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Next: Summary and Conclusions Up: Implementation of DFPT Algorithm Previous: Molecular Polarisabilities   Contents

IR Spectroscopy

Thus far, the calculations have indicated that DFPT can reliably and accurately determine the normal mode frequencies and dielectric properties of crystalline and molecular systems. However, if one is to use it to obtain IR spectra, then the accuracy and reliability of the method must be investigated. To this end, in this section, the spectra of a series of representative molecules is obtained and compared to previous calculations and measurements.

The IR spectrum for methane is shown in figure 4.15. In table 4.9, the normal mode frequencies and the corresponding relative IR intensities are presented.

Figure 4.15: IR spectrum of methane as obtained using DFPT. A Gaussian broadening has been applied, as in all the subsequent spectra presented.
\includegraphics[scale=0.4,angle=-90]{methane_IR_intensity.ps}


Table 4.9: IR intensities for methane. The values given are relative intensities, with the most intense assigned the value `1'.
  $\omega $ ( $\mbox{cm}^{-1}$)
Mode 1$T_{2}$ 2$T_{2}$
Present Work 1306 3112
Porezag and Pederson [103] 1283 3090
Experiment [103] 1357 3158
  $I^{IR}$
Mode 1$T_{2}$ 2$T_{2}$
Present Work 1 0.69
Porezag and Pederson [103] 0.70 1
Experiment [103] 0.57 1


The normal modes are in better agreement with experiment than the results of Porezag and Pederson [103], using a Gaussian basis set. The ratio of intensities between the two theoretical results shows good agreement, and these are in reasonable agreement with the experimental values. One noteworthy feature is that the calculations presented in this work disagree over the ordering of the IR intensities for the two IR-active modes. Porezag and Pederson attribute this to a systematic error in the LDA; however, their results are obtained within the GGA using localised basis sets. This makes this contention difficult to support, and suggests that instead, it may be related to the basis set used, and some fortuitous cancellation of errors. Such a contention is supported by the work of Yamaguchi et al. [161] who have demonstrated that within Hartree-Fock theory, the ordering of the peaks may alter: for example, the minimal STO-3G basis set predicts that the ratio of intensities between the 1$T_{2}$ and 2$T_{2}$ peaks is 0.0699:1; the 6-311 + + G(3d,3p) basis set, conversely, predicts a reversal of the ordering, with the intensity ratio being 1:0.249. it can therefore be seen that the inclusion of polarisation effects in the basis set can radically alter the intensities and ordering, even within the same ab initio method.

In table 4.10 the normal modes and relative IR intensities of ethane are presented. The agreement can be seen to be mixed; although the normal mode frequencies are, on the whole, determined accurately, the relative intensities are not. All the values agree that the $3E_{u}$ mode is the most intense, although in some cases the intensities calculated in this work disagree by an order of magnitude. The full spectrum is shown in figure 4.16.


Table 4.10: Wavenumbers of IR-active modes and relative intensities for ethane. The values given are relative intensities, with the most intense assigned the value `1'.
Mode $\omega $ ( $\mbox{cm}^{-1}$) $I^{IR}$
  Present Ref. [103] Experiment CASTEP Ref. [103] Experiment
$1A_{1u}$ 309 297 303 0.0 0.0  
1$E_{u}$ 821 800 822 0.195 0.074 0.050
1$A_{1g}$ 1003 998 1016 0.0 0.0  
1$E_{g}$ 1197 1177 1246 0.0 0.0  
1$A_{2u}$ 1384 1354 1438 0.002 0.012 0.020
2$A_{1g}$ 1390 1361 1449 0.0 0.0  
2$E_{u}$ 1472 1456 1526 0.016 0.166 0.125
2$E_{g}$ 1475 1457 1552 0.0 0.0  
2$A_{2u}$ 2988 2973 3061 0.271 0.518 0.411
3$A_{1g}$ 2985 2969 3043 0.0 0.0  
3$E_{g}$ 3043 3022 3175 0.0 0.0  
3$E_{u}$ 3067 3055 3140 1.000 1.000 1.000


Figure 4.16: IR spectrum of ethane as obtained using DFPT.
\includegraphics[scale=0.4,angle=-90]{ethane_IR_intensity.ps}

In table 4.11, the relative IR intensities of water are presented. The full spectrum is given in figure 4.17. Examining the relative intensities, it is apparent that, except for the HF/STO-3G results, the ab initio methods agree with the experimental ordering of the peaks. This reinforces the point made with regard to methane concerning the effects of the basis set: the failure to have a sufficiently large basis set may cause incorrect ordering and/or intensities, irrespective of the methods employed. The results also suggest that the effects of correlation are not significant upon determining the relative intensities, at least for water. An important feature to note is that the different methods do not predict that the peaks occur at the same frequencies; the values for the frequencies in table 4.11 are those obtained in this work. Note that in all cases the theoretical methods (save HF/STO-3G) display a tendency to overestimate the intensities of the IR peaks.


Table 4.11: IR intensities for water. The values given are relative intensities, with the most intense assigned the value `1'. Theoretical and experimental values are from Yamaguchi et al. [161]. Frequencies are as obtained in this work.
$\nu $ Relative Intensities
(cm$^{-1}$) Present HF HF MP2 Expt.
    STO-3G 6-311++G(3d,3p) 6-31+G(d)  
1598 1 0.163 1 1 1
3773 0.103 1 0.159 0.110 0.0371-0.0506
3883 0.904 0.677 0.945 0.664 0.860-0.891


Figure 4.17: IR spectrum of water as obtained using DFPT.
\includegraphics[scale=0.4,angle=-90]{water_IR_graph.ps}

In table 4.12 the relative IR intensities of ammonia are presented. The full spectrum is shown in figure 4.18. Again, Hartree-Fock theory with the STO-3G basis set fails to predict the correct ordering, in addition to predicting inaccurate relative intensities. The DFPT results obtained in this work are in broad agreement with the experimental spectrum, save for a slight overestimation of the intensity of the peak at 1627 $\mbox{cm}^{-1}$, and a large overestimation occurring at 3532 $\mbox{cm}^{-1}$. It does however, in common with the HF/6-311++G(3d,3p) and MP2/6-31+G(d) results calculate that this peak should be larger than the peak at 3401 $\mbox{cm}^{-1}$ by an order of magnitude; this is in contrast to the experimental results, which suggest that the ordering of these two peaks should be reversed, with comparable intensities.


Table 4.12: IR intensities for ammonia. The values given are relative intensities, with the most intense assigned the value `1'. Theoretical and experimental values are from Yamaguchi et al. [161]. Frequencies are as obtained in this work.
$\nu $ Relative Intensities
(cm$^{-1}$) Present HF HF MP2 Expt.
    STO-3G 6-311++G(3d,3p) 6-31+G(d)  
979 1 1 1 1 1
1627 0.262 0.041 0.203 0.263 0.088-0.232
3401 0.031 0.099 0.008 0.003 0.028-0.034
3532 0.124 0.124 0.062 0.057 0.018-0.022


Figure 4.18: IR spectrum of ammonia as obtained using DFPT.
\includegraphics[scale=0.4,angle=-90]{ammonia_IR_graph.ps}

In table 4.13 the relative intensities of the IR peaks of hydrazoic acid are presented. The full spectrum is presented in figure 4.19. The calculations of Shen and Durig [162] at the MP2/6-31G* level of theory disagree with the other calculations presented as to the ordering of the IR peaks; these authors introduce a scaling factor in order to compensate and restore the ``correct'' ordering. Again, these results illustrate the importance of choice of basis set. The present work displays a tendency to underestimate the intensities of the peaks compared to the other methods used. The B3LYP results are of interest, as these are obtained within a density functional framework; the intensity of the the $\mbox{NH}_{3}$ stretch observed at 3444 $\mbox{cm}^{-1}$ in this work is grossly underestimated, whilst, in contrast, the intensities of the other peaks are overestimated in comparison with the results in this work. This is a further indication of the effects of choice of basis set upon IR intensities.


Table 4.13: IR intensities for hydrazoic acid. The values given are relative intensities, with the most intense assigned the value `1'. Theoretical and experimental values are from Shen and Durig [162]. The MP2/6-31G* results quoted from this work are the unscaled values. Frequencies are as obtained in this work.
$\nu $ Relative Intensities
(cm$^{-1}$) Present MP2/6-31G* MP2/6-311+G** B3LYP/6-31G*
3444 0.210 0.301 0.292 0.072
2174 1 0.891 1 1
1133 0.529 1 0.765 0.676
567 0.022 0.125 0.102 0.048


Figure 4.19: IR spectrum of hydrazoic acid as obtained using DFPT.
\includegraphics[scale=0.4,angle=-90]{HN3_IR_graph.ps}

A more complicated system is benzene. In table 4.14 the normal modes of benzene are presented. Good agreement with experiment and other theoretical calculations is observed for most frequencies, although larger discrepancies of around 100 $\mbox{cm}^{-1}$ with regard to the experimental frequencies are observed for the higher frequency modes. It is noteworthy that the results of Clarke et al. [163] share this shortcoming, in contrast to those of Pulay et al. [164]. This warrants some comment. The values due to Pulay et al. [164] are Hartree-Fock results using a 4-21 basis set; although this basis set is adequate for calculations of total energies, it systematically overestimates the force constants in benzene, an effect that is due to the neglect of electron correlation inherent in Hartree-Fock methods, and the truncated basis set used. To remedy this, Pulay et al. introduce empirical scaling factors that reproduce the high frequency modes accurately; thus it is slightly misleading to consider these results to be truly ab initio. The results of Clarke et al. are obtained via a classical trajectory study. This takes as its potential energy surface the force constants of Pulay et al. [164], with the CH stretch diagonal terms discarded, and the addition of a Morse potential. Interestingly, discarding these terms appears to lead to the same level of discrepancy between the calculated and experimental high frequency modes as is found with DFPT. This, coupled with the fact that the modes in question are C-H stretches, suggests that the issue here is anharmonicity, which is not dealt with at all in the DFPT scheme implemented in this work, based as it is upon a harmonic approximation.


Table: Benzene normal mode frequencies ( $\mbox{cm}^{-1}$) and assignments: op denotes out of plane modes.
Assignment PFB set 1 [164] Clarke et al.[163] Experiment [165] Present
op 402   398 402
  607 606 606 612
op 667   673 681
op 701   707 722
op 843   846 849
op 969   967 980
op 996   990 996
  983 993 993 986
  997 1009 1010 1014
  1036 1033 1037 1046
  1162 1140 1146 1175
  1183 1179 1178 1187
  1297 1305 1309 1331
  1365 1479 1350 1372
  1482 1479 1482 1491
  1607 1602 1599 1598
  3051 3135 3057 3127
  3061 3145 3056 3135
  3080 3163 3064 3150
  3095 3178 3073 3159


In figure 4.20 the calculated IR spectrum of benzene is illustrated. This is in good agreement with the experimental spectrum, with the maximum intensity peak occurring at around 680 $\mbox{cm}^{-1}$. This is due to an out-of-plane wagging motion of all the C-H bonds. A peak at 1500 $\mbox{cm}^{-1}$ is correctly predicted, although its relative intensity with respect to the peak at 3200 $\mbox{cm}^{-1}$ is slightly wrong; the peak at 1500 $\mbox{cm}^{-1}$ should be slightly more intense than that at 3200 $\mbox{cm}^{-1}$, in contrast to what is predicted. The peak at 1500 $\mbox{cm}^{-1}$ is due to stretching of the C-C bonds in the aromatic ring. However, the spectrum does not include the smaller intensity peaks that should occur between 1500 and 200 $\mbox{cm}^{-1}$.

Figure 4.20: IR spectrum of benzene as obtained using DFPT.
\includegraphics[scale=0.4,angle=-90]{benzene_IR_intensity.ps}

The calculations of IR spectra presented here seem to suggest that although DFPT can be used to obtain IR spectra, the results should be treated with caution. Certainly, it correctly predicts which modes are silent or IR-active, and in general predicts relative intensities in good agreement with other high level methods. There are, however, a minority of cases in which it does however, give inconsistent results as to the relative intensities, sometimes failing to predict these (as in ethane), or at other times correctly obtaining the intensities, but getting the ordering of the peaks wrong (as in methane).

The results presented, in particular the IR spectra of water, methane and hydrazoic acid, indicate that the relative intensities are extremely sensitive to the choice of basis set; indeed, even when using the same method in two calculations, a different basis set may lead to vastly different values of the intensities and/or the ordering of the peaks altering. It is therefore possible that the problems encountered with regard to methane and, to a lesser extent, ethane, could be due to problems with the basis set used, rather than to any inherent problem with the theoretical method used. One could perhaps expect such problems with localised basis sets, especially using sets that are too small to include polarisation effects; it seems more surprising that such problems could occur using plane waves. It is, of course, possible to suggest that the problems could arise, at least in part, from failures of the harmonic approximation inherent in determining the molecular normal modes. However, the evidence would appear to refute this; for example, in the case of methane, the normal mode frequencies obtained in this work are in closer agreement with experimental values than those of Porezag and Pederson [103] and yet the ordering of the IR peaks is not correct. This therefore suggests, in conjunction with Porezag and Pederson's use of the PW GGA [103,48] and a localised Gaussian basis set, that it is indeed a basis set problem. Furthermore, it also implies that one may have a basis set that is well converged with respect to groundstate properties and normal modes, and yet not be appropriate for accurate determination of IR spectra. This can probably be attributed to the fact that one is dealing with a derivative of the groundstate dipole moment. Calculations have also been carried out to verify that the size of the supercell chosen does not affect the IR spectra, provided that the supercell is sufficiently large that calculations may be converged with regard to groundstate and structural properties (i.e. forces are smaller than 0.01 eV/$\mbox{\AA}$).

The question of whether a mode is IR-active or not is largely down to the symmetry of the effective charge tensor and the dynamical matrix eigenvectors; as this is independent of the quality of the calculation, it is not surprising that this is correctly predicted. It would therefore seem that while the DFPT spectra can be used to correctly indicate where IR peaks will occur, that conclusions based upon the predicted intensities should be treated with caution.


next up previous contents
Next: Summary and Conclusions Up: Implementation of DFPT Algorithm Previous: Molecular Polarisabilities   Contents
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