1 Introduction

Due to the short pulse widths (10–15 s), the interaction times with materials are much shorter than the lattice thermal equilibrium times (10–12 s). Compared to long pulse width lasers (nanosecond, picosecond), the femtosecond energy is concentrated in a very small area that is not easily transferred out of focus. This reduces thermal effects in the material, making femtosecond laser processing a form of “cold processing” [1, 2]. It has therefore been used in the precision manufacture of special explosive components and in the disposal of waste ammunition [3, 4]. The absorption of photons is very important for the effective removal of energetic materials.

There is a wide range of laser absorption in explosives. The selectivity of different molecular structures can be determined by studying the absorption mechanisms. With the right technical advice on laser parameters, femtosecond laser ablation can be used to process energetic materials with high efficiency, precision and safety. The absorption mechanism of energetic materials for lasers of different wavelengths varies, mainly including absorption in two bands, the ultraviolet to the visible and the near infrared. Short-wavelength high-energy UV–vis laser (i. e., ≈ 100 nm to ≈ 750 nm) induces valence electron transitions with electron carriers [5]. After losing electrons, excited molecules are highly ionised and can cause photochemical reactions between other molecules.

Long-wavelength, low-energy IR (i. e., ≈ 750 nm to ≈ 1000 μm) lasers induce vibrational excitations that intensify the thermal motions of explosive molecules and trigger bond breaking of the corresponding functional groups [6]. The result is a thermochemical reaction. There is no strict boundary between the absorption mechanisms, such as near-IR laser (i. e., ≈750 nm to ≈ 2.5 μm). In this region, the absorption mechanism could be complex, in the form of valence electron transitions and molecular vibrations [7].

The molecular structure influences the absorption mechanism. When a conjugated structure was present, such as butadiene (CH2=CH–CH=CH2), benzene (C6H6), naphthalene (C10H8), and anthracene (C14H10) [8], a strong effect would occur in the outer electron orbitals of the molecule, and the energy required for valence-electron transitions would be reduced. This causes the absorption peak to be “redshifted” [9]. Conjugated explosives can thus absorb low laser energy for valence electron transitions. When the vibrational frequencies of hydrogen-containing groups (e.g., O–H, N–H, and C–H) coincide with the frequency of near-IR laser, an explosive molecule was able to absorb laser energy in the form of molecular vibrations.

Absorption spectra can be calculated and analysed using first-principles calculations. These calculations can describe electronic transitions and molecular vibrations because they are based on quantum mechanics. Thus, absorption mechanisms of energetic materials can be characterised. In general, the UV–vis absorption spectrum can be obtained by analysing electronic transitions using the first-principles method of time-dependent density functional theory (TDDFT). In 2006, Zheng et al. [10] calculated the UV–vis spectra of polyazacyclic energetic compounds via TDDFT and found strong UV absorption over 193–298 nm. In 2013, Cooper et al. [11] used TDDFT for six explosives (RDX, HMX, PETN, TNT, TATP (triacetone triperoxide) and HMTD (hexamethylene triperoxide diamine)) to obtain UV–vis absorption spectra. All six explosives had strong absorption peaks.

For IR laser, the first-principles calculations of vibrational excitations are generally used to obtain absorption spectra. In 2010, Karabacak et al. [12] obtained the IR spectrum of 2-aminoterephthalic acid based on first-principles vibrational frequency analysis. In 2016, Yuan et al. [13] obtained the IR spectrum of RDX based on first-principles vibrational calculations and found that it had a strong absorption peak.

However, there have been few reports on the near-IR absorption spectra in the transition region by first-principles calculations. Most of the current high-power femtosecond laser sources operate in the near-IR, and the absorption mechanisms of energetic materials in this region are unclear. Here, the absorption spectra of the explosives β-HMX, TATB, RDX, PETN, and TNT have been calculated using first-principles DFT methods and vibrational frequency analysis. The electronic transitions were analysed, and spectral parameters such as excitation energy, electric dipole moment, and oscillator strengths were calculated. UV–vis and IR absorption spectra were obtained, and the near-IR absorption characteristics of the explosives were analysed.

2 Calculation method and model

2.1 Calculation method of UV–vis spectra

The electronic transitions could be analysed by quantum calculations, which involved solving the Schrödinger equation. Solving the Kohn–Sham equation was an important step in obtaining an approximate solution to the Schrödinger equation. In 1984, Runge et al. [14] added time-dependent terms to the solution of the Kohn–Sham equation, and established TDDFT. The theory is well suited to time-dependent problems in multi-particle systems and can describe non-perturbative quantum physical processes occurring for electrons under the action of very strong, time-dependent external fields with both computational accuracy and efficiency. It has therefore been widely used to calculate energy level transitions.

TDDFT can accurately describe transitions in explosives and has been widely used to calculate valence electron transitions. Here, TDDFT was used to calculate the energy level transitions in the five explosives. By solving the time-dependent Kohn–Sham equation via cyclic iterations, parameters such as vertical excitation energies [15] \(\left( {E_{Exc} } \right)\), transition dipoles, [16] and corresponding oscillator strengths [17] \(\left( {f_{TOS} } \right)\) were obtained. The vertical excitation energy is the electron transition energy when the molecular geometric configuration remains unchanged during the transition. It has an accuracy of 0.1–0.3 eV and is expressed by:

$$E_{exc} = E_{i} - E_{j} { = }\frac{hc}{\lambda }$$
(1)

where \(E_{Exc}\) is the excitation energy required for the electron to transition to enter an excited state, and \(E_{i} - E_{j}\) is the energy difference between the excited state \(E_{i}\) and the ground state \(E_{j}\). The electric dipole moment between the nucleus and the electron can be described by [16]:

$$\mu = q_{0} \cdot {\mathbf{r}}$$
(2)

where \(\mu\) is the electric dipole moment, \(q_{0}\) is the electron charge, and \({\mathbf{r}}\) is the vector from the electron centre to the nuclear centre. As the electrons absorb laser photons and transition from the ground state to the excited state, the dipole moment changes accordingly. The change \(\Delta \mu\) can be expressed by:

$$\Delta \mu = q_{0} \cdot ({\mathbf{r}}_{i} - {\mathbf{r}}_{j} )$$
(3)

where \({\mathbf{r}}_{j}\) and \({\mathbf{r}}_{i}\) are the ground state and excited state vectors, respectively.

The oscillator strength [17] is given by:

$$f_{TOS} = f_{ij} = \frac{2}{3}(E_{i} - E_{j} )\left| {\left\langle {i\left| { - r} \right|j} \right\rangle } \right|^{2}$$
(4)

where, \((E_{i} - E_{j} )\) is the energy difference between the excited and ground states. \(\left\langle {i\left| { - r} \right|j} \right\rangle\) is the change in the electric dipole moment, and the oscillator strength \(f_{TOS}\) is a dimensionless quantity used to measure the transition strength between the molecular excited states. Large oscillator strengths typically indicate strong absorption peaks and significant absorption in the excited states. A UV–vis absorption spectrum can be calculated from the excitation energy and the oscillator strength.

In a UV–vis spectrum, the abscissa and ordinate represent the wavelength (\(\lambda\)) and absorption intensity (\(I_{UV - vis}\)) respectively, for the valence electron transitions, as given by [18, 19]:

$$\lambda { = }\frac{ch}{{E_{Exc} }}$$
(5)
$$I_{UV - vis} \propto f_{TOS}$$
(6)

where \(c\) is the speed of light, \(h\) is Planck’s constant, \(E_{Exc}\) is the excitation energy, and \(f_{TOS}\) is the oscillator strength. The UV–vis spectra calculated from Eqs. (5) and (6) were discrete points. A continuous curve similar to the experimental data can be obtained by “broadening” as shown in Fig. 1.

Fig. 1
figure 1

Schematic diagram of UV–vis spectrum calculation

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The UV–vis spectra is closer to experimental results when Gaussian broadening is used, which is given by:

$$G(\omega ) = \frac{1}{{c\sqrt {2\pi } }}e^{{ - \frac{{(\omega - \omega_{1} )^{2} }}{{2c^{2} }}}}$$
(7)

where, \(c = ({\text{FWHM}})/2\sqrt {2\ln 2}\), and \({\text{FWHM}}\) is the full-width at half-maximum.

2.2 Molecular vibration calculations for IR spectra

IR laser causes vibrational excitations, and different chemical bonds absorb in different infrared regions. The molecular harmonic approximation model [20] can be used to calculate vibrational frequencies to obtain IR absorption spectra. In the model, covalent bonds are approximated as springs connecting two atoms. In this case, the elastic potential energy of a harmonic oscillator can be used to characterize the stretching energy of the bond. In the calculations, each atom moves a small distance dx, dy, and dz in the x-, y-, and z-directions, causing a change in the second derivative of the energy, as given by [21, 22]:

$${\mathbf{D}}_{ij} = \frac{{\delta^{2} E}}{{\delta {\mathbf{R}}_{i} \delta {\mathbf{R}}_{j} }}$$
(8)

where \(\delta\) is the differential operator, \(E\) is the potential energy function, \(\delta {\mathbf{R}}\) is the differential of the atomic displacement given by \(\delta {\mathbf{R}} = (dx,dy,dz)\),\({\mathbf{D}}_{ij}\) is the second order derivative of the energy, and the second order tensor of the mathematical expansion is the Hessian matrix describing the local curvature of the potential function.

The vibrational frequency and intensity of the molecule can be calculated from the eigenvectors and eigenvalues of the Hessian matrix [23,24,25,26].

The vibrational frequency is given by:

$$\tilde{v}_{i} { = }\sqrt {\frac{{\lambda_{i} }}{{4p^{2} c^{2} }}}$$
(9)

where \(\tilde{v}_{i}\) is the frequency, \(\lambda_{i}\) is the eigenvalue of the Hessian matrix \({\mathbf{D}}_{ij}\), and \(c\) is the speed of light. The vibrational intensity is given by:

$$A(\omega ) \propto \int {\left\langle {\dot{\mu }(\tau )\dot{\mu }(t + \tau )} \right\rangle }_{r} e^{ - i\omega t} dt$$
(10)

where, \(\mu\) is the dipole moment, \(\omega\) is the frequency, and \(\left\langle {\dot{\mu }(\tau )\dot{\mu }(t + \tau )} \right\rangle\) is the autocorrelation function of the dipole moment \(\mu\). The Lorentz function can be used to broaden the infrared spectrum, and is given by:

$$L(\omega ) = \frac{{{\text{FWHM}}}}{2\pi }\frac{1}{{(\omega - \omega_{i} )^{2} + 0.25 \times {\text{FWHM}}^{{2}} }}$$
(11)

2.3 Calculation model and details

The gas phase monomolecular models of the explosives were first constructed and then TDDFT and vibrational frequencies were calculated. The monomolecular structures of β-HMX [27], TATB [28], RDX [29], PETN [30], and TNT [31] were obtained from crystalline X-ray diffraction data. Material Studio was used to place a single molecule in a 24 × 24 × 24-Å3 simulation box. As shown in Fig. 2, C, H, N, and O atoms were represented by grey, white, blue, and red colours, respectively. The simulation was represented by a purple box.

Fig. 2
figure 2

Schematic diagram of gas-phase monomolecular model of β-HMX, TATB, RDX, PETN and TNT

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CP2K [32] software was used to perform quantum-chemistry calculations. The input files were generated using the Multiwfn [33] program. First, the truncation energy was determined with a convergence test. The energies of β-HMX at different cut-off energies, where β-HMX was stable, and the number of lattices from levels 1 to 4 are listed in Table 1 and Fig. 3. When the cut-off energy was increased to 1100 Ry, it met the accuracy requirement of the calculations. When the truncation energy exceeded 1100 Ry, the number of four-level grids increased and the calculation time increased significantly. Therefore, a cut-off energy of 1100-Ry satisfied both calculation accuracy and speed and was used in the subsequent calculations.

Table 1 Cut-off energy, molecular energy, and number of meshes at levels 1–4 of β-HMX
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Fig. 3
figure 3

Curves of energy, number of lattices, and cut-off energy for β-HMX

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The DFT PBE functional and the DZVP-MOLOPT-SR-GTH basis set were then used to optimize the geometric structures of the monomolecular models to obtain steady-state structures with the lowest energies. The Broyden-Fletcher-Goldfarb-Shanno method was used for its fast calculation speed, and Kohn–Sham diagonalization was used to calculate the wave functions.

The predicted explosive structures were reasonable. For example, the β-HMX structure was not significantly distorted. The trend of HMX energy in structure optimisation is shown in Fig. S1. The energy of the HMX molecule was reduced to a minimum value within 10 ionic steps. The quantitatively calculated predicted and experimental values [34] of bond lengths and bond angles of HMX molecules are given in Table S1. The predicted bond lengths and angles deviated slightly from the experimental values, with the largest error was in C–H bond lengths. The predicted bond lengths of C1–H2 (1.090 Å) and C2–H4 (1.089 Å) were quantitatively similar, while there was a slight difference from the experimental values of C1-H2 (0.830 Å) and C2–H4 (1.070 Å). Thus, the quantitative predictions of C-H bond lengths were more accurate than the experimental measurements. Except for the C–H bond lengths, the deviations between the predictions and experimental measurements of β-HMX bond lengths and angles were small, with an error range of 0.206–6.667%. Thus, both the calculated β-HMX structure and the gas-phase monomolecular model were reasonable.

After the structural optimizations, quantum-chemistry calculations of the electronic transitions and vibrational absorptions were performed as follows. For the electronic transitions, TDDFT was used with a PBE functional and the aug-TZV2P-GTH basis set. The upper limit for the number of excited states was 40, and the cut-off energy was 1100 Ry. The excitation energy, electric dipole moment, and vibronic intensity parameters were obtained, and the relationship between the optical absorption mechanism and the energy level orbital characteristics of the explosives were analysed. A Python script was used to broaden the function for discrete excitation energy and oscillator strength data points to obtain UV–vis absorption spectra.

For vibrational excitation, quantitative calculations were performed using the vibrational frequency method, with the PBE functional and the TZV2P-MOLOPT-GTH basis set. The wave functions were calculated by Kohn–Sham diagonalization to obtain the frequencies and intensities. The of discrete frequency and intensity data points were functionally broadened using open-source post-processing Molden software [35, 36] to obtain IR absorption spectra for the explosives.

3 Results and discussion

3.1 UV–vis absorptions of explosives

3.1.1 Analysis of excitation energies, electric dipole moments, and oscillator strengths

The excitation energy is the energy required for an electron to transition to enter an excited state. The oscillator strength indicates the probability of the electronic transition. These parameters, together with the electric dipole moment, can describe the characteristics of the energy levels and provide laser excitation mechanisms for explosives and provide laser-excitation mechanisms of explosives.

The transition absorption mechanisms were closely related to the molecular structures, and explosives with conjugated structures were more likely to be electronically excited. The conjugation reduced the energy difference between the highest occupied orbital and the lowest unoccupied orbital, allowing the use a lower energy laser. The excitation energies and oscillator strengths for 40 excited states of the five explosives are shown in Fig. 4. Tables S2–S6 showed the excitation energies, electric dipole moments and vibronic strengths of the five explosives HMX, TATB, RDX, PETN and TNT, respectively. In the table, only 40 excited states of low orbitals (S1–S40) were given because the energy of the excited state S40 is more than 6.5 eV, which corresponds to a photon wavelength less than 190 nm and belongs to the far ultraviolet wavelength, which is not the wavelength range of interest in this paper. The electric dipole moment is a vector quantity, that characterizes the change in molecular polarity caused by the change in the orbital of the electron energy level. The excitation energies increased in Fig. 4a with the number of excited state levels, and were not continuous, but had several discrete values. β-HMX, TATB, RDX, PETN, and TNT had S1 excitation energies of 4.43 eV, 3.67 eV, 4.42 eV, 4.72 eV, and 3.66 eV, respectively. The transition to the S1 excited state was the lowest for the explosives to absorb laser photons. More energy was needed for transitions to higher excited states. In Fig. 4b, the oscillator strengths varied greatly, and those for the S8 and S9 excited states in TATB were greater than those for the other explosives.

Fig. 4
figure 4

Excitation energy and oscillator strength of β-HMX, TATB, RDX, PETN and TNT

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The electric dipole moment is a vector characterizing the change in molecular polarity caused by the change in the electronic energy level, and its unit is a.u. (1 a.u. = 8.478 × 10–30 C m). The energy level diagrams and electric dipole moment distributions for the five explosives are shown in Fig. 5, where the vertical coordinates were excited state energies. The horizontal coordinate of the energy level diagram was the energy level search path, and that for the electric dipole moment distribution was the magnitude of the dipole moment. The components of the dipole moment vector in the x-, y-, and z-directions are represented by blue, red, and green colours, respectively.

Fig. 5
figure 5

Energy level diagram and electric dipole moment distribution of β-HMX, TATB, RDX, PETN and TNT

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As shown in Fig. 5, the outer electrons jumped to different energy levels after absorbing photons of different energies. Shorter laser wavelengths increased the orbits of the outer electrons because higher absorbed energies allowed for stronger transitions. In order to analyse the structural characteristics of excited-state explosives, the excited state orbitals for the maximum electric dipole moments for β-HMX, TATB, RDX, PETN, and TNT were marked and listed in Table 2. When an electron jumps to the excited state orbitals, the dipole moments of different excited states changed differently. The change in the electric dipole moment reflects the change of the spatial configuration of the explosive molecules. The larger the electric dipole moment of the explosive molecules, the stronger the corresponding polarity of the explosive molecules, and configuration of the explosive molecules would be significantly distorted.

Table 2 Orbits of the maximum dipole moment of five kinds of explosive molecules
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3.1.2 UV–vis spectral analysis

After further processing of the data obtained in Sect. 3.1.1 using the methods in Sect. 2.1, the UV–vis absorption mechanisms and spectra of β-HMX, TATB, RDX, PETN were shown in Fig. 6a–e. The red line was the intensity of the oscillators corresponding to the different excited states of the explosives; the blue line was the UV–vis absorption spectrum obtained by Gaussian broadening. The comparison between the calculated value and the reported experimental UV–Vis absorption spectrum of β-HMX, TATB, RDX, PETN and TNT was shown in Fig. 6f. It is worth noting that all reported experimental UV–vis absorption spectrum curves were measured in the solution state, which may be the reason for a significant redshift compared to the calculated curves for HMX, RDX and TATB.

Fig. 6
figure 6

UV–Vis absorption spectrum of β-HMX, TATB, RDX, PETN and TNT

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The UV–vis absorption regions of β-HMX, TATB, RDX, PETN, and TNT were concentrated at 150–350 nm. The spectra of β-HMX, RDX, and PETN were concentrated over the range 170–225 nm with strong absorption peaks. TATB and TNT had a broader absorption region, with peaks in the range of 230–300 nm. The molecular skeletons of TATB and TNT consisted of benzene rings that had E and B absorption bands in the UV region. The 180–200 nm E band was further divided into E1 and E2 bands, where the E1 band was the π → π* transition of the isolated vinyl group on the benzene ring, and the E2 band was the π → π* transition of the conjugated diene group on the benzene ring. The B absorption band had a small peak at 250–300 nm characteristic of aromatic compounds, and was attributed to the benzene ring.

The calculated and experimentally measured maximum absorption peaks of the explosives are listed in Table 3. The calculated 180-nm RDX maximum absorption peak differed from the 227-nm experimental value. The experimentally measured maximum absorption peak was in agreement with the second absorption peak (approximately 227 nm) predicted by the calculation. The positions of the two largest peaks calculated by experiment and calculation were in agreement. On the other hand, the calculated maximum absorption peaks of β-HMX and TNT were more consistent with experimental results [38]. This indicated that the spectral characteristics of the explosives obtained by the TDDFT calculation method in this article were relatively reliable.

Table 3 Calculated estimate and experimentally measured values of the maximum UV–Vis absorption peak of explosives
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Therefore, the β-HMX, TATB, RDX, PETN and TNT explosives could absorb laser photons via the valence electron leap mechanism. They could absorb 150–350 nm laser wavelengths and undergo electronic transitions to excited states. In the excited state, the outer electrons were further away from stable orbits, making the bonding groups more likely to break. This resulted in unstable structures for the explosives, that could trigger photochemical reactions. However, the absorption of the laser transitions by the explosive molecules did not include near-infrared radiation at 810 nm.

3.2 IR vibrational excitation of explosives

The calculated IR spectra for β-HMX, TATB, RDX, PETN, and TNT are shown given in Fig. 7. The exocyclic groups have been labelled, with R representing the molecular backbone, and the vertical coordinate unit is A = L/(mol·cm). The IR absorption regions of the explosives were concentrated in the frequency range 0–3500 cm−1 (wavelengths > 2857 nm). The maximum frequency ranges of the vibrational absorption peaks for the different explosives were similar, with a frequency range of 2800–3500 cm−1 (wavelengths of 2857–3571 nm).

Fig. 7
figure 7

IR absorption spectrum of β-HMX, TATB, RDX, PETN and TNT

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Figure 8 showed the comparison of the calculated and reported experimental vibrational absorption spectrum for β-HMX [39], TATB [40], RDX [41], PETN [42] and TNT [43]. The calculated IR results were generally consistent with the literature, indicating the reliability of the calculation.

Fig. 8
figure 8

Comparison between calculated value and reported experimental infrared absorption spectrum of β-HMX [37], TATB [40], RDX [41], PETN [42] and TNT [43]

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Statistical analyses were performed on the exocyclic groups of β-HMX, TATB, RDX, PETN and TNT to analyse the vibrational absorption response in the IR absorption band, as shown in Table 4. The IR absorption derived from hydrogen-containing exocyclic groups, and the vibrational groups for the explosives were slightly different. In TATB, the R-NH2 bond contained two N–H bonds, and there were two parallel double absorption peaks in the 2800–3500 cm−1 region. While β-HMX, RDX, PETN, and TNT had C–H bonds with strong absorption peaks at 3125–3571 nm. The C–H2 groups on β-HMX, PETN, and TNT had two C–H structures with double peaks in the range 2800–3500 cm−1, while RDX molecules had only one C-H structure with one absorption peak in this region.

Table 4 Exocyclic groups of explosives and their corresponding absorption bands
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To induce strong vibrational absorption in explosives, the frequency of laser photons must to be close to the vibrational frequency of the infrared functional group of the explosive molecules (e.g., C–H or R-NH2, etc.), thus inducing a strong resonance absorption effect in the material. β-HMX, TATB, RDX, PETN and TNT could absorb laser energy via vibrational excitations, with frequency distributions ranging over 2800–3500 cm−1 (2857–3571 nm). Thus, an IR laser in this wavelength band could trigger vibrational excitations.

However, the absorption frequencies of the infrared spectrum of explosives were far from the 810-nm (12,516 cm−1) near-IR band laser, which is widely used for femtosecond laser machining. Therefore, it is difficult for the near-IR laser to induce resonant absorption of explosives molecules.

4 Conclusions

Here, first-principles techniques were used to determine the mechanisms for vibrational and electronic transition absorption of various explosives. The absorption mechanisms were comprehensively analysed using vibrational frequency analysis and time-dependent density functional theory. The calculations indicated that the explosives would be selective with respect to laser wavelengths.

Different explosives had different UV–vis absorption characteristics. Under certain wavelengths of laser, the explosive molecules undergo a transition phenomenon due to the absorption of photon energy, resulting in the excited state of the explosive molecules. Then the outer electrons of each atom of the explosives in the excited state were farther away from the bonding orbitals, making the internal bonding groups of the explosive easier to break, causing instability in the chemical structure of the explosive molecule and triggering photochemical reactions. The spectra for β-HMX, RDX, and PETN were concentrated in the 170–225 nm range with strong absorption peaks. In contrast, TATB and TNT had broader absorption ranges over 230–300 nm, with strong absorption peaks. The IR absorption ranges of all five explosives were concentrated at wavelengths > 2857 nm (0–3500 cm−1), and the maximum frequencies of the vibrational absorption peaks were similar, with wavelengths ranging over 3125–3571 nm (2800–3500 cm−1). These bands were far from the 810 nm near-IR femtosecond lasers. Therefore, single-photon electronic transition and vibrational excitation with these lasers would not be the main pathway by which these explosives absorbed near-IR femtosecond laser photons.