J. Phys. Chem. 1993,97, 10964-10970
10964
Electron Correlation in Tetrapyrroles. Ab Initio Calculations on Porphyrin and the Tautomers of Chlorin Jan Almltif,'~f*# Thomas H. Fischer,s Paul G. Gassman,? Abhik Ghosh,? and Marco Hiiserl Institut f i r Physikalische Chemie, Universitiit Karlsruhe (TH), Kaiserstrasse 12, 7500 Karlsruhe 1, Germany; Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455; Interdisziplinijres Projektzentrum f i r Supercomputing, ETH- Zentrum, 8092 Ziirich, Switzerland; and Minnesota Supercomputer Institute, 1200 Washington Ave. S., Minneapolis, Minnesota 55415 Received: June 7, 1993'
Ab initio calculations including geometry optimization at the Hartree-Fock, MP2, and LDF levels have been carried out for free-base porphyrin and for the tautomers of free-base chlorin, using polarized basis sets. The Hartree-Fock approximation artificially favors frozen resonance structures with alternating single and double bonds for these compounds. This incorrect behavior is completely reversed when correlation effects are accounted for. Correct, delocalized structures of tetrapyrroles are obtained with the MP2 and LDF levels of approximation.
I. Introduction
H
WH
Interest in the theoretical study of porphyrins and hydroporphyrins hascontinuedundiminishedoverthe past threedecades.14 However, because of the size of these molecules, most theoretical investigationsin this area have been carried out at relatively low levels of theory, e.g., molecular mechaniq2 semiempirical quantum chemical methods? and ab initio methods with small basis sets.&-h Methods, algorithms, and computer technology have now matured to a point where these systems can be studied with more sophisticated appr~aches.~i-~ Direct techniques have been developed (TURBOMOLE5 at the University of Karlsruhe; DISCO, University of Minnesota"), which allow high-quality ab initio calculations on large molecules such as porphyrins. These methods have enabled us to carry out ab initio geometry optimization calculations on porphyrin and chlorin. The geometry of porphyrinoid molecules is a subject of considerablecurrentinterest. Buckled porphyrinshave been found in proteins, and the macrocycle buckling may have important consequences for enzyme mechanisms.' A large number of syntheticporphyrins exhibit a variety of distortionsfrom planarity. Crystallographic approaches to understanding the intrinsic conformation of porphyrins are complicated by unpredictable crystal packing effects. In contrast, accurate ab initio geometry optimizations, if feasible, are ideally suited for studying the intrinsic geometries of these large molecules without interference from intermolecular interactions. In principle, such calculations can be used to analyze the influence of peripheral substituents, macrocycle oxidation level, and the nature of the complexedmetal on the geometry of the macrocycle. This paper represents the first step in this line of research and describes the ab initio optimizationof the geometries of two of the simplest tetrapyrroles, viz., unsubstituted porphyrin and chlorin. We report here, at various levels of theory, the optimized geometriesof porphyrin and chlorin,the relative stabilityof chlorin tautomers, and some results on the tendency of the chlorin macrocycle to deviate from planarity. We have also carried out calculations of singlet and triplet instabilities to establish the tendency of the SCF wave functions to break symmetry. A significant conclusion from this investigation is that electron correlation plays a critical role in determining the properties of University of Minnesota. Supercomputer Institute.
t Minnesota
I ETH.
A Universiat a Abstract
Karlsruhe.
published in Advance ACS Absrructs, October 1, 1993.
0022-365419312097-10964304.0010
"
HT H'
'H
Figure 1. Two equivalent, nonnvitterionic resonance structures of frecbase porphyrin. Each structure has Ch symmetry.
these molecules. A quantitative understandingof thesecorrelation effects is crucial. Figures 1 and 2 show the nonzwitterionic resonance structures of porphyrin and the transs tautomers, A and B, of chlorin, respectively. Both the A and the B forms of chlorin have been experimentallyobserved. The high-energy form can be generated by photoisomerizationof the lower form9and is metastable at the temperature of liquid helium.lo It is generally agreedl1-l4that the cis tautomers of porphyrin and chlorin are of significantly higher energy than the trans tautomers, and, therefore),we have only considered the trans tautomers in the present study. In the A form of chlorin, the central N-H.-H-N axis is parallel to the reduced C& bond, while in the B form, it is perpendicular to the reduced C& , bond. Proton NMR studies on substituted chlorins have suggested A to be the predominant tautomer.l**$b Indirect evidence from cry~tallographyl~ and from l3C NMRIS also supports the predominance of the A form for a variety of substitution patterns. Accordingly,the B form has not been extensively characterized, and thecomputed resultsshould therefore beofinterest. Asimilar photoisomerization is known for porphyrin,I6 in addition to the thermal reaction path, but this tautomerism is degenerate as a consequence of molecular symmetry. 11. Methods
All-electron ab iniriocalculationswere carried out for porphyrin and the A and B tautomers of trans-chlorin using (semi)direct SCF and MP2 schemes as implemented in the codes DISCO6 and TURBOMOLE.5 In the MP2 calculations,all electronshave been correlated unless otherwise stated. The Gaussian basis sets used in these calculations are summarized in Table I. Local density functional (LDF) calculations were performed with the program DMOLlS using double-numerical ("DND") basis sets with d functions on C and N. 0 1993 American Chemical Societv
Electron Correlation in Tetrapyrroles
The Journal of Physical Chemistry, Vol. 97, No. 42, 1993 10965
H
H
-H
H-
-H
H-
H
H
Figure 2. Nonzwitterionic resonance structures of chlorin tautomers A and B.
TABLE I: Basis Sets Used in the Calculations. For Basis Sets Containing d-Type Functions, Only Five Functions Were Used; the Totally Symmetric Component Was hojected out of the Basis basis for N/C/H(central)/H(peripheral) symbol primitive contracted DZlg 7s4p, 7s4p, 4s, 4s 3s2p, 3s2p, 2s, 2s DZPlb-' 7~3pld,7~3pld,3 ~ l p39 , 3~2pld,3~2pld,2slp, 29 DZP2" 7s4pld, 7s4pld, 4slp, 4slp 3s2pld, 3s2pld, Zslp, 2slp TZDPd 9s5p2d, 9s5p2d, 5s2p, 5slp 5s3p2d, 5s3p2d, 3s2p, 3slp The s and p exponents were obtained from ref 17b; the exponents of the polarization functions for C, N, and H were 0.8, 1.0, and 0.8, rtspectively. The primitivefunctionswereobtained from the compilation of van Duijneveldt (ref 17a). For basis set DZP1, the exponents of the polarization functions for C, N, and H were 0.55, 0.817, and 0.727, respectively. The s and p exponents were obtained from ref 17c; the polarization function exponents for C, N, and H were 0.46 and 1.39 for C, 0.58 and 1.73 for N, 0.46 and 1.39 for the central H,and 0.8 for all other H. The theoretical level of a particular calculation is described in subsequent sections by a notation such as MP2/DZP2(SCF/ DZPl), which denotes a calculation carried out at the MP2 level using the DZP2 basis set, with a molecular geometry obtained from an optimization at the SCF level using the DZPl basis set. Molecular geometries were optimized using analytic gradie n t ~ . For ' ~ molecules containing multiple fused rings, geometry optimization is often a traumatic procedure. Using "natural internal coordinates",2@,ba prudent estimate of the initial nuclear Hessian,z&andefficient optimizationtechniques,m it was possible toconvergethegeometriesof thesesystemsinabout 6-10 geometry steps. 111. Results and Discussion
A. Structure of the Porphyrin Molecule and the Adequacy of the Computational Methods. The crystallographic structure of porphyrin13 shows an essentially planar molecule possessing approximate symmetry (with mirror planes bisecting the pyrrole rings). This contrasts sharply with the large bond length alternation obtained by Foresman et a1.4k at the RHF (spinrestricted Hartree-Fock) SCF/STO-3G and SCF/3-21G levels of theory. They found a frozen resonance form with alternating single and double bonds; the molecular point group was Cb with the C2 axis passing through the central hydrogen atoms. They speculated that basis sets which do not include polarization functionsmay overestimatethe stabilityof frozen resonance forms; accordingly, an SCF/6-31G* optimization might lead to a equilibrium geometry without bond length alternation. The issue of bond length alternation in porphyrin has been addressed previously.3i The porphyrin molecule contains an [ 181annulene substructure, and it is worthwhile to briefly review the situation in [ 1Ilannuleneand polyacetylene(an infinite annulene). While RHF theory yields a bond-alternating structure of D3h symmetry for [ 18]annulene,21inclusion of electron correlation at
theMP2/6-31G(SCF/6-31G) levelof theoryfavorsa delocalized D6h structure in agreement with experimental evidence. At the RHF level of theory, a polarized basis set does not lower the energy of the Dbh form of [ 18lannulene to below that of the D3h form. The tendency of RHF theory to lead to bond alternation in annulenes may be traced to a "singlet instability",22present at least in the case of some empirical Hamiltonians. Empirical Hamiltonians of the PPP type23 have also been used in full configuration interaction studies of annulenes with equal C-C bonds;24the latter study predicts strong electron correlation (with non-negligible connected quadruple excitations) as the ring size grows. The asymptotic limit would be polyacetylene. But instead of being a linear metal, polyacetylene is a semiconductor owing to a symmetry-breaking Peierls distortion;2s the bond length alternation has been experimentallydetermined to be AR = 0.080.09 A.Z6 However, it has not yet been possible to prepare nearperfect polyacetylene~amples,2~ and there is still some uncertainty in the magnitude of the symmetry-breaking distortion. As expected, RHF theory yields pronounced bond alternation: AR = 0.10 A.28 MP2/6-31GS* calculations on polyacetylene favor a bond-alternating structure with AR = 0.08 A.2sb In contrast, local density functional calculations yield the linear metallic structureZ9a or obtain very little bond alternation.29b Overall, the ab initio results on [ 18lannulene and polyacetylene indicate that MP2 theory is likely to provide a meaningful description of the bond length alternation in porphyrin and its derivatives while SCF and LDF approaches may predict too much or too littlebond alternation, respectively. Semiempirical(AM1) calculation~3~ indicate that UHF theory produces a qualitatively correct equilibrium structure for porphyrin. This indicates that triplet instabilities are also important in tetrapyrroles, as is the case in many other ?r-sy~tems.~~JO Since UHF calculations produce a poorly defined mixture of spin states, a further analysis along those lines would not be straightforward and has not been pursued here. To determine the molecular point group symmetry of porphyrin at the SCF level of theory, we performed an SCF/DZl(SCF/ DZ1) force constant calculation with the molecular geometry restricted to D2h. The results indicate that the D2h equilibrium geometryisasaddlepinton theSCF/DZl energysurface. There is only one imaginary frequency of B3, symmetry which leads to a Cb structure with pronounced bond alternation; this basically reproduces the results obtained with small basis sets by Foresman et u Z . ~ ~The imaginary frequency shows the direction of spontaneous distortion, and the real low-lying modes indicate directions along which the molecule would easily distort. The lowest in-plane mode (BI, symmetry at 119 cm-1) is illustrated in Figure 3 and corresponds to a distortion toward an internally hydrogen-bonded structure. It has been pointed out previously that this frequency is unusually low as a result of intramolecular hydrogen bonding.& All other vibrations below 150 cm-1 are out-of-plane modes: the B1, mode at 6 1 cm-1 is the "saddle distortion" of the porphyrin system and the A, mode at 75 cm-1 is a "ruffled distortion mode",
Alml6f et al.
10966 The Journal of Physical Chemistry, Vol. 97, No. 42, 1993
Figure 3. Distortion of the porphyrin molecule, corresponding to the low-frequency B1, mode of vibration ( v = 119 cm-I).
TABLE II: Eigenvalues (in Atomic Units) of the SCF Stability Matrices (Siglet, Nonreal, TripletP for Free-Base Porphyrin Restricted to & Symmetry (SCF/DZl(SCF/ DZ1)). For Each Irreducible Representation, the Lowest Two Eigenvalues Are Given symmetrp singlet nonrcal triplet A, BI, B28 B3,
Au B1u
B2u B3u ,
0.185 0.202 0.174 0.205 0.233 0.305 0.230 0.307 0.235 0.291 0.232 0.297 0.039 0.197 0.033 0.177
0.174 0.205 0.165 0.192 0.229 0.298 0.210 0.300 0.23 1 0.295 0.209 0.291 0.095 0.141 0.095 0.140
Porphyrin (D2,,):SCFlTZDP
0.039 0.109 0.013 0.033 0.220 0.285 0.178 0.284 0.223 0.283 0.176 0.279 4.084 0.059 4 . 0 31 0.028
a We use the convention described in ref 32, with the zaxis perpendicular to the molecular plane and the x axis aligned with the N-H bonds. The representations BI,, Bzu,and B3,, are symmetric with respect to rotations about the I, y , and x axes, respectively.
whereas the B1, mode at 107 cm-I is the umbrella mode of the porphyrin system.31 In keeping with the scope of this paper, this discussion on vibrational frequencies is restricted to modes associated with spontaneous or low-energy distortions from the D2h StfUCtUre. As expected, explicit calculations in C2h symmetry show no distortion from full D2h symmetry, and calculations with no other constraint than the planarity of the molecule led to the C, form (with the C2 axis passing through the central hydrogens). To ensure that no discontinuities in the potential surface arise as symmetry is lowered from D2h to Cb,we calculated the lowest eigenvalues of the SCF stability matrice~.~Z~O The results, shown in Table 11, indicate that there is no singlet instability in D2h porphyrin. A different result would have implied the immediate necessity to use a multiconfigurational wave function. The eigenvalues of the SCF singlet stability matrix are rather small (0.04 and 0.03 au for the representationsB2, and B3,, respectively), indicating that configuration interaction may be important. The triplet instability, which demonstrates the existence of a spinunrestrictedHartree-Fockdeterminant of lower energy," is typical for extended *-systems. The negative value of -0.084 au is twice that found for benzene.30 It should be noted, however, that the situation for porphyrin is not nearly as extreme as in, e.g., ozone (open form), where the corresponding SCF/DZl value is -0.22 au.
\
11.46
H Porphyrin (&):
SCFlTZDP
Figure 4. Geometry of porphyrin obtained at the SCF/TZDP level of approximation with D a and Cb symmetry constraints.
TABLE II1: Total Energies (E)of & and C', Optimized Structures of Porphyrin and Average Bond Length Alternation (AR)in the Diaza[l8b~uleneSubstructure of Porphyrin (C2J calculation SCF/DZl (SCF/DZl) SCF/DZPZ(SCF/DZPZ) SCF/TZDP(SCF/TZDP) MPZ/DZPZ(SCF/DZPZ) MPZ/DZPZ(MPZ/DZP2) LDF(SCF/DZPZ) LDF(LDF)
-982.070 99 -982.522 82 -983.485 96 -985.858 45 -985.864 9 -981.441 58 -98 1.44168
-982.073 97 -982.528 40 -983.492 87 -985.827 08
-
-98 1.428 34 -
-82.3
0.062 0.075 0.082 -
-34.8
-
7.8 14.6 18.1
-
-
-
-
The influence of the one-particle basis set on the extent of symmetry breaking (as evidenced by the bond length alternation) in porphyrin is studied in Table 111. The SCF/TZDPgeometries obtained with and without D2h symmetry constraints are shown in Figure 4. The bond alternation increases as the basis set limit is approached, contrary to what has been suggested by Foresman et D2h symmetry is restored, however, if electron correlation is accounted for by second-order perturbation theory, as found in a geometry optimization carried out at the MP2/DZP2 level, started at the SCF/DZP2 Cb equilibrium geometry. The
Electron Correlation in Tetrapyrroles H
The Journal of Physical Chemistry, Vol. 97, No. 42, 1993 10967
H
SCF Porphyrin (D2,,): MPP/SVP
111.4O
H
Porphyrin (D2,,): LDFlDND Figure 5. Geometry of porphyrin obtained at the MP2/DZP2 and LDF/ DND levels of approximation with the symmetry constrained to Cb.
converged MP2/DZP2 equilibrium geometry is depicted in Figure 5a. The agreement with the geometry inferred from X-ray analysisl3 is excellent; discrepancies in bond lengths are only of the order of 0.01-0.02 A and would be reduced even further if corrected for basis set deficiencies. The local density approximation leads to a molecular geometry close to the MP2/DZP2 equilibrium geometry (Figure 5b), but the stabilization energy of the delocalized D2h structure relative to the bond-alternating CZ,structure is only half the value predicted by MP2 theory (see Table 111). It still remains to be settled which of these two estimates, MP2 or LDF, is the more reliable one. To obtain further insight into the correlation effects in porphyrin, we have calculated potential curves along the bond length alternation coordinate AR of the diaza[ 1Slannulene subsystem in porphyrin. For each value of AR,all other degrees of freedom have been re-optimized at the SCF/DZl level of theory. In addition, we have also calculated energies at these geometries using several other methods. The results given in Figure 6 show the MP2 potential curves as steep linear functions everywhere except in the immediate vicinity of the minimum. The LDF potential curve is considerablyless steep. The dramatic increasein correlationenergy upon going from CZ,to D2h geometry (SCF/DZl) is also reflected by changes in the first-order wave
Figure 6. Energy of porphyrin as a function of bond length alternation AR (in pm) in the diaza[ 18lannulenesubsystem. For each value of AR, all other degrees of freedom have been re-optimized at the SCF/DZl level.
function: the T2 cluster amplit~des3~ representing simultaneous excitations out of the two highest occupied a-orbitals increase by 50% on average. Thus, it seems plausible that near-degeneracy electron correlation effects may be important in porphyrin. However, any all-electron (or all-valence)treatment oftheelectron correlation beyond second-orderperturbation theory is at present prohibitive for systems of this size. It might at first appear sufficient in this situation to correlate the a-electron systemonly; however, a comparisonof the a-MP2/ DZP2 and all-electron MP2/DZP2 potential curves reveals that correlation descriptions restricted to a-electrons miss half of the correlation stabilization (cf. Figure 6 ) , in agreement with CI results obtained at a D2h ge~metry.~Ll It isn't clear at this point whether this is an inherent property of the porphyrin system or if it originates in the poor reference state provided by Hartree-Fock. In the latter case, multiconfigurational SCF (MCSCF)33a4or even coupled cluster CCSD(T) approaches34restricted to the a-system would rectify the situation. With regard to possible multiconfigurational approaches, a previous analysis of spin coupling in benzene deserves attention?3c in particular as UHF theory leads to an unphysical description consisting of approximately alternating a and 0 spin densities on adjacent carbon atoms in porphyrin.3' In summary, the RHF method is not adequate for calculating equilibriumgeometries of tetrapyrroles and related systems unless symmetry constraints can be used to force an averaging of the appropriate resonance structures. In that case, qualitatively correct behavior may be expected for geometrical degrees of freedom that are not strongly coupled to the bond alternation mode. Both MP2 and LDF theories describe bond alternation in tetrapyrroles in a qualitatively correct manner even if symmetry constraints are not applied. Concerning the quantitative energetics of bond alternation, not all questions have been resolved. B. Relative Stability of Chlorin Tautomers and Macrocycle Buckling. The optimized geometries for chlorin A and chlorin B, obtained under C, symmetry constraints, are shown in Figures 7 and 8, respectively. At the LDF level of theory, the computed geometry for the stable tautomer A is in good agreement with existing crystallographic data. The bond distances in the LDF structures of both forms show extensive a delocalization, and the differencesin bond distancesdue to the tautomerism do not exceed 0.03 A. On comparing the SCF and LDF results, it is clear that the RHF results for chlorin B suffer from artificial localization of the a-electron system, as in porphyrin. Since tautomer B has only one nonzwitterionic resonance form which has the maximum symmetry possible for the molecule,
Almlbf et al.
10968 The Journal of Physical Chemistry, Vol. 97, No. 42, 1993
I
\1.52
H-
H
)=(:
125.8"
106.2"
H
125.3"
6
H
H
1.52
H
1.39
Chlorin B (Czv):SCFlDZ
Chlorin A (CJ: SCF/DZ 106.38+
107.7"
po H
113.1" 110.7' 102.90
I
\1 S O
1.39
Chlorin A (&): LDFlDND Figure 7. Geometries for chlorin A, optimized at the SCF and LDF levels of approximation.
Chlorin B (CJ: LDFlDND Figure 8. Geometriesfor chlorin B, optimized at the SCF and LDF levels of approximation.
symmetry constraints are not effective for reducing the excessive bond localization inherent in the RHF method. This was verified in (SCF/DZI) calculations on the A and B forms, using no other constraints than the planarity of the macrocycle. As expected, the geometry optimizations converged to structures of C, and Cb symmetry for A and B, respectively. In both the A and B forms, the C,-N-C, angle in the reduced ring exceeds the C,-N-C, angle in the opposite ring by &So. In addition, in both the A and B forms, the C,-N-C, angles in the N-protonated pyrrole rings are significantly wider than those in the N-deprotonated pyrrole rings. This is a well-known trend in both porphyrin and hydroporphyrin molecular s t r u ~ t u r e s , and ~~J~ it is used diagnostically to determine the positions of the internal protons in free-base tetrapyrroles even when they have not been actually located in the crystallographic structure determination. Many crystal structures of chlorins have thus shown in an indirect fashion that the N-H-H-N axis is parallel to the reduced C&B bond. The calculations further confirm that the C,-C, distance is significantly longer than the C,-C,, distance. Table IV lists the total energies of the two (planar) tautomers of chlorin. If the geometries have been obtained by the LDF method, an approach which has proven satisfactory for porphyrin, the A tautomer, Le., the one generally preferred based on
TABLE IV: Total Energies of the Planar (CZ,) Chlorin Tautomers A and B method LDF(LDF) SCF/DZPZ(LDF) MP2/DZP2(LDF) SCF/DZPI(SCF/DZPl) MP2/DZPl (SCF/DZPl)
EA- EB
EA (hartree) EB (hartrce) (kJ/mol) -982.641 -982.630 -29 -983.681 -11 -983.693 -981.042 -987.021 -39 -983.690 -983.695 +13 -986.815 -986.856 41
experimental informati0n,~IJ4is clearly favored at all levels of theory employed. Electron correlation further stabilizes the A form. However, if C , constrained RHF equilibrium geometries are used, the A and B tautomers are not treated on an equal footing: the A form is unduly destabilized at the RHF level, as it cannot relax into a bond-alternating structure. There are no significantly buckled (or nonplanar) conformers of chlorin A or B at any of the levels of theory used here. The potential energy surface is extremely flat in some degrees of freedom corresponding to out-of-plane distortion; a torsion of the -CH&Hrbond in the pyrroline ring by 6 O increases the energy by less than 0.25 kJ/mol. Although the matter is not chemically significant,it may be worthwhile to point out that it was impossible
Electron Correlation in Tetrapyrroles
The Journal of Physical Chemistry, Vol. 97,No. 42, 1993 10969
TABLE V: Eigenvalues of the SCF Singlet Stability Matrix for Chlorin A Restricted to C’, Symmetry (SCF/DZl(SCF/ DZ1)). The Lowest Two Eigenvalues Are Given for Each Irreducible Representation (in Atomic Units) representation
eigenvalue
Ai
0.078 0.171 0.249 0.267 0.220 0.243 0.048 0.191
A2
Bi B1
to characterize the minimum of the LDF potential surface due to limited numerical accuracy of the software used. Consistent with our observations, an SCF/DZl(SCF/DZl) force constant calculation on chlorin A at Cb symmetry indicates very soft macrocycle out-of-plane modes; viz., a saddle distortion31 at 56 cm-I, the ruffled distortion mode” at 99 cm-1, and the umbrella mode at 108 cm-1. Unlike any vibrational mode in porphyrin, chlorin A possesses an additional mode at 49 cm-l, which describes a localized torsion of the saturated C& bond and which has virtually no effect on the planarity of the macrocycle. The internal hydrogen-transfer mode, similar to the one depicted in Figure 3, appears at 116 cm-l in chlorin A. Macrocycle buckling in chlorins may easily arise as a result of steric forces from bulky substituents or from a central ion too small to be optimally coordinated by a planar ~ h l o r i n . * ~ , ~ 5 With respect to the RHF stability problem in chlorins, similar behavior is observed in chlorin A as in porphyrin. The imaginaryfrequency B2 mode leading to bond alternation occurs at 801i cm-I at the SCF/DZl level of theory. The lowest eigenvalue of the SCF singlet stability matrix is 0.048 au (see Table V). This indicates a smaller degree of possible near-degeneracy correlation effects than in porphyrin. Presumably, this tendency continues with further hydrogenation of the porphyrin system.
IV. Conclusion We have optimized the geometries of free-base porphyrin and of the tautomers of free-base chlorin using ab initio SCF and MP2 methods and the local density approximation. We conclude that the Hartree-Fock approximation is generally inadequate for determining the structures of tetrapyrroles. This is not a basis set problem but one that can be remedied with simple schemes of electron correlation, such as the LDF or MP2 methods. The difficultyconsists of the RHF method preferring frozen resonance forms with alternating single and double bonds over thedelocalized structures. This problem could be removed through the use of an unrestricted Hartree-Fock formalism but at the expense of an unphysical spin-polarization in the A system. At correlated levels, porphyrin and both tautomers of chlorin possess planar and nonalternating geometries compatible with the highest possible point group symmetry. Geometrical degrees of freedom which do not couple with the bond alternation mode seem to be well described at the RHF level of theory. The vibrational modes of lowest frequency (about 50-100 cm-l at the SCF/DZl level) correspond to out-of-plane modes characterized by the attributes “saddle”, “ruffled”, and “umbrella”; then follows the internal hydrogen-transfer mode. In comparison to porphyrin, chlorin A is more flexible with respect to buckling and has an additional low-frequency torsion mode localized at the saturated C& , bond.
Acknowledgment. We are grateful to Dr. F. Haase at the Humboldt Universitit in Berlin for supplying us with his latest versions of the TURBOMOLE MPGRAD program and to H. Weiss and M. Ehrig, Universitit Karlsruhe, for valuable support with the LDF geometry optimizations (internal coordinate
interface). The work was supported by the Minnesota Supercomputer Institute, the National Science Foundation (J.A., P.G.G.), the Deutsche Forschungsgemeinschaft (T.H.F.), a University of Minnesota Graduate School Dissertation Fellowship (A.G.), and the Fonds der Chemischen Industrie (M.H.).
References and Notes (1) Fora recentreviewoftheoreticalstudiesoftetrapyrroles,see:Hanson,
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