摘要:

T. Marinkovic� and S. Oss* Vibrational modes of CH bonds in n-paraffin molecular chains: an algebraic description Department of Physics, University of Trento and Istituto Nazionale di Fisica della Materia, 38050 Povo, Trento, Italy Received 13th February 2002, Accepted 8th March 2002 Published on the Web 26th March 2002 We apply the one-dimensional vibron model to compute energies and infrared intensities of CH vibrational modes (stretching and bending states) of n-paraffin molecules. We consider the possible onset of interactions between different kinds of bending motions as well as that of anharmonic resonances between CH stretching modes and CH bending overtone/combination vibrations. 1 Introduction Quite a massive amount of information exists in the study of general spectroscopic features of molecular chains of arbitrary length, such as n-paraffins, (CH3)–(CH2)n–(CH3).It is quite a long time since infrared spectra of a series of such molecules (bending modes of C20H42–C30H62) were recorded at lowtemperature1 and described with a model inclusive of intermolecular effects.2 Long paraffin chains and polyethylene (infinite chains) were compared in terms of normal mode coordinate calculations by using perturbation methods.3–5 Specific analyses of crystalline normal paraffins (C3H8–C30H62) taking into account their detailed conformations were carried out, mainly at low temperatures and in the region of mixed bending mode excitation,6–8 while liquid IR spectra were obtained and reported in ref.4 and 9. More recently, the possible onset of anharmonic (Fermi) resonances in the region of infrared active CH stretching modes was considered.10 Studies are available dealing with more detailed spectroscopic features of weak IR modes.11 Raman spectra are also considered to some extent, see for example ref. 12 and 13. As a practical and alternative route to traditional approaches and with the aim of providing a simple, yet general description of such kinds of molecules, we present here an algebraic treatment of the full set of CH vibrational modes. We obtain energies and infrared intensities of CH stretching and bending degrees of freedom. Novel aspects that will be addressed here are in particular (i) the quantitative inclusion of specific interactions between different kinds of bending modes, (ii) the general features of combination/overtone bands, and (iii) a first, possible route leading to the description of anharmonic (Fermi) resonances involving different families of vibrational modes.The algebraic method adopted in this work is the first practical application of actual interest based on the general idea introduced in ref. 14. This paper is structured as follows. In Section 2 we briefly recall the most important aspects of algebraic techniques and their application to the present case study. In Section 3 we obtain energies and infrared transition intensities of fundamental CH stretching modes of n-paraffins, inclusive of endeffects induced by CH3 terminal groups.In Section 4 we extend the Hamiltonian operator to describe fundamental CH bending modes and we introduce more complex terms devoted to describe twist/rock and bend/wag interactions. In Section 5 we describe the anharmonic (Fermi-like) resonance mechanisms involving CH fundamental stretches and CH overtone/ combination bending modes in the 3000 cm21 energy region. ai ^Ciz Xn i<j~1 2 Algebraic model Algebraic techniques are nowadays adopted as an alternative approach to address several, important problems of molecular spectroscopy.15 Yet, it was only recently suggested that Liealgebraic methods can also be used in the treatment of vibrational modes of molecular chains and of polymers (infinite length chains).14,16 In this paper we show that a simple, onedimensional model can be adopted to describe stretching and bending of CH bonds in n-paraffin molecules.Such chains are the starting point for more complex case studies such as the extension to polymer chains. The basic aspects of algebraic models for molecular spectroscopy have already been presented and discussed in several papers17–19 and books.15 Here, we limit ourselves to recall those aspects needed to extend the algebraic model to molecular chains of arbitrary length, as suggested in ref. 14. We construct an algebraic Hamiltonian operator to obtain energies and wave functions of CH stretching/bending modes of the n-paraffin molecule shown schematically in Fig. 1. The one-dimensional algebraic Hamiltonian for n coupled oscillators is18 ^H~E0zXn i~1 66 PhysChemComm, 2002, 5(10), 66–75 This journal is # The Royal Society of Chemistry 2002 Paper (1) lijM ^ij , aij ^Cijz Xn i<j~1 where the operators C �i, C �ij and M�ij generate the anharmonic, Morse-like sequence of vibrational energies and states, as Fig.1 Bond numbering, lengths, angles and labelling of inter/intrasite couplings in the n-paraffin molecule. DOI: 10.1039/b201553fwidely explained elsewhere.18 More specifically, one obtains anharmonic energies in terms of the eigenvalues (2) Sv i ^Ci viT~{4vi(vi{N), in which the anharmonicity is given by 1/N (N is the vibron number for the U(2) representation and is strictly related to the total number of bound states) and vi ~ 1, 2,…, N are the labels for the vibrational quantum numbers in the (local, uncoupled) basis |v1v2…vnm.Coupling between oscillators are introduced through the operators C �ij (cross-anharmonicity) and M�ij (Majorana, off-diagonal interactions), whose expectation values are given by (3) and Sviz1,vj{1 ^Mij Svivj ^Cij vivjT~{4(vizvj)(2N{vi{vj)=N =N: ~{ (4) vivjT p........................................................ vj(viz1)(N{vi)(N{vjz1) M ^~ M ^X2n i~1 d~ aiz1az ^Mibiz1b ), ^Fc~Xn{1 (M ^iaiz1bzM ^ibiz1a ) i~1 Xn{1 i~1H ^CH2(S)~a ^Czl ^Mzfd ^Fdzfc ^Fczsd ^Sdzsc^Sc, in which we introduce the operators C iaib ^i , Xn i~1 ( ^Mi ( ^Mi C ^~ F^S ^d~Xn{2 i~1 3 CH stretching modes in n-paraffin molecules, v~1 We extend the simple Hamiltonian (1) to account for the various intrasite and intersite couplings in the n-paraffin molecule.First, we consider CH stretching modes (S). Such modes are taken as completely uncoupled from other vibrational species, in a first approximation. If one disregards terminal CH3 sites, the Hamiltonian operator can be written as (5) (6) (7) aiz2az ^Mibiz2b ), ^Sc~Xn{2 (M ^iaiz2bzM ^ibiz2a ), i~1 see again Fig. 1. The operator C �is used to generate 2n-fold degenerate sets of levels representing uncoupled, local vibrational modes of CH stretches. The operator M�introduces non-diagonal, intrasite interactions between pairs of equivalent CH stretches. Its effect is such that two fundamental (v ~ 1) energies (corresponding to symmetric/antisymmetric combinations of localized vibrational states) are associated to each site.The remaining operators in eqn. (6) account for intersite, anharmonic interactions involving the 2n CH2 groups. More specifically, the operators F�d and F�c refer to first-neighbor direct- and first-neighbor cross-interactions, respectively. The operators S �d and S �c are used to introduce second-neighbor couplings. The action of intersite couplings is that of removing the n-fold degeneracy associated with both symmetric and antisymmetric combinations of local vibrational modes in the fundamental polyads of levels.In the simplest case of first-neighbor interactions only (sd~sc~0 in eqn. (5)), it is well-known1 that the n-degenerate levels (v ~ 1) split according to the dispersion law E(k)~E0zF cos kp nz1 , k~1,2, . . . ,n, where E0, F can be written in terms of the parameters a, l, fd, fc of eqn. (5). More explicitly, in the model for CH stretches, the Hamiltonian matrix for the fundamental polyad of vibrational modes with first-neighbor interactions only is given by the tridiagonal form H(I) S(v~1)~6 CH 2D T 666664 in which (b) v v v1 (a) 1 where e is the energy of the local CH mode, e ~ anC �m ~ 24a(N 2 1). It is convenient to rearrange the local basis in the (b)m. We introduce the symmefk T D T T P P 0 l e 2(b)…v (a) k n v(+)T~ v(a)T+ v(b)T, v D~ e l , T~ fd fcized states given by c nk , in terms of which the matrix (8) becomes block-diagonal, CHS(v~1)~ H(1z) 01 where 1 H(I) 2 e(+) f (+) 6 f (+) e(+) f (+) 0 66664 and f H(+)~6 Ek(+) E(k)~E S(v~1)~ H1’(z) e(+)~e+l, f (+)~fd+fc: So, it is possible to obtain explicit dispersion laws for the fundamental polyads of CH stretches in terms of the blocked Hamiltonian (11), whose eigenvalues are given by ~e(+)z2f (+) cos kp nz1 , k~1,2, .. . ,n: We see that this matrix formulation gives a simple, direct physical significance to the dispersion parameter F of the general expression (7).One can thus expect different dispersions in the energy manifolds for symmetric (1) and antisymmetric (2) modes as a function of the relative magnitudes of first-neighbor, direct- and cross-interaction terms, f (¡) ~ d¡fc. For example, a small dispersion of antisymmetric levels could be explained in terms of comparable values of direct- and cross-interaction contributions to first-neighbor couplings. At the same time, symmetric and antisymmetric dispersion curves should be different as long as the cross-interaction terms give a non-negligible contribution. We discuss in the following some aspects concerning a direct comparison with experiments. If one includes second-neighbor interactions in the Hamiltonian operator, the dispersion law (7) becomes more complex but, for relatively large values of n it can be approximated by16 0zF cos kp nz1zS cos n2zkp1 , k~1,2, .. . ,n: (15) It can be shown that, for realistic values of F, S parameters, when n > 8 the relative error of computed energies remains as low as 0.1%. In our model for CH stretches, the Hamiltonian matrix for the fundamental polyad (v ~ 1) after the introduction of the symmetrized basis states (10) can be written as H(II) CH 0 " # 0 1 in which PhysChemComm, 2002, 5(10), 66–75 0 3 (8) (9) 7777P D T T D775 fd (10) (11) (12) " # 0 H({) 0 3 f (+) 777775 P P P e(+) f (+) 7 f (+) e(+) (13) (14) (16) H’({) 67e(+) f (+) s(+) f (+) e(+) f (+) s(+) s(+) f (+) (17) 261 s(+) 66 777s(+) f (+) e(+) f (+) 3 P P s(+) P P f (+) s(+) 7 s(+) f (+) e(+) H’(+)~666664 77775with (18) (19) (20) e(+)~e+l, f (+)~fd+fc, s(+)~sd+sc and eigenvalues Ek(+)%e(+)z2f (+) cos kp nz1z2s(+) cos n2zkp1 , k~1,2, .. . ,n: Once again, we see how the present matrix formulation leads to a simple, direct physical explanation of the first- and secondneighbor parameters of the observed dispersion law in the general form of eqn. (7). We consider in the following more specific numerical aspects of this result. The fact that eqn. (19) holds only in the limit of large n, implies that a fitting procedure over CH stretching energies of n-paraffin molecules will lead to f (¡) and s(¡) values which will be adopted to describe dispersion laws of the polyethylene molecule (infinite length chain).In order to understand the actual contents of this procedure, we consider its application to the study of CH stretching modes in n-paraffin molecules. Experimental values are available for n ~ 3,…, 10 as well as for polyethylene (n A ‘),3,4,13 in which the discrete sequence of energy levels is replaced by a continuous dispersion curve. In Fig. 2 we show a partial collection of experimental and computed energies for a number of n-paraffins. The Hamiltonian operator is based on the algebraic parameters listed in Table 1. This table contains also further parameters devoted to account for end-effect associated with the terminal CH3 groups, which in fact are the most relevant cause of deviations from the dispersion law, eqn.(19). Observed and computed levels are in fair agreement. As suggested by the labelling introduced in Fig. 1, the most important terms in this respect are such that the complete Hamiltonian operator can be written as CHS~^HCH2(S)z^HCH3-Iz^HCH3-IIz^HCH2-CH3 H ^ in which the operators for CH3 sites are given by Fig. 2 Comparison between experimental and computed symmetric CH stretches in a number of paraffin molecules (n ~ 3,…, 10) as a function of the normalized site index k. Full circles, selected experimental data; open squares, computed energies. A similar plot can be drawn for antisymmetric CH stretches. 68 PhysChemComm, 2002, 5(10), 66–75 Table 1 Parameters of the algebraic Hamiltonian operator adopted to describe CH stretching modes in n-paraffin molecules.All values in cm21, except N which is dimensionless. The meaning of algebraic terms is illustrated in Fig. 1 as well as in eqn. (5), (20)–(22) Parameter value Term 43 2961.00 230.00 6.20 0.10 20.20 20.10 3015.00 227.00 NalfddcCH3 bc fc ssalla 230.00 0.03 0.07 0.08 H ^ ft fbd fbc CH3-I=II~aCH3 i~a,b,cC X ^izlbcM ^bczla(M ^bazM ^ca) (21) and the interaction term (limited to first-neighbor couplings) is given by H ^CH2-CH3~ft(M ^aazM ^ab)zfbd(M ^bazM ^cb) (22) zfbc(M ^bbzM ^ca) with a similar expression for the CH3(II) site at the opposite end of the chain.It is expected that in long chains end-effects will give very small contributions to the vibrational spectrum. Since we present here results for relatively small values of n, this approximation cannot be adopted and we need to account for explicit contributions caused by the terminal CH3 groups. An important feature of vibrational modes in regular chains, such as n-paraffins, is that they are dispersed according to an alternating sequence of irreducible representations of the molecular point group. In n-paraffins the symmetry groups are C2v (n odd) and C2h (n even). Such an alternating sequence originates characteristic sequences in the infrared (as well as Raman) spectra, as discussed in the following.We observe that in polyethylene the CH stretching dispersion laws (associated with symmetric/antisymmetric modes in the fundamental polyad) lead to a splitting (at k ~ 0) by y74 cm21 with the center of gravity located at y2880 cm21. We thus have a sensible way to calibrate the most important parameters of the CH2 Hamiltonian operator, i.e. the parameters a and l in eqn. (5), according to the splitting scheme shown in Fig. 3. The slightly different behavior of symmetric/ antisymmetric dispersion curves implies, following eqn. (15)– (19), that the cross-interaction part of the first-neighbor coupling (F�c) should be taken as a relatively small term in Fig. 3 Splitting of CH2 stretching levels in the n-paraffin molecule. The parameters a, N and l are related to the direct Casimir, irreducible representation of U(2) and Majorana intrasite terms in the Hamiltonian operator, respectively.the Hamiltonian operator.At the same time, we observe that a good fit is obtained with a simple dispersion law in the form of eqn. (14). We thus understand that second-neighbor couplings involving CH stretching motions are not of particular significance in these kinds of molecules (at least at the present level of confidence, as shown in Table 1). Starting from the algebraic Hamiltonian operator, one can also obtain eigenfunctions in the representative local basis. For example, the simple case of three CH2 sites (5-paraffin molecule), excluding terminal CH3 groups, gives rise to six vibrational modes whose splitting scheme and eigenfunctions are schematically shown in Fig.4. a (23) ({^tkz^tbk )z^teex , a 3.1 Transition intensities Algebraic methods allow a relatively straightforward treatment of dipole/quadrupole operators to compute infrared/Raman transition intensities.15 In the present work we show how to construct the basic dipole operator for CH modes, whilst a more refined treatment of these kinds of problems will be addressed in a subsequent paper. The general recipe for calculating infrared spectra of simple molecular chains has been given in ref. 14. Here, we extend such a computation to the more realistic case of n-paraffins, arbitrary n, inclusive of first- and second-neighbor interactions, where required, as well as of terminal CH3 groups.In the specific case of CH stretches in an n-paraffin molecule, the Cartesian components of the dipole operator can be written (see Fig. 1) as^x~sin a T k~1 ^y~cos a T Xn k~1 Xn T (24) ({1)kz1(^tkz^tbk )z^teey k ^tk j j0T&tk exp ({bvk): ^z~^tz where we introduce local (bond) dipole operators t�k, whose expectation values in the local basis can be approximated as15 Sv In this last expression, tk (dipole strength) and b (intensity curve slope) are arbitrary parameters to be adjusted in comparison with experimental values. We notice that, in eqn. (23), dipole operators are defined in terms of certain geometrical Fig. 4 Schematic representation of level splitting and wave functions of CH stretching modes in the 5-paraffin molecule.Drawings not on scale. Table 2 Comparison between computed and observed energies and infrared intensities of CH stretching modes in the n-paraffin molecule, n ~ 7 Symmetry ABABABBBAAABBBAA1 11 12 211111 22222 a 2878 2875 2965 2965 2965 2965 2850 —2852 —2866 —2920 ——2930 Values in cm21. bValues in cm21, from ref. 4. Assignments are in certain cases only temptative due to lack of sufficiently high resolution spectra. cValues in arbitrary units, from ref. 8. dValue normalized to the experimental intensity. parameters, such as equilibrium angles for CH bonds (angle a in Fig.1). We have obtained such values with a numerical energy minimization (Hartree–Fock, basis 6-31G*) based on commercial software (PC Spartan PRO, #Wavefunction, Inc.). The operators t�ee (j ~ x,y,z) are devoted to describing the dipole components of the terminal CH3 groups (ee signifies end-effects). Their explicit form can be easily obtained once again by considering the explicit geometry of CH3 groups and their Cartesian projections along the chosen axes. If one considers only fundamental transitions (v ~ 0 A 1), the parameter b is not required to compute the IR spectrum (it concurs to define relative intensities of overtone/combination modes, v w 1). We have computed IR intensities of n-paraffins (n ~ 3, …, 22). A sample of such a computation is reported in Table 2 and shown in Fig.5 for the molecules with n~7, 10, 15. In this figure, we have smoothed out the intensity peaks with a Lorentzian shape (FWHM ~ 1.5 cm21). As a general feature of these spectra, we observe that for larger and larger n the relative contribution of CH3 groups becomes smaller and smaller in comparison with that of CH2 groups. Intensity (calc.) Intensity (exp.)c Energy (exp.)b Mode Energy (calc.)a —0.38 3.25 ——3.25 ——2.00 1.18 0.32 3.81 0.68 0.00 3.99 0.00 0.00 2.34 0.16 0.01 0.04 0.51 6.95d 2883 2884 2965 2966 2969 2970 2851 2863 2846 2857 2867 2906 2917 2927 2911 2923 CH CH CH CH CH CH CH CH CH CH CH CH CH CH CH CH ————6.95 —— 0.00 0.00 3333332222222222 j 69 PhysChemComm, 2002, 5(10), 66–75Fig. 5 Simulated infrared spectra of selected n-paraffins (n~7, 10, 15) in the region of CH stretching modes.The continuous line is based on a Lorentzian shape (FWHM ~ 1.5 cm21). This is simply due to the fact that for larger n the number of CH2 groups is correspondingly larger while one has always two terminal CH3 groups. The comparison with experimental data is however feasible with caution since only liquid or crystalline paraffins have been measured in the IR regime. It would be very interesting to develope a series of IR/Raman measurements in the gaseous phase as well. Yet, the general features observed in experiments are fairly well reproduced in terms of the present, simple dipole operators in the algebraic basis obtained with the Hamiltonian operator discussed in this work.A more definitive treatment of this problem (at present under study) will require the inclusion of medium effects (such as intermolecular couplings in a liquid or crystalline environment). 4 CH bending modes in n-paraffin molecules, v ~ 1 It is now possible to extend the Hamiltonian operator to include CH bending degrees of freedom. Such modes are usually considered as belonging to two distinct families. In the first family, one has ‘‘in-plane’’ bends, in which CH bond motions stay in a plane parallel to the plane containing the (rigid) carbon chain.In the second family, one has ‘‘out-ofplane’’ motions, in which CH bonds move perpendicularly to the above-mentioned plane. As a further distinction, one considers symmetric/antisymmetric motions with respect to CH bond exchange within a single CH2 site. As a consequence, in-plane motions lead to ‘‘wagging’’ (symmetric) and ‘‘twisting’’ (antisymmetric) modes. Out-of-plane motions are divided into ‘‘scissor’’ (symmetric) and ‘‘rocking’’ (antisymmetric) modes. In Fig. 6 we show a schematic representation of 70 PhysChemComm, 2002, 5(10), 66–75 Fig. 6 Schematic representation of bending CH modes in the n-paraffin molecule. Green arrows refer to in-plane motions (twist, symmetric and wag, antisymmetric species); red arrows refer to out-of-plane motions (scissor, symmetric and rock, antisymmetric species).twist/wag and scissor/rock vibrations. A further kind of bending-like mode occurs as soon as one accounts for CH bending modes of terminal CH3 groups coupled through the whole molecular chain. One can in fact observe (at quite low frequency, y200 cm21) torsion-like vibrations associated with symmetric/antisymmetric in-phase, out-of-plane bends of CH3 bonds. Even though we have accounted for such modes in our model (by means of quite a typical projection method for separating spurious modes from non-spurious ones), they will not be discussed in detail in the present work. We focus our attention to the description of bending modes in terms of an algebraic Hamiltonian operator which turns out to be very similar to that previously described for CH stretches.The main differences stand in the parameters adopted and, quite obviously, in the physical interpretation of energies and wave functions. Even though experimental information concerning the dispersion of bending modes in polyethylene is available,3,13 here we consider once again only discrete dispersion curves computed for n-paraffin molecules. The Hamiltonian operators for bending modes are in the form of eqn. (20)–(22). We consider two distinct sets of operators for in-plane and out-ofplane bending vibrations. We are able to calibrate these operators in such a way that bending modes for n-paraffins (arbitrary n) are computed in fair agreement with the available experimental values.More specifically, we observe in polyethylene that out-of-plane (scissor/rocking) vibrations are separated by y750 cm21 (zero phase-angle), while for in-plane (twisting/wagging) vibrations the splitting is y5 cm21. This means that (at zero phase-angle) out-of-plane modes are much less coupled than in-plane modes. This can be explained in terms of strong/weak intrasite interactions, respectively, as easily shown in the framework of the present simple algebraic formulation. As already done for CH stretches, the two-fold sets of degenerate levels then split by the action of intersite anharmonic couplings. The algebraic parameters for first- and second-neighbor couplings as well as for direct- and crossexcitation schemes are reported in Table 3.Contrarily to CH stretches, CH bending motions are such that direct-and crossterms for first-neighbor interactions are of comparable magnitude. Moreover, second-neighbor terms acquire a nonnegligible role (mostly for out-of-plane benders). We think that this can be simply explained in terms of the obviously different geometric nature of stretching/bending motions (CH bending modes ‘‘disturb’’ motions of adjacent sites, while stretching modes do not). In Table 4 we report a sample of computed and observed energy values of bending modes for n-paraffin molecules, n ~ 4, 5, 6. Calculations are also available for infrared intensities. These have been performed in terms of algebraic wave functions and of bond-like dipole operators associated with bending modes.The explicit forms of such operators are very similar to those adopted for CH bond operators, eqn. (23), and are not reported here. Table 4 shows that the agreement between theory and experiment is oqualitatively settled, at least for a number of levels whose assignments are, in our opinion, far from being satisfying ones.Table 3 Parameters of the algebraic Hamiltonian operator adopted to describe CH bending modes in n-paraffin molecules. All values in cm21, except N which is dimensionless. The meaning of algebraic terms is the same as that adopted for CH stretches, see text Term3 NalfddcCH bc a5H12 6H14 afc ssallft fbd fbc As previously mentioned, we think that it would be interesting to acquire new experimental data in the gaseous state mainly in the intermediate energy range (CH bending modes, 700– 1500 cm21).Another possibly important improvement to the present treatment will be achieved by including skeletal CC vibrations along with their couplings with CH benders. It is well-known that certain CC vibrational modes in n-paraffins are dispersed according to ‘‘acoustical’’ laws.8,13 Such aspects require quite a deep extension of the algebraic model which will be presented in a subsequent paper. Table 4 Algebraic analysis of bending vibrational modes of n-paraffin molecules, n ~ 4, 5, 6. Different kinds of vibrations are denoted by r (rock), t (twist), w (wag), s (scissor).All values in cm21 Molecule C4H10 CCFrom ref. 3. Value (out-of-plane) 180 1106 300 232 31 211 12 1450 30 3000.01 0.01 0.01 Mode u ggu uu1g g 22 22 2211 1 11u gu gug guu g g u g ug r A r B t B w B t A w A s A s B r B r A r B t A t B t A w B w A w B s A s B s A r A r B r A r B t A w A t B t A w B t B w A w B s A s B s A s Bu Value (in-plane) 180 1250 22 235 210 23 22 1350 20 22.5 0.01 0.09 0.01 Exp.a Calc. 733 …… 1293 1257 … 1455 1459 731 857 1196 1220 1246 1384 1436 1436 725 759 958 … 1229 1240 1258 1336 1380 1449 1453 1458 716 771 895 1185 1222 1256 1270 1326 1384 1396 1436 1436 721 … 798 … 996 …… 1225 1242 … 1302 1333 1450 1452 1462 1463 712 734 813 917 1180 1189 1206 1237 1242 1261 1297 1353 1394 1397 1436 1436 (25) 4.1 In-plane and out-of-plane mode interactions In this work we also address a possible route to improve the agreement between observed and computed infrared intensities of CH bending modes.As suggested in the past,7 a more complete description of the vibrational dynamics of polymer chains must also account for in-plane/out-of-plane mode interactions. A general solution to this problem in the framework of algebraic models would require the adoption of two/three dimensional techniques.19,17 Even though feasible, such methods go definitely beyond one goal of this work, i.e.to show that in/out bending mode interactions, even when simply described, lead to a better effective description of these molecules. We show in Fig. 7 computed spectra in the energy region of twisting/wagging modes (1100–1400 cm21) for n~7, 11. Coupling terms are introduced in the usual algebraic fashion, as illustrated in Fig. 8. We consider, in the algebraic Hamiltonian operator, the (anharmonic) interactions for the CH2 site k given by H ^(k) I=O~d( ^Mka,IP ,ka,OPz ^Mkb,IP,kb,OP )zc ^Mka,IP,kb,OP zQd( ^Mka,IP ,kz1a,OPz ^Mkb,IP,kz1b,OP ) zQc( ^Mka,IP ,kz1b,OPz ^Mkb,IP,kz1a,OP ) d( ^Mka,IP ,kz2a,OPz ^Mkb,IP,kz2b,OP ) zs zsc( ^Mka,IP ,kz2b,OPz ^Mkb,IP,kz2a,OP ) In this expression, d, c refer to intrasite, direct- and cross-in/out Fig.7 Simulated infrared spectrum of n-paraffin molecules (n ~ 7, 11) in the bending twist/wag region. In- and out-of-plane modes are coupled through anharmonic interactions. We observe a small amount of infrared activity of otherwise dark twisting modes associated with their coupling with rocking modes. 71 PhysChemComm, 2002, 5(10), 66–75Fig. 8 Schematic representation of the relevant terms in the algebraic model for describing interactions between in- and out-of-plane bending modes. In this figure we plot all terms associated with a single out-ofplane local bender.Similar connections should then be drawn for each CH bond. coupling, respectively. Parameters Qd, Qc, sd, sc are used to give first-, second-neighbor, direct (d) and cross (c) intersite in/out couplings, respectively. We believe that, according to Fig. 7, our calculations give a minimum support to the possibility of accounting for in/out couplings in n-paraffin molecules. Indeed, it is only by virtue of such interactions that twisting modes gain a small, yet non-zero infrared activity, as possibly shown in a few experimental data.6 In Fig. 7 we see that with the inclusion of in/out-of-plane interactions at a non-negligible level (the corresponding parameters in eqn. (25) are d ~ 20 cm21, c ~ 215 cm21, Qd ~ 4 cm21, Qc ~ 23 cm21, sd ~ 2 cm21, sc~22 cm21) the spectrum is heavily distorted in this energy region.Yet, the difficult assignment of vibrational levels does not allow a non-ambiguous choice of reliable parameters. More definitive and precise results will be accessible when more careful experimental data will be available in this energy range. 5 First overtone and anharmonic resonances in n-paraffins One of the main advantages of the (one-dimensional) algebraic formulation is that anharmonicity is included in the Hamiltonian operator from the beginning. In order to consider anharmonic effects in a systematic way, it is necessary to include higher overtone/combination bands in the computation of vibrational spectra. As a particularly important and interesting feature of CH spectra, the possible onset of anharmonic (Fermi-like) resonances is a well-known fact.There is very little experimental information10 concerning possible evidence of anharmonic resonant couplings among fundamental CH stretching modes (located at y3000 cm21) and the region of first overtone/combination bands of bending (scissor/wagging) modes in n-paraffin molecules. In this paper we show that with our model it is a relatively straightforward task to describe these kinds of interactions in a systematic way. As a side result, we derive and briefly discuss the most important features of the intriguing patterns of levels of the second vibrational polyad of a molecular chain. First, we obtain the pattern of vibrational energy levels for v ~ 2 in absence of intersite interactions (such couplings produce the dispersion of levels which will be considered in the following).End-effects are also banished from this first level of approximation. So, we limit ourselves to treat n CH2 sites in a chain of finite, arbitrary length (n should not be confused with the dimension of the n-paraffin molecule, which in fact has two CH3 extra sites). The Hamiltonian matrix can be diagonalized in analytical form, as suggested in ref. 16. The spectrum consists of n(2n 1 1) levels which can be initially grouped in two distinct blocks of degenerate energies. These are associated with basis states in the form |0…020…0m and |010…010…0m, 72 PhysChemComm, 2002, 5(10), 66–75 respectively. Their energies are given by (20)~2a(1{2=N), (26) (27) e~E e’~E(11)~2a(1{1=N), where we make use of the parameters a, N of eqn. (5).In particular, we observe that e and e’ differ by the amount 2a/N, which becomes smaller and smaller as NA‘ (harmonic limit). By inclusion of intrasite (Majorana) anharmonic couplings, the n(2n 1 1) modes further split into six families of energies whose values and degeneracy are given by e ½n-fold e’ ½n(n{1)-fold 2 2 eze’+1q........................... ½n-fold each (e{e’)2z8g2 e’+2g’ ½n(n{1)=2-fold each In these expressions we introduce the parameters g, g’ corresponding to the expectation values of the Majorana operator (direct intrasite interaction) mixing vibrational states whose two quanta of excitation are (i) both in a single local mode, (|0…020…0m) and (ii) one state has two quanta in a single local mode and the other one have them distributed over two local modes, (|010…010…0m).Such expectation values are explicitly given by g~ld[2(N 2 1)/N], g’~l, in which l is the parameter associated to the Majorana intrasite interaction. The pattern of vibrational energies for v ~ 2, excluding intersite interactions (dispersion of levels), is shown schematically in Fig. 9. We now include intersite couplings (first- and secondneighbor interactions). As a result, the above-mentioned six manifolds of levels become dispersed according to simple laws in the site index k, as already discussed in Section 4 and shown in eqn. (15). More detailed information concerning analytical expressions of v ~ 2 dispersion laws for first-neighbor Hamiltonian operators can be found elsewhere.16 In Fig.10 we show the dispersion of CH bending/deformation modes of the molecular chain with n ~ 7 (only first-neighbor couplings are included and end-effects are not accounted). It is possible to compute symmetry characters of single v ~ 2 vibrational states. As a general feature, we obtain that each manifold of levels in the v~2 region has a unique character with respect to reflection operations of the molecular point group. Within each manifold, one observes an alternating sequence of characters Fig. 9 Energy levels for the first vibrational polyad (v ~ 2) of CH bending modes of n-paraffin molecules.CH3 terminal groups are not included. The scheme is obtained by taking into account of intrasite (Majorana, l) and anharmonicity (N) effects only. Degeneracies of vibrational polyads (in square brackets) are lifted as soon as intersite couplings (first-neighbor or higher order terms) are incuded in the computation. States relevant for the onset of the 1 : 2 anharmonic resonance with v ~ 1 CH stretches are those in the 3000 cm21 energy region.Fig. 10 Dispersion of first overtone/combination levels (v ~ 2) in the molecular chain with n ~ 7 CH2 sites. Terminal CH3 groups are not included in the calculation. Thick horizontal lines: only intersite (Majorana, strength ~ 250 cm21) terms included; empty circles: firstneighbour couplings added to the Hamiltonian operator (strength: direct, 8 cm21; cross, 4 cm21).with respect to rotation operators of the molecular point group. As a consequence, vibrational states (in both v ~ 1 and v ~ 2 energy regions) will present an alternating sequence of infrared activity. A portion of bending overtone/combination bands is located at about 3000 cm21, i.e. close to fundamental CH stretching modes. As already pointed out, this fact allows one to consider an anharmonic coupling scheme in which molecular states belonging to the above-mentioned families (CH stretching fundamental modes and CH bend overtone/combination bands) can be found in relatively strong interaction (if they belong to the same symmetry species of the molecular point group). So, one expects to observe in this energy region vibrational modes with mixed (stretch/bend) character as a consequence of such an anharmonic (1 : 2) resonance. As a typical experimental evidence of the onset of anharmonic resonances, one expects to observe (i) energy levels more or less displaced with respect to their location in the uncoupled spectrum and (ii) changes in infrared/Raman intensities of those modes involved in the resonant mechanism. More specifically, one could expect that v ~ 2 uncoupled bending modes would be very weak (or completely dark) in the infrared spectrum, at least when compared with v ~1 CH modes in the same energy region.As soon as such modes get coupled with active fundamental CH stretches some infrared brightness can be shared with CH bending overtones.This is schematically shown in Fig. 11, in which we see that only those modes with the same symmetry character are coupled via resonant mechanisms. So, in the present case study, only A1/B2 (for n odd) and A1g/B1u (for n even) CH stretches can interact with bending overtones in the 3000 cm21 energy region. In fact, bending overtones in such a region behave symmetrically with respect to point group reflection operations. We introduce an algebraic realization of the Fermi-like operators suited to describe anharmonic interactions between bending overtone/combination modes and stretching fundamental modes in n-paraffin molecules. Such operators are usually written in terms of creation/annihilation components.15 The general form of non-zero matrix elements is vS{1,vBz2T (28) SvS,vB ^R1:2 vS(vBz1)(vS{NS)(vB{NBz1) p................................................................, ~f where we generically denote local stretching/bending vibrational quantum numbers as vs/vB.The Fermi operator, R�1 : 2, is such Fig. 11 Coupling scheme in a CH2 molecular chain (n ~ 9) under the action of a 1 : 2 anharmonic (Fermi) resonance. Empty circles represent non-interacting manifolds of CH stretching fundamentals (v ~ 1) and CH bending combination/overtone levels (v~2). Full triangles give the amount of wave function mixing with the 1 : 2 coupling (strength 35 cm21). Only symmetric modes get mixed and can eventually share their infrared brightness with CH benders (see also the enlarged insert).that one quantum of stretching excitation is exchanged in favor of two quanta of bending excitation. The explicit form of the operator R�1 : 2 for n-paraffin molecules will depend on the specific interaction terms chosen to describe resonant couplings among vibrational mode in distinct CH2 sites and terminal CH3 sites. In Fig. 12 the coupling scheme is shown. We choose to include only intrasite resonances CHS ~ 1/CHB ~ 2. One can then consider two kinds of interactions. In the first one, one quantum of CH stretch is exchanged with two quanta of CH bend within the same bond in the same site. The corresponding non-zero matrix element will be obtained in correspondence with basis states involving CHS and CHB modes according to (29) S10 .. . 0; 00 . . . 0 ^Rd 00 . . . 0; 20 . . . 0T: In the second kind of coupling we choose to exchange one quantum of CH stretch with two quanta of CH bend distributed over both CH bonds within the same CH2 site. The corresponding non-zero matrix element will be obtained in Fig. 12 Schematic representation of anharmonic 1 : 2 stretch/bend couplings in a single CH2 site of the n-paraffin molecule. Left side: ‘‘direct’’ interaction associated with the exchange of one quantum of CH stretching and two quanta of adjacent CH out-of-plane bending motion (operator Rd); right side: ‘‘cross’’ interaction, associated with the exchange of one quantum of CH stretching and one quantum of both adjacent and opposite CH out-of-plane bends (Rc).73 PhysChemComm, 2002, 5(10), 66–75correspondence with basis states according to (30) S10 . . . 0; 00 . . . 0 ^Rc 00 . . . 0; 110 . . . 0T: The associated operators can be written as ^ (31) R S S Bb a Bai i i bi i~1 and ^ ^ d~ i~1 Rc~k ^ ^ Jz J{2 Xnh i z^Jz J^{2 zh:c: (32) B B B S a b b a i Bai i Sbi ^ ^ Jz J{ i Xnh i (33) ^ J{iz^Jz J{ J^{ zh:c:, see also Fig. 12. In the expressions (31) and (32) we make use of pseudospin J (¡) operators which create/annihilate one anharmonic quantum in the local mode k15 (subscripts S and B refer to stretching and bending modes, respectively). The Hamiltonian operator adopted to describe stretching/bending manifolds inclusive of their anharmonic resonant couplings can thus be written as H ^~^HCHS(v~1)zH ^CHB(v~2)zrdR ^dzrcR ^c: This operator is diagonalized in the direct product basis including both families of vibrational modes (CH stretching fundamentals and out-of-plane bending overtone/combination bands, as already hinted to in eqn. (29) and (30)).In Fig. 13 we show the effect of Fermi-like operators on the simulated infrared spectrum of CH stretches in n-paraffin molecules (Fig. 13 should be compared with the first two panels of Fig. 5). Fig. 13 Computed infrared spectra of n-paraffin molecules (n ~ 7, 11) inclusive of anharmonic 1 : 2 resonances between fundamental CH stretches and first overtone/combination bands of bending CH vibrations.Fermi operators have equal strengths, rd ~ rc ~ 30 cm21. 74 PhysChemComm, 2002, 5(10), 66–75 Strengths in eqn. (33) are fixed to rd ~ rc ~ 30 cm21. In particular, we see that the resonant coupled bending overtones are located in the energy region of non-symmetrical infrared active CH stretches. As a consequence, the computed spectrs such that one of the most relevant evidences of resonant couplings (borrowing of infrared activity by dark overtones) is partially hindered by modes which are not involved in the resonant mechanism (because of their ‘‘wrong’’ symmetry). Yet, we systematically observe in paraffins a light asymmetry in the peak associated with infrared active CH stretches (in the high-energy region of the spectrum, y2970 cm21).We think that a possible explanation of such a feature can be found in the onset of a resonant coupling mechanism as that here presented. In order to sustain a sensible comparison with observed spectra, it is needed to account in the present computation for end-effects caused by terminal CH3 groups. The extension of resonant mechanisms to such groups is straightforward in the present framework. The general conclusions remain unchanged: as said above, the masking effect operated by non-resonant modes (non-symmetrical infrared active CH stretching states) is a dominant feature of these spectra. The detailed composition of wave functions in the resonant energy region can be used to assign the proper quantum numbers of the involved vibrational modes.Such an analysis will be presented in a forthcoming paper in which we will also consider, in a more systematic way, the possibility of determining a general coupling scheme eventually leading to a comprehensive (possibly in analytical form) model to be applied to polyethylene. Intramolecular redistribution of vibrational energy (IVR), with the associated time-dependent analysis, will be also considered in this same framework. 6 Conclusions In this paper we have presented the complete algebraic analysis of CH modes of n-paraffin molecules. The one-dimensional vibron model is applied for the first time in a systematic way to obtain energies (dispersion laws) and infrared intensities of CH stretching and bending modes in paraffin chains of finite, arbitrary length.The inclusion of lower energy modes of the molecular skeleton (CC modes) along with their couplings with CH states will be presented in a forthcoming paper. The present model is extended to include spectral information concerning the first overtone/combination band region. The results are applied to the first systematic (even if preliminary) study of the possible onset of anharmonic (Fermi-like) resonant mechanisms involving CH stretching fundamental states and CH bending overtone/combination bands. We think that such mechanisms could explain certain otherwise unclear features of several observed infrared and Raman spectra. We hope that this paper will stimulate further experimental investigation (mainly in the gas phase) of molecular chains such as paraffins. This would eventually lead to a more detailed, consistent explanation of the dynamic behavior of several polymeric aggregates. Acknowledgement We thank F. Iachello for his inspiring guidance and useful discussions during this work. References 1 R. G. Snyder, J. Mol. Spectrosc., 1960, 4, 411. 2 R. G. Snyder, J. Mol. Spectrosc., 1961, 7, 116. 3 M. Tasumi, T. Schimanouchi and T. Miyazawa, J. Mol. Spectrosc., 1962, 9, 261. 4 J. H. Schachtschneider and R. G. Snyder, Spectrochim. Acta, 1963, 19, 117.5 J. H. Schachtschneider and R. G. Snyder, J. Polym. Sci., 1963, 7C, 99. 6 R. G. Snyder and J. H. Schachtschneider, Spectrochim. Acta, 1963, 19, 85. 7 H. Matsuda, K. Okada, T. Takase and T. Yamamoto, J. Chem. Phys., 1964, 41, 1527. 8 R. G. Snyder, J. Chem. Phys., 1965, 42, 1744. 9 R. G. Snyder, J. Chem. Phys., 1967, 47, 1316. 10 R. G. Snyder, S. Hsu and S. Krimm, Spectrochim. Acta, 1977, 34A, 395. 11 R. P. Wool and R. S. Bretzlaff, J. Polym. Phys., 1986, 24B, 1039. 12 D. A. Cates, H. L. Strauss and R. G. Snyder, J. Phys. Chem., 1994, 98, 4482. 13 D. I. Bower and W. F. Maddams, The Vibrational Spectroscopy of Polymers, Cambridge University Press, Cambridge, 1989. 14 F. Iachello and P. Truini, Ann. Phys. (NY), 1999, 276, 120. 15 F. Iachello and R. D. Levine, Algebraic Theory of Molecules, Oxford University Press, Oxford, 1994; S. Oss, Adv. Chem. Phys., 1996, XCIII, 455; A. Franck and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics, Wiley, New York, 1994. 16 F. Iachello and A. Del Sol Mesa, J. Math. Chem., 1999, 25, 345. 17 F. Iachello, Chem. Phys. Lett., 1981, 78, 581. 18 F. Iachello and S. Oss, Phys. Rev. Lett., 1991, 66, 2776. 19 F. Iachello and S. Oss, J. Chem. Phys., 1996, 104, 6956. 75 PhysChemComm, 2002, 5(10), 66–75

ISSN:1460-2733    

DOI:10.1039/b201553f

出版商:RSC    

年代:2002

数据来源: RSC