Intermolecular Dynamics in Crystalline Iron Octaethylporphyrin

Sep 13, 2008 - Current address: Department of Physics, Idaho State University, Pocatello, Idaho 83209. , §. Department of Chemistry and Biochemistry,...
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12656

J. Phys. Chem. B 2008, 112, 12656–12661

Intermolecular Dynamics in Crystalline Iron Octaethylporphyrin (FeOEP) Valeriia Starovoitova,†,‡ Graeme R. A. Wyllie,§,| W. Robert Scheidt,§ Wolfgang Sturhahn,⊥ E. Ercan Alp,⊥ and Stephen M. Durbin*,† Department of Physics, Purdue UniVersity, West Lafayette, Indiana 47907, Department of Chemistry and Biochemistry, UniVersity of Notre Dame, Notre Dame, Indiana 46556, and AdVanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: July 14, 2008

The new technique of nuclear resonance vibrational spectroscopy (NRVS) has increased the range and quality of dynamical data from Fe-containing molecules that when combined with Raman and infrared spectroscopies impose stricter constraints on normal mode simulations, especially at lower frequencies. Going beyond the usual single molecule approximation, a classical normal-mode analysis that includes intermolecular coupling and the full crystalline symmetry is found to produce a better fit with fewer free parameters for the heme compound iron octaethylporphyrin (FeOEP), using NRVS data from polycrystalline material. Off-diagonal force constants were completely unnecessary, indicating that their role in previous single molecule fits was just to emulate intermolecular coupling. Sound velocities deduced from the calculated phonon dispersion curves are compared to NRVS measurements to further constrain the intermolecular force constants. The NRVS data by themselves are insufficient to rigorously determine all unknown force constants for molecules of this size, but the improved crystal model fit indicates the necessity of including intermolecular interactions for normal-mode analyses. 1. Introduction The normal modes of vibration in a molecule are determined by the set of interatomic force constants associated with the molecule’s equilibrium structure. Normal mode frequencies are measured with a variety of experimental techniques such as Raman and infrared spectroscopies,1-3 with gas phase, liquid, and even solid specimens, and the experimental spectra are compared to those predicted by empirical force fields4,5 or from density functional theory,6-10 for example, in order to refine the force fields or confirm the validity of the particular density functionals employed. This is typically a highly underdetermined problem, as is clearly seen with empirical force fields: the number of force field components connecting various sets of atoms in the molecule is substantially larger than the number of parameters needed to fit the data. Nonetheless, by applying constraints from similar molecules and other chemical considerations, reasonable ranges for the force constants can be arrived at, allowing useful modeling of molecular dynamics. Such calculations typically assume that any intermolecular interactions are weak, so that only a single molecule need be considered. The normal modes of an infinite crystal, on the other hand, also are a function of the phase shift of a given atomic motion with respect to translation from one unit cell to the next, a consequence of Bloch’s theorem. This phase shift is expressed by the factor exp(ik · R) in the normal mode eigenfunction, where k is the wave vector and R the unit cell translation vector. The complete set of normal modes is calculated as a function * Corresponding author. † Department of Physics, Purdue University. ‡ Current address: Department of Physics, Idaho State University, Pocatello, Idaho 83209. § Department of Chemistry and Biochemistry, University of Notre Dame. | Current address: Department of Chemistry, Concordia College, Moorhead, Minnesota 56562. ⊥ Advanced Photon Source, Argonne National Laboratory.

of k, and the dependence of these normal modes on wave vector yields the dispersion curves where a given normal mode (now referred to as a phonon) might be dispersed over a continuous band of energies for the range of k vectors encompassed by the reciprocal unit cell. Molecular crystals are ones where the molecules that condense to form a crystal retain strong intramolecular interactions but with weak intermolecular interactions. For molecular crystals one expects a close correspondence between the normal modes of the single molecule and the phonon bands of the crystal. Nuclear resonance vibrational spectroscopy (NRVS) is a synchrotron-based technique that measures the Fe-weighted vibrational density of states (VDOS) for iron-containing materials, based on the resonant interaction of 57Fe nuclei with X-rays.11-16 VDOS spectra have been obtained from a variety of heme proteins and iron-porphyrin complexes, and several have been compared to normal mode calculations.4,5,17 Normal mode refinement to NRVS data is especially stringent because the computed normal modes must match the measured VDOS in both frequency and Fe amplitude, on an absolute scale. NRVS measurements of heme model compounds have been obtained mostly from polycrystalline material,8-10 and from single crystals in a few special cases. 18-21 Despite that, all normal mode calculations have assumed an isolated single molecule, neglecting any intermolecular interactions associated with crystal packing. Two issues commonly arise in the normal mode refinements to NRVS data on heme compounds: the fits are poorer for energies below about 50 cm-1, and the empirical force field approach generally requires the introduction of off-diagonal coupling constants that couple one kind of motion at a particular location in the molecule to another motion at a different location. Although necessary for a good fit, these off-diagonal terms have no clear physical origin. The disagreement at low energies has been attributed to the neglect of intermolecular coupling.5,15

10.1021/jp806215r CCC: $40.75  2008 American Chemical Society Published on Web 09/13/2008

Crystalline Iron Octaethylporphyrin

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Figure 1. Structure of iron octaethylporphyrin (FeOEP). (a) The four-coordinated iron atom is centered among four nitrogen atoms, with the remainder all carbon atoms; hydrogen atoms are omitted. Atom labels are used for the internal coordinates in the normal mode calculations (Table 1). (b) Intermolecular coordination in the (100) crystal plane. Dashed lines show the shortest nearest neighbor distances to the central Fe atoms, and the ethyl-ethyl intermolecular distances. These form the initial set of intermolecular couplings for the crystal normal-mode analysis.

In this work we calculate the full lattice dynamics of a simple heme molecular crystal, iron octaethylporphyrin (FeOEP), using an empirical force field that explicitly includes intermolecular force constants. We compare these results with a previously published single molecule normal-mode analysis for FeOEP refined to NRVS data and to Raman and infrared peaks,22 to demonstrate what essential dynamics are neglected by the standard single molecule approach. Including lattice dynamics leads to better fits to the low energy density of states and the elimination of certain high energy artifacts, while producing equally good fits to the Raman and infrared lines. The lattice calculations are done without the offdiagonal couplings necessary for the single molecule fits, showing that these were a consequence of leaving out the intermolecular interactions. We also show how comparison of calculated and measured sound velocities allows for improved refinement of the intermolecular couplings. While density functional calculations (DFT) are normally restricted to single molecules,6,23 some molecular crystals have been a subject for periodic DFT as well. Calculations have been reported for benzoic acid and compared with inelastic neutron scattering (INS) measurements on powder molecular crystals.24 Later, inelastic X-ray scattering (IXS) data were obtained and compared to INS data25 and to dispersion curves obtained from periodic DFT calculations. Full lattice dynamics results with periodic DFT26 were also described for benzene molecular crystal powder specimens. The application of NRVS to lattice modes in molecular crystals has been discussed recently27 for the special case where the vibrational spectrum can be separated into “lattice phonons” and narrow “local molecular modes.” We consider below the general case of normal modes in molecular crystals obtained by refinement to NRVS data. Recent work has also extended such lattice calculations to a protein crystal.28 2. Crystal Model and Normal Mode Calculations Sample preparation and experimental methods are described in a previous report.22 The NRVS technique utilizes the Mo¨ssbauer transition in the 57Fe nucleus to observe shifts in resonant X-ray absorption to measure the Fe vibrational density of states (VDOS). All data were obtained at the Advanced Photon Source X-ray synchrotron at Argonne National Laboratory. Empirical force field normal mode calculations for a molecular crystal are very similar to those for a single molecule, except for the addition of intermolecular force constants and the consequences of periodic symmetry. From Bloch’s theorem, if an eigenvector has a given atomic displacement within one

molecular unit cell, the atomic displacement in another unit cell differs by a phase shift exp(ik · R), where k is a wave vector within the Brillouin zone and R the unit cell translation vector. Choosing a particular wave vector we compute the normal modes, then repeat for a set of wave vectors that span the Brillouin zone. The final Fe VDOS is the average of each individual Fe VDOS over all values of k. The FeOEP molecule has 41 non-hydrogen atoms, with no ligand attached to the four-coordinated Fe atom. The X-ray structure determined a point group of Ci, having only inversion symmetry, and the P1j space group with one molecule per unit cell.29 Figure 1 shows the stacking of molecules within a [100] plane, with four dashed lines originating on the Fe atom denoting the nearest neighbor distances between the Fe and C atoms of the adjacent molecules. In the purely phenomenological approach we have adopted, we assign intermolecular force constants to these “bonds.” Since prior single molecule normalmode analysis demonstrated the importance of peripheral substituents to Fe motion, we also assigned force constants to the nearest-neighbor intermolecular ethyl-ethyl pairs, as indicated by dashed lines in Figure 1. The number of such pairs turns out to be especially important for fitting the measured speed of sound. There are three atoms (one Fe atom and two C atoms) in the porphyrin core of the neighboring molecule that form four such bonds with distances about 3.5Å, in addition to four bonds between four pairs of ethyl carbon atoms. Thus we have chosen seven (three core plus four ethyl) extra atoms to form eight (four core plus four ethyl) intermolecular bonds. It would seem that the original molecule of 41 (non-hydrogen) atoms needs to be extended by at least 7 more atoms and their associated force constants, but the crystal unit cell consists of the same single molecular unit of 41 atoms. The calculation needs to be restructured to include the force constants of the intermolecular bonds, plus the symmetry of the crystal by way of the phase factor exp(ik•R). Using the standard approach to classical normal-mode analysis,30 the Euler-Lagrange equation for a set of coupled oscillators is: 3N

q¨j +

∑ fijqi ) 0

(1)

i)1

where the qi are mass-weighted coordinates of the 3N oscillators (N ) number of atoms). Note the similarity between eq 1 and

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Starovoitova et al.

TABLE 1: Best-Fit Force Constants for Single Molecule and Crystal Models of FeOEPa internal coordinate

1 2 3 4 5 6 7 8 9

stretch (mdyne/A) Fe-NP AC-NP AC-BC AC-MC BC-BC BC-XC XC-YC Stretch_core_i Stretch_ethyl_i

10 11 12 13 14 15 16 17 18 19 20 21 22 23

bend (mdyne/rad) AC-NP-Fe NP-Fe-NP-90 NP-Fe-NP-180 BC-AC-NP MC-AC-NP AC-NP-AC AC-BC-BC AC-BC-XC AC-MC-AC BC-AC-MC BC-BC-XC BC-XC-YC Bend_core_i Bend_ethyl_i

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

torsion (mdyn/rad) BC-AC-NP-Fe Fe-NP-AC-MC AC-NP-Fe-NP-90 AC-NP-Fe-NP-180 BC-BC-AC-NP NP-AC-NP-XC AC-MC-AC-NP AC-NP-AC-BC AC-NP-AC-MC AC-BC-BC-AC AC-BC-BC-XC AC-BC-XC-YC AC-MC-AC-BC BC-BC-AC-MC BC-BC-XC-YC MC-AC-BC-XC XC-BC-BC-XC Tor_core_i Tor_ethyl_i

out-of-plane bend (mdyne/rad) 43 Fe-NP-NP-NP 44 NP-Fe-AC-AC 45 AC-NP-BC-MC 46 BC-AC-BC-xC 47 OOP_core_i

single molecule

sets of two, three, or four adjacent oscillators. This mapping leads to the equation

crystal best fit

3N

q¨j +

constrained

1.13 5.4 4.4 4.3 4.1 7.8 4.1

0.73 10.9 9.1 10.6 9.5 10.8 7.6 0.1 0.2

0.73 6.6 5.4 3.8 5.1 5.4 4.7 0.1 0.2

0.11 0.12 0.06 0.04 0.32 0.99 0.43 0.17 0.06 0.17 0.05 0.12

0.22 0.12 0.16 0.04 0.12 2.5 0.43 0.17 0.06 0.17 0.05 0.08 0.02 0.01

0.22 0.12 0.16 0.04 0.12 2.5 0.43 0.17 0.06 0.17 0.05 0.08 0.02 0.01

0.012 0.006 0.01 0.02 0.015 0.015 0.03 0.02 0.001 0.016 0.003 0.002 0.015 0.003 0.021 0.003 0.072

0.012 0.006 0.01 0.02 0.015 0.015 0.03 0.02 0.001 0.016 0.003 0.002 0.015 0.003 0.021 0.003 0.072 0.05 0.01

0.012 0.006 0.01 0.02 0.015 0.015 0.03 0.02 0.001

1.19 1.19 1.19 1.19

0.42 0.59 0.65 0.52 0.57

0.42 0.59 0.65 0.52 0.57

0.003 0.002 0.015 0.003 0.021 0.003 0.072 0.05 0.01

a Stretch, bend, torsion, and out of plane internal coordinates are listed for single molecule and crystal normal mode calculations. Atom labels for internal coordinates are based on Figure 1 (inset). Intermolecular internal coordinates connecting atoms on adjacent molecules are included at the end of each subsection, e.g. “Stretch_core_i.” The column labeled “best fit” are for the unconstrained crystal calculation; the “constrained” column is for the normal mode analysis with stretch constants limited to less than 7 mdyne/A.

Hooke’s law: the coefficients fij are equivalent to spring constants. It is more useful, however, to expand the force constants into so-called internal coordinates, which describe specific stretch, bend, out-of-plane, and torsional motions of

∑

i,t,t′)1

BjtT′ftt′Btiqi ) 0

(2)

or in matrix notation

|B†FB - ω2I| ) 0

(3)

This formulation is convenient because all details of the structure and internal coordinates are contained in the B matrix, all of the force constants for the internal coordinates are in the F matrix. The equations are solved for the eigenfrequencies ωi and the corresponding eigenvectors. The single molecule B matrix, Bsm, has n rows for the internal coordinates and m columns for the three vector components of the N atoms (m ) 3N), a matrix of size 334 × 123. Conceptually the molecule is then enlarged to include seven additional atoms from neighboring molecules (one Fe and two C from the core, and four C from ethyls). The B matrix is then extended to include the new stretch, bend and torsional internal coordinates, for a new matrix Bext of size 410 × 144. This extended matrix must be mapped back onto the same number of atoms as in the single molecule, since there is one molecular unit per unit cell in the crystal. The mapping is given by: ext Bij ) Bijsm + Bi,j+3N

1 e j e 21 ) Bijsm

21 < j e 3N (4)

for a crystal B matrix of size 410 × 123. This makes the dynamics of one of the additional atoms from a neighboring molecule equivalent to that same atom in the original molecule, and reduces the B matrix back to the correct size for 41 atoms per molecular unit. Finally, the wave vector k is incorporated as a phase factor in the B matrix,

b · b)B Bijcrystal ) exp(ik r ij

(5)

where b rj are the position coordinates of the atom corresponding to the jth column. 3. Results Normal mode calculations commenced with the best-fit empirical force constants determined by the single molecule analysis of NRVS data,22 with the addition of trial values for the intermolecular force constants. After sampling the entire Brillouin zone, results were averaged to produce the Fe-weighted VDOS to compare with the NRVS data. The 410 force constants far exceeds the number of parameters needed to fit the measured VDOS as well as the Raman and IR lines used in the refinement. Following the protocol described in previous work22 on FeOEP, we first reduced the number of force constants, breaking them into identical groups based on symmetry. The FeOEP molecule possesses only inversion symmetry, but we initially impose D4h symmetry on the force constants. After optimizing the fit with D4h symmetry, inversion symmetry was reimposed and force constants fine-tuned to achieve the best fit shown in Figure 2. At all stages we required matching to the Raman and infrared lines; Figure 3 shows that both the crystalline and the single molecule results fit these lines equally well. We also find that adjusting force constants to fit the high frequency Raman and infrared lines does not by itself lead to good fits for the low frequency VDOS; substantially more refinement is needed to match the more constraining NRVS data.

Crystalline Iron Octaethylporphyrin

Figure 2. Fe-weighted vibrational densities of state from crystal normal-mode analysis compared with NRVS measurement. The “unconstrained” curve is the best fit result; the “constrained” curve is the best fit when artificially limiting the stretch force constants to values less than 7 mdyne/A.

Figure 3. Agreement of calculated normal modes with Raman and infrared peaks for high frequency “marker” lines. Plotted is the percent deviation of the calculated high frequency normal modes from the published Raman (ref 1) and infrared (ref 3) values versus the measured frequency: best-fit unconstrained values are the square symbols (9), constrained values are the triangles (2), and the earlier single molecule results (ref 22) are the diamonds ([). The numbers labeling some of the points are the reported normal mode assignments from the optical measurements, e.g. “19” refers to the ν19 normal mode.

A comparison of force constants used for the single molecule and crystal models is presented in Table 1. The torsion constants are identical, while the out-of-plane crystal constants are smaller by more than half. The most important terms for a good fit are the bend constants, where there are some significant differences between the models. The stretch constants, however, are consistently larger for the crystal model, increasing from about 5 mdyne/A to about 10 mdyne/A. The lower value is more consistent with other dynamical models of similar molecules, so we repeated the crystal normal mode calculation with an additional constraint of keeping the stretch constants below about 7 mdyne/A. A resultant VDOS for both the constrained and

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Figure 4. Comparison of calculated Fe vibrational density of states (VDOS) to the NRVS data above 200 cm-1. The Fe VDOS from the unconstrained crystal (thick line) and single molecule (thin line) normal mode calculations are both in fairly good agreement the NRVS result (dotted line) for modes between 200 and 400 cm-1. The single molecule calculation, however, predicts a strong peak at 450 cm-1 that is not seen in the data nor in the crystal model, and thus appears to be a consequence of neglecting intermolecular coupling.

unconstrained normal-mode analyses is also displayed in Figure 2 and the force constants are listed in Table 1. While the unconstrained spectrum gives a better fit especially at low frequencies, the constrained result fits most of the data reasonably well. While we were unable to obtain as good a fit with the smaller stretch force constants, we cannot prove that no such solution exists. This is a consequence of the underdetermined character of the calculation, and will require additional experimental constraints to resolve. In Figures 4 and 5, we compare the unconstrained best fit to the previous single molecule results for the VDOS. The spectrum can be divided into three regions: low frequencies below 100 cm-1, the central region from 100 to 450 cm-1, and high frequencies above 450 cm-1. Modes in the central region are largely core vibrations that prove to be relatively insensitive to the addition of intermolecular interactions. This is consistent with the previous observation that artificially increasing the ethyl masses on the single molecule had little impact in this frequency range.22 The single molecule normal-mode analysis also produced a high frequency mode associated with out-of-plane ethyl motion at 450 cm-1, not seen in the data. This clearly was an artifact of the single molecule approximation that is absent in the crystal model. Poor fits to NRVS data at low frequencies have previously been attributed to the neglect of interactions with neighboring molecules. Figure 5 shows both the single molecule best fit to FeOEP and the crystal best fit, showing the significant improvement possible from the inclusion of only a few intermolecular coupling constants and crystalline periodicity. Even better fits are likely with more complete modeling of all physically relevant interactions between molecular units. The crystal model without off-diagonal force constants yields a better fit than the single molecule model that includes them. This strongly suggests that these terms, which have no clear physical origin, merely served to emulate the crystal field effects that are explicitly neglected in single molecule models. The improved fit is not due to an increase in the number of fitting parameters: more off-diagonal terms were needed for the single molecule best-fit VDOS than the number of additional inter-

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Figure 5. Low frequency comparison of crystal and single molecule models. Top panel compares the crystal Fe VDOS calculation to the NRVS data for frequencies below 200 cm-1; lower panel is the same for the single molecule calculation. Significantly improved agreement is seen for the crystal calculation, indicating that intermolecular coupling is especially important for low frequencies.

molecular force constants for the crystal calculation. That is, a crystal calculation replaces ad hoc off-diagonal terms with fewer but physically realistic force constants and produces a better fit. Normal mode analysis (NMA) for a crystal is done by calculating the VDOS at different points of the Brillouin zone and then calculating the total VDOS weighting each point of the zone according to the amplitude of the iron atom vibration.

Starovoitova et al. While in single molecule NMA each particular mode has a distinct frequency, in crystal NMA each particular mode can have slightly different frequencies at different points of the Brillouin zone (i.e., dispersion) that can result in some broadening of the VDOS. To investigate the dispersion effect, the frequency of each mode was calculated as a function of the wavevector k. As can be seen from Figure 6 only a few low frequency modes exhibit dispersion comparable to or larger than the instrumental broadening of 8 cm-1. Further analysis shows that dispersion contributes little to the observed peak widths in the Fe VDOS. Sampling the Brillouin zone at as few as 27 equally spaced points was sufficient for obtaining a fairly complete VDOS; the output shown in Figures 4 and 5, however, were obtained at 183 ) 5832 distinct values of k. The dispersion curves in Figure 6 each are the output of 180 equally spaced values of k. Each mode for a given wave vector k is specified by a frequency and its eigenvector, the amplitude and phase of the motion of each atom in the unit cell. At k ) 0, these modes will be nearly identical to the single molecule modes, except for the effect of the several new intermolecular force constants that have been added (and the elimination of all off-diagonal force constants). The single molecule modes were discussed in some detail previously,22 and the generally poor overlap with the standard “core” porphyrin modes would certainly apply to the crystal modes. A perfectly flat band in the crystal dispersion curves (Figure 6) means that the atomic motions for that mode in one unit cell has negligible effect on the neighboring cells; a band that is not flat indicates that the motions for that mode are strongly affected by interactions with neighboring cells. Crystal normal-mode analysis refined to NRVS data provides not only a best-fit force field, but can also reveal elastic properties that are determined by the dispersion curves, such as the velocity of sound:

V)

ω k

(6)

This equation allows one to find the speed of sound in any chosen direction as a slope of the dispersion curve (mode frequency ω vs wave vector k) as it approaches the origin.

Figure 6. Calculated phonon dispersion curves for FeOEP. The three panels display the dispersion curves for modes below 90 cm-1 for the [001], [010], and [100] reciprocal lattice directions.

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The average speed of sound can be extracted directly from the measured VDOS, where the PHOENIX program for computing the VDOS from the raw data utilizes the Debye limit at the very lowest frequencies13,31-33

D(E) )

mFe 2π Fν p 2

3 3

E2

(7)

where D(E) is the density of states, mFe the mass of 57Fe nucleus, E the energy, F the density, and V the average velocity of sound, which can be written as

ν)

(

3Vt3Vl3 (2Vl3 + Vt3)

)

1/3

(8)

with Vl and Vt being the longitudinal and transverse velocities. The NRVS-determined value for FeOEP is 1204 m/s, consistent with values for similar materials.34 The average sound velocity in FeOEP obtained from the dispersion curves computed from the initial calculation (4 ethyl and 4 core intermolecular bonds) was only 234 m/s. Although this small number of intermolecular bonds is sufficient to get a good fit to the overall VDOS, it is lacking in the very low frequency limit. Increasing the strength of these force constants had little effect on the speed of sound, so increasing the number of intermolecular bonds was explored. It was found that while changing the number of core bonds increased only the transverse speed of sound in [001] direction, had little impact on longitudinal speeds, and significantly degraded the VDOS fit in the high frequency region. Increasing the number of ethylethyl bonds from 4 to 8, however, increased the speed from 234 to 504 m/s, and 16 bonds further improved it to 843 m/s. These results demonstrate the relative importance of the ethylethyl bonds as opposed to Fe-C bonds for elastic constants. 4. Conclusions Normal mode analysis for a molecular crystal produces a better fit to the NRVS vibrational density of states for FeOEP than the usual single molecule approximation, without an increase in fitting parameters. Systematic uncertainties are observed, specifically for the stretch force constants, a reminder that even this enlarged data set is not sufficient by itself to solve for all force constants. A minimal set of intermolecular bonds was successful in achieving improved fits across the frequency range, but additional ethyl-ethyl couplings are needed for good agreement with the measured speed of sound. Since good fits are obtained without the use of off-diagonal coupling constants, these ad hoc terms apparently serve only to emulate the physical intermolecular couplings. The effect of dispersion on the vibrational density of states for this molecular crystal was found to be insignificant compared to the NRVS instrumental broadening except for a few low frequency bands. Calculating normal modes for refinement to the NRVS vibrational density of states from polycrystalline or crystal specimens is thus more effective with a crystal model that includes intermolecular coupling. Acknowledgment. We thank Jianguang Guo for help with the computer program, and E. W. Prohofsky for wide-ranging advice. This work was supported by the National Science Foundation through the Award no. PHY-9988763, by the National Institutes of Health Grant GM-38401, and by the Purdue Research Foundation. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.

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