CCSD(T) Study of Dimethyl-Ether Infrared and Raman Spectra

Oct 15, 2011 - M. Villa. †. Departamento de Química, UAM-I Purísima y Michoacan, s/n, CP ... Experimentales, Universidad de Huelva, 21071 Huelva, ...
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CCSD(T) Study of Dimethyl-Ether Infrared and Raman Spectra M. Villa† Departamento de Química, UAM-I Purísima y Michoacan, s/n, CP 09340, Mexico D.F.

M. L. Senent* Departamento de Química y Física Teoricas, Instituto de Estructura de la Materia, IEM-CSIC, Serrano 121, Madrid 28006, Spain

R. Dominguez-Gomez‡ Departamento de Ingeniería Civil, Catedra de Química, E.UIT Obras Publicas, Universidad Politecnica de Madrid, Spain

O. Alvarez-Bajo§ and M. Carvajal|| Departamento de Física Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain ABSTRACT: CCSD(T) state-of-the-art ab initio calculations are used to determine a vibrationally corrected three-dimensional potential energy surface of dimethyl-ether depending on the two methyl torsions and the COC bending angle. The surface is employed to obtain variationally the lowest vibrational energies that can be populated at very low temperatures. The interactions between the bending and the torsional coordinates are responsible for the displacements of the torsional overtone bands and several combination bands. The effect of these interactions on the potential parameters is analyzed. Second order perturbation theory is used as a help for the understanding of many spectroscopic parameters and to obtain anharmonic fundamentals for the 3N  9 neglected modes as well as the rotational parameters. To evaluate the surface accuracy and to verify previous assignments, the calculated vibrational levels are compared with experimental data corresponding to the most abundant isotopologue. The surface has been empirically adjusted for understanding the origin of small divergences between ab initio calculations and experimental data. Our calculations confirm previous assignments and show the importance of including the COC bending degree of freedom for computing with a higher accuracy the excited torsional term values through the Fermi interaction. Besides, this work shows a possible lack of accuracy of some available experimental transition frequencies and proposes a new assignment for a transition line. As an example, the transition 100 f 120 has been computed at 445.93 cm1, which is consistent with the observed transition frequency in the Raman spectrum at 450.5 cm1.

’ INTRODUCTION Astrophysical surveys contain many spectral lines whose identification requires complete molecular catalogs.1 These lines derive from sources where very well-known molecules coexist with new uncharacterized species and with previously detected molecules not fully described. This is the case of dimethyl-ether (DME), for which only the vibrational ground state has been well explored.27 DME is an astrophysical relevant molecule first detected in emission from the Orion Nebula by Snyder et al. in 1974.8 Later on, it was found to be abundant in star-forming regions where highly excited rotational lines have been observed.9 DME represents the most abundant ether. It is generally accepted that ethers and alcohols coexist in gas phase sources where they are involved in common chemical reactions, coming from gasgrain processes which play an important role in their formation.10,11 Many facts concerning their abundances in hot core regions are expected to be clarified with the new ALMA observatory that will r 2011 American Chemical Society

be a key instrument for their studies.12 For this reason, various laboratories have demonstrated a great interest in DME spectroscopic studies. They try to describe well the millimeter and submillimeter regions for isotopic varieties containing deuterium and 13C, focusing on the monodeuterated species. In addition, the rotational spectrum of the most abundant isotopologue needs to be recorded at the first excited vibrational states that can be populated at hot core temperatures. The expectation of these new experiments has motivated the present study performed with state-of-the art ab initio calculations. From the theoretical point of view, DME is a nonrigid molecule whose potential energy surface contains nine minima.1315 The two methyl group torsions that intertransform and connect Received: July 1, 2011 Revised: October 14, 2011 Published: October 15, 2011 13573

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The Journal of Physical Chemistry A the multiple minima by tunneling effects show very low energy levels. The two corresponding fundamentals lie below 250 cm1. A third large amplitude mode, the COC-bending, has been observed in the Raman spectrum at 412 cm1. In 1977, Groner and Durig13 recorded and assigned the far-infrared region spectra of various isotopic DME varieties. For their assignments, they used an effective two-dimensional Hamiltonian adequate for the tunneling splitting description. One year before, in 1976, the microwave spectrum was recorded,2 and more recently, the submillimeter spectrum corresponding to the vibrational ground state has been measured.57 In 1995, some of us performed two ab initio studies of DME using M€ollerPlesset theory and a double-ζ basis set.14,15 One of these studies14 was performed with a two-dimensional model (2D) first developed for the study of acetone.16 The two independent variables were the two methyl group torsions, whereas the remaining 3N  8 vibrational coordinates were optimized in all the conformations selected for the potential energy surface determination. Generally, this procedure describes well the interactions among the torsions and the other vibrational degrees of freedom if they are relatively small. However, in DME, the coupling between the COC bending and the torsions was found to be too large to be accurately described by the relaxation, obliging us to solve, in a second paper, a three-dimensional Hamiltonian (3D) where the COC bending coordinate was added explicitly.15 We concluded that, whereas the fundamental frequencies calculated with the 2D are enough accurate, overtones and several combination bands involving excited torsional levels are displaced by non-negligible Fermi interactions. The actual computational resources that allow us to perform more accurate calculations than those of 1995 and the fresh interest coming from the development of Herschel and the ALMA observatory are our motivations for recovering the theoretical DME study determining a new three-dimensional potential energy surface (3D-PES) using highly correlated ab initio methods. The present paper focuses on the new surface calculation that we test, using it for the study of the most abundant DME isotopologue, 12CH316O12CH3, for which available experimental data can be used to discuss the accuracy. In the future, once new experimental data will be available, we will use this surface to predict many spectroscopic properties of various isotopologues to help assignments. For this purpose, we will combine recent techniques previously employed for acetic acid and methyl-formate isotopologues.17,18 The new potential energy surface has been calculated with CCSD(T)/aug-cc-pVTZ, optimizing the geometry of 126 molecular conformations. Methodological improvements are included to improve old results (i.e., we use a more proper definition of the methyl group torsional coordinates and add the zero point vibrational energy correction to contemplate isotopic effects). Vibrational levels are determined variationally, although we also use perturbation theory as a help to understand the molecular spectroscopic properties.

’ COMPUTATIONAL DETAILS All the ab initio calculations have been performed with the Gaussian 09 package.19 The geometries and the harmonic force field have been calculated with coupled cluster theory using both single and double substitutions.20 To obtain very accurate energies, single point calculations were performed by adding triple excitations noniteratively (CCSD(T)) on the CCSD

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Figure 1. DME at the equilibrium geometry. Independent coordinates α, θ1, and θ2.

Table 1. Structural Parameters (in Å, degrees), Dipole Moment (in Debyes), and Rotational Constants (in MHz) Corresponding to the Minimum Energy Geometry of DME Calculated with CCSD/aug-cc-pVTZ O1C2dO1C3

1.4090

H4C2dH7C3

1.0879

H5C2dH6C2dH8C3dH9C3

1.0962

C3O1C2

111.4

H4C2O1dH7C3O1

107.5

H5C2O1dH6C2O1dH8C3O1dH9C3O1 H4C2O1C3dH7C3O1C2

111.2 180.0

H5C2O1H4dH9C3O1H7

119.4

H6C2O1H4dH8C3O1H7

119.4

μ (MP2/aug-cc-pVTZ)

1.4882

Ae

Be

Ce

38973.4024

10145.2871

8961.8951

geometries.21 To describe well long-range effects, such as the intramolecular nonbonding interactions, the augmented aug-ccpVTZ set was employed in all the computations.22 The vibrational levels corresponding to the three vibrations (the two torsions and the COC bending) have been calculated variationally from a CCSD(T)/CCSD three-dimensional, vibrationally correcte potential energy surface and the ENEDIM code.23 To determine spectroscopic parameters for the 3N  9 remaining coordinates, the full-dimensional anharmonic spectroscopic analysis was performed with second order perturbation theory implemented in FIT-ESPEC.23 The starting point was a CCSD quadratic force field and cubic and quartic force fields determined with second order M€ollerPlesset theory (MP2).24 The vibrational corrections of the 3D-PES are also determined with MP2.

’ RESULTS AND DISCUSSION DME Molecular Structure. At the ground electronic state, the most abundant isotopologue of DME shows a C2v geometry. One hydrogen atom of each methyl group is anti to the other carbon atom (H4C2O1C3dH7C3O1C2 = 180°; see Figure 1). In Figure 1, our three independent coordinates, θ1, θ2, and α, corresponding to the two methyl group torsions and the COCbending modes, are also shown. These coordinates are defined below. At the equilibrium geometry used as reference, θ1 = 0°, θ2 = 0°, and α = 0°. The structural parameters and the rotational 13574

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Table 2. Torsional Energy Barrier V060 and 2D Relative Maximum V6060 (in cm1) Dependence with the Bending Coordinate (in degrees) MP4/6-31G(d,p) α

V060

V6060

6

1260

3012

3

1124

2509

0 3

1006 905

6 9

CCSD(T)/aug-cc-pVTZ

V6060  2V060

V060

V6060

492

1190

2898

261

1046

2390

2082 1734

70 76

921 816

819

1454

184

748

1238

258

n

n

n

n

n

þ

n

n

n

n

518

1210

2919

499

298

1065

2410

280

1966 1618

124 14

939 832

1984 1634

105 30

729

1343

115

742

1353

132

656

1127

185

668

1133

204

1

∑ ∑ ∑ ∑ fijkl qi qjqkql i ¼ 1 j ¼ 1 k l ¼ 1 24 1

V6060  2V060

V6060

∑ ∑ fij qi qj þ t∑¼ 1 j∑¼ 1 k∑¼ 1 6 fijk qi qj qk t ¼1 j¼1 2 1

V6060  2V060

V060

constants calculated with CCSD/aug-cc-pVTZ and the MP2/ aug-cc-pVTZ dipole moment are shown in Table 1. At the equilibrium geometry, the COC bending angle was evaluated to be 111.4° and the rotational constants to be Ae = 38973.4024 MHz, Be = 10145.2871 MHz, and Ce = 8961.8951 MHz. In principle, they can be compared with the experimental values for the vibrational ground state (A0 = 38788.18 MHz, B0 = 10056.48 MHz, and C0 = 8886.83 MHz).5,7 The MP2 dipole moment, μ = 1.4882 D, differs from the experimental one of μ = 1.302 D given in ref 7. DME is a nonrigid molecule whose most abundant isotopologue can be classified in the G36 Molecular Symmetry Group (ref 16 and references therein). The internal rotation of the two methyl groups intertransforms nine minima through feasible energy barriers. Table 2 shows the one-dimensional V060 and the two-dimensional V6060 barriers and the V6060  2V060 difference as a function of the COC bending coordinate (V060 = E(α,0,60)  E(α,0,0); V6060 = E(α,60,60)  E(α,0,0)), and E(0,0,0) is the energy of the reference equilibrium geometry. V6060  2V060 represents a measure of the two methyl group interactions. When the COC bending angle is augmented, the nonbonding interactions among different methyl group hydrogen atoms decrease. For small values of α, V6060  2V060 > 0, whereas, for large values, V6060  2V060 < 0. All the interaction terms of the torsional Hamiltonian depend strongly on α. Full Dimensional Anharmonic Analysis. A preliminary anharmonic analysis using second order perturbation theory (PT2) has been performed for evaluating the number of internal coordinates that could be considered as independent variables in the variational calculation of the CH3 torsional levels. Although PT2 represents a deficient theoretical model for nonrigid molecules, it is taken as a first approximation for the determination of the torsional frequencies as well as for the analysis of possible Fermi interactions among our three independent modes and the remaining vibrations. At the same time, this theory allows us to obtain the fundamental frequencies for the remaining vibrational modes. For this analysis, the quadratic, cubic, and quartic force field terms expressed in V ðq1 , q2 , :::, qn Þ ¼

CCSD(T)/aug-cc-pVTZ+ZPVE

ð1Þ

have been evaluated using CCSD and MP2 calculations and the subroutines implemented in Gaussian.19 We use the FIT-ESPEC code23 to transform the force field in normal coordinates

Table 3. Harmonic (ω) and anharmonic (ν) fundamental frequencies of DME-h6 (in cm‑1) MP2/aug-cc-pVTZ C2v A1

A2

B1

B2

a

mode

ω

ν

CCSD/aug-cc-pVTZ

ν + Δν

a

ω

ν

υ1

3179

3043

3050

3147

3014

υ2

3024

2888

2827

3011

2755

υ3

1533

1489

1490

1537

1492

υ4

1498

1461

1461

1511

1474

υ5

1275

1244

1244

1291

1260

υ6

958

931

931

971

946

υ7 υ8

417 3090

410 2961

405 2978

421 3059

413 2958

υ9 υ10

1507 1173

1461 1149

1461 1151

1511 1180

1465 1156

υ11

210

202

202

202

189

υ12

3082

2954

2950

3055

2939

υ13

1518

1471

1470

1521

1474

υ14

1201

1173

1173

1212

1183

υ15

260

246

246

252

235

υ16

3178

3043

3045

3145

3013

υ17 υ18

3019 1519

2903 1471

2898 1471

3002 1522

2956 1475

υ19

1467

1434

1434

1478

1445

υ20

1211

1176

1176

1231

1198

υ21

1131

1105

1105

1139

1113

Δν = Fermi displacements (emphasized in bold if Δν > 10 cm1).

produced by Gaussian, in a force field defined in Cartesian and internal coordinates. All the 3N  6 frequencies are localized and identified in Table 3. Bands displaced by strong Fermi interactions are emphasized in bold type. The CCSD calculations are quite time demanding; consequently, the quadratic force field is considered at the CCSD level, whereas the cubic and quartic fields are evaluated with MP2. The MP2 anharmonic effects included in the CCSD case are a good approximation due to the small value of the anharmonic correction. Then, the anharmonic frequencies of Table 3 were determined from MP2 theory (4th column) and combining CCSD and MP2 theories (7th column). Harmonic frequencies allow us to compare MP2 and CCSD theories. In general, for the hydrogen stretching modes and the torsional modes, CCSD frequencies are lower than the MP2 ones whereas for the remaining modes CCSD produces higher values than MP2. Anharmonic fundamentals follow the same tendency 13575

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Table 4. Predicted Spectroscopic Properties Using Second Order Perturbation Theory MP2/aug-cc-pVTZ CCSD/aug-cc-pVTZ

obs7

Fermi Displacements of Torsional Bands (in cm‑1) 2ν11 (torsion)

401 f 398

370 f 370

2ν15 (torsion) ν7 (COC bending)

491 f 498 410 f 405

467 f 479 413 f 401

Equilibrium Rotational Constants (in MHz) Ae

38549.07

38973.40

Be

10192.86

10145.29

Ce

8974.20

8961.90

following our experience in this type of studies, the calculated variation of the rotational constant with the vibrational energy (α-rovibrational parameters) is trustworthy. 3D-Variational Calculations. The energy levels corresponding to the two large amplitude vibrations and the COC bending mode of DME have been determined by solving variationally the following three-dimensional Hamiltonian:26 ! ! 3 3 ∂ ∂ ^ 3D ¼  ∑ ∑ H Bij þ V ðα, θ1 , θ2 Þ ∂qi ∂qj i j þ V 0 ðα, θ1 , θ2 Þ þ V ZPVE ðα, θ1 , θ2 Þ

i, j ¼ α, θ1 , θ2

ð2Þ

Rotational Constants (in MHz) A0

38405.04

38966.84

38788.18

B0

10038.45

10000.01

10056.48

C0 Aν7

8851.91 39030.36

8846.44 39517.12

8886.83

Bν7

10014.68

10030.27

Cν7

8817.27

8810.80

Aν11

38417.92

38205.88

Bν11

10009.19

9987.19

Cν11

8828.64

8801.94

Aν15

38419.36

38539.24

Bν15 Cν15

9967.78 8824.86

9961.91 8806.22

The first term, which depends on the Bij parameters (the G matrix elements in cm1), is a 3D-kinetic energy operator T3D; V(α,θ1,θ2), V0 (α,θ1,θ2), and VZPVE(α,θ1,θ2) represent the potential energy surface, the “Podolsky pseudopotential”, and the zero point vibrational energy correction, respectively (ref 26 and references therein). As the 3D-PES transforms as the totally symmetric representation of the G36 Molecular Symmetry Group,1416 we use the following analytical expression for the potential energy surface: V ðα, θ1 , θ2 Þ ¼

4

þ

Centrifugal Distortion Constants (in MHz) ΔJ

0.0095

0.0091

0.0091

ΔK

0.3346

0.3328

0.3420

ΔJK

0.0313

0.0292

0.0269

δJ

0.0019

0.0018

0.0018

δK

0.0087

0.0117

0.0138

rule, with the exception of ν7 (COC bending) and ν17 (H-asymmetric stretching), strongly displaced by Fermi interactions. With MP2, ω7 and ν7 are calculated to be 417 and 405 cm1, whereas, with CCSD, they are calculated to be 421 and 401 cm1 (see Tables 3 and 4). The tendency rule infringement, which produces an overestimation of the Fermi displacements, may be explained from the mix of levels of theory used for anharmonic CCSD calculations. To our knowledge, DME fundamentals of the medium and high frequency modes have been measured in the solid phase,25 except for the ν6, ν17, and ν21 modes, which were measured in the gas phase.6 We cannot compare our results with available experimental data, especially when possible errors are too small. The COC bending (ν7) interacts strongly with the 2ν15 torsional overtone (see Table 4) and increases it an amount of 712 cm1. Perturbation theory confirms the need for a 3-dimensional (3D) model to obtain torsional excited CH3 vibrational states above the fundamentals. Fortunately, interactions with the remaining 3N  9 modes are negligible. A 3D model is sufficient. In Table 4, some rotational constants and centrifugal distortion constants, derived from second order perturbation theory, are shown. For the vibrational ground state, they are compared with experimental data.7 As far as we know, there are not published experimental data for excited vibrational levels. There is a good agreement for the centrifugal distortion constant, and

2

2

∑ ∑ ∑ ½AMNL αM cosð3Nθ1 Þ cosð3Lθ2 Þ M¼0 N ¼0 L¼0 BM11 αM sinð3θ1 Þ sinð3θ2 Þ ∑ M¼0

ð3Þ

In this equation AMNL = AMLN. Formally identical expressions can be employed for V0 , VZPVE, and Bθ1θ2. Although the kinetic term T3D is totally symmetric (A1) (ref 16 and references therein), some kinetic parameters present low symmetry properties in several conformations. The Hamiltonian parameters have been determined from the energies and optimized geometries of a set of 126 selected conformations. They have been chosen for different values of the COC bending angle (104.676° f 119.676°, Δα = 3°), and the H4C2O1C3 and H7C3O1C2 dihedral angles (0, 30, 90, 150, 180, 30, 90, and 150°). The remaining 3N  9 parameters were allowed to be relaxed. Whereas the geometries were calculated with CCSD/aug-cc-pVTZ, the energies were obtained by performing single point CCSD(T)/aug-cc-pVTZ calculations on the partially optimized geometries. For each molecular structure, the G matrix elements and V0 were computed with the MATRIZG subroutine of the ENEDIM code from the internal coordinates,23 and the ZPVE correction was determined from the MP2/aug-cc-pVTZ harmonic frequencies ωi: EZPVE ¼

i ¼ 3N  6

∑ i¼n þ 1

ωi 2

In this equation, the sum neglects two torsional modes and the COC bending modes (n = 3). V is isotopically invariable, but VZPVE, V0 , and Bij are different for the different isotopologues. For this reason, the effective potential VEF (VEF = V + VZPVE + V0 ) and the effective barriers depend on the nuclear masses. The vibrational correction effect on the DME-h6 potential parameters is shown in Table 2. For α = 0, V060 varies from 921 to 939 cm1 and V6060 from 1966 to 1984 cm1 when ZPVE is considered. The correction augments the barrier height, allowing prediction of a displacement of the 13576

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spectra to high frequencies and a reduction of the torsional splittings. The geometry optimization is a partial way to consider vibrational interactions. Thus, during the motions of the independent variables (torsions and COC bending), the remaining coordinates are forced to lie at their minima. Later on, once the ZPVE correction is added, they are fixed at their zero point vibrational energy. This procedure leads to accurate results but introduces numerical errors that need to be minimized. For example, the tree dihedral angles H4C2O1C3 (D11), H5C2O1C3 (D12), and H6C2O1C3 (D13) of each methyl group (or H7C2O1C3 (D21), H8C2O1C3 (D22), and H9C2O1C3 (D23)) are not identically treated. One is fixed (D11 or D21), and the other two are relaxed, producing an artificial distortion and symmetry breaking. To obtain a proper definition of torsional coordinates, we used linear combinations of internal coordinates transforming as the C3v symmetry group representation for defining the vibrational coordinate:27 θ1 ¼ ðD11 þ D12 þ D13  2πÞ θ2 ¼ ðD21 þ D22 þ D23  2πÞ

Table 5. Expansion Coefficients for the CCSD(T)/aug-ccpVTZ Potential Energy Surface, V(α,θ1,θ2), the Vibrationally Corrected VEF(α,θ1,θ2) (VEF = V + V0 + VZPVE), and the Refined VREF Surfaces (in cm1/degreesM)a coeff

a

Not totally symmetric linear combinations are used to define four deformation coordinates: 1 β1 ¼ ðD11  D12 Þ 2 1 β3 ¼ ðD21  D22 Þ 2 1 1 β2 ¼  ðD11  D12 Þ þ D13 4 2 1 1 β4 ¼  ðD21  D22 Þ þ D23 4 2

ð5Þ

To obtain the 3D-PES, the total electronic energies of the 126 structures are fitted to a 7D-PES, given by the function of eq 3 plus some additional terms depending on the deformation coordinates: Eα, θ1 , θ2 , β1 , β2 , β3 , β4 ¼

1

4

½CMi αM sinð3θ1 Þ sinð3θ2 Þβi  ∑ ∑ M¼0 i¼1 þ

1

4

4

∑ ∑ ∑ ½CijαM sinð3θ1Þ sinð3θ2Þβiβj M¼0 i¼1 j¼i þ 1 ð6Þ

The Cij coefficients corresponding to the additional terms depending on βi are very small, and they are neglected.17 It has to be pointed out that errors derived from the combination of different levels of theory (CCSD(T), CCSD, and MP2 used for energy, geometry, and vibrational corrections. respectively) are really small (