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Ground-state spectral features of molecular clusters RDX excited at THz frequencies
Vibrational Spectroscopy 64 (2013) 62–67
Contents lists available at SciVerse ScienceDirect
Vibrational Spectroscopy journal homepage: www.elsevier.com/locate/vibspec
Ground-state spectral features of molecular clusters RDX excited at THz frequencies L. Huang a , A. Shabaev b,∗ , S.G. Lambrakos a , L. Massa c a b c
Naval Research Laboratory, Washington, DC 20375, USA George Mason University, Fairfax, VA 22030, USA Hunter College, CUNY, New York, NY, USA
a r t i c l e
i n f o
Article history: Received 27 January 2012 Received in revised form 16 October 2012 Accepted 16 October 2012 Available online 24 October 2012 Keywords: THz spectra Vibrational resonances DFT calculations
a b s t r a c t Calculations are presented of ground state resonance structure at THz frequencies for molecular clusters of the high explosive RDX using density functional theory (DFT). The spectral features of this resonance structure are due to coupling of resonance modes for ground state excitation. In particular, the coupling among ground state resonance modes provides a reasonable molecular level interpretation of spectral features associated with THz excitation of molecular clusters. THz excitation is associated with frequencies that are characteristically perturbative to molecular electronic states, in contrast to frequencies that can induce appreciable electronic state transition. Owing to this characteristic of THz excitation, one is able to make a direct association between local oscillations about ground-state minima of molecules, either isolated or comprising a cluster, and THz excitation spectra. The DFT software GAUSSIAN was used for the calculations of ground state resonance structure presented here. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Infrared absorption of molecular ground states is within the frequency range of the electromagnetic spectrum spanning the band 0.1–100 THz. However, the part of the spectrum 0.1–10 THz, which until recently was unexplored, has traditionally been called the “terahertz gap” owing to the lack of efficient sources for generating frequencies in this range. For the past two decades, however, there has been extensive advancement in the design of THz systems, which has been motivated by the development of new THz sources, detectors and applications for noninvasive detection [1–4]. A significant characteristic of THz excitation is that it is associated with frequencies not involving electronic state transitions. Of course, the practical aspect of the IR absorption character of THz excitation for detection is that noninvasive detection methodologies can be developed, which do not damage materials under examination. In addition, the perturbative character of THz excitation with respect to molecular states has significant implications with respect to its numerical simulation based on density functional theory (DFT). It follows that, owing to the absorption character of THz excitation, which is characteristically that of vibrational fine structure, one is able to make a direct association between local oscillations about ground-state minima of a given molecule and THz excitation spectra.
∗ Corresponding author. 0924-2031/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.vibspec.2012.10.003
With respect to THz excitation of materials, the procedure of using response spectra calculated using DFT, which is associated with ground state resonance structure, for the direct construction of permittivity functions is well posed owing to the physical characteristic of THz excitation. In particular, it is important to note that the constructing permittivity functions using response spectra calculated using DFT is physically consistent with the characteristically linear response associated with THz excitation of molecules. Accordingly, one observes a correlation between the advantages of using THz excitation for detection of explosives (and ambient materials) and those for its numerical simulation based on DFT. Again, this follows from the fact that THz excitation is associated with frequencies that are characteristically perturbative to molecular states, in contrast to frequencies that can induce appreciable electronic state transitions. Density functional theory has been successfully used to investigate the vibrational spectra of energetic materials in the form of single molecules and molecular crystals [5–11]. These calculations provide detection signatures for various forms of materials which can be encountered in various detection scenarios [12,13]. Calculations of resonant structure associated with isolated molecules help to identify intramolecular vibrational modes in the absorption spectra of various materials. Additionally, calculations of resonance structure associated with the solid state, which are in agreement with experiment, show several new modes in the region below 3 THz, which can be identified with intermolecular vibrations [6–8]. With respect to interpretation based on theory, the results of
L. Huang et al. / Vibrational Spectroscopy 64 (2013) 62–67
calculations and analysis provides new insight into the chemical origins of the intricate electromagnetic response features of molecules and molecular complexes including the delicate interplay between intra-molecular degrees of freedom and relatively weak intermolecular interactions. A significant aspect of using response spectra calculated by DFT for the construction of dielectric response functions is that it adopts the perspective of computational physics, according to which a numerical simulation represents another source of “experimental” data. This perspective is significant in that a general procedure may be developed for construction of dielectric response functions using DFT calculations as a quantitative initial estimate of spectral response features for subsequent adjustment with respect to additional information such as experimental measurements and other types of theory based calculations. That is to say, for the purpose of simulating many electromagnetic response characteristics of materials, DFT is sufficiently mature for the purpose of generating data complementing, as well as superseding, experimental measurements [6,11,13]. The application of density functional theory (DFT) and related methodologies for the determination of electromagnetic response characteristics is important for the analysis of parameter sensitivity with respect to parametric representations of excitation spectra. That is to say, many characteristics of the electromagnetic response of a given material may not be detectable, or in general, not relevant for detection. Accordingly, sensitivity analyses concerning the electromagnetic response of layered composite systems can adopt the results of simulations using DFT, and related methodologies, to provide realistic limits on detectability that are independent of a specific system design for detection. In addition, analysis of parameter sensitivity based on atomistic response characteristics of a given material, obtained by DFT, provide for an “optimal” fit of experimental measurements for the construction of permittivity functions. It follows that within the context of parameter sensitivity analysis, data obtained by means of DFT represents a true complement to data obtained by means of experimental measurements. In what follows, calculations are presented of ground state resonance structure associated with molecular clusters of the high explosive RDX. RDX is among the class of industrial or military explosives, in contrast to homemade explosives, that are of primary concern for detection in realistic environments. In addition, this explosive has a characteristic spectrum in the THz range of frequencies that is potentially distinguishable from other materials [3]. A schematic representation of the molecular geometry of RDX is shown in Fig. 1. The coordinates corresponding to an equilibrium configuration of a single isolated molecule of RDX, which have been calculated using DFT, are given in Ref. [14]. It is significant to note that, with respect to practical application, molecular clusters represent a separate regime for using DFT calculations for construction of dielectric response functions. In addition, with respect to theoretical understanding, the dielectric response of clusters provides insight concerning the nature of the transition of dielectric response to that of solids or systems having long range order. This type of insight can in turn assist in understanding the nature of the difference between the intermolecular and intramolecular vibrations.
2. Calculation of spectra using density functional theory The DFT software GAUSSIAN09 (G09) can be used to compute an approximation of the IR absorption spectrum of a molecule [15]. This program calculates vibrational frequencies by determining second derivatives of the energy with respect to the Cartesian nuclear coordinates, and then transforming to mass-weighted coordinates at a stationary point of the geometry [16]. The IR
63
Fig. 1. (A) Single molecule of RDX and (B) unit cell and positions of molecules for clusters of 2, 4 and 6 RDX molecules after geometry optimization.
absorption spectrum is obtained using DFT to compute the ground state electronic structure in the Born–Oppenheimer approximation using Kohn–Sham density functional theory [17–21]. GAUSSIAN uses specified orbital basis functions to describe the electronic wavefunctions and density. For a given set of nuclear positions, the calculation directly gives the electronic charge density of the molecule, the potential energy V, and the displacements in Cartesian coordinates of each atom. The procedure for vibrational analysis followed in GAUSSIAN is that described in Ref. [22]. Ref. [23] presents a fairly detailed review of this procedure. All vibrational frequencies for the optimized geometries of RDX clusters discussed here are positive, as must occur for true minimum geometries. For the calculations presented here geometry optimization and vibrational analysis was effected using the DFT model B3LYP [24,25] and basis function 6-311G(2d,2p) [26,27]. According to the specification of this basis function, (2d,2p) designates polarization functions having 2 sets of d functions for heavy atoms and 2 sets
64
L. Huang et al. / Vibrational Spectroscopy 64 (2013) 62–67
of p functions for hydrogen atoms [28]. A schematic representation of the molecular geometry of a single isolated molecule of RDX is shown in Fig. 1A. Schematic representations of molecular geometries of molecular clusters consisting of 2, 4 and 6 molecules are shown in Fig. 1B–D, respectively. It is significant to note that the relative positions of molecules associated with each of the molecular clusters is according to crystallographic structure conditions that are for a bulk material. In particular, the crystal structure of RDX, whose CCDC (Cambridge Crystallographic Data Centre) reference code is CTMTNA, has been investigated by Choi and Prince [29]. The space group for this crystal structure is Pbca (symmetry operators are x, y, z; 1/2 − x, −y, 1/2 + z; 1/2 + x, 1/2 − y, −z; −x, 1/2 + y, 1/2 − z; −x, −y, −z; 1/2 + x, y, 1/2 − z; 1/2 − x, 1/2 + y, z; x, 1/2 − y, 1/2 + z), and the unit cell constants are a = 13.182, b = 11.574, c = 10.709, ˛ = 0.00, ˇ = 90.00, = 90.00 and density 1.806 g/cm3 . The ground-state oscillation frequencies and IR intensities for different molecular clusters of RDX, corresponding to their relaxed equilibrium configurations, are calculated by DFT according to the frozen phonon approximation. In the cases of molecular clusters of 2, 4 and 6 molecules, these values are given in Tables 3–5, respectively, of Ref. [33]. In the case of 2-molecule clusters, a total of 120 frequencies (from 14 to 3221 cm−1 ) were calculated (see Table 3 of Ref. [33]). In the case of 4-molecule clusters a total of 246 frequencies (from 5.9 to 3227 cm−1 ) were calculated (see Table 4 of Ref. [33]). In the case of 6-molecule clusters a total of 372 frequencies (from 4 to 3238 cm−1 ) were calculated (see Table 5 of Ref. [33]). Fig. 2 shows spectra of a single molecule of RDX in comparison with the spectra of 2-, 4- and 6-molecule clusters. The DFT model and basis function used for these calculations are the same as those used for the single isolated molecule of RDX. The programs ConQuest and Mercury [30] were used for searching the Cambridge Structural Database (CSD), for visualizing database entries associated with RDX, and constructing the molecular clusters. The construction of any given cluster required consideration of the interactions between a single molecule and its neighborhood. Accordingly, hydrogen bonding between molecules is important for establishing a stable configuration for any given cluster.
3. Discussion Computational experiments using DFT, concerning molecular clusters of RDX, are presented in this section. These results include the relaxed or equilibrium configuration of a single isolated molecule of RDX (see Table 1 in Ref. [33]), ground-state oscillation frequencies and IR intensities for this configuration, which are calculated by DFT according to the frozen phonon approximation (see Table 2 in Ref. [33]). The ground state resonance structure for a single isolated molecule of RDX is adopted as a reference for analysis of spectral features associated with molecular clusters of different sizes. Calculated IR intensities for a single isolated molecule, 2molecule cluster, 4-molecule cluster and 6-molecule cluster, respectively, are shown in Fig. 2. Referring to Fig. 2, one notes continuous spectra consisting of a superposition of essentially Lorenzian functions of various heights and widths, which have been constructed using discrete spectra. Although the primarily purpose of this type of construction within GAUSSIAN is for the purpose of enhanced visualization of spectral features, it is significant to note that this operation represents an estimation of the characteristic scaling and width of resonances contributing to the dielectric response [13]. For qualitative comparison of spectral features this type of zeroth-order estimate should be sufficient. For the construction of permittivity functions to be used for quantitative simulations, it is more appropriate, however, to assume the characteristic scaling and widths of DFT calculated resonances
Fig. 2. Discrete spectra for a single isolated molecule of RDX and molecular clusters containing 2, 4 and 6 molecules.
as adjustable parameters, i.e., parameters to be assigned values according to additional information. A qualitative comparison of spectral features is shown in Fig. 2. Referring to this figure, it can be observed that in general each molecular cluster is characterized by a relatively distinct spectrum consisting of its own set of characteristic features. Again, hypothesizing with respect to using this type of information for construction of models based on effective medium theory, a composite material could be characterized by a non-interacting uniform “weighted” distribution of finite-size molecular clusters within a host material, where the weighting is according to the relative fraction of clusters within the distribution consisting of a given number of molecules. Initial consideration is given to what extent various average features of DFT calculated spectra for molecular clusters of different
L. Huang et al. / Vibrational Spectroscopy 64 (2013) 62–67
65
Fig. 3. Qualitative comparison of DFT calculated spectra for 1-, 2-, 4- and 6-molecule clusters of RDX and experimentally measured spectrum for dielectric response of RDX solid [31].
sizes, as well as experimentally measured spectra, may be correlated with each other. Referring to Fig. 2, it can be seen that cluster resonances form bands grouped near the original high frequency molecular resonances. A new band of resonances can be observed to form at the low end of the frequency spectrum. Interestingly, the average features of the high and low frequency bands may be analyzed using a relatively simple coupled-oscillators model. It follows that one is able to associate with molecular clusters, by their nature, two separate interaction regimes where stiff and soft springs are to be associated with intramolecular and intermolecular modes, respectively. That is to say, more tightly bound atoms which comprise a given molecule, interact with atoms associated with other molecules comprising the cluster. It could be argued in retrospect that the existence of a commingling of stiff and soft spring interactions is intuitively obvious in the case of molecular clusters (for certain types of molecules). Proceeding with our comparison of spectra, we note that studies have emphasized the observation that calculated spectra corresponding to a single isolated molecule are not strongly correlated with that of the bulk-lattice. In particular, Allis et al. [7] have emphasized that calculated spectra for the dielectric response of a bulk lattice are more appropriate with the use of periodic boundary conditions for representation of intermolecular influences. Relative to this point, however, the DFT calculations of spectra presented here tend to indicate a relatively interesting result, which has implications with respect to both understanding the nature of coupling between inter- and intra-molecular modes for explosives, as well as the use of calculated spectra for modeling dielectric response for practical applications, i.e., simulation of methods for explosives detection. Specifically, the DFT calculated spectra for molecular clusters show progressively higher correlation with that of the bulk with increasing size of the cluster within the neighborhoods of the resonances labeled A–C in Fig. 4. This supports the notion that farfield coupling of intramolecular modes to intermolecular modes is dominant only within a restricted range of molecular cluster size. In addition, this implies that with respect to the modeling of bulk lattice response for estimating dielectric response, for the purpose of simulating detection systems, DFT calculated spectra corresponding to molecular clusters should provide a reasonable estimate of bulk dielectric response characteristics. Spectra of clusters in comparison with experimentally measured spectra [31,32] are shown in Fig. 3. Referring to these figures, initial consideration is given to what extent various average features of the DFT calculated spectra for different size molecular
clusters, as well as experimentally measured spectra, are correlated with each other. Referring to Fig. 3, one observes on average a noticeable correlation between DFT calculated spectra for molecule clusters of RDX and experimentally measured spectra for dielectric response of RDX solid. The comparison of spectra shown in Fig. 3 also supports the notion that certain resonance features, which are associated with finite-size molecular clusters, are preserved within clusters of increasing size, as well as in solid form, thus implying a persistence of these modes after coupling to intermolecular influences. The persistence of certain intramolecular modes, which is irrespective of molecular cluster size, i.e., whether the molecule is isolated, part of a cluster or within a bulk lattice, is demonstrated by comparison of DFT calculated and experimentally measured spectra. A comparison of DFT calculated frequencies corresponding to prominent absorptions, which have been observed experimentally for RDX in solid state is shown in Table 1. It is significant to note that these experimental measurements show good correlation with those of Ref. [34]. Referring to Table 1, one can observe a high correlation between DFT calculated spectra for different cluster sizes and experimentally determined absorption lines at the specific frequencies indicated. Next, we consider what average spectral features can be correlated with the size of a molecular cluster. Again referring to Fig. 3, which compares both DFT calculated and experimentally measured spectra [12,10], it can be observed that as the size of the cluster increases, there is a relative increase in the presence of resonance structure at lower frequencies. An examination of this trend in spectral features can be made more quantitative by consideration of the DFT calculated spectra in terms of their discrete delta-function representations as shown in Fig. 2 for a single isolated molecule and 2-, 4- and 6-molecule clusters of RDX. Referring to this figure, one observes an increase in the number of absorption lines for the molecule clusters relative to the discrete spectrum associated with a single isolated molecule. Again, referring to this figure, it can be
Table 1 Comparison of DFT calculated and experimentally measured frequencies at prominent absorptions for RDX solid [31]. Exp (cm−1 )
27
35
46
51
66
74
103
1-RDX 2-RDX 4-RDX 6-RDX
– 26 17–22 21
– 35 33 35
42 49 46 49
57 59 54 59
62 – 64 66
– 87 73 74
94 100 84–94 92
66
L. Huang et al. / Vibrational Spectroscopy 64 (2013) 62–67
significant structure at lower frequencies for solid materials, which is physically consistent with intermolecular vibrations. 4. Conclusion Calculations of ground state resonance structure at THz frequencies for molecular clusters of RDX using density functional theory have been presented. Average spectral features associated with this resonance structure can be interpreted qualitatively as coupling between stiff and soft resonance modes. Comparison of DFT calculated spectra for a single isolated molecule of RDX and DFT calculated spectra for a molecular cluster of RDX indicates that certain spectral features associated with the isolated molecule can be observed to persist within the spectrum for the molecular cluster. Comparison of DFT calculated spectra for molecular clusters of RDX and experimentally measured spectra indicates the persistence of certain intramolecular modes, which is irrespective of molecular cluster size, ranging from molecules that are isolated, part of a cluster or within a bulk lattice. Finally, the DFT calculations presented in this study represent a specific set of computational experiments, which can be compared to laboratory measurements, as well as other computational experiments. Accordingly, comparison of the results of this study to DFT calculations using other functionals, e.g., PBE0 and CAMB3LYP, could provide further insight concerning the nature of numerical artifacts manifesting as spectral features. Acknowledgment This work was supported by the Office of Naval Research. References
Fig. 4. Qualitative comparison of DFT calculated spectra for single isolated molecule and 6-molecule cluster of RDX with experimentally measured spectrum for dielectric response of RDX vapor [32].
observed that certain spectral features associated with the isolated molecule can be observed to persist within the spectrum for the molecular cluster. Referring to Fig. 4, one observes on average a noticeable correlation between DFT calculated spectra for molecule clusters of RDX and experimentally measured spectra for the dielectric response of RDX vapor. A comparison of DFT calculated frequencies corresponding to prominent absorptions, which have been observed experimentally for RDX in vapor state, is shown in Table 2. Referring to this table, one can observe a high correlation between DFT calculated spectra for different cluster sizes and experimentally determined absorption lines at the specific frequencies indicated. Finally, a comparison of the ground state response structure of the spectra shown in Figs. 3 and 4 (Tables 1 and 2) indicates
Table 2 Comparison of DFT calculated and experimentally measured frequencies at prominent absorptions for RDX vapor [32]. Exp (cm−1 )
1272
1602
1-RDX 2-RDX 4-RDX 6-RDX
1310 1306 1314 1316
1663–1701 1677 1696 1673
[1] B. Ferguson, X.-C. Zhang, Nat. Mater. 1 (2002) 26. [2] J.F. Federici, B. Schulkin, F. Huang, D. Gary, R. Barat, F. Oliveira, D. Zimdars, Semicond. Sci. Technol. 20 (2005) S266. [3] D. Abbott, X.-C. Zhang, Proc. IEEE 95 (2007) 1509. [4] X. Xie, J. Dai, M. Yamaguchi, X.-C. Zhang, Int. J. High Speed Electron. Syst. 17 (2007) 261. [5] B.M. Rice, C.F. Chabalowski, J. Phys. Chem. 101 (1997) 8720. [6] Y. Chen, H. Liu, Y. Deng, D. Schauki, M.J. Fitch, R. Osiander, C. Dodson, J.B. Spicer, M. Shur, X.-C. Zhang, Chem. Phys. Lett. 400 (2004) 357. [7] D.G. Allis, D.A. Prokhorova, T.M. Korter, J. Phys. Chem. A 110 (2006) 1951. [8] D.G. Allis, T.M. Korter, Chem. Phys. Chem. 7 (2006) 2398. [9] J. Chen, Y. Chen, H. Zhao, G.J. Bastiaans, X.-C. Zhang, Opt. Express 15 (2007) 11763. [10] M.R. Leahy-Hoppa, M.J. Fitch, X. Zheng, L.M. Hayden, R. Osiander, Chem. Phys. Lett. 434 (2007) 227. [11] J. Hooper, E. Mitchell, C. Konek, J. Wilkinson, Chem. Phys. Lett. 467 (2009) 309. [12] M.R. Leahy-Hoppa, M.J. Fitch, R. Osiander, Anal. Bioanal. Chem. 395 (2009) 247. [13] A. Shabaev, S.G. Lambrakos, N. Bernstein, V.L. Jacobs, D. Finkenstadt, Appl. Spectrosc. 65 (2011) 409. [14] L. Huang, A. Shabaev, S. G. Lambrakos, N. Bernstein, V. Jacobs, D. Finkenstadt, L. Massa, Ground State Resonance Structure of Some Typical High Explosives Calculated by Density Functional Theory, Naval Research Laboratory Memorandum Report NRL/MR/6390-11-9320, 2011. [15] M.J. Frisch, et al., Gaussian 09, Revision A. 1, Gaussian Inc., Wallingford, CT, 2009. [16] A. Frisch, M.J. Frisch, F.R. Clemente, G.W. Trucks, Gaussian 09 User’s Reference, Gaussian, Inc., Wallingford, CT, 2009, pp. 105–106. [17] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [18] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [19] R.O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 689. [20] W.W. Hager, H. Zhang, Pacific J. Optim. 2 (2006) 35. [21] R.M. Martin, Electronic Structures Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 2004, p. 25. [22] E.B. Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. [23] J.W. Ochterski, Vibrational Analysis in Gaussian, [email protected], 1999. [24] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [25] B. Miehlich, A. Savin, H. Stoll, H. Preuss, Chem. Phys. Lett. 157 (1989) 200. [26] A.D. McLean, G.S. Chandler, J. Chem. Phys. 72 (1980) 5639. [27] T. Clark, J. Chandrasekhar, G.W. Spitznagel, P.V.R. Schleyer, J. Comput. Chem. 4 (1983) 294. [28] M.J. Frisch, J.A. Pople, J.S. Binkley, J. Chem. Phys. 80 (1984) 3265. [29] C.S. Choi, E. Prince, Acta Crystallogr. B 28 (1972) 2857.
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[33] L. Huang, A. Shabaev, S.G. Lambrakos, L. Massa, Ground State Resonance Structure of Molecular Clusters of HMX Calculated by Density Functional Theory for THz Frequencies, Naval Research Laboratory Memorandum Report NRL/MR/6390-11-9338, 2011. [34] F. Huang, B. Schulkin, H. Altan, J.F. Federici, D. Gary, R. Barat, D. Zimdars, M. Chen, D.B. Tanner, Appl. Phys. Lett. 85 (2004) 5535.
Contents lists available at SciVerse ScienceDirect
Vibrational Spectroscopy journal homepage: www.elsevier.com/locate/vibspec
Ground-state spectral features of molecular clusters RDX excited at THz frequencies L. Huang a , A. Shabaev b,∗ , S.G. Lambrakos a , L. Massa c a b c
Naval Research Laboratory, Washington, DC 20375, USA George Mason University, Fairfax, VA 22030, USA Hunter College, CUNY, New York, NY, USA
a r t i c l e
i n f o
Article history: Received 27 January 2012 Received in revised form 16 October 2012 Accepted 16 October 2012 Available online 24 October 2012 Keywords: THz spectra Vibrational resonances DFT calculations
a b s t r a c t Calculations are presented of ground state resonance structure at THz frequencies for molecular clusters of the high explosive RDX using density functional theory (DFT). The spectral features of this resonance structure are due to coupling of resonance modes for ground state excitation. In particular, the coupling among ground state resonance modes provides a reasonable molecular level interpretation of spectral features associated with THz excitation of molecular clusters. THz excitation is associated with frequencies that are characteristically perturbative to molecular electronic states, in contrast to frequencies that can induce appreciable electronic state transition. Owing to this characteristic of THz excitation, one is able to make a direct association between local oscillations about ground-state minima of molecules, either isolated or comprising a cluster, and THz excitation spectra. The DFT software GAUSSIAN was used for the calculations of ground state resonance structure presented here. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Infrared absorption of molecular ground states is within the frequency range of the electromagnetic spectrum spanning the band 0.1–100 THz. However, the part of the spectrum 0.1–10 THz, which until recently was unexplored, has traditionally been called the “terahertz gap” owing to the lack of efficient sources for generating frequencies in this range. For the past two decades, however, there has been extensive advancement in the design of THz systems, which has been motivated by the development of new THz sources, detectors and applications for noninvasive detection [1–4]. A significant characteristic of THz excitation is that it is associated with frequencies not involving electronic state transitions. Of course, the practical aspect of the IR absorption character of THz excitation for detection is that noninvasive detection methodologies can be developed, which do not damage materials under examination. In addition, the perturbative character of THz excitation with respect to molecular states has significant implications with respect to its numerical simulation based on density functional theory (DFT). It follows that, owing to the absorption character of THz excitation, which is characteristically that of vibrational fine structure, one is able to make a direct association between local oscillations about ground-state minima of a given molecule and THz excitation spectra.
∗ Corresponding author. 0924-2031/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.vibspec.2012.10.003
With respect to THz excitation of materials, the procedure of using response spectra calculated using DFT, which is associated with ground state resonance structure, for the direct construction of permittivity functions is well posed owing to the physical characteristic of THz excitation. In particular, it is important to note that the constructing permittivity functions using response spectra calculated using DFT is physically consistent with the characteristically linear response associated with THz excitation of molecules. Accordingly, one observes a correlation between the advantages of using THz excitation for detection of explosives (and ambient materials) and those for its numerical simulation based on DFT. Again, this follows from the fact that THz excitation is associated with frequencies that are characteristically perturbative to molecular states, in contrast to frequencies that can induce appreciable electronic state transitions. Density functional theory has been successfully used to investigate the vibrational spectra of energetic materials in the form of single molecules and molecular crystals [5–11]. These calculations provide detection signatures for various forms of materials which can be encountered in various detection scenarios [12,13]. Calculations of resonant structure associated with isolated molecules help to identify intramolecular vibrational modes in the absorption spectra of various materials. Additionally, calculations of resonance structure associated with the solid state, which are in agreement with experiment, show several new modes in the region below 3 THz, which can be identified with intermolecular vibrations [6–8]. With respect to interpretation based on theory, the results of
L. Huang et al. / Vibrational Spectroscopy 64 (2013) 62–67
calculations and analysis provides new insight into the chemical origins of the intricate electromagnetic response features of molecules and molecular complexes including the delicate interplay between intra-molecular degrees of freedom and relatively weak intermolecular interactions. A significant aspect of using response spectra calculated by DFT for the construction of dielectric response functions is that it adopts the perspective of computational physics, according to which a numerical simulation represents another source of “experimental” data. This perspective is significant in that a general procedure may be developed for construction of dielectric response functions using DFT calculations as a quantitative initial estimate of spectral response features for subsequent adjustment with respect to additional information such as experimental measurements and other types of theory based calculations. That is to say, for the purpose of simulating many electromagnetic response characteristics of materials, DFT is sufficiently mature for the purpose of generating data complementing, as well as superseding, experimental measurements [6,11,13]. The application of density functional theory (DFT) and related methodologies for the determination of electromagnetic response characteristics is important for the analysis of parameter sensitivity with respect to parametric representations of excitation spectra. That is to say, many characteristics of the electromagnetic response of a given material may not be detectable, or in general, not relevant for detection. Accordingly, sensitivity analyses concerning the electromagnetic response of layered composite systems can adopt the results of simulations using DFT, and related methodologies, to provide realistic limits on detectability that are independent of a specific system design for detection. In addition, analysis of parameter sensitivity based on atomistic response characteristics of a given material, obtained by DFT, provide for an “optimal” fit of experimental measurements for the construction of permittivity functions. It follows that within the context of parameter sensitivity analysis, data obtained by means of DFT represents a true complement to data obtained by means of experimental measurements. In what follows, calculations are presented of ground state resonance structure associated with molecular clusters of the high explosive RDX. RDX is among the class of industrial or military explosives, in contrast to homemade explosives, that are of primary concern for detection in realistic environments. In addition, this explosive has a characteristic spectrum in the THz range of frequencies that is potentially distinguishable from other materials [3]. A schematic representation of the molecular geometry of RDX is shown in Fig. 1. The coordinates corresponding to an equilibrium configuration of a single isolated molecule of RDX, which have been calculated using DFT, are given in Ref. [14]. It is significant to note that, with respect to practical application, molecular clusters represent a separate regime for using DFT calculations for construction of dielectric response functions. In addition, with respect to theoretical understanding, the dielectric response of clusters provides insight concerning the nature of the transition of dielectric response to that of solids or systems having long range order. This type of insight can in turn assist in understanding the nature of the difference between the intermolecular and intramolecular vibrations.
2. Calculation of spectra using density functional theory The DFT software GAUSSIAN09 (G09) can be used to compute an approximation of the IR absorption spectrum of a molecule [15]. This program calculates vibrational frequencies by determining second derivatives of the energy with respect to the Cartesian nuclear coordinates, and then transforming to mass-weighted coordinates at a stationary point of the geometry [16]. The IR
63
Fig. 1. (A) Single molecule of RDX and (B) unit cell and positions of molecules for clusters of 2, 4 and 6 RDX molecules after geometry optimization.
absorption spectrum is obtained using DFT to compute the ground state electronic structure in the Born–Oppenheimer approximation using Kohn–Sham density functional theory [17–21]. GAUSSIAN uses specified orbital basis functions to describe the electronic wavefunctions and density. For a given set of nuclear positions, the calculation directly gives the electronic charge density of the molecule, the potential energy V, and the displacements in Cartesian coordinates of each atom. The procedure for vibrational analysis followed in GAUSSIAN is that described in Ref. [22]. Ref. [23] presents a fairly detailed review of this procedure. All vibrational frequencies for the optimized geometries of RDX clusters discussed here are positive, as must occur for true minimum geometries. For the calculations presented here geometry optimization and vibrational analysis was effected using the DFT model B3LYP [24,25] and basis function 6-311G(2d,2p) [26,27]. According to the specification of this basis function, (2d,2p) designates polarization functions having 2 sets of d functions for heavy atoms and 2 sets
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of p functions for hydrogen atoms [28]. A schematic representation of the molecular geometry of a single isolated molecule of RDX is shown in Fig. 1A. Schematic representations of molecular geometries of molecular clusters consisting of 2, 4 and 6 molecules are shown in Fig. 1B–D, respectively. It is significant to note that the relative positions of molecules associated with each of the molecular clusters is according to crystallographic structure conditions that are for a bulk material. In particular, the crystal structure of RDX, whose CCDC (Cambridge Crystallographic Data Centre) reference code is CTMTNA, has been investigated by Choi and Prince [29]. The space group for this crystal structure is Pbca (symmetry operators are x, y, z; 1/2 − x, −y, 1/2 + z; 1/2 + x, 1/2 − y, −z; −x, 1/2 + y, 1/2 − z; −x, −y, −z; 1/2 + x, y, 1/2 − z; 1/2 − x, 1/2 + y, z; x, 1/2 − y, 1/2 + z), and the unit cell constants are a = 13.182, b = 11.574, c = 10.709, ˛ = 0.00, ˇ = 90.00, = 90.00 and density 1.806 g/cm3 . The ground-state oscillation frequencies and IR intensities for different molecular clusters of RDX, corresponding to their relaxed equilibrium configurations, are calculated by DFT according to the frozen phonon approximation. In the cases of molecular clusters of 2, 4 and 6 molecules, these values are given in Tables 3–5, respectively, of Ref. [33]. In the case of 2-molecule clusters, a total of 120 frequencies (from 14 to 3221 cm−1 ) were calculated (see Table 3 of Ref. [33]). In the case of 4-molecule clusters a total of 246 frequencies (from 5.9 to 3227 cm−1 ) were calculated (see Table 4 of Ref. [33]). In the case of 6-molecule clusters a total of 372 frequencies (from 4 to 3238 cm−1 ) were calculated (see Table 5 of Ref. [33]). Fig. 2 shows spectra of a single molecule of RDX in comparison with the spectra of 2-, 4- and 6-molecule clusters. The DFT model and basis function used for these calculations are the same as those used for the single isolated molecule of RDX. The programs ConQuest and Mercury [30] were used for searching the Cambridge Structural Database (CSD), for visualizing database entries associated with RDX, and constructing the molecular clusters. The construction of any given cluster required consideration of the interactions between a single molecule and its neighborhood. Accordingly, hydrogen bonding between molecules is important for establishing a stable configuration for any given cluster.
3. Discussion Computational experiments using DFT, concerning molecular clusters of RDX, are presented in this section. These results include the relaxed or equilibrium configuration of a single isolated molecule of RDX (see Table 1 in Ref. [33]), ground-state oscillation frequencies and IR intensities for this configuration, which are calculated by DFT according to the frozen phonon approximation (see Table 2 in Ref. [33]). The ground state resonance structure for a single isolated molecule of RDX is adopted as a reference for analysis of spectral features associated with molecular clusters of different sizes. Calculated IR intensities for a single isolated molecule, 2molecule cluster, 4-molecule cluster and 6-molecule cluster, respectively, are shown in Fig. 2. Referring to Fig. 2, one notes continuous spectra consisting of a superposition of essentially Lorenzian functions of various heights and widths, which have been constructed using discrete spectra. Although the primarily purpose of this type of construction within GAUSSIAN is for the purpose of enhanced visualization of spectral features, it is significant to note that this operation represents an estimation of the characteristic scaling and width of resonances contributing to the dielectric response [13]. For qualitative comparison of spectral features this type of zeroth-order estimate should be sufficient. For the construction of permittivity functions to be used for quantitative simulations, it is more appropriate, however, to assume the characteristic scaling and widths of DFT calculated resonances
Fig. 2. Discrete spectra for a single isolated molecule of RDX and molecular clusters containing 2, 4 and 6 molecules.
as adjustable parameters, i.e., parameters to be assigned values according to additional information. A qualitative comparison of spectral features is shown in Fig. 2. Referring to this figure, it can be observed that in general each molecular cluster is characterized by a relatively distinct spectrum consisting of its own set of characteristic features. Again, hypothesizing with respect to using this type of information for construction of models based on effective medium theory, a composite material could be characterized by a non-interacting uniform “weighted” distribution of finite-size molecular clusters within a host material, where the weighting is according to the relative fraction of clusters within the distribution consisting of a given number of molecules. Initial consideration is given to what extent various average features of DFT calculated spectra for molecular clusters of different
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Fig. 3. Qualitative comparison of DFT calculated spectra for 1-, 2-, 4- and 6-molecule clusters of RDX and experimentally measured spectrum for dielectric response of RDX solid [31].
sizes, as well as experimentally measured spectra, may be correlated with each other. Referring to Fig. 2, it can be seen that cluster resonances form bands grouped near the original high frequency molecular resonances. A new band of resonances can be observed to form at the low end of the frequency spectrum. Interestingly, the average features of the high and low frequency bands may be analyzed using a relatively simple coupled-oscillators model. It follows that one is able to associate with molecular clusters, by their nature, two separate interaction regimes where stiff and soft springs are to be associated with intramolecular and intermolecular modes, respectively. That is to say, more tightly bound atoms which comprise a given molecule, interact with atoms associated with other molecules comprising the cluster. It could be argued in retrospect that the existence of a commingling of stiff and soft spring interactions is intuitively obvious in the case of molecular clusters (for certain types of molecules). Proceeding with our comparison of spectra, we note that studies have emphasized the observation that calculated spectra corresponding to a single isolated molecule are not strongly correlated with that of the bulk-lattice. In particular, Allis et al. [7] have emphasized that calculated spectra for the dielectric response of a bulk lattice are more appropriate with the use of periodic boundary conditions for representation of intermolecular influences. Relative to this point, however, the DFT calculations of spectra presented here tend to indicate a relatively interesting result, which has implications with respect to both understanding the nature of coupling between inter- and intra-molecular modes for explosives, as well as the use of calculated spectra for modeling dielectric response for practical applications, i.e., simulation of methods for explosives detection. Specifically, the DFT calculated spectra for molecular clusters show progressively higher correlation with that of the bulk with increasing size of the cluster within the neighborhoods of the resonances labeled A–C in Fig. 4. This supports the notion that farfield coupling of intramolecular modes to intermolecular modes is dominant only within a restricted range of molecular cluster size. In addition, this implies that with respect to the modeling of bulk lattice response for estimating dielectric response, for the purpose of simulating detection systems, DFT calculated spectra corresponding to molecular clusters should provide a reasonable estimate of bulk dielectric response characteristics. Spectra of clusters in comparison with experimentally measured spectra [31,32] are shown in Fig. 3. Referring to these figures, initial consideration is given to what extent various average features of the DFT calculated spectra for different size molecular
clusters, as well as experimentally measured spectra, are correlated with each other. Referring to Fig. 3, one observes on average a noticeable correlation between DFT calculated spectra for molecule clusters of RDX and experimentally measured spectra for dielectric response of RDX solid. The comparison of spectra shown in Fig. 3 also supports the notion that certain resonance features, which are associated with finite-size molecular clusters, are preserved within clusters of increasing size, as well as in solid form, thus implying a persistence of these modes after coupling to intermolecular influences. The persistence of certain intramolecular modes, which is irrespective of molecular cluster size, i.e., whether the molecule is isolated, part of a cluster or within a bulk lattice, is demonstrated by comparison of DFT calculated and experimentally measured spectra. A comparison of DFT calculated frequencies corresponding to prominent absorptions, which have been observed experimentally for RDX in solid state is shown in Table 1. It is significant to note that these experimental measurements show good correlation with those of Ref. [34]. Referring to Table 1, one can observe a high correlation between DFT calculated spectra for different cluster sizes and experimentally determined absorption lines at the specific frequencies indicated. Next, we consider what average spectral features can be correlated with the size of a molecular cluster. Again referring to Fig. 3, which compares both DFT calculated and experimentally measured spectra [12,10], it can be observed that as the size of the cluster increases, there is a relative increase in the presence of resonance structure at lower frequencies. An examination of this trend in spectral features can be made more quantitative by consideration of the DFT calculated spectra in terms of their discrete delta-function representations as shown in Fig. 2 for a single isolated molecule and 2-, 4- and 6-molecule clusters of RDX. Referring to this figure, one observes an increase in the number of absorption lines for the molecule clusters relative to the discrete spectrum associated with a single isolated molecule. Again, referring to this figure, it can be
Table 1 Comparison of DFT calculated and experimentally measured frequencies at prominent absorptions for RDX solid [31]. Exp (cm−1 )
27
35
46
51
66
74
103
1-RDX 2-RDX 4-RDX 6-RDX
– 26 17–22 21
– 35 33 35
42 49 46 49
57 59 54 59
62 – 64 66
– 87 73 74
94 100 84–94 92
66
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significant structure at lower frequencies for solid materials, which is physically consistent with intermolecular vibrations. 4. Conclusion Calculations of ground state resonance structure at THz frequencies for molecular clusters of RDX using density functional theory have been presented. Average spectral features associated with this resonance structure can be interpreted qualitatively as coupling between stiff and soft resonance modes. Comparison of DFT calculated spectra for a single isolated molecule of RDX and DFT calculated spectra for a molecular cluster of RDX indicates that certain spectral features associated with the isolated molecule can be observed to persist within the spectrum for the molecular cluster. Comparison of DFT calculated spectra for molecular clusters of RDX and experimentally measured spectra indicates the persistence of certain intramolecular modes, which is irrespective of molecular cluster size, ranging from molecules that are isolated, part of a cluster or within a bulk lattice. Finally, the DFT calculations presented in this study represent a specific set of computational experiments, which can be compared to laboratory measurements, as well as other computational experiments. Accordingly, comparison of the results of this study to DFT calculations using other functionals, e.g., PBE0 and CAMB3LYP, could provide further insight concerning the nature of numerical artifacts manifesting as spectral features. Acknowledgment This work was supported by the Office of Naval Research. References
Fig. 4. Qualitative comparison of DFT calculated spectra for single isolated molecule and 6-molecule cluster of RDX with experimentally measured spectrum for dielectric response of RDX vapor [32].
observed that certain spectral features associated with the isolated molecule can be observed to persist within the spectrum for the molecular cluster. Referring to Fig. 4, one observes on average a noticeable correlation between DFT calculated spectra for molecule clusters of RDX and experimentally measured spectra for the dielectric response of RDX vapor. A comparison of DFT calculated frequencies corresponding to prominent absorptions, which have been observed experimentally for RDX in vapor state, is shown in Table 2. Referring to this table, one can observe a high correlation between DFT calculated spectra for different cluster sizes and experimentally determined absorption lines at the specific frequencies indicated. Finally, a comparison of the ground state response structure of the spectra shown in Figs. 3 and 4 (Tables 1 and 2) indicates
Table 2 Comparison of DFT calculated and experimentally measured frequencies at prominent absorptions for RDX vapor [32]. Exp (cm−1 )
1272
1602
1-RDX 2-RDX 4-RDX 6-RDX
1310 1306 1314 1316
1663–1701 1677 1696 1673
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