# A Scheme for Ultrasensitive Detection of Molecules with Vibrational Spectroscopy in Combination with Signal Processing

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## Abstract

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## 1. Introduction

_{5}gas is only 20 ppm; it is desirable to be able to detect harmful molecules at much smaller concentrations than the LC50. While multiple detection techniques exist, they all have their limitations, specifically, in sensitivity and selectivity. For example, detection of molecules by light absorption/photoluminescence properties has been proposed [1] but is very non-selective, as many molecules absorb or luminesce in similar bands. Some methods require a liquid sample, which may not be available; for example, when detecting molecules in the atmosphere or at the border of a territory where release may occur. As stated in a relatively recent review [2], “<available> devices have several limitations, such as low specificity and inability to detect all CW agents. Definitive identification of an agent can be carried out onsite in a mobile analytical laboratory or in an off-site laboratory, and this will generally take many hours. Clinical symptoms and signs in exposed individuals may be the most useful indicators of the likely agent.” The last sentence highlights how important detection is and that available methods are still deficient. One certainly does not want to be in a situation where “clinical symptoms and signs in exposed individuals may be the most useful indicators of the likely agent”.

^{−1}[15]; many of those transitions are IR inactive, but many are and have a small intensity. The complexity of the entire vibrational spectrum considered as a signal means that it is a unique molecular fingerprint. This signal complexity (and therefore a sharp autocorrelation function) also means that its recovery in the presence of noise (thermal noise in the spectrometer or signals due to external radiation or presence of other species in the environment) could be efficient with signal processing techniques [19] such as matched filtering. This, in turn, means that selectivity and sensitivity could be drastically improved vs existing techniques. To use this idea, one must have accurate benchmarks (reference spectra) of vibrational spectra in a wide excitation energy range to program the matched filter. The reliance of the detector on multiple spectral lines in a wide frequency range means that anharmonicity should be considered when using computed spectra as a reference.

## 2. Methods

^{−1}(Gaussian width) for further processing. The calculations were done in a vacuum.

**x**′ indexed by k) with a filter

**h**(another vector) that is parallel with the signal, maximizing the inner product. This is achieved when $\mathit{h}=\alpha {\mathit{R}}_{\xi}^{-1}\mathit{x}$, where ${\mathit{R}}_{\xi}$ is the covariance of the noise and α a normalization constant. The vector

**x**is the useful (expected) signal component of the input

**x**′, which is deteriorated by the noise $\mathbf{\xi}$: ${\mathit{x}}^{\prime}=\mathit{x}+\mathbf{\xi}$. For white noise assumed here, we can put $\mathit{h}=\mathit{x}$ and matched filtering becomes equivalent to correlating the received signal with the expected signal with the output

**y**proportional to the signal’s autocorrelation function. The technique is, therefore, the more powerful the more complex the shape of

**x**is i.e., the sharper its autocorrelation function, ideally approaching the delta function for very complex-shaped signals. The quality of the detection deteriorates if there is a mismatch between

**h**and

**x**. A popular example is relatively easy radar detection of stealth aircraft using complex-shaped, broadband signals. In signal processing applications, k usually indexes time intervals.

**x**; it plays the role of a benchmark. It is mixed with white noise and is detected using a spectrum computed with different levels of theory as

**h**. We also test performing the detection in the entire relevant spectral range (set here from 0 to 4000 cm

^{−1}) vs. a frequency window (from 680 to 2000 cm

^{−1}where the fundamentals corresponding to vibrations of most bonds and angles lie).

^{−1}using 2

^{16}points (k values) up to 4000 cm

^{−1}. Under spectral broadening of 1 cm

^{−1}used here, this provided converged results (no changes were detected with a finer discretization). Note that 0.061 cm

^{−1}is a digitization resolution and the 1 cm

^{−1}broadening models the spectrum resolution. Both are easily achievable in measurements and data processing. The white noise was subjected to the same broadening. Unsigned noise was added to the spectrum for computational simplicity. The peak in the autocorrelation function was detected by looking for any elements of

**y**which exceed N × σ where σ is the standard deviation of

**y**and N is a chosen parameter. The detection limit is defined as a noise level with which the rate of false negatives reaches 50%. False positives, in this case, are due to a finite probability of satisfying the criterion ${y}_{k}>N\times \sigma $ with the noise in the absence of the signal

**x**. To reduce the rate of false positives, the peak search was performed in the window of width 300 and 80 cm

^{−1}(when using the full spectrum range and the window 680–2000 cm

^{−1}, respectively) around the mid-point of

**y**. When

**h**=

**x**(i.e., when the same level of theory is used for the signal and for the filter), the peak was exactly at the mid-point. When different levels of theory are used, the peak may be off-center; the peak detection window was chosen for the worst mismatch among all cases. The rate of false positives was computed by using noise-only inputs. With each instance of noise (for both signal-containing and noise-only inputs), 50 numeric experiments were performed. The corresponding SNR is defined as the ratio of the root mean square of the signal

**x**(the spectrum) and the standard deviation of the noise, both under the same broadening of 1 cm

^{−1}. With 50 experiments, the statistical spread of SNR estimates were within 0.1, which is sufficient for our purposes.

## 3. Results and Discussion

^{−40}esu

^{2}cm

^{2}, are dropped in the subsequent discussion as they are unimportant and make no difference to the conclusions as long as all calculations are done at the same scale and with the same broadening, which is the case). At noise levels on the order of 500, the spectrum is completely unusable for assignment. This case corresponds to a value of SNR as defined above of about 0.25. Examples of correlation functions (filter output

**y**) of the benchmark spectrum with spectra computed at different levels of theory are shown in Figure 4. The top pair of plots show the autocorrelation function which possesses a sharp peak in the middle, which is due to the complex shape of

**x**. The delta function–like nature of the autocorrelation function of a vibrational spectrum of a polyatomic molecule makes it well suited for applying matched filtering and therefore can permit detection in the presence of high levels of noise. The following three pairs of panels show the effect of errors in the computed spectrum (which programs the filter

**h**) on the quality of the peak in

**y**due to the use of the harmonic approximation, of a less accurate exchange-correlation functional, and of a small basis set, respectively. The final pair of plots (at the PBE/harmonic/6-31G level) show the combined effect of these sources of error. Note that the application of Equation 1 results in the range of the abscissa values for

**y**which is double that of the original spectrum, i.e., 8000 cm

^{−1}, with the autocorrelation peak centered at 4000 cm

^{−1}.

^{−1}with respect to the position to the autocorrelation function’s peak. Qualitatively, a similar effect is observed when using a smaller basis (other parameters being equal). All lower-level approximations lead to the appearance of strong off-center peaks. The combined effect of all these sources of error (bottom pair of plots) is a peak height reduced by about a factor of 20 and shifted by more than 100 cm

^{−1}, with multiple satellite peaks, some exceeding in height the central peak. Overall, the results shown in Figure 4 suggest that anharmonicity treatment is important as well as proper choices of functional and basis set, and that one must allow for a “window” on the order of ±200 cm

^{−1}for peak detection.

^{−1}. In this case, components of

**x**′ and of

**h**outside this range were set to zero. The resulting correlation functions (filter output

**y**) of the benchmark spectrum with spectra computed at different levels of theory are shown in Figure 5. The use of a spectral window reduced the peak height in the autocorrelation function by about a factor of two but it also reduced the degree of relative deterioration of the height and position of the main peak with approximate computational schemes. For example, the use of the harmonic approximation (other parameters being equal) reduced the peak height by about a factor of 2½. The use of the PBE functional (other parameters being equal) caused the peak’s shift but hardly reduced its height. The use of a small basis (other parameters being equal), however, reduced the peak height by about a factor of four and had a worse effect than in the case of a full spectral range. The combined effect of all these sources of error is a peak weakened by an order of magnitude with multiple and strong satellite peaks. It may appear, therefore, based on Figure 5 that using a limited spectral range may be beneficial if the filter is programmed with a computed spectrum using a sufficiently complete basis set.

^{−1}or a detection window of 680–2000 cm

^{−1}. The detection thresholds were determined by comparing the rate of positives with the rate of false positives (on noise-only inputs) and correspond to 50% probability of detection. Different N were tried in the peak detection criterion ${y}_{k}>N\times \sigma $. This criterion was applied in the window of ±150 (when using full spectral range) or ±40 cm

^{−1}(when using a window 680–2000 cm

^{−1}), based on the results of Figure 4 and Figure 5. Examples of these calculations when programming

**h**with the spectrum computed at the B3LYP/anharmonic/6-311G+(2d,2p) level are shown in Figure 6 for the case of full spectrum detection and in Figure 7 for detection in the spectral window.

^{−1}). Detection in a narrower window, in spite of some advantages listed above, results in higher required SNR, albeit reliable detection should still be possible with SNR < 0.5. This highlights the fact that it is really in a wide spectral range that the molecular vibrational spectrum has unique, fingerprint-like quality.

^{−1}, which we used for the systematic tests described above was chosen to balance resolutions typically achieved in the gas phase (sub-cm

^{−1}) and at surfaces and in other aggregate states (typically 1–4 cm

^{−1}) [3,4,5]. We conducted selective calculations with other broadening values to confirm that the conclusions are not skewed by the choice of a specific broadening value. For example, when using a broadening of 2 cm

^{−1}, the detection limit when using a spectrum computed at the B3LYP/harmonic/6-311G+(2d,2p) level to program the filter is, in terms of SNR (using full spectral range): 0.32 when using a 5σ criterion and 0.39 using a 6σ criterion, i.e., somewhat higher as expected due to less sharp features of the spectrum and of the autocorrelation function with larger broadening (lower spectral resolution). We also conducted a test of the effect of the presence of another species in the environment on the detection capability of the filter: we mixed in the input the vibrational spectrum of a water molecule. For consistency, the H

_{2}O spectrum was computed at the same level of theory as the reference A232 input spectrum and broadened the same way (by 1 cm

^{−1}) as the input spectrum and the noise. This test ignored ro-vibrational contributions which are important in the case of a light molecule like water, it is therefore indicative of the effect of the presence of another species without being quantitative. The intensities in the water spectrum were multiplied by 10 to reflect the fact that concentrations of ambient species may be much higher than that of the target molecule. The detection limit when using a spectrum computed at the B3LYP/harmonic/6-311G+(2d,2p) level to program the filter is, in terms of SNR (using full spectral range): 0.23 when using a 5σ criterion and 0.28 using a 6σ criterion, i.e., on the same order as that without water. In general, it is expected that the selectivity should remain high in the presence of other well-structured signals in the input such as well-resolved spectra of other molecules, although of course in applications, specific studies should be conducted for specific environments where particular species are present.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Sample Availability: Not available. |

**Figure 1.**The molecular structure of the A232 nerve agent. Atom color scheme: C, brown; H, pink; N, blue; O, red; P, violet; F, green.

**Figure 2.**IR spectra of the A232 molecule computed with different levels of theory. Spectra are Gaussian broadened by 1 cm

^{−1}(Gaussian width). Some spectra are plotted in the negative for better visibility. The intensities are in 10

^{−40}esu

^{2}cm

^{2}.

**Figure 3.**The IR spectrum of A232 computed at the B3LYP/anharmonic/6-311G+(2d,2p) level and summed with the white noise of amplitudes (clockwise from top left) (

**A**) 0, (

**B**) 500, (

**C**) 1500, and (

**D**) 1000.

**Figure 4.**The correlation function of the reference spectrum (anharmonic spectrum computed with B3LYP/6-311G+(2d,2p)) with spectra computed at different levels of theory, using the full spectral range 0–4000 cm

^{−1}. From top to bottom: (

**A**) B3LYP/anharmonic/6-311G+(2d,2p) (i.e., autocorrelation), (

**B**)_B3LYP/harm/6-311G+(2d,2p), (

**C**) PBE/anharmonic/6-311G+(2d,2p), (

**D**) B3LYP/anharmonic/6-31G, and (

**E**) PBE/harmonic/6-31G. The top panel in each pair of plots is for the entire signal

**y**and the bottom panel is zoomed around the center, where a peak is expected.

**Figure 5.**The correlation function of the reference spectrum (anharmonic spectrum computed with B3LYP/6-311G+(2d,2p)) with spectra computed at different levels of theory within a detection window of 680–2000 cm

^{−1}. From top to bottom: (

**A**) B3LYP/anharm/6-311G+(2d,2p) (i.e., autocorrelation), (

**B**) B3LYP/harm/6-311G+(2d,2p), (

**C**) PBE/anharm/6-311G+(2d,2p), (

**D**) B3LYP/anharm/6-31G, and (

**E**) PBE/harm/6-31G. The top panel in each pair of plots is for the entire signal

**y**and the bottom panel is zoomed around the center, where a peak is expected.

**Figure 6.**The fraction of positive detection outcomes (detection criterion y > N × σ satisfied) for different levels of noise and N, for detection in the full spectral range of 0–4000 cm

^{−1}. The top panel is for the inputs consisting of the molecular spectrum and noise and the bottom panel for noise-only inputs. The filter is programmed with a spectrum computed at the B3LYP/anharm/6-311G+(2d,2p) level.

**Figure 7.**The fraction of positive detection outcomes (detection criterion y > N × σ satisfied) for different levels of noise and N, for detection in the full spectral window 680–2000 cm

^{−1}. The top panel is for the inputs consisting of the molecular spectrum and noise and the bottom panel for noise-only inputs. The filter is programmed with a spectrum computed at the B3LYP/anharm/6-311G+(2d,2p) level.

**Table 1.**Detection levels (noise levels and corresponding SNR values in parentheses) when using spectra computed at different levels of theory to program the filter. The reference spectrum computed at the highest level of theory is used to model the signal deteriorated by noise. Lower level approximations vs. the reference spectrum are highlighted in bold. The results are shown for the most promising peak detection criteria (N × σ).

Method | Detection Window | Full Spectrum | ||
---|---|---|---|---|

N × σ Detection Criteria | ||||

4σ | 5σ | 5σ | 6σ | |

Detection Threshold (SNR) | ||||

B3LYP/anharmonic/6-311G+(2d,2p) | 819 (0.17) | 608 (0.22) | 1600 (0.08) | 1317 (0.10) |

B3LYP/harmonic/6-311G+(2d,2p) | 533 (0.25) | 388 (0.34) | 684 (0.20) | 465 (0.29) |

B3LYP/anharmonic/6-31G | 250 (0.56) | 143 (0.97) | 869 (0.16) | 672 (0.20) |

PBE/anharmonic/6-311G+(2d,2p) | 373 (0.37) | 304 (0.46) | 700 (0.20) | 522 (0.26) |

PBE/harmonic/6-31G | 190 (0.72) | 156 (0.85) | 357 (0.38) | 221 (0.62) |

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## Share and Cite

**MDPI and ACS Style**

Tan, Y.B.; Tay, I.R.; Loy, L.Y.; Aw, K.F.; Ong, Z.L.; Manzhos, S.
A Scheme for Ultrasensitive Detection of Molecules with Vibrational Spectroscopy in Combination with Signal Processing. *Molecules* **2019**, *24*, 776.
https://doi.org/10.3390/molecules24040776

**AMA Style**

Tan YB, Tay IR, Loy LY, Aw KF, Ong ZL, Manzhos S.
A Scheme for Ultrasensitive Detection of Molecules with Vibrational Spectroscopy in Combination with Signal Processing. *Molecules*. 2019; 24(4):776.
https://doi.org/10.3390/molecules24040776

**Chicago/Turabian Style**

Tan, Yong Boon, Ian Rongde Tay, Liang Yi Loy, Ke Fun Aw, Zhi Li Ong, and Sergei Manzhos.
2019. "A Scheme for Ultrasensitive Detection of Molecules with Vibrational Spectroscopy in Combination with Signal Processing" *Molecules* 24, no. 4: 776.
https://doi.org/10.3390/molecules24040776