Neural network consistent empirical physical formula construction for DFT based nonlinear vibrational spectra intensities of N-(2-methylphenyl) and N-(3-methylphenyl) methanesulfonamides

Abstract

Vibrational intensities are both experimentally measured and theoretically estimated important physical quantities which are directly related to distributions of the electric charges in a molecule. In this paper, as a novel approach, by a layered feedforward neural network (LFNN), empirical physical formulas (EPFs) were constructed for density functional theory (DFT) vibrational spectra intensities of N-(2-methylphenyl) and N-(3-methylphenyl) methanesulfonamides. The spectral data was obtained from our previous study. Although the DFT spectral data was inherently extremely difficult-to-fit (sparse frequency intervals, highly nonlinear and sharply fluctuating intensities), still the optimally constructed LFFN-EPFs succeeded in fitting this data to medium and higher level of satisfaction. Moreover, LFNN–EPFs test set (i.e. yet-to-be measured experimental data) intensity predictions were also moderate to higher level. This briefly means that the general tendency of the intensity data was consistently estimated by the LFNN to an acceptable degree. In conclusion, provided that vibrational spectral data measured over sufficiently dense frequency intervals are available for any unknown molecule of significant complexity, suitable LFNN–EPFs can be constructed. Then, by various mathematical tools such as differentiation, integration, minimization, these vibrational LFNN–EPFs can be used to estimate the electronic charge distributions of the molecule. Moreover, these estimations can be compared and combined with those of theoretical DFT atomic polar tensor calculations to contribute to the identification of the molecule.

Introduction

Vibrational spectral intensities are both experimentally measured and theoretically estimated important physical quantities which are directly related to distributions of the electric charges in a molecule [1]. The pioneering early studies [2], [3] to interpret a wealth of experimental intensity data were later followed by works dealing with mainly the formal aspects of the interpretation of IR intensities [4], [5]. After the formal matters were resolved reasonably, attempts into the physical meaning of intensity curves enabled to correlate some intensity patterns with charge distributions inside the molecule [6], [7]. Very briefly, the measured vibrational intensities I_i of the ith normal mode are related to the atomic charges by the molecular dipole moment ($\overrightarrow{M}$) derivatives with respect to associated normal coordinate Q_i. More details into intensity-atomic charge relationship are given in Section 2.1. Here it suffices to say that, thanks to over five decade of efforts into this relationship, recent ab initio calculations have accurately predicted IR (infrared) intensity related molecular charge distributions, in very good agreement with experimental data [8], [9], [10], [11].

Nevertheless, for the molecules of very complex structures, typical IR or Raman spectral intensities are naturally highly nonlinear and of complex pattern. In this case, it may be more difficult to perform appropriate ab initio intensity calculations in order to compare with the experimental data. Moreover, the identity and three dimensional (3D) structure of the chemical compound may be totally unknown before a set of appropriate spectral experiments are done. Indeed, although 40 million chemical compounds are known, experimental data for 3D structure are available only for 250,000 compounds. The largest database for IR spectra contains only 220,000 spectra. In other words, data for 3D structure and IR spectra are available only for 0.5% of all known chemical molecules [12]. Briefly, satisfactory ab initio vibrational intensity calculations cannot be made for the most molecules because of their unknown detailed spatial structures. Therefore, it would be helpful if the experimenter had a kind of explicit form of empirical physical formula (EPF) for this highly nonlinear vibrational intensity pattern which experimentally obtained for the molecule of unknown detailed 3D structure. An appropriate EPF by using a layered feedforward neural network (LFNN) [13] can indeed be consistently constructed, as we previously theoretically [14] and experimentally [15], [16] showed. Actually, the LFNN, as we mention more in Section 2.2, is a universal nonlinear function approximator [17].

In this paper, as a novel approach, by a LFNN, we constructed consistent EPFs for nonlinear density functional theory (DFT) vibrational spectra intensities of N-(2-methylphenyl) (2MPMSA) and N-(3-methylphenyl) methanesulfonamides (3MPMSA). The spectral data was obtained from our previous study [18]. Before explicitly mentioning the novelty in this paper, some preliminary remarks are necessary. There have been a number of previous works [19], [20], [21], [22], [23], [24] which applied neural networks to molecular vibrational intensities. Among these works, the very systematic and pioneering studies of Gasteiger and coworkers [19], [20], [21] into deriving 3D molecule structures from their corresponding IR spectra must be mentioned with a particular stress. They successfully use radial distribution function (RDF) code for the simulation of an IR spectra by a suitable counterpropagation (CPG) neural network [25]. As a reverse operation, they also employ CPG neural network to obtain a 3D structure from a large database by using IR spectra as an input for the neural network. However, they do not particularly aim to construct an explicit mathematical functional form for the experimental IR spectra pattern. We can now state explicitly the novelty in this paper. This paper is the first to construct consistent explicit form of LFNN–EPFs for vibrational intensities, particularly for methanesulfonamides. Although the DFT spectral data was inherently extremely difficult-to-fit (sparse frequency intervals, highly nonlinear and sharply fluctuating intensities or activities), still the optimally constructed LFFN-EPFs succeeded in fitting this data to medium and higher level of satisfaction. Moreover, LFNN–EPFs test set (i.e. yet-to-be measured experimental data) intensity predictions were also moderate to higher level. This briefly means that the general tendency of the spectral intensity data were consistently estimated by the LFNN to an acceptable degree. We conclude that, provided that sufficient (non-sparse) vibrational spectral data available for any given molecule of significant complexity, suitable LFNN–EPFs can be constructed. Then, by various mathematical tools such as differentiation, integration, minimization, these LFNN–EPFs can be used to obtain further vibrational intensity related molecular structural parameters.