Calculation of the vibrational frequency and line strength versus applied field of carbon monoxide

Chemical Physics 15 1 ( 1991) 37-43 North-Holland

Calculation of the vibrational frequency and line strength versus applied field of carbon monoxide * Jo& Luis And&s, Miquel Duran, Agusti Lled6s and Juan Bert& Departament de Quimica. Universitat Autbnoma de Barcelona, 08193 Bellaterra, Catalonia, Spain Received 2 May 1990, in final form 2 1 October 1990

Analytic determination of dipole moment derivatives with respect to nuclear displacements (and hence infrared intensities) are discussed for closed-shell wave functions under the influence of an external uniform electric field. The method is applied to the ground state of the CG molectde. The calculated vibrational Stark tuning rate and the effect of the applied field on the line intensity are compared with experimental data.

1. Iatroductlon The effects of uniform electric fields on infrared spectra have been determined using in situ vibrational spectroscopy [ l-31. Likewise, frequency shifts in vibrational frequencies of multiple-band molecules have been observed [ 41 by applying strong fields in special infrared cells. Frequency shifts in vibrational modes of aromatic molecules have been detected [ 51 using Raman spectroscopy. Changes caused by an applied electric field in the vibrational frequency of adsorbed carbon monoxide [ 6 ] and the azide ion [ 7 ] have also been measured. Finally, band shifts and line splittings have been observed [ 8 ] for methane adsorbed in zeolite at trapping sites. Theoretical studies of the changes in vibrational frequencies of molecules in a uniform electric field have been carried out using a variety of methods [ 9 1. On one hand, one can solve the field-dependent Hamiltonian for nuclear motion [ lo] and obtain vibrational frequency vs. electric field. On the other hand, one can include the electron-field interaction term [ 111 in the Hartree-Fock one-electron Hamiltonian, and obtain fielddependent harmonic frequencies through diagonalization of the mass-weighted Cartesian second derivative matrix. * A contribution from the “Grup de Quimica Quitmica de Catahmya de I’Institutd’Estudis Catalans”.

Change in the intensity of the vibrational line caused by the application of an electric field [ l-3,1 2 ] has also been observed experimentally. The variation of the infrared cross section of oriented CO with applied field has been reported by Kunimatsu [ 12 1. Intensities of adsorbate infrared-active modes may be modified [ 1 ] by altering the electric field across the electrochemical double layer. Lambert [ 21 has found that the line intensity decreases with increasing magnitude of the field in CO adsorbed on Ni( 100) and in CO adsorbed in the electrochemical double layer. Beden et al. [ 3 ] have also found modifications in signal intensities upon application of various amplitudes of potential in an electrochemically modulated infrared reflectance spectroscopy (EMIRS) study of electrosorbed formic acid at a platinum electrode. The appearance of forbidden bands, i.e., the increase in line intensities, has been found for the field-induced spectrum of methane [ 8 ] adsorbed in zeolites. From a theoretical point of view, the intensity of adsorption of an infrared spectrum is obtained from the derivative of the molecular dipole moment. In a harmonic oscillator model [ 9 ] the integrated adsorption coefficient of a normal mode depends on the square of the dipole moment derivative with respect to the nuclear motion in the normal mode. In recent years there has been an increasing number of analytic calculations of dipole moment derivatives to compute infrared intensities (see ref. [ 131

0301-0104/91/S 03.50 0 1991- Elsevier Science Publishers B.V. (North-Holland)

38

J.L. Andres ei al. / Vibrationalfrequency and bnestrength vs.field of CO

and references therein), which are outnumbered by calculations using numerical differentiation of the dipole moment by finite differences. The theory of analytic determination of ap/ilR has been extended [ 9,14- 19 ] from Hartree-Fock to MC-SCF and other post-SCF wavefunctions. It is obvious that the dipole moment of a field-free molecule changes [ 91 when an external electric field is applied; the electron density relaxes and the geometry changes. For instance, the dipole moment of the methane molecule is no longer zero when a uniform electric field is applied, because its geometry changes [ 111 form T,, symmetry to C,, symmetry. In this paper we discuss the computation of analytic dipole moment derivatives of a Hartree-Fock wavefunction under the effect of an external uniform electric field. This wavefunction, despite its simplicity, gives quite good molecular properties. The ultimate purpose of the present paper is to obtain the band intensity vs. field strength. The carbon monoxide molecule has been chosen because, to our knowledge, it is the only system for which quantitative experimental data for changes in infrared intensities under external uniform electric fields are available, and because it is the system where the vibrational Stark effect has been studied most. We shall next discuss the computational approach and the theoretical model, followed by its application to the carbon monoxide (CO) molecule in an electric field. The band shifts and band intensity changes will be discussed in the third section.

2. Computational approach The intensity of an infrared band of a field-free chemical system corresponding to a normal mode Qi, within the so-called double harmonic approximation, is given [ 91 by A -‘-

NAd,

-ap

2

12trJcZ(1 aQ,o'

(1)

where N* is the Avogadro number, d, is the degeneracy of the normal mode, co is the dielectric constant, and c is the speed of light. The only remaining term in eq. ( 1) to be determined is the derivative of the unperturbed dipole moment with respect to the nu-

clear displacements along the normal mode (ap/ aQi)o. Theoretical ab initio methods can furnish such an information. An applied electric field can give an induced infrared spectrum (band intensity changes and new bands appear), because the dipole moment derivatives are modified by the applied field, i.e., the intensities of the induced spectrum are proportional to the square of (ap/aQ,),, where F stands for the strength of the applied field. The dipole moment can be split into a nuclear part and an electronic part. The nuclear part follows classical physics and carries no difficulty. For the electronic part, we have followed the formulation of Yamaguchi et al. [ 141 to compute analytically the dipole moment derivatives with respect to nuclear displacements, which are equivalent to computing the crossed second derivatives of the energy with respect to a nuclear displacement and an electric field. We will hereafter summarize the main changes required in tieldfree expressions of analytic second derivatives to take into account the effect of the electric field in closedshell RHF wavefunctions. Extension to UHF and general ROHF wavefunctions is straightforward. The effect of a uniform electric field on a SCF wavefunction is taken into account in the one-electron Hamiltonian (electron-field interaction) and in the nuclear repulsion term (nucleus-field interaction). Thus, the expression of the one-electron Hamiltonian turns out to be [ 9,111 h*=T+

V-F*(r-C)=h-F*(r-C)

)

(2)

where F is the uniform electric field, r is the vector of electron coordinates, and C is the origin for dipole moment integrals, which is usually taken to be the origin of coordinates. With respect to a field-free SCF calculation, this perturbed Hamiltonian gives rise to new orbital energies and molecular orbitals, thus changing the electron density. The analytic expression for the electronic part of dipole moment derivatives with respect to a nuclear displacement (a) is equivalent [ 141 to the second derivative of the energy with respect to a field f (at zero intensity) and a nuclear displacement (a): --a a&cF =2 C h:f+4 da

afaa

da

,

d.o.

-2 1 elms:-4 I

all

C 1 Vie; J

/

d.o. all

1 c e,u~S;. 1

J

(3)

J.L. Andreset al. / Vibmtionaljkquenc.y and line strength vs.j?eld ofC0

For the meaning of the different terms, see ref. [ 141. One must distinguish between the fieldf, which is only formally used (at strength zero) to compute field-free dipole moment derivatives, and the external unifo~ field F, which will be introduced shortly after. A formula equivalent to equation (3) is da --4 ~ESCF =2 c /$I-4

af aa

,

da. all

c c U$h$.

(4)

, j

Both formulae yield the same results. However, from a practical point, eq. (3) requires the terms U$ (derivatives of M.O. coefficients Cwith respect to a field j) , that are found by solving the CPHF (coupled-perturbed Hartree-Fock) equations, which for a closedshell wavefunction are

(51 where ej and ej are orbital energies, AQ,w depends oniy on meld-independent two-electron terms, and

In contrast, eq. (4) requires the solution of the CPHF equations for the U$ terms (derivatives of M.O. coefficients C with respect to nuclear displacements a ) , that are obtained by solving the CPHF equations (eq. 5, with f replaced by a), where B&, contains contributions from several one- and two-electron terms(seeeqs.(5)and(6)ofref. [ll]) B&=

C C&, zlu

“‘$t *) + other terms

,

(7)

where h is the field-free one-electron Hamiltonian operator. Although the first formula (eq. 3) requires the solution of only three sets of linear equations because an electric field has three components, the second formula (eq. 4) requires the solution of 3N sets of equations (three for each atom). However, the second expression is more compact. Moreover, dipole moment derivatives (i.e., infrared intensities) are computed usually after second derivatives with respect to nuclear displacements (harmonic frequencies) have been calculated, which require themselves the calculation of U$, so the second formula can make use of their availability. When an external uniform electric field F is intro-

39

duced, some changes must be made in formulae (3) and (4) to account for the perturbing field. First, a new SCF calculation must be carried out including the effect of the field. Further, in eq. (2 ), the Fg term ~rn~F~,~u~the~o~~tor~rn~~* (eq. (2). Thus, this term must also include the derivatives of the dipole moment integrals. Moreover, the U$ term in eq. (4 ) must have been computed using the modified CPHF equations, with the B$,$ term, which is similar to that of eq. (7) with the inclusion of the derivatives of dipole moment integrals appearingin c?/da(plh*l v). In the end, the required changes in formulae for analytic dipole moment derivatives when an external uniform electric field is applied result in changing F$ and B& The results reported in the present paper have been computed after these changes were made in our code [ 111 for analytic energy second derivatives with respect to nuclear displacements. The results computed anal~i~lly have been checked against finite differences of dipole moments, and no significant differences have been found.

3. Results and discussion In this section ~~u~atio~ of the infrared spectrum of carbon monoxide is discussed and compared with previous theoretical and experimental data. First we focus on frequencies; second, we deal with intensities; finally, a short discussion of the basis sets used is given. The present study uses two basis sets for the carbon monoxide molecule: a double-{quality basis set of H~na~-Dunning [ 20,2 1) (labelled DZ) and the same basis set supplemented by d functions on each atom, with Gaussian exponents (w,=O.75 and (Y,,= 0.85 ( labelled DZP ). The calculations were carried out at the one-configuration SCF level of theory. This level does not give the correct sign of the dipole moment for the field-free CO molecule, but results below will show this level to be adequate for studying changes in molecular properties caused by a uniform electric field. Tables 1 (DZ basis set) and 2 (DZP basis set) present the CO bond lengths, dipole moments and total energies for several strengths of the uniform electric field. Regarding the definition of the applied

40

J.L. Andres et al. / Vibrational frequency and lrne strength vs. field of CO

Table I Bond length R (in A), dipole moment p (in D), total energy E scF (in hartree), harmonic frequency u (in cm-‘), dipole moment derivative (in D/A), and infrared intensity I (in km/mol) for the CO moleculae under an applied electric field of intensity F (in au, I au=5 14x IO” V/m). These calculations have been carried out with the basis set labelled DZ. A positive dipole moment means the polarity C-O+ F

R

P

&CF

”

-@la

I

- 0.04 -0.03 -0.02

1.161 1.154 1.148

- 1.910 - 1.546 -1.185

- 112.70396 - 112.69716 -112.69179

2080 2134 2183

7.22 6.76 6.29

322 281 243

-0.01 0.00 0.01 0.02 0.03 0.04

1.143 1.138 1.134 1.130 1.127 1.124

-0.828 -0.472 -0.116 0.239 0.596 0.955

-

2226 2265 2299 2329 2355 2377

5.81 5.33 4.84 4.34 3.83 3.30

208 175 144 116 90 67

112.68783 112.68528 112.68412 I 12.68436 112.68601 112.68906

Table 2 Bond length R (in A), dipole moment p (in D), total energy E scF (in hartree), harmonic frequency Y (in cm-’ ), dipole moment derivative (in D/A), and infrared intensity I (in km/mol) for the CO moleculae under an applied electric field on intensity F (in au, 1 au= 5.14~ IO” V/m). These calculations have been carried out with the basis set labelled DZP. A positive dipole moment means the polarity C-O+ F

R

P

&CF

v

-dMR

I

-0.04

1.136 1.131 1.126 1.121 1.117 1.114 1.111 1.108 1.106

- 1.545 - 1.201 -0.860 -0.519 -0.179 0.161 0.503 0.847 1.193

-

2260 2307 2350 2389 2424 2455 2483 2506 2526

6.96 6.52 6.08 5.62 5.16 4.69 4.21 3.72 3.21

298 262 228 195 164 136 109 85 64

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04

field, a negative field is defined as that increasing the C+O- character, whereas a positive field is defined as that increasing the C-O+ character. A positive dipole moment is defined for the C-O+ polarity. Moreover, the dipole moment derivatives with respect to the C-O bond length are reported. Finally, the fundamental C-O vibrational frequency along with its intensity are given. As has been pointed out by Pancir, the number of null vibrational frequencies of a molecule is decreased [ 221 by two when an electric field is introduced. In other words, there is a loss of rotational invariance about the two axes perpendicular to the electric field. Therefore, for a linear molecule like CO, the five null frequencies in absence of any field are reduced to three when a field is applied, and two new lines are created by the perturba-

112.77292 112.76752 112.76347 112.76076 112.75938 112.75935 112.76065 112.76331 112.76732

tion. For the CO molecule, the intensity of such lines is negligible, so we do not report their frequencies, which never exceed 100 cm- ’ . However, preliminary calculations for other systems have shown that nonnegligible intensities may be found. We shall deal with this point again later. It is interesting here to compare our field-free results with those arising from other studies. Previous theoretical studies on the effect of a uniform electric field on field-free CO frequencies have been carried out by Bauschlicher [ 231 and Lambert [ 241. In turn, a basis set dependence study of the infrared spectra of field-free CO has been carried out by Yamaguchi et al. [ 251. This last study found that with no applied field the DZ bond length is 1.138 A and the DZP one is 1.117 A, as compared to the experimental value of

J.L. Andres et al. / Vibrational frequency and line strength vs.field of CO

1.128 A. The harmonic vibrational frequency is 2265 cm- ’ (DZ) and 2424 cm-’ (DZP) compared to the experimental of 2143 cm-‘. (For experimental values of parameters, see references in refs. [ 141 and [ 231.) Likewise, IR intensities computed with both basis sets are also larger than found experimentally, and both basis sets yield incorrect dipole moments. It is worth mentioning that high-level correlated methods have been very recently applied [ 26-281 to free carbon monoxide. Finally, for field-free CO, the large basis set CASSCF study by Bauschlicher [ 23 ] found 1.138 A, 2 147 cm-’ and 0.112 D (compared to the experimental value of 0.1222 D ) . This is a very good quality calculation that can be used as a benchmark for these quantities. Let us consider our field-induced results. In general, a free molecule aligns along an electric field in such a way that its dipole moment is increased. For carbon monoxide, experience shows that there is a shift in electron charge from oxygen to carbon, so correct alignment of this molecule corresponds to positive values of the field. The stability of the alignment is related to the non-existence of imaginary frequencies [ 111. Only if all frequencies are real, the system is stable under rotation about the axes perpendicular to the field. In fact, for CO we calculated that for negative fields the new non-zero frequencies are always real, whereas for positive fields they are imaginary for the 0.01 and 0.02 (DZ) and 0.01 (DZP) strengths and real otherwise. The switch from imaginary to real frequencies for positive fields means that application of a field improves the adequacy of the one-configuration treatment. With regards to the behavior of bond lengths with respect to the strength of the applied field, their trend is in qualitative agreement with theoretical calculations of Bauschlicher [ 231 and other calculations involving adsorbed CO. The first author found a shortening of the CO bond length upon increase of the field strength. On the other hand, Mehandru [ 291 found with the ASED-MO theory that when a negative potential is applied to a C-adsorbed CO molecule, the CO bond lengthens, and shortens if it is positive. This is in good agreement with our results for positive fields. From a quantitative point of view, the changes in our computed bond lengths are larger than those computed by Bauschlicher: he found that the CO bond length shortens only by 0.00 1 A when a field of

41

strength 0.01 au is applied, whereas our DZP calculations for positive fields predict such shortening to be 0.004 A. Regarding the harmonic vibrational frequencies of the carbon-oxygen bond, tables 1 and 2 show that for negative fields they decrease, whereas for positive fields they increase. This is in qualitative agreement with previous results [ 231 found with non-analytical methods. It is especially interesting to investigate the Stark tuning rate &, i.e., the derivative of the frequency with respect to the electric field at strength zero. Our results for the DZ basis set (positive fields) yield 8,=5.43x 1O-7 cm-‘/(V/cm), whereas for the DZP basis set they yield &=4.94X 10m7cm-‘/ (V/cm) (in the interval from Fz0.00 to F=0.04 au). Larger values are obtained if a smaller interval is taken. A very recent experimental value for this quantity [2] is &=5.09f 1.00x 10m7 and 4.94x 10m7 cm-‘/ (V/cm). Moreover, Bauschlicher’s [ 231 theoretical calculation predicts gvEto be 3.63.8x 10e7 cm-‘/(V/cm). Previous works (see ref. [2] ) give similar values. Therefore, although the wavefunction used in this study gives the wrong sign of the dipole moment of field-free CO, it reproduces quite well the experimental Stark tuning rate. The shifts in vibrational frequencies are actually not exactly linear [ 1,231 with the strength of the applied field. Let us turn to the line intensity change upon introduction of the field. An interesting discussion on this subject about theoretical and experimental data is found in ref. [ 11. The available experimental data of Lambert [2] estimate that the intensity of the line decreases upon increase of the magnitude of the electric field applied. Likewise, the EMIRS study of Beden et al. [ 3 ] also shows a decrease in the signal amplitudes with the increase in applied potential. Therefore, there is a good qualitative agreement with our results, since for both basis sets we have computed that the line intensities will decrease when the field strength is increased (positive fields). The changes in the intensity of IR bands [9] are intimately related with the square of the derivative of the dipole moment. The agreement between the field-free dipole moment derivative computed with the DZP basis set and the experimental value (3.093 D/A) is not very good. However, it is still very interesting to compare how the line intensity varies with

42

J.L. Andres et al. / Vibrational frequency and hne strength vs. field of CO

the applied field. Although the change in dipole moment with the strength of the applied field is almost linear, the variation in intensity of the C-O stretching band is not. This situation is similar to the nonlinearity of the variation in frequency. This notwithstanding, Lambert [ 21 determines experimentally the fractional change in IR cross section gsE to be =-5.5+5.8x10-9cm/V.Fromthedataoftable2 (DZP basis set, positive fields) and considering the fractional change between F=O.OOand F=0.04 au, the effect of the field strength on the line intensity is 6SE= -4.577xlop9 cm/V, which is well within the range given by Lambert. The same calculation starting from the data of table 1 (DZ basis set) yields almost the same value, &=4.670X 10u9 cm/V. The difference between the results calculated with each basis set can be ‘attributed to the different equilibrium bond lengths calculated for each basis set, which turns into a fair difference in harmonic frequency. Anyway, it must be remarked that the uncertainty of Lambert’s measurement is quite large. Comparison of the calculated values with the experimentally observed effect in electrochemical experiment gives good agreement [ 30,3 11. However, the cause of the experimentally observed change of line strength with electric potential is not clear [ 30331, because bonding of CO to the metal may change, and because CO may shift between different sites. The effect calculated in the present paper is largely due to a change in ap/aRat fixed R,and not to change in R,.This is supported by the calculated value of the dipole moment derivative at the field-free equilibrium bond length, which has a value of 6.04 Debye/ A if a field of 0.02 au is applied. This value is much larger than the field-free value of 5.16 Debye/A. Likewise, at the equilibrium bond length when a field of 0.02 au is applied, the field-free and F=0.02au values are 5.19 and 6.08 D/A, respectively. Thus, the dipole moment derivative is not altered much by the change in bond length, but is modified to a large extent by application of the field. A deeper insight is needed here on this effect: in eq. ( 18 ) of ref. [ 21, it is stated that s, E=-

+L, a20

(8)

where gSEis the fractional change in IR cross section, and cl*, and u2,, are terms of a double Taylor series

expansion of the potential energy, the first index referring to normal mode displacements from equilibrium Q, and the second index referring to electric field F.Using a,, and a2o determined from rotational-vibrational spectra of gas phase CO, ref. [ 2 ] predicts 8sE= 1.3X lo-” cm-‘/V, which is lower by more than two orders of magnitude than the limit set by the experiment. In the derivation of eq. ( 18) of ref. [ 21, crossed terms in E and Q were omitted from the derivation of the dipole moment formula. If inclusion of such terms is made, a formula similar to eq. ( 18) of ref. [ 21 is obtained: 6

=zd4+w!!2

SE qaF (

aQaF’ >

(9)

where q is the dipole moment derivative with respect to bond length. If the coefficients of the Taylor expansion are used, one obtains s,,=-2(2-22). With respect to eq. (8)) this formula includes the additional term -2alz/a, ,, where aI2 stands for the polarizability derivative, i.e., the change of dipole moment derivative with the applied field. As seen above, the change in ap/aQ with applied field is very important, so the second term in the last equation should not be disregarded. This may be the origin of the low value obtained for Ss, in ref. [ 21. It is interesting to note here that the values of the dipole moment derivatives computed under the influence of a positive and negative uniform electric field cannot be used to calculate polarizability derivatives with respect to nuclear coordinates, as opposed to the finite differences method suggested by McIver and Komornicki [ 341. This approach requires that no geometry reoptimization be performed, whereas in the present work optimal geometries have been recomputed for each value of the intensity of the applied field. The same thing can be said for the finite difference approach [ 341 to compute dipole moment derivatives and dipole moments. A last point to be addressed concerns the basis sets used in this work. In a previous paragraph, we have already carried out a discussion on the character of the two new non-zero frequencies for each basis set. Furthermore, inspection of tables 1 and 2 shows that

J.L. Andres et al. / Vibrationalfreguency

for positive fields, where CO is correctly aligned with respect to the electric field, line intensities decrease together with a huge increase in dipole moment with the correct sign. However, for negative fields line intensities increase together with an increase of dipole moment in the wrong direction. It is worth pointing out that the DZP basis set gives a much smaller dipole moment for free CO than the DZ basis set. In fact, introduction of a field of strength 0.01 au already changes the sign of the dipole moment, whereas its sign for the DZ basis set is changed when the strength of the applied field reaches a value of 0.02 au. All in all, the polarized basis set behaves much better than the non-polarized basis set. Really, the double-c plus polarization basis sets have been shown [ 271 to provide useful information for qualitative classification of infrared intensities of field-free molecules, in particular in carbon monoxide. 4. Conclusions We have shown in this paper that the analytic computation of the line intensities of field-perturbed spectra is feasible. A Stark tuning rate very close to experiment has been obtained, whereas changes in line intensities follow the experimental trend, and relative changes agree almost quantitatively with experiment. We have shown that the Stark tuning rate and the infrared cross section change do not vary very much with the basis set size. However, electron correlation may be important in determining induced spectra. For that purpose, calculations are under way to account for the effect of electron correlation. Furthermore, extension to polyatomic molecules is also being carried out. These new studies will be the subject of further reports. Acknowledgement

This project has been supported by the Spanish Direcci6n General de Investigaci6n Cientifica y Ttcnica under Project No. PB86-0529. References [ I] K. Ashley and S. Pons, Chem. Rev. 88 (1988) 673. [2] D.K. Lambert, J. Chem. Phys. 89 ( 1988) 3847. [3] B. Beden, A. Bewick and C. Lamy, J. Electroanal. Chem. 148 (1983) 147.

and line strength vs. field of CO

43

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