Structures and conformational dynamics of monomethylated derivatives of acrolein: A quantum-chemical study O.S. Bokareva, V.A. Bataev * , I.A. Godunov Moscow State University, Chemistry Department, 119991 Moscow, Russian Federation article info Article history: Received 25 June 2009 Received in revised form 30 July 2009 Accepted 6 August 2009 Available online 11 August 2009 Keywords: Methyl vinyl ketone Methacrolein Crotonaldehyde Internal rotation Anharmonic approximation Non-rigid molecules abstract For all monomethylated derivatives of acrolein: methyl vinyl ketone CH 3 C(O)CH@CH 2 , methacrolein CHOAC(CH 3 )@CH 2 , trans- and cis-crotonaldehydes CHOACH@CHACH 3 in the ground electronic state, the conformer energy differences, barriers to internal rotation, geometric parameters of minima and tran- sitions states corresponding to the barriers of internal rotation were studied by means of various quan- tum-chemical methods (B3LYP, MP2, QCISD, CCSD(T), CASSCF and others). The conformer energy differences were also estimated using the extrapolative technique VFPA. The vibrational frequencies were calculated at MP2/6-311G(d,p) level in harmonic and different anharmonic approximations. The coupling of two internal rotation motions was investigated by constructing one- and two-dimensional potential energy surface sections and by solving respective vibrational problems. Ó 2009 Published by Elsevier B.V. 1. Introduction Molecules of a,b-unsaturated carbonyls contain two common chromophore groups (C@C and C@O) in the positions providing their mutual influence. The structure, conformational dynamics, and properties of a,b-unsaturated carbonyls in the ground and excited electronic states are of theoretical and practical interest for photochemistry, photophysics, biochemistry, polymer and environmental sciences (see, e.g. [1–4]). Recently, we found anomalously strong coupling of internal rotation around ordinary CAC bond and non-planar distortion of carbonyl fragment HCCO for the simplest representative of this class acrolein (CH 2 @CHCHO) in the lowest excited electronic states of n,p* type [5]. The present work opens a quantum-chemical investigation of the structure of all possible monomethylated derivatives of acrolein (R 1,2,3,4 @H) (see Fig. 1) where more compli- cated cases of coupling may occur. We begin this work with the study of the ground electronic state. Molecules under study are methyl vinyl ketone (R 1–3 @H, R 4 @CH 3 ), methacrolein (R 1,2,4 @H, R 3 @CH 3 ), d-trans-crotonaldehyde (R 1,3,4 @H, R 2 @CH 3 ), and d-cis-crotonaldehyde (R 2–4 @H, R 1 @CH 3 ) (in this work, d- denotes cis- and trans-isomers relative to the double C 2 @C 3 bond). The planar s-trans and s-cis (but non-s-gauche) conformations were proved experimentally to be stable for acrolein [6], acryloyl fluoride [7] and chloride [8], acrylic acid [9], and methyl vinyl ketone [10] (s- denotes cis- and trans-isomers relative to the ordin- ary C 1 AC 2 bond). However, in some cases, the data on the existence and structures of second conformers are either absent or incom- plete, e.g., there are no reliable data for the second conformer of d-trans-crotonaldehyde. In addition, there are no data on the struc- ture of d-cis-crotonaldehyde conformers though this compound is quite stable and can be obtained from d-trans-crotonaldehyde by irradiation [11]. The NMR [11] and electronic absorption [12] spec- tra of d-cis-crotonaldehyde were reported. The structure of methyl vinyl ketone was investigated with microwave [13,14] and vibrational spectroscopy [10,15–18]. According to [10,14–16,18], this molecule existed as s-trans and s-cis-conformers; s-trans is lower in energy by 200–280 cm 1 [10,17]. For s-trans-conformer, the geometric parameters were determined [14]. In [10,13,14,17], the parameters of one-dimen- sional potential functions of asymmetric (about central C 1 AC 2 bond) and symmetric internal rotation (rotation of the methyl top) were found. In [14], the coupling of these internal rotations was also studied and was shown to be essential for both potential and kinematic parts of Schrödinger equation. A notable disagree- ment in the assignment of vibrational bands in two most complete studies [10,18] should be noted for stretching CAH and CAC modes, planar and non-planar deformations, symmetric (s(CH 3 )) and asymmetric (s(CAC)) torsion modes. For methacrolein, s-trans-conformer was shown to be stable, while in a number of works [18,19] the evidences of existence of 0166-1280/$ - see front matter Ó 2009 Published by Elsevier B.V. doi:10.1016/j.theochem.2009.08.004 * Corresponding author. Tel./fax: +7 495 939 36 89. E-mail address: [email protected](V.A. Bataev). Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 Contents lists available at ScienceDirect Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Transcript
Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
For all monomethylated derivatives of acrolein: methyl vinyl ketone CH3C(O)CH@CH2, methacroleinCHOAC(CH3)@CH2, trans- and cis-crotonaldehydes CHOACH@CHACH3 in the ground electronic state,the conformer energy differences, barriers to internal rotation, geometric parameters of minima and tran-sitions states corresponding to the barriers of internal rotation were studied by means of various quan-tum-chemical methods (B3LYP, MP2, QCISD, CCSD(T), CASSCF and others). The conformer energydifferences were also estimated using the extrapolative technique VFPA. The vibrational frequencies werecalculated at MP2/6-311G(d,p) level in harmonic and different anharmonic approximations. The couplingof two internal rotation motions was investigated by constructing one- and two-dimensional potentialenergy surface sections and by solving respective vibrational problems.
� 2009 Published by Elsevier B.V.
1. Introduction
Molecules of a,b-unsaturated carbonyls contain two commonchromophore groups (C@C and C@O) in the positions providingtheir mutual influence. The structure, conformational dynamics,and properties of a,b-unsaturated carbonyls in the ground andexcited electronic states are of theoretical and practical interestfor photochemistry, photophysics, biochemistry, polymer andenvironmental sciences (see, e.g. [1–4]).
Recently, we found anomalously strong coupling of internalrotation around ordinary CAC bond and non-planar distortion ofcarbonyl fragment HCCO for the simplest representative of thisclass acrolein (CH2@CHCHO) in the lowest excited electronic statesof n,p* type [5]. The present work opens a quantum-chemicalinvestigation of the structure of all possible monomethylatedderivatives of acrolein (R1,2,3,4@H) (see Fig. 1) where more compli-cated cases of coupling may occur. We begin this work with thestudy of the ground electronic state. Molecules under study aremethyl vinyl ketone (R1–3@H, R4@CH3), methacrolein (R1,2,4@H,R3@CH3), d-trans-crotonaldehyde (R1,3,4@H, R2@CH3), andd-cis-crotonaldehyde (R2–4@H, R1@CH3) (in this work, d- denotescis- and trans-isomers relative to the double C2@C3 bond).
The planar s-trans and s-cis (but non-s-gauche) conformationswere proved experimentally to be stable for acrolein [6], acryloyl
Elsevier B.V.
.ev).
fluoride [7] and chloride [8], acrylic acid [9], and methyl vinylketone [10] (s- denotes cis- and trans-isomers relative to the ordin-ary C1AC2 bond). However, in some cases, the data on the existenceand structures of second conformers are either absent or incom-plete, e.g., there are no reliable data for the second conformer ofd-trans-crotonaldehyde. In addition, there are no data on the struc-ture of d-cis-crotonaldehyde conformers though this compound isquite stable and can be obtained from d-trans-crotonaldehyde byirradiation [11]. The NMR [11] and electronic absorption [12] spec-tra of d-cis-crotonaldehyde were reported.
The structure of methyl vinyl ketone was investigated withmicrowave [13,14] and vibrational spectroscopy [10,15–18].According to [10,14–16,18], this molecule existed as s-trans ands-cis-conformers; s-trans is lower in energy by 200–280 cm�1
[10,17]. For s-trans-conformer, the geometric parameters weredetermined [14]. In [10,13,14,17], the parameters of one-dimen-sional potential functions of asymmetric (about central C1AC2
bond) and symmetric internal rotation (rotation of the methyltop) were found. In [14], the coupling of these internal rotationswas also studied and was shown to be essential for both potentialand kinematic parts of Schrödinger equation. A notable disagree-ment in the assignment of vibrational bands in two most completestudies [10,18] should be noted for stretching CAH and CACmodes, planar and non-planar deformations, symmetric (s(CH3))and asymmetric (s(CAC)) torsion modes.
For methacrolein, s-trans-conformer was shown to be stable,while in a number of works [18,19] the evidences of existence of
Fig. 1. s-trans-Conformers of acrolein derivatives.
methyl vynil ketone
C3 C4
C1C2
H’
H’’
O
H1
H2
H3
1.4901.341
1.2211.084
1.083
1.083
1.515
1.0881.092
124.4
119.6121.2
121.8
121.2
121.5
108.7
Hv
s-trans
s-cis
H’’ C3
H’
C2 C1
O
C4
H2
H3
H1Ha
1.4961.342
1.215
1.108
1.0831.083
1.501
1.086
1.090
118.8 124.9
120.3
119.8
121.6
124.6
110.9
d-trans-crotonaldehyde
d-cis-crotonaldehyde
HaH’’ C3
H’
C2 C1
OC4
H2
H3
H1
1.4841.343
1.216
1.107
1.0851.083
1.497
1.087
1.088
117.5
123.8
121.2
120.6
122.0
125.2110.9
s-trans
Hv
C3
H’
C2 C1
O
C4
H2H3
H1
Ha
1.4731.343
1.216
1.109
1.085
1.0891.498
1.090
1.093
120.5
122.9122.2
117.9
125.3121.0
111.3
s-trans
Hv
H’’ C3
C2 C1
O
C4
H2H3
H1
Ha
1.4741.3461.218
1.103
1.084
1.087
1.502
1.088
1.093
125.2
122.9
120.0
117.1
127.9
120.1
113.3
s-trans
Hv
C3
H’
C2 C1
OC4
H2H3
H1
Ha
1.4841.343
1.217
1.1061.085
1.087
1.4971.093
1.091
121.4 124.5
121.2
117.2
125.0
120.2
111.6
s-cis
C3
C4
C1C2
H’
H’’O
H1
H2
H3
Hv
1.499
1.340 1.218
1.085
1.0841.082
1.513
1.0881.093
121.0 121.8
121.2
119.6
121.8
122.2
110.1
s-cis
s-cis
H’’
Hv
O
H3
C3
C2 C1
C4 H2
H1
Ha
1.222
1.4761.351
1.1081.087
1.089
1.499
1.0901.092
124.7 125.1
119.2
116.8
126.4
120.1
111.5
methacrolein
Fig. 2. Equilibrium geometries of conformers of monomethylated acroleins as predicted by CCSD(T)/cc-pVTZ. For s-cis-conformer of d-cis-crotonaldehyde, the MP2/6-311++G(d,p) results are presented.
O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 255
Table 1Calculated and experimental values for conformer energy differences DE and barriers to internal rotation V (in cm�1).
Fig. 3. Minimal energy paths corresponding to rotation of methyl group in the regionof d(CCCO) = 0� of d-cis-crotonaldehyde. Numbers denote the groups of methods (seetext). Designations of stationary points are in accordance with Fig 4d,e and f.
256 O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
the second conformer were not found. Later [20], a number ofvibrational frequencies for s-trans and s-cis-conformers, parame-ters of potential functions of internal rotation, conformer energydifference, and barrier heights were determined. The geometricstructure of s-trans-conformer was obtained [20] from data onrotational constants of d0- and d1-methacrolein [19,20]. The funda-mental frequencies found in [18,20] were in general agreementwith each other.
According to vibrational spectrum studies [21] of d-trans-cro-tonaldehyde, there was s-trans-conformer; the second (s-cis ors-gauche) conformer existed only in fluid phases. In microwave stud-ies [22,23], the geometry and barrier to the symmetric internal rota-tion were determined for s-trans-conformer. The rough estimationsof conformer energy difference (�600 cm�1) and barrier to asym-metric internal rotation (�5700 cm�1) were reported [21]. Thevibrational spectra were investigated in several works [17,18,21]but the assignments of some frequencies were not consistent.
As it was stated above, no data on d-cis-crotonaldehyde struc-ture were reported.
Thus, for above mentioned molecules, the geometric parameters,vibrational frequencies were not reliably determined. In some cases,values for conformer energy differences and barrier heights werenot found. The assignments for vibrational frequencies of methylvinyl ketone [10,18] and d-trans-crotonaldehyde [17,18,21] indifferent works did not agree with each other.
Theoretical investigations of target molecules [4,20,24–28] arenot-systematic and do not allow to build the complete picture ofstructure; the most of studies were carried out with the rough ap-proaches. To note is the absence of theoretic investigations ofd-cis-crotonaldehyde.
The aim of the present work was to determine geometric struc-ture of conformers, vibrational frequencies, and values for con-former energy differences and barrier heights and to studyPotential Energy Surface (PES) structure and coupling of large-amplitude motions of monomethylated acroleins by means of abinitio methods including high-level correlation approaches.
2. Methods
The investigation was carried out using density functionaltheory with B3LYP and PBE0 functionals, perturbation theory
MP2, quadratic configuration interaction QCISD [29], coupled clus-ters CCSD(T) [30], complete active space SCF CASSCF and multi-reference perturbation theory CASPT2 [31,32]. The 6-311G(d,p)[33] and cc-pVTZ [34] basis sets were utilized with the latter usedonly for geometric optimizations. The MP2/6-311G(d,p) methodwas the main to investigate the form of the PES since it combinedsufficient accuracy and computational efficiency. The multi-config-urational techniques (CASSCF and CASPT2) were applied to inves-tigation of the ground state, since they were the mainapproaches to study excited states (to be published later).
For calculations by CASSCF and CASPT2, the following activespaces were chosen (denoted as (m,n), m electrons distributed overn orbitals):
� (6,5) included all bonding and anti-bonding p-orbitals ofC@CAC@O frame and non-bonding orbital of oxygen atom nO;
� (8,7) included all orbitals of (6,5) space and, additionally, rC@O
and r*C@O orbitals.
Table 2Calculated (MP2/6-311G(d,p)) and experimental vibrational frequencies for methyl vinyl ketone (in cm�1).
Assignment s-trans s-cis
Harm. Anharm. Expt. IR gas [10] Raman, gas [10] IR and Raman, gas [18] Harm. Anharm. Expt. IR gas [10] IR and Raman, gas [18]
The equal numeric indexes mean that the corresponding vibrations are mixed.a For solid phase [10].b 1721 from Raman spectra [10].c For liquid phase [10].d From Raman spectra [10].e 772 from Raman spectra [10].f From [10].g 273 from Raman spectra [10].
O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 257
Table 3Calculated (MP2/6-311G(d,p)) and experimental vibrational frequencies for methacrolein (in cm�1).
Assignment s-trans s-cis
Harm. Anharm. IR [18] IR [20] Raman [20] Harm. Anharm. Expt. IR [20] Expt. Raman [20]
The equal numeric indexes mean that the corresponding vibrations are mixed.a For solid phase [20].b For liquid phase [20].
258 O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
The choice of active space was based on the investigations ofelectronic structure of acrolein in the ground [35] and 1,3(n,p*)[5] and 1,3(p,p*) [36] excited electronic states.
The frozen-core approximation was used in all post-SCF calcu-lations with the exception of MP2.
Conformer energy differences for methyl vinyl ketone, methac-rolein, and d-trans-crotonaldehyde, along with barriers to symmet-ric internal rotation of methyl vinyl ketone conformers, were alsoestimated using extrapolative technique Valence Focal-Point Anal-ysis (VFPA) (see, e.g. [37]) using cc-pVNZ (N = D, T, Q, 5) [34] basissets. The obtained values were corrected on core-core and core-valence electron correlation DCORE (CCSD(T)/cc-pCVTZ), non-adia-batic effects (diagonal Born–Oppenheimer correction) DDBOC (HF/cc-pVTZ), and zero point vibrational energy DZPE (MP2/6-311G(d,p)).
Since two internal rotations are intramolecular motions withlarge amplitude, the one- (1D) and two-dimensional (2D) quantumanharmonic vibrational problems were considered. The approachis described elsewhere [38,39]. Vibrational Schrödinger equationwith Hamiltonian
HðuÞ ¼ � ddu
FðuÞ dduþ VðuÞ ð1Þ
in the 1D case and
Hðua;usÞ ¼ �X
i;j
@
@uiBijðua;usÞ
@
@ujþ Vðua;usÞ ð2Þ
(where indices i and j are a or s) in 2D case was solved variationally.The coordinates of internal rotation were denoted as u (ua ¼dðC3C2C1OÞ for asymmetric rotation and the averaged dihedral angle
Table 4Calculated (MP2/6-311G(d,p)) and experimental vibrational frequencies for d-trans and s-
The equal numeric indexes mean that the corresponding vibrations are mixed; frequenca For liquid phase.b From IR spectrum of solution.c For d-trans-crotonaldehyde, these vibrations can be denoted as d(CH0), c(CH0), and ad For solid phase.
us ¼ dðHCCCÞav ¼ 13ðdðH1CCCÞ þ dðH2CCCÞ þ dðH3CCCÞÞ for symmet-
rical rotation); F(u) and B(ua,us) were kinematic functions chosento assure the division of vibrations and rotation, according to theEckart conditions; V was the potential function.
Potential and kinematic functions were approximated by thefollowing rows:
VðuÞ ¼ 12
X
k
Vkð1� cos kuÞ; ð3Þ
Vðua;usÞ ¼X
K
X
L
VCCKL cosKua cosLusþ
X
K
X
L
VSSKL sinKua sinLus;
ð4Þ
FðuÞ ¼X
n
Fn cos nu; ð5Þ
Bijðua;usÞ¼X
K
X
L
CCCKL cosKua cosLusþ
X
K
X
L
CSSKL sinKua sinLus:
ð6Þ
The variables ua and us changed in the ranges 0� 6 ua 6 180�and 0� 6 us 6 60� with step of 15� and 10�, respectively.
The quantum-chemical calculations were performed usingMOLPRO 2008.1 [40], ACES II [41], and Gaussian 03 [42] programpackages.
cis-crotonaldehydes (in cm�1).
d-cis-Crotonaldehyde
s-cis s-trans s-cis s-cis2
t. Raman [21] Harm. Anharm. Harm. Anharm. Harm. Harm.
Table 5Calculated (MP2/6-311G(d,p)) harmonic and anharmonic, and experimental torsiontransition energies for methyl vinyl ketone (in cm�1).
Transition 1D 2D Harm. Anharm. Expt. [10]
s(CAC)s-trans 0 ? 1 102 101 113 111 116.0
1 ? 2 101 101 108 109.02 ? 3 101 101 104.0
s-cis 0 ? 1 90 85 88 87 87.01 ? 2 91 87 89 84.0
s(CACH3)s-trans 0 ? 1 116 129 149 135 125
1 ? 2 94 113 129s-cis 0 ? 1 129 120 135 126 121
1 ? 2 113 101 107
O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 259
3. Results and discussion
3.1. Geometries
The main geometric parameters of all four molecules underinvestigation calculated with CCSD(T)/cc-pVTZ are presented inFig. 2 (for s-cis-conformer of d-cis-crotonaldehyde, the MP2/6-311++G(d,p) geometries are presented instead, for explanations seebelow). The data obtained with other methods (B3LYP/6-311G(d,p),MP2/6-311G(d,p), QCISD/cc-pVTZ, CASSCF/6-311G(d,p), andCASPT2/cc-pVTZ) are similar to those of CCSD(T)/cc-pVTZ and givenin Tables S1–S4 of Supplementary Material. According to the resultsof all methods, methyl vinyl ketone, methacrolein, and d-trans-crotonaldehyde exist as s-trans and s-cis-conformers (CS point groupsymmetry), which agrees with experiments [10,14–16,18,20,21].
In general, all methods give similar results. Extension of the ac-tive space within CASSCF significantly increases the C@O distancefor methyl vinyl ketone and C@C distance for other molecules(Tables S1–S4). For methyl vinyl ketone and methacrolein, the re-sults of MP2/6-311G(d,p) are in better agreement with CCSD(T)/cc-pVTZ than QCISD/cc-pVTZ ones; it might be due to casual compen-sation of method and basis set errors.
The geometries of s-trans and s-cis-conformers within onemethod are very similar; the main differences are observed forC1AC2 distance (by �0.01 Å) and valence angles of heavy atomframe (by �1–4�). For methacrolein and d-trans-crotonaldehyde,the changes of valence angles are less pronounced.
The experimental data were insufficient to determine the com-plete sets of geometric parameters of s-trans-conformers ofmonomethylated acroleins. That is why the majority of parameterswere the same that those obtained for more simple related mole-cules [13,14,19,22,23]. In this case, the comparison of calculatedand experimental rotational constants (but not geometric parame-ters) is more correct, see Tables S5–S7. The CCSD(T)/cc-pVTZ andMP2/6-311G(d,p) results are very close to the experimental data.To make the comparison of experimental (constrained r0-structureobtained by microwave spectroscopy) and theoretic parametersmore correct we calculated both equilibrium re and vibrationallyaveraged rz structures according to perturbation theory [43]. InTables S1–S3, the increments (rz � re) are listed (MP2/6-311G(d,p)); the corresponding calculated rotational constants arepresented in Tables S5–S7. It can be seen that increments are rathersmall and rotational constants calculated from rz and re structuresare similar which justifies the comparison with experiment.
The situation is trickier for d-cis-crotonaldehyde. All methodspredict the existence of s-trans-conformer similar to other mole-cules. However, different methods give somewhat different struc-tures of PES in the neighborhood of d(CCCO) = 0�. In general,methods could be conveniently divided into three groups. The firstgroup of methods (RHF, B3LYP, PBE0 with 6-311G(d,p) and 6-311++G(d,p) bases, QCISD/cc-pVTZ, and CASSCF/6-311G(d,p)) pre-dicts one s-cis-minimum (d(CCCO) = 0� and d(HCCC)av = 0�; CS
group symmetry) as for other molecules under study. The secondgroup (MP2 and CASPT2 with 6-311G(d,p) basis) gives two differ-ent minima denoted as s-cis and s-cis2 with symmetry plane andparameters (0�;0�) and (0�;60�) for d(CCCO) and d(HCCC)av, respec-tively; the barrier height between these minima is about 20 cm�1.The third group (MP2, CASSCF, CASPT2 with 6-311++G(d,p)) givestwo equivalent minima without symmetry plane with parameters(�12�;80�) and (12�;40�); the barrier is about 10 cm�1. Such situa-tion sensitive to method/basis combination might be explained bydifferent extent of accounting for spatial and electronic effects forCH3 and CHO fragments within different methods. It is the s-cis-conformer of d-cis-crotonaldehyde where these fragments comevery close to each other.
3.2. Conformer energy differences and barriers to internal rotation
Experimental and calculated conformer energy differences DE(including VFPA results, for details see Tables S8–S10) for methylvinyl ketone, methacrolein, and d-trans-crotonaldehyde are pre-sented in Table 1. For all four molecules, the geometric parametersof transition states (see Table S13) corresponding to barriers toasymmetric (Va) and symmetric (Vtrans
s and Vciss ) internal rotations
were obtained. Values for barrier heights are given in Table 1.For methyl vinyl ketone, the values for Vtrans
s and Vciss were also
extrapolated to complete basis set within VFPA (Tables S11 andS12). Experimentally determined values of barrier heights arefound from potential functions of internal rotation of molecules.The determination of such functions is associated with inverseproblem solution. The accuracy and reliability of ‘‘experimental”potential function are defined by amount and ‘‘quality” of experi-mental data used and also by other factors. In general, the determi-nation of ‘‘experimental” values for potential barriers toasymmetric internal rotation Va is more complicated task than thatof symmetric rotation Vs or conformer energy difference DE.
According to the B3LYP/6-311G(d,p) results for methyl vinylketone, the s-cis-conformer is more stable; MP2, CASSCF (withboth active spaces) give very close conformer energies; QCISD,CCSD(T), and CASPT2 give the order of conformers in accordancewith experiments [10,17] (Table 1). Extension of active space with-in CASSCF practically does not change the DE. The VFPA resultincluding all the corrections is DE = 141 cm�1 in good agreementwith experimental data 200–280 cm�1 [10,17] taking into accountexperimental error of 100–150 cm�1. The calculated values forVtrans
s are in very good agreement with experiments [10,13,14](Table 1). MP2/6-311G(d,p) gives 1900 cm�1 value for Va which is2.5 times higher than experimental [10].
For methacrolein, all methods (including VFPA) give similarestimations of DE: 900–1200 cm�1 in good agreement with exper-imental value [20], see Table 1. The heights of barriers Va and Vtrans
s
obtained by MP2/6-311G(d,p) are also in accordance with experi-ments [19,20].
For d-trans-crotonaldehyde, different methods give value forDE in the range 500–750 cm�1 (Table 1) with rough experimentalestimation being 600 cm�1 [21]. The Vtrans
s value agrees with exper-iments well [21–23]. The height of Va barrier according to MP2 isalmost two times lower than experimental estimation [21].
For d-cis-crotonaldehyde, the three cases corresponding tothree groups of methods (The division in groups is rather conven-tional and consistent with Section 3.) are illustrated on the exam-ple of the 1D Minimal Energy Paths (MEP) in Fig. 3. All methods ingeneral predict that s-cis-minimum (minima) is higher in energythan s-trans by 130–600 cm�1 with exception of B3LYP and PBE0with 6-311G(d,p) basis, where two conformers are very close in
260 O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
energy. For the first group of methods, the barrier Vciss is the highest
(100–200 cm�1). In the second group, both minima are very closein energy and are separated by very low barrier (not more than20 cm�1). The lowest barrier is observed for the third group of
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Fig. 4. 2D PES sections along coordinates corresponding to asymmetric ua (d(CCCO)) anmethacrolein, (c) d-trans-crotonaldehyde, (d–f) d-cis-crotonaldehyde. Methods: (a–c andMn are minima, TSn are first order saddle points and max is second-order saddle point
methods (less than 10 cm�1). Actually, this means that for all threegroups there is only one s-cis-conformer due to low barriers forsecond and third groups. However, such potential bumps belowground vibrational level may be detected with microwave
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d symmetric us (averaged d(HCCC)) internal rotations. (a) Methyl vinyl ketone, (b)e) MP2/6-311G(d,p), (d) RHF/6-311++G(d,p), (f) MP2/6-311++G(d,p). Energy in cm�1..
Table 6Calculated (MP2/6-311G(d,p)) harmonic and anharmonic, and experimental torsiontransition energies for methacrolein (in cm�1).
Table 7Calculated (MP2/6-311G(d,p)) harmonic and anharmonic, and experimental torsiontransition energies for d-trans-crotonaldehyde (in cm�1).
Transition 1D 2D Harm. Anharm. Expt. [21]
s(CAC)s-trans 0 ? 1 147 133 125 122 122
1 ? 2 146 132 122s-cis 0 ? 1 145 131 141 134
1 ? 2 145 130 134
s(CACH3)s-trans 0 ? 1 163 186 197 186 173a
1 ? 2 150 178 175s-cis 0 ? 1 166 197 202 194
1 ? 2 154 189 193
a Solid phase.
O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 261
spectroscopy through the unusual change of rotational constantswith vibrational quantum number (see [44] and references there-in). According to RHF and MP2 calculations with 6-311G(d,p) and6-311++G(d,p) bases, value for Va is 2080–2180 cm�1 and forVtrans
s is 340–440 cm�1.
3.3. Vibrational frequencies
The experimental, harmonic, and anharmonic (according tovibrational perturbation theory including cubic and quartic forceconstants [43]) vibrational frequencies calculated at MP2/6-311G(d,p) level are presented in Tables 2–4. The assignment offrequencies was carried out on the base of potential energy distri-bution (PED) analysis using program SPECTRUM described in [45].
3.3.1. Methyl vinyl ketoneIn general, for both conformers, the assignment proposed in
[10,18] is in good agreement with the results of anharmonic calcu-lation. For a number of frequencies, the differences are not morethan 10 cm�1 (see Table 2).
Our results confirm the assignment of m11 and m12 for s- trans-conformer from [10,16,17] but not from [18]. On the contrary,for m16 and m17, the calculated results agree with [18] but not with[10] (see Table 2). Our results also confirm the assignment of [18]for modes m14, m18, and m25 and for mode m24 of [10]. The experimen-tal range of band position assigned to m22 is 965–997 cm�1 whilethe anharmonic value is 1016 cm�1 which agrees with assignmentproposed in [17] (997 cm�1).
For s-cis-conformer, the assignment of modes m13 and m16 madein [18] (but not in [10]) agrees with our results. The assignmentsproposed in [10,17] should be accepted for m22 and m23; in [18]the assignment is the opposite. The calculated frequencies m12
and m25 are close to data of [18] and m24 to data of [10] (Table 2).According to anharmonic approach (including 2D variational
approach, see section Large-amplitude vibrations and Table 5),for s-trans-conformer, the values 125 cm�1 (m26) and 116 cm�1
(m27) [10] and for s-cis-conformer 121 cm�1 (m26) and 87 cm�1
(m27) [10] should be accepted.
3.3.2. MethacroleinIn general, the assignment of vibrational bands for s-trans-con-
former made in [18,20] is in good agreement with the results ofanharmonic calculation (Table 3). According to our results, for m8,m9, m23, m24, the assignment of [18] and, for m22, of [20] should be ac-cepted. The anharmonic torsion m26 frequency 133 cm�1 is in goodagreement with [20] (gas phase IR spectrum).
For s-cis-conformer, the general agreement of calculated andmeasured frequencies should be mentioned (Table 3). For m13,m20, m23, and m25, frequencies, the differences are less than20 cm�1, in other cases they are 20–70 cm�1. According to anhar-monic calculation (including 2D), the frequency of symmetric tor-sion is higher than that of asymmetric one, whereas in [20] it is viceversa.
3.3.3. d-trans-CrotonaldehydeFor a number of vibrations of s-trans-conformer, the good
agreement of calculated results with data of [18,21] is observed.The calculation agrees with assignment of [18] for m13 - m18 butnot of [21] (Table 4). Anharmonic frequency 930 cm�1 for m15 isin agreement with 928 cm�1 band found in [21] but assigned toCAC stretching mode. The calculated m22 frequency poorer agreeswith experiment.
The frequencies for s-cis-conformer are given in Table 4; thereare no experimental data for this conformer. To note is the similar-ity of s-cis and s-trans frequencies (Table 4).
3.3.4. d-cis-CrotonaldehydeThe anharmonic frequencies were calculated only for s-trans-
conformer while for s-cis and s-cis2 only harmonic frequenciesare given (Table 4). The anharmonic frequencies for s-cis-con-former are not presented since the free internal rotation occursand anharmonic approach using perturbation theory is inapplica-ble in such cases [43]. The majority of frequencies are very similarfor three minima (MP2/6-311G(d,p)), see Table 4. According to ourcalculations, the assignment of m13 - m15 is different for these threeconformers. The most notable differences in vibrational frequen-cies of d-cis and d-trans-crotonaldehydes are observed for m11,m16, m24, and m25, see Table 4.
3.4. Large-amplitude vibrations
To solve variational vibrational problems, 1D and 2D PES sec-tions were constructed with MP2/6-311G(d,p) for all moleculesunder investigation. For d-cis-crotonaldehyde, HF/6-311++G(d,p)and MP2/6-311++G(d,p) were also used to investigate differentcases of PES structure in the region of s-cis-conformation (seeSection 3.2). 2D PES sections obtained are presented in Fig. 4;1D sections and stationary points (minima Mn, transition statesof first TSn and second order max) are also plotted. Geometriesof stationary points are collected in Fig. 2 and Tables S1–S4 andS13. The frequencies of torsion transitions obtained with solvinganharmonic variational problems are presented in Tables 5–8(for the A symmetry of vibrational states; the E symmetry statesare not presented).
The 2D-approach made it possible to evaluate the coupling ofsymmetrical and asymmetrical internal rotations, and to obtaindata on the energies of vibrational levels. The following aspectswere taken into account:
262 O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
– form of vibrational wave functions;– curvature of MEPs corresponding to internal rotations;– the values for non-diagonal elements of kinetic energy matrices;– the energies of vibrational levels within 1D and 2D approaches.
The assignment of lowest vibrational levels to different con-formers and determination of vibrational quantum numbers ta
and ts for asymmetric and symmetric internal rotations were car-ried out on the base of analysis of localization and nodal surfacesstructure of vibrational wave functions. In Fig. 5 some examplesof wave functions for d-cis-crotonaldehyde are plotted. ‘a’ and ‘b’are (1, 0) and (1, 1) levels for s-trans conformer; ‘c’ and ‘d’ are(0, 0) and conventionally (1, 0) for s-cis conformer. In parentheses,the first number corresponds to ta, the second corresponds to ts.The 2D wave functions of the lowest vibrational levels of all mole-cules are localized near the minima on PES, examples correspond-ing to s-trans and s-cis-conformers of s-cis-crotonaldehyde see inFig. 5a and b. The exception is the s-cis conformer of d-cis-croton-aldehyde, where the wave function of the lowest vibrational levelis delocalized in the neighborhood of u1 = 0� depending on thegroup of methods (Fig. 5c and d). Such strong delocalization evi-dences practically free internal rotation (for second and thirdgroup of methods) for d-cis-crotonaldehyde in the region of u1 = 0�.
The 2D sections of PES (Fig. 4) allow to conclude on the couplingof vibrations in the potential part of the Schrödinger equation withHamiltonian (2). The deviation of MEPs from straight lines evi-dences the notable coupling of two large-amplitude motions. Assee in Fig. 4c, for d-trans-crotonaldehyde, this coupling is the weak-est, since different MEPs are practically straight and form theangle 90�. For methyl vinyl ketone and methacrolein, this couplingis notable but rather weak (Fig. 4a and b). In case of d-cis-crotonal-dehyde, the coupling in the potential part is strong for the s-cis-conformer area. This coupling is reflected in notable curvatureand angle between MEPs about 90 (Fig. 4d–f). Besides that, thereare six minima instead of three for second and third group of meth-ods (see Geometries) (Fig. 4e and f). In Fig. 4f, there is PES area be-tween equivalent minima (M2) where MEPs coincide. Such PESstructure shows the necessity of solving 2D variational problem.
Values for non-diagonal kinematic coefficients B12 fromHamiltonian (2) approximately reflect the coupling of two internalrotations in kinetic part of Schrödinger equation. The analysis oftwo-dimensional surfaces showed B12 to be only slightly depen-dent on u2. For methyl vinyl ketone and methacrolein, the degreeof coupling in s-trans-conformer area is two times stronger thanin s-cis area (–0.7 and 0.3 cm�1, respectively). The strongest cou-pling is observed for d-trans-crotonaldehyde where axes of internalrotations are almost parallel, with this coupling being two timesstronger for s-trans-conformer (–2.0 cv–1 and –1.2 cm�1). For
Table 8Calculated harmonic and anharmonic torsion transition energies for d-cis-crotonaldehyde
Method Transition MP2/6-311G(d,p)
1Da 2D Harm.a
s(CAC)s-trans 0 ? 1 121 118 117
1 ? 2 119 113s-cis 0 ? 1 83 112 132 (167)
1 ? 2 93 112
s(CACH3)s-trans 0 ? 1 117 124 147
1 ? 2 94 94s-cis 0 ? 1 10 (11) 46 49 (63)
1 ? 2 - -
a The frequencies for s-cis2-conformer (MP2/6-311G(d,p)) are given in parentheses if
d-cis-crotonaldehyde, the values B12 are comparable but differ insign (0.5 cm�1 for s-trans and �0.6 cm�1 for s-cis).
The additional characteristic of the coupling of two motions isthe comparison of transition energies obtained within 1D and 2Dapproaches. The results of 1D and 2D vibrational problems,harmonic, anharmonic (perturbation theory), and experimentaltorsion transition energies are collected in Tables 5–8 (only A-levels). For methyl vinyl ketone and methacrolein, the couplingin potential and kinetic parts is weak and the results of 1D and2D approaches only slightly differ (except for s(CH3) for s-trans-methyl vinyl ketone). In the case of d-trans-crotonaldehyde, thecontribution of kinematic effects dominates: the results withintwo approaches differ by 10-30 cm�1. with the coupling in poten-tial being very weak. For d-cis-crotonaldehyde, the coupling inpotential is very strong for s-cis conformer area and the differencesin results are obviously due to potential effects.
The following conclusions can be made from comparison of tor-sion transition energies with experiments: in general, good agree-ment of obtained values is observed, the agreement of 2D results isnot always better than 1D ones (see Tables 5–7). To note is generaldisagreement of calculated torsion frequencies of s-cis-methacro-lein with experimental data [20] (Table 6).
4. Conclusions
Equilibrium geometries of conformers of all monomethylsubsti-tuted acroleins were obtained by various ab initio methods includinghigh-level approximations (CCSD(T)/cc-pVTZ). d-cis-Crotonalde-hyde was investigated for the first time. The calculated rotationalconstants (MP2/6-311G(d,p) and CCSD(T)/cc-pVTZ) demonstrategood agreement with experimental values. Thus, the theoretical re-sults obtained are more reliable than partially fitted experimentalstructures. These results allow to add the experimental informationon the structure of s-cis-conformers of methyl vinyl ketone, methac-rolein, and d-trans-crotonaldehyde.
Values for conformer energy differences and barrier heightswere calculated for all molecules with MP2/6-311G(d,p) and con-former energy differences DE were also extrapolated to CBS usingVFPA. For d-cis-crotonaldehyde, the free internal rotation wasproved to appear in the s-cis-area.
The results of anharmonic calculation along with potentialenergy distribution analysis allow to cancel some inconsistenciesin the experimental assignments for methyl vinyl ketone andd-trans-crotonaldehyde.
The coupling of two internal rotations is analyzed by construc-tion of 1D and 2D PES sections. The very strong coupling in the po-tential for d-cis-crotonaldehyde was found. The comparison of
Fig. 5. 2D vibrational wave functions for d-cis-crotonaldehyde (MP2/6-311++G(d,p)). (a) and (b) are (1, 0) and (1, 1) levels for s-trans conformer; (c) and (d) are (0, 0) andconventionally (1, 0) for s-cis conformer.
O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264 263
transition energies obtained within 1D and 2D approachesevidence the notable coupling of torsion motions for d-trans- andd-cis-crotonaldehydes. For the former, this coupling is determinedby kinematic and, for the latter, by potential effects.
It could be mentioned that for this class of molecules it is ofpractical and theoretical interest to investigate their structures inthe excited electronic states that makes it possible to predictphotochemical and photophysical properties. Such investigationswill be published in the next articles.
Acknowledgements
The authors were thankful to Dr. A.V. Abramenkov for his pro-gram package for solving multi-dimensional variational vibrationproblems.
This work has been financially supported by Russian Founda-tion for Basic Research (Grant No. 07-03-00090).
The authors are grateful to the Joint SuperComputer Center(JSCC) for computing facilities. The authors also thank theUkrainian-American laboratory of computational chemistry andpersonally Prof. O.V. Shishkin who has made it possible to realizeab initio computations with Gaussian 03.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.theochem.2009.08.004.
References
[1] D.A.C. Compton, in: J.R. Durig (Ed.), Vibrational Spectra and Structure, Elsevier,Amsterdam, 1981, p. 255.
[2] M. Olivucci, M.A. Robb, F. Bernardi, in: J. Waluk (Ed.), Conformational Analisysof Molecules in Excited Electronic States, John Wiley & Sons, New York, 2000,p. 297.
[3] F. Aquilante, V. Barone, B.O. Roos, J. Chem. Phys. 119 (2003) 12323.[4] J. García, J.A. Mayoral, L. Salvatella, X. Assfeld, M.F. Ruiz-López, J. Mol. Struct.:
THEOCHEM 362 (1996) 187.[5] O.S. Bokareva, V.A. Bataev, V.I. Pupyshev, I.A. Godunov, Int. J. Quant. Chem. 108
(2008) 2719.[6] C.E. Blom, G. Grassi, A. Bauder, J. Am. Chem. Soc. 106 (1984) 7427.[7] J.J. Keirns, R.F. Curl Jr., J. Chem. Phys. 48 (1968) 3773.[8] J.E. Katon, W.R. Feairheller Jr., J. Chem. Phys. 47 (1967) 1248.[9] K. Bolton, N.L. Owen, J. Sheridan, Nature 218 (1968) 266.
[10] J.R. Durig, T.S. Little, J. Chem. Phys. 75 (1981) 3660.[11] D.E. McGreer, B.D. Page, Can. J. Chem. 47 (1969) 866.[12] A.F. Moskvin, O.P. Yablonsky, L.F. Bondar, Theor. Expt. Khim. 11 (1966) 635.[13] P.D. Foster, V.M. Rao, R.F. Curl Jr., J. Chem. Phys. 43 (1965) 1064.[14] A.C. Fantoni, W. Caminati, Chem. Phys. Lett. 133 (1987) 27.[15] K. Noack, R.N. Jones, Can. J. Chem. 39 (1961) 2225.[16] A. Krantz, J. Am. Chem. Soc. 94 (1972) 4022.[17] A.J. Bowles, W.O. George, W.F. Maddams, J. Chem. Soc. B 7 (1969) 810.[18] H.-I. Oelichmann, D. Bougeard, B. Schrader, J. Mol. Struct. 77 (1981) 179.[19] M. Suzuki, K. Kozima, J. Mol. Spectrosc. 38 (1971) 314.[20] J.R. Durig, J. Qiu, B. Dehoff, T.S. Little, Spectrochim. Acta A 42 (1986) 89.[21] J.R. Durig, S.C. Brown, V.F. Kalasinsky, W.O. George, Spectrochim. Acta A 32
(1976) 807.[22] E.A. Hsu, W.H. Flygare, Chem. Phys. Lett. 4 (1969) 317.[23] M. Suzuki, K. Kozima, Bull. Chem. Soc. Jpn. 42 (1969) 2183.[24] J. de Smedt, F. Vanhouteghem, C. van Alsenoy, H.J. Geise, B. van der Veken, P.
Coppens, J. Mol. Struct. 195 (1989) 227.[25] Y. Wang, J. de Smed, I. Coucke, C. van Alsenoy, H.J. Geise, J. Mol. Struct. 43–59
(1993) 299.[26] S. Thakur, V.P. Gupta, Ind. J. Pure Appl. Phys. 36 (1998) 567.[27] N. Tezer, R. Ozkan, J. Mol. Struct.: THEOCHEM 546 (2001) 79.[28] S. Pandit, D. De, B.R. De, J. Mol. Struct.: THEOCHEM 819 (2007) 160.[29] C. Hampel, K.A. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 (1992) 1.[30] M.J.O. Deegan, P.J. Knowles, Chem. Phys. Lett. 227 (1994) 321.[31] B.O. Roos, K. Andersson, Chem. Phys. Lett. 245 (1995) 215.[32] H.-J. Werner, Mol. Phys. 89 (1996) 645.[33] R. Krishnan, J.S. Binkley, R. Seeger, J.A. Pople, J. Chem. Phys. 72 (1980) 650.[34] T.H.J. Dunning, J. Chem. Phys. 90 (1989) 1007.[35] O.S. Bokareva, V.A. Bataev, I.A. Godunov, Russ. J. Phys. Chem. 83 (2009) 81.[36] O.S. Bokareva, V.A. Bataev, V.I. Pupyshev, I.A. Godunov, Spectrochim. Acta A 73
(2009) 654.[37] A.G. Császár, W.D. Allen, H.F.I. Schaefer, J. Chem. Phys. 108 (1998) 9751.[38] A. Niño, C. Muñoz-Caro, D.C. Moule, J. Phys. Chem. 98 (1994) 1519.[39] V.A. Bataev, V.I. Pupyshev, A.V. Abramenkov, I.A. Godunov, Russ. J. Phys. Chem.
264 O.S. Bokareva et al. / Journal of Molecular Structure: THEOCHEM 913 (2009) 254–264
[40] H.-J. Werner, P.J. Knowles, R. Lindh, F.R. Manby, M. Schütz, P. Celani, T. Korona,A. Mitrushenkov, G. Rauhut, T.B. Adler, R.D. Amos, A. Bernhardsson, A. Berning,D.L. Cooper, M.J. O. Deegan, A.J. Dobbyn, F. Eckert, E. Goll, C. Hampel, G. Hetzer,T. Hrenar, G. Knizia, C. Köppl, Y. Liu, A.W. Lloyd, R.A. Mata, A.J. May, S.J.McNicholas, W. Meyer, M.E. Mura, A. Nicklass, P. Palmieri, K. Pflüger, R. Pitzer,M. Reiher, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, T. Thorsteinsson, M.Wang, A. Wolf, MOLPRO, version 2008.1, a package of ab initio programs, 2008.
[41] J.F. Stanton, J. Gauss, J.D. Watts, P.G. Szalay, R.J. Bartlett with contributionsfrom A.A. Auer, D.E. Bernholdt, O. Christiansen, M.E. Harding, M. Heckert, O.Heun, C. Huber, D. Jonsson, J. Jusélius, W.J. Lauderdale, T. Metzroth, C. Michauk,D.P. O’Neill, D.R. Price, K. Ruud, F. Schiffmann, M.E. Varner, J. Vázquez and theintegral packages MOLECULE (J. Almlöf and P.R. Taylor), PROPS (P.R. Taylor),and ABACUS (T. Helgaker, H.J. Aa. Jensen, P. Jørgensen, and J. Olsen). For thecurrent version, see http://www.aces2.de.
[42] M.J.T. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R.Cheeseman, J.A. Montgomery, T. Vreven Jr., K.N. Kudin, J.C. Burant, J.M.Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N.
Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J.Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li,J.E. Knox, H.P. Hratchian, J.B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R.Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W.Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G.Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D.Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S.Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I.Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A.Nanayakkara, M. Challacomxbe, P.M.W. Gill, B. Johnson, W. Chen, M.W.Wong, C. Gonzalez, J.A. Pople, Gaussian 03, Gaussian, Inc., Wallingford, CT,2004.
[43] V. Barone, J. Chem. Phys. 120 (2004) 3059.[44] R.G. Ford, J. Chem. Phys. 65 (1976) 354.[45] A.G. Yagola, I.V. Kochikov, G.M. Kuramshina, Y.A. Pentin, Inverse Problems of
