Molecular orbital anharmonic estimates for the infrared spectrum of CO2

Spectrochimica Acta Part A 58 (2002) 2621– 2632 www.elsevier.com/locate/saa

Molecular orbital anharmonic estimates for the infrared spectrum of CO2 Harley P. Martins Filho * Departamento de Quı´mica, Uni6ersidade Federal do Parana´, CP 19081, CEP 81531 -990 Curitiba, PR, Brazil Received 5 September 2001; received in revised form 11 December 2001; accepted 19 December 2001

Abstract The vibrational spectrum of CO2 up to second overtones has been calculated at four different ab initio levels using second order perturbation theory equations in a simplified manner, in which just a few cross-terms suitable for numerical estimation are considered in the Taylor series representing the potential energy and dipole moment functions. The series coefficients are obtained through polynomial regression of estimated single point energy and dipole values for a few distorted geometries along each normal coordinate. The effect of Fermi resonance on near-degenerate energy levels was also taken into account through the usual first order perturbation equations. MP2/6-31G(extended) frequency estimates have a root mean square error of just 32.14 cm − 1. This accuracy is achieved partly due to the underestimation of the harmonic frequencies, which compensates for the neglect of the cross-term ij anharmonic constants. The ii constants which depend on cubic and quartic energy coefficients are reasonably well estimated at all ab initio levels. The energy coefficient isbb responsible for the magnitude of the Fermi resonance is estimated with a maximum error of just 13%. Despite the inclusion of anharmonicities, errors for band intensities are still much larger than for the frequencies. Both electrical and mechanical anharmonicities may be equally important to the band intensity. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Vibrational spectra; Overtones; Perturbation methods; Molecular property surfaces; Fermi resonance

1. Introduction A complete estimation of a vibrational spectrum demands knowledge of high-order potential energy and dipole moment hypersurfaces for the molecule in question, as well as the theoretical tools for linking the hypersurface features to the

* Fax: + 55-21-41-3613186 E-mail address: [email protected] Filho).

(H.P.

Martins

spectrum parameters. A suitable approach for deriving spectrum parameters from property hypersurfaces is the application of second-order Canonical Van Vleck Perturbation Theory to the harmonic vibrational wavefunctions and transition moment equations [1,2]. Nielsen [3] first derived the equations for calculating band center frequencies in terms of the energy surface expressed as a Taylor series expansion. His work and further developments have been comprehensively reviewed by Mills [1]. Concerning the band intensities, Willetts et al. [4] were the first to

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H.P. Martins Filho / Spectrochimica Acta Part A 58 (2002) 2621–2632

derive general analytic equations for dipole transitions and integrated intensities of fundamentals, first overtone and combination transitions. Bludsky´ et al. [5] have redone the work of Willetts et al. [4], finding that errors remained in the treatment of resonating states. The theory was extended to examine higher excited states by McCoy and Sibert [6], but with no analytic equations for the band frequencies or transition moments. The energy and dipole moment hypersurfaces that are required as data for the perturbation methods are usually expressed as Taylor series expansions about the normal coordinates. The series coefficients represent energy or dipole derivatives with respect to the normal coordinates and are obtained through either a multivariate polynomial regression of estimated values of the properties in a grid of points or by central differences procedures. Most of the molecular orbital packages calculate only the Hessian matrix (second order energy derivatives) and the first derivatives of the dipole moment. Analytic procedures for calculating higher (and mixed) derivatives have also been developed [7], but their implementation in the most common computer packages has not appeared yet. A complete spectrum estimation according to the perturbation methods at their highest development is hampered by the complexity of the formulae and the high computational cost of evaluating energy and dipole moments for large geometric grids necessary to derive accurate high-order derivatives. The purpose of this work is to study the accuracy of a simplified perturbation treatment in which some energy cross terms and all dipole cross terms in the expansions considered in the treatment are neglected, thus reducing the size of the geometrical grid needed to characterize the energy and dipole functions. Instead of calculating the derivatives with respect to all Cartesian coordinates, as is usually done, each particular normal coordinate is displaced through suitable geometrical distortions and the corresponding single point energy and dipole values are used for obtaining the individual energy and dipole functions, through polynomial regression. This is a very simplified procedure, which may be interesting if we want to focus on just a particular mode

of a large molecule, because it allows separate analysis of the modes with much fewer calculations than in complete analyses. The molecule of CO2 provides a challenging basis for probing the accuracy of the method because of the well-known Fermi resonance between the symmetric stretch and bend modes and the remarkable difficulty in estimating accurate band intensities through harmonic dipole moment functions, at any ab initio level. The force field of CO2 has been extensively investigated. Maslen et al. [8] calculated a fourth-order energy hypersurface for CO2 at MP2/DZP level through finite differences of analytic second derivatives of the energy. The harmonic frequencies are estimated to within 3% from experiment and the anharmonic constants to within 34%. Csa´ sza´ r [9] calculated a complete sixth-order force field for CO2 using a finite difference formalism to obtain higher-order force constants from analytic first and second-order geometric derivatives of the electronic energy at SCF/TZ2P and CCSD(T)/QZ2P (electron correlation at coupled cluster level with single and double excitations and perturbative estimation of the effects of connected triple excitations) levels. For the last level, the average deviation from the experimental values derived by Cihla and Che´ din [10] is only 2.1 and 5.8% for the quadratic and cubic constants, respectively. Martin et al. [11] calculated the quartic force field of CO2 at CCSD(T) level with a basis set of spdfg quality, yielding fundamental and overtone frequencies in excellent agreement with experiment. No estimates have been reported up to date for the intensities of the two infrared active bands of CO2 taking into account anharmonic factors in the energy and dipole functions. The most recent estimates, within the double harmonic approximation, show that even at CCSD(T) level with large and heavily polarized basis sets (TZ2P [12] and TZ(2df,2pd) [13]) the deviation from experiment is still 20% on average. Halls and Schlegel [14,15] recently reported calculations for the intensities of a group of 12 molecules, including CO2, through several methods of density functional theory (DFT). For CO2, the calculations with hybrid functionals (B3-LYP, B3-P86 and B3-PW91) and a 6-31G(d) basis set give better estimates than

H.P. Martins Filho / Spectrochimica Acta Part A 58 (2002) 2621–2632

conventional ab initio methods as MP2, with estimates deviating − 35% on average from the experimental value. The methodology used in this work was already tested for H2O [16], with calculations giving similar results to those of a complete perturbation treatment, with the exception of the fundamental intensity for the bend mode. Here it is tested for the problematic features of the double bond CO.

2. Theory and calculations

2.1. The relation of band frequencies to energy surfaces The general dependence of the molecular vibrational energy levels on features of the energy function is usually expressed by the empirical term formula [1]

      

= %…s s + s

+ % ss% s ] s%

1 + %…t (t +1) 2 t 1 s + 2

+%st s + s,t

 

ss =

8… 2s − 3… 2s% 1 1 kssss − %i 2sss% 16 16 s% …s%(4… 2s − … 2s%)

tt =

1 1 8… 2t − 3… 2s ktttt − %i 2tts 16 16 s …s (4… 2t − … 2s ) −





(3)

1 8… 2t − 3… 2t% %i 2ttt% 16 t% …t%(4… 2t − … 2t%)

(4)

Since it is intended in this work to neglect crossterms in the energy expansions, the constants ss%, tt% and st, which mix different normal modes and depend exclusively on these cross-terms, will be neglected in Eq. (1). However, even the singlemode anharmonic constants ss and tt depend upon cubic cross-terms isss%, istt and ittt%, and therefore these terms will be the only ones considered in the present methodology. Accounting for these approximations, Eq. (1) for CO2 reduces to G(s, b, a )

   

= …s s +

G(s, k, t,k)

 

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1 1 + …b (b + 1)+ …a a + 2 2

+ ss s +

1 2

2

+ bb (b + 1)2 + aa a +

1 2

2

(5)

1 s% + 2

1 (t +1) + % tt%(t +1)(t% +1) 2 t ] t% (1)

where the  are the anharmonic constants and the indexes s and t refer to non-degenerate and degenerate normal coordinates, respectively. For the calculation of the , the molecular electronic energy is expanded in a Taylor series in dimensionless normal coordinates to fourth order: V (cm − 1) 1 1 1 = %…i q 2i + % iijkqi qj qk + % k qqq q 2 i 6 i, j,k 24 i, j,k,l ijkl i j k l (2) On applying standard perturbation theory to the vibrational wavefunctions in which the cubic and quartic terms of Eq. (2) are the perturbing terms, the following equations are found for the constants  [1]:

where indexes s, b and a refer to the symmetric stretch, bend and antisymmetric stretch modes, respectively. The symmetry constraint that the energy function must be invariant with respect to the symmetry operations of the molecular point group greatly reduces the number of valid constants i and k in Eq. (2). For CO2 (point group D h) just the cubic constants isss, isbb and isaa and the quartic constants kssss, kbbbb, kaaaa, kssbb, kssaa and kbbaa are non-zero. In our approach, only the last three fourth-order cross-terms will be neglected. These terms would influence the neglected anharmonic constants st and ss%. The problem of accidental near-degeneracy between vibrational levels of CO2 was first recognized by Fermi [17] and treated in a practical fashion by Dennison [18]. Because the harmonic frequencies for the symmetric stretch and bend modes bear the relationship …s $ 2…b, there are near-degenerate vibrational levels corresponding to the conditions 2s + b = 2, 3, 4…. As a consequence of the resonance between these levels, they

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are ‘‘pushed’’ apart and the corresponding infrared bands share intensity, though in the case of CO2 the bands have no intensity due to symmetry constraint. Dennison’s treatment [18] employs standard perturbation theory for degenerate levels in which a perturbing term involving Z= …s − 2…b must be added to take account of the fact that the levels are not exactly degenerate. The perturbed levels are obtained upon solving a secular determinant in which the diagonal matrix elements are the energy of the unperturbed levels and the off-diagonal matrix elements depend on the only energy term in Eq. (2) connecting the vibrational modes involved, isbb. According to Nielsen [19] the off-diagonal elements can be calculated through s, b, l H1 s −1, b +2, l



= −



isbb 1/2 s [(b +2)2 −l 2]1/2 23/2

(6)

Only vibrational levels of the bend mode with the same value of the associated angular quantum number l can be mixed on a secular determinant for a Fermi resonance. As an example, the levels corresponding to 2s +b =2 are (1, 00), (0, 20) and (0, 22), in the notation (s,  lb). Of these, only the levels (1, 00) and (0, 20) couple through resonance (same value of l) while the level (0, 22) is left unperturbed. An energy coefficient isbb of significant magnitude cannot be accepted as a perturbing term in the perturbation method used to derive Eqs. (3) and (4). Therefore, the term 12isbbqs (q 2b1 +q 2b2) must be dropped from the energy perturbing function, which means that a corresponding term in the summation of Eq. (4) for bb must be dropped too, namely i 2sbb(8… 2b −3… 2s )/…s (4… 2b −… 2s ). This term would become infinitely large because of the near-degeneracy between …s and 2…b.

2.2. The relation of band intensities to dipole moment surfaces The total intensity or integrated absorption of a vibration–rotation band relative to a transition %’ %% of normal mode i is given by [20]

%¦ A %¦ = (8y 3NA/3hc)w %¦ (i ) 2 i i R

(7)

where R’’’(i ) is the vibrational matrix element of the molecular dipole moment for the transition % ’ %%. The dipole moment operator Ph is usually expanded in a Taylor series in dimensionless normal coordinates qi. For the x component of the dipole moment, for example, 1 Px (q)= P 0x + %P 1x(i )qi + %P 2x(i, j )qi qj 2 i, j i 1 + % P 3x(i, j, k)qi qj qk 6i, j,k

(8)

The simultaneous determination of the anharmonic Cs (based on first order harmonic wavefunctions) and calculation of the vibrational matrix elements of Eq. (8) is the subject of the work of Willetts et al. [4] and Bludsky´ et al. [5], in which contact transformation is used for application of second order perturbation theory. In Ref. [16], it was proposed that the cross-terms in the dipole expansion of Eq. (8) could be neglected without significant loss of accuracy. In this approach, we can separate the analysis in terms of each normal mode, using for each one the equations for R%%%(i ) derived by Herman and Schuler for diatomic molecules [21]. These equations reduce to simple linear combinations of the dipole derivatives of the form 2 %¦ R%¦(i, h)= P 1h(i )R%¦ 1 (i, h)+ P h(i, i )R2 (i, h)

+ P 3h(i, i, i )R%¦ 3 (i, h)

(9)

where h= x, y, z. The R%¦ j (i, h) terms with j=1 and 2 are functions of the single-mode energy coefficients iiii and kiiii. Herman and Schuler report the expressions for calculating these terms for fundamentals and first and second overtones [21], which we will not present here. When symmetry constraints are applied to the dipole function, it is found that only the bend (Pu) and the antisymmetric stretch (S+ u ) modes are infrared active. As the anharmonic terms in the dipole expansions must conform to these symmetry species too, the quadratic coefficients P 2h(i, i ) in Eq. (8) are found to be forbidden.

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2.3. Molecular orbital estimation of energy and dipole surfaces The procedure for determination of the property derivatives starts with a standard infrared harmonic spectra calculation, as implemented in Gaussian 98W [22], which furnishes harmonic frequencies and the normal coordinates (L matrix, normalized to unity). The correctly normalized normal coordinates, which relate coordinates in a.m.u.1/2 A, to Cartesian coordinates in A, [23], are obtained through the transformation [24] Lij = cij

,



%mi c 2ij

1/2

(10)

i

where the cij are the normalized-to-unity elements of the Gaussian 98W L matrix and mi is the mass of the nucleus associated with Cartesian coordinate Xi. For each of the vibrational modes of CO2, three displacements were considered, both positive and negative. Those displacements were transformed in Cartesian displacements using the L matrix and single point calculations were performed for each geometry. A regression of the energy values on a fourth-order polynomial in the normal coordinate and a regression of the dipole moment values on a third-order polynomial in the same coordinate are performed. Since the energy is an even function with respect to the bend and antisymmetric stretch modes while the dipole moment function is odd, only one direction of displacement for each of these modes was considered. Thus, just 12 single point calculations were needed for the three individual normal mode functions. The maximum displacements allowed correspond to stretching the bonds to 10% of their equilibrium values or bending the molecule to 160°. These limits correspond to the amplitudes for the fourth energy level ( =3) of the stretch modes and the fifth energy level (= 4) of the bend mode. This allows estimate of fundamental, first and second overtone frequencies, as well as combination band frequencies up to 2s +b =4. For estimation of energy cross-term derivatives isbb and isaa we note that the term ijii in the general expansion of Eq. (2) represents the deriva-

2625

tive #3E/#q 2i #qj. In the energy regression for a normal coordinate i, the coefficient of the secondorder term represents the derivative 12…i = 12(#2E/ #q 2i ). So, if one differentiates 12…i with respect to the normal coordinate j one should obtain 12(#3E/ #q 2i #qj )= 12iiij. For differentiating 12…i numerically, two energy regressions are made for coordinate i, one corresponding to coordinate qj set to zero and the other corresponding to a small displacement of qj. For the estimation of isbb, for example, an additional energy regression for the bend mode was done in which the displaced geometries included the effect of a slight displacement in the symmetric stretch mode. As explained above, just one direction of displacement for the bend coordinate is needed, independent of the displacement of the symmetric stretch mode. Therefore, six additional single point calculations have been performed (for isbb and isaa ) to complete the description of the simplified energy function considered in this work for CO2, leading to a total of 18 calculations. Calculations have been done at four levels of ab initio methods for which harmonic results of good accuracy had already been published. Electron correlation at MP2 level [25] was used with the standard basis set 6-31+ G(d) and the 631G(extended) basis set of Simandiras et al. [26], which is based on the usual 6-31G set [27] with specific polarization and diffuse functions added to bias the calculation in favor of dipole moment calculations. For polarizations, hd(O) = 1.20 and 0.40 and hd(C) = 0.80 and 0.266. For diffuse functions (sp shells), hsp(O) = 0.094 and hsp(C) = 0.056. The highest ab initio level tested by Thomas et al. [13] which we could perform on Gaussian 98 [22] was the use of electron correlation at CISD level [28] with the basis set TZ+ 2P, which consists of the TZ set of Dunning–Huzinaga [29] with two sets of polarization functions (hd(O) = 1.70 and 0.425; and hd(C) = 1.50 and 0.375). From the work of Halls and Schlegel [14,15] on the use of DFT methods, the best harmonic results for CO2 come from the use of Becke’s three-parameter exchange functional (B3) [30] with the Lee–Yang–Parr gradient-corrected correlation functional (LYP) [31], associated with the basis set 6-31G(d).

H.P. Martins Filho / Spectrochimica Acta Part A 58 (2002) 2621–2632

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For mathematical procedures, we have used Mathviews® [32] and Mathcad® [33]. All codes were installed on a Pentium II 300 MHz microcomputer with 256 MB of RAM.

metric stretch, but this is partly due to the larger magnitude of its harmonic frequency. The RMS errors taking into account all frequencies for all modes are 74.09 cm − 1 (1.3%) for MP2/6-31 + G(d), 32.14 cm − 1 (0.7%) for MP2/6-31G extended, 189.56 cm − 1 (7.4%) for CISD/TZ + 2P and 84.35 cm − 1 (1.6%) for B3-LYP/6-31G(d). The smallest errors come from the MP2/6-31G (extended) estimates and this reflects the better estimation of the harmonic frequencies, since the corresponding estimated anharmonic constants are not the best set from Table 1. Actually, the harmonic frequencies are underestimated at this level, which compensates for the lack of the negative neglected cross-term anharmonic constants ij which would decrease the frequency estimates (see below). The same trend for underestimation was observed in calculations with the same basis set for H2O [4,16]. The MP2/6-31 + G(d) estimates for the basic vibrational parameters are very similar to those of MP2/6-31G(extended) calculations, with the exception of the antisymmetric stretch harmonic frequency, which is overestimated. This is what causes the MP2/6-31 + G(d) general RMS error to rise much above the RMS error for the MP2/6-31G(extended) estimates. The estimates more apart from the experimental data come from

3. Results and discussion For all energy regressions the standard deviation was less than 10 − 6 Hartree, with total ranges of energy variation of the order of 10 − 2 Hartree. For all dipole moment regressions, the standard deviation was less than 10 − 5 Debye, with total ranges of dipole of the order of 10 − 1 Debye. The anharmonic coefficients i and k and the basic vibrational parameters … and  are shown in Table 1. The band center frequencies derived from the data of Table 1 are shown in Tables 2 and 3, as well as the experimental values as reported by Cihla and Che´ din [10]. To aid the comparison of theoretical estimates for several frequencies simultaneously, we also report root mean square (RMS) errors for each ab initio level at the bottom of Tables 2 and 3. The fundamental band intensities are reported in Table 4. As can be seen in Tables 2 and 3, the absolute errors are systematically larger for the antisymTable 1 Harmonic frequencies and anharmonic constants for CO2 (cm−1)a Exp. (Ref. [10])

MP2/6-31+G(d)

MP2/6-31Gext

B3-LYP/6-31G(d)

CISD/TZ+2P

…s …b …a

1354.329 672.660 2397.776

1322.73 649.24 2419.38

1321.78 649.51 2384.67

1372.55 640.66 2438.68

1428.96 701.80 2452.21

ss sb sa bb ba aa

−3.014 −5.058 −19.048 1.521 −12.616 −12.597

−2.51 0 0 3.23 0 −10.70

−2.60 0 0 4.36 0 −9.90

−2.94 0 0 5.53 0 −13.59

−2.78 0 0 5.85 0 −11.97

−43.019 1.407 75.369 2.587 −252.351 6.598

−43.811 1.952 73.419 2.152 −236.180 6.358

−43.792 1.891 79.165 2.908 −235.316 6.783

−45.214 1.766 83.706 3.685 −259.208 6.559

−45.214 1.726 85.098 3.898 −264.635 7.597

isss kssss ibbs kbbbb iaas kaaaa

a Since Cihla and Che´ din [10] make use of an energy expansion of the form of Eq. (2) with the numerical factors adsorbed into the constants, we present numerical values for the constants i and k according to this use.

a

MP2/ 6-31Gext 13.58 (0.7%)

1996.16

CISD/ TZ+2P 178.31 (8.3%)

−8.45 (−1.3) −10.81 (−0.8) −7.12 (−0.4)

Errorsa

−17.06 (−0.6) −26.23 (−1.0) −26.51 (−1.0)

−11.12 (−0.4) −18.17 (−0.7)

−16.40 (−0.8) −20.14 (−1.0)

B3-LYP/ 6-31+G(d) 23.79 (1.1%)

2013.97

1333.92

662.60

MP2/ 6-31Gext

2522.00

2652.73

2806.46

2576.21

2777.04

1915.52

2077.63

1268.60

−4.78 (−0.7) −1.21 (−0.1) 10.69 (0.5)

Errorsa

9.31 (0.3) −18.49 (−0.7) −26.28 (−1.0)

16.31 (0.59) −8.83 (−0.3)

0.77 (0.0) −16.95 (−0.9)

−6.29 (−0.45) −16.81 (−1.3)

Errorsa

Absolute (cm−1) and relative (%) errors for the estimated values, relative errors in parentheses.

RMS error (cm−1)

MP2/ 6-31+G(d) 17.20 (0.9%)

2003.28

(0, 33)

RMS errors Ab Initio level

1335.13

(0, 22) 1324.32

658.93

667.38

2521.77

2548.28

(0, 11)

2644.99

2671.22

MP2/ 6-31+G*

2780.09

2566.87

2585.04

2797.15

2749.61

1912.33

1932.47

2760.73

2060.46

2076.86

Unperturbed energy le6els of the bend mode Level Observed (Ref. [10])

(2, 00), (1, 20) and (0, 40)

(1, 22) and (0, 42)

(1, 11) and (0, 31)

1268.99

1285.41

−15.16 (−1.1) −16.42 (−1.3)

1381.90

1373.03

(1, 00) and (0, 20)

1388.19

MP2/ 6-31Gext

Perturbed energy le6els of the symmetric stretch and bend modes Coupled levels Observed MP2/ Errorsa (Ref. [10]) 6-31+G*

2193.10

1450.37

719.34

CISD/ TZ+2P

2736.25

2876.38

3049.92

2800.09

3021.20

2079.18

2256.66

1375.22

1498.55

CISD/ TZ+2P

51.96 (7.8) 115.24 (8.6) 189.82 (9.5)

Errorsa

252.77 (9.0) 205.16 (7.7) 187.97 (7.4)

215.05 (8.3) 260.47 (9.4)

179.80 (8.7) 146.71 (7.6)

110.37 (7.9) 89.81 (7.0)

Errorsa

2004.89

1325.54

657.24

B3-LYP/ 6-31+G*

2553.30

2716.48

2845.21

2591.23

2796.29

1930.16

2098.65

1283.45

1408.77

B3-LYP/ 6-31+G*

Table 2 Estimated and observed vibrational energy levels (s,  lb) relative to the zero point energy (cm−1) for the symmetric stretch and bend modes of CO2

−10.14 (−1.5) −9.59 (−0.72) 1.61 (0.1)

Errorsa

48.06 (1.7) 45.26 (1.7) 5.02 (0.2)

35.56 (1.3) 6.19 (0.2)

21.79 (1.0) −2.31 (−0.1)

20.58 (1.5) −1.97 (−0.2)

Errorsa

H.P. Martins Filho / Spectrochimica Acta Part A 58 (2002) 2621–2632 2627

4673.31

6972.55

(2)

(3)

a

MP2/ 6-31Gext 42.89 (0.8%)

7129.67

4774.52

2397.97

MP2/ 6-31+G*

CISD/ TZ+2P 172.66 (3.4%)

48.84 (2.1) 101.21 (2.2) 157.22 (2.3)

Errorsa

B3-LYP/ 6-31+G(d) 130.91 (2.6%)

7035.23

4709.95

2364.88

MP2/ 6-31Gext 15.74 (0.7) 36.64 (0.8) 62.68 (0.9)

Errorsa

Absolute (cm−1) and relative (%) errors for the estimated values, relative errors in parentheses.

RMS error (cm−1)

MP2/ 6-31+G(d) 111.59 (2.2%)

2349.14

(1)

RMS errors Ab Initio level

Observed (Ref. [10])

Level (a)

7212.97

4832.59

2428.27

CISD/ TZ+2P 79.13 (3.4) 159.28 (3.4) 240.42 (3.4)

Errorsa

Table 3 Estimated and observed vibrational energy levels relative to the zero point energy (cm−1) for the antisymmetric stretch mode of CO2

7152.89

4795.78

2411.49

B3-LYP/ 6-31+G*

62.35 (2.6) 122.48 (2.6) 180.34 (2.6)

Errorsa

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Table 4 Estimated and observed fundamental infrared intensities (km mol−1) for the antisymmetric stretch and bend modes of CO2 and the harmonic and anharmonic contributions for the transition dipole moment R (Debye) Harmonic Bend (observed [34,35]: 24 km mol−1) MP2/6-31+G(d) 25.64 MP2/6-31G (ext) 21.60 CISD/TZ+2P 40.48 B3-LYP/6-31G(d) 30.70

Anharmonic

24.62 20.84 39.92 30.16

Antisymmetric stretch (observed [36]: 548 km mol−1) MP2/6-31+G(d) 569.11 569.18 MP2/6-31G (ext) 564.57 563.23 CISD/TZ+2P 833.82 809.16 B3-LYP/6-31G(d) 551.21 536.10 a

Errorsa

Rharm

Ranharm×103

0.62 −3.16 15.92 6.16

(2.6) (−13.2) (66.3) (25.7)

0.12490 0.11440 0.15043 0.13708

−2.81 −2.39 −1.63 −1.77

21.18 15.23 261.16 −11.90

(3.9) (2.8) (47.7) (−2.2)

0.30513 0.30602 0.36660 0.29908

2.59 2.22 −2.00 −1.27

Absolute (km mol−1) and relative (%) errors for the estimated values, relative errors in parentheses.

the CISD/TZ +2P calculations and this is due mainly to the overestimation of all the harmonic frequencies, though the estimates for bb and isbb at this level are also the least accurate from Table 1. As can be seen in Table 1, the best estimate (MP2/6-31 +G(d)) to the small anharmonic constant bb is 112% above the experimental value while the least accurate estimate is almost four times larger than the experimental value. The constant ss, though small too, has much better theoretical estimates, the least accurate being just 17% below the experimental value. This may be due to the fact that bb depends only on the quartic coefficient kbbbb while ss depends on the quartic coefficient kssss and the cubic coefficient isss, and the last contributes a larger part. We see in Table 1 that though the absolute errors for the estimates of kbbbb are of the same magnitude as those for isss, the relative error is much bigger for the former ( − 17 to 51%), thus the estimates for ss should be more accurate than those for bb. However, the above mentioned relative errors for bb do not match those for the coefficient kbbbb, which suggests a discrepancy in the treatment of Fermi resonance by Cihla and Che´ din [10]. In fact, even the high-quality calculations of Martin et al. [11] yield an estimate for bb which is 57% above the experimental value, while the errors for the other constants are much smaller. The same general reasoning applies to the constant aa,

which also depends on a cubic and a quartic energy coefficient and has good theoretical estimates too. A crucial factor in judging the accuracy of our methodology is to assess the importance of the neglected cross-term anharmonic constants ij to the observed frequencies, which can be afforded by analysis of the experimental data. By comparison of several experimental spectrum parameters with sophisticated ab initio force field estimates, Martin et al. [11] and Csa´ sza´ r [9] indicate the experimental results of Cihla and Che´ din [10] as the most reliable. In Table 1, we present their results. It can be seen that the experimental crossterm ij anharmonic constants are on average larger than the diagonal ii constants, which would point to the non-validity of omitting the first ones. However, the relative importance of these constants also depends on their coefficients in Eq. (1), and the weight of the cross-term constants, defined by the  values, is never larger than that of the diagonal ones for CO2. For the energy of a combination level involved in Fermi resonance such as (1, 1, 0), for example, the cross-term constants have a relative weight of 43% and the diagonal ones 57%. For the first excited state (0, 1, 0) of the degenerate bend coordinate, those weights are 33 and 67%, respectively; while for the stretches they are 44 and 56%, respectively. The weight of the diagonal constants increase for higher combination levels and pure overtone lev-

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els due to their quadratic coefficients in Eq. (1), therefore making more and more valid the omission of the cross-term constants. Nevertheless, for CO2 the magnitude of the last make them important for the lowest levels. As an example, the absolute contribution of the cross-term constants to the observed fundamental frequency of the symmetric stretch, calculated with experimental parameters, is −14.58 cm − 1 (1.1% of the harmonic frequency) while that of the diagonal constant is − 6.03 cm − 1 (0.4% of the harmonic frequency). For the fundamental frequency of the bend mode, the contribution of the cross-term constants is bigger and opposite in sign than that of the diagonal constant. On the other hand, the second overtone frequency of the antisymmetric stretch has a contribution of −66.42 cm − 1 (0.9% of the harmonic frequency) from the cross-term constants and −151.16 cm − 1 (2.1% of the harmonic frequency) from the diagonal one. Despite the importance of the cross-term constants, the results are generally satisfactory, even for the frequencies associated with the bend mode. This is partly due to the importance of the correction for the Fermi resonance, which depends upon the large isbb coefficient, above the other anharmonic corrections. The observed frequency separation provoked by the removal of near-degeneracies range from 102.78 cm − 1 for the levels (1, 00, 0) and (0, 20, 0) (7.7% of the mean unperturbed frequencies) to 248.87 cm − 1 for the levels (2, 00, 0), (1, 20, 0) and (0, 40, 0) (9.3% of the mean unperturbed frequencies). The best estimates for isbb come from the MP2 calculations, but even the worst (CISD/TZ +2P) value is only 13% in error relative to the experimental value, so the large resonance correction is well predicted at all ab initio levels. From the fundamental intensity estimates in Table 4, it can be seen that no ab initio level gives the best estimates for both bands simultaneously, but the MP2/6-31 +G(d) level yields best estimates on average. The relative errors, with the exception of that for MP2/6-31 + G(d) level, are bigger for the least intense band, as it could be expected. As for the frequencies, the least accurate estimates are the CISD/TZ +2P ones.

All calculations predict a slight decrease in intensity from the harmonic value to yield the corrected ‘‘anharmonic’’ value. Anharmonic contributions to the intensities come from the electrical anharmonicity (dipole high-order derivatives) present in the R estimate through Eq. (9) and the mechanical anharmonicity carried both by the frequency estimate explicitly appearing in Eq. (7) and by the cubic and quartic energy coefficients needed for the R estimates. In order to isolate the frequency estimate in the analysis, we present in Table 4 the R estimates as well as its additive harmonic and anharmonic contributions. Actually, since the P 2h(i, i ) derivatives are symmetry-forbidden for both infrared-active modes, the anharmonic contribution is defined by just the last term in Eq. (9). In this term, R%%% 3 (i, a) is a constant [21] depending only upon the transition considered (2.12132 for the fundamentals). Therefore, the anharmonic contributions to R are defined just by electric anharmonicities, for the CO2 molecule. For the bend mode, Ranharm is negative for all calculation levels, subtracting 1.1– 2.3% of the Rharm value in the R estimate, depending on the calculation level. On the other hand, the frequency estimates are 1.5–2.6% above the respective harmonic estimates. Since the R estimate is squared in the intensity estimate, the decrease in intensity for the anharmonic estimates is seen to be determined by the electric dipole anharmonicities. For the antisymmetric stretch, Ranharm is so small compared to Rharm (0.4–0.8% of the last, depending on the calculation level) that even its sign changes depending on the ab initio level. The frequency estimates are 0.8–1.1% below the respective harmonic values and therefore they dominate the observed decrease in intensity from the harmonic to the anharmonic estimate. However, both electrical and mechanical anharmonicities are very small for this mode. The good accuracy of the estimates (except for the CISD/TZ +2P calculation) is due therefore to accurate harmonic estimates for this mode. 4. Conclusions We do not intend to compare our results to those reported by, for example, Martin et al. [11]

H.P. Martins Filho / Spectrochimica Acta Part A 58 (2002) 2621–2632

or Csa´ sza´ r [9], since they use heavily correlated methods, very large basis sets and complete energy functions which demand calculations in large geometric grids. Our goal is to establish a methodology for semiquantitative estimation of infrared band frequencies and at least qualitative (within 20% of experiment) estimation of band intensities for application in large systems or parts of them. The frequency estimates at MP2/ 6-31G(extended) level reported here satisfy this goal, but the accuracy of the calculations for other systems depend upon the underestimation of harmonic frequencies at this ab initio level, since the neglect of the cross-term constants ij which cause the frequencies to decrease must be compensated in some way. In our previous calculations for H2O at the same level [16], we had similar results. Since the harmonic frequencies for H2O are much higher than those for CO2, the absolute errors for the estimated band frequencies are larger than those reported here, but the relative RMS error, not reported in Ref. [16], was just 1.6%. For the fundamental band frequencies of H2O, the contribution of the cross-term constants is roughly equal to that of the diagonal constants, which was not discussed in Ref. [16]. For all overtone frequencies, the contribution of the diagonal constants is larger. The quality of the results for CO2 was shown to depend considerably on the ab initio level and the basis set. The CISD/TZ +2P estimates, for example, are the least accurate, but Thomas et al. [13] showed that the correlation level which yields best harmonic frequencies combined with basis set TZ+ 2P is CCSD (coupled-cluster with single and double excitations). Since Gaussian 98W does not perform populational analysis at CCSD level, we have not tried this calculation level for CO2. The anharmonic constants  which depend on cubic and quartic coefficients of the energy functions are well estimated at all ab initio levels, while the constant which depend only on a quartic coefficient is not. This is partly due to a lower relative accuracy of the quartic coefficient estimate. The anharmonic energy coefficient isbb responsible for the magnitude of the Fermi reso-

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nance is well estimated at all ab initio levels and therefore the near-degeneracy removal is the most satisfying feature of the present calculation, though the theoretical perturbation treatment employed is not new. The accuracy of the intensity estimates depends primarily on the accuracy of the harmonic estimates. The anharmonic contributions are small and their theoretical estimates may vary much more than the harmonic contribution, for different ab initio levels. Nevertheless, almost all calculations consistently predict that the anharmonic estimate is lower than the harmonic one by no more than 4%. The analysis of the anharmonic dipole contribution estimates and the frequency estimates involved in the intensity estimate shows that both electrical and mechanical anharmonicities may be important to the estimate. Considering its simplicity, our methodology for spectra estimate seems to be useful for the proposed goals. For property functions calculations, the inclusion of electron correlation at MP2 level with a medium-sized basis set such as 6-31G(extended) or 6-31+ G(d) seems to be the optimal ab initio choice in terms of accuracy and computational demands.

Acknowledgements We gratefully acknowledge CNPq for complete financial support and The Mathwizards, Inc. for the code of Mathviews®.

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