##
1Infrared intensities of furan, pyrrole and thiophene: beyond the double harmonic approximation
Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj1

1

Infrared intensities of furan, pyrrole and thiophene: beyond the double harmonic approximation

Type: Object |
Advantage: None |
Novelty: New |
ConceptID: Obj1

2

Infrared intensities of the fundamental, overtone and combination transitions in furan, pyrrole and thiophene have been calculated using the variational normal coordinate code MULTIMODE.

Type: Object |
Advantage: None |
Novelty: None |
ConceptID: Obj1

3

We use pure vibrational wavefunctions, and quartic force fields and cubic dipole moment vector surfaces, generated by density functional theory.

Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met1

4

The results are compared graphically with second-order perturbation calculations and with relative intensities from experiment for furan and pyrrole.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

## Introduction

5

This paper is a continuation of our previous studies of vibrational transitions in five-membered aromatic ring compounds.

^{1}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

6

We have found that both perturbative and variational approaches, based on the fourth degree Taylor expansion of the potential, yield energies of vibrational levels for fundamental and overtone transitions in excellent agreement with experimental data.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

7

The next logical step is to expand the dipole moment vector as a Taylor series in normal coordinates and calculate anharmonic contributions to the infrared intensities.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

8

This approach was tested recently for water and formaldehyde.

^{2}
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac1

9

While for small molecules there are many avenues along which to pursue infrared intensities of higher quality, for larger molecules (with more than five atoms) expansion in normal coordinates appears to be the only practical approach.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

10

A number of scientists are now recognising the importance of comparing theoretical spectra with observation.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac2

12

The paper is organised as follows.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac3

13

In Section 2 we discuss the potential and dipole fields, variational and perturbation approaches used to evaluate the transition dipole moments and calculation of infrared intensities from these moments.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met1

14

In Section 3 we present and discuss the results for furan, pyrrole and thiophene.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

15

In Section 4 we summarise the findings and outline future directions.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

## Methodology

16

To study molecular vibrations it is necessary to solve the Schrödinger equation for nuclear motion.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac4

17

While there are many different coordinate systems in which the kinetic and potential energy operators can be expressed, for larger molecules the only expansion of the kinetic energy operator, which is not prohibitively complicated, is in terms of normal coordinates.

^{5}
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

18

Therefore it is advantageous to express the potential energy also in terms of the normal coordinates.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

19

If analytical second derivatives

*φ*_{jk}are calculated at the equilibrium geometry and at the geometries displaced along the normal coordinates, cubic and semi-diagonal quartic force constants,*φ*_{ijk}and*φ*_{iijk}, can be obtained by numerical differentiation: If infrared intensities are desired, also dipole moment field needs to be evaluated.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

20

Since the dipole moment is a derivative of the energy with respect to the electric field, the equivalent of the quartic expansion of the potential is a cubic expansion of the dipole surfaces.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

21

The second and third derivatives of dipole moment,

*d*_{αij}and*d*_{αiij}(where*α*is a Cartesian component of the dipole moment vector*d*_{αj}referred to Eckart axes), are calculated in a manner analogous to the cubic and quartic force constants: Potential and dipole moment surfaces were calculated by the Density Functional Theory (DFT) approach using the B97-1 functional^{6}and a TZ2P basis set.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod1

22

Calculations were performed using the Cadpac program package.

^{7}
Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met2

23

The step size δ

*Q*_{i}was chosen such that it would correspond to an energy step of 1 m*E*_{h}on the harmonic potential surface.
Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met2

24

A fuller discussion of the finite difference calculation, the effects of the step size, as well as values for the anharmonic constants may be found in .

^{ref. 1}
Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met2

25

The variational calculations were performed using the program MULTIMODE.

^{8–10}
Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met3

26

This program performs vibrational self-consistent force field (VSCF) calculations, followed by the truncated Configuration Interaction (CI) expansion in virtual VSCF functions.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met3

27

This has an advantage of explicitly treating all resonances among states included in the CI expansion.

Type: Method |
Advantage: Yes |
Novelty: Old |
ConceptID: Met3

28

Terms coupling up to three normal coordinates were used in the VSCF Hamiltonian.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met3

29

The CI expansion included all single, double, triple and quadruple VCI excited states with the sum of vibrational quantum numbers less than or equal to four.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met3

30

Finally, the matrix elements

*R*_{αij}were evaluated using the dipole moment surface*D*_{α}(**) and the CI vectors***Q**Ψ*(**):***Q**R*_{αij}= ∫*Ψ*_{i}(**)***Q**D*_{α}(**)***Q**Ψ*_{j}(**)***Q**d***Second-order perturbation calculations were carried out using the program SPECTRO.***Q*^{11}
Type: Method |
Advantage: None |
Novelty: Old |
ConceptID: Met4

31

While this program is capable of treating low-level resonances by removing the divergent terms and explicitly solving 2 × 2 CI problems (or by providing matrix elements for larger CI problems), second-order perturbation theory does not handle resonances involving states with the sum of vibrational quantum numbers being larger than 2.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met4

32

While there are many resonances present in the systems studied, none of the resonances are very strong.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met4

33

Therefore, we decided not to include any resonances in the present perturbation calculations with SPECTRO.

Type: Method |
Advantage: None |
Novelty: None |
ConceptID: Met4

34

For more information about the programs MULTIMODE and SPECTRO and their use, see for example

^{refs. 1 and 2}.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac5

35

The infrared intensities of a

*v*_{i}→*v*′_{j}transitions were evaluated using the usual expression: where*N*_{A}is Avogadro's constant,*ν*_{a}is the wavenumber of the transition, and*N*_{v}is the fraction of molecules in the state*v*.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

36

If we assume an equilibrium distribution, then for the transitions originating from the ground vibrational state the term (

*N*_{v}–*N*_{v′}) tends to 1.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

37

Evaluating the fraction and converting to practical units leads to a working formula with

*ν*_{a}in cm^{–1}and*R*_{αij}in debye.
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

38

This formula does not sum over hot and stimulated emission bands, which can all be treated explicitly.

Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

39

It also allows us to treat all kinds of transitions on an equal footing, which is especially important for variational calculations which mix transitions of various characters (

*e.g.*fundamental, overtone, combination).
Type: Model |
Advantage: None |
Novelty: None |
ConceptID: Mod2

## Results and discussion

40

The stick spectra representing vibrational transitions from the ground vibrational state, obtained by variational and perturbational calculations for furan and pyrrole, are compared with scaled experimental results of Mellouki

*et al*^{12}. in Figs. 1 and 2, respectively.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res1

41

Since, to our knowledge, there is no consistent set of experimental infrared intensities for thiophene available, Fig. 3 compares only calculated results.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs1

42

The peaks corresponding to fundamental transitions

*ν*_{i}are labeled by*i*.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs1

43

Transitions for which experimental intensities were not available are denoted with a star.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs1

44

It can readily be seen that the agreement between the stick spectra calculated by variational and second-order perturbation approaches is, in general, very good, especially for furan and thiophene.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs2

45

For pyrrole there are some disagreements, most notably for the states involving

*ν*_{16}.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs2

46

As observed previously

^{1}this mode shows large negative anharmonicity and is probably not adequately described by a quartic Taylor expansion.
Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res2

47

Also the density of states is higher for pyrrole, which leads to an increase in the number and strength of resonances.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res3

48

It seems that inclusion of resonances would be necessary for a proper perturbation description of intensities in pyrrole; however, this would eliminate much of the simplicity of the second-order perturbative approach, and this is one of its most attractive features.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res3

49

(A referee has observed that it is difficult to see from Figs. 2(a) and (b) that the variational treatment represents an improvement over the perturbational approach.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res4

50

We agree, but we anticipate that as the molecules become larger, as a force field becomes more complicated with more complicated resonances, the difference between the variational and perturbational spectra will become more marked.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res4

51

We underline a fact that only the variational approach can include all resonances, and therefore it must be recommended when it is possible.)

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con1

52

In the region above 2000 cm

^{–1}there are more states shown in the variational calculation panels than in the corresponding second-order perturbation calculation and experiment panels.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs3

53

This is due to the fact that MULTIMODE yields all states (as long as they are included in the CI expansion), while SPECTRO produces only states with the sum of vibrational quantum numbers less than or equal to two.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res5

54

Higher excited states can be produced, but the quality of the results is not the same.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res5

55

Also, only states which were properly assigned are included in the experimental panel.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res5

56

One of the consequences of the higher number of states and explicit treatment of all couplings in MULTIMODE are apparent in the lower intensities for high-frequency C–H and N–H stretches obtained from MULTIMODE, as compared with those obtained from SPECTRO.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res6

57

In fact, all of these modes are strongly coupled to other modes, resulting in several modes each with lower intensity.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res6

58

However, if a stick spectrum is replaced by gaussians with finite width, the summed intensities are comparable.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res6

59

Comparison with experiment is complicated by the lack of firm experimental data.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res7

60

Mellouki

*et al*^{12}. published relative intensities for furan and pyrrole.
Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac6

61

However, the authors caution that the data are not accurate.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac6

62

For comparison with our stick spectra we have scaled the relative intensities of

^{ref. 12}by factors of 22.2 for furan and 163 for pyrrole, respectively.
Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs4

63

Despite the above mentioned uncertainties in the experimental spectrum, it can be seen that the correspondence between theoretical and experimental results is good for these two molecules.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs5

64

It can be observed that overtone and combination bands obtained from MULTIMODE are shifted towards higher wavenumbers with respect to those obtained from SPECTRO.

Type: Observation |
Advantage: None |
Novelty: None |
ConceptID: Obs6

65

This is due to the relatively small size of the CI.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res8

66

To calculate the wavenumbers of overtones it is necessary to include states with the sum of vibrational quantum numbers larger than four.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res9

67

A proper description of combination bands further requires the inclusion of quintuple excitations in the CI.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res10

68

Nevertheless, the intensities of overtone and combination bands are in reasonable agreement with those obtained from SPECTRO and with the scaled experimental results.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res10

## Conclusions

69

In this paper we have demonstrated the success of the variational code MULTIMODE to predict integrated band intensities for molecules.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con2

70

All one needs are quartic potential energy surfaces and cubic dipole surfaces, expanded in normal coordinates.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con2

71

An

*ab initio*or DFT quantum chemistry code which calculates analytic second derivatives is also required.
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con2

72

The quality of our results is immediately apparent by comparing Fig. 1(a) (MULTIMODE) with Fig. 1(c) (experiment), for furan, and likewise Figs. 2(a) and (c) for pyrrole.

Type: Result |
Advantage: None |
Novelty: None |
ConceptID: Res11

73

We have every reason to expect that the MULTIMODE predicted spectra (Fig. 3(a)) for thiophene would be close to observation.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con3

74

The new feature of our science is its automatic capability of working beyond the double harmonic approximation, thus including anharmonicity, overtones and combination bands in our calculations.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con4

75

Of course we have only looked at the wider detail.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con4

76

Our earlier studies show that frequencies determined by DFT are in error by 1%, and the calculated intensities are also only very approximate, and without fine structure.

Type: Background |
Advantage: None |
Novelty: None |
ConceptID: Bac7

77

Even so, we believe that these qualitative spectra will be useful in molecular identification, at least.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con5

78

A caveat to these results is that ideally we should perform as large a CI as possible within MULTIMODE in order to ensure that the overtones and combination bands are fully converged.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con6

79

The larger the molecule, the more mixing is present.

Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con6

80

Finally, we are in the process of calculating ro-vibrational wavefunctions, from which band spectra can be obtained (for low

*J*).
Type: Conclusion |
Advantage: None |
Novelty: None |
ConceptID: Con7