Approximate molecular orbital anharmonic parameters for the infrared spectrum of H2O

Spectrochimica Acta Part A 57 (2001) 83 – 93 www.elsevier.nl/locate/saa

Approximate molecular orbital anharmonic parameters for the infrared spectrum of H2O Harley P. Martins, Filho * Departamento de Quı´mica, Uni6ersidade Federal do Parana´, CP 19081, CEP 81531 -990, Curitiba, PR, Brazil Received 2 May 2000; accepted 20 June 2000

Abstract The vibrational spectrum of H2O was calculated at MP2/6-31G(extended) and MP2/6-311G* levels taking into account anharmonicities through a simple approach to second-order perturbation theory in which molecular energy and dipole moment are expanded as Taylor series in normal coordinates with no cross terms, to simplify calculations. The series coefficients are obtained separately for each normal coordinate through polynomial regression of calculated single point property values corresponding to a few distorted molecular geometries. The energy coefficients are used to calculate the harmonic frequencies and the xii anharmonicity constants, and so the band origins. For the band intensities, second-order perturbation theory equations derived earlier for diatomic molecules are used for each mode. Estimated frequencies have accuracy equivalent to those of previous complete perturbation calculations at the same ab initio levels, being at most 2.6% above the experimental values for the MP2/6-31G(extended) level. The fundamental intensity estimates are equivalent to those for the complete treatments, with the exception of that at MP2/6-31G(extended) level for the bending mode, which is 7% above the experimental value. Estimated overtone intensities by both complete treatments and the simple approach may still differ in magnitude from the experimental values, though to a lesser extent for the formers. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Vibrational intensities; Overtones; Perturbation methods; Molecular property surfaces

1. Introduction A suitable approach for estimating vibrational spectra beyond the ‘double harmonic’ approximation is the application of second-order Canonical Van Vleck perturbation theory to the molecular vibrational wavefunctions and transition moment * Fax: + 55-21-413613186. E-mail address: [email protected] (H.P. Martins, Filho).

equations [1,2]. Nielsen [3] first suggested that contact transformation would be a suitable method for application of second order perturbation theory for evaluation of transition moments for many-normal mode polyatomic molecules. The theory was slowly developed by various groups, including Nielsen’s own [4], Secroun’s [5], Overend’s [6] and Geerlings’ [7]. By means of explicit transformation of the dipole moment operator, Willetts et al. [8] were the first to derive analytic equations for dipole transitions and inte-

1386-1425/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 1 4 2 5 ( 0 0 ) 0 0 3 4 6 - 2

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grated intensities of fundamentals, first overtone and combination transitions. The formulae are so complex that they were derived using a computer algebra program. In fact, owing to the complexity of the work, many correction reports were made by the various groups involved as the theory developed. Bludsky´ et al. [9] have redone Willetts’ work, finding that errors remained in the treatment of resonating states. Willetts’ equations are implemented in the program SPECTRO [10], which furnish virtually all vibrational spectroscopic parameters for a molecule, given its potential energy and dipole derivatives in a Cartesian space-fixed coordinate system. The theory was extended to examine higher excited states by McCoy and Sibert [11], but with no analytic equations for the transition moments. In all perturbation methods, the features of the molecular energy and dipole moment surfaces with respect to the normal modes enter as parameters which must be estimated usually by ab initio methods. In other words, derivatives of molecular energy and dipole moment with respect to normal modes must be estimated. These derivatives are usually evaluated with respect to Cartesian coordinates and then converted to normal coordinate space. Most of the molecular orbital packages calculate the Hessian matrix (second order energy derivatives), which yields harmonic frequencies and the L matrix for converting derivatives from one coordinate system to the other. Most of them also calculate the first derivatives of the dipole moment needed for the estimation of the harmonic intensities. Analytic procedures for calculating the derivatives are implemented for many kinds of wavefunctions, including those at high electron correlation treatment levels, like MP2 or CI. Analytic procedures for calculating higher (and mixed) derivatives have also been developed [12], but their implementation in the most common computer packages has not appeared yet. The purpose of this work is to study the accuracy of a simplified perturbation treatment in which cross terms in the energy and dipole expansions considered in the treatment are neglected. In doing so, simpler perturbation equations derived earlier by Herman and Schuler [13] for diatomic

molecules can be used for each vibrational mode separately. Instead of calculating the derivatives with respect to all Cartesian coordinates, as is usually done, each particular normal coordinate will be 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 one wants to concentrate in just a particular mode of a large molecule, for it allows separate analysis of the modes with fewer calculations than in complete analyses. Complete analyses for H2O were done at MP2/6-31G(extended) level by Willetts et al. [8] and Bludsky´ et al. [9]. The H2O spectrum will be estimated through the proposed procedure at the same ab initio level and its accuracy will be judged against their results. The most recent calculation of overtone intensities in the H2O spectrum was done by Kjaergaard et al. [14], who used a harmonically coupled anharmonic oscillator model in which Morse oscillators are used to represent the vibrational wavefunctions of the bond coordinates of H2O. These wavefunctions are parameterized with selected observed peak positions. The dipole moment is expanded in a Taylor series in terms of the bond coordinates and the series coefficients are obtained from ab initio MP2/6-311G** single point dipole grid values using finite-difference procedures. The results of this work provide a basis from a different approach of taking into account anharmonicities for comparison with those here, if the spectrum parameters are also calculated at the MP2/6-311G** level.

2. Theory and calculations

2.1. The relation of spectral data to energy and dipole surfaces The total intensity or integrated absortion A y%y%% of a vibration–rotation band is given by [15,16] A y%y¦ = (8p 3NA /3hc)n y%y¦ Ry%y¦ 2

(1)

H.P. Martins, Filho / Spectrochimica Acta Part A 57 (2001) 83–93

where Ry%y%% is the vibrational matrix element of the molecular dipole moment for the transition y% ’ y%%: Ry%y¦ 2 = and y%y¦ a

R

2 % Ry%y¦ a

(2)

a = x,y,z

&

= C*y%Pa Cy¦ dt

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Px = P 0x + P 1xq+ P 2xq 2 + P 3xq 3 V (cm

)= k0q + bq + gq

−1

2

3

(6)

4

(7)

y%y%%

The equations for the R reduce to simple linear combinations of the dipole derivatives in the form 1 y%y¦ 2 y%y¦ 3 y%y¦ Ry%y¦ x = P xR1 (x)+ P xR2 (x)+ P xR3 (x)

(3)

Since there is no analytic expression for the dipole moment operator P, the latter is expanded in a Taylor series in dimensionless normal coordinates q. For the x component of the dipole moment, for example, 1 Px = P 0x + % P 1x(a)qa + % P 2x(a, b)qaqb 2 ab a 1 % P 3 (a, b, c) (4) 6 a,b,c x Maintaining just the linear terms in this series constitutes the electric harmonic approximation. The Cs are anharmonic vibrational wavefunctions corresponding to a potential energy surface of the form +

V (cm − 1) 1 1 = % vaq 2a + % ba,b,cqaqbqc 6 a,b,c 2 a 1 + % g qqqq (5) 24 a,b,c,d a,b,c,d a b c d Truncating the series in the quadratic terms constitutes the mechanical harmonic approximation. The simultaneous determination of the Cs (based on first order harmonic wavefunctions) and calculation of the integrals of Eq. (3) is the subject of the work of Willetts et al. [8] and Bludsky´ et al. [9], where contact transformation is used for application of second order perturbation theory. Since cross-terms between normal modes will be neglected, one can separate the analysis in terms of each normal mode, using for each one the equations derived by Herman and Schuler for diatomic molecules [13]. In this case, contact transformation is not needed and the Taylor expansions reduce to (numerical factors are incorporated into the constants in Herman and Schuler’s notation)

(8)

The Ry%y¦ i (x) terms are functions of the energy derivatives b and g. Herman and Schuler give the expressions for calculating these terms for fundamentals (actually, any transition y% −y%% =1), first overtones (y% −y%%= 2) and second overtones (y%− y%%= 3) [13]. Higher overtones demand higher-order Taylor expansions and perturbation treatments to be employed. Besides the Ry%y%% term, there is another important term in Eq. (1) which is affected by mechanical anharmonicity (derivatives b and g), the band origin frequency n y%y%%. Since its value may be very different from the harmonic frequency, it might determine the accuracy of the intensity estimation. For the frequency estimation, the dependence of the vibrational levels on anharmonic factors is usually expressed in the form [17] G(ya, yb, . . .) = % va (ya + 1/2) + % xab (ya + 1/2)(yb + 1/2) a5b

a

(9)

From this equation, expressions for the n y%y%% may be derived. The anharmonicity constants xab are related to the potential energy derivatives through [1]



8v 2a − 3v 2b 1 1 gaaaa − % b 2aab vb (4v 2a − v 2b) 16 16 b 1 1 b b xab = gaabb − % aac bbc 4 4 c vc

xaa =



1 − % b 2abcvc (v 2c − v 2a − v 2b)/Dabc 2 c

(10)

(11)

where Dabc = (va + vb + vc )(va − vb − vc ) (− va + vb + vc )(−va − vb + vc )

(12)

In Eq. (11) terms relative to Coriolis rotation– vibration interaction are not shown. For the eval-

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uation of the n y%y%% it would be simpler to neglect cross-terms, but even the single-mode anharmonicity constants xaa depend upon cross-terms baab, as can be seen in Eq. (10). It was decided to consider these terms for calculating the xaa but not for evaluating Ry%y%%, which would require the complete equations of Willetts et al. [8]. The constants xab are neglected. In this way, the following expressions were used for the band frequencies of each normal mode a: 6 10 = v+2x 6 20 = 2v+ 6x n 30 =3v+12x

(13)

2.2. Molecular orbital estimation of energy and dipole surfaces The procedure for obtaining the property derivatives starts with a geometry optimization of the molecule followed by a standard infrared harmonic spectra calculation, as implemented in Gaussian 98W [18]. In doing so, the Hessian matrix is calculated analytically through the procedure first suggested by Pulay [19] and (weighted for inverse nuclear masses) diagonalized, yielding harmonic frequencies and the normal coordinates (normalized to unity). For the proposed procedure, the correctly normalized normal coordinates, which relate coordinates in a.m.u.1/2A, to Cartesian coordinates in A, [20], must be obtained. This is easily done by the transformation [21]





Lij = cij / % mi c 2ij

1/2

(14)

i

where the cij are the normalized-to-unity elements of the eigenvectors and mi is the mass of the nucleus associated with Cartesian coordinate Xi. Since Gaussian 98W [18] presents the cij elements with just two significant figures, the Hessian matrix is required to be printed in the frequency calculation output and re-diagonalize independently, for better accuracy. Having established the normal coordinates, one can then attribute numerical values to just one of them and calculate the respective Cartesian displacements using the L matrix. For each molecular

geometry so obtained, a single point calculation is done which yields energy and dipole values. A regression of energy on a fourth-order polynomial in the normal coordinate and a regression of dipole moment on a third-order polynomial in the same coordinate are performed. The regression coefficients, which are derivatives, are transformed to the dimensionless normal coordinate space through the relation Qi = (h/4p 2cvi )1/2qi and the derivatives are applied in the Herman–Schuler equations [13]. For each of the three vibrational modes of H2O, six single point calculations were then performed for six distorted geometries. Although this number may seem too small, the results are of good quality, as will be seen. The maximum allowed distortion corresponded to stretching the bonds to 10% of their equilibrium values (for the stretching modes) or bending the HOH angle to 10% of its equilibrium value. For the anharmonic constants xii, estimation of energy cross-derivatives baab is required. In the energy regression for a normal coordinate a, the coefficient of the second-order term is the derivative k0,aa = 1/2(( 2E/(q 2a). In the general expansion of Eq. (5), the term baab means the derivative ( 3E/(q 2a(qb. So, if k0,aa is differentiated with respect to the normal coordinate b one should obtain 1/2(( 3E/(q 2a(qb )= 1/2baab. To differentiate k0,aa numerically, two energy regressions are made for coordinate a, one corresponding to coordinate qb set to zero and the other corresponding to a small value of qb. Since each normal coordinate must have its energy and dipole functions differentiated with respect to two other coordinates, two more regressions were done for each normal mode, which requires further 2 ×6= 18 single point calculations for each mode. The adimensional displacements of the other two normal coodinates that influenced the last two energy regressions corresponded to Cartesian mean displacements of 0.005 A, . For comparison with the results of Kjaergaard et al. [14] and Willetts et al. [8], the calculations were done with inclusion of electronic correlation at the MP2 level [22,23] and employed the 6311G* standard basis set and the 6-31G(extended) basis set of Willets et al. [8], in which polarization and diffuse functions are added to bias the calcula-

H.P. Martins, Filho / Spectrochimica Acta Part A 57 (2001) 83–93

tion in favor of dipole moment calculations. The polarization functions are d shells on oxygen (exponents 1.2 and 0.4) and p shells on hydrogen (exponents 0.75 and 0.25). The diffuse functions are an sp shell on oxygen (exponent 0.094) and an s shell on hydrogen (exponent 0.054). For mathematical procedures, Mathviews® [24] and Mathcad® [25] have been used. All codes were installed on a Pentium II 300 MHz microcomputer with 256 MB of RAM. Geometry optimization and harmonic spectrum calculation cost less than 5 min of CPU time and each single point calculation uses no more than 30 s of CPU time.

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taken as the sum of all observed rotational lines associated with that transition. In Tables 1–3 differences from the experimental values of Ref. [26] are also included for the theoretical estimates. Displacements of the antisymmetric stretching normal coordinate cause changes in the y and z components of the dipole moment (the other two normal coordinates only change the z component). In this case separate regressions for the y and z components are performed and the band intensities for this mode are calculated by separate computation of Ry%y¦ 2 for each component and summing the results.

3.1. Symmetric stretching mode (Table 1) 3. Results and discussion All energy and dipole moment regressions corresponding to Eqs. (6) and (7) had correlation coefficients better than 0.9999. The spectrum parameters derived from the regressions coefficients are shown in Tables 1 – 3, as well as the ones of Willetts et al. [8], Kjaergaard et al. [14] and experimental values. Willetts et al. [8] do not report calculated parameters relative to the overtones, so the comparison between the MP2/631G(extended) estimates and theirs will be based in fundamental frequencies and intensities. Since the spectrum calculation method of Kjaergaard et al. [14] takes the experimental frequencies as parameters, it is not an estimation method for the frequencies. So, the comparison between the MP2/6-311G** estimates and theirs will be based just in intensities, both fundamentals and overtones. The estimates for the overtone frequencies can only be compared with the experimental values. Since there are estimates for all parameters with both basis sets, the basis sets accuracies can also be compared. The experimental parameter values were chosen among the most recently reported [26– 30]. The most recent ones are taken from the HITRAN database [26], where the intensities are reported as oscillator strengths of rotational lines in highly resolved spectra. Though they are all given with three significant figures, their accuracies range from 6% for the most intense lines to 50% for the least intense ones [31]. The intensity of a given vibrational transition is

The analysis begins with the frequencies. The MP2/6-31G(extended) harmonic frequency is in very good agreement with the experimental value and should be equal to that of Willetts et al. [8], since they used finite-difference procedures also. The difference must be due to the numerical approximations in the procedures. The MP2/631G(extended) estimate for the fundamental frequency n 10 is also in very good agreement with the experimental value (less than a 2% difference). It differs from that of Willetts et al. [8] owing to neglect of the xij constants, of which x13 is the biggest (symmetric-antisymmetric stretching interaction). In this case, however, their neglect does not lead to significant error. The estimates for the two overtone frequencies are in good agreement with the experimental values. The largest relative error (2.6%) occurs for the second overtone, as could be expected. It is still a low figure, though it represents a 280 cm − 1 absolute difference from the experimental value. The MP2/6-311G** frequency estimates are much less accurate. The experimental fundamental intensity values for this band agree well with each other, the only exception being the most recent one of 3.00 km mol − 1. The MP2/6-31G(extended) double harmonic calculated value is almost 50% above this last experimental value. The anharmonic estimate is in good agreement with that of Willetts et al. [8], and both are close to the most recent experimental value, considering that the band is weak.

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The estimate of Kjaergaard et al. [14] is calculated from a theoretical MP2/6-311G** dipole expansion and parameterized wavefunctions based on Morse oscillators. These wavefunctions require experimental band origin values as parameters for their determination. Though an intensity estima-

tion partially based on experimental data could be expected to be more accurate, the estimate is not, being 40% above the experimental value. Since the MP2/6-311G** estimate is almost equal to theirs, the main source of error must be the quality of the MP2/6-311G** dipole surface.

Table 1 H2O spectrum parameters for the symmetric stretching mode (A1) calculated at MP2 level for 6-31G(extended) and 6-311G** basis sets and experimental values Harmonic frequency, anharmonic constant and band center frequencies (cm−1)

6-31G(extended) Willetts et al. [8] 6-311G** Experimental

v

x

n 10

Da

D (%)

3808.61 3815.3 3904.76 3832.0 [27]

−45.80 −43.07 −46.74 −42.58 [27]

3717.01 3638.8 3811.29 3657.053 [26]

60.00 −18.25 154.24

1.6 −0.5 4.2

n 20

Da

D (%)

n 30

Da

D (%)

7342.42 7529.09 7201.54

140.88 327.55

2.0 4.6

10876.20 11153.40 10599.686

276.51 553.71

2.6 5.2

Double harmonic

Anharmonic

Da

D (%)

4.43 4.54 6.19

3.46 3.26 4.13 4.19 3.00 2.24 2.18 2.21

0.46 0.26 1.13 1.19

First overtone

Da

D (%)

Second overtone

Da

D (%)

0.892 2.03 0.821 0.368

0.524 1.662 0.453

142 452 123

0.0319 0.0454 0.0139 0.0127

0.0192 0.0327 0.0012

O6ertone band center frequencies (cm−1)

6-31G(extended) 6-311G** Experimental [26]

Fundamental intensities (km mol−1)

6-31G(extended) Willetts et al. [8] 6-311G** Kjaergaard et al. [14] Experimental [26] Experimental [28] Experimental [29] Experimental [30]

15.3 8.7 37.7 39.7

O6ertone intensities (km mol−1)

6-31G(extended) 6-311G** Kjaergaard et al. [14] Experimental [26]

Transition matrix element partition (6 -31G(extended)) for the fundamental band (Debye)

This work Willetts et al. [8] a

Double harmonic

Anharmonic

−0.0216 −0.0217

2.40×10−3 2.86×10−3

Difference from experimental value of Ref. [26].

151 257 9.4

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89

Table 2 H2O spectrum parameters for the bending mode (A1) calculated at MP2 level for 6-31G(extended) and 6-311G** basis sets and experimental values Harmonic frequency, anharmonic constant and band center frequencies (cm−1)

6-31G(extended) Willetts et al. [8] 6-311G** Experimental

v

x

n 10

Da

D (%)

1633.22 1634.3 1667.51 1648.9 [27]

−14.01 −15.45 −14.36 −16.81 [27]

1605.20 1583.6 1638.79 1594.75 [26]

10.45 −11.2 44.04

0.7 −0.7 2.8

n 20

Da

D (%)

n 30

Da

D (%)

3182.38 3248.87 3151.630

30.7 97.24

1.0 3.1

4731.55 4830.22 4666.793

64.76 163.43

1.4 3.5

Double harmonic

Anharmonic

Da

D (%)

63.42 63.92 51.23

68.43 65.24 61.44 58.6 64.0 53.6 63.9 66.9

O6ertone band center frequencies (cm−1)

6-31G(extended) 6-311G** Experimental [26]

Fundamental intensities (km mol−1)

6-31G(extended) Willetts et al. [8] 6-311G** Kjaergaard et al. [14] Experimental [26] Experimental [28] Experimental [29] Experimental [30]

4.43 1.24 −2.56 −5.4

6.9 1.9 −4.0 −8.4

O6ertone intensities (km mol−1)

6-31G(extended) 6-311G** Kjaergaard et al. [14] Experimental [26]

First overtone

Da

D (%)

Second overtone

Da

D (%)

2.17 5.66 0.757 0.456

1.714 5.204 0.301

376 1141 66.0

1.95×10−4 2.12×10−3 2.36×10−3 2.38×10−3

−2.2×10−3 −2.6×10−4 −2.0×10−5

−91.8 −10.9 −0.8

Transition matrix element partition (6 -31G(extended)) for the fundamental band (Debye)

This work Willetts et al. [8] a

Double harmonic

Anharmonic

0.1245 0.1249

5.60×10−3 3.29×10−3

Difference from experimental value of Ref. [26].

The MP2/6-311G** estimate for the first overtone intensity can be seen to be one order of magnitude larger than the experimental value, while that for the second overtone is four times the experimental value. The estimates of Kjaer-

gaard et al. [14] are much better, though that for the first overtone is 120% above the experimental value. The high accuracy for the second overtone may be fortuitous, of course, for their estimation of the fundamental intensity (which in principle

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90

should be better) is very poor. The MP2/631G(extended) estimates are much closer to the experimental values, showing the strong influence of basis set in the results.

3.2. Bending mode (Table 2) The MP2/6-31G(extended) calculated harmonic frequency is in very good agreement with the

Table 3 H2O spectrum parameters for the antisymmetric stretching mode (B1) calculated at MP2 level for 6-31G(extended) and 6-311G** basis sets and experimental values Harmonic frequency, anharmonic constant and band center frequencies (cm−1)

6-31G(extended) Willetts et al. [8] 6−311G** Experimental

v

x

n 10

Da

3940.32 3947.2 4012.38 3942.5 [27]

−53.80 −47.93 −51.10 −47.56 [27]

3832.71 3758.9 3910.17 3755.930 [26]

n 20

Da

D (%)

n 30

Da

D (%)

7557.81 7718.14 7445.07

112.74 273.07

1.5 3.7

11 175.30 11 423.90 11 032.406

142.89 391.49

1.3 3.5

Double harmonic

Anharmonic

Da

D (%)

58.88 59.82 33.44

56.86 56.66 32.50 33.6 43.3 44.6 48.2 39.8

13.56 13.36 −10.8 −9.7

31.3 30.8 −24.9 −22.4

First overtone

Da

D (%)

Second overtone

Da

D (%)

0.488 0.119 0.0554 0.0220

0.466 0.097 0.0334

2118 441 152

0.0128 0.00843 0.0289 0.0115

0.0013 −0.0031 0.0174

11.3 −26.7 151

76.78 2.97 154.24

D (%) 2.0 0.1 4.1

O6ertone band center frequencies (cm−1)

6-31G(extended) 6-311G** Experimental [26]

Fundamental intensities (km mol−1) Fundamental

6-31G(extended) Willetts et al. [8] 6-311G** Kjaergaard et al. [14] Experimental [26] Experimental [28] Experimental [29] Experimental [30]

O6ertone intensities (km mol−1)

6-31G(extended) 6-311G** Kjaergaard et al. [14] Experimental [26]

Transition matrix element partition (6 -31G(extended)) for the fundamental band (Debye)

This work Willetts et al. [8] a

Double harmonic

Anharmonic

−0.0772 −0.0777

−6.91×10−4 2.12×10−4

Difference from experimental value of Ref. [26].

H.P. Martins, Filho / Spectrochimica Acta Part A 57 (2001) 83–93

experiment value, as is that of Willetts et al. [8]. The MP2/6-31G(extended) estimate for the fundamental frequency n 10 is also in very good agreement with the experimental value. The accuracy of this estimation is essencially the same of Ref. [8]. The MP2/6-31G(extended) estimates for the overtone frequencies are in better agreement with the experimental values than those for the symmetric stretching mode. The maximum relative error is only 1.4% and occurs again for the second overtone, as could be expected. The absolute difference in this case is 65 cm − 1. The MP2/6311G** estimates are less accurate. The different experimental values for the fundamental intensity are more disperse than those for the symmetric stretching mode. The most recent value, 64.0 km mol − 1 is very close to the mean value. This band is the strongest and all the theoretical estimates are reasonable, as can be seen in Table 2. For this mode the MP2/631G(extended) theoretical estimate is substantially different from that of Willetts et al. [8] (7 and 2% above the experimental values, respectively), which indicates the influence of cross-term derivatives for this mode. For the estimation of intensity A 10, both an estimate for n 10 and for R10 are required in Eq. (1), and it was seen that the estimate for n 10 is close to that of Willetts et al. [8]. Any difference in accuracy for A 10 must come from the estimate of R10. In Tables 1 – 3 a partition of R10 into a double harmonic term and an anharmonic term is also presented, corresponding to the first term in the summation of Eq. (8) and to the rest of the summation, respectively. The latter term is influenced by both energy and dipole moment derivatives. In Table 2, it is seen that the anharmonic term for R10 is larger than that of Willetts et al. [8], which shows the importance of cross-term dipole derivatives for this mode. Kjaergaard et al. [14] underestimate the fundamental intensity for this mode by 8% and the MP2/6311G** estimate is very close to theirs. The MP2/6-311G** estimate for the first overtone intensity is one order of magnitude larger than the experimental value while the estimate for the second overtone intensity is only 10% smaller than the experimental value, which must be fortuitous. The estimates of Kjaergaard et al. [14] are

91

much better, though the agreement for the first overtone is still not good. Their estimate for the second overtone intensity is almost exact, which is probably fortuitous considering the errors for other overtone intensity estimates.

3.3. Antisymmetric stretching mode (Table 3) The MP2/6-31G(extended) calculated harmonic frequency is in excellent agreement with the experimental value, as do that of Willetts et al. [8] too. The MP2/6-31G(extended) estimate for the fundamental frequency n 10 is in reasonable agreement with the experimental value, but that of Willetts et al. [8] is almost exact, which indicates a considerable influence of the cross-term energy derivatives. The accuracies of the MP2/6-31G(extended) overtone frequency estimates are very good, the maximum relative error being a 1.5% difference for the first overtone frequency. This last figure corresponds to a 113 cm − 1 absolute difference from the experimental value. Again, the MP2/6-311G** frequency estimates are less accurate. The different experimental values for the fundamental intensity are somewhat disperse, but the value measured most recently, 43.3 km mol − 1 is very close to the mean value. All four theoretical estimates are poor. The MP2/6-31G(extended) theoretical estimate is almost exactly the same as that of Willetts et al. [8], which means that neglect of cross-term derivatives is not significant for this mode. Kjaergaard et al. [14] again underestimate the fundamental intensity and the MP2/6-311G** estimate is very close to theirs. Despite the underestimation, both the MP2/6-311G** estimates are as far from the experimental value as those for MP2/6-31G(extended) calculations. The estimation of overtone intensities is again very poor. The estimates of Kjaergaard et al. [14] for both overtones are 150% above the experimental values. The MP2/6-311G** theoretical value for the first overtone is 5.5 times the experimental value and that for the second overtone is in good agreement with the experimental value (only 27% smaller). This last figure may be fortuitous, of course. For this mode, the experimental relative intensities are atypical, since the intensity of the first overtone is three orders of magnitude smaller than that of the fundamental and the

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intensity of the second overtone is of the same magnitude of the first overtone.

4. Conclusions The frequency estimation at the MP2/631G(extended) level by this procedure is satisfying both for fundamentals and overtones, but this requires consideration of cross terms baab. If estimates for the fundamental frequencies within 2% of the experimental values are required, the neglect of the other energy cross terms is not significant. The basis set influence, however, may be determinant, since the errors for the MP2/6311G** estimates in Tables 2 and 3 may reach 5.2% (second overtone of the symmetric stretching mode). For fundamental intensity estimation, one should expect the neglect of dipole derivatives to cause slight deviations from the results of complete treatments. However, the influence of basis set is much more important to the accuracy of the estimates. The MP2/6-311G** estimates for the overtone intensities vary greatly in accuracy. Though there are estimates close to the experimental value (e.g. for the second overtone of the bending mode), half of them are one order of magnitude above the experimental values. The results for the 631G(extended) basis sets are better, but the remarks above still apply. The estimates of Kjaergaard et al. [14] are much better, being consistently in the same order of magnitude as the experimental values. Discrepancies from these last may still reach 150%, however. This shows that cross-term dipole derivatives must be very important in the calculation of overtone intensities. Indeed, Kjaergaard et al. [14] take into account these derivatives, having performed single point calculations for a grid of 76 geometry points in order to determine their dipole surface, whereas just 18 calculations were needed for the simplified estimates in this work (the dipole information of the additional 36 calculations was not used for the dipole function determinations). Considering its simplicity, the proposed methodology for spectra estimation seems useful

to spectra estimation in which semiquantitative accuracy for frequency estimation and qualitative accuracy for fundamental intensity estimation are required. This accuracy must be tested for different levels of ab initio theory and larger wavefunctions.

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

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