Identification of Unknown Peaks in Mass Spectra Based on the Characteristic Isotopic Pattern via the Hilbert–Noda Matrix

Reagents

1,2-dibromoethane (CP grade) was obtained from Shanghai Chemical Reagent Station Central Chemical Factory. Chloroform, high-performance liquid chromatography (HPLC) grade, was purchased from Beijing Tong Guang Fine Chemicals Company.

Instruments

In thermogravimetric analysis-mass spectrometry (TGA-MS) experiments, the mass spectra of 1, 2-dibromoethane and chloroform have been measured on a TGA quadrupole mass spectrometer (TGA-8000, Clarus SQ8  T, Perkin Elmer). In the experiment, the temperature of the samples was elevated at a rate of 20 °C min⁻¹ and high-purity N₂ at a flow rate of 20 mL/min and 40 mL/min were used as sample and balance purging gas, respectively. The emitted gas of the samples was transferred to the Quadrupole Mass spectrometer via high-purity N₂. The mass spectra of the sample were measured under an EI mode, and the total ion chromatograms (TIC) of each sample were recorded. The mass spectra obtained via this experiment are low-resolution spectra, and the difference in the mass-to-charge ratio (m/z) between adjacent data points is 1.

The high-resolution GC-MS of 1,2-dibromoethane was performed on a Hybrid Quadrupole-Orbitrap GC-MS System (Q-Exactive GC, Thermo). The data acquisition software was TraceFinder 5.0, and the data analysis software was Xcalibur 4.0. The samples were injected into a TG-5HT, 30  m × 0.25  µm GC column for chromatography, the flow rate was 1.0 mL/min of helium, and the temperature gradient was set as 40 °C for 2 mins; to 350 °C at 15 °C/min and held for 2.25 mins. The injector was heated to 250 °C and was on split mode with a split ratio of 1:10. Electron ionization (EI)-MS spectra were collected at 70  eV.

Data Processing

The assignment of some unknown peaks in mass spectra was accomplished via programs written by our group using the Matlab platform.

An Approach to Assigning Unknown Peaks to a Molecule/Fragment Containing Isotopic Elements with a Specific Composition

Herein, we use a fragment containing three chlorine atoms (RCl₃⁺) as an example to describe the basic idea of the approach in this paper. Chlorine has two stable isotopes (³⁵Cl and ³⁷Cl). As a result, four isotopologues are produced:

A fragment containing three ³⁵Cl atoms. The formula and the nominal m/z value of this species are R³⁵Cl₃⁺ and M, respectively.

Based on the above result, a characteristic vector to describe the theoretical intensities of the four isotopologues based on the natural abundance of the isotopes of ³⁵Cl/³⁷Cl and the chemical composition of RCl₃⁺ is constructed as Eq. 4. Then, we inspect a mass spectrum. The mass spectrum can be expressed as y(x), where x is the mass-to-charge ratio. We have x∈[X_min, X_max], where X_min and X_max are respectively the lowest and the largest mass to charge ratio observed in the mass spectrum. For any given x, a testing vector b → (x) is constructed via Eq. 5.

b ⇀ ( x ) = ( y ( x ) y ( x + 2 ) y ( x + 4 ) y ( x + 6 ) )

(5)

Via vector a → _RCl3 and vector b → (x), an auxiliary spectrum (R(x)) for RCl₃⁺ in the given mass spectrum is generated via Eq. 6.

R RCl 3 + ( x ) = | a → RCl 3 | ⋅ | b → ( x ) | | ( a → RCl 3 ) T N b → ( x ) |

(6)

For a specific x = m₀, if the profiles of a → and b → (m₀) are the same, that is:

b → ( m 0 ) = k a → RCl 3

(7)

( a → RCl 3 ) T N b → ( m 0 ) = ( a → RCl 3 ) T N ( k a → RCl 3 ) = k ( a → RCl 3 ) T N a → RCl 3 = 0

(8)

As shown in Eq. 6, |( a → _RCl3)^TN b → (x)| appears in the denominator. If the value of |( a → _RCl3)^TN b → (x)| is zero, the corresponding R(m₀) will become infinity. In a real-world experiment, however, noise is inevitably present in every data point. Hence, Eq. 7 can be modified as Eq. 9.

b → ( m 0 ) = k a → RCl 3 + δ →

(9)

Thus, ( a → _RCl3)^TN b → (m₀) can be expressed as

( a → RCl 3 ) T N b → ( m 0 ) = k ( a → RCl 3 ) T N ( a → RCl 3 ) + ( a → RCl 3 ) T N δ →

(10)

( a → RCl 3 ) T N b → ( m 0 ) = | δ → | ( a → RCl 3 ) T N [ δ → | δ → | ]

(11)

The module δ → is quite small, the final value of |( a → _RCl3)^TN b → (m₀)| is quite close to zero. Even if |( a → _RCl3)^TN b → (m₀)| is indeed zero, protective approaches are adopted to turn |( a → _RCl3)^TN b → (m₀)| into a small but nonzero value in the corresponding program, thereby preventing the occurrence of a divide-by-zero error. When the near-zero value is present in the dominator, the corresponding value of R_RCl3(m₀) becomes very large. Thus, y(m₀) is highlighted in the auxiliary spectrum. The presence of highlighted peaks at x = m₀ in the auxiliary spectrum serves as an alarm, suggesting the peak at x = m₀ arises from RCl₃ based on the corresponding characteristic isotopic pattern in the mass spectrum.

Concerning the auxiliary spectrum, the following issues need to be discussed:

When the module of b → (x) is quite small, the corresponding peak will be incorrectly magnified even if a significant difference exists between the profiles of b → (x) and that of a → _RCl3. To avoid this undesirable situation, the module of b → (x) (| b → (x)|) is placed in the numerator of Eq. 6, thereby cancelling the contribution of | b → (x)| from the final values of R_RCl3(x). In a similar manner, the contribution of | a → _RCl3| in R_RCl3 + (x) has also been cancelled by putting | a → _RCl3| in the numerator of Eq. 6.

To address this problem, a modified auxiliary spectrum is generated via the following procedure.

The n-dimensional characteristic vector a → , which is expressed as Eq. 12, is reconstructed into n-1 two-dimensional sub-vectors (₁ a → , ₂ a → , …, _n-1 a → ).

a ⇀ = ( a 1 a 2 … a n )

(12)

i a → = ( a 1 a i + 1 )

(13)

The n-dimensional testing vector b → (x) at x from the given mass spectrum, which is expressed as Eq. 14, is also reconstructed into n-1 2-dimensional sub-vectors (₁ b → (x), ₂ b → (x), …, _n-1 b → (x).

( b → x ) = ( y ( x ) y ( x + m 1 ) ⋮ y ( x + m n − 1 ) )

(14)

The modified auxiliary spectrum is calculated via Eq. 16.

R ( x ) = n − 1 ∑ i = 1 n − 1 | ( i a → ) T N 2 ( i b → ( x ) ) | | i a → | ⋅ | i b → ( x ) |

(16)

N 2 = ( 0 1 − 1 0 )

(17)

The reason we use Eq. 16 to construct the modified auxiliary spectrum is based on the following consideration:

a → =k b → (x) is equivalent to the following statement. For every i, where i∈{1, 2, …, n – 1}, we have _i a → =k·_i b → (x)

Based on the above consideration, we try to use the auxiliary spectra generated via Eq. 6 and Eq. 16 to identify unknown peaks from fragments containing bromine atom/atoms in the mass spectrum of 1,2-dibromoethane (CH₂BrCH₂Br).

The mass spectra of 1,2-dibromoethane were obtained from a TGA-MS experiment. In the experiment, 3230 mass spectra were acquired. Figure 1 presents the obtained 3230 mass spectra via a contour map.

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Figure 1.

A contour map of the 3230 mass spectra obtained from the TGA-MS experiment.

Herein, we try to use the auxiliary spectra generated via Eq. 6 and Eq. 16 to identify the peaks of fragments containing different amounts of bromine atoms.

Identification of Fragments Containing a Single Bromine Atom

Figure 2a is a typical auxiliary spectrum of 1,2-dibromoethane. From the experimental result, we have never found any peak whose m/z value is larger than 200. Figure 2a only focuses on the mass spectrum whose m/z values are below 200.

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Figure 2.

(a) A typical auxiliary spectrum of 1,2-dibromoethane obtained from the TGA-MS experiment. (b) An auxiliary spectrum generated via Eq. 6 to highlight the peaks for possible fragments of CmHnBr⁺. (c) The corresponding statistically auxiliary spectrum obtained from the 3230 auxiliary spectra.

Firstly, we try to identify the fragments containing a single bromine atom. Bromine has two stable isotopes (⁷⁹Br and ⁸¹Br). The corresponding isotopologues are (C_mH_n⁷⁹Br⁺ and C_mH_n⁸¹Br⁺). Since the nominal mass difference between C_mH_n⁷⁹Br⁺ and C_mH_n⁸¹Br⁺ is 2, the corresponding nominal m/z values for C_mH_n⁷⁹Br⁺ and C_mH_n⁸¹Br⁺ are M and M + 2, respectively.

It should be pointed out that C_mH_nBr⁺ contains carbon atoms and hydrogen atoms, which also have stable isotopes (¹²C, ¹³C for a carbon atom and ¹H, ²H for a hydrogen atom). Thus, the contributions from the isotopologues, including ¹²C_m-2¹³C₂¹H_n⁷⁹Br⁺, ¹²C_m-1¹³C₁¹H_n-1²H₁⁷⁹Br⁺, and ¹²C_m¹H_n-2²H₂⁷⁹Br⁺, which have contributions to the peak whose m/z value is M + 2, should also be considered.

However, the natural abundances of ²H and ¹³C are significantly lower than those of ¹H and ¹²C. Moreover, the maximal number of carbon atoms in the fragment containing a single Br atom is less than five in the mass spectrum of 1,2-dibromoethane. In Section 1 of the Supplemental Material, we have calculated the contents of ¹²C_m-2¹³C₂¹H_n⁷⁹Br⁺, ¹²C_m-1¹³C₁¹H_n-1²H₁⁷⁹Br⁺, and ¹²C_m¹H_n-2²H₂⁷⁹Br ⁺ . The calculation results demonstrate that the contents of ¹²C_m-2¹³C₂¹H_n⁷⁹Br⁺, ¹²C_m-1¹³C₁¹H_n-1²H₁⁷⁹Br⁺, and ¹²C_m¹H_n-2²H₂⁷⁹Br⁺ are very low compared with C_mH_n⁷⁹Br⁺ and C_mH_n⁸¹Br ⁺ . Thus, the spectral contribution of ¹²C_m-2¹³C₂¹H_n⁷⁹Br⁺, ¹²C_m-1¹³C₁¹H_n-1²H₁⁷⁹Br⁺, and ¹²C_m¹H_n-2²H₂⁷⁹Br⁺ can be safely neglected, and C_mH_n⁷⁹Br⁺ and C_mH_n⁸¹Br⁺ are the dominant species among the possible isotopologues. According to the analysis presented in Section 2 of the Supplemental Material, we have: On the other hand, b → (x) can be extracted from the mass spectrum as Eq. 19.

Then, we try to generate an auxiliary spectrum to help us to identify fragments containing a single bromine atom. Since both a → and b → (x) are two-dimensional vectors, the auxiliary spectrum is produced via Eq. 6, and the obtained auxiliary spectrum is displayed in Figure 2b. In Figure 2b, several peaks whose m/z values are 79, 80, 107, 120, and 135 are highlighted. However, we found that most highlighted peaks (peaks at 79, 80, 120, and 135) in Figure 2b are too weak to be observed in the original mass spectrum shown in Figure 2a. When we magnify the original mass spectrum, the corresponding peak pairs can indeed be observed (Figure S1, Supplemental Material). We notice that some highlighted peaks in Figure 2b are quite weak in the original mass spectrum. The interference of noise in the original mass spectrum might cause b → (x) to align coincidentally with a → _CmHnBr+. Consequently, some peaks might be false-positively highlighted in the auxiliary spectrum under the interference of random noise.

Fortunately, both TGA-MS and GC-MS experiments produce multiple mass spectra. This experimental setup provides a chance to address the problem that highlighted peaks in an auxiliary spectrum actually originated from accidental orthogonality caused by noise. Herein, the following statistical approach has been adopted.

We generate an auxiliary spectrum for each of the 3230 original mass spectra. For every m/z value (denoted as x), a statistical function S(x) is generated. For each given mass-to-charge ratio (x), the value of S(x) is the number of auxiliary spectra whose R(x) value is larger than a given threshold (herein, the threshold is tentatively set as 20). The selection of the 20 as a threshold is based on the following consideration: according to Eq. 6 and Eq. 16, the better the match between the pattern of a → and the pattern of b → (x), the higher the value of R(x) is. Theoretically speaking, the threshold value should be high. However, the high value of the threshold often results in some real peaks failing to be highlighted because of the interference of noise in the original mass spectra. If the threshold value is too low, many undesirable false-positive peaks are produced. After a try-and-error process, the threshold value is tentatively set as 20.

For convenience, the S(x) is called the statistically auxiliary spectrum hereafter in this manuscript. The resultant S(x) for C_mH_nBr⁺ is displayed in Figure 2c. We notice that some peaks, which are weak in R(x) shown in Figure 2b, become quite strong in S(x) shown in Figure 2c. For example, the peak at m/z = 79, which is quite weak in Figure 2b, turns out to be a dominant peak in Figure 2c. On the other hand, the peak at m/z = 172 is quite strong in Figure 2b, while the peak at m/z = 172 turns out to be buried in the background in Figure 2c. The statistical nature of the result shown in Figure 2c can effectively suppress the possibility that these peaks are originated from coincidental self-orthogonality. Thus, we believe the peaks highlighted in Figure 2c are reliable and indeed originated from fragments containing a single bromine atom. In Figure 2c, five peaks (the m/z values of these peaks are: 79, 80, 107, 121, and 135) are highlighted in at least 1500 auxiliary spectra.

According to the above results, a tentative assignment on the above five peaks is proposed, and the assignments are listed in Table I.

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Table I.

A tentative assignment of the five highlighted peaks observed in the statistically auxiliar spectrum shown in Figure 2c for the fragments containing a single Br atom.

Peak positions in the statistically auxiliary spectrum	Tentative assignment	Peak positions for the corresponding doublet in the high-resolution mass spectra		The difference in peak positions between the doublet
79	Br+	78.9176	80.9157	1.9981
80	HBr+	79.9254	81.9236	1.9982
107	C2H4Br+	106.9489	108.9468	1.9979
121	C3H6Br+	120.9645	122.9626	1.9981
135	C4H8Br+	134.9805	136.9783	1.9978

Wu et al. have investigated the mass spectra of 1,2-dibromoethane under an intense femtosecond laser.^78,79 In these investigations, they have observed the mass peaks arising from Br⁺, whose m/z is 79, and C₂H₄Br⁺, whose m/z value is 107. These results support our assignment of the two highlighted peaks, whose m/z values are 79 and 107 in Figure 2c.

To validate the above results, the following experiments are carried out: We have recorded the mass spectra of 1,2-dibromoethane via GC coupled with high-resolution mass spectra. Firstly, we examine the mass spectra obtained from the elution peak of 1,2-dibromoethane. The highlighted peak in Figure 2c at m/z = 107 corresponds to the doublet at m/z = 106.948989 and m/z = 108.946758 in the high-resolution mass spectrum. The mass difference between the doublet is 1.997769, which matches the mass difference between ⁷⁹Br and ⁸¹Br perfectly. This result confirms that the fragments for the highlighted peak at m/z = 107 in Figure 2c indeed contain a single bromine atom. Moreover, the exact mass of the peaks whose m/z are 106.9490 and 108.9468 confirms that the two peaks can be safely assigned to C₂H₄⁷⁹Br⁺ and C₂H₄⁸¹Br⁺, respectively. The above results support the assignment of the peak whose m/z is 107 in Figure 2c as a fragment of C₂H₄Br ⁺ . Via a similar manner, the peaks whose m/z are 79 and 80 are confirmed to be from Br⁺ and HBr⁺ fragments.

Then, we come to the highlighted peak at 121 and 135 in Figure 2c. In the mass spectra obtained from the elution peaks of 1, 2-dibromoethane, we cannot find the doublets corresponding to the highlighted peak around 121 and 135. We have also queried the database of high-resolution spectra. From the records of the mass spectra of 1,2-dibromoethane, no peaks at 121 and 135 are observed. The above results clearly indicate that the two peaks are not from the fragments of 1,2-dibromoethane. From the tentative assignment listed in Table I, the peaks may arise from C₃H₆Br⁺ and C₄H₈Br⁺, respectively. The assignments suggest the fragments are produced by merging different fragments in the EI process. However, such events can hardly take place in a high-vacuum environment.

Thus, we need to explain why the peaks at 121 and 135 are clearly observed in Figure 2c. As a matter of fact, more than 2000 highlighted peaks at 121 and 135 are found in the 3230 auxiliary spectra. When we examined the GC/high-resolution data of 1,2-dibromoethane, the doublets for the highlighted peaks at 121 and 135 are indeed found. The mass difference between the doublet and the exact mass of these peaks supports that the two peaks at 121 and 135 in Figure 2c are originated from C₃H₆Br⁺ and C₄H₈Br⁺, respectively. However, the fragments are not produced from the eluted peak of 1, 2-dibromoethane from the GC process, but arise from low-abundance impurities in the sample. Thus, the identified peaks at 121 and 135 via the approach described in this paper really belong to the fragments C₃H₆Br⁺ and C₄H₈Br⁺ originated from the impurity of the 1,2-dibromoethane sample.

In the work on the mass spectra of 1,2-dibromoethane, Wu et al.^78,79 demonstrated that a fragment of CH₂Br⁺ whose m/z value is observed is 93 in the mass spectrum. Such a fragment contains a single bromine atom, and the peak for this fragment should be highlighted in the auxiliary spectrum shown in Figure 2b. However, the peak at 93 was not observed in Figure 2b. In the corresponding results of statistical analysis (S(x)) shown in Figure 2c, the peak at 93 was not highlighted, either. Figure S2 (Supplemental Material) displays the histogram of R(93) of the 3230 auxiliary spectra. The values of most mass spectra are below the threshold (the value of the threshold is set as 20). This is the reason the peak at 93 was not highlighted in the auxiliary spectra in Figure 2c. To find why the values of R(93) are not high enough, we examined the original mass spectra. Some typical mass spectra are present in Figure S3. In these mass spectra, the peaks at 93 and 95 are present. However, the peaks at 93 are stronger than the peak at 95. The values of y(93)/y(95) are larger than 120%. That is to say, the corresponding b → (93) does not match well with a → _CmHnBr+, which is shown in Eq 17, is the reason the peak at 93 is not standout in the auxiliary spectra. A possible reason for the above phenomenon is that some impurity in the sample has a peak around 93. In the low-resolution mass spectra obtained via the TGA-MS experiment, the peak for CH₂Br⁺ at 93 and the peak from the impurity around 93 merge into a single peak. Consequently, the intensity of the peak at 93 is considerably stronger than the peak at 95 in the original mass spectra, resulting in a poor match between b → (93) and a → _CmHnBr+. To test whether this hypothesis is correct or not, we return the high-resolution mass spectra. Figure S4 presents the Total ion chromatogram of the 1,2-dibromoethane sample. The strong elution peak at t = 4.60  min is from 1,2-dibromoethane. In addition, there are several elution peaks that are originated to impurity that can be observed. Figure S5 is a typical mass spectrum from the elution peak of 1,2-dibromoethane at t = 4.60  min. In the spectrum, the doublet from CH₂Br⁺ at 92.9333 and 94.9313 can be clearly visualized. The intensities of the two peaks are almost the same. The mass difference between the doublet and the exact mass of the two peaks confirms that CH₂Br⁺ appears in the mass spectrum of 1,2-dibromoethane. Figure S6 displays a typical mass spectrum from the elution peak of impurity at t = 5.63  min. In the spectrum, a strong peak at 92.9156 is present. There is no strong corresponding peak around 95 for the peak at 92.9156. Additionally, a weak doublet for CH₂Br⁺ can be observed at 92.9331 and 94.9312. The peak positions for the peaks at 92.9333 and that at 92.9156 are quite close. In the low-resolution TGA- MS experiment, the peak at 93.9156 and the peak at 92.9333 merge into a single peak. Consequently, the overlapping for the peak of CH₂Br⁺ at 92.9333 and the peak at 91.9156 from the impurity is the reason the peak for CH₂Br⁺ was not identified by the approach described in this paper.

Identification of Fragments Containing Two Bromine Atoms

Subsequently, we try to identify fragments containing two bromine atoms (The formula of the fragment is C_mH_nBr₂⁺) from the mass spectrum. The corresponding isotopologues are (C_mH_n⁷⁹Br₂⁺, C_mH_n⁷⁹Br⁸¹Br⁺, and C_mH_n⁸¹Br₂⁺). The corresponding m/z values for C_mH_n⁷⁹Br₂⁺, C_mH_n⁷⁹Br⁸¹Br⁺, and C_mH_n⁸¹Br₂ ⁺ are M, M + 2, and M + 4, respectively.

Carbon atoms and hydrogen atoms also have stable isotopes. Thus, isotopologues containing multiple ¹³C, ²H atoms may also have a spectral contribution to the peaks whose m/z are M + 2 and M + 4. Here, the situation of ²H and ¹³C is similar to the discussion for fragments containing one bromine atom in 1 of the Supplemental Material. The spectral contribution from isotopologues containing multiple ¹³C, ²H atoms can be safely neglected.

In this case, we only need to consider the following three isotopologues: C_mH_n⁷⁹Br₂⁺, C_mH_n⁷⁹Br⁸¹Br⁺, and C_mH_n⁸¹Br₂⁺. The natural abundance of ⁷⁹Br and ⁸¹Br is, respectively, a(⁷⁹Br) = 0.5069 and a(⁸¹Br) = 0.4931. Thus, the percentage of C_mH_n⁷⁹Br₂⁺ fragment can be calculated as C 2 2 a² = 0.5069² = 0.2569; the percentage of C_mH_n⁷⁹Br⁸¹Br⁺ fragment is C 2 1 ab = 2 × 0.5069 × 0.4931 = 0.4999; and the percentage of C_mH_n⁸¹Br₂⁺ fragment is C 2 0 b² = 0.4931² = 0.2431. Therefore, a → _CmHnBr2+ is present in Eq. 20.

a → C m H n B r 2 + = ( [ a ( 79 Br ) ] 2 2 [ a ( 79 Br ) ] [ a ( 81 Br ) ] [ a ( 81 Br ) ] 2 ) = ( 0.2569 0.4999 0.2431 )

(20)

Then, we try to use the auxiliary spectra to identify fragments containing two bromine atoms in the mass spectrum. For comparison, Figure 3a presents a typical mass spectrum of 1,2-dibromoethane.

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Figure 3.

(a) A typical mass spectrum of 1,2-dibromoethane. (b) An auxiliary spectrum generated via Eq. 6 to highlight the peaks for possible fragments of CmHnBr₂⁺. (c) An auxiliary spectrum generated via Eq. 16 to highlight the peaks for possible fragments of CmHnBr₂⁺. (d) The corresponding statistically auxiliary spectrum obtained from the 3230 auxiliary spectra produced with Eq. 6. (e) The corresponding statistically auxiliary spectrum obtained from the 3230 auxiliary spectra produced with Eq. 16.

Since the dimension of a → is 3, the corresponding auxiliary spectra for C_mH_nBr₂⁺ are generated via Eq. 6 and Eq. 16, respectively. The auxiliary spectrum produced via Eq. 6 (denoted as R₁(x)) is present in Figure 3b, and the auxiliary spectrum generated via Eq. 16 (R₂(x)) is present in Figure 3c.

We notice that the original peaks that are highlighted in Figure 3b and Figure 3c are too weak. The interference of noise in the original mass spectrum might make the relative intensities of the peaks at M, M + 2, and M + 4 happen to be close to those of the element of vectors showing in Eq. 19. Consequently, some peaks may be false-positively highlighted in the auxiliary spectra via the accidental orthogonality. To avoid the production of false positive peaks, the statistically auxiliary spectra mentioned in the previous paragraphs are adopted. The statistically auxiliary spectrum produced from the 3230 R₁(x) is denoted as S₁(x), and the statistically auxiliary spectrum produced from the 3230 R₂(x) is denoted as S₂(x). The resultant S₁(x) and S₂(x) are displayed in Figure 3d and Figure 3e, respectively. Since the accidental orthogonality can be effectively avoided via the statistical approach, we focus on Figure 3d and Figure 3e. Compared with Figure 3e, more peaks are highlighted in Figure 3d. To find out the reason for the difference in the highlighted peak between Figure 3d and Figure 3e, we selected a peak at 192 as an example. The peak was considerably highlighted in Figure 3d, but was not highlighted in Figure 3e. A typical mass spectrum is shown in Figure S7. If the highlighted peak is really from a fragment of C_mH_nBr₂⁺, the corresponding triplet should appear at 192, 194, and 196. The ratio of intensities among the three peaks should be roughly 1:2:1. In Figure S7, the peaks at 192 and 194 are present; however, the peak at 196 turns out to be absent. Moreover, the ratio of the intensity between the peaks at 192 and 194 is far from 1 : 2. Thus, the peak at 192 in Figure 3d was false-positively highlighted via Eq. 6 as the dimensions of both a → _CmHnBr2+ and b → (x) are larger than 2. In this case, the results from auxiliary spectra generated via Eq. 16 and the corresponding statistically auxiliary spectra are more reliable. In Figure 3e, four peaks are identified, and the m/z values for the four peaks are 158, 171, 185, and 186. A tentative assignment of the above four peaks to a fragment containing two Br atoms is listed in Table II. Wu et al. have investigated the mass spectra of 1,2-dibromoethane under an intense femtosecond laser (800  nm). In these investigations, they have observed the mass peaks arising from Br₂⁺, whose m/z is 158.^78,79 This result supports the assignment in Table II on the auxiliary peak at 158.

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Table II.

A tentative assignment of the four highlighted peaks observed in the statistically auxiliar spectrum shown in Figure 3e for the fragments containing two Br atoms.

Peak positions in the statistically auxiliary spectrum	Tentative assignment	Peak positions for the corresponding triplet in the high-resolution mass spectra			The difference in peak positions between the adjacent peaks in the triplet
158	Br2+	157.8357	159.8339	161.8318	1.9982	1.9979
171	CHBr2+	170.8436	172.8413	174.8391	1.9977	1.9978
185	C2H3Br2+	184.8589	186.8578	188.8552	1.9989	1.9974
186	C2H4Br2+	185.8669	187.8649	189.8638	1.9980	1.9989

For each of the highlighted peaks shown in Figure 3e, the corresponding triplets in the original mass spectrum are weak, but indeed occur. In addition, we have recorded the mass spectra of 1,2-dibromoethane via a GC coupled with high-resolution mass spectrometric experiment. For each highlighted peak in Figure 3e, the corresponding triplets are observable in the mass spectra obtained from the elution peak of 1,2-dibromoethane. In the triplet, the mass difference between the adjacent peaks is around 1.998, which matches the mass difference between ⁷⁹Br and ⁸¹Br. Moreover, the exact mass of the peaks confirms the assignments of the fragment containing two bromine atoms shown in Table II.

Concerning the Peak from RBr²⁺

Wu et al claimed the observation of the peak for Br²⁺ in the spectrum of 1,2-dibromoethane under the excitation of an intense femtosecond laser (800  nm).^78,79 We use the approach described in this paper to check whether peaks for C_mH_nBr²⁺ occur or not. Figure 4a displays a typical example of the original mass spectrum of 1,2-dibromoethane. The vector for C_mH_nBr²⁺ is shown in Eq. 22. Since C_mH_nBr²⁺ possesses two positive charges, the corresponding b → (x) is modified as Eq. 23.

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Figure 4.

(a) A typical example of the original mass spectrum of 1,2-dibromoethane. (b) The auxiliary spectrum generated via Eq. 6 to highlight the peaks for possible fragments of CmHnBr^2 + . (c) The corresponding statistically auxiliary spectrum.

Since the dimensions of both a → _CmHnBr2+ and b → (x) are 2, the auxiliary spectrum is generated via Eq. 6. Figure 4b displays the resultant auxiliary spectrum. If Br²⁺ is present, the highlighted peak should be observed at m/z = 39 in Figure 4b. However, no highlighted peak for Br²⁺ is present at 39. To remove the interference from noise, the statistically auxiliary spectrum is also constructed and presented in Figure 4c. No highlighted peak is found at m/z = 39 in Figure 4c, either. To find what happens on the peak for the so-called Br²⁺, we return the original mass spectrum. Figure S8 is the histograms of the values of y(39) for ⁷⁹Br²⁺ and y(40) for ⁸¹Br²⁺ in the 3230 mass spectra. If Br²⁺ is present, the values of y(39) should be roughly the same as the values of y(40). As a matter of fact, the value of y(40) is about twice that of y(39). Moreover, the profiles of the histograms of y(39) and y(40) are obviously different. We have also inspected the result from GC coupled with high-resolution mass spectrometry. No doublet for Br²⁺ can be found. The above results clearly demonstrate that Br²⁺ does not occur in the mass spectrum of 1,2-dibromoethane.

In addition, we found that no peaks are highlighted in Figure 4c. As a matter of fact, the strongest peak in the statistically auxiliary spectrum shown in Figure 4c is the peak at m/z = 105, and the intensity of the peak is only 693. This result indicates that only the peak at 105 is highlighted in 693 mass spectra among the 3230 mass spectra. Since the peak is highlighted in a low percentage of the mass spectra (693/3230 < 25%). At present, we do not believe the peak at 105 is from fragments of C_mH_nBr²⁺, which is present in the mass spectra of 1,2-dibromoethane. According to the literature, Electron ionization (EI) is unlikely to produce fragments that carry multiple positive charges. This may be the reason no peak from either Br²⁺ or C_mH_nBr²⁺ can be found in the mass spectrum of 1,2-dibromoethane in the present work.

Identification of Fragments Containing a Single Chlorine Atom, Two Chlorine Atoms, and Three Chlorine Atoms in the Mass Spectra of Chloroform

Herein, we use the approach to analyze the mass spectra of chloroform as a second real-world example. In the TGA-MS experiment, 429 mass spectra are collected. A typical mass spectrum is present in Figure S9a.

The first task is to identify the fragments containing a single chlorine atom. The corresponding isotopologues are (C_mH_n³⁵Cl⁺ and C_mH_n³⁷Cl⁺). Since the nominal mass difference between C_mH_n³⁵Cl⁺ and C_mH_n³⁷Cl⁺ is 2, the corresponding nominal m/z values for C_mH_n³⁵Cl⁺ and C_mH_n³⁷Cl⁺ are M and M + 2, respectively.

The situation of ²H and ¹³C is similar to the discussion for fragments containing one bromine atom in 1 of the Supplemental Material. The spectral contribution from isotopologues containing multiple ¹³C, ²H atoms can be safely neglected. C_mH_n³⁵Cl⁺ and C_mH_n³⁷Cl⁺ are the dominant species among the possible isotopologues. Thus, we have:

a ⇀ C m H n CI + = β ( a ( 35 Cl ) a ( 37 Cl ) ) = β ( 0.7577 0.2423 )

(24)

Subsequently, we used the a → _CmHnCl+ and b → (x) to produce the corresponding auxiliary spectra. In addition, 429 auxiliary spectra are generated from the 429 original mass spectra from the TGA-MS experiment, so that the corresponding statistically auxiliary spectrum is obtained and shown in Figure S9b. In Figure S9b, the peaks whose m/z values are 35, 36, 47, 49, 112, 119, and 120 are highlighted. According to the results, these peaks are from fragments containing a single chlorine atom, a tentative assignment of these peaks is summarized in Table S2.

The second task is to identify the fragments containing two chlorine atoms. Similar to the case of fragments containing a single chlorine atom, the spectral contribution from isotopologues containing ¹³C, ²H can be neglected. Thus, the resultant isotopologues are (C_mH_n³⁵Cl₂⁺, C_mH_n³⁵Cl³⁷Cl⁺, and C_mH_n³⁷Cl₂⁺), and the nominal m/z values for the three isotopologues are M, M + 2, and M + 4. Thus, we have:

a ⇀ C m H n Cl 2 + = ( [ a ( 35 Cl ) ] 2 2 [ a ( 35 Cl ) ] [ a ( 37 Cl ) ] [ a ( 37 Cl ) ] 2 ) = ( 0.5741 0.3672 0.0587 )

(25)

The third task is to identify the fragments containing three chlorine atoms. Similar to the case of fragments containing a single chlorine atom, the spectral contribution from isotopologues containing ¹³C, ²H can be neglected. Thus, the resultant isotopologues are (C_mH_n³⁵Cl₃⁺, C_mH_n³⁵Cl₂³⁷Cl⁺, C_mH_n³⁵Cl³⁷Cl₂⁺, and C_mH_n³⁷Cl₃⁺), and the nominal m/z values for the three isotopologues are M, M + 2, M + 4, and M + 6. Thus, we have:

a ⇀ CmHnCl 3 + = ( [ a ( 35 Cl ) ] 3 3 [ a ( 35 Cl ) ] 2 [ a ( 37 Cl ) ] 3 [ a ( 35 Cl ) ] 2 [ a ( 37 Cl ) ] 2 [ a ( 37 Cl ) ] 3 ) = ( 0.4350 0.4173 0.1335 0.0142 )

(26)

( b ⇀ x ) = ( y ( M ) y ( M + 2 ) y ( M + 4 ) y ( M + 6 ) )

(27)

We used the a → _CmHnCl3+ and b → (x) to produce the corresponding auxiliary spectra. In addition, 429 auxiliary spectra are generated from the 429 original mass spectra from the TGA-MS experiment, so that the corresponding statistically auxiliary spectrum is obtained and shown in Figure S9d. In Figure S9d, the peaks whose m/z values are 117 and 118 are highlighted. According to the results that these peaks are from fragments containing three chlorine atoms, the two highlighted peaks can be assigned to CCl₃⁺ and CHCl₃⁺, respectively (Table S2).

Most assignments shown in Table S2 are supported by the results of Lago et al. ⁸⁰ However, the following issues should be noted: (i) The peak at 112 in Figure S9b is tentatively assigned to C₆H₅Cl⁺, which may be originated from impurity of the chloroform sample. (ii) The peak at 96 in Figure S9c is preliminarily attributed to C₂H₂Cl₂⁺, which may also belong to an impurity of the sample. (iii) We believe that the peaks at 119 and 120 in Figure S9d are the peaks from CCl₃⁺ and CHCl₃⁺, respectively. The above two fragments contain three chlorine atoms. The corresponding isotopologues form a quartet. According to Eq. 26, the ratio between the second peak and the third peak is {3[a(³⁵Cl)]²[a(³⁷Cl)]}/{3[a(³⁵Cl)][a(³⁷Cl)]²} = a(³⁵Cl)/a(³⁷Cl). This value happens to be the same as the ratio of the intensities of the doublet for the isotopologues for the fragment containing a single chlorine atom. Consequently, the fragments CCl₃⁺ and CHCl₃⁺ are coincidentally highlighted at 119 and 120 in Figure S9b.4). The highlighted peaks in Figure S9b, Figure S9c, and Figure S9d are quite weak and difficult to be observed in the original mass spectra shown in Figure S9a. However, the corresponding doublets, triplets, and quartets indeed occur in the original mass spectra. In Figure S10, the magnified doublets, triplets, and quartets in the original mass spectra are present.