3.4 TÉCNICAS DE PROCEDIMIENTO, ANÁLISIS Y DISCUSIÓN DE RESULTADOS
3.4.1 PROCESAMIENTO Y ANÁLISIS DE ENTREVISTAS APLICADAS
250 200 ^150 H 100 50 0 350 300 250 200 150 1 0 0 -i 50 0 A, 20 40 60 80 100 fre q u e n c y (cm ‘) 120 140 160 0 20 40 60 80 100 fre q u e n c y (cm ') 120 140 160 6.2.3.2 Pyrazine, C4N2H4
Figure 6-3. Crystal structure o f pyrazine. Figures have been prepared using the Cerius2 software.t^-^*
a) ab-plane b) bc-plane
...
... ---
...
W hile the structure and dynamics o f aromatic hydrocarbons have been extensively studied and the intermolecular interactions can be m odelled quite successfully using simple model potentials, the usefulness o f pure exp-6 models is limited for polar interactions. Pyrazine is a non-polar azabenzene, but the local polarity around the nitrogen atoms gives a different character to the intermolecular
L attice D ynam ica] Studies o f M o le c u lar C rystals w ith A p p lication to Po ly m o rp h ism and Structure P rediction
G r a e m e M. D a y
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 1 1 4 interactions than in hydrocarbons. Pyrazine crystallises with the N-N axis o f all the molecules aligned along the longest unit cell axis, a, and the planes o f molecules offset by a/2 form a 47° angle (Figure 6-3). There are two major electrostatic interactions - offset ti-tt stacking approxim ately along b, and C-H " N interactions with their major com ponent along a. As well as these electrostatic interactions, there are also fairly short C-H H-C contacts between molecules translated along c.
As with naphthalene, the symmetry o f this m olecular crystal offers advantages for experimental and computational studies. There are two molecules in the unit cell, so only nine optic modes in the vibrational spectrum (Ag, Au, B,u, 2xB|g, B]», 2xB2g, Bsg) and, with the pyrazine molecules on crystallographic inversion centres, mixing o f the translational and rotational vibrational m otions is forbidden. Hence, the analysis o f the calculated eigenvectors and characterisation o f the experim ental spectra is simplified.
Table 6-5a. Observed and calculated cell constants for pyrazine.
Structure -^latt
(kJ/mol) a (A) b (A ) c ( A ) Vol (A^) RMS error ®
Expt (RT)t6-5] 56.3 ± 0.5 [6.47] 9.32 3.82 5.91 210.1 - Expt (184K)[6-6] 9.32 3.73 5.85 203.6 - UNI 61.3 9.50 (+1.86%) 3.70 (-0.82%) 5.38 (-8.08%) 189.0 4 .8 1 % ” W 84+ESP 53.2 7.53 (-19.27%) 3.69 (-1.17%) 7.56 (+29.23%) 208.0 (+2.14%) 20.22% ” W 99+ESP 49.5 • 8.43 3.81 (+2.21%) 6.62 (+13.11%) 204.6 (+0.45%) 9.45% ” W 84+DMA 54.3 9.67 (+3.67%) 3.76 (+0.77%) 5.70 207.5 (+1.88%) 2.60% W 99+DM A A 1 9.54 1 (+2.32%) 3.83 (+2.60%>) 5.53 202.2 (-0.72%) 3.72%
" Root mean square error in lengths o f lattice vectors a, b ^ The original symmetry is lost and final (P2\!c) structure (W 84+ESP), a = 73.89° (W99+ESP)
and c, compared to 184K structure, has a = 87.73° (UNI), a = 82.21°
The relaxed structures (Table 6-5a & b) show the importance o f the electrostatic interactions in determ ining the structure o f this crystal. Using the DMA based potentials, electrostatic interactions account for 35-40% o f the total lattice energy (about 20 kJ/m ol) and over tw o thirds o f this results from anisotropic terms in the potential (charge-dipole and higher). All but the m odels using distributed multipole electrostatics fail to reproduce the lattice structure satisfactorily, with large errors in the a and c lattice vectors and significant m olecular reorientations, destroying much o f the
L attice D ynam ical S tudies o f M olecular Crystals w ith A pplication to Polym orphism and Structure Prediction
G raem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 115 crystal's symmetry. The distributed multipole potentials reproduce the tc-tc stacking distance o f 2.33
A to within 0.32 A (W 84+DM A) and 0.13 A (W 99+DMA), while the others expand this interaction by 0.9 - 1.2 A. The point-charge models also fail to reproduce the orientation o f the C-H-- N interaction - the W 84+DM A and W99+DMA models reproduce the experimental C - H - N angle o f 153° to within 3°, while the angle becomes much more acute with the ESP models (106° with W 84+ESP, 137° with W 99+ESP). The UNI potential reproduces the weak hydrogen bond adequately, but fails to model the K~n interaction properly.
Table 6-5b. M olecular displacements o f pyrazine during lattice energy relaxation.
Fully relaxed I Fixed cell
RMS translation (A) RMS rotation (deg.) F 1 RMS
I
translation (A) RMS rotation (deg.) UNI 0 .2 0 15.25 136.6 0 .0 0 17.28 W 84+ESP 0.84 47.87 1931.6 0 .0 0 18.84 W 99+ESP 0.58 61.81 1517.0 0 .0 0 43.35 W84+DMA 0.13 2.07 23.1 0 .0 0 1.89 W99+DMA 0.14 5.14 50.2 0 .0 0 0.37Frequencies calculated at the lattice energy minima for the DM A models are com pared to the lowest tem perature observed spectra (Table 6-7). The far-infrared spectrum was m easured at 77K by Gerbaux and Hadnit^-^1 and both By frequencies were identified. The Raman spectrum has been studied more e x t e n s i v e l y l ^ - ^ A 8 - 5 0 ] g^d the full characterisation o f the librational frequencies, using
isotope shifts and polarisation measurements, was reported by Hieida and c o - w o r k e r s , w h o m easured frequencies down to 4.2K. To our knowledge, the optically inactive A„ mode has not been observed experimentally, even in inelastic neutron scattering experiments.I^-^ï
The modes calculated from the simpler potentials cannot be directly compared to experim ent because o f the large distortions o f the lattice and m olecular rearrangements. Q uasi-harm onic calculations, using the 300K lattice constants, were performed for all o f the model potentials and these are compared to room tem perature spectra (Table 6-7b). By enforcing the Pmnn sym m etry, the structure relaxed to second order transition points w ith the sim pler models. At these structures, the eigenvectors are comparable to those o f the true minima (W 84+DM A, W 99+DM A) and the calculations show the instability to molecular rotation (V2, V4).
Lattice D ynam ical S tu d ies o f M o lecu lar C rystals w ith A pplication to Polym orphism and S tructure Prediction
G raem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 116 Table 6-6. Qualitative description o f the normal modes o f pyrazine, using RUDOLPh.t^-^'^1
symmetry mode description
V | symmetric libration o f all molecules about their N-N axes
B]g
V 2 symmetric libration o f all molecules about an axis approxim ately 2 0° out o f the m olecular plane, perpendicular to the N -N axis
V 3
symmetric libration o f all molecules about an axis perpendicular to that in V2 (roughly perpendicular to the m olecular plane)
B2g
V 4
anti-symmetric libration o f all molecules about an axis approxim ately 2 0° out o f the m olecular plane, perpendicular to the N-N axis
V 5
anti-symmetric libration o f all molecules about an axis perpendicular to that in V4 (roughly perpendicular to the m olecular plane)
Bsg V 6 anti-symmetric libration o f all m olecules about their N-N axes
A« V 7
out-of-phase translation o f each molecule along a, in the direction o f the N-N molecular axes
Biu Vg
out-of-phase translation o f each molecule along c, roughly perpendicular to the molecular planes
Biu V9 out-of-phase translation o f each molecule along b
The lowest frequency Bzg (V4) mode bends the C-H -N interaction out o f the plane o f the ring, with alm ost no change in the N H separation. Thus, the restoring force is almost entirely due to the orientational dependence o f the interaction. The lowest B,g vibration (V2) is also a C-H N bending mode, w hile also shearing the close C-H --H-C contacts. Like V4, V; is unstable w ithout anisotropic electrostatics, indicating that the higher order Coulomb interactions contribute an important part o f the restoring force. The higher frequency pair o f B,g and B2g m otions (vg, V5) distort the hydrogen bonding in the plane o f the ring. In the translational modes ( V 7 , V g , V 9) , both the n-n and C-H H-C contacts are unchanged, so distortions o f the weak C-H- -N hydrogen bonds are essentially isolated. However, unlike the librational motions, these modes also change the (C)-H---N separation. The Ag (vi) and B]g (v^) librations about the molecular N-N axes stretch the (C-)H---H(-C) close contact distances and distort the face-to-face ring contacts.
Lattice D ynam ical S tudies o f M olecular C rystals w ith A pplication to Polym orphism and S tructure Prediction
Graem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 117 Table 6-7a. Harmonic k = 0 phonon frequencies calculated for pyrazine at the minimum in
Frequencies in cm '.
M ode Expt. (Low T) W84+DMA W 99+DM A
Translational modes (77 K)
V? (Au) not observed 123.3 123.0
Vg (Biu) 54 (IR) 52.2 54.8
Vg (Bzu) 79 (IR) 65.0 69.8
Librational modes (4.2K, Raman)
V1 ( Ag) 128 147.3 145.4 V2(B|g) 57 54.2 64.8 V] (Big) 88 76.8 73.6 V4 (B^g) 58 48.1 51.1 V5 (Bzg) 98 111.3 117.1 V6 (Bag) 136 161.1 161.3 RMS error 13.9% 14.4%
Table 6-7b. Quasi-harmonic k = 0 phonon frequencies calculated for pyrazine at the fixed room temperature lattice. Frequencies in cm '.
Mode
Expt “ (room temp.)
UNI W 84+ESP W99+ESP W 84+DM A W 99+DM A
Translational modes
V? (Au) not observed 120.8 (d4) 137.0 (d4) 119.5 (d4) 141.5 (d4) 122.1(d4)
Vg (B iu) 48 53. 8 (d4) 64.7 (d4) 62.5 (d4) 69.1 (d4) 61.4 (d4) Vg (B%u) 6 8 67.8 (d4) 75.7 (d4) 64.8 (d4) 70.1 (d4) 64.7(d4) Librational modes Vi(Ag) 104 103.2 86 .8 104.0 111.1 101.1 V2(B ig) 46 -6 2 .4 -6 0 .0 -7 3 .7 26.8 58.6 V3(B ig) 72 97.1 82.7 104.2 90.4 67.4 V4 (Bag) 47 -3 6 .4 -3 0 .4 -4 1 .7 10.2 27.6 V5 (B2g) 78 34.0 63.1 59.3 113.6 117.2 V6 (Bag) 115 115.6 106.9 122.7 138.7 129.4 RMS error 27.5% " 19.4% " 24.1% " 40.3% 27.4%
“ Room tem perature IR experimental frequencies from [6.7], 300K Raman frequencies from [6.8]. ^ Excluding unstable modes.
These results support earlier computational studies o f the pyrazine crystal,[^-^’^^’^^l which found that C-H " N dipole-dipole interactions were necessary to describe the structuret^-^^1 and
Lattice D ynam ical S tudies o f M olecular Crystals w ith A pplication to P olym orphism and Structure Prediction
Graem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations___________________________ 118 vibrationst^-^’^^l satisfactorily. Gamba and Bonadeol^-^'l also calculated the relative importance o f each part o f the model potential to the dynamical matrix, ( k = 0) , and found that the electrostatic contribution is most important for the lower frequency librational B,g (vz) and Bzg (V4) modes, which we find are unstable without anisotropic atom-atom electrostatics. Using the DM A m odels, both o f these modes are quite well described at the minimum in lattice energy, but, in the QH approxim ation, all but the W 99+DM A B|g frequency are much too soft. In our calculations, the N - H-C charge- charge attractive energy lessens when the lattice is relaxed, while the higher m ultipole interactions strengthen significantly compared to the QH structure. So, in the relaxed lattice, the directionality o f the interaction is optimised, but the fixed unit cell hinders optimal orientation o f the w eak hydrogen bonds in the QH calculations.
The higher frequency Big/Bzg pair (V3 V5) are fairly well reproduced with distributed multipoles, but V5 is far too soft with the isotropic potentials, indicating that the higher order electrostatics are important for in-plane bending as well. The B„ modes (vg Vg), are reproduced very well with all m odels, showing that, with the stretching as well as bending component, atom -atom anisotropy is m uch less important. The fully relaxed DMA models reproduce the low tem perature v, and v& frequencies well and the softening o f these modes in the QH approximation gives very good agreem ent with the room temperature observations. The exp-6 contributions to the dynam ical matrix are dom inant for this (C)-H H-(C) stretching m odel^-^'l and the W99 model gives noticeably better results than the W 84 model, due to the improved aromatic carbon parameters and shift o f the hydrogen repulsion along the C-H bonds.
6.2.3.3
Hexamethylenetetramine(-'di
2))
The elastic constants o f hexamethylenetetramine (Figure 5-7, Chapter 5) were studied in the previous chapter. As with pyrazine, the molecule has local polarity, but no net dipole m om ent and no strong hydrogen bond donors. The shortest intermolecular contacts are (C-)H-- H (-C) at ju st under 2.7Â and C-H N at about 2.8 A. These are both longer that the sum o f the van der W aals radii, so neither is expected to be particularly strong. The body-centred cubic packing suggests that hexam ethylenetetram ine behaves as if it were spherical, but the sum o f the weak interactions provides enough anisotropy in the molecule-molecule interactions to keep the crystal orientationally ordered up
L attice D ynam ical Studies o f M olecular C rystals G raem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 119 to at least 340K, above which there is significant broadening o f the librational mode in the neutron scattering spectrum .t^ *
Unlike pyrazine, the structure is reproduced satisfactorily using simple potentials (Table 5-13, Chapter 5), raising the question o f whether accurate potentials are needed to reproduce the dynam ical properties. In the study o f the effect o f model potential on elastic constants (Section S.2.3.6), the UNI potential seriously exaggerated the stiffness o f the crystal, while the W84 and W 99 potentials gave errors which are more typical o f those resulting from thermal effects. As with the structure, the elastic stiffness is nearly unaffected by replacing atomic charges w ith distributed multipoles.
Due to the high symmetry, there is only one (triply degenerate) optic interm olecular mode, at approxim ately 60cm '% which is inactive in both the IR and Raman spectra. Dolling and Powellt^-^^1 have m easured the dispersion curves o f all o f the intermolecular modes via coherent inelastic neutron scattering at lOOK and 298K (Table 6-8). The large gap between the frequency o f this librational mode and our lowest calculated intramolecular mode (as 375.6cm '', M P2/6-31G(ûf,/7)) justifies our use o f the rigid body formalism.
Table 6-8. Harm onic and quasi-harmonic (lOOK and 298K) k = 0 librational lattice frequency o f hexamethylenetetramine. Frequencies in cm '.
Exptt^'^ n UNI W84+ESP W 99+ESP W84+DMA W 99+DM A
lOOK 63.7 (±2)
Fully relaxed lattice
95.3 78.8 74.1 66.4 56.6
lOOK fixed lattice (a = 6.949Â)
74.1 87.0 72.7 73.7 56.1
298K 58.4 (±2)
298K fixed lattice (a = 7.024Â)
62.8 77.0 64.9 64.9 50.2
While the frequency calculated with the UNI potential is about 50% too large (Table 6-8), consistent w ith the elastic stiffness, those models using explicit electrostatics are in good agreem ent with the observed frequency. The improvement when point charges are replaced with distributed multipoles is slightly greater than that associated with changing the exp-6 parameters. However, some o f these effects are due to the differences in relaxed lattice parameters. Although these differences in lattice parameters are only about 1%, they are on the same magnitude as observed changes in the lattice structure over a large temperature range. The equilibrium lattice o f the W84 models nearly corresponds to room temperature (a = 7.024Â), while the W99 parameter corresponds approxim ately
Lattice D y n am ical S tudies o f M olecular C rystals w ith A pplication to Polym orphism and S tructure Prediction
G raem e M. Day
Chapter 6. Validation o f Phonon and Thermodynamical Calculations 120