# Harmonic Vibrational Frequency Simulation of Pharmaceutical Molecules via a Novel Multi-Molecular Fragment Interception Method

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}and 18.35 cm

^{−1}for Finasteride, 15.95 cm

^{−1}and 26.46 cm

^{−1}for Lamivudine, and 12.10 cm

^{−1}and 25.82 cm

^{−1}for Repaglinide. Additionally, this work provides comprehensive vibrational frequency calculations and assignments for Finasteride, Lamivudine, and Repaglinide, which have never been thoroughly investigated in previous research.

## 1. Introduction

^{−1}[3,4]. With advancements in testing technology, Fourier transform infrared (FT–IR) spectroscopy emerged as a potent device for distinguishing and identifying an array of diverse samples, and, indeed, all forms of samples could be characterized thereby. It is a prompt and refined modality for sample characterization whereby the chemical configuration can be analyzed as simple molecules, revealing specific absorption bands in the FT–IR spectra [5,6].

## 2. Results and Discussion

#### 2.1. Traditional Single-Molecular Vibration Analysis

^{−1}and 29.80 cm

^{−1}, 52.17 cm

^{−1}and 109.50 cm

^{−1}, 26.48 cm

^{−1}and 57.51 cm

^{−1}for the single-molecular simulation; and 16.99 cm

^{−1}and 27.14 cm

^{−1}, 50.56 cm

^{−1}and 101.75 cm

^{−1}, 29.39 cm

^{−1}and 44.34 cm

^{−1}for the scaled spectra of FIN, LAM, and REP, respectively. Therefore, after scaling, the infrared spectrum of LAM exhibited greater similarity to the experimental data, while the opposite was observed for FIN and REP. Thus, for macro-molecules, particularly those with complex configurations, the scale factor may not be entirely applicable. Nonetheless, at hydrogen-bonding sites, the scaled vibrations outperformed the single-molecular simulations. In the case of FIN (Table 1), the N-H and C=O stretching were calculated as 3517.12 cm

^{−1}, 3514.26 cm

^{−1}, 1695.16 cm

^{−1}, 1693.63 cm

^{−1}for the single-molecular simulation and 3481.95 cm

^{−1}, 3479.12 cm

^{−1}, 1678.21 cm

^{−1}, 1676.69 cm

^{−1}for the scaled spectra, respectively. In these bands, the scaled vibrations exhibited better fit to the experimental spectra, which was consistent with the behaviors observed in LAM and REP (Table 1). This result correlates with a slight reduction in the value of the RMSE after scaling. Consequently, the scale factor was primarily employed to adjust the vibrations of hydrogen-bonded functional groups, and had minimal impact on the overall molecule, particularly in the skeleton vibration and fingerprint region.

#### 2.2. Central-Molecular Simulation Analysis

^{−1}and 21.81 cm

^{−1}, 20.36 cm

^{−1}and 30.40 cm

^{−1}, 23.12 cm

^{−1}and 52.21 cm

^{−1}for FIN, LAM, and REP, respectively, whereas the corresponding values for the scaled spectra were 16.99 cm

^{−1}and 27.14 cm

^{−1}, 50.56 cm

^{−1}and 101.75 cm

^{−1}, 29.39 cm

^{−1}and 44.34 cm

^{−1}, as listed in Table 2. The significant reduction observed in the RMSE of LAM was primarily due to the precise calculation of the infrared vibration at the hydrogen-bonding site (ν

_{as}N

_{3}H

_{3–4}) by the central-molecular model. Therefore, the central-molecular model for vibrational simulation was found to be more accurate than the single-molecular and scaled results for all three pharmaceutics. Moreover, for the hydrogen-bonded functional groups, the central-molecular model was also found to yield better fits to the experimental spectra.

_{3}-H

_{3}stretching was 3383 cm

^{−1}, which was calculated as 3620.42 cm

^{−1}for scaled spectra and 3484.85 cm

^{−1}for the central-molecular calculation, respectively. For REP (Table 2), the experimental value of O

_{4}H

_{36}peak was at 3428 cm

^{−1}, which corresponded to 3162.17 cm

^{−1}and 3566.16 cm

^{−1}for the central-molecular and scaled spectra, respectively. In general, the accuracy of the central-molecular model was similar to that of the scale factor method. To overcome this limitation, a new multi-molecular fragment interception method was proposed.

#### 2.3. Multi-Molecular Fragment Interception Simulation Analysis

^{−1}, 20.36 cm

^{−1}, and 23.12 cm

^{−1}for FIN, and 8.21 cm

^{−1}, 15.95 cm

^{−1}, and 12.10 cm

^{−1}for the multi-molecular fragment model. In addition, the RMSE values were also calculated and compared with the MAE values. The RMSE values for FIN were 29.80 cm

^{−1}, 21.81 cm

^{−1}, and 18.35 cm

^{−1}for the single-molecular, central-molecular, and multi-molecular fragment models, respectively. For LAM, the RMSE values were 109.50 cm

^{−1}, 30.40 cm

^{−1}, and 26.46 cm

^{−1}, and for REP, the RMSE values were 57.51 cm

^{−1}, 52.21 cm

^{−1}, and 25.82 cm

^{−1}. Therefore, the utilization of the multi-molecular fragment model resulted in a considerable reduction in the MAE and RMSE values. These results indicate that the multi-molecular fragment interception model outperformed both the single-molecular and central-molecular models in terms of ability to accurately predict vibrational frequencies. Therefore, the multi-molecular fragment interception model for vibrational simulation was more accurate than the other models described above for all three pharmaceutics. Additionally, the frequencies calculation with the multi-molecular fragment model for hydrogen-bonded functional groups also demonstrated the best performance relative to empirical spectra. For instance, for FIN in Table 3, the C

_{3}-O

_{1}and C

_{19}=O

_{2}stretching vibrations were calculated to be 1664.17 cm

^{−1}and 1662.83 cm

^{−1}for the central-molecular model, and 1681.01 cm

^{−1}and 1665.21 cm

^{−1}for the multi-molecular fragment model, respectively. The experimental frequencies for these two vibrations were 1688 cm

^{−1}and 1668 cm

^{−1}, indicating that the multi-molecular fragment interception model was a better fit to the experimental spectrum. Similar conclusions could be drawn for the REP and LAM molecules.

#### 2.4. Absolute Error Analysis

^{−1}, 8.96 cm

^{−1}and 57.04 cm

^{−1}, 18.94 cm

^{−1}and 28.36 cm

^{−1}, 13.36 cm

^{−1}in the functional group and non-functional group regions, respectively. For LAM and REP, the improvement of computational accuracy in the functional group regions had a more significant effect on reducing the overall MAE, whereas for FIN, the contribution of both functional group and non-functional group vibrations to the overall error was similar. Interestingly, for all three drug molecules, the magnitude of MAE reduction in the multi-molecular fragment model in the functional group region (LAM > REP > FIN) was consistent with the variational trend of intermolecular hydrogen bonds or interactions. This indicates that, for multi-molecular models, the more intermolecular hydrogen bonds present, the greater the reduction in errors observed in functional group vibrations. This may be due to the retention of hydrogen bonding and conjugation information in the multi-molecular model, leading to higher computational accuracy at the hydrogen-bonding connection sites during vibrational frequency calculation.

#### 2.5. Computational Time Comparison

## 3. Materials and Methods

#### 3.1. Materials

#### 3.2. Preparation of Drug Crystalline Forms

#### 3.3. FT–IR

^{−1}. Samples were prepared in KBr pellets by grinding Ca. 1 mg of the drug with KBr and the resolution was 2 cm

^{−1}.

#### 3.4. Computer Details

^{−5}Hartree, 0.002 Hartree Å

^{−1}and 0.005 Å, respectively. The vibrational frequencies of the single-molecular and multi-molecular fragment were simulated at the same level of theory as the geometrical optimizations, and the MAEs between the calculated and experimented frequencies were compared to validate the accuracy of the new method. All calculations were conducted by the DMol3 module in Material Studio 2017 program package [45], utilizing a Dell Precision 7670 Workstation.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

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**Figure 1.**The infrared spectra of experimental, simulated single molecule, and simulated single molecule after scaling for (

**a**) FIN, (

**b**) LAM, and (

**c**) REP.

**Figure 2.**The minimum multi-molecular repeating units of FIN, LAM, and REP (Hydrogen bonds are illustrated by blue dashed lines). N atoms are colored blue, O atoms are colored red, H atoms are colored white, C atoms are shown in gray, and S atoms are shown in yellow.

**Figure 3.**The infrared spectra of experimental, simulated central-molecular and simulated single-molecular models after scaling for (

**a**) FIN, (

**b**) LAM, and (

**c**) REP.

**Figure 4.**The structural comparison of REP molecule after optimization with single-molecular and multi-molecular repeating units. The multi-molecular optimized structure is colored yellow.

**Figure 6.**The infrared spectrum of experimental, simulated multi-molecular fragment, simulated central molecule, simulated single molecule and scaled single molecule for (

**a**) FIN, (

**b**) LAM, and (

**c**) REP.

**Figure 7.**Absolute errors in functional group (FG) and non-functional group (NFG) vibrations of FIN, LAM, and REP.

**Figure 8.**The spatial configuration and atom numbering schemes of Finasteride, Lamivudine, and Repaglinide.

**Table 1.**A portion of observed and calculated vibrational frequencies with their respective dominant normal modes for FIN, LAM, and REP

^{a}.

Name | Assignment | Exp | Single Molecular | Single Molecular after Scaled by Scale Factor |
---|---|---|---|---|

FIN | νN_{2}H_{35} | 3429 | 3517.12 | 3481.95 |

A:νN_{1}H | 3349 | 3514.26 | 3479.12 | |

νC_{13}H_{14} | 2914 | 2924.11 | 2894.87 | |

νC_{3}=0_{1}; νC_{1}=C_{2}; βN_{1}H_{36}; | 1688 | 1695.16 | 1678.21 | |

νC_{19}=0_{2}; βN_{2}H_{35}; | 1668 | 1693.63 | 1676.69 | |

βC_{8,16}H_{9,19}; | 1277 | 1268.44 | 1255.76 | |

ρC_{14–15}H_{15–18}; γN_{2}C_{19} | 766 | 766.83 | 759.16 | |

MAE (for all data) ^{b} | 12.99 | 16.99 | ||

RMSE (for all data) ^{c} | 29.80 | 27.14 | ||

LAM | ν_{as}N_{3}H_{3–4} | 3383 | 3656.99 | 3620.42 |

ν_{s}N_{3}H_{3–4} | 3328 | 3518.43 | 3483.25 | |

νC_{1}O_{1} | 1651 | 1701.03 | 1684.02 | |

νC_{3}N_{2}; βN_{3}H_{3} | 1498 | 1531.51 | 1516.19 | |

γC_{2}H_{1}; ρC_{8}H_{9–10} | 1030 | 1041.08 | 1033.67 | |

ρN_{3}H_{3–4}; βC_{5}H_{5}; γC_{4,6}H_{2,6};γC _{8}H_{9–10}; ring prckering vibration; | 538 | 533.74 | 528.40 | |

MAE (for all data) ^{b} | 52.17 | 50.56 | ||

RMSE (for all data) ^{c} | 109.50 | 101.75 | ||

REP | νO_{4}H_{36} | 3428 | 3640.80 | 3566.16 |

νN_{2}H_{35} | 3307 | 3551.09 | 3478.29 | |

νC_{12,16}H | 2804 | 2863.56 | 2804.86 | |

νC_{25}O_{3}; βO_{4}H_{36} | 1686 | 1747.65 | 1711.82 | |

νC_{17}O_{1}; βN_{2}H_{35}; γC_{18}H_{8} | 1635 | 1699.94 | 1665.09 | |

νC_{7,18}H;N_{2}H | 1300 | 1299.66 | 1273.02 | |

γN_{2}H_{35}; ωC_{18}H_{8–9}; γC_{7–11}H | 474 | 475.74 | 465.99 | |

MAE (for all data) ^{b} | 26.48 | 29.39 | ||

RMSE (for all data) ^{c} | 57.51 | 44.34 |

^{a}Frequencies are in cm

^{−1};

^{b}MAE in cm

^{−1};

^{c}RMSE in cm

^{−1}; ν: stretching; β: in-plane bending; γ: out-of-plane bending; ρ: in-plane rocking.

**Table 2.**Observed and calculated vibrational frequencies and their dominant normal modes for FIN, LAM, and REP

^{a}.

Name | Assignment | Exp. | Central Molecular | Single Molecular after Scaled by Scale Factor |
---|---|---|---|---|

FIN | νN_{2}H_{35} | 3429 | 3522.08 | 3481.95 |

A:νN_{1}H | 3349 | 3437.05 | 3479.12 | |

νC_{13}H_{14} | 2914 | 2920.99 | 2894.87 | |

νC_{3}=0_{1}; νC_{1}=C_{2}; βN_{1}H_{36}; | 1688 | 1664.17 | 1678.21 | |

νC_{19}=0_{2}; βN_{2}H_{35}; | 1668 | 1662.83 | 1676.69 | |

βC_{8,13,16}H_{9,14,19} | 1225 | 1224.87 | 1218.55 | |

βN_{1}H_{36}; βC_{6,10}H_{6–7,10–11}; δasC_{17,18}H_{20–25} | 890 | 882.29 | 872.55 | |

MAE (for all data) ^{b} | 9.34 | 16.99 | ||

RMSE (for all data) ^{c} | 21.81 | 27.14 | ||

LAM | ν_{as}N_{3}H_{3–4} | 3383 | 3484.85 | 3620.42 |

ν_{s}N_{3}H_{3–4} | 3327.9 | 3331.83 | 3483.25 | |

νC_{1}O_{1} | 1615 | 1671.64 | 1684.02 | |

βN_{3}H_{3}; νC_{3}N_{2} | 1498 | 1467.85 | 1516.19 | |

βC_{5,7}H_{5,8}; ωC_{6}H_{6–7};γC _{2}H_{1} | 1184 | 1182.63 | 1151.56 | |

ωC_{6}H_{6–7}; νC_{6}S_{1} | 752 | 757.16 | 750.60 | |

MAE (for all data) ^{b} | 20.36 | 50.56 | ||

RMSE (for all data) ^{c} | 30.40 | 101.75 | ||

REP | νO_{4}H_{36} | 3428 | 3162.17 | 3566.16 |

νN_{2}H_{35} | 3307 | 3424.00 | 3478.29 | |

νC_{25}O_{3}; βO_{4}H_{36} | 1686 | 1666.48 | 1711.82 | |

νC_{17}O_{1}; βN_{2}H_{35}; γC_{18}H_{8} | 1635 | 1621.51 | 1665.09 | |

νC_{7,18}H; νN_{2}H | 1300 | 1304.87 | 1273.02 | |

γN_{2}H_{35}; γC_{2,3,4,5}H_{6,7,5,4}; γC_{12–16}H; νC_{1}N_{2} | 619 | 610.53 | 597.38 | |

MAE (for all data) ^{b} | 23.12 | 29.39 | ||

RMSE (for all data) ^{c} | 52.21 | 44.34 |

^{a}Frequencies are in cm

^{−1};

^{b}MAE in cm

^{−1};

^{c}RMSE in cm

^{−1}; ν: stretching; β: in-plane bending; γ: out-of-plane bending; δ: formation; ω: out-plane rocking.

**Table 3.**Vibrational frequencies and dominant normal modes of multi-molecular fragment model and central molecular model for FIN, LAM, and REP

^{a}.

Name | Assignment | Exp. | Multi-Molecular Fragment Model | Central Molecular |
---|---|---|---|---|

FIN | νN_{2}H_{35} | 3429 | 3520.79 | 3522.08 |

A:νN_{1}H | 3349 | 3399.51 | 3437.05 | |

νC_{3}=0_{1}; νC_{1}=C_{2}; βN_{1}H_{36}; | 1688 | 1681.01 | 1664.17 | |

νC_{19}=0_{2}; βN_{2}H_{35}; | 1668 | 1665.21 | 1662.83 | |

βC_{8,13,16}H_{9,14,19} | 1225 | 1225.5 | 1224.87 | |

γN_{1}H_{36}; ρC_{15}H_{17–18}; δ_{as}C_{17}H_{20–22} | 600 | 589.69 | 580.71 | |

MAE (for all data) ^{b} | 8.21 | 9.34 | ||

RMSE (for all data) ^{c} | 18.35 | 21.81 | ||

LAM | ν_{as}N_{3}H_{3–4} | 3383 | 3393.37 | 3484.85 |

ν_{s}N_{3}H_{3–4} | 3327.9 | 3232.27 | 3331.83 | |

νC_{1}O_{1} | 1651 | 1653.55 | 1671.64 | |

βN_{3}H_{3}; νC_{3}N_{2} | 1498 | 1498.19 | 1467.85 | |

γC_{2}H_{1}; ρC_{8}H_{9–10} | 1030 | 1033.46 | 1031.56 | |

γN_{3}H_{3}; γC_{5}H_{5} | 752 | 757.55 | 757.16 | |

MAE (for all data) ^{b} | 15.95 | 20.36 | ||

RMSE (for all data) ^{c} | 26.46 | 30.40 | ||

REP | νO_{4}H_{36} | 3428 | 3415.14 | 3162.17 |

νN_{2}H_{35} | 3307 | 3395.98 | 3424.00 | |

νC_{25}O_{3}; βO_{4}H_{36} | 1686 | 1689.41 | 1666.48 | |

νC_{17}O_{1}; βN_{2}H_{35}; γC_{18}H_{8} | 1635 | 1632.85 | 1621.51 | |

βC_{7–12,14}H; βN_{2}H_{35} | 1112 | 1125.59 | 1117.81 | |

γN_{2}H_{35}; γO_{4}H_{36}; γC_{26}H; ring prckering vibration | 763 | 758.78 | 743.32 | |

MAE (for all data) ^{b} | 12.10 | 23.12 | ||

RMSE (for all data) ^{c} | 25.82 | 52.21 |

^{a}Frequencies are in cm

^{−1};

^{b}MAE in cm

^{−1};

^{c}RMSE in cm

^{−1}; ν: stretching; β: in-plane bending; γ: out-of-plane bending; δ: formation; ρ: in-plane rocking.

Drug | Process | Single Molecule Model (h) | Central Molecule Model (h) | Multi-Molecule Fragment Model (h) |
---|---|---|---|---|

FIN | Configuration optimization | 0.27 | 4.18 | 4.18 |

Frequency calculation | 2.51 | 2.44 | 4.10 | |

LAM | Configuration optimization | 0.10 | 15.40 | 15.40 |

Frequency calculation | 0.44 | 0.48 | 3.26 | |

REP | Configuration optimization | 0.26 | 26.21 | 26.21 |

Frequency calculation | 2.45 | 2.42 | 7.76 |

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## Share and Cite

**MDPI and ACS Style**

Wang, L.; Zhang, P.; Geng, Y.; Zhu, Z.; Yuan, S.
Harmonic Vibrational Frequency Simulation of Pharmaceutical Molecules via a Novel Multi-Molecular Fragment Interception Method. *Molecules* **2023**, *28*, 4638.
https://doi.org/10.3390/molecules28124638

**AMA Style**

Wang L, Zhang P, Geng Y, Zhu Z, Yuan S.
Harmonic Vibrational Frequency Simulation of Pharmaceutical Molecules via a Novel Multi-Molecular Fragment Interception Method. *Molecules*. 2023; 28(12):4638.
https://doi.org/10.3390/molecules28124638

**Chicago/Turabian Style**

Wang, Linjie, Pengtu Zhang, Yali Geng, Zaisheng Zhu, and Shiling Yuan.
2023. "Harmonic Vibrational Frequency Simulation of Pharmaceutical Molecules via a Novel Multi-Molecular Fragment Interception Method" *Molecules* 28, no. 12: 4638.
https://doi.org/10.3390/molecules28124638