J. Phys. Chem. 1996, 100, 7007-7013
7007
Vibrational Assignment and Definite Harmonic Force Field for Porphine. 1. Scaled Quantum Mechanical Results and Comparison with Empirical Force Field Pawel M. Kozlowski,* Andrzej A. Jarze¸ cki,* and Peter Pulay* Department of Chemistry, UniVersity of Arkansas, FayetteVille, Arkansas 72701 ReceiVed: December 6, 1995X
The ground state geometry, force field, and vibrational intensity data of free-base porphine were determined by the scaled quantum mechanical (SQM) method, using a force field calculated by density functional theory. Calculations were carried out with the Becke-Lee-Yang-Paar composite exchange-correlation functional (B3-LYP) and with the 6-31G* basis set. The resulting force field was transformed to nonredundant internal coordinates and scaled by a set of optimized scaling factors. There is very good agreement between the calculated and experimental frequencies and intensities and between experimental and simulated infrared (IR) spectra. The assignment of the harmonic frequencies is discussed and reassignment of some modes present in the IR spectra is suggested. Comparison between the empirical and scaled quantum mechanical force fields shows that the empirical field is qualitatively correct for the dominant terms but neglects many of the smaller but still significant force constants. The good reproduction of the b3u fundamentals, including the “Kekule” vibration is conclusive proof for the D2h symmetry of the molecule.
1. Introduction Free-base porphine (21H,23H-porphine, Figure 1) is the prototype of porphyrins, which play important roles in biology,1,2 and therefore its spectroscopic characterization is of considerable interest to a large community of chemists and biochemists. The vibrations of porphine are understood better than any other molecule of its size, due to its high symmetry and abundant experimental data. In particular, the novel photooriented spectra of Radziszewski et al.3-7 established definite vibrational assignments for a large number of bands. Although much is already known about the vibrations of this class of molecules at an undeniably useful level, a definite characterization of the harmonic force field is still far from complete. There are two main reasons for this situation. Accurate quantum mechanical calculations were prohibited by the size of such systems. Inclusion of correlation energy is essential for the correct prediction of a delocalized structure with D2h symmetry,8,9 rather than C2V broken symmetry. Recent advances in computer technology and software development have overcome this problem, and ab initio vibrational studies for porphine are now possible. Development of a purely empirical force field suffers from known problems, one of which is associated with the large number of adjustable parameters.10-12 As a result, for molecules the size of porphine it is virtually impossible to develop a unique force field from experimental data without assistance from theory. The aim of the present work is to provide an accurate harmonic force field for porphine using a combination of theoretical and experimental data. This is accomplished by applying the scaled quantum mechanical (SQM) force field method13 to a force field calculated by density functional theory. It has been demonstrated recently that this combination leads to accurate results.14,15 We used this method for the force field of azulene16 and in our preliminary investigation of the in-plane force field for porphine.9 The plan of the present paper is the following. First, we describe the SQM force field, and then we compare the experimental and calculated infrared spectra and discuss the assignment. Finally, we compare the SQM with X
Abstract published in AdVance ACS Abstracts, April 1, 1996.
0022-3654/96/20100-7007$12.00/0
Figure 1. Numbering of the porphine skeleton. The 18-membered conjugated ring is shown in bold.
the latest empirical force field. The second part of this series deals with the analysis of non-resonance Raman spectra. We are in the process of extending our study to resonance Raman spectra of porphyrins and force field development for metalloporphyrins. 2. The Scaled Quantum Mechanical Force Field The details of geometry optimization and frequency calculations are given in our previous paper9 and are summarized here for the convenience of the reader. Both the geometry and frequencies have been obtained using density functional theory with B3-LYP nonlocal exchange-correlation functional and with the 6-31G* basis set as it is implemented in the Gaussian 92/ DFT suite of programs.17 The geometry was optimized under D2h symmetry restrictions, and all normal frequencies at the optimized geometry were real. The calculated Cartesian force constants at the optimized geometry were transformed to natural internal coordinates19 generated by the TX90 program.18 In previous work we used a set of 112 redundant coordinates, 4 more than the 108 internal degrees of freedom. Those 4 © 1996 American Chemical Society
7008 J. Phys. Chem., Vol. 100, No. 17, 1996
Kozlowski et al.
TABLE 1: Nonredundant Natural Internal Coordinates of Porphinea
among the symmetry species of D2h symmetry as follows
optimized transferable coordinate scale factorsb scale factorsc
Γ ) 19ag + 18b1g + 9b2g + 8b3g + 8au + 10b1u + 18b2u + 18b3u (2)
R1, ..., R28 r1, r2 r3, ..., r12 ζ1, ..., ζ18 τ1, ..., τ20 β1, β2 γ1, ..., γ14 β3, ..., β15
0.922 0.920 0.920 0.990 0.935 0.876 0.976 0.950
0.9337 0.8820 0.9122 0.9880 0.9619 0.9405 0.9486 0.9340
description C-N and C-C stretch N-H stretch C-H stretch rings, symmetrized bendings rings, symmetrized torsions NH rocking in plane CH and NH wagging CH rocking in place
a The supporting information contains the exact definition of the internal coordinates. A detailed description of natural internal coordinates is contained in ref 19. b Reference 14. c For internal consistency, the scale factors are given to 4 decimals, although only 3 are significant.
redundant coordinates were associated with the deformations of the macrocycle. In the present work we omitted them, and the vibrational analysis has been performed with 108 nonredundant coordinates. These are briefly summarized in Table 1; their full definition, needed to characterize the force field, is given in the supporting information. To correct systematic errors in the quantum mechanical force field and to take into account approximately the effects of anharmonicity we used the SQM method.13 The calculated force constants were scaled according to the formula
Fij ) (cicj)1/2F′ij
(1)
where F′ij denotes the unscaled harmonic force constants and ci and cj the scale factors. In the previous study we used only the transferable scaling factors optimized previously on 20 organic molecules.14 No experimental data associated with porphine itself were used. However, we found some disagreement between the calculated and observed frequencies. In particular, the δNH modes were too low, due to the low scale factor used for N-H rocking.9 Therefore, we refined the scaled factors to improve the agreement between the calculated and observed frequencies. In the refinement procedure we used unlabeled porphine (1) and five of its isotopomers: porphine-d2 (1-d2), porphine-d4 (1-d4), porphine-d8 (1-d8), porphine-d12 (1-d12), and porphine-15N4 (1-15N4). We included only these experimental values in our fitting procedure which could be assigned without doubt using largely the recent and most reliable values of Radziszewski et al.6,7 The infrared active experimental frequencies used in the scaling procedure are given in Tables 2-4: the Raman ones are given in the second part of this series. The results of the refinement are presented in Table 1. For most vibrations the optimized scale factors agree well with the transferable factors derived earlier.14 The largest discrepancy occurs for factors associated with N-H stretching and in-plane N-H rocking. We pointed out previously9 that some transferable scale factors associated with N-H vibration have been optimized in amines and may not be fully appropriate for porphine. As an aid for their assignment, Radziszewski et al.7 have calculated the force field of porphine at the Hartree-Fock level with the 3-21G* basis. After a simple uniform scaling, this force field also reproduces the experimental spectra quite well, except in b3u species where it gives an imaginary frequency for the “Kekule” vibration, i.e., the alternative contraction and expansion of the bonds along the conjugated pathway (bold lines in Figure 1). 3. Comparison with Experimental IR Spectra The 108 fundamental modes of porphine can be classified
In our previous study9 we used the IUPAC recommendations in labeling the symmetry species.20 This corresponds to yz being the plane of the molecule and the N-H bonds coinciding with the z direction. Many experimental groups use a different labeling scheme where xy is the plane of the molecule. This alternative convention has the advantage of corresponding directly to the notation used in D4h metalloporphyrins. In this paper we changed the symmetry labeling to the conventional one. Among the symmetry species, modes of ag, b1g, b2u, and b3u symmetry occur in the molecular plane, while the remainder are out-of-plane vibrations. The in-plane vibrations have been discussed in our previous work.9 In the meantime, Radziszewski et al.7 published a detailed study of the polarized infrared spectra of photooriented matrix-isolated porphine and its isotopomers and revised the symmetry assignment of a number of bands. With one exception discussed below, the new assignments agree with our calculations and were adopted in the SQM procedure. The experimental and calculated fundamental frequencies and IR intensities for b1u, b2u, and b3u in porphine and its isotopomers are summarized in Tables 2-4, respectively. When there is a unique correspondence between the normal modes in different isotopomers, we lined them up in Tables 2-4, but this was not possible in all cases; in these cases the fundamentals are simply listed by decreasing frequency. The comparison of experimental IR absorption spectra with simulated spectra is shown in Figure 2a-f. In general the simulated IR spectra agree excellently with experiment, with only a few discrepancies. The fact that we were able to fit several hundred well-characterized fundamental frequencies with only eight scale factors is a strong argument that the SQM force field is essentially correct. The mean absolute deviation of the frequencies below 1700 cm-1 is 4.2 cm-1 for porphine-d0, including all well established experimental frequencies (see part II for the Raman active vibrations). Most of our results strongly support the reassignments proposed by Radziszewski et al.7 Small isotope shifts are sensitive indicators of the quality of the force field. Most of our calculated shifts agree well with the matrix data.7 The few cases where there is larger discrepancy all involve relatively weak bands participating in Fermi resonance or site splitting, such as the 977 cm-1 b2u and 1134 cm-1 b3u fundamentals. In the following, we discuss some problematic or ambiguous assignments separately by symmetry species. Unless otherwise noted, the discussion refers to the porphine-d0 isotopomer. The porphine skeleton has approximate D4h symmetry. This leads to quasi-degeneracy between b2u and b3u vibrations and also between b2g and b3g ones. This quasi-degeneracy is quite obvious in both the calculated and the experimental spectra. If the quasi-degeneracy is close, it can interfere with the polarization measurements. As discussed below, this is probably what happened in a few cases in the polarization measurements of ref 7. The b2u and b3u fundamentals, forming quasi-Eu degenerate pairs are as follows (calculated frequencies, b2u first): (351,350); (745,736); (782,785); (945,965); (1053,1049); (1409,1408). 3.1. b1u Frequencies. All fundamental vibrations in this out-of-plane species fall below 860 cm-1. The weak band at 938 cm-1 must be an overtone. The strong band at 852 cm-1 agrees very well with the one calculated at 853 cm-1. The two bands at 785 cm-1 (weaker) and 773 cm-1 (stronger) in the experimental spectrum come out in calculations at 786 and 776 cm-1, respectively, but the intensity of the bands is reversed
Vibrational Assignment for Porphine
J. Phys. Chem., Vol. 100, No. 17, 1996 7009
TABLE 2: Experimental and Calculated b1u Fundamental Frequencies (cm-1) and IR Intensities (km/mol) for FBP and Its Isotopomers ∆(15N)a
1-d0
1-d2
1-d4
1-d8
1-d12
exp
calc
int
exp
calc
exp
calc
int
exp
calc
int
exp
calc
int
exp
calc
int
852 785 773 731 691 639 335 219
853 786 776 727 697 639 331 204 91 55
145.4 140.1 53.2 11.6 14.2 2.2 5.6 1.8 7.7 0.0
0 2 -1 0 5
0 1 2 0 5 6 2 5 1 0
853 540b 762
853 535 762 778 697 647 331 204 89 55
141.5 60.9 83.7 0.0 12.6 2.8 5.7 2.2 7.3 0.0
799
801 779 747 699 647 627 302 204 90 55
286.9 2.7 30.7 7.4 11.0 2.2 5.8 1.7 7.7 0.0
847 753 722
846 761 723 705 559 539 328 199 85 52
75.0 163.8 15.5 5.0 39.0 0.1 4.7 1.7 7.0 0.0
764 745 709 635 563
767 746 710 633 559 539 299 199 85 52
187.6 18.9 18.0 8.9 38.7 0.0 5.1 1.6 7.0 0.0
a
693
745 699 647 627
566
∆(15N) ) (1-d0) - (1-15N4). b This band appears to be misclassified in ref 7, Table 2. See text.
TABLE 3: Experimental and Calculated b2u Fundamental Frequencies (cm-1) and IR Intensities (km/mol) for FBP and Its Isotopomers ∆(15N)a
1-d0
1-d2
1-d4
1-d8
1-d12
exp
calc
int
exp
calc
exp
calc
int
exp
calc
int
exp
calc
int
exp
calc
int
3124 3112 3045
3109 3107 3059 1595 1547 1491 1409 1355 1252 1225 1156 1053 981 945 782 745 351 283
46.7 2.0 13.0 16.4 22.6 2.1 10.3 4.5 0.3 57.8 0.0 35.6 5.7 89.3 0.3 25.5 8.4 0.1
1 -2 0
0 0 0 1 1 5 6 5 4 6 4 0 8 13 1 1 1 3
3124 3112 3045
3109 3108 3059 1585 1546 1476 1403 1352 1250 1171 1093 1052 863 945 739 746 350 275
46.8 2.0 13.0 9.8 22.8 6.2 5.0 7.5 4.8 15.2 18.6 31.4 7.2 93.2 0.4 29.2 8.3 0.1
3122 3110 2258
3109 3107 2258 1585 1543 1484 1369 1338 1240 1204 1058 988 958 924 768 695 349 280
45.0 2.1 7.9 17.5 23.8 2.4 2.4 17.8 23.2 25.6 26.0 25.9 33.3 50.4 1.0 17.8 8.6 0.2
2338
2321 2296 3059 1589 1525 1456 1404 1315 1227 1204 1063 877 782 933 709 761 333 275
17.0 3.1 14.7 10.3 24.9 14.7 18.8 0.9 45.7 16.1 0.0 26.8 23.6 66.4 4.9 11.0 7.4 0.0
2324 2263 2258
2321 2296 2258 1578 1519 1448 1355 1313 1212 1108 952 866 763 919 752 678 332 273
17.3 3.5 7.1 10.4 27.5 20.4 14.6 2.8 53.8 0.2 8.6 5.8 18.1 83.1 4.9 10.2 7.6 0.0
1540 1490 1406 1357 1255 1228 1156 1054 977b 951 745 357
3 6 4 2 10 4 1 3 13 1
1479 1352 1255 1172c 1098 1053 950 744
1542 1479 1347 1339 1241 1203 1062 996 961 923 694 357
3046 1521 1447 1402 1324 1225 1204 881 784 938 710
1522 1447 1352 1310 1211 952 870 918 752 677
a ∆(15N) ) (1-d ) - (1-15N ). b This band consists of two closely spaced lines with weighted average 982 cm-1. c There is a medium strong band 0 4 in the observed spectrum at 1172 cm-1, the polarization of which is given as b3u in Table 2 of ref 7 but as b2u in Figure 1.
TABLE 4: Experimental and Calculated b3u Fundamental Frequencies (cm-1) and IR Intensities (km/mol) for FBP and Its Isotopomers ∆(15N)a
1-d0
1-d2
1-d4
1-d8
1-d12
exp
calc
int
exp
calc
exp
calc
int
exp
calc
int
exp
calc
int
3324 3128 3088 3042 1522 1507 1412 1396 1287 1177 1134 1043 994 971 780 723 357 310
3330 3124 3089 3059 1522 1512 1408 1400 1286 1206 1138 1049 996 968 785 726 350 312
63.7 18.2 8.3 14.9 5.0 1.1 26.0 6.4 1.8 3.4 20.5 43.0 0.4 56.4 1.9 31.3 6.5 2.9
8 0
8 0 0 0 0 2 4 2 1 4 8 0 13 12 1 1 1 3
2475 3128 3118 3042
2451 3124 3089 3059 1522 1512 1403 1397 1285 1202 1138 1049 996 952 784 725 349 312
43.3 18.5 8.3 14.8 5.0 1.0 11.4 25.6 1.2 5.6 22.2 41.9 0.0 48.3 1.8 31.8 6.8 2.7
3324 3117
3330 3124 3089 2258 1512 1493 1404 1357 1267 1183 993 1050 923 979 775 681 349 308
63.8 17.7 7.2 10.3 3.0 7.9 4.9 25.3 14.4 3.5 2.2 35.5 10.7 57.5 2.1 26.2 6.9 2.6
3324 2336
3330 2327 2279 3059 1512 1491 1403 1355 1211 1189 1084 956 882 773 760 700 332 303
63.6 9.0 5.8 16.7 3.4 4.1 31.8 5.1 2.1 9.1 16.5 44.0 3.9 46.7 1.6 11.8 3.1 4.8
0 3 1 2 4 1 5 2 0 12 2 1
1406 1394 (1172)b 1134 1043 957 723
2266 1505 1498 1409 1352 1264 1161 990 1044 981 770 680
3040 1515 1406 1359 1210 1162 1078 959 884 773 700
exp 2334 2269 2260 1500 1471 1369 1350 1173 1114 969 920 775 741 667
calc
int
3330 2327 2280 2258 1494 1474 1369 1342 1189 1123 969 924 868 766 748 667 331 300
63.6 9.2 7.4 8.5 0.5 15.5 10.5 25.0 7.8 6.3 34.9 23.4 0.1 27.7 6.5 18.4 3.7 4.2
a ∆(15N) ) (1-d0) - (1-15N4). b See footnote to Table 3. The calculations have fairly large error for this fundamental (see text). Using the calculated isotope shift of 4 cm-1 and the -d0 frequency of 1177 cm-1, we estimate that this fundamental should occur in the experimental spectrum at 1173 cm-1 and is overlapping with the 1172 cm-1 b2u band.
(the 786 cm-1 band is stronger than the 776 cm-1 one). There are two possible explanations for this discrepancy. The SQM force field may reverse the order of these frequencies, or the calculated dipole moment derivatives may be in error. The
second possibility is more likely, as we were not able to reverse the positions of the two bands in the scaling procedure, and there is also a similar intensity discrepancy in the 1-d2 isotopomer at 535 cm-1. To investigate the basis set dependence
7010 J. Phys. Chem., Vol. 100, No. 17, 1996
Figure 2. Comparison of the experimental IR spectrum of porphine (used with permission) with the SQM simulated spectrum. Symmetries are indicated by triangles (b1u), dots (b2u), and circles (b3u) in the experimental spectra and by 1 for b1u, 2 for b2u, and 3 for b3u in the simulated spectra. In the simulated spectra 1 - τ (τ ) transmittance) was plotted, adjusted so that the strongest level had 5% transmittance, (a) 1-d0, (b) 1-15N4, (c) 1-d2, (d) 1-d4, (e) 1-d8, and (f) 1-d12.
of the intensities we performed calculations on the pyrrole molecule using the B3-LYP level of theory and two different basis sets [6-31G* and 6-311++G(3p,3d)], but no significant dependence on basis set was found. The experimental band at 731 cm-1 agrees well with the calculated one at 727 cm-1. The same is true for 691 cm-1 (experiment) and 697 cm-1 (calculation) as well as for 639 cm-1 (experiment) and 639 cm-1 (calculation). A comparison of Table 2 and Figure 1 of ref 7 indicates that the moderately strong band listed for porphine-
Kozlowski et al. d0 at 540 cm-1 belongs to the -d2 isotopomer and can be excluded as a fundamental of -d0. We predict no fundamental in this range for -d0 but a strong band, the N-H out-of-plane wagging, in -d2. The largest disagreement we found was for the 335 cm-1 band assigned by Radziszewski et al. to b2u symmetry.7 According to our SQM force field this band is probably of b1u symmetry (see next section). The 91 cm-1 b1u band was not observed. 3.2. b2u Frequencies. None of the weak bands above 1600 cm-1 up to 3000 cm-1 are fundamentals nor are the weak bands at 1535, 1365, 1216, 1222, 1220, 990, and 986 cm-1. The medium band calculated at 1595 cm-1 is not listed in Table 2 of ref 7, although apparently there is a peak at about this position in the experimental spectrum. The strong bending fundamental of water near 1600 cm-1 could have interfered with the observation of this band.21 We cannot fully reconcile our calculations with the experimental data of ref 7 for this vibration. Our calculations predict a medium intensity fundamental above the strong one at 1547 cm-1 (calculated in -d0), but the experimental spectra show at most weak peaks in this region. Perhaps our vibrational forms are not accurate in this case. It is quite difficult, in general, to describe two closely spaced vibrations of the same symmetry. The experimental band at 944 cm-1 might be a combination enhanced by a Fermi resonance from the strong band at 951 cm-1. The moderately strong band predicted 863 cm-1 in the -d2 isotopomer may be hidden under the b1u band (calculated, 853; observed, 853 cm-1) which is predicted to be almost 20 times more intense. Alternatively, this may be the band observed at 866 cm-1 if we assume that the proximity of the very strong 853 cm-1 band influenced the polarization measurements or enhanced the intensity of a b1u combination through Fermi resonance. Both the positions and the intensities in the experimental spectrum, compared to the calculations, suggest reassignment of the 357 cm-1 experimental band as two overlapping bands of symmetry b2u and b3u (from calculations both are at 351 cm-1), although this appears to contradict to the polarization results of Radziszewski et al.7 This was also suggested by ab initio calculations of these authors7 and by empirical force field results.12 However, these calculations were not considered conclusive enough to challenge the experimental results. Our higher quality calculations are less easy to disregard. 3.3. b3u Frequencies. For the b3u species, the weak bands at 1138 and 1143 cm-1 are not fundamentals according to the calculations, nor are the weak bands at 1028, 1032, 956, and 964 cm-1. The weak band at 735 cm-1 is not very likely a fundamental either. The band at 357 cm-1 assigned experimentally to b1u symmetry is reassigned to overlapping b2u and b3u fundamentals, as already discussed above. The medium band predicted at 924 cm-1 in the -d4 isotopomer is hidden by the accidentaly degenerate stronger b2u one (calculated at 924 cm-1, observed at 923 cm-1). There is one significant discrepancy in this species: the fundamental calculated at 1206 cm-1 in -d0 is observed at 1177 cm-1. This band consists mainly of deformations of the pyrrole rings. Although this is a weak band, it is observed consistently about 25 cm-1 lower than calculated in all isotopomers and thus the possibility of experimental misassignment can be excluded. The “Kekule” vibration, i.e., the alternating contraction and expansion of the bonds, is not a pure normal vibration. It corresponds most closely to the strong b3u infrared fundamental calculated at at 726 cm-1 (experiment, 723) and is also contributing to the weak IR band at 785 cm-1 (experiment, 780). The good reproduction of these bands, and in general the low-
Vibrational Assignment for Porphine
J. Phys. Chem., Vol. 100, No. 17, 1996 7011
TABLE 5: Corrected Values for the Empirical Force Field of Ref 12 no.
type
Table V
filesa
5-13 5′-13′ 4-14 4′-14′ 4′-12′ 4′-15′ 4-16b 4′-16b 12-12 12′-12′ 2-4′ 2′-4 3-16b 3′-16b 8-8 8′-8′
II II V/3 V/3 V/1 V/1 V/1 V/1 VII VII III III V/1 V/1 VI VI
0.00 0.00 -0.05 -0.07 -0.12 -0.08 0.08 0.08 0.15 0.25 -0.10 0.15 -0.15 0.20 -0.10 0.10
0.15 0.25 -0.07 -0.08 -0.13 -0.09 0.10 0.10 0.00 0.00 0.15 -0.10 -0.14 -0.24 -0.08 -0.08
a Li and Zgierski, private communication. b The definition of 16 is equivalent to that of 16′.
frequency part of the b3u vibrations, is, in our opinion, conclusive proof for the D2h symmetry of the free-base porphine. 4. Comparison with an Empirical Force Field There has been continuous interest in developing empirical force fields for porphyrins and metalloporphyrins,10-12 although, to quote Radziszewski et al.,7 “the problem is woefully underdetermined for a purely empirical fitting procedure”. It is of interest to determine the accuracy of these force fields. We have therefore compared the latest empirical in-plane force field of Li and Zgierski12 with our results. To perform such a comparison, the two force fields must be expressed in the same set of nonredundant coordinates. Most empirical force fields, including those of Li and Zgierski,12 are given in a highly redundant coordinate system. Many workers in this field appear to be unaware of how to transform quadratic force fields between different coordinate systems (including redundant ones), so we summarize it here.23 According to this derivation, the transformation matrix U, which yields the force constants F (expressed in coordinates q), from F ˜ (expressed in terms of q˜ ), ˜ U, is given by U ) B ˜ BT(BBT)-1 where according to F ) UTF 22 B and B ˜ are the B matrices corresponding to the two sets of coordinates. This method, and the corresponding computer code FCT, has been in use in the group of P. Pulay since the early 1970s. The set q˜ must be nonredundant in this formalism, although we have developed a method to define quadratic force constants uniquely in redundant coordinate sets.24 The geometry used in our study differs from the geometry used by Li and Zgierski, who used the X-ray structure of Tulinsky et al.25 The only important difference is in the N-H distance. Li and Zgierski accepted the X-ray N-H distance of 0.86 Å. This is, however, implausible; we estimate that about 100 kcal/mol is needed to shorten the calculated bond length of 1.015 Å to 0.86 Å. We transformed the force field of Li and Zgierski to the common set of coordinates using their geometry.12 Initially, we were unable to reproduce the frequencies of Li and Zgierski, due to a number of typographical errors in Table 5 of ref 12, showing that it is not trivial to define and communicate a force field for larger molecules. We were able to find and eliminate these errors (see Table 5) with the assistance of authors of ref 12; the corrected empirical force field reproduces the frequencies of Li and Zgierski (at their geometry) with a maximum deviation of 10 cm-1. Most frequencies agree within 2 cm-1. The results of the comparison between the empirical and SQM force fields are shown in Figures 3 and 4. For convenience of
Figure 3. Comparison of the SQM diagonal force constants (x) with the empirical force field values of Li and Zgierski (y) (ref 12). (a) Stretchings (aJ/Å2); the best linear fit is given by y ) 1.857 + 0.677x with a correlation coefficient of 0.914. (b) Deformations (aJ/rad2); the best linear fit is given by y ) -0.093 + 1.155x with a correlation coefficient of 0.993.
presentation, we visualize them in the form of correlation graphs. It has been found that the largest elements of the empirical force field were in much better agreement with the SQM results than the smaller ones. In the latter, a sign error in the off-diagonal elements is quite common, especially for stretch-stretch and stretch-deformation couplings. There are many small offdiagonal elements assumed to be zero in the empirical model and a few significant ones as well (elements which exceed about 10% of the geometric mean of the corresponding diagonal constants in the SQM force field). It is interesting to note that the stretching force constants in the transformed force field differ from their values in the empirical force field even though the stretching coordinates are identical in both cases. This is caused by the fact, often not appreciated, that, at least in principle, eVery force constant depends on the definition of all coordinates, not only the ones it is associated with. Compliance constants (elements of the inverse force constant matrix) do not suffer from this general dependence. The most significant deviation of the empirical force field from the SQM one (which we assume is an accurate representation of the true force field) occurs for the C6-C7 diagonal stretching force constant which is 5.814 aJ/Å in the empirical force field (after transformation to our coordinates) but only 4.998 aJ/Å in the SQM force field. We have plotted
7012 J. Phys. Chem., Vol. 100, No. 17, 1996
Kozlowski et al.
Figure 5. Force constant-bond length relationship for the C-C bonds. b, SQM force field and DFT optimized geometry; O, the force field of Li and Zgierski (ref 12, transformed to the coordinates of Table 1) and X-ray geometry.
Figure 4. Comparison of the SQM coupling force constants (x) with the empirical force field values of Li and Zgierski (y) (ref 12). (a) Stretch-stretch couplings (aJ/Å2); the best linear fit is given by y ) 0.002 + 0.829x with a correlation coefficient of 0.863. (b) Stretchdeformation couplings (aJ/Å‚rad); the best linear fit is given by y ) 0.883x with a correlation coefficient of 0.818. (c) Deformationdeformation couplings (aJ/rad2); the best linear fit is given by y ) -0.001 + 1.203x with a correlation of 0.982.
the C-C force constants against the bond lengths in Figure 5. This shows clearly that the high value in the empirical force field is implausible. To illustrate the SQM force field results, the stretch-stretch couplings along the conjugated pathway are plotted in Figure 6. These force constants show a regular sign alternation, and interactions across the ring are still quite significant (e.g., the coupling between bond 5 (atoms 5-6) and 14 (atoms 15-16) is 0.160 aJ/Å). By contrast, bonds which are not part of the 18-membered conjugated ring (e.g. the bond between atoms 7-8) do not have long-range couplings. 5. Summary and Conclusions We have determined a scaled quantum mechanical harmonic force field, on the basis of nonlocal density functional calculations, which reproduce the observed and reliably assigned fundamental frequencies of porphine and its isotopomers to high accuracy (the mean absolute deviation is less than 5 cm-1). The
Figure 6. Stretch-stretch coupling force constants along the conjugated pathway (bold ring in Figure 1) in aJ/Å2. The numbering of the bonds is as follows: bond 1 is 1-2 in Figure 1, bond 2 is 2-3, ..., bond 6 is 6-22, etc. The force constant Fij (i * j) is plotted as a function of j. The value of i is indicated by the arrow.
infrared intensities are also well reproduced. Existing empirical force fields, as expected for a molecule of this size, are only qualitatively correct. The calculations show that the experimentally observed 723 cm-1 b3u band should be assigned to the “Kekule” vibration and prove thus conclusively that freebase porphine has D2h symmetry. Coupling force constants, especially between stretching modes, are quite significant even across the macroring.
Vibrational Assignment for Porphine Acknowledgment. This work was supported by the U. S. National Science Foundation under Grant No. CHE-9319929. We thank Professors X.-Y. Li and M. Z. Zgierski for useful comments and for assistance with the correction and interpretation of the empirical force field and Professor J. Michl for clarifying some points about the experimental spectra. We are thankful to the American Chemical Society for permission to reproduce the experimental IR spectrum of porphine. Supporting Information Available: Cartesian coordinates, nonredundant internal coordinates, and SQM force constants (21 pages) for free base porphine (Figure 1). Ordering information is given on any current masthead page. References and Notes (1) Dolphin, D., Ed. The Porphyrins; Academic Press: New York, 1978/79; Vols. 1-7. (2) Gouterman, M., Ed. Porphyrins: Excited States and Dynamics; ACS Symposium Series 321; American Chemical Society: Washington, DC, 1986. (3) Radziszewski, J. G.; Burkhalter, F. A.; Michl, J. J. Am. Chem. Soc. 1987, 109, 61. (4) Radziszewski, J. G.; Waluk, J.; Michl, J. Chem. Phys. 1989, 136, 165. (5) Radziszewski, J. G.; Waluk, J.; Michl, J. J. Mol. Spectrosc. 1990, 140, 373. (6) Radziszewski, J. G.; Waluk, J.; Nepras, M.; Michl, J. J. Phys. Chem. 1991, 95, 1963. (7) Radziszewski, J. G.; Nepras, M.; Balaji, V.; Waluk, J.; Vogel, E.; Michl, J. J. Phys. Chem. 1995, 99, 14254. (8) Almo¨f, J.; Fischer, T. H.; Gassman, P. G.; Ghosh, A.; Ha¨ser, M. J. Phys. Chem. 1993, 97, 10964.
J. Phys. Chem., Vol. 100, No. 17, 1996 7013 (9) Kozlowski, P. M.; Zgierski, M. Z.; Pulay, P. Chem. Phys. Lett. 1995, 247, 379. (10) Gladkov, L. L.; Solovyov, K. N. Spectrochim. Acta 1985, 41A, 1437; 1986, 42A, 1. (11) Li, X.-Y.; Czernuszewicz, R. S.; Kincaid, J. R.; Su, Y. O.; Spiro, T. G. J. Phys. Chem. 1990, 94, 31. Li, X.-Y.; Czernuszewicz, R. S.; Kincaid, J. R.; Stein, P.; Spiro, T. G. J. Phys. Chem. 1990, 94, 47. (12) Li, X.-Y.; Zgierski, Z. J. Phys. Chem. 1991, 95, 4268. (13) Pulay, P.; Fogarasi, G.; Pongor, G.; Boggs, J. E.; Vargha, J. A. J. Am. Chem. Soc. 1983, 105, 7073. (14) Rauhut, G.; Pulay, P. J. Phys. Chem. 1995, 99, 3093. (15) Rauhut, G.; Pulay, P. J. Am. Chem. Soc. 1995, 117, 4167. (16) Kozlowski, P. M.; Rauhut, G.; Pulay, P. J. Chem. Phys. 1995, 103, 5650. (17) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Wong, M. W.; Foresman, J. B.; Robb, M. A.; Head-Gordon, M.; Replogle, E. S.; Gomperts, R.; Andres, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzales, J. S.; Martin, P. L.; Defrees, D. J.; Baker, D. J.; Stewart, J. J. P.; Pople, J. A. Gaussian G92/DFT; Gaussian Inc.: Pittsburgh, PA, 1993. (18) Pulay, P. TX90, Fayetteville, AR, 1990. Pulay, P. Theor. Chim. Acta 1979, 50, 229. (19) Fogarasi, G.; Zhou, X.; Taylor, P. W.; Pulay, P. J. Am. Chem. Soc. 1992, 114, 8191. (20) Report on Notation for the Spectra of Polyatomic Molecules. J. Chem. Phys. 1955, 23, 1997. (21) Michl, J. Private communication. (22) Wilson, E. B., Jr.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955 (New edition, Dover: New York, 1980). (23) Fogarasi, G.; Pulay, P. In Vibrational Spectra and Structure; Durig, J. R., Ed.; Elsevier: Amsterdam, 1985; Vol. 14, pp 125-219. (24) Pulay, P. To be published. (25) Chen, B. M. L.; Tulinsky, A. J. Am. Chem. Soc. 1972, 94, 4144. Tulinsky, A. Ann. N.Y. Acad. Sci. 1973, 206, 47.
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