# Modelization of vibrational spectra beyond the harmonic approximation from an iterative variation–perturbation scheme: the four conformers of the glycolaldehyde

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DOI: 10.1007/s00214-012-1183-1

- Cite this article as:
- Carbonniere, P. & Pouchan, C. Theor Chem Acc (2012) 131: 1183. doi:10.1007/s00214-012-1183-1

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## Abstract

This paper presents the computed anharmonic frequencies and IR intensities in the mid-infrared region for the four conformers of glycolaldehyde (Cis cis, Trans trans, Trans gauche and Cis trans forms). The fundamental transitions and their connected overtones and combination bands through strong anharmonic couplings (Fermi resonances) are provided. The results are stemmed from an iterative variational–perturbational resolution of the vibrational problem implemented in the VCI-P code. The four potential electronic surfaces are built as a Taylor series truncated to the fourth order around each minimum geometry. The second derivatives with respect to the normal coordinates were computed at the CCSD(T)/cc-pVTZ level, while the third and fourth derivatives were estimated with the B3LYP/6-31 + G(d,p) model chemistry. For the most stable Cc form, an average deviation of about 10 cm^{−1} is obtained with respect to the unambiguous experimental values. Furthermore, some of the transitions observed in the CH stretchings region were reassigned. The theoretical values calculated for the Tt and Tg forms are compared to the experimental data obtained from the irradiation of the Cc conformer isolated in Ar matrix with an IR source.

### Keywords

Ab initioDFTAnharmonic vibrational spectraGlycolaldehydeIR intensities## 1 Introduction

The vibrational fingerprint of a molecular system that infrared (IR) or Raman spectroscopy allows to observe is a widely used information for substance recognition and for the understanding of its behavior within a chemical environment. Thus, it is also used for applications such as structural and conformational analysis [1–3] reactivity monitoring [4, 5] and molecular design [6].

The subtle interplay between the different effects that induce the experimental picture of such fingerprint makes the identification task non-trivial, particularly when the molecular system presents strong anharmonic couplings giving rise to IR (or Raman)-active overtones and combination bands. From a theoretical point of view, the need to go beyond the harmonic approximation for an accurate modelization of a vibrational spectrum requires to face the computational cost that makes the treatment impracticable in its most complete formalism.

In the framework of the time independent formalism, several models have been developed and implemented to explicitly take into account anharmonicity. For small molecules (3–5 atoms), converged rovibrational levels can be obtained by fully variational methods, *that is,* the discrete variable representation (DVR) [7] or the vibrational configuration interaction (VCI) [8] methods. For larger molecules, four approximate schemes have been proposed. The vibrational self-consistent field (VSCF) approach represents the total vibrational wave function by a separable product of single-mode wave functions optimized separately using an effective mean field potential [9]. For a better accuracy, the correlation between modes is commonly treated by (i) the vibrational Moller–Plesset perturbation theory (VMP) [10] that is computationally cheap but overestimates the strong anharmonic complings, (ii) a vibrational configuration interaction (VCI) [8] that treats properly the strongest interactions but proves much more time consuming, (iii) the vibrational mean field configuration interaction (VMFCI) [11] that plays on the partitioning of the vibrational modes and encompasses both VSCF and VCI as particular case and (iv) the vibrational coupled cluster (VCC) [12] level of theory. Several strategies to strongly reduce the computational cost of such treatment were proposed in the literature [13–16] (see Ref. [16] for a more detailed description). A possible way which is used in the present study is to take advantage of both perturbative and variational approaches [17]. This variational–perturbational scheme in its most advanced version has been implemented in the VCI-P code [16] that uses small VCI matrices to treat the strongest interactions while the myriads of the weakest interactions are treated perturbationally (see the section computational details and method). Such a recipe was used to investigate the vibrational properties of the four conformers of glycolaldehyde (CH_{2}OHCHO).

In the present study, we report the computed anharmonic fundamental transitions and their connected overtones or combination transitions through Fermi resonances of the four conformers of glycolaldehyde at least for two reasons (i) because of the presence of strong anharmonic couplings in the CH stretching region in which the assignment of the vibrational transitions appears non-univocal for the Cc and Tt conformers and (ii) because no experimental data are reported for the Tg and Ct forms.

## 2 Computational details and method

Computations at the CCSD(T) [26] level of theory were performed with the CFOUR [27] program, while Gaussian 09 [28] was used for the DFT computations. The anharmonic vibrational treatment relies on a hybrid [29] quartic force field in which the second derivatives are computed at the CCSD(T)/cc-pVTZ [30] level of theory, while the third and fourth derivatives are computed at the DFT level using the B3LYP functional [31, 32] with a valence double zeta Pople basis set including diffuse and polarization functions, 6–31+G(d,p) [33]. This model chemistry was chosen since it has been previously shown [34, 35] that for the prediction of harmonic and anharmonic force constants and in the case of small organic systems, it is able to approach the results obtained using the more expensive CCSD(T)/cc-pVTZ level of theory by an average value of 10 cm^{−1} on the fundamental transitions.

The analytic model of the potential functions is determined as follows [36]: from a minimum energy structure of a given system, a quartic force field is built in which the third and fourth derivatives are computed by 6 N–11 (N = number of atoms) numerical differentiation of analytical second derivatives. In the quartic approximation, the potential provided by the Gaussian code is a Taylor series in normal coordinates limited to the 3-mode interactions. From a technical point of view, it can be shown [36] that the best setup is obtained using a step size of 0.01 Å for the numerical differentiation of harmonic frequencies, tight convergence criteria for structural optimizations and fine grids for integral evaluation (that is, at least 99 radial and 590 angular points).

*q*

_{i}and

*p*

_{i}are, respectively, the dimensionless normal coordinates and their conjugate momenta.

*V*(

*q*1

*,…, qM*) is a polynomial expansion of the PES in terms of normal coordinates

*q*

_{i}truncated to the fourth order The last term represents the major component of the rotational contribution to the anharmonicity in which

*B*

_{α}is the rotational constant of the system with respect to the Cartesian axis

*α,*and \( \zeta_{ij}^{\alpha } \) is the Coriolis constant that couples

*q*

_{i}and

*q*

_{j}through the rotation about the

*α*axis.

The anharmonic vibrational treatment is based on an iterative variational–perturbational scheme implemented in the VCI-P code [16]. This code computes the anharmonic wavenumbers and the anharmonic IR intensities of the fundamental transitions, overtones and combination bands which are IR (or Raman) actives through strong anharmonic couplings (Fermi or Darling Dennisson resonances).

The process is ended at the L^{th} iteration when the convergence on the VCI-P energy of the state *i* is reached. It has been set to 0.2 cm^{−1} in the present study. Note that in the above expression, \( E_{i,n}^{VCI} \) corresponds to the VCI energy obtained by the diagonalization of an active space containing the N_{strong} configurations \( \Upphi_{k}^{{n_{k} }} \)^{,}\( \Upphi_{k}^{{n_{k} }} \) being the n_{k}th excitations of the configuration *i*. The second term represents the perturbational contribution of the weakest configurations \( \Upphi_{j}^{{n_{j} }} \)interacting with the state \( \Uppsi_{i}^{{n_{i} }} \)determined variationally at the iteration *L*-*1*.

Note that any configuration \( \Upphi_{k}^{{n_{k} }} \) for which \( c_{ik}^{2} \) is greater than a threshold (4% in the present study) is considered as a strong resonant configuration. As consequence, their multiexcitations are generated from the procedure detailed above up to the convergence of their VCI-P energy. This ensures for each CI matrix the orthogonality between the state \( \Uppsi_{i}^{{n_{i} }} \) and its connected states through Fermi resonances for which the corresponding anharmonic intensities can be determined more safely.

*d*

_{ij}(where

*α*is a Cartesian component of the dipole moment vector

*d*

_{j}referred to the Eckart axes) are calculated in a manner analogous to the cubic and quartic force constants detailed above:

*i*and a final state

*j*is then evaluated as follows:

## 3 Results and discussion

Calculated wavenumbers (cm^{−1}) for the glycolaldehyde in its Cc form with the VCI-P method from a CCSD(T)/cc-pVTZ//B3LYP/6-31G(d,p) hybrid quartic force field

Description | Mode | Experimental | Calculations | ||||||
---|---|---|---|---|---|---|---|---|---|

Gas phase | Ar | Ar | ν | ν | I | Description | ν | ||

OH stretch | ν | 3,549 | 3,541 | 3,542 | 3,542 | 3,570 | 54 | 81% ω | 3,624 |

as. CH | ν | 2,881 | 2,895 | 2,855 | 2,876 | 2,898 | 13 | 74% ω | 2,921 |

ν | 2,649 | 2,650 | 1 | 83% ω | |||||

ν | 2,503 | 2,496 | 1 | 83% ω | |||||

2ν | 2,798 | 2,913 | 14 | 58% 2ω | |||||

ν | 2,882 | 2,883 | 9 | 54% ω | 2,836 | ||||

2ν | 2,874 | 11 | 32% 2ω | ||||||

s. CH | ν | 2,840 | 2,845 | 2,895 | 2,999 | 2,967 | |||

2ν | 2,794 | 6 | 55% 2ω | ||||||

ν | 2,689 | 2,681 | 2 | 76% ω | |||||

CH stretch | ν | 2,820 | 2,907 | 2,845 | 2,822 | 2,854 | 19 | 49% ω | 2,870 |

2ν | 2,831 | 2,816 | 14 | 43% 2ω | |||||

ν | 2,775 | 2,767 | 5 | 62% ω | |||||

2ν | 2,710 | 2,712 | 2,713 | 2,720 | 2,707 | 6 | 78% 2ω | 2,718 | |

C=O stretch | ν | 1,754 | 1,747 | 1,747 | 1,770 | 1,761 | 108 | 64% ω | 1,753 |

2ν | 1,706 | 1,697 | 1,707 | 1,702 | 1,716 | 33 | 58% 2ω | 1,711 | |

2ν | 1,441 | 1,412 | 10 | 73% 2ω | |||||

CH | ν | 1,456 1,425 | 1,443 1,429 1,421 | 1,430 1,424 | 1,460 | 1,454 | 11 | 81% ω | 1,462 |

CH | ν | 1,378 | 1,399 | 1,400 | 1,426 | 1,421 | 27 | 92% ω | 1,420 |

ip CH bend | ν | 1,356 | 1,367 | 1,366 | 1,367 | 1,363 | 31 | 94% ω | 1,378 |

ip OH bend, CH | ν | 1,275 | 1,267 | 1,268 | 1,279 | 1,273 | 41 | 94% ω | 1,265 |

CH | ν | 1,229 | – | 1,229 | 1,225 | 3 | 95% ω | 1,235 | |

2ν | 2,209 | 2,229 | 2,233 | 1 | 86% 2ω | 2,222 | |||

C–O stretch | ν | 1,112 | 1,110 | 1,110 | 1,120 | 1,120 | 80 | 93% ω | 1,116 |

CH | ν | 1,130 | – | 1,087 | 1,081 | 0 | 95% ω | 1,090 | |

CC stretch | ν | 859 | 858 | 856 | 865 | 865 | 49 | 91% ω | 867 |

O=CC bend, CCO bend | ν | 752 | 749 | 751 | 757 | 758 | 9 | 95% ω | 740 |

CH | ν | – | – | 722 | 713 | 0 | 96% ω | 715 |

Globally, the VCI-P results achieve an average convergence of 11 cm^{−1} with the overall of the non-ambiguous experimental values (values reported in Table 1 excepted ν_{7}, ν_{13}, ν_{15}, ν_{16}, as discussed). This mean absolute deviation (M.A.D) is confronted to the VPT2 results [25] stemmed from a MP2/6-311G(d,p) quartic force field for which a mean deviation of about 17 cm^{−1} is observed on the same set.

The experimental values in the 1,800–700 cm^{−1} region are particularly well reproduced by the VCI-P approach, namely for the remarkable case of Fermi resonance that occurs between the C=O stretching and the first overtone of the C–C stretching mode. Indeed, a gap of 45–50 cm^{−1} is observed between the two transitions both experimentally and theoretically with an intensity ratio \( I_{{\nu_{C = O} }} :I_{{2\nu_{C - C} }} \) of about 5:1 in gas phase [23], 2:1 in Ar matrix [24] and 3:1 from our calculations.

Here, the most ambiguous assignment concerns the fundamental transition of the CH_{2} scissoring mode for which two bands of about the same intensity [23, 24] were observed. From the experiment in gas phase [23], the two values (1,425 and 1,456 cm^{−1}) were assigned to the fundamental transition even though it was pointed out that one of the two positions could correspond to the first overtone of ν_{4} (CH_{2} rocking mode). From low-temperature experiments in Ar matrix, a triplet is reported in Ref. [24] (1,421–1,429–1,443 cm^{−1}) invoking possible site effects in this material, while the Ref. [25] mentions only a doublet (1,424–1,430 cm^{−1}). These observations are consistent with our calculations concerning both the positions and the intensity ratio. Thus, the values around 1,430 and 1,450 cm^{−1} are assigned to the 2ν_{4} and ν_{13} (CH_{2} scissoring), respectively.

More problems are encountered in the region above 2,600 cm^{−1} since the symmetric CH_{2} and CH stretching modes are affected by strong anharmonic couplings, giving rise to several IR-active overtones and combination bands. Starting from the CH stretching mode region, the presence of a Fermi resonance between this mode (ν_{15}) and the first overtone of the op CH bending mode (2ν_{11}) was observed in the three experiments [23–25], but different values of ν_{15} were reported (2,820, 2,907 and 2,845 cm^{−1}), while the value of 2ν_{11} is determined around 2,710 cm^{−1}. The theoretical description reveals a more subtle interplay involving four transitions, in particular between the 2ν_{12} and the ν_{15}, calculated at 2,816 and 2,854 cm^{−1}, respectively, for which the intensities are of the same magnitude. Thus, the experimental value of 2,820 cm^{−1} corresponds probably to the 2ν_{12} transition. Note, therefore, that the deviation between theory and experiment on the 2ν_{11}, ν_{15} and 2ν_{12} are of 9, 4 and 3–5 cm^{−1}, respectively.

_{2}stretching mode (ν

_{16}) is not straightforward. According to our theoretical description, the harmonic character ω

_{16}is widely spread within the states 2ν

_{13}, ν

_{12}+ ν

_{13}and 2ν

_{4}+ ν

_{13}and no major component of ω

_{16}appears, according to the form of the PES used. However, Fig. 2a that displays the anharmonic spectrum of the Cc form shows a multiplet shape in the CH stretching region very close to the pictures reported in Ref. [20, 23].

If we turn our attention on the asymmetric CH_{2} and OH stretching modes, the observed deviation is about 3–17 cm^{−1} and 21–29 cm^{−1} with the VCI-P results, respectively, while it is about 5–19 cm^{−1} and 0–7 cm^{−1} with the VMP2 results. Here again [38], in the absence of Fermi resonance, the VMP2 treatment is closer to the experimental counterpart for a mode having a well-defined Morse potential even though the analytical form of the PES used has no Morse behavior far from the local minimum (quartic force field). Nevertheless, the VCI-P wavenumber, based on such a force field, is closer to the experimental value than its VCI counterpart (see Eq. 2) that yields some discrepancies of 20–34 cm^{−1} and 39–50 cm^{−1} for the asymmetric CH_{2} stretching mode and OH stretching mode, respectively.

^{−1}, the very strong absorptions (ν

_{C–C}, ν

_{C–O}, ip OH bending) are in good agreement with the theoretical computations, while the very weak absorption (op CH bending) is too far from its theoretical counterpart. In the region above 2,600 cm

^{−1}, the experimental transition observed at 2,935 cm

^{−1}probably does not correspond to ν

_{a,CH2}since it appears 72 cm

^{−1}higher than the corresponding theoretical value and the observed ν

_{OH}mode at 3,668 cm

^{−1}is overestimated by 72 cm

^{−1}by our computations. According to Table 3, it should be noted that the experimental values pointed out here (2,935 and 3,668 cm

^{−1}) are closer to their theoretical counterpart belonging to the Tg form which is expected very close in energy to the Tt form.

Calculated wavenumbers (cm^{−1}) for the glycolaldehyde in its Tt form with the VCI-P method from a CCSD(T)/cc-pVTZ//B3LYP/6-31G(d,p) hybrid quartic force field

Description | Mode | exp | ν | ν | Iν | Description |
---|---|---|---|---|---|---|

OH stretch | ν | 3,668 | 3,707 | 3,740 | 36 | 82% ω |

as. CH | ν | 2,940 | 2,903 | 2,921 | 15 | 74% ω |

ν | 2,621 | 2,625 | 1 | 83% ω | ||

ν | 2,553 | 2,534 | 1 | 65% ω | ||

2ν | 2,849 | 2,921 | 8 | 64% 2ω | ||

2ν | 2,882 | 10 | 34% 2ω | |||

s. CH | ν | 2,935 | 2,971 | 2,863 | 10 | 33% ω |

CH stretch | ν | 2,810 | 2,807 | 2,823 | 35 | 34% ω |

2ν | 2,795 | 2,783 | 9 | 52% 2ω | ||

ν | 2,748 | 2,743 | 20 | 67% ω | ||

2ν | 2,701 | 2,686 | 9 | 76% 2ω | ||

C=O stretch | ν | 1,747 | 1,767 | 1,770 | 145 | 91% ω |

CH | ν | 1,439 | 1,472 | 1,458 | 12 | 76% ω |

2ν | 1,441 | 1,420 | 3 | 64% 2ω | ||

CH | ν | 1,410 | 1,406 | 6 | 91% ω | |

ip CH bend | ν | 1,353 | 1,357 | 1,354 | 4 | 94% ω |

CH | ν | – | 1,220 | 1,217 | 0 | 94% ω |

ip OH bend, CH | ν | 1,203 | 1,214 | 1,203 | 68 | 93% ω |

op CH bend | ν | 1,129 | 1,088 | 1,083 | 2 | 95% ω |

C–O stretch | ν | 1,065 | 1,082 | 1,083 | 63 | 90% ω |

CC stretch | ν | 998 | 996 | 994 | 46 | 92% ω |

CH | ν | – | 728 | 719 | 0 | 95% ω |

O=CC bend, CCO bend | ν | 538 | 539 | 540 | 6 | 91% ω |

Calculated wavenumbers (cm^{−1}) for the glycolaldehyde in its Tg and Ct forms with the VCI-P method from a CCSD(T)/cc-pVTZ//B3LYP/6-31G(d,p) hybrid quartic force field

Tg | Ct | |||||
---|---|---|---|---|---|---|

Description | Mode | CC//B3 | Iν | Mode | CC//B3 | Iν |

OH stretch | ν | 3,703 | 27 | ν | 3,715 | 36 |

as. CH | ν | 2,965 | 7 | ν | 2,883 | 20 |

ν | 2,650 | 1 | ||||

ν | 2,624 | 1 | ||||

ν | 2,531 | 1 | ||||

s. CH | ν | 2,918 | 16 | ν | 2,857 | 29 |

2ν | 2,859 | 2 | 2ν | 2,884 | 5 | |

ν | 2,683 | 9 | ||||

2ν | 2,822 | 1 | 2ν | 2,912 | 10 | |

4ν | 2,757 | 5 | ||||

ν | 2,830 | 5 | ||||

CH stretch | ν | 2,794 | 23 | ν | 2,710 | 62 |

2ν | 2,718 | 12 | 2ν | 2,790 | 49 | |

2ν | 2,763 | 1 | ||||

ν | 2,682 | 13 | ||||

C=O stretch | ν | 1,764 | 137 | ν | 1,790 | 151 |

2ν | 1,707 | 10 | ||||

CH | ν | 1,446 | 7 | ν | 1,449 | 16 |

2ν | 1,411 | 5 | ||||

CH | ν | 1,372 | 13 | ν | 1,416 | 7 |

ip CH bend | ν | 1,361 | 16 | ν | 1,384 | 7 |

ip OH bend, CH | ν | 1,312 | 3 | ν | 1,233 | 1 |

CH | ν | 1,182 | 15 | ν | 1,184 | 68 |

C–O stretch | ν | 1,079 | 66 | ν | 1,129 | 26 |

CH | ν | 1,049 | 32 | ν | 1,080 | 2 |

CC stretch | ν | 1,023 | 29 | ν | 860 | 35 |

O=CC bend, CCO bend | ν | 712 | 5 | ν | 718 | 0 |

CH | ν | 519 | 2 | ν | 713 | 56 |