perform well when the SR CCSD reference does not provide a good description of the ground state due to a quasi-degeneracy. However, a renormalization of triple corrections, as introduced by Piecuch and Kowalski,17-19 makes it possible to largely overcome this problem (see ref 20 for an exhaustive list of references). Nonetheless, it would be highly desirable to have a genuine MR CC approach that can handle highly degenerate situations (for a discussion of relative merits of SR vs MR CC approaches, †
Part of the “Klaus Ruedenberg Festschrift”. * To whom correspondence should be addressed. ‡ Also at Department of Chemistry and Guelph-Waterloo Center for Graduate Work in Chemistry (GWC)2sWaterloo Campus, University of Waterloo, Waterloo, ON, Canada N2L 3G1.
see ref 21). In its original formulation, the genuine MR SU CC method15 relies on a complete model space (CMS). In actual applications, this requirement restricts the method to very small model spaces, involving only one or two occupied and virtual active molecular orbitals (MOs). For this and other reasons (e.g., the likely presence of intruder states when using a CMS) the MR SU CC approach has been employed in a very limited number of studies. These were generally based on model spaces involving only two active MOs, namely, the highest occupied MO (HOMO) and the lowest unoccupied MO (LUMO).22-32 In all such studies, the emphasis was on obtaining a good description of the ground state by accounting for the presence of a quasi-degeneracy via the MR formalism rather than on the description of the excited states. As is well-known, the low-lying excited states arise primarily via single excitations (and only in special cases via double excitations; cf., e.g., ref 16), which are associated with a large active space, rather than via excitations spanning a small model space. Of course, when we employ a large enough set of active MOs, any model space that is spanned by some proper subset of excited configurations can be regarded as a truncated version of a suitable CMS and may thus be referred to as an incomplete model space (IMS) (for various types of IMSs and their use in MR CC theories, see, e.g., refs 33-37 and references therein). However, we prefer to use the term GMS in order to emphasize the arbitrariness, at least in principle, of the set of configurations spanning the model space, the choice of which is not necessarily based on the grouping of orbitals into the core, active (valence), and virtual (excited) subsets, but rather on the role these configurations play in the excited states of interest. The abovementioned GMS SU CCSD approach employs such a general model space, while preserving size extensivity13 as well as the intermediate normalization. The use of a GMS is also essential if we wish to avoid the detrimental effect of intruder states. The key feature of the GMS SU CC method is the handling of those cluster amplitudes that are associated with the so-called internal excitation operators, namely, those that interrelate configurations spanning a chosen model space. In order to preserve the intermediate normalization, we require an appropriate cancellation of such internal amplitudes by the corresponding
10.1021/jp911602k 2010 American Chemical Society Published on Web 02/15/2010
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disconnected amplitudes of the same rank (cf. also ref 15). Thus, only those amplitudes that are associated with external excitations (relative to a chosen GMS) are explicitly accounted for and evaluated by solving the SU CC equations. A preliminary test of the performance of the GMS SU CCSD method on small molecular systems not only showed the feasibility of MR CC calculations with model spaces having a rather large dimension (up to and equal to 14) but also yielded excellent results.12,16,38,39 Thus, for example, in the case of the cc-pVDZ model of the water molecule, when a comparison can be made with almost full CI energies, the discrepancies are extremely small, amounting, on the average, to less than 0.1 eV (and much less for the lowest lying states). Once larger basis sets are used, an excellent agreement with experiment and some high level CI results is found as well.38 Needless to say that for a realistic description of excited states at the MR CC level, particularly for the higher-lying ones (namely, those lying above the lowest state of a given symmetry), we have to employ not only genuine MR approaches but also sufficiently large basis sets involving both polarization and diffuse functions, as well as molecule-centered Rydberg functions. In such cases some excited states will involve more than one dominant configuration, implying an even larger number of leading Slater determinants. It is thus important to investigate how well the GMS SU CCSD method performs in such circumstances and how reliable are the results it yields when compared with experimental observations. In this study, we apply the GMS CCSD method to investigate excited states of the two medium-sized molecules: furan and pyrrole. Note that no MR CC approach of any kind has ever been applied to molecules of this size. Furan and pyrrole are five-membered, six π-electron, aromatic-ring molecules. Their low-lying excited states are associated with excitations from the π-bonding MOs into various virtual MOs of valence or Rydberg type. Even when we restrict our attention to vertical excitation energies, a proper account of the excited states of furan and pyrrole represents a very challenging task, primarily due to a strong valence-Rydberg mixing, requiring a sophisticated account of electron correlation effects. There has been a great deal of interest in the excited states of furan and pyrrole, resulting in a large number of both experimental40-53 and theoretical54-62 studies, some combining theory with experiment. Thanks to these extensive studies it is possible to make a thorough comparison with the existing theoretical and experimental data and thus assess the performance of our GMS CCSD approach. We must keep in mind, however, thatsdespite the enormous progress in the experimental techniquessthe detailed nature of some of the low-lying electronic states of furan and pyrrole has yet to be firmly established, since its spectra involve broad bands having a complex structure thanks to the presence of a large number of valence and Rydberg excited states with a closely spaced energies. Thus, the band maxima and vertical excitation energies may differ by as much as 0.2 eV, affecting a comparison with theoretical results. In this regard, our work not only enables an assessment of the performance of our MR SU CCSD method in handling of excited states with a strong valence-Rydberg mixing but also can meaningfully contribute to our knowledge of the excited states of these molecular systems. In section II we briefly outline the methods employed, namely, the aforementioned GMS SU CCSD method, as well as its externally corrected version, referred to by the acronym (N,M)-CCSD. The computational details are then described in
Li and Paldus section III, and the results and their discussion are presented in section IV. Finally, our results are summarized in section V. II. Method The SU CC methods are based on the Jeziorski and Monkhorst Ansatz for the wave operator.15 The wave function for the jth state |Ψj〉 has the form M
|Ψj〉 )
∑ cijeT(i)|Φi〉
(1)
i)1
where T(i) designates the cluster operator that is associated with the reference |Φi〉. The operator T(i) is then expressed as a linear combination of the excitation operators Gl(k)(i) relative to the ith reference |Φi〉, with the coefficients tl(k)(i) designating the corresponding cluster amplitudes
T(i) )
∑ Tk(i) ) ∑ tl(k)(i)Gl(k)(i) k
(2)
l,k
Here the superscript (k) designates the excitation level relative to the reference |Φi〉, while the subscript l labels the individual excitations and implicitly defines k (so that we can drop the superscript (k) once we know the explicit form of l). In practical applications, the cluster operators T(i) are truncated at the oneand two-body level
T(i) ) T1(i) + T2(i)
(3)
leading to the SU CCSD method. Choosing a CMS as the reference or model space, one includes configurations Gl(k)(i)|Φi〉 given by all possible distributions of a given number of electrons among a given number of MOs. The corresponding cluster amplitudes tl(k) are then obtained by solving the appropriate SU CC equations, whose general form is15
〈Gl(i)Φi |e-T(i)HeT(i) |Φi〉 )
∑ 〈Gl(i)Φi|e-T(i)eT(j)|Φj〉Hji(eff)
j(*i)
(4) Here Hji(eff) designate the matrix elements of the effective Hamiltonian. The cij coefficients in Ansatz (1) are given by the components of the eigenvectors of the effective Hamiltonian (cf., e.g., refs 4-6, 63, and 64). In the presence of a spatial symmetry of the nuclear framework, the effective Hamiltonian will factorize into distinct blocks assuming that the reference configurations belong to different symmetry species of the pertinent point group. In order that the SU CCSD method will provide a good description of the states of interest, the references |Φi〉, spanning the model space, must represent leading configurations (determinants) for these states. In the case of furan and pyrrole, the leading determinants of the relevant low lying singlet and triplet states are singly excited determinants involving appropriate occupied and virtual MOs. To handle simultaneously more than one excited state of a given symmetry, a number of singly excited determinants must be employed as references in order to yield a corresponding number of excited states. Clearly, the model space spanned by such reference configurations is not generally a CMS, but an IMS. Hence, we have to rely on the
Electronic Excitations in Furan and Pyrrole GMS version of the SU CCSD method, taking special precautions, as represented by the so-called C-conditions for the internal amplitudes.11-14 In brief, the GMS SU CCSD method requires a solution of CC equations for the external amplitudes, eq 4, while simultaneously accounting for the internal amplitudes via the C-conditions. The GMS SU CCSD method employs only one- and twobody cluster operators relative to each reference. However, as in the SR CC case, higher-than-pair clusters can play a nonnegligible role. In order to account for the most important higher-than-pair clusters, we can exploit a concept of the externally corrected (ec) CC theory.4,5,65,66 For this purpose we employ some external source, in particular the modest-size MR CISD wave functions, and extract a small but important subset of t(3) and/or higher-order cluster amplitudes via the cluster analysis67 and use them subsequently to correct the GMS SU CCSD equations. In this way we arrive to a natural extension of the reduced multireference (RMR) CC method68 to the MR case. Specifically, we first carry out an N-reference CISD and extract higher-than-pair cluster amplitudes from the M pertinent MR CISD wave functions (clearly, M e N; for the cluster analysis of the MR CI wave functions, see ref 67). These higherthan-pair cluster amplitudes are then used to correct the M-reference SU CCSD equations, similarly as in the RMR CCSD case. This approach is referred to as the (N, M)-CCSD method and has been described in detail in ref 69. The usefulness of the (N,M)-CCSD approach in handling of the low-lying excited states can be rationalized by the following example: A single excitation a f i of a closed-shell ground state gives rise to a singlet and a triplet excited states which can be described by a two-reference (2R) SU CCSD approach that is based on the reference determinants |(...)aıj| ≡ |(...)aRiβ| and |(...)aji|. Here the symbol (...) indicates the doubly occupied part of the configuration. Suppose, next, that in this singlet and/ or triplet state another single excitation b f j or, equivalently, determinants |(...)bjj| and |(...)bjj|, also play an important role. In such a case, it is sensible to employ a (4,2)-CCSD approach, in which the 2R SU CCSD is externally corrected by a fourreference (4R) CISD. Here the 4R configurations are the aforementioned four determinants. Should another single excitation be also important, we may employ a (6,2)-CCSD approach, etc. The main reason for using an (N,M)-CCSD method (M < N) rather than an N-reference SU CCSD method is to avoid the intruder states, which are more likely to occur in the latter approach. In the (N,M)-CCSD approach, M determinants are employed as references in the SU CC method, so that the dimension of the effective Hamiltonian is M. The effect of the remaining (N - M) determinants, which are also important, but not included in the M-reference model space, are taken into account indirectly via the higher-than-two-body terms that are determined from the MR CISD wave functions. Clearly, these additional (N - M) configurations or determinants may act as intruders should we use the N-reference SU CCSD method. III. Computational Details A proper choice of reference determinants represents the key step in carrying out GMS SU CCSD calculations. The lowlying excited states of furan and pyrrole are dominated by single excitations. In general, we are faced with two basic cases, (a) and (b), as shown in Figure 1. In case (a), each single excitation couples to a triplet and a singlet and their energies are nearly degenerate, the triplet energy being usually lower than that of the singlet. This case arises in most situations. In case (b), there
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8593
Figure 1. A schematic representation of the two basic types of the excited-state energies of the triplet (T) and singlet (S) configurations arising via single excitations.
exists one single excitation yielding the triplet state with a significantly lower energy than that of the singlet. Moreover, there are several (a)-type states lying between these triplet and the singlet states. Of course, theoretically, an even more complicated situation may arise. To represent singlet-excited states, we must employ low-spin MS ) 0 reference determinants. By use of the spin-orbital formalism, this will also generate the MS ) 0 components of triplet states. For each a f i excitation, there are two MS ) 0 determinants, namely, |(...)aıj| and |(...)aji|. Here, as well as in the following text, the symbol (...) designates all doubly occupied MOs, which, in the case of furan and pyrrole, always include MOs 1 through 16, as well as the MO 17 (or 18) if the single excitation is from 18 (or 17). Another way to generate triplet-excited states is to use highspin MS ) 1 reference determinants. This naturally yields only the triplets. With a single excitation, a f i is then associated one high-spin determinant |(...)ai| ≡ |(...)aRiR|. Note that while the MS ) 0 and MS ) 1 components of a triplet are degenerate at the exact full CI level, this is not the case at the SD-truncated excitation level within the spin-orbital formalism. Usually, the energies obtained with the MS ) 1 references are better than those resulting from the MS ) 0 references, since in the latter case both the triplets and singlets are handled simultaneously, so that the final results are influenced by an intrinsic balance involving the states of different spin symmetry. When all the low-lying states considered are of type (a) (cf. Figure 1), then the a1 f i1 excitation can be used in a tworeference (2R) MS ) 0 calculation or in a one-reference MS ) 1 calculation. (For the totally symmetric states of the A1 symmetry, the Hartree-Fock reference must also be included in a MS ) 0 calculation, yielding a three-reference model space.) When we wish to generate more than one state, both a1 f i1 and a2 f i2 excitations may be used in a four-reference (4R) MS ) 0 calculation or in a 2R MS ) 1 calculation. Likewise, we can add a3 f i3 to perform a six-reference (6R) MS ) 0 calculation or a three-reference (3R) MS ) 1 calculation, and so on. Although using more references will generate more states, the energies are very much independent of how many references are employed.16,38 Thus, if a1 f i1 and a3 f i3 are used in a 4R calculation, while the a2 f i2 excitation is left out, then the second lowest singlet and triplet states will be missing, leading possibly to a wrong ordering of states. A more complex situation arises when we are faced with the type (b) case of Figure 1. Clearly, in the high-spin MS ) 1 case one should use the a1 f i1 determinant in a SR calculation, or {a1 f i1, a2 f i2} in a 2R calculation, or {a1 f i1, a2 f i2, a3 f i3} in a 3R calculation, and so on. Our experience indicates that the same selection of excitations should be employed in
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Li and Paldus
TABLE 1: List of Rydberg-Type RHF MOs for Furan and Pyrrole (as ordered by their orbital energies) by Their Symmetry Speciesa furan
pyrrole
a1 a2 b1 b2 a1 a2 b1 b2
a
23(s), 30(s/pz), 32(s), 34(s/d), 36(s/d), 39(pz), 42(s/d), 43(s/d), 46(s) 26(dxy), 35(dxy), 45(dxy) 22(px), 28(dxz), 31(px), 37(dxz), 40(px), 44(dxz) 20(py), 29(py), 33(dyz), 38(py), 41(dyz), 50(py) 23(s), 25(s/d), 28(s/d), 29(pz), 32(s), 34(s/d), 36(s/d), 38(pz), 42(d), 43(s/d), 46(s), 48(pz) 37(dxy) 22(px), 26(dxz), 31(px), 35(dxz), 40(px), 44(dxz) 21(px), 24(dyz), 30(py), 33(dyz), 39(py), 41(dyz)
The angular momentum nature of each orbital is indicated in parentheses.
MS ) 0 calculations. Indeed, when we use a1 f i1 in a 2R calculation, we obtain the lowest triplet and a higher-lying, but not the lowest, singlet. On the other hand, if we use a2 f i2 in a 2R calculation, the iteration of CC equations may not converge due to the intruder state problem, or will yield poor results for both the triplet and the singlet. An important point to emphasize is the fact that all three excitations {a1 f i1, a2 f i2, a3 f i3} must be employed and (for a given example) a 6R approach must be used in order to obtain accurate results for both the triplets and the singlets. In section IV we shall present a few examples involving different selections of individual excitations. As the preceding discussion indicates, a proper choice of references is essential for a correct description of excited states. This in turn requires a correct qualitative understanding of the manifold of excited states of interest. This may be achieved with the help of a suitable low-level theory description. Because the low-lying excited states of furan and pyrrole are dominated by single excitations, we can employ for this purpose a simple CIS (configuration interaction with singles) method. Note that even when doubly excited states play an important role (see, e.g., Table 1 of ref 16), we can achieve the same goal by relying on CISD assuming that we employ a reasonably small model space. In any case, this approach is by no means a “black box” type universal prescription for a proper choice of relevant references and requires a specific attention depending on the case at hand. Since we have also performed EOM CCSD calculations, we can also identify important configurations for the description of low-lying excited states by the size of their amplitudes. In the case of triplets, it is helpful to compare the MS ) 0 and MS ) 1 results. In the present study, we use up to four excitations for each symmetry species, resulting in up to eight MS ) 0 references (or up to nine references (9R) for states having A1 symmetry, in which case we also have to include the Hartree-Fock reference) and four MS ) 1 references. Concerning other computational details, we employ B3LYP/ 6-311G(dp) geometries for the five-membered furan ring, C4OH4, and pyrrole ring, C4NH5. These structures are employed for all states considered, yielding vertical excitation energies. The excitation energies are then computed using the correlation consistent cc-pVDZ basis sets for the C, O, N, and H atoms, augmented by diffuse functions with exponents ζs ) 0.031 and ζp ) 0.039 on carbon, ζs ) 0.036 and ζp ) 0.047 on nitrogen, and ζs ) 0.041 and ζp ) 0.055 on oxygen, respectively. No diffuse functions are used for the hydrogen atom. To describe Rydberg states, we use three diffuse functions of each s, p, and d type located at the center of mass of the molecule. The exponents of these diffuse functions are ζs ) 0.005858, 0.003346, 0.002048, ζp ) 0.009988, 0.005689, 0.003476, and ζd ) 0.014204, 0.008077, 0.004928 and are based on the rule given in refs 61 and 70.
We employ five Cartesian components of the d-functions, giving rise to 137 MOs for furan and 142 MOs for pyrrole. The valence electrons are correlated in all CC calculations, while the 1s core orbitals are kept frozen. In all calculations, including those using the MS ) 1 reference determinants for the excited triplets, we employ restricted Hartree-Fock (RHF) MOs, optimized for the ground state, as rendered by the GAMESS codes.71 In the case of furan, each reference implies about 0.9 million SD-excited determinants. Thus, e.g., an eight-reference (8R) GMS SU CCSD calculation involves about 7.4 million cluster amplitudes. For pyrrole, the same calculation requires about 10% more amplitudes than in the case of furan. The model spaces employed are characterized in the following text by listing the pertinent excitations a f i, implying an excitation from the MO a to i, as well as the corresponding MS ) 0 or MS ) 1 references. Consequently, a brief description of the nature of these MOs is in place. For both furan and pyrrole, the occupied MOs 17 (b1 symmetry) and 18 (a2 symmetry) are π bonding orbitals. In the case of furan, the MOs 49(b1), 51 (a2), 60 (b1), and 63 (a2) are valence π* antibonding orbitals. Similarly, in the case of pyrrole, the MOs 50 (b1), 52 (a2), and 64 (a2) are valence π* antibonding orbitals. Other MOs that will be referred to latter on have a Rydberg character and are listed by their symmetries in Table 1. For the sake of a comparison, we also generated the EOM CCSD results17-19,72 using the GAMESS codes.71 Since the GMS SU CCSD and EOM CCSD results employ the same basis sets and geometries, they tend to agree rather well with one another. On the other hand, when comparing these excitation energies with other theoretical results that are available in the literature, we must keep in mind that they were obtained with different basis sets and geometries. The computational cost of GMS SU CCSD generally exceeds that of EOM CCSD, even though both scale essentially as n6. Clearly, GMS depends linearly on the dimension of the model space employed: Using M references, the scaling goes approximately as Mn6. IV. Results and Discussion A. Furan. The single reference (SR) CCSD ground state energy is -229.38778 au. With a nine-reference (9R) GMS SU CCSD approach to compute the excited states of A1 symmetry (see below), the resulting energy for the ground 11A1 state is -229.38800 au. The ground state energies obtained with smaller model spaces lie in between these two values, separated by only 0.22 mEh ≈ 0.006 eV. For the sake of simplicity and in view of a very small difference between these two ground state energies, we employ the 9R GMS SU CCSD value as the energy zero when reporting the MR GMS SU CCSD excitation energies even for M < 9. The singlet and triplet excitation energies for different symmetry species, relative to the ground state, as obtained with
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TABLE 2: GMS SU CCSD Excitation Energies (in eV) of Furan As Obtained with a Different Number of References M and Both the Low-Spin MS ) 0 and the High-Spin MS ) 1 Determinants E (MS ) 0, triplet) excitationsa A1 symmetry 17 f 49 + 18 f 51 + 18 f 35 + 17 f 40 A2 symmetry 18 f 32 + 18 f 30 + 18 f 42 + 18 f 36 B1 symmetry 18 f 38 + 18 f 33 + 17 f 32 + 18 f 20 B2 symmetry 18 f 49 + 18 f 31 + 18 f 28 + 18 f 22 a
M
1
2
3 5 7 9
6.26 5.73 5.73 5.74
7.14 7.13 7.14
2 4 6 8
5.90 5.90 5.90 5.90
6.60 6.60 6.60
2 4 6 8
6.45 6.45 6.42 6.42
7.15 7.14 7.13
2 4 6 8
4.34 4.36 4.35 4.35
6.67 6.65 6.66
3
7.36 7.43
6.93 6.93
7.37 7.36
7.53 7.47
E (MS ) 0, singlet) 4
1
2
8.15
7.41 6.88 6.89 6.89
7.38 7.45
7.20
5.94 5.94 5.94 5.94
6.61 6.61 6.61
7.71
6.49 6.49 6.46 6.46
7.16 7.15 7.14
7.71
6.67 6.53 6.52 6.51
6.90 6.86 6.87
E (MS ) 1, triplet)
3
4
M
1
2
3
4
8.18
8.36 8.36 8.37
1 2 3 4
6.12 5.53 5.53 5.53
7.00 7.00 7.00
7.35 7.34
8.08
7.21
1 2 3 4
5.89 5.89 5.90 5.89
6.60 6.60 6.60
6.91 6.91
7.19
7.71
1 2 3 4
6.45 6.45 6.42 6.41
7.13 7.12 7.12
7.36 7.35
7.70
7.72
1 2 3 4
4.25 4.26 4.25 4.26
6.62 6.62 6.64
7.45 7.45
7.70
7.00 7.00
7.40 7.39
7.56 7.50
For each excitation a f i, there are two MS ) 0 reference determinants |(...)aRiβ| and |(...)aβiR| and one MS ) 1 determinant |(...)aRiR|.
the GMS SU CCSD method and various model spaces are given in Table 2. The dimension M of each model space involved is implied by the excitations considered that are listed in the first column. As additional excitations are included, the dimension M of the model space increases accordingly and the SU CCSD method yields additional, higher-lying excited states, while the lower-lying ones are steadily improved in this process. The results in Table 2 include the corresponding excitation energies obtained with the low-spin MS ) 0 determinants, yielding both singlets and triplets, as well as those using the high-spin MS ) 1 determinants, producing triplets only (listed in the rightmost part of Table 2). Thus, considering m single excitations, the number of references M equals 2m in the MS ) 0 case (or 2m + 1 for the A1 symmetry block) and m in the MS ) 1 case. The states of A2, B1, and B2 symmetry are of type (a) (see section III and Figure 1). Thus, for the MS ) 0 model spaces, the 2R approach yields the lowest triplet and singlet of similar energy. Adding two reference determinants, the 4R approach yields additional triplets and singlets of similar energy and has little effect on the energy of the lower lying states. This behavior continues up to the 8R approach, when the effective Hamiltonian possesses a more or less 2 × 2 × 2 × 2 block-diagonal structure, which is reflected in the eigenvectors cij (cf., eq 1), the largest c-coefficients in the corresponding blocks being about (0.7. The energies of triplets obtained with the MS ) 1 references agree reasonably well with those resulting from the MS ) 0 calculations (see below). Due to a stronger coupling between different excitations, the states belonging to the A1 symmetry species are of type (b) of Figure 1. Considering first the results obtained with the MS ) 1 references, we find the SR CCSD (i.e., M ) 1 or 1R) energy by about 0.6 eV higher than the energies obtained with larger model spaces. In fact, already the 2R result yields a stable value which does not change with further increase of the model space. This reflects the fact that there is a strong interaction between the lowest two triplet configurations associated with the excitations 17 f 49 and 18 f 51. For the 13A1 state, the c-coefficients corresponding to these two excitations are 0.82 and -0.57. For the 23A1 state, these coefficients are 0.79 and 0.61. The 3R and
TABLE 3: Two or Three Most Important Excitations in the Singlet Excited States of Furan state 1
2 A1 3 1 A1 4 1 A1 5 1 A1 1 1 A2 2 1 A2 3 1 A2 4 1 A2 1 1 B1 2 1 B1 3 1 B1 4 1 B1 1 1 B2 2 1 B2 3 1 B2 4 1 B2
important excitations 0.56(17 0.70(18 0.69(17 0.54(17 0.70(18 0.70(18 0.67(18 0.70(18 0.62(18 0.70(18 0.69(17 0.63(18 0.64(18 0.65(18 0.68(18 0.69(18
f f f f f f f f f f f f f f f f
49), 35), 40), 49), 32), 30), 42), 36), 38), 33), 32), 20), 40), 40), 37), 22),
0.40(18 0.44(18 0.55(17 0.43(18 0.56(18 0.70(18 0.60(18 0.70(18 0.41(18 0.60(18 0.57(17 0.31(18 0.54(18 0.50(18 0.57(18 0.66(18
f f f f f f f f f f f f f f f f
51), 45), 31) 51), 46), 39) 34) 43) 29), 41) 46), 38) 49), 49), 28), 40)
0.36(18 f 63) 0.40(18 f 26) 0.38(18 f 63) 0.50(18 f 23)
0.37(18 f 50) 0.50(17 f 23) 0.41(18 f 31) 0.42(18 f 31) 0.39(18 f 49)
4R approaches generate more triplets, but have no effect on the energies of the first two triplets. A similar behavior is found when we employ MS ) 0 references. Again, the 3R results are rather unsatisfactory, differing from the final 9R results by about 0.5 eV. Employing two excitations (i.e., five references (5R)), we obtain two lowest triplets and, in addition to the ground state (not shown in Table 2), the second (21A1) and the fifth (51A1) singlets. (Note that in Table 2 the excitation energy of 8.36 eV is listed in the column associated with the 51A1 state). Adding another excitation, we generate an additional low-lying triplet and four singlets, namely, 11A1 ≡ X1A1, 21A1, 31A1, and 51A1. The 9R approach then yields four lowest triplets and five lowest singlets. Again, using model spaces with dimension M > 3 generates, of course, additional excited states, but has only a marginal effect on the lower-lying excited states. Comparing the triplet state excitation energies obtained with the MS ) 1 and MS ) 0 references, we see that they differ by about 0.1-0.2 eV. As already mentioned the former ones are to be preferred (see also below). In order to comprehend the nature of the excited state wave functions, we have to consider both the c-coefficients in the
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TABLE 4: Comparison of Experimental and Theoretical Excitation Energies (in eV) of Furan, As Obtained with Various Theoretical Methods expa
state 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 1 2 1 2 d
1
A1 A1 1 A1 1 A1 1 A2 1 A2 1 A2 1 A2 1 B1 1 B1 1 B1 1 B1 1 B2 1 B2 1 B2 1 B2 3 A1 3 A1 3 A2 3 A2 3 B1 3 B1 3 B2 3 B2 1
7.28 5.91 6.61 6.47 7.38 7.52 6.04 6.75 7.43 7.53 5.15 6.5
3.99 6.5
GMS CCSD
EOM CCSD
SAC-CIb
CC3/CCSDc
CASPT 2d
MRDCIe
ADC(2)f
6.89 7.45 8.18 8.37 5.94 6.61 7.00 7.21 6.46 7.14 7.39 7.71 6.51 6.87 7.50 7.72 5.53 7.00 5.89 6.60 6.41 7.12 4.26 6.64
6.90 7.46 8.06 8.43 6.02 6.70 7.08 7.32 6.54 7.22 7.37 7.77 6.53 6.92 7.61 7.82
6.79 7.36 7.99 8.14 5.99 6.66 7.04 7.27 6.45 7.14 7.45 7.67 6.40 6.82 7.51 7.71 5.63 7.10 5.98 6.68 6.52
6.61 7.58 8.20 8.26 6.11 6.80 7.12 7.39 6.64 7.32 7.52 7.90 6.35 6.94 7.72 7.94
6.16 7.31 7.74
6.63 7.75 8.15 8.58 5.95 6.41 7.15 7.41 6.63 6.99 7.14 7.65 6.88 6.66 7.71 7.82 5.28 6.65 8.68 8.73 9.08
6.70 7.22 7.71 7.82 5.86 6.50 6.89 7.11 6.35 6.98 7.05 7.48 6.37 6.73 7.35 7.53 5.50 6.79 5.81 6.49 6.31
3.93 6.31
4.31 6.54
a Experimental data and assignment are taken from ref 55. From ref 56. e From ref 59. f From ref 62.
6.04 6.48 7.13 5.15 5.86 6.42
4.39 6.66 b
5.92 6.59 7.00 7.22 6.46 7.15 7.21
3.99
From ref 55. c From ref 57 CC3 for valence and CCSD for Rydberg states.
TABLE 5: A Comparison of (6,2)-CCSD and 2R SU CCSD Excitation Energies (in eV) of Furan for the Lowest-Lying States of a Given Symmetry state
excitationa
1 3 A2 1 1 A2 1 3 B1 1 1 B1 1 3 B2 1 1 B2
18 f 32 18 f 38 18 f 49
(6,2)-CCSDb
2R SU CCSD
5.92 5.94 6.43 6.48 4.42 6.38
5.90 5.94 6.45 6.49 4.34 6.67
expc. 5.91 6.47 3.99 6.04
a In the MR SU CCSD approach each excitation a f i brings two MS ) 0 references, namely, |(...)aRiβ| and |(...)aβiR|. b Six-references are used in MR CISD. See the text for details. c From ref.55
MR expansion, eq 1, indicating the relative importance of individual references, and the cluster amplitudes associated with each reference. For example, while for states of the A2, B1, and B2 symmetry, the c-coefficients are more or less blocked and equal approximately to (0.7 in each block associated with the singlet and triplet state for each excitation, as indicated above, several cluster amplitudes are often found to be very large, their value being close to 1. For example, in the 8R SU CCSD wave function for the 21A2 state, the t1 amplitude associated with the 30 f 39 excitation equals 1. By employing this amplitude in the CC expansion, eq 1, and realizing that the appropriate cij coefficient has the value of 0.7, we can formally write an approximate wave function for this state as follows
|Ψ(21A2)〉 ≈ 0.7|18 f 30〉 + 0.7|18 f 39〉
(5)
where each excitation can be either of the R to R or β to β type, so that the kets |a f i〉 represent in fact two determinants. In other words, the 21A2 state represents an equal mixture of the 18 f 30 and 18 f 39 excitations. Important excitations for each excited singlet state are summarized in Table 3. These wave functions correspond to the 8R or 9R SU CCSD level of theory and are intermediately
normalized. As expected, these excited states represent a mixture of various excitations. We must emphasize that each excitation listed in Table 3 represents two MS ) 0 determinants. Thus, unlike the ground state that is described by a single dominant determinant, these excited states involve generally quite a few important determinants. Clearly, when the two excitations, say a f i1 and a f i2, have the same weight and are thus equally important, either one of them can be used as a reference. Thus, for example, when we use the excitation a f i1 as a reference, then the cluster amplitude in the excitation operator Gi1 f i2 should be approximately equal to 1, so that when it acts on the reference |a f i1〉 it produces the second excitation a f i2. In Table 4 we compare our GMS SU CCSD excitation energies with their experimental values, as well as with other theoretical results, including those obtained with the EOM CCSD, symmetry-adapted-cluster CI (SAC-CI),55 LR CC,57 complete-active-space second-order perturbation theory (CASPT2),2,56 multireference CISD (MRCID),59 and polarization propagator methods, the latter referred to as the second-order algebraic-diagrammatic construction [ADC(2)].62 The experimental assignments follow those given in the SAC-CI paper.55 Since both the GMS and EOM CCSD results were obtained using the same geometry and basis sets, their comparison is the most meaningful one. Indeed, we find an excellent agreement
Electronic Excitations in Furan and Pyrrole
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8597
TABLE 6: The GMS SU CCSD Excitation Energies (in eV) of Pyrrole As Obtained with a Different Number of References M of Both the Low-Spin MS ) 0 and the High-Spin MS ) 1 Determinants E (MS ) 0, triplet) excitationsa A1 symmetry 17 f 50 + 18 f 52 + 17 f 40 + 17 f 35 A2 symmetry 18 f 32 + 18 f 38 + 18 f 34 + 18 f 36 B1 symmetry 18 f 39 + 17 f 32 + 18 f 33 + 17 f 38 B2 symmetry 18 f 50 + 18 f 40 + 18 f 35 + 18 f 22 a
M
1
2
3 5 7 9
5.91 5.68 5.68 5.68
6.59 6.59 6.58
2 4 6 8
5.06 5.07 5.07 5.07
5.81 5.80 5.80
2 4 6 8
5.81 5.73 5.74 5.75
5.93 5.93 5.92
2 4 6 8
4.69 4.69 4.70 4.70
5.97 5.97 5.97
3
6.85 6.86
6.37 6.36
6.43 6.38
6.71 6.71
E (MS ) 0, singlet) 4
1
2
7.52
7.02 6.53 6.53 6.53
6.85 6.85
6.44
5.09 5.10 5.09 5.10
5.82 5.81 5.81
6.61
5.84 5.78 5.79 5.79
5.97 5.97 5.96
6.44 6.40
7.02
5.96 5.96 5.96
6.62 6.61
6.51 6.80 6.90 6.87
E (MS ) 1, triplet)
3
4
M
1
2
3
4
7.47
7.79 7.78 7.86
1 2 3 4
5.83 5.53 5.53 5.53
6.42 6.41 6.41
6.86 6.86
7.51
6.45
1 2 3 4
5.06 5.07 5.07 5.07
5.81 5.81 5.80
6.38 6.34
6.42
6.65
1 2 3 4
5.80 5.73 5.73 5.74
5.92 5.92 5.91
6.42 6.38
6.61
7.06
1 2 3 4
4.52 4.53 4.52 4.53
5.97 5.97 5.96
6.70 6.70
7.01
6.39 6.38
See the footnote to Table 2.
TABLE 7: Two or Three Most Important Excitations in the Singlet Excited States of Pyrrole state 1
2 A1 3 1A 1 4 1A 1 5 1A 1 1 1A 2 2 1A 2 3 1A 2 4 1A 2 1 1B 1 2 1B 1 3 1B 1 4 1B 1 1 1B 2 2 1B 2 3 1B 2 4 1B 2
important excitation 0.56(17 0.71(17 0.63(17 0.51(17 0.67(18 0.62(18 0.58(18 0.59(18 0.57(18 0.56(17 0.64(18 0.62(17 0.60(18 0.53(18 0.53(18 0.58(18
f f f f f f f f f f f f f f f f
50), 40), 35), 50), 32), 38), 34), 36), 39), 32), 33), 38), 40), 35), 50), 22),
0.43(18 0.65(17 0.47(17 0.44(18 0.49(18 0.59(18 0.49(18 0.59(18 0.54(18 0.37(18 0.56(18 0.55(17 0.40(18 0.43(18 0.35(18 0.35(18
f f f f f f f f f f f f f f f f
52), 31) 26), 52), 23), 29) 42) 43) 30), 39), 41), 29), 31) 50), 22), 40),
0.37(18 f 64) 0.42(17 f 44) 0.37(18 f 64) 0.46(18 f 46)
0.41(17 0.35(18 0.38(18 0.37(17
f f f f
32) 30) 24) 48)
0.40(18 f 26) 0.33(18 f 62) 0.31(18 f 31)
between these two CC-based approaches, rendering almost identical results. This is not surprising, since the excited states considered are dominated by single excitations, in which case the EOM CCSD method is known to perform very well. When compared with the experimental values, the GMS CCSD energies give a slightly better agreement (by about 0.1 eV on the average) and are also very similar to the SAC-CI results.55 As mentioned in section III, the MOs 49, 51, 60, and 63 are valence π* antibonding orbitals, while other MOs appearing in Table 3 have a Rydberg character. Hence, 11B2 and 21A1 are valence excited states, and the 51A1 state has also a somewhat valence character. Nonetheless, as implied by the important excitations listed in Table 3, these valence states possess a strong Rydberg character. Other singlet excited states are mostly Rydberg in nature. The largest discrepancy (0.47 eV) between the GMS CCSD and theoretical excitation energies is found for the 11B2 state (6.51 vs 6.04 eV). In order to find out if this discrepancy arises due to the higher-than-doubly excited clusters, as well as for illustrative purposes, we have also applied the externally corrected (N,M)-CCSD approach to the lowest singlet and triplet
states of the A2, B1, and B2 symmetry. These results are given in Table 5. As is apparent from Table 3, there are several excitations of roughly the same importance characterizing the 11A2, 11B1, and 11B2 states (and, similarly, for the corresponding triplet states). For this reason, we have applied the (6,2)-CCSD method, in which a six-reference MR CISD is employed to account for the effect of higher-than-pair clusters. The six MS ) 0 reference determinants used in MR CISD correspond to the three most important excitations listed in Table 3, namely, to 18 f 32, 18 f 23, 18 f 46 for the 11, 3A2 states, 18 f 38, 18 f 29, 18 f 50 for the 11, 3B1 states, and 18 f 49, 18 f 40, 18 f 31 for the 11, 3B2 states. Moreover, in the (6,2)-CCSD calculation, we used a threshold of 3 × 10-5 in a truncation of the size of the MR CISD expansion based on PT amplitudes (for details, see ref 73). Comparing the (6,2)-CCSD and 2R GMS CCSD energies in Table 5, we find that the largest improvement occurs for the 11B2 state. In this case, the (6,2)CCSD energy is lower than the 2R GMS CCSD one by 0.29 eV and is thus closer to the experimental value by the same amount. For the other two states, the improvement over GMS CCSD is insignificant. B. Pyrrole. Pyrrole, being very similar to furan, has been treated in an analogous manner as furan, and the results are also presented in a parallel fashion. We also keep the discussion of this species brief, since more detailed explanations have already been given in the preceding section A. The energy of the 11A1 ground state that has been used in the computation of excitation energies is that obtained with a 9R SU CCSD approach, amounting to -209.56365 au, which is slightly lower than the CCSD energy of -209.56360 au. The GMS CCSD excitation energies, as obtained with various reference spaces are given in Table 6. The A2 and B1 states are of type (a) (cf. Figure 1). For each excitation, the triplet is lower than the singlet. The triplet energies obtained with the MS ) 1 and MS ) 0 references are essentially identical. In contrast, the A1 and B2 states are of type (b) (cf. Figure 1). In the case of A1 symmetry species, the single excitation (i.e., 3R) SU CCSD based on the MS ) 0 references yields poor results for both the lowest triplet and singlet states. The 5R
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TABLE 8: Comparison of the Excitation Energies (in eV) of Pyrrole, As Obtained with Various Theoretical Methods, with Experiment55 state 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 4 1 2 1 2 1 2 1 2 a
expa
1
A1 A1 1 A1 1 A1 1 A2 1 A2 1 A2 1 A2 1 B1 1 B1 1 B1 1 B1 1 B2 1 B2 1 B2 1 B2 3 A1 3 A1 3 A2 3 A2 3 B1 3 B1 3 B2 3 B2 1
7.0-7.1 5.22
5.7 6.42 6.5-6.7 6.2-6.5 6.78 7.0-7.1 5.1
4.2
GMSCCSD
EOMCCSD
SAC-CIb
CC3c
CASPT 2d
ADC(2)e
6.53 6.85 7.47 7.86 5.10 5.81 6.38 6.45 5.79 5.96 6.40 6.65 5.96 6.61 6.87 7.06 5.53 6.41 5.07 5.80 5.74 5.91 4.53 5.96
6.55 6.88 7.51 7.86 5.18 5.88 6.45 6.54 5.84 5.97 6.47 6.66 6.02 6.65 6.92 7.14
6.41 6.64 6.86 7.26 5.11 5.81 6.38 6.44 5.80 6.05 6.39 6.68 5.88 6.48 6.76 6.99 5.60
6.37 6.77 6.94 7.60 5.10 5.86 6.43 6.50 5.85 5.99 6.47 6.72 5.98 6.63 6.91 7.66
5.92 6.54 6.65
6.42 6.54 6.66 7.12 5.03 5.71 6.25 6.37 5.68 5.77 6.27 6.43 5.86 6.48 6.71 6.88 5.55
5.08 5.83 6.42 6.51 5.85 5.97 6.42 6.62 5.78 6.00 6.53 5.16
5.08
5.04
5.82
5.82
4.58
4.27
5.00 5.70 5.63 5.72 4.59
Experimental data and assignments are taken from ref 55. b From ref 55. c From ref 58. d From ref 56. e From ref 61.
TABLE 9: GMS SU CCSD Excitation Energies (in eV) Obtained with Different Choices of Model Spaces, See the Text for Details excitations
E (MS ) 0, triplets)
M
E (MS ) 0, singlets)
Pyrrole (states with B2 symmetry) selection 1 18 f 50 + 18 f 40 + 18 f 35 selection 2 18 f 40 + 18 f 35 + 18 f 50
2 4 6
4.69 4.69 4.70
5.97 5.97
2 4 6
NC NC 4.70
NC 5.97
6.71
5.96 5.96
6.62
6.51 6.80 6.90
6.71
NC NC 5.96
NC 6.62
6.90
8.15
7.41 6.88 6.89 6.89
7.38 7.45
8.18
8.36 8.36 8.37
8.15
7.41 7.37 7.29 6.89
7.53 7.62 7.45
8.19 8.18
8.37
Furan (states with A1 symmetry) selection 1 17 f 49 + 18 f 51 + 18 f 35 + 17 f 40 selection 2 17 f 49 + 18 f 35 + 17 f 40 + 18 f 51
3 5 7 9
6.26 5.73 5.73 5.74
7.14 7.13 7.14
3 5 7 9
6.26 6.19 6.22 5.74
7.32 7.10 7.14
7.36 7.43
8.16 7.43
approach generates two lowest triplets and the 11A1 ground state, as well as 21A1 and 51A1 excited states (thus, skipping the 31A1 and 41A1 states; cf. Figure 1). The excited singlets 21A1 and 51A1 have again heavy weights in both 17 f 50 and 18 f 52 excitations. The 7R approach then yields three lowest triplets and the 11A1, 21A1, 31A1, and 51A1 states, still skipping over the 41A1 state. Finally, the 9R approach yields four lowest triplets and all five lowest singlets. Similarly, in the case of the B2 symmetry species, the 2R MS ) 0 GMS CCSD yields the lowest triplet and the third singlet (31B2) states. The 4R approach generates the two lowest triplet and the singlet 11B2 and 31B2 states (skipping the 21B2 state). The 6R approach then gives three lowest triplets and three lowest singlets. Adding the fourth excitation, which is of type (a), yields the four lowest triplets and four lowest singlets.
For both A1 and B2 symmetry species, the triplet energies obtained with MS ) 1 references are clearly superior to those obtained with the MS ) 0 references. The largest difference amounts to 0.17 eV for the 13B2 state. The leading excitations for the excited singlet states are presented in Table 7. These wave functions correspond to 8R or 9R GMS CCSD and are intermediately normalized. The coefficients characterizing these wave functions represent again the combination of both the c-coefficient of the SU Ansatz, eq 1, and of the largest cluster amplitude. Clearly, these excited states represent a mixture of various excitations. Since each excitation implies two determinants, these excited states involve a number of important configurations. In Table 8, we compare the resulting GMS CCSD excitation energies with the experimental values, as well as with other
Electronic Excitations in Furan and Pyrrole theoretical results, obtained with the EOM CCSD, SAC-CI,54,55 LR CC,58 CASPT2,56 and ADC61 methods. Again, the GMS CCSD results are almost the same as the EOM CCSD ones or represent a slight improvement of the order of 0.1 eV. We have also assigned the experimental excitation energy of 6.2-6.5 eV to the 21B2 state, rather than to the 11B2 state, as done in ref 55. As mentioned in section III, the MOs 50, 52, and 64 are valence π* antibonding orbitals, while other MOs are Rydberg orbitals. Hence the excited states 21A1, 51A1, 21B2, and 31B2 are valence excited states, even though they have a strong Rydberg admixture, as implied by important excitations listed in Table 7. Other singlet excited states are of a Rydberg nature. C. Choice of References: A Comparison. When we rely on a spin-nonadapted, spin orbital formalism, it is important to distinguish two kinds of states, referred to above as those of type (a) and (b) and schematically represented in Figure 1. In the presence of the type (b) states, as in the case of the A1 states of furan and B2 states of pyrrole, we have essentially two options how to build larger and larger MS ) 0 model spaces. The approach used above in sections A and B, which we shall refer to as “selection 1”, is based on the sequence of low-lying triplet states. Thus, the smallest model space, based on the excitation we shall label as E1 (≡a1 f i1), yields the lowest triplet, adding excitation E2 then brings in the second lowest triplet, excitation E3 the next lowest triplet, etc. In this way, however, the singlets generated by such model spaces do not necessarily arise in the order of their excitation energies, as we have seen in Tables 2 and 6 in the case of the above-mentioned symmetry species involving the (b)-type states. One can thus ask, particularly when we are primarily interested in singlets, if we can employ another sequence of excitations that would yield properly ordered singlet states according to their increasing energy. Referring to this choice as “selection 2”, we thus employ first a 2R space based on, say, excitation E2, then add excitation E3, and eventually the excitation E1. Consider, first, the results obtained for the B2 states of pyrrole, presented in Table 9. The leading excitations yielding the three lowest triplets are, respectively, 18 f 50, 18 f 40, and 18 f 35, representing thus selection 1. On the other hand, the leading excitations yielding the three lowest-lying singlets are, respectively, 18 f 40, 18 f 35, and 18 f 50, implying selection 2. Of course, once all three excitations are included, the resulting 6R approach is identical in both cases, generating three lowest triplets and singlets. Yet, a very different outcome occurs in calculations with smaller model spaces. On the basis of selection 1, the 2R approach yields the lowest triplet and the third singlet, while the 4R approach yields two lowest triplets and the first and the third singlets. However, based on selection 2, meaningful convergence cannot be achieved when using either the 2R or the 4R model space. We next turn our attention to the A1 states of furan, also given in Table 9. Here we are faced with an extended type (b) case (cf. Figure 1), since there is one excitation of type (a) that is situated below the type (b) ones, so that the 3R approach is identical in both selections 1 and 2. With the 5R model space, selection 1 generates two lowest triplets and the first (i.e., the ground state), second, and the fifth singlets, while selection 2 yields some approximate energies that are considerably removed from their correct final values. The 7R model space then yields similar results as the 5R space. Finally, when all four excitations are employed, selections 1 and 2 yield identical results. The above two examples clearly indicates that selection 1 is to be preferred.
J. Phys. Chem. A, Vol. 114, No. 33, 2010 8599 V. Conclusions We have employed the GMS SU CCSD method to investigate the electronically excited states of furan and pyrrole using a sufficiently extended basis set including diffuse functions and molecule-centered Rydberg functions yielding quantitatively meaningful results. When compared with the corresponding EOM CCSD results that employ the same geometry and basis sets, the GMS CCSD energies are in most cases essentially identical with the EOM CCSD ones, or about 0.1 eV (i.e., a few mEh) better when compared with the experimental values. This may have been expected, since at the SD-level of the theory, the EOM CCSD method performs well for excited states dominated by single excitations. Only when doubly excited configurations intervene, the EOM CCSD method is lacking and requires an extension by triples, while the SU CCSD method still performs well.16 In a broad sense, methods that are capable of handling electronically excited states belong to two distinct classes, depending on whether they determine individually the energies and wave functions of each excited state first and then evaluate the excitation energies as the energy differences or, in the second case, if they compute directly the energy differences. The typical representatives are the CI and Green function approaches, respectively. Clearly, the GMS SU CCSD method, together with MR CI, belong to the first class, while the EOM, LR CC, and SAC-CI methods belong to the second class. Both classes yield equivalent results in the exact, i.e., the full CI, limit. However, at the SD-level of the theory, they only provide similar results for well-behaved molecular systems with excited states dominated by monoexcitations. This is the case for the low-lying excited states of both furan and pyrrole, in which case the relevant single excitations involve the occupied MOs 17 or 18 and virtual MOs i, with i . 17 or 18 (cf. Tables 2,3 and 6,7 for furan and pyrrole, respectively). The relevant most important virtual MOs are valence π* antibonding orbitals (41, 51, 60, 63 for furan and 50, 52, and 64 for pyrrole; see section III), the remaining virtuals having primarily Rydberg character (see Table 1). We have also seen that each excited state involves quite a few important determinantal configurations. Our results demonstrate that one can employ this type of general reference in the GMS CCSD theory to handle the relevant excited states. The key is to employ appropriate references, so that no low-lying states are missed. Since the excited state wave functions involve several important configurations, the choice of references is not unique. Obviously, if two excitations are equally important, either one can be used as a reference. Such alternate choices will yield closely lying, yet not identical, results. In view of this multiconfigurational character, the convergence of the GMS SU CCSD equations may be slower, thus requiring a larger number of iterations when solving the SU CC equations than is the case for the CCSD ground state. On the whole, the presented results clearly demonstrate the capability and effectiveness of the GMS SU CCSD method in handling of the exited states, simultaneously with the ground state, even for medium size molecular systems, yielding highly accurate results that can be meaningfully compared with the experimental data and eventually used for their prediction. Acknowledgment. The continued support (J.P.) by the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.
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