A CASSCF theoretical study of the vibrational frequencies and structure of formaldehyde, acetaldehyde and acetone valence excited states

Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69 www.elsevier.com/locate/theochem

A CASSCF theoretical study of the vibrational frequencies and structure of formaldehyde, acetaldehyde and acetone valence excited states Celestino Angeli*, Stefano Borini, Lara Ferrighi, Renzo Cimiraglia Dipartimento di Chimica, Universita` di Ferrara, Via Borsari 46, I-44100 Ferrara, Italy Received 18 November 2004; revised 15 December 2004; accepted 15 December 2004

Abstract The equilibrium geometries of the singlet and triplet n/p*, p/p* and s/p* valence states of the formaldehyde, acetaldehyde and acetone molecules have been obtained performing a full geometry optimization at the single state CASSCF level. The harmonic vibrational frequencies have been computed analytically at the same level of theory. A common strategy for the various states and molecules has been used in order to allow the comparison of the results. The geometrical structure and the harmonic frequencies of two states of acetaldehyde (S2 and 3(s/p*)) and two of acetone (3(p/p*) and 3(s/p*)) are described for the first time. For the 3(s/p*) state of formaldehyde the first determination of the harmonic frequencies is reported. The strategy here adopted has allowed the identification of various trends for the substitution, on the carbonyl chromophore, of the hydrogen atom with the methyl group. q 2004 Elsevier B.V. All rights reserved. Keywords: Carbonyl molecules; Excited state equilibrium geometry; Excited state vibrational frequencies; Ab initio methods; CASSCF

1. Introduction The study of the properties of molecules in electronically excited states represents an important part of modern quantum chemistry. In general these studies focus on the determination of the energy of the relevant states, often at the ground state (GS) equilibrium geometry, with a view to understanding the vertical part of the electronic spectrum. A relatively smaller number of works is dedicated to the calculation of other properties of excited states, such as, for instance, the properties of the equilibrium geometry, and only in few cases the study of the full potential energy surface has been faced. The reasons for such a situation rest in the additional difficulties found in the description of the excited states with respect to the GS due to the open-shell electronic nature often encountered in such states. This forces one to use multireference (MR) methods, for which ‘black box’ computational implementations are not easily available.

* Corresponding author. Tel.: C39 532 291323; fax: C39 532 240709. E-mail address: [email protected] (C. Angeli). 0166-1280/$ - see front matter q 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2004.12.017

This paper is devoted to the study of the valence excited states of small carbonyl molecules. The carbonyl chromophore has been chosen for the particular role it plays in photochemistry and because the representative molecules for this category, formaldehyde, acetaldehyde and acetone, have long attracted the attention of experimental and theoretical chemists. For instance, the first 1A2 excited state in formaldehyde, which can be qualitatively described as an excitation of one electron from the oxygen lone pair orthogonal to the CO axis to the p* orbital, 1(n/p*), is important because of its role [1] in the photochemical decomposition of H2CO in H2CCO and HCHCO. In aliphatic ketones the excitation to the S1, n/p*, state leads primarily to the cleavage of one of the CC bonds in a position, producing an acyl and an alkyl radical. This process has a fundamental importance in the photochemistry and is known as Norrish Type-I reaction [2,3]: for a recent review of such reaction in acetone the reader is referred to Ref. [4]. The work here reported is part of a study performed in our laboratory with the aim to help the interpretation of the electronic spectrum of small carbonyl molecules. In a twin paper [5] the adiabatic excitation energies for

56

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

the molecules and states here considered have been studied using the n-electron valence state perturbation theory [6–9] (NEVPT2). In this approach, which belongs to the multireference perturbation theory family, the zero-order Hamiltonian has a bielectronic nature and the perturber functions are obtained applying suitable excitation operators to the zero-order CASSCF wavefunction. NEVPT2 is endowed with some desirable formal properties such as, for instance, size-consistency and absence of intruder states and it has proved to be a reliable tool in the study of the electronic structure of molecules. From the theoretical point of view these molecules are interesting because their small dimension enables one to use high level ab initio methods. Therefore, for a certain number of cases, the results here obtained can be compared with different theoretical determinations in order to assess their quality level. Formaldehyde is surely the most studied carbonyl molecule and the bibliography regarding it is very large. An extended review of the experimental works on this molecule has been presented by Moule and Walsh [10] in 1975 and successively by Clouthier and Ramsay [11] in 1983. Many theoretical calculations have been performed on formaldehyde [12–27], often concentrating on the vertical transitions. The excited states of the acetaldehyde molecule have also been studied both experimentally and theoretically [16–18,28–32], but only in few cases the geometrical structure and the frequencies of the excited states have been investigated. Similarly, the excited states of acetone have been less studied than in formaldehyde from the theoretical point of view. The published works have concentrated on the first excited singlet [33–37], which is known to be, as in the other small carbonyls, the 1(n/p*) state. The S1 and T1 energy surfaces have been investigated by different authors [3,38–40] in connection with the Norrish Type-I reaction. Despite the large number of theoretical works on these three molecules, various valence states have never been characterized and even for the states considered a study of all the molecules based on a common strategy appear to be missing. The use of different theoretical approaches, for which different approximation levels have been used, makes it difficult to compare the data reported in the literature. This work tries to remedy this deficiency. The rest of this paper is organized as follows: in Section 2 the details of the calculations are presented, in Section 3 the equilibrium geometries for the ground state and for the relevant excited states are discussed and in Section 4 the harmonic vibrational frequencies are reported and compared with experiments and previously published values.

2. Computational details The computational strategy used in this work is the same reported in Ref. [5]. The atomic natural orbital (ANO) basis

set [41] has been used with the contraction scheme C,O (14s9p4d)/[4s3p1d], H (8s4p)/[2s1p]. The absence of diffuse functions, needed to describe the Rydberg states which are known to play an important role in the electronic spectra of small carbonyl molecules, is due to the fact that the present study focuses the attention on the valence state equilibrium geometries, where Rydberg states are expected to have a marginal influence [21]. The equilibrium geometries of the ground state (GS) and of the first three valence singlet (1(n/p*), 1(p/p*) and 1(s/p*)) and triplet (3(n/p*), 3(p/p*) and 3(s/p*)) excited states have been computed at the single state CASSCF level using the DALTON program [42] optimizing all the geometrical parameters. For all the three carbonyl compounds here considered, the molecular skeleton in the GS (H–CO–H for formaldehyde, C–CO–H for acetaldehyde and C–CO–C for acetone) lies in the yz plane with the CO group in the z axis. For acetaldehyde and acetone the hydrogen atoms lying on the yz plane are at the maximum distance from the z axis. In the GS the molecular symmetry group is C2v for formaldehyde and acetone and Cs for acetaldehyde. It is well known that for these molecules the symmetry of the GS is no longer maintained in the n/p* and s/p* excited singlet and triplet states, due to the pyramidalization occurring at the carbonyl carbon atom therefore leading to the Cs symmetry group for formaldehyde and acetone and to the C1 symmetry group for acetaldehyde. For the p/p* singlet and triplet states the situation is less clearly defined and we shall discuss the equilibrium geometry for such states in detail in Section 3. The strategy here adopted is to have a unique active space for all the states under investigation, consisting of the ny lone pair of the oxygen atom and the s, p and p* CO orbitals. The term ‘s CO orbital’ indicates here the highest occupied molecular orbital with s symmetry with respect to the CO axis: at the equilibrium geometry of the excited states this orbital can be seen as a mixing of the nz lone pair of the oxygen atom and of the bonding sCO orbital of the GS (with a larger component on the latter). Given that in the excited states an elongation of the CO bond is expected (an electron is promoted to the p* antibonding orbital) the s* orbital has been also included in the active space, leading to an active space defined by six electrons in five orbitals (hereafter called CAS(A)). In the case of the formaldehyde and acetone the presence of the xz symmetry plane for all states helps in defining the active space and ensures that the active orbitals maintain their nature in the optimization process. In the case of acetaldehyde, problems have been encountered in the CASSCF calculation of the GS and p/ p* states, where the O ny lone pair is replaced by the O nz orbital, since the latter brings a larger contribution to the correlation energy than the O ny orbital. Such a problem does not occur in the n/p* excited states where the electronic nature of the state imposes the presence of the O ny orbital in the active space. The enlargement of the active space including one more ‘doubly occupied’ a 0 orbital does

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

not solve the problem because the latter turns out to be the sCC orbital. In order to avoid this bias among the states, the O nz and the sCC orbitals have been added to the active space for acetaldehyde leading to an active space with 10 electrons in seven orbitals (CAS(B)). For formaldehyde, a second kind of active space has been used following Dallos et al. [43], containing all the valence orbitals (bonding, nonbonding and antibonding; 3–7 a1, 1–2 b1 and 1–3 b2) and electrons. This space will be indicated hereafter with CAS(C). Methodological works [44,45] have recently shown that multiple solutions exist for the CASSCF equations for a given active space (defined by the number of orbitals for each irrep and the number of electrons) corresponding to different choices in the nature of the active orbitals and that geometry optimization can be performed for each solution. Multiple solutions for the same state have been found in the present work, usually differing only in the nature of the active orbitals with occupation number close to two or zero. In all these cases, the equilibrium geometries corresponding to the different solutions are very similar. The solutions presented in the following sections are those for which the orbitals defining the active space have the correct nature. It is worth noticing that this choice does not always match the minimum energy criterion. For excited state equilibrium geometries, where the molecular skeleton is non-planar, the Cs (formaldehyde and acetone) and C1 (acetaldehyde) symmetry groups have been used and the motion of the CO bond out of the CR2 plane (wagging) is described by the out-of-plane bending angle (q) between the CO axis and the CR2 plane. The vibrational frequencies have been computed analytically using the harmonic approximation and are here reported without any scaling factor. The normal mode assignment has been done looking at the dominant contributions of the atom displacements. From the knowledge of the harmonic frequencies, the zero point energy (ZPE) of the various states has been obtained. The ZPE values have been used in Ref. [5] in order to take into account the vibrational contribution to the adiabatic transition energies.

3. Equilibrium geometries 3.1. Ground state The CASSCF optimized geometries for the ground state are reported in Table 1 together with experimental and some recently published theoretical results. The CASSCF geometrical parameters show an overall agreement with experiment and with other theoretical data. From the inspection of Table 1 one notes that the carbonyl frame has a structure very similar in the three molecules.

57

Table 1 Formaldehyde, acetaldehyde and acetone ground state equilibrium geometry Method Formaldehyde CASSCF(CAS(A))c CASSCF(CAS(C))c Experimental [46] Acetaldehyde CASSCF(CAS(B))c Experimental [47] Acetone CASSCF(CAS(A))c Experimental [48]

RCO

RCHa

1.220 1.210 1.208

1.090 1.118 1.116

1.222 1.213

1.092 1.106

1.224 1.222

:XCYb

RCC

:OCC

117.3 115.1 116.5 1.501 1.504

116.3 114.9

124.2 124.0

1.509 1.507

117.2 117.1

121.4 121.4

˚ , angles in degrees. Distances in A a In acetaldehyde H is the aldehydic hydrogen. b Formaldehyde XZH, YZH; acetaldehyde XZH, YZC; acetone XZC, YZC. c This work.

3.2. The n/p* states The equilibrium geometries for the n/p* triplet and singlet states are reported in Tables 2–4 for the formaldehyde, acetaldehyde and acetone, respectively. The 1(n/p*) state of formaldehyde is experimentally well characterized and many spectroscopical parameters are Table 2 Equilibrium geometries for the formaldehyde

1

(n/p*) and

3

(n/p*) states of

Method

RCO

RCH

:HCH

qa

1 (n/p*) excited state CASSCF(CAS(A))b CASSCF(CAS(C))b CASSCF(6e5o)/6-311CCG** [49] CASSCF(10e/9o) [50] MRD-CI [51] MR-CISD [43] MR-AQCC [43] P-EOM-MBPT2 [35] EOM-CCSD [52] EOM-CCSD [53] CASPT2 [21] EOM-CCSD [54] DFT/BLYP [32] Experimental [55] Experimental [56]

1.387 1.358 1.382 1.364 1.335 1.338 1.325 1.340 1.324 1.311 1.352 1.321 1.329 1.323 1.323

1.078 1.109 1.076 1.107 1.116 1.080 1.090 1.093 1.096 1.095 1.092 1.105 1.102 1.098 1.103

119.9 116.8 120.3 117.5 120.2 119.5 117.4

37.5 37.7 36.8 37.2 34.5 34.3 34.9

118.8 120.6 116.6 115 118.8 118.1

30.3 32.5 30.8 28.0 38 34.0 34.0

3 (n/p*) excited state CASSCF(CAS(A))b CASSCF(CAS(C))b CASSCF(6e5o)/6-311CCG** [49] CCSD(T) [57] MRD-CI [22] MP2 [22] CIS [28] CIS [58] Experimentalc

1.365 1.335 1.360 1.314 1.313 1.318 1.248 1.256 1.307

1.079 1.112 1.077 1.090 1.100 1.092 1.096 1.092 1.084

119.3 113.9 119.7 115.6 116.3 117.3 111.5 112.8 121.8

39.3 43.7 38.5 41.4 40.0 40.8 51 43.1 41.1

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work. c Refs. [11] and [59] as cited in Ref. [22].

58

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

Table 3 Equilibrium geometries for the 1(n/p*) and 3(n/p*) states of acetaldehyde Method

RCO

RCC

RCHald

:OCC

:CCHald

1.383 1.334 1.359 1.256

1.501 1.524 1.494 1.531

1.080 1.102 1.095 1.089

114.5 118 116.0 118.3

120.3 117

qa

1

(n/p*) excited state CASSCF(CAS(B))b DFT/BLYP [32] P-EOM-MBPT2 [35] CIS [28] Exp. [60] Exp. [29] Exp. [61] 3 (n/p*) excited state CASSCF(CAS(B))b UMP2 [30] UMP2 [28] UMP2 [29] CIS [28] Exp. [60]

38.4 35

117.0 35.7 37.8 26

1.32 1.360 1.323 1.338

1.506 1.509 1.505

1.082 1.094 1.094

114.8 114.5 114.2

119.1 127.6 119.1

1.254

1.544

1.097

116.8

113.1

40.2 39.7 39.7 x48 42.2

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work.

known [56,63,64]. The equilibrium geometry obtained with the CAS(A) and CAS(C) active spaces are in reasonable agreement with the experimental determinations: as for the bond lengths, RCO has an appreciable error (0.064 and ˚ for the two spaces, respectively) while RCH is well 0.035 A described with CAS(C) and slightly too short with CAS(A). This behaviour for RCO was already found at CASSCF level in Refs. [49,50] and is common to all the excited states of formaldehyde here considered: the CASSCF(CAS(A)) description gives too large values, only partially improved with the CAS(C) active space. Dallos et al. [43] have found that increasing the basis set size, the RCO distance decreases and one can therefore expect an improvement using a larger basis set. Moreover, one can expect that performing the geometry optimization including the dynamical correlation energy could further decrease the CO distance. The out-ofplane bending angle is found to be 37.5 and 37.78 in the present work, 3.5 and 3.78 higher than the experimental value. For the 3(n/p*) state similar considerations can be done. In this case there are fewer theoretical works in literature. The results here obtained confirm that the n/p* triplet state is more bent than the singlet [10,28] and has a shorter CO bond length. In the case of acetaldehyde, the state is much less characterized experimentally: only the out-of-plane bending angle has been determined for the two n/p* states and for the singlet the CO bond length is also known. The values here obtained compare well with these results. The theoretical knowledge of the equilibrium structure of the two n/p* states is less complete than for formaldehyde, being essentially based on single reference methods. The comparison with the analogous states in formaldehyde shows that the CO and CHald bond lengths and the out-ofplane angle are very similar for the two molecules.

For acetone no experimental data exist, but some CASSCF geometry optimizations for both the n/p* triplet and singlet have been published. For the singlet also EOMCCSD [35] and DFT/BLYP [32] studies have been carried out. The results here obtained agree with these calculations, the main difference being for the 1(n/p*) state where

Table 4 Equilibrium geometries for the 1(n/p*) and 3(n/p*) states of acetone Method 1 (n/p*) excited state CASSCF(CAS(A))b CASSCFc CASSCFd CASSCFe CASSCFf CASSCFg P-EOM-MBPT2 [35] DFT/BLYP [32] 3 (n/p*) excited state CASSCF(CAS(A))b CASSCFd CASSCFe CASSCFf

RCO

RCC

:CCC

qa

1.401 1.355 1.399 1.394 1.394 1.401 1.376 1.341

1.499 1.521 1.507 1.504 1.504 1.500 1.502 1.524

121.3 117.3

39.6 43.1 42.3 41.7 41.2 36.3 35.6 36

1.381 1.376 1.370 1.371

1.502 1.512 1.510 1.509

120.4

120.3 120.7 121.5 122.3 119

119.4 119.5

40.2 49.7 41.8 43.6

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work. c CASSCF with 10 electrons in 11 orbitals using the 6-311G** basis set, Ref. [36]. d CASSCF with eight electrons in seven orbitals using a C[3s2p1d]/ H[2s1p] basis set, Ref. [62]. e CASSCF with eight electrons in seven orbitals using the 6-31G* basis set, Ref. [39]. f CASSCF with 10 electrons in eight orbitals using the 6-31G(d) basis set, Ref. [3]. g CASSCF with six electrons in five orbitals using the 6-31G(d,p) basis set, Ref. [33].

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

the DFT/BLYP [32] and the CASSCF [36] results give a shorter CO bond length, a longer CC bond length and a smaller :CCC angle. As concerns the comparison with the smaller molecules, one can note that all the geometrical parameters show modest variations changing the carbonyl substituents (considering CAS(A) for formaldehyde). The only exception is the CO bond length in acetone which is slightly larger than in formaldehyde and acetaldehyde for both states. The present results show that the excitation process, apart from the lengthening of the CO bond and from the pyramidalization, leads to a moderate increase of the :RCR 0 (R, R 0 ZH, CH3) and to a modest shortening of the CHald bond length while the CC bond length does not change very much. 3.3. The p/p* and s/p* states The CASSCF equilibrium geometry for the p/p* and s/p* states of formaldehyde, acetaldehyde and acetone are reported in Tables 5–7, respectively. The equilibrium geometry of the 1(p/p*) state of formaldehyde has been subject of debate. The Walsh rules [67] indicate that all the n/p*, p/p* and s/p* states Table 5 Equilibrium geometries for the S2 (the 1(n/p*) and 1(s/p*) configurations are mixed), 3(p/p*) and 3(s/p*) states of formaldehyde Method

RCO

RCH

:HCH

qa

S2 exited state CASSCF(CAS(A))b (I) CASSCF(CAS(C))b (I) CASSCF(CAS(A))b (II) CASSCF(CAS(C))b (II) MRD-CI [65] EOM-CCSD [53] CASPT2 [21] CIS [58] MR-CISD [43] MR-AQCC [43]

1.564 1.535 1.356 1.257 1.528 1.583 1.492 1.460 1.505 1.495

1.082 1.112 1.140 1.264 1.080 1.095 1.094 1.073 1.082 1.088

116.2 113.2 91.0 55.7 111.3 119.6 118.4 124.4 117.5 118.9

48.3 44.6 76.5 66.8 w44 0.0 46.2 0.0 45.8 45.4

3 (p/p*) excited state CASSCF(CAS(A))b CASSCF(CAS(C))b MRD-CI [22] MP2 [22] CIS [58] Experimentalc

1.474 1.465 1.476 1.451 1.408 1.423

1.077 1.102 1.075 1.082 1.072

120.0 119.0 119.9 121.3 119.2

35.2 32.2 38.1 28.9 39.4

(s/p*) excited state CIS [28] MRD-CI [51]

1.489 1.529

1.079 1.080

121.9 115.1

46 42.1

3 (s/p*) excited state CASSCF(CAS(A))b CASSCF(CAS(C))b MRD-CI [22]

1.499 1.483 1.455

1.073 1.101 1.090

129.4 128.7 133.5

46.9 45.2 35.5

Table 6 Equilibrium geometries for the S2 (the 1(p/p*) and 1(s/p*) configurations are mixed), 3(p/p*) and 3(s/p*) states of acetaldehyde Method

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work. I and II indicate two different minima found on the S2 energy surface. c Electron-impact spectroscopy, Ref. [66] as cited in Ref. [22].

RCO

S2 excited state CASSCF (CAS(B))b 1.624 CIS [28] 1.486 3 (p/p*) excited state CASSCF (CAS(B))b 1.478 3 (s/p*) excited state CASSCF (CAS(B))b 1.523

:OCC

:CCHald

qa

1.448 1.083 1.472 1.079

110.7 112.8

118.2 121.8

32.8

1.489 1.078

114.3

120.4

37.3

1.482 1.074

110.9

129.9

45.5

RCC

RCHald

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work.

should have bent equilibrium geometry, as actually has been experimentally and theoretically demonstrated for the n/ p* state (Table 2). For the p/p* state no experimental data exist regarding the equilibrium geometry. Excited state SCF (SCFX) [68], two configuration SCF (TCSCF) [68], CI of singles (CIS) [58,28], multireference singles and doubles CI (MRD-CI) [12,69] and CASPT2 [21] calculations have found that the equilibrium geometry for the 1(p/p*) state is planar. In a recent work Dallos et al. [43] performing full geometry optimization using high level MR methods, have shown that it has a pyramidal equilibrium geometry and that the structure optimized under planar constraints is a saddle point, therefore confirming the predictions of the Walsh rules. Moreover, Dallos et al. have found that the 1(p/p*) and 1(s/p*) energy surfaces exhibit a conical intersection and that the 1(p/p*) state at its equilibrium geometry has a strong component on the n2 GS electronic configuration (this is the origin of the predissociated character of all VUV absorption bands), and a large component of the 1(s/p*) state. A pyramidal minimum in the two 1A 0 energy surface with a mixed p/p* and s/p* character was also found by other authors [12,21]. In the present calculations, two minima have been found on the S2 potential energy surface of formaldehyde: they are Table 7 Equilibrium geometries for the S2 (the 1(p/p*) and 1(s/p*) configurations are mixed), 3(p/p*) and 3(s/p*) states of acetone Method

1

59

S2 excited state CASSCF(CAS(A))b CASSCFc 3 (p/p*) excited state CASSCF(CAS(A))b 3 (s/p*) excited state CASSCF(CAS(A))b

RCO

RCC

:CCC

qa

1.648 1.642

1.453 1.453

123.8 123.7

44.6 45.4

1.483

1.493

121.1

40.4

1.445

1.494

120.6

40.1

˚ , angles in degrees. Distances in A a Out-of-plane bending angle. b This work. c CASSCF with six electrons in six orbitals using the 6-311CG(d,p) basis set, Ref. [40].

60

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

reported in Table 5 and indicated with the notations I and II. In other cases, multiple solutions have been found which differ only for the nature of the active orbitals with occupation number close to two or zero and describe therefore the same geometrical minimum, giving almost the same equilibrium geometry. On the contrary, in this case the two solutions are related to different geometrical structures. Structure I confirms the results of Dallos et al.: the molecule has a pyramidal structure with an out-of-plane bending angle of 45–488 and the wavefunction has a p/p* and s/p* mixed character. This mixing makes the label here used, p/p*, less meaningful than for the other states and this state would be better termed S2. The CASSCF RCO bond distance for the I minimum with both the CAS(A) and CAS(C) active spaces is slightly too long with respect to the MR-CISD [43] ones, while the RCH is slightly too large with CAS(C) and is well described with CAS(A). The :HCH angle is always lower than in the MR-CISD calculations. The second minimum (II) has a peculiar structure: the RCO distance is shorter than for the I structure (with CAS(C) it is close to the value found for the GS), the RCH bond length is the longest found for all the states here studied and the :HCH angle is the smallest. The molecule is strongly bent with an out-of-plane angle as large as 76.58 (CAS(A)) and 66.88 (CAS(C). This structure, which has never been characterized, can be reasonably excluded as a relevant structure in the interpretation of the absorption spectrum and it will not be described in more detail in the following. The triplet p/p* state has also a pyramidal equilibrium geometry with a shorter RCO bond length and a less bent structure than the singlet state. The geometrical parameters agree with previous published results, apart from the already discussed characteristic of the RCO bond distance which turns out to be slightly too long with respect to the experimental value. In this case no mixing is noticed with the s/p* configuration and the electronic configuration p/p* provides a good representation. The s/p* singlet and triplet states of formaldehyde have received much less attention than the n/p* and p/ p* states. In this work no minimum has been found on the S3 energy surface, thus confirming the results of Dallos et al. [43]. Also in this case, the triplet is exempt from the p/ p*/s/p* mixing. It has a pyramidal structure with a larger out-of-plane bending angle than the 3(p/p*) state. In the case of acetaldehyde and acetone, for the p/p* states only two theoretical determinations of the equilibrium geometry have been published: the first regards the 1(p/ p*) state of acetaldehyde [28] while the second regards the 1 (p/p*) state of acetone [40]. The 3(p/p*) and the s/ p* states have not been studied. No experimental data exist in these cases. The results here obtained show that, as in formaldehyde, the second singlet excited state has a mixed p/p* and s/ p* nature in both molecules: therefore it is called hereafter S2. In acetaldehyde the CASSCF approach gives for this state a geometrical structure quite similar to the CIS one [28], apart

from the CO bond length for which CIS is known to give too short values. The equilibrium geometry here obtained for acetone is very close to the one published in Ref. [40] and confirms that the molecule is bent with an out-of-plane angle similar to the one found for formaldehyde. The CO bond length is remarkably longer and the :CCC angle is larger than in formaldehyde. In both triplets no mixing has been found between the p/p* and s/p* electronic configurations. The structure of the 3(p/p*) state for acetaldehyde and acetone is here reported for the first time. As for the S3 state, our calculations confirm the results on acetone of Diau et al. [40], who have found that no stationary point exists on the S3 surface, any attempt to find it always leading to the S2 minimum. The comparison of the three molecules reveals that the 3 (p/p*) state has a geometrical structure similar for all molecules, only the out-of-plane bending angle showing a moderate variation. This behaviour was also found for the n/p* states. On the contrary, the S2 and 3(s/p*) states present different structures when changing the carbonyl substituents. In the case of the S2 state the CO bond length and the :RCR 0 (R, R 0 ZH, CH3) angle increase when the hydrogen atoms are substituted with the methyl group, while the CC and CHald bond lengths remain almost unchanged. The pyramidalization angle does not have a clear trend, being large for formaldehyde, small for acetaldehyde and intermediate for acetone. For the 3(s/ p*) state the behaviour is less clear: one can again note that the CC and CHald bond lengths show small variations and that the :RCR 0 and pyramidalization angles are very close for formaldehyde and acetaldehyde. In all cases the excitation process leads to a CO bond lengthening larger than for the n/p* states: this can be easily understood noting that in the p/p* and s/p* states the electron is promoted from a bonding to an antibonding orbital.

4. Harmonic vibrational frequencies 4.1. Ground state The harmonic vibrational frequencies for the ground state are reported in Tables 8–10 for the three molecules, respectively, together with the most meaningful published values. For the ground state of the three molecules here considered the experimental fundamental and harmonic vibrational frequencies have been published and the assignment of the normal modes is known without ambiguity. The comparison of the results here obtained with these data allows some general considerations to be drawn. It is well known that for HF calculations the computed harmonic frequencies are consistently too high and they must be scaled by a factor less than one for the comparison with experimental harmonic frequencies due to

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

61

Table 8 Vibrational frequencies (cmK1) and ZPE (eV) for the ground state of formaldehyde Method

Asym CH str. (b2)

Sym CH str. (a1)

CO str. (a1)

CH2 sciss. (a1)

CH2 rock (b2)

Out-of-pl. bend (b1)

ZPE

CAS(A)a CAS(C)a CAS(C)b CAS(C)c MR-CISDd MR-AQCCd CCSDe DFTf Experimentg Experimenth Experimenti,j

3236 3031 3006 2999 3102 3011 3042 2947 2997 3009 2843

3140 2795 2789 2780 3024 2946 2963 2890 2978 2944 2782

1755 1788 1791 1797 1778 1773 1787 1810 1778 1764 1746

1590 1558 1564 1567 1564 1543 1531 1531 1529 1563 1500

1302 1294 1304 1311 1285 1272 1259 1264 1299 1288 1249

1223 1214 1224 1224 1212 1170 1167 1200 1191 1191 1167

0.759 0.724 0.724 0.724 0.742 0.726 0.728 0.722 0.730 0.729 0.700

a b c d e f g h i j

Theoretical harmonic frequencies, ANO basis set (C,O (14s9p4d)/[4s3p1d], H (8s4p)/[2s1p] contraction scheme), this work. Theoretical harmonic frequencies, ANO basis set (C,O (14s9p4d3f)/[5s4p2d1f], H (8s4p3d)/[3s2p1d] contraction scheme), this work. Theoretical harmonic frequencies, cc-pVQZ basis set (C,O (5s9p5d3f1g)/[8s7p5d3f1g], H (6s3p2d1f)/[4s3p2d1f] contraction scheme), this work. Theoretical harmonic frequencies, Ref. [43]. Theoretical harmonic frequencies, Ref. [54]. Theoretical harmonic frequencies, Ref. [70]. Experimental harmonic frequencies from stimulated emission spectroscopy, Ref. [71]. Experimental harmonic frequencies, Ref. [72]. Experimental fundamental frequencies, Ref. [72]. FT-IR gas phase measurement of fundamental frequencies, Ref. [73].

the lack of electronic correlation (static and dynamic) of the HF wavefunction. In the CASSCF wavefunction part of the static electronic correlation is in principle taken into account, so a closer agreement with experiment could be expected. Actually this is true for the normal modes involving bonds for which the valence bonding and

antibonding orbitals are included in the active space. In the calculations here reported, this is always true only for the CO stretching. In the particular case of formaldehyde with CAS(C), where all the valence bonding and antibonding orbitals are active, the above consideration is true for all bonds. For instance the CH bonds in formaldehyde with

Table 9 Vibrational frequencies (cmK1) and ZPE (eV) for the ground state of acetaldehyde Symm. A

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

A 00

ZPE a b c d e f g

Approx. mode

CAS(B)a

DFTb

DFTc

Exp.d

Exp.e

CH3 str CH3 str CH str CO str CH3 bend CH3 bend CH bend CH3 rock CC str CO bend CH3 str CH3 bend CH bend CH3 rock CH3 torsion

3272 3169 3146 1777 1583 1520 1487 1206 955 525 3220 1595 1203 837 167 1.591

3142 3030 2878 1802 1466 1424 1384 1132 886 511 3079 1475 1136 778 158 1.505

3136 3021 2871 1808 1460 1420 1377 1133 886 510 3075 1469 1128 776 152 1.501

3138 3056 2842 1774 1457 1412 1400 1146 907 521 3073 1470 1130 781 187 1.506

3014 2923 2716 1743 1433 1395 1352 1114 867 509 2964 1448 1111 764 150 1.457

CASSCF harmonic frequencies, this work. DFT B3LYP/6-311CCG(2d,2p) harmonic frequencies, Ref. [70]. DFT B3LYP/6-311CCG** harmonic frequencies, Ref. [74]. Experimental harmonic frequencies, Ref. [75]. Experimental fundamental frequencies, Ref. [76]. Fundamental frequencies, Ref. [61]. Fundamental frequencies, Ref. [77]. 142 and 144 cmK1 correspond to v 00 15(E) and v 00 15(A), respectively.

Exp.

00001742f

00000764f 142–144 g

62

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

Table 10 Vibrational frequencies (cmK1) and ZPE (eV) for the ground state of acetone Symm. A1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

A2

B2

B1

ZPE a b c d e

Approx. mode

CAS(A)a

HFb

DFTb

DFTc

Exp.d

Exp.e

CH3 s str CH3 s str CO str CH3 a bend CH3 s bend CH3 rock CC s str CCC s bend CH3 a str CH3 a bend CH3 rock CH3 torsion CH3 a str CH3 a str CH3 a bend CH3 a bend CC a str CH3 rock CO bend CH3 a str CH3 a bend CH3 rock CO bend CH3 torsion

3274 3173 1785 1587 1519 1164 837 397 3217 1589 978 71 3273 3166 1584 1528 1329 982 551 3225 1612 1202 511 156

3286 3180 1975 1592 1524 1182 840 404 3221 1591 972 51 3285 3172 1584 1533 1343 973 578 3231 1614 1232 535 150

3146 3041 1778 1476 1398 1090 783 383 3086 1472 892 59 3145 3034 1465 1394 1237 892 537 3093 1496 1122 495 141

3020 2914 1733 1414 1330 1041 750 364 2960 1410 848 22 3018 2907 1404 1332 1183 848 515 2967 1431 1075 471 133

3004 2937 1731 1435 1355 1072 777 385 2972 1426 872 112 3018 2920 1410 1364 1216 891 530 2972 1454 1090 484 130

3132 3062 1760 1471 1389 1099 797 395 3097 1462 894 115 3147 3044 1446 1398 1247 914 543 3099 1491 1118 496 133

2.400

2.421

2.273

2.175

2.204

2.278

CASSCF harmonic frequencies, this work. HF and DFT B3LYP 6-311CCG(2d,2p) harmonic frequencies, Ref. [70]. DFT B3LYP/6-311G** harmonic frequencies scaled by a factor 0.9614, Ref. [36]. Experimental fundamental frequencies, Ref. [78]. Experimental harmonic frequencies estimated using the ratio harmonic/observed values of acetaldehyde, Ref. [78].

CAS(A) are treated in a ‘single reference’ (similar to HF) manner therefore neglecting the s/s* correlation. For formaldehyde (Table 8) the CAS(A) frequencies are in acceptable agreement with the experimental harmonic frequencies, apart from the symmetric and asymmetric stretchings of the CH bonds for which too high (x200 cmK1) values are found. The use of a larger active space has the effect to improve the agreement with experimental data, in particular lowering the two CH stretchings, but whereas the asymmetric stretching is now very close to experiment, the symmetric one shows a considerable error (x150 cmK1). The source for such an error can be located in the incompleteness of the basis set or in the limited account of the correlation effects brought by the CASSCF wavefunction. In order to clarify this point, larger basis sets (ANO with more orbitals and cc-pVQZ) have been used: the results (reported in Table 11) show quite modest variations with respect to those obtained with the C,O (14s9p4d)/[4s3p1d], H (8s4p)/[2s1p] contraction of the ANO basis sets. The completeness of the basis set is therefore excluded as the source of the discrepancy on the symmetric CH stretching frequency, which can be reasonably ascribed to the lack of dynamical correlation in the CAS(C) wavefunction. This result is confirmed by the comparison with other theoretical data reported in Table 8: indeed all

methods take into account the dynamic correlation and their agreement with experiment is higher than in the present case. For acetaldehyde and acetone similar considerations can be done. The CASSCF frequencies are all too high with respect to the experimental harmonic frequencies with the exception of the CO stretching, for which a good agreement is found. The value of the CO stretching frequency in all molecules is similar. 4.2. The n/p* p/p* and s/p* states The vibrational frequencies computed in this work for the excited states, together with the experimental and some relevant theoretical data are reported in Tables 11–13 for formaldehyde, acetaldehyde and acetone, respectively. For the excited states experimental information on the vibrational frequencies is scanty and the theoretical approach can be of great help for the interpretation and assignment of the vibrational structure of spectra. In the case of acetaldehyde and acetone the few published experimental data are not reported in respective tables, but they are commented in the text for the sake of clarity. Let us first note that for excited states, especially in the case of acetaldehyde and acetone, the assignment of a description label to the normal modes is not an easy task (and in some cases

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

63

Table 11 Vibrational frequencies (cmK1) and ZPE (eV) for the excited states of formaldehyde Method

Asym CH str. (a 00 )

Sym CH str. (a 0 )

CO str. (a 0 )

CH2 sciss. (a 0 )

CH2 rock (a 00 )

Out-of-pl. bend (a 0 )

3254 2919 3025 3102 3073 2847

1098 1128 1184 1187 1249 1173

1514 1356 1353 1413 1408 887 1290d 1429

1059 976 936 933 934 904 0683e

839 771 691 565 720

0.692 0.630 0.640 0.646 0.656

3243 2868 2700 3019 2871i

1153 1194 1251 1249 1218j

1509 1322 1405 1333

1031 929 778 895

857 842 721 767 0524k

0.692 0.628 0.596 0.644

3238 2911 3121 2932 3191

717 690 815 818 737

1443 1366 1329 1351 1495

985 960 987 963 938

1241 1113 1141 2204 –

0.682 0.625 0.661 0.703

3261 2999 2938

998 1022 1164 0887o

1548 1460 1464

1160 1121 1159

832 693 765

0.695 0.646 0.596

3271 2954

807 756

1422 1306

1147 1048

961 876

0.689 0.626

ZPE

1

(n/p*) excited state CAS(A)a 3392 CAS(C)a 3018 3134 MR-AQCC [43]b P-EOM-MBPT [35]b 3223 EOM-CCSD [79]b 3190 Exp.c 2968 Exp.d Exp f 3 (n/p*) excited state 3377 CAS(A)a CAS(C)a 2974 CISg 2765 CCSD(T)h 3127 Exp. S2 (mixed p/p* and s/p*) excited state CAS(A)a 3383 CAS(C)a 3037 3266 MR-AQCCl CISm 3075 CIDSn 3328 3 (p/p*)excited state 3406 CAS(A)a CAS(C)a 3125 CISm 3044 Exp. 3 (s/p*)excited state CAS(A)a 3501 CAS(C)a 3157 a

Theoretical harmonic frequencies, this work. Theoretical harmonic frequencies. c Experimental fundamental frequencies from Ref. [64]. d Experimental fundamental frequencies by tentative assignment of a cold band, Ref. [80]. e Experimental fundamental frequencies defined as the energy between the average of v(0C,0K)4v(1C,1K), Refs. [81,28]. f Experimental fundamental frequencies by assignment of the bands recorder in Ref. [64] using calculated transition intensities, Ref. [82]. g Theoretical harmonic frequencies scaled using scaling factors defined for the ground state (0.875–0.908), Ref. [28]. The two frequencies 1251 and 1405 cmK1 are assigned in Ref. [28] to the CH2 scissor and CO stretch, respectively. h CCSD(T) theoretical harmonic frequencies using TZ2P(f,d) basis. i Ref. [83] as cited in Ref. [28]. j Ref. [84] as cited in Ref. [28]. k Ref. [66] as cited in Ref. [28]. l Theoretical harmonic frequencies, Ref. [43]. m Theoretical harmonic frequencies scaled using scaling factors defined for the ground state (0.875–0.908), Ref. [28]. n The CISD wavefunction is generated from three reference configurations, Ref. [68]. o Ref. [66] as cited in Ref. [28]. b

meaningless). Indeed the fact that the molecules have a lower symmetry than the GS and a distorted geometry, with elongated bonds, can make the identification of a normal mode with a group motion difficult because both mixing of the GS normal modes and vibrations involving large portions of the molecule are often observed. In the following the labels used for the ground state have been maintained, but they must be intended in general only as an approximate description of the nuclear motion. For formaldehyde, the effect of the n/p* excitation is to lower the CO stretching, the CH2 rocking and the out-ofplane bending while the CH stretchings show modest

variations (the symmetric stretching is slightly increased). This behaviour is common to both the singlet and the triplet state and the agreement with the experimental value is good. Moreover, let us note that the singlet and triplet states have very similar theoretical harmonic frequencies. For the CH2 scissoring mode in the singlet state, particular care must be exerted in analyzing the experimental data. As described by Dallos et al. [43], the original value of 887 cmK1 obtained analyzing hot bands [64], has been modified to 1290 cmK1 by Hardwick and Till [80] using a tentative assignment of a cold band and to 1429 cmK1 by van Dijk et al. [82] on the basis of theoretical transition

64

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

Table 12 Vibrational frequencies (cmK1) and ZPE (eV) for the excited states of acetaldehyde Symm.

A0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ZPE

Approx. mode CH str CH3 str CH3 str CH3 str CH3 bend CH3 bend CH3 bend CH bend CH3 rock CC str CH3 rock CO str CH bend CO bend CH3 torsion

1

3

(n/p*) a

(n/p*)

b

a

3

S2 c

b

a

b

(p/p*) a

3

(s/p*)

CAS(B)

CIS

CAS(B)

UHF

CIS

CAS(B)

CIS

CAS(B)

CAS(B)a

3282 3249 3218 3154 1607 1596 1521 1393 1178 1140 1092 943 698 399 203 1.530

3313 3271 3202 3187 1672 1617 1607 1508 1277 1123 1092 968 556 408 202 1.550

3259 3246 3221 3156 1609 1598 1518 1388 1200 1121 1098 960 711 397 200 1.530

3316 3279 3247 3180 1663 1651 1576 1422 1196 1159 1114 939 673 388 184 1.549

3319 3283 3199 3055 1660 1616 1612 1497 1240 1128 1012 924 670 375 202 1.537

3284 3261 3209 3023 1591 1519 1462 1330 1232 1063 891 601 697 339 292 1.481

3302 3330 3190 3120 1615 1543 1506 1265 1210 1093 1046 811 763 404 234 1.515

3305 3241 3206 3138 1597 1587 1529 1420 1190 1141 1093 890 692 404 220 1.528

3349 3267 3223 3154 1600 1566 1503 1291 1119 1174 1036 764 669 402 212 1.508

The numbers in italic indicate strongly coupled modes. a CASSCF harmonic frequencies, this work. b Ref. [28]. c Ref. [85].

intensities. The CAS(C) value here obtained (1356 cmK1) is between the two experimental values, but considering that the anharmonicity of the potential usually gives fundamental frequencies lower than the harmonic ones, the CAS(C) result is in a better agreement with the 1290 cmK1 value. The same consideration has been provided by Dallos et al. [43]. For the p/p* and s/p* triplet and S2 states only one frequency has been experimentally measured (CO stretching for the 3(p/p*) state). The 3(s/p*) state has never been studied theoretically and for the 3(p/p*) state only one CIS [28] calculation has been published. The results obtained with CAS(A) and CAS(C) are reported for such states in Table 11. The comment hereafter is based on the CAS(C) values. One promptly notes the sizeable lowering of the CO stretching in the S2 (690 cmK1) and 3(s/p*) (756 cmK1) with respect to the n/p* states (1128 cm K1 for singlet, 1194 cmK1 for triplet) and with respect to the GS (1788 cmK1). The lowering is less pronounced for the 3 (p/p*) state (1022 cmK1). This behaviour is understandable if one considers that for the p/p* and s/p* states there is a hole in a bonding orbital, while in the n/p* state the hole is in a non-bonding orbital. A common characteristic of all the states reported in Table 11 is the quite high value for the CH stretchings, where a sizeable difference with respect to the GS is found. Another interesting feature of the data reported in Table 11 is the value of the out-of-plane bending frequency for the S2 singlet (1113 cmK1): it is closer to the ground state value (1214 cmK1) than to the n/p* one (771 cmK1 for singlet, 842 cmK1 for triplet) while for the p/p* and s/p* triplets the opposite occurs. The particular behaviour of this singlet could be a manifestation

of the avoided crossing of the p/p* and s/p* surfaces which leads to the mixing of the two electronic configurations. Finally in all states the CH2 scissoring has a frequency similar to the one found for the n/p* states, only the 3(p/ p*) state showing a difference larger than 100 cmK1. For the acetaldehyde molecule, the previously published calculations of the excited state harmonic frequencies regard only the 1(n/p*), 3(n/p*) and S2 states and are all performed with single reference based methods. For these states, those here reported are the first example of multireference values, while for the 3(p/p*) and 3(s/ p*) states this work presents the first theoretical study. First of all, one notes that the occurrence of quasi degeneracy of the vibrational levels (and therefore their mixing) appears in various cases: a similar result was also obtained by Hadad et al. [28]. The aldehydic CH stretching is the highest for all states, with an increase, with respect to the GS, in the range of 100–200 cmK1. In two cases (3(n/p*) and S2) this mode becomes quasi degenerate with one of the CH3 stretchings, leading to a mixing of the two modes. The CH3 stretchings show very modest variations with respect to the GS: the only exception is given by the S2 state, where the lowest frequency is x120 cmK1 lower than in the GS. In the n/p* states, the CO stretching is strongly mixed with other modes (CH3 rocking and CC stretching): the resulting frequencies are in the range 950–1200 cmK1. For the other states, the frequency associated to such mode is very low (601, 890 and 764 cmK1 for the S2, 3(p/p*) and 3(s/p*) states, respectively) with respect to the GS value (1777 cmK1). The CH ‘in-plane’ bend (1487 cmK1 in the GS) is lowered by x100 cmK1 in the n/p* states, by x160 cmK1 in the S2 state and by x60 and 200 cmK1 in

Table 13 Vibrational frequencies (cmK1) and ZPE (eV) for the excited states of acetone Symm.

A0

ZPE

CH3 str CH3 str CH3 str CH3 bend CH3 bend CH3 bend CO str CH3 rock CH3 rock CCC str CO bend CCC bend CH3 torsion CH3 str CH3 str CH3 str CH3 bend CH3 bend CH3 bend CCC str CH3 rock CH3 rock CO bend CH3 torsion

1

3

(n/p*)

(n/p*)

3

S2

(p/p*)

3

(s/p*)

CAS(A)a

CASb

CASc

CAS(A)a

CASc

CAS(A)a

CASd

CAS(A)a

CAS(A)a

3245 3212 3142 1615 1604 1531 1337 1192 1008e 816 401 350 201 3242 3210 3138 1597 1590 1520 1308 1047 1017 530 181 2.358

3019 2986 2928 1400 1481 1408 1226 1113 942 768 395 320 196 3021 2992 2927 1483 1470 1399 1294 962 952 360 170 2.183

3313 3278 3186 1735 1734 1544 1053 1455 1153 838 407 371 234 3307 3269 3178 1577 1576 1531 1349 1360 1200 434 192 2.435

3245 3212 3142 1617 1606 1530 1352 1197 1024 826 411 348 203 3242 3211 3139 1597 1591 1517 1292 1042 1023 400 179 2.353

3317 3282 3189 1745 1745 1539 1077 1465 1147 847 398 370 235 3314 3276 3184 1578 1578 1524 1323 1360 1199 446 192 2.438

3265 3206 3062 1599 1533 1478 1263 1122 640e 881 393 366 292 3261 3202 3058 1600 1523 1483 1344 1057 782 307 287 2.294

3269 3209 3048 1526 1482 1219 1604 1174 906 639 440 352 261 3263 3204 3043 1606 1524 1481 1341 1046 798 336 238 2.294

3248 3210 3132 1608 1596 1531 1287 1174 1004 793 389 369 212 3243 3205 3128 1592 1584 1527 1341 1043 995 417 190 2.345

3253 3215 3136 1617 1584 1530 1234 1196e 981e 791 400 348 141 3247 3207 3132 1600 1565 1515 1356 1030 984 397 202 2.335

The numbers in italic indicate strongly coupled modes. a CASSCF harmonic frequencies, this work. b CASSCF/6-311G** harmonic frequencies scaled by a factor 0.93, Ref. [36]. c CASSCF with eight electrons in seven orbitals using a C[3s2p1d]/H[2s1p] basis set, Ref. [62] The authors have not indicated the symmetry of the vibration, here a tentative labeling is given. d CASSCF(6,6)/6-311CG(d,p) harmonic frequencies, Ref. [40]. e The harmonic frequency partially describe the CO stretch.

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

A 00

1 2 20 4 21 5 3 6 22 7 23 8 24 13 9 14 15 16 10 17 18 11 19 12

Approx. mode

65

66

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

the 3(p/p*) and 3(n/p*) states, while for the ‘out-ofplane’ bending (1203 cmK1) the lowering is much larger (frequencies in the range 669–711 cmK1 are found according to the state). As for the various CH3 bending and rocking modes, one notes that they do not show sizeable variations, with the exception of the lowest CH3 bending for the S2 state (where a lowering of almost 60 cmK1 is found) and from the lowest CH3 rocking which shows an increase in all states (from 837 to 1036–1098 cmK1) apart from S2 (where the increase is only of x50 cmK1). The CC stretching also shows an increase of the vibrational frequency passing from the GS (955 cmK1) to the excited states: the increase is in a close range (170–220 cmK1) for all states apart from the S2 one for which it is 108 cmK1. It is important to note that, for the n/p* states, the lowest CH3 rocking and the CC stretching are badly defined due to the large mixing with other modes, and therefore the above considerations have to be regarded as only indicative. Finally, for the CH3 torsion, for all the states there is a modest increase from 167 cmK1 in the GS to 200–220 cmK1, apart again the case of the S2 state (292 cmK1). From the experimental point of view, some information is available for the n/p* states. For the singlet, one of the CH3 deformations has been assigned [86] to 1254 cmK1, the CO stretching to 1119 [61] and 1173 [86] cmK1, the CO out-of-plane bend to 374 [87] and 482 [86] cmK1 and finally the CH3 torsion to 202 [77], 187 [61] and 186 [88] cmK1. The comparison of these data with the results of this work leads to the following considerations: † for the CH3 deformation the CAS(B) value is 1521 cmK1, with a difference of 267 cmK1 with the experimental one. In the GS the difference was smaller (100–150 cmK1) for the three values indicated in Ref. [86] in the analogous modes (1433, 1352 and 1114 cmK1); † the experimental CO stretching seems to be far from the present assignment. It is worth noticing again that in the calculation on this state and on the n/p* triplet, the CO stretching has a component over four different normal modes. Three of them are closer to the experimental values; † for the out-of-plane CO bending the comparison is difficult: it is well known that along this normal coordinate the system shows a double minimum potential and the vibrational wavefunction is delocalized in both minima. The values here reported, obtained using the harmonic approximation in one minimum, appear to be consistent with the experimental assignment; † for the CH3 torsion the agreement is good. For the triplet n/p* state four vibrational frequencies have been measured experimentally [89]: 905, 518, 364, and 179 cmK1. These values agree well with the present 1098, 711, 397 and 200 cmK1, remembering that experimental fundamental and theoretical harmonic frequencies are compared.

As for the acetone molecule, let us note that in the calculations of the excited states, the geometry optimization has been performed in the Cs symmetry group: this allows the molecular skeleton to pyramidalize, but imposes a constraint for the rotation of the methyl groups. Nevertheless the optimized geometries have been confirmed to be true minima (possibly not the global minimum) by a vibrational analysis. Let us note that while for the 1(n/p*), 3 (n/p*) and S2 states CASSCF calculations have been published, for the 3(p/p*) and 3(s/p*) states the one here reported is the first theoretical study. Some general considerations can be drawn for all the states. First of all, as in acetaldehyde, in some cases difficulties have been found in the assignment of the vibrational modes. In one case the mixing of two modes of the GS was strong, due to the large coupling among them. The CH stretching and the CH3 bending modes show higher frequencies than in the ground state, while the CH3 bending frequencies do not show sizeable variations. The asymmetric and symmetric CC stretchings remain almost unchanged. The CH3 torsions have higher frequencies for all states, especially for the S2 singlet, probably due to a sterical effect originated by the pyramidalization of the carbonyl carbon atom. In all the excited states the CCC bend has a slightly lower frequency than in the GS. As far as the CO bending modes are concerned, remarkably lower frequencies are found for all states apart from 1(n/p*) for the out-ofplane (A 00 ) mode. This behaviour is particularly pronounced for the S2 state where a lowering of more than 200 cmK1 is found for the out-of-plane bend. On the other hand, for the in-plane (A 0 ) bend the lowering is larger than 150 cmK1 for several states. The most interesting mode is, however, the CO stretching for which a marked lowering is expected given the nature of the excited states. The lowering with respect to the GS (1785 cmK1) is of the order of 400–600 cmK1 according to the state and is in general less pronounced than in formaldehyde and acetaldehyde, particularly for the S2, 3 (p/p*) and 3(s/p*) states where in both molecules frequencies lower than 1000 cmK1 have been found (for instance in formaldehyde the reported values were 717, 998 and 807 with CAS(A) for the three states, respectively). In acetone this frequency ranges from 1234 cmK1 in the 3(s/ p*) state to 1352 cmK1 in the 3(n/p*) one. Some disagreement with previously published results is observed: the values obtained by Sakurai and Kato [62] for the two n/p* states (1053 and 1077 cmK1) are lower by almost 300 cmK1 with respect to those here computed. Note that the value obtained by Liao et al. [36] for the 1(n/p*) state (1226 cmK1) is scaled by a factor 0.93 (1318 cmK1 before scaling) and therefore is in excellent agreement with our result (1337 cmK1). In the case of the 1(n/p*) state, refined analysis of the vibrational structure of the spectra have allowed the determination of some vibrational frequencies.

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

In 1985, Baba et al. [86] have given a detailed vibronic analysis of the fluorescence excitation spectrum in a supersonic nozzle beam assigning the following frequencies: CO stretching, 1121 cmK1; CO out-of-plane bending, 373 and 578 cm K1 for the 1C and 1K levels, respectively; CCC stretching, 757 (sym) and 1294 (asym) cmK1; CH3 rocking or deformation, 955 cmK1 (tentative assignment). In 1992, Zuckermann et al. [90] performed a high resolution (0.01 cmK1) laser-induced fluorescence study giving the following assignment: CH3 torsions, 156 and 172 cmK1; CCC bending, 373 cmK1; CO out-of-plane bending, 333 cmK1; CO in-plane bending, 177 or 465 cmK1 with two different assignment strategies. The comparison of these values with the ones obtained in this work shows that the CCC symmetric and asymmetric stretching and the CO out-of-plane modes agree with the assignment of Baba et al. [86]. Moreover, the tentative assignment for the 955 cmK1 frequencies is confirmed (CH3 rocking). The CASSCF CO stretching is notably higher than the experimental values (x200 cmK1), indicating that the CASSCF potential energy surface is not well described along the CO stretch and/or large anharmonic effects can be expected for this mode. With respect to the results of Zuckermann et al. [90], while the CH3 torsions are in good agreement, the assignment of the CCC bending and of

67

the CO out-of-plane bending are probably reversed and for the CO in-plane bending the ‘second possibility’ discussed in Ref. [90] (465 cmK1) better agrees with the value here obtained (530 cmK1). Gwaltney and Bartlett [35] support the same conclusion on the basis of P-EOM-MBPT(2) calculations.

5. Adiabatic transition energies For the sake of completeness, the NEVPT2 adiabatic transition energies (with the ZPE correction) computed for all states and molecules using the equilibrium geometries and the harmonic frequencies obtained in this work are reported in Table 14 together with the known experimental values. A detailed comparison of the theoretical results with the experimental ones has been reported in Ref. [5]: here we limit ourselves to noting that a general good agreement between theory and experiment is found, with the relevant exception of the 3(p/p*) state where differences in the range 0.3–0.7 eV are found for all molecules. Such a disagreement can be explained supposing that the lowest observed peak is not assigned to the 0)0 vibrational transition, but rather to a n)0 one [5,22,66] (nR1, the vibrational levels concern the v2 CO stretch normal coordinate).

Table 14 Adiabatic transition energies (eV) (all states computed at the CASSCF equilibrium geometry with the ZPE correction included) for the n/p*, p/p* and s/p* singlet and triplet excited states of formaldehyde, acetaldehyde and acetone Method Formaldehyde NEVPT-SC (CAS(C)) NEVPT-PC (CAS(C)) Experimental Acetaldehyde NEVPT-SC (CAS(B)) NEVPT-PC (CAS(B)) Experimental Acetone NEVPT-SC (CAS(A)) NEVPT-PC (CAS(A)) Experimental a b c d e f g h i j k l

1

3

S2

3

3.48 3.48 3.50a

3.11 3.08 3.12a

7.55 7.52

4.40 4.42 4.70b, 4.83c

7.10 7.07

3.61 3.62 3.69d,e, 3.56f 3.58g

3.39 3.39 3.29d 3.38g

7.05 7.02

4.41 4.43 5.08f

6.97 6.96

3.70 3.70 3.75h, 3.77i 3.8k

3.45 3.44 3.47j 3.44–3.54l

6.52 6.51

4.41 4.42 5.15h 5.3k

7.18 7.14

(n/p*)

(n/p*)

Optical spectroscopy results from Refs. [11,59]. Estimated in Ref. [22] from a polynomial fit of the electron-impact data for the v2 progression in Ref. [66]. Lowest observed peak in the v2 progression assigned to v 0 Z1 in Ref. [66]. Ref. [87]. Ref. [91]. Ref. [92]. Ref. [61]. Electron-impact data from Ref. [92]. Fluorescence excitation spectra in Ar supersonic nozzle beam, Ref. [86]. Experimental determination of Refs. [93,94], as reported in Ref. [4]. Electron-energy-loss spectroscopy in condensed phase at 18 K, Ref. [95]. Experimental estimation of the 0–0 band, Ref. [96].

(p/p*)

3

(s/p*)

68

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69

6. Conclusions This paper has reported a CASSCF study of the equilibrium geometrical parameters and of the harmonic vibrational frequencies of three singlet and triplet valence excited states of formaldehyde, acetaldehyde and acetone. Particular attention has been paid to a common strategy for the different states and molecules in order to allow a comparison of the results and to identify, if possible, trend behaviours. As far as the equilibrium geometry is concerned, the comparison of the structure of the same state for the different molecules has allowed us to show the effect of the methyl groups and to reveal common trends. The results obtained compare well with the available experimental and high-level theoretical data and allow some controversial assignment to be clarified. For some states the results here presented are the first theoretical characterizations. The main contributions of this work can be summarized as follows: † the first theoretical description of the harmonic frequencies of the 3(s/p*) state of formaldehyde; † the first theoretical description of the geometrical structure and of the harmonic frequencies for the 3 (p/p*) and 3(s/p*) states of acetaldehyde and acetone; † the first multireference study of the vibrational frequencies of the 3(p/p*) state of formaldehyde and of the n/p* states (singlet and triplet) of acetaldehyde; † the first example of a common approach to all the valence states of three small carbonyl molecules, thus allowing a meaningful comparison of the results.

Acknowledgements This research has been supported by local funding of the University of Ferrara.

References [1] C.B. Moore, J.C. Weisshaar, Annu. Rev. Phys. Chem. 34 (1983) 525. [2] W.A. Noyes Jr., Photochemistry and Reaction Kinetics in: P.G. Ashmore, F.S. Dainton, T.M. Sugden (Eds.), Cambridge University Press, Cambridge, 1967, pp. 1–25. [3] E.W.-G. Diau, C. Ko¨tting, A.H. Zewail, Chem. Phys. Chem. 2 (2001) 273. [4] Y. Haas, Photochem. Photobiol. Sci. 3 (2004) 6. [5] C. Angeli, S. Borini, L. Ferrighi, R. Cimiraglia, submitted for publication. [6] C. Angeli, R. Cimiraglia, S. Evangelisti, T. Leininger, J.-P. Malrieu, J. Chem. Phys. 114 (2001) 10252. [7] C. Angeli, R. Cimiraglia, J.-P. Malrieu, Chem. Phys. Lett. 350 (2001) 297. [8] C. Angeli, R. Cimiraglia, J.-P. Malrieu, J. Chem. Phys. 117 (2002) 9138.

[9] C. Angeli, S. Borini, M. Cestari, R. Cimiraglia, J. Chem. Phys. 121 (2004) 4043. [10] D.C. Moule, A.D. Walsh, Chem. Rev. 75 (1975) 67. [11] D.J. Clouthier, D.A. Ramsay, Annu. Rev. Phys. Chem. 34 (1983) 31. [12] M.R.J. Hachey, P.J. Bruna, F. Grein, J. Phys. Chem. 99 (1995) 8050. [13] J.L. Whitten, M. Hackmeyer, J. Chem. Phys. 51 (1969) 5584. [14] R.J. Buenker, S.D. Peyerimhoff, J. Chem. Phys. 53 (1970) 1368. [15] E.R. Davidson, L.E. McMurchie, in: E.C. Lim (Ed.), Excited States vol. 5, Academic Press, New York, 1982. [16] S.R. Gwaltney, R.J. Bartlett, Chem. Phys. Lett. 241 (1995) 26. [17] S.R. Gwaltney, M. Nooijen, R.J. Bartlett, Chem. Phys. Lett. 248 (1996) 189. [18] K.B. Wiberg, R.E. Stratmann, M.J. Frisch, Chem. Phys. Lett. 297 (1998) 60. [19] C. Adamo, G.E. Scuseria, V. Barone, J. Chem. Phys. 111 (1999) 2889. [20] A.I. Krylov, C.D. Sherrill, M. Head-Gordon, J. Chem. Phys. 113 (2000) 6509. [21] M. Merchan, B.O. Roos, Theor. Chim. Acta 92 (1995) 227. [22] P.J. Bruna, M.R.J. Hachey, F. Grein, J. Phys. Chem. 99 (1995) 16576. [23] V.N. Staroverov, E.R. Davidson, Chem. Phys. Lett. 296 (1998) 435. [24] T. Muller, H. Lischka, Theor. Chem. Acc. 106 (2001) 369. [25] H. Lischka, M. Dallos, R. Shepard, Mol. Phys. 100 (2002) 1647. [26] J. Pitarch-Ruiz, J. Sanchez-Marin, A.S. De Meras, D. Maynau, Mol. Phys. 101 (2003) 483. [27] C. Angeli, S. Borini, R. Cimiraglia, Theor. Chem. Acc. 111 (2004) 352. [28] C.M. Hadad, J.B. Foresman, K.B. Wiberg, J. Phys. Chem. 97 (1993) 4293. [29] A. Nin˜o, C. Mun˜oz-Caro, D.C. Moule, J. Phys. Chem. 98 (1994) 1519. [30] C. Mun˜oz-Caro, A. Nin˜o, D.C. Moule, J. Mol. Struct. (Theochem) 315 (1994) 9. [31] M. Head-Gordon, R.J. Rico, M. Oumi, T.J. Lee, Chem. Phys. Lett. 219 (1994) 21. [32] I. Frank, J. Hutter, D. Marx, M. Parrinello, J. Chem. Phys. 108 (1998) 4060. [33] O. Setokuchi, M. Sato, S. Matuzawa, Y. Simizu, J. Mol. Struct. (Theochem) 365 (1996) 29. [34] O. Setokuchi, Y. Simizu, J. Mol. Struct. (Theochem) 401 (1997) 29. [35] S.R. Gwaltney, R.J. Bartlett, J. Chem. Phys. 110 (1999) 62. [36] D.W. Liao, A.M. Mebel, M. Hayashi, Y.J. Shiu, Y.T. Chen, S.H. Lin, J. Chem. Phys. 111 (1999) 205. [37] A.B. Rocha, C.E. Bielschowsky, Chem. Phys. Lett. 337 (2001) 331. [38] O. Setokuchi, S. Matuzawa, Y. Shimizu, Chem. Phys. Lett. 284 (1998) 19. [39] D. Liu, W.-H. Fang, X.-Y. Fu, Chem. Phys. Lett. 325 (2000) 86. [40] E.W.-G. Diau, C. Ko¨tting, T.I. Solling, A.H. Zewail, Chem. Phys. Chem. 3 (2002) 57. [41] P.-O. Widmark, P.-A. Malmqvist, B.O. Roos, Theor. Chim. Acta 77 (1990) 291. [42] T. Helgaker, H.J. Aa. Jensen, P. Jørgensen, et al., DALTON, A Molecular Electronic Structure Program, Release 1.2 2002. [43] M. Dallos, T. Muller, H. Lischka, R. Shepard, J. Chem. Phys. 114 (2001) 746. [44] C. Angeli, C.J. Calzado, R. Cimiraglia, S. Evangelisti, D. Maynau, Mol. Phys. 101 (2003) 1937. [45] T. Leininger, C. Angeli, S. Evangelisti, R. Cimiraglia, D. Maynau, Chem. Phys. Lett. 371 (2003) 49. [46] K. Takagi, T. Oka, J. Phys. Soc. Jpn 18 (1963) 1174. [47] M.D. Harmony, V.W. Laurie, R.L. Kuczkowski, R.H. Schwendeman, D.A. Ramsay, F.J. Lovas, W.J. Lafferty, A.G. Maki, J. Phys. Chem. Ref. Data 8 (1979) 619. [48] R. Nelson, L. Pierce, J. Mol. Spectrosc. 18 (1965) 344. [49] V.A. Bataev, V.I. Pupyshev, I.A. Godunov, J. Mol. Struct. 480/481 (1999) 263. [50] T. Nakajima, S. Kato, J. Chem. Phys. 105 (1996) 5927. [51] M.R.J. Hachey, P.J. Bruna, F. Grein, J. Mol. Spectrosc. 176 (1996) 375.

C. Angeli et al. / Journal of Molecular Structure: THEOCHEM 718 (2005) 55–69 [52] K.K. Baeck, J. Chem. Phys. 112 (2000) 1. [53] K. Wiberg, A. de Oliveira, G. Trucks, J. Phys. Chem. A 106 (2002) 4192. [54] J.E. Del Bene, S.R. Gwaltney, R.J. Bartlett, J. Phys. Chem. A 102 (1998) 5124. [55] M. Kreglewski, J. Mol. Struct. 60 (1980) 105. [56] P. Jensen, P.R. Bunker, J. Mol. Spectrosc. 94 (1982) 114. [57] Y. Yamaguchi, S.S. Wesolowski, T.J. Van Huis, H.F. Schaefer III, J. Chem. Phys. 108 (1998) 5281. [58] J.B. Foresman, M. Head-Gordon, J.A. Pople, M.J. Frisch, J. Phys. Chem. 96 (1992) 135. [59] D.J. Clouthier, D.C. Moule, Top. Curr. Chem. 150 (1989) 167. [60] H. Liu, E.C. Lim, C. Mun˜oz-Caro, A. Nin˜o, R.H. Judge, D.C. Moule, J. Chem. Phys. 105 (1996) 2547. [61] L.M. Hubbard, D.F. Bocian, R.R. Birge, J. Am. Chem. Soc. 103 (1981) 3313. [62] H. Sakurai, S. Kato, J. Mol. Struct. (Theochem) 461 (1999) 145. [63] K.K. Innes, J. Mol. Spectrosc. 99 (1983) 294. [64] V.A. Job, V. Sethuraman, K.K. Innes, J. Mol. Spectrosc. 30 (1969) 365. [65] M. Hachey, P.J. Bruna, F. Grein, J. Chem. Soc. Faraday Trans. 90 (1994) 683. [66] S. Taylor, D.G. Wilden, J. Comer, Chem. Phys. 70 (1982) 291. [67] A.D. Walsh, J. Chem. Soc. 1953 (1953) 2306. [68] W.D. Allen, H.F. Schaefer III, J. Chem. Phys. 87 (1987) 7076. [69] M.R.J. Hachey, F. Grein, Chem. Phys. Lett. 256 (1996) 179. [70] W.O. George, B.F. Jones, Rh. Lewis, J.M. Price, J. Mol. Struct. 550/551 (2000) 281. [71] D.E. Reisner, R.W. Field, J.L. Kinsey, H.-L. Dai, J. Chem. Phys. 80 (1984) 5968. [72] J.L. Duncan, P.D. Mallinson, Chem. Phys. Lett. 23 (1973) 597.

69

[73] T. Nakanaga, S. Kondo, S. Sae¨ki, J. Chem. Phys. 76 (1982) 3860. [74] H.T. Kim, S.L. Anderson, J. Chem. Phys. 114 (2001) 3018. [75] K.B. Wiberg, Y. Thiel, L. Goodman, J. Leszczynski, J. Phys. Chem. 99 (1995) 13850. [76] H. Hollenstein, H.H. Gunthard, Spectrochim. Acta 27A (1971) 2027. [77] H. Gu, T. Kundu, L. Goodman, J. Phys. Chem. 97 (1993) 7194. [78] H. Hollenstein, H.H. Gunthard, J. Mol. Spectrosc. 84 (1980) 457. [79] J.F. Stanton, J. Gauss, N. Ishikawa, M. Head-Gordon, J. Chem. Phys. 103 (1995) 4160. [80] J.L. Hardwick, S.M. Till, J. Chem. Phys. 70 (1979) 2340. [81] V.T. Jones, J.B. Coon, J. Mol. Spectrosc. 31 (1969) 137. [82] J.M.F. van Dijk, M.J.H. Kemper, J.H.M. Kerp, H.M. Buck, J. Chem. Phys. 69 (1979) 2453. [83] J.C.D. Brand, C.G. Stevens, J. Chem. Phys. 58 (1973) 3331. [84] G.W. Robinson, V.E. DiGiorgio, Can. J. Phys. 36 (1958) 31. [85] J.S. Yadav, J.D. Goddard, J. Chem. Phys. 84 (1986) 2682. [86] M. Baba, U. Nagashima, I. Hanazaki, Chem. Phys. 93 (1985) 425. [87] M. Noble, E.K.C. Lee, J. Chem. Phys. 81 (1984) 1632. [88] M. Noble, E.C. Apel, E.K.C. Lee, J. Chem. Phys. 78 (1983) 2219. [89] N.N. Yakovlev, I.A. Godunov, Can. J. Chem. 70 (1992) 931. [90] H. Zuckermann, Y. Haas, M. Drabbels, J. Hainze, L. Meerts, J. Reuss, J. van Bladel, Chem. Phys. 163 (1992) 193. [91] H. Liu, C. Lim, R.H. Judge, D.C. Moule, J. Chem. Phys. 102 (1995) 4315. [92] K.N. Walzl, C.F. Koerting, A. Kuppermann, J. Chem. Phys. 87 (1987) 3796. [93] L.D. Waits, R.J. Horwitz, J.A. Guest, Chem. Phys. 155 (1991) 149. [94] S.W. North, D.A. Blank, J.D. Gezelter, C.A. Longfellow, Y.T. Lee, J. Chem. Phys. 102 (1995) 4447. [95] M. Lepage, M. Michaud, L. Sanche, J. Chem. Phys. 112 (2000) 6707. [96] R.F. Borkman, D.R. Kearns, J. Chem. Phys. 44 (1966) 945.