12076
J. Phys. Chem. 1996, 100, 12076-12085
Modeling the Soret-Resonant Raman Intensities of Metalloporphyrins and Heme Proteins. 1. Nickel Porphine Thomas Rush III, Ranjit Kumble,† Arka Mukherjee, Milton E. Blackwood, Jr., and Thomas G. Spiro* Department of Chemistry, Princeton UniVersity, Princeton, New Jersey 08544 ReceiVed: March 4, 1996; In Final Form: May 16, 1996X
A framework is developed for modeling the resonance Raman (RR) intensities of metalloporphyrins, with a view toward rationalizing the enhancement patterns observed in the spectra of heme proteins. The geometry of the S2 excited state of nickel(II) porphine is computed using INDO/1s methods, and the structural changes resulting from S0-S2 photoexcitation are projected onto the ground-state normal modes to calculate the intensity of each Raman-active vibration. The RR intensities derive mainly from expansion of the CRCm and CβCβ bonds in the excited state, with the relative intensities strongly influenced by the phasing between CRCm and CβCβ stretching coordinates. Analysis of the ν8 overtone shows the INDO predicted geometry changes to be about 25% too large. Results are compared at successive levels of approximation, demonstrating that inclusion of displacements along bending coordinates in the excited state are essential, as are frequency-dependent scaling factors which are determined from the absorption spectrum by the transform approach to RR scattering. Finally, the activation of non-totally symmetric modes by an A-term mechanism is modeled by distortion of the excited state along a b1g coordinate. Enhancement of the experimentally observed non-totally symmetric modes is correctly predicted, although quantitative modeling of their intensity requires the inclusion of nonCondon coupling.
I. Introduction Resonance Raman (RR) spectroscopy is a useful probe of ground- and excited-state properties of chromophores within protein environments.1-3 Its ability to selectively enhance the vibrational modes of an absorbing chromophore without interference from the vibrations of the protein permits the evaluation of geometric, environmental, and electronic properties of the chromophore in functional states of the protein. Ground state properties of the chromophore are probed by analyzing the vibrational frequencies of the RR spectrum. The positions of the vibrational modes in the spectrum are sensitive to the ground state structure and environment. The RR intensities, on the other hand, reflect the projection of the resonant excited state’s geometry onto the ground state normal-mode coordinates; therefore, a detailed analysis of the intensities of a RR spectrum will also provide information about the properties of the resonant electronic excited state. Since RR intensities are generally more sensitive to the chromophore’s structure and environment than the vibrational frequencies, a detailed understanding of the enhancement pattern of the vibrational modes of a chromophore in a protein may help in elucidating important chromophoreprotein interactions. RR spectroscopy has played a key role in the investigation of heme proteins. The sensitivity of heme vibrational frequencies to changes in oxidation state, ligation state, and spin state has allowed extensive application of RR spectroscopy to probe the structure and environment of the heme group within proteins.4-6 On the other hand, determinants of the RR intensity patterns of these modes and their sensitivity to different environments have not been established. Previous heme RR intensity analyses have been directed at testing the applicability of various theoretical approaches, and at investigating the origins of spectral broadening,7 mechanisms of RR enhancement,8 and * To whom correspondence should be addressed. † Present address: Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323. X Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00660-0 CCC: $12.00
inter- and intrastate coupling phenomena.9 In this work, we seek to connect the distribution of RR intensity among the normal modes to the electronic structure of the molecule. According to current RR theories, knowledge of the ground state normal-mode structure, the structure of the resonant excited state, and the extent of electron-nuclear coupling is required to evaluate the resonant enhancement of Raman active modes.2,3 A detailed understanding of the normal-mode structure of symmetric porphyrins and hydroporphyrins (chlorins and bacteriochlorins) is now available.10-12 The extent of electronnuclear coupling has also been extensively investigated in several heme containing proteins by Champion and coworkers.13-15 However, the RR enhancement patterns are not well understood because, apart from preliminary results reported by Warshel et al.,2 a connection has not been made to the geometry of the resonant excited states. The origins of resonant enhancement can be examined using either of two approaches. The first involves experimental determination of the excited-state structure by measurement and analysis of RR cross sections, to obtain the excited-state origin shift along each Raman-active mode.3 If the ground-state force field is available, the origin shifts may be decomposed into individual internal coordinate displacements, enabling determination of excited-state bond lengths and angles. However, a fundamental difficulty stems from the inability to establish the sign of origin shifts. A large number of possible structures (all of which are consistent with the observed RR intensities) may be obtained from different sign permutations for the origin shifts, and additional constraints are required to select the correct geometry.3,16e In the second approach, the electronic properties of the excited state are computed by ab initio or semiempirical methods and are used to estimate the geometry; in combination with the ground state force field, RR intensities are calculated and compared against the observed spectrum.16-18 The observed intensities are thus rationalized from a perspective which takes into account the electronic wave function of the resonant state. The analysis can then be refined with more rigorous modeling © 1996 American Chemical Society
Raman Intensities of Metalloporphyrins and Heme Proteins
J. Phys. Chem., Vol. 100, No. 29, 1996 12077 Here, ωL is the incident laser frequency, nm is the index of refraction of the sample, Ωk is the vibrational frequency of the kth vibrational mode, Fk is the depolarization ratio for the kth mode, the quantity njk is the Bose-Einstein factor ({exp(ωL/ kT) - 1}-1), ∆k is the dimensionless displacement of the kth normal mode’s equilibrium position upon excitation (a measure of the Condon electron-nuclear coupling), Ck is the non-Condon electron-nuclear coupling term for the kth mode, and
φ(ωL) ) P∫dω I(ω)(ω - ωL)-1 + iπI(ωL) Figure 1. Structure of nickel(II) piorphine (NiP), and a comparison of the bond order differences (S2 - S0) computed using INDO and SPMO methods (SPMO values in parentheses).
techniques which provide greater quantitative accuracy. Once a basic correlation between the electronic and vibrational properties is established for a simple model system, the analysis can then be elaborated to take into account perturbations of the electronic and vibrational structure which result from effects such as symmetry lowering, peripheral substitution, or interactions within a protein environment. With recent advances in force field development for naturally occurring chromophores,20 isolated molecule intensity analyses can be further extended to better understand proteins containing heme, chlorophyll, or bacteriochlorophyll chromophores. RR intensity analysis could help to address issues which are currently of great interest regarding electronic and structural interactions and excited-state dynamics of such systems. In the present study, we model the enhancement of Ramanactive vibrations for a simple metalloporphyrin, the nickel(II) complex of porphine (NiP, Figure 1) with excitation in its Soret absorption band. The structural properties expected for the resonant S2 excited state21 are considered in combination with the ground-state force field to compute the relative intensities of vibrations observed in the Soret-resonant Raman spectra. In this fashion, the relative influences of the electronic and vibrational structure of metalloporphyrins in determining the observed RR intensities are examined. The RR spectrum is computed using a Kramers-Kronig transform procedure7,22-26 which formally bypasses the extensive summation over intermediate vibrational levels required in the traditional sum-overstates approach.3,27 An important goal of this study is to test the validity of approximations which have been suggested in previous treatments for the computation of RR intensities.16 In a subsequent report, the extension of this model to account for substituent-, metal-, and ligand-dependent effects on metalloporphyrin RR intensities will be presented, and the relevance of these results with regard to heme protein spectra will be discussed. II. Theory The transform theory of RR intensities22-24 has been derived from both the time-correlator formalism of Hizhnyakov and Tehver25 and the traditional sum-over-states approach. The basic expression for the Stokes (+) and anti-Stokes (-) scattering of the kth totally symmetric Raman fundamental of a molecule of D4h symmetry in the harmonic limit is28
( ) ( )
(
)
dσ( 3 + 3Fk nm2 k (ωL) 1 1 ωL(ωL - Ωk)3 njk + ( × ) dΩ 3 - 4Fk 4 2 2 |(
∆k2 2
1/2
{φ(ωL) - φ(ωL - Ωk)} + Ck{φ(ωL) + φ(ωL - Ωk)}|2 (1)
(2)
where P denotes the principle value of the integral and I(ωL) represents the normalized absorption, given by
I(ωL) ≡ [∫dω′ R(ω′)/ω′]-1[R(ωL)/ωL]
(3)
For non-totally symmetric modes having B1g or B2g symmetry, the quantity 3 - 4Fk in the denominator of eq 1 is replaced by the factor 5. Recently, Rush and Peticolas have shown the feasibility of calculating accurate resonance Raman spectra by combining ab initio molecular orbital calculations with the transform approach to resonance Raman scattering.17,18 Their expression for the RR intensity of the kth vibration at excitation frequency ωL, including the ω4 term
Ik(ωL) ∝ ωL(ωL - Ωk)
( )[( )
3
1
xΩk
2
5.8065
3N-6
(∑ j)1
Lkj-1∆Rje)
]
2
×
|φ(ωL) - φ(ωL - Ωk)|2 (4)
assumes that only one excited state contributes to the intensities, the potential surfaces are harmonic, there is no inhomogeneous broadening of the electronic transition, the temperature is low enough that all of the transitions originate from the V ) 0 state, there is a constant transition length, and both Dushinsky rotations and non-Condon effects are negligible. This formula can easily be extended to include both higher temperature and non-Condon contributions. In the equation, Lkj-1 are the elements of the L inverse matrix determined from the solution of the ground state normal-mode eigenvalue problem,29 and ∆Rej are the changes in the internal coordinates upon excitation of the molecule into the relevant excited state. It should be noted that the most serious limitation to this formula, and the transform theory in general, is that the accuracy of a direct transform is compromised if the electronic transition is broadened primarily by an inhomogeneous mechanism.28 In light of the analysis of spectral broadening mechanisms for the Soret band of heme proteins by Champion and co-workers,46 we conclude that the Soret transition of nickel porphine is predominantly broadened by a homogeneous mechanism and that the transform approach described here is valid. Since the real part of φ(ωL) is simply a Kramers-Kronig transform of the absorbance at frequency ωL (i.e., φ(ωL) ) Transform[I(ωL)] + iI(ωL)), the value of |φ(ωL) - φ(ωL-Ωk)|2 may easily be determined experimentally. The procedure has been described in earlier publications on the transform theory.22-26 Implementation of eq 4 requires (1) a ground state normal-mode analysis of the molecule under investigation, (2) determination of the changes in the equilibrium values of its internal coordinates upon excitation, and (3) a Kramers-Kronig transform of the absorption spectrum.12,13 In the same spirit as the Rush and Peticolas reformulation of the transform formula for the fundamental (0 f 1) RR intensity of a symmetric vibration, we have reformulated the expressions
12078 J. Phys. Chem., Vol. 100, No. 29, 1996
Rush et al.
for the RR intensity of the overtone and combination bands of symmetric vibrations:25
Ik0f2(ωL) ∝ ωL(ωL - 2Ωk)3
( )[( ) 1
xΩk
8
5.8065
]
3N-6
4
( ∑ Lkj-1∆Rje) × j)1
|φ(ωL) - 2 φ(ωL - Ωk) + φ(ωL - 2Ωk)|2 (5) 0f1,0f1 Ix+y (ωL) ∝ ωL(ωL - Ωx - Ωy)3 ×
( )[( ) 1
xΩx
4
5.8065
] [( ) 2
3N-6
( ∑ Lxj-1∆Rje) j)1
xΩy
5.8065
]
3N-6
2
( ∑ Lyj-1∆Rje) × j)1
|φ(ωL) - φ(ωL - Ωx) - φ(ωL - Ωy) + φ(ωL - Ωx - Ωy)|2 (6) No additional information is required to implement these formulae. Equation 4 has been used in simplified forms in the past to reduce the computational demand of calculating resonance Raman intensities. These include approximating the ∆R’s by considering changes in bond lengths only, and by approximating the normalized transform scaling factors (|φ(ωL) - φ(ωL Ωk)|2) with just the square of the vibrational frequency, Ωk.16 In the former approximation, ∆Q (the change in the equilibrium value of the normal coordinate upon excitation) is obtained by summing over the B bond stretching coordinates only, on the assumption that these terms have the largest contributions to the summation: 3N-6
∆Qk )
∑ j)1
B
3N-6
j)1
j)B+1
Lkj-1∆Rje ) ∑Lkj-1∆Rje +
∑
Lkj-1∆Rje ≈ B
Lkj-1∆Rje ∑ j)1 Therefore, eq 4 becomes
Ik(ωL) ∝ ωL(ωL - Ωk)3
( )[( ) 1
xΩk
2
5.8065
]
B
(7)
2
(∑Lkj-1∆Rje) × j)1
|φ(ωL) - φ(ωL - Ωk)|2 (8)
This equation is easier to evaluate because the ∆R’s for the bond stretching coordinates can be estimated from relatively simple ab initio or semiempirical molecular orbital calculations. In the latter approximation, the transform term is dropped from eq 4 and the scaling is assumed to go as the frequency of the vibration (Ωk) squared:
Ik(ωL) ∝ ωL(ωL - Ωk)
( )[( )
3
1
xΩk
2
5.8065
3N-6
( ∑ Lkj j)1
-1
∆Rje)
]
2
(Ωk)2 (9)
This approximation was derived from both the sum-over-states and the time-dependent RR formalisms. In the sum-over-states RR formalism, this approximation is derived by assuming that the V ) 0, 1 Franck-Condon terms will dominate when there is a small shift between the ground and excited state potential
surfaces (∆).16a Using time-dependent formalism, on the other hand, it has been demonstrated that this approximation is valid in the limit of “short-time” dynamics and is appropriate under certain specific conditions (e.g., preresonant excitation or ultrafast excited-state dynamics).45 Since eq 9 is currently still in use, it is also one of our intentions to help support the finding that serious quantitative deficiencies in the calculated spectra can potentially result from naive use of this approximation.26 III. Experimental and Computational Methods A. Synthesis and Spectroscopy of Ni Porphine. The experimental spectra in this investigation were recorded on solutions of Ni porphine (NiP) dissolved in carbon disulfide. NiP was prepared by refluxing porphine, which was purchased from Midcentury Chemical Co. (Posen, IL), with nickel(II) acetate in dimethylformamide (DMF). The UV/visible absorption spectrum was recorded on a Hewlett-Packard HP 89532A UV/visible spectrometer. Solutions (0.5 mM) were prepared in carbon disulfide due to the high solubility of NiP in this solvent: for RR scattering experiments, the solution was contained in a 5 mm diameter NMR tube. RR spectra were recorded using excitation at 406.71 nm from a Kr+ ion laser (Coherent INNOVA). A 135° backscattering geometry was employed, and scattered radiation was collected and focused into a double spectrometer (SPEX 1422) equipped with a photomultiplier tube (Hamamatsu); spectra were recorded in 1 cm-1 increments with acquisition times of 2 s per step. Care was taken to minimize self-absorption effects by placing the focus of the incident laser beam close to the solution-glass interface within the NMR tube. This was achieved by spinning the tube at a fast rate such that the porphyrin solution formed a thin film against the tube: alignment of the incident laser on this film thus minimized the path length of the scattered radiation along the collection axis. Using this configuration, the relative band intensities were found to be virtually independent of concentration, showing self-absorption to be unimportant. B. Computational Methods. In this paper we report and compare calculations of the RR spectrum of NiP using several approximate methods to obtain the information needed for the use of equation (4). These include using our own intermediate neglect of differential overlap (INDO) molecular orbital calculations for the determination of ∆R, the screened potential molecular orbital (SPMO) calculations of Sekino and Kobayashi30 for the determination of ∆R, the frequency squared scaling approximation (eq 9), and the “bond length change only” approximation (eq 8). All of these calculations were performed on an SGI Iris Indigo and with the use of a recently developed scaled quantum mechanical force field for Ni porphine.31 Our INDO calculation of the Ni porphine molecular orbitals and transition dipole moments was performed using the program ARGUS.32 The ARGUS program employs the INDO/S semiempirical Hamiltonian33 for the determination of molecular orbitals and is parametrized to enable treatment of aromatic systems and first-row transition metal complexes. Thompson et al. have previously shown the usefulness of the INDO/S method in studying the excited states of the bacteriochrophyll b dimer in the photoreaction center of Rhodopseudomonas Viridis.32 Using ARGUS, bond order differences (∆b) between the S2 excited state and S0 ground state of NiP were computed,34 then converted to bond length changes (∆R) by assuming the linear relationship35,36
∆Rj ) -0.18(∆b)j
(10)
Since only bond length changes are available in this manner,
Raman Intensities of Metalloporphyrins and Heme Proteins TABLE 1: Predicted Geometry Changes for the S2 Excited State of NiP internal coordinate Cβ-Cβ (Å) CR-Cm (Å) CR-N (Å) CR-Cβ (Å) N-CR-Cmc (rad) pyrrole ring 1c,d (rad)
scaled scaled ∆R, INDO ∆R INDO ∆R, SPMO ∆R SPMO valuesa valuesa,b calc calcb +0.0080 +0.0074 +0.0016 -0.0020 -0.0050 +0.0025
+0.0065 +0.0060 +0.0013 -0.0016 -0.0041 +0.0020
+0.0025 +0.0028 +0.0005 -0.0020 -0.0025 +0.0005
+0.0060 +0.0067 +0.0012 -0.0048 -0.0060 +0.0012
a As described in section IVB. b Calculated from the data presented in: Sekino, H.; Kobayashi, H. J. Chem. Phys. 1987, 86, 5045. c Determined from the INDO calculation and spectra optimization. d Pyrrole deformation 1 is an in-plane ring deformation with the following unnormalized definition: +1.000 000 (CR-N-CR) - 0.809 017 (N-CR-Cβ) + 0.309 017 (CR-Cβ-Cβ) +0.309 017 (CR-Cβ-Cβ) -0.809 017 (N-CR-Cβ).
J. Phys. Chem., Vol. 100, No. 29, 1996 12079 in order to also obtain full geometry changes upon excitation, the NiP molecule was built in the BIOGRAF molecular mechanics program (BIOGRAF 3.2.1, Molecular Simulations Inc.) with the calculated bond lengths used as a constraint. The excited state equilibrium angles were then measured using an internal feature of the program and varied manually until the smallest error was found between the calculated and experimental spectra. This refinement was very straightforward due to the fact that the symmetric modes only depend on two angular coordinates (see Tables 1 and 2). The determination of L-1, which is also needed in eq 1, was accomplished by taking the inverse of the eigenvector matrix from the normal-mode calculation of NiP23 using the Mathematica program (Mathematica 2.2.1, Wolfram Industries). The integrity of the L-1 matrix was tested by confirming that LL-1 ) I. After the measurement of the absorption spectrum, the only other necessary information needed for the calculation of the
TABLE 2: Calculated Normal Modes and ∆ Values for the Soret Resonance Raman Analysis of NiP eigenvector inverse elementsc
mode
frequency (cm-1)
obsd freq (cm-1)a
internal coordinate contributionsb
magnitude
sign
ν8
362
369
ν7
733
732
ν16
741
732
ν15
998
1003
ν6
999
995
ν17
1063
1060
ν9
1068
1066
ν13
1193
1185
ν4
1385
1376
ν12
1394
ν3
1468
1459
ν11
1520
1505
ν2
1584
1574
ν10
1658
1650
36% CR-Cm stretch 33% pyrrole ring ip 1e 14% CR-Cβ stretch 11% N-CR-Cm bend 56% N-CR-Cm bend 28% CR-N stretch 8% CR-Cβ stretch 6% CR-Cm stretch 68% pyrrole ring ip 1 15% CR-Cm stretch 8% CR-N stretch 58% CR-Cβ stretch 26% CR-N stretch 14% pyrrole ring ip 1 44% pyrrole ring ip 1 34% CR-Cβ stretch 19% CR-N stretch 80% Cβ-H wag 16% Cβ-Cβ stretch 80% Cβ-H wag 13% Cβ-Cβ stretch 41% Cm-H wag 26% CR-Cβ stretch 18% CR-N stretch 11% pyrrole ring ip 1 36% CR-Cβ stretch 36% N-CR-Cm bend 25% CR-N stretch 40% Cm-H wag 38% CR-N stretch 8% CR-Cβ stretch 6% Cβ-Cβ stretch 56% Cβ-Cβ stretch 28% CR-Cm stretch 10% CR-N stretch 68% Cβ-Cβ stretch 18% Cβ-H wag 8% pyrrole ring ip 1 27% Cβ-Cβ stretch 25% CR-Cm stretch 18% pyrrole ring ip 1 12% Cβ-H wag 12% CR-N stretch 6% CR-Cβ stretch 70% CR-Cm stretch 10% Cm-H wag 8% pyrrole ring ip 1
2.655 2.466 1.223 1.631 0.888 0.974 0.442 0.426 1.083 0.665 0.430 0.840 0.677 0.375 0.731 0.646 0.551 0.253 0.734 0.255 0.691 0.248 0.419 0.394 0.205 0.406 0.293 0.360 0.221 0.476 0.174 0.295 0.957 0.451 0.246 0.940 0.077 0.151 0.533 0.365 0.245 0.064 0.259 0.120 0.570 0.082 0.125
++++++++ ++++ ++++++++ -------++++++++ ++++++++ -------++++++++ -+-+ -++--++-++--+++--++--+ +--++--+ -+-+ ++++ --------------++++++++ -+-+ -++--+++++ ++++ -++--+++--++--+ -+-+ --------------++++++++ ++++ -++--+++--++--+ -+-+ ---++++++++ -------+-+++++++++ -+-+ ----------++++ -++--++++++++++ ++++++++ +--++--+ ----+-+
final scaled ∆ valuesd +0.738
-0.001
+0.049 +0.023 +0.073 +0.035 +0.140 -0.013
+0.121 -0.015
-0.055 -0.037 -0.176
-0.0345
a From ref 11a. b Defined as LLt, where Lt is the transpose of L. c The L-1 elements are given in order as they appear for the corresponding internal coordinate (the internal coordinate found in the same row of the table) proceeding clockwise around the porphyrin ring. d As described in section IVC. e Pyrrole deformation 1 is an in-plane ring deformation with the following unnormalized definition: +1.000 000 (CR-N-CR) 0.809 017 (N-CR-Cβ) + 0.309 017 (CR-Cβ-Cβ) + 0.309 017 (CR-Cβ-Cβ) - 0.809 017 (N-CR-Cβ).
12080 J. Phys. Chem., Vol. 100, No. 29, 1996
Rush et al.
Figure 2. Space-filling electron density maps of the frontier molecular orbitals of nickel(II) porphine as predicted by an ab initio CIS calculation. (The maps were made with the QCPE program MOLDEN.)
TABLE 3: Predicted Soret Resonant Raman Normalized Band Areas for NiP
mode
exptl
ν8 ν6 ν9 ν13 ν4 ν3 ν11 ν2 ν10
10.000 0.499 0.593 0.250 1.048 0.106 0.257 1.964 0.414
INDO full geometry
SPMO full geometry
10.000 0.250 1.169
10.000 1.553 1.504
1.082 0.126
3.527 0.001
2.285
3.069
INDO Jahn-Teller distorted 10.000 0.320 1.250 0.012 1.048 0.223 0.104 2.312 0.090
Soret RR intensities through the use of eq 4 was the transform of the normalized Soret absorption spectrum. This was accomplished by first resolving the individual Soret band from the full absorption spectrum with a least squares fitting routine, normalizing it to 1.0, and then using our own modified version of the program KRONIG37 to take its Kramers-Kronig transform. Finally, the RR intensities were calculated using eq 4. The calculated spectra were then plotted using a constant full width at half-maximum of 10 cm-1. It is important to note though that the comparison between calculated and experimental intensities was performed through the use of normalized integrated band areas as shown in Table 3 and not by a simple comparison of intensity values from these plots. IV. Results and Discussion A. Molecular Structure in the S2 Excited State. Absorption properties of cyclic tetrapyrroles in the visible and nearultraviolet regions are described by Gouterman’s four-orbital model,21a in which the electronic transitions involve excitations from the two highest occupied (π) molecular orbitals (HOMOs) into the two lowest unoccupied (π*) molecular orbitals (LUMOs). In the case of fourfold symmetric metalloporphyrins, the two LUMOs comprise a degenerate pair of eg(π*) symmetry, while the two HOMOs are nearly degenerate and are of a1u(π) and a2u(π) symmetry (Figure 2). The (a1u f eg) and (a2u f eg) excitations are of nearly equal energy and are both of Eu symmetry; they are strongly mixed by two-electron interactions to produce transitions which are in-phase and out-of-phase
Figure 3. Absorption spectrum of NiP in a solution of carbon disulfide.
combinations of the individual excitations. The x-polarized states can be expressed in the form
S2(x) ) c1(a2u f eg) + c2(a1u f eg)
(11)
S1(x) ) c2(a2u f eg) - c1(a1u f eg)
(12)
where c1 and c2 are normalized coefficients defining the contribution of the individual excitations to the electronic wave function of each state. Transition dipoles for the (a1u f eg) and (a2u f eg) excitations are approximately equal, and therefore the intensity deriving from each excitation adds up for the transition to the higher (S2 or B) state and cancels for the transition to the lower (S1 or Q) state. The intense Soret (B) absorption band, typically observed in the 24 000-27 000 cm-1 region, corresponds to the S0-S2 transition, while the weaker Q-band observed in the 17 000-2000 cm-1 region corresponds to the S0-S1 transition (see Figure 3). The INDO calculation for the S2 wave function of NiP yields c1 ) 0.81 and c2 ) 0.58, indicating a larger contribution from the (a2u f eg) excitation. These results are entirely consistent with the earlier conclusion of Spellane et al.,21b who have determined the (a2u f eg) excitation to lie at considerably higher energy than the (a1u f eg) excitation for metal complexes of porphine (judged from trends in absorption and emission spectra). This leads to the expectation of greater (a1u f eg) character for the S1 state and a larger (a2u f eg) contribution to the S2 state in porphine complexes, as predicted by the INDO calculation. Based on differences in the nodal patterns of the a1u and a2u π orbitals (Figure 2), different structural consequences are predicted for the (a1u f eg) and (a2u f eg) excitations.38 Effects of the former excitation are confined to the inner pyrrole (CRN and CRCβ) bonds, while the latter excitation is expected to cause a large expansion of the outer ring (CRCm and CβCβ bonds). Consistent with a larger contribution from the (a2u f eg) excitation in NiP, the calculated bond order differences between the S2 and S0 states (Figure 1) are dominated by expansion of the CRCm and CβCβ bonds. A similar trend is calculated using the results from an ab initio screened potential molecular orbital
Raman Intensities of Metalloporphyrins and Heme Proteins
Figure 4. Transform scaling of the calculated NiP resonance Raman spectrum. (A, top) NiP’s Soret absorption and its Kramers-Kronig transform. (B, bottom) A comparison of the transform scaling factors and the frequency squared scaling factors which are normalized to the 1576 cm-1 ν2 peak.
(SPMO) study by Sekino and Kobayashi30,39 (Figure 1). The ∆R’s predicted by the INDO and screened potential methods (as described in section IIIb) are listed in Table 1. B. RR Intensities of Nickel(II) Porphine. The force field chosen for the calculation of the RR intensities in this paper was the DFT scaled quantum mechanical force field (performed at the B3-LYP level with a 6-31G* basis set for H, C, and N, and a VTZ basis set for Ni) described by Kozlowski et al.31 The predicted frequencies and coordinate contributions (LLt) for the relevant in-plane modes are listed in Table 2 (Lt ) transpose of L). The nonredundant coordinates of this force field consist of bond stretches, pyrrole ring deformations, porphine ring deformations, hydrogen waggings, and a few independent angle bends (see ref 31 for a complete description). This newer force field gives slightly better agreement with experimental frequencies and intensities than does the best current empirical force field.11 We note that the empirical force field employs a redundant set of internal coordinates and therefore requires the application of the relationship ∆Q ) (LtL)-1 (Lt∆R) in order to calculate the RR intensities. As described in the Experimental and Computational Methods section, the final information needed to calculate the RR intensities with the explicit use of eqs 4-6 are the values of the normalized Soret absorption spectrum and its Kramers-
J. Phys. Chem., Vol. 100, No. 29, 1996 12081
Figure 5. Comparison of the experimental RR spectrum of NiP at 406.7 nm with the calculated spectra using full geometry changes. (A) The experimental spectrum of 1 mM NiP in CS2 taken with 406.7 nm excitation. The CS2 peaks are labeled with an *. (B) The calculated spectrum using full geometry changes dictated by the INDO MO calculation. (C) The calculated spectrum using full geometry changes dictated by the screened potential MO calculation. The dotted curve in both (B) and (C) are the overtone intensities after the geometry scaling (as described in text). All spectra are normalized to the 1576 cm-1 ν2 peak.
Kronig transform, at the excitation frequency (ωL), at the difference between the excitation frequency and the vibrational frequency (ωL - Ωk), and at the difference between the excitation frequency and 2 times the vibrational frequency (ωL - 2Ωk) (or (ωL - Ωx - Ωy) for the case of combintion bands). Figure 4a shows the experimental normalized Soret absorption band with its calculated Kramers-Kronig transform. From these two curves, we then determined the values of the transform scaling term in eq 4 as described above. Figure 4b shows these scaling factors as a function of wavenumber, which have been normalized to the NiP ν2 vibrational frequency (1576 cm-1). Also shown in Figure 4b is the normalized scaling curve obtained when using the frequency-squared approximation (eq 9). The two curves are quite different. The transform curve increases rapidly at low wavenumber and then levels out at higher wavenumber, in contrast to the parabolic trend expected for the frequency-squared approximation. The consequences of these scalings are described below. Soret-resonant Raman intensities of the totally symmetric modes were then computed from eqs 4 and 5 using the inverse L matrix from the normal-mode calculation of Kozlowski et al.,31 the full geometry changes computed by INDO or SPMO methods, and the transform scaling factors. The experimental and calculated RR spectra are scaled to the 1576 cm-1 peak and compared in Figure 5. The agreement is reasonable between observed and calculated spectra using either the INDO or the SPMO excited-state geometries, although the average error (the
12082 J. Phys. Chem., Vol. 100, No. 29, 1996 average [(experimental I - calculated I)/experimental I] × 100) in the calculated intensities is lower for the INDO calculation (25.76 vs 110.9%). We note that the SPMO calculation was not specifically performed for a nickel(II) complex of porphine and that the ground-state geometry differs slightly from the one we have used for the INDO calculation. The calculations illuminate the factors which determine the relative enhancement of Raman-active vibrations. The strongest features in the Soret RR spectrum of NiP, ν8 and ν2, derive most of their intensity from the expansion of CRCm and CβCβ bonds in the excited state, while lower enhancements of ν4, ν6, and ν9 reflects the smaller distortions of the inner pyrrole (CRCβ and CRN) bonds. In rationalizing the observed relative intensities, it is necessary to consider the direction of displacements along each internal coordinate in the excited state (Table 1) together with the phasing of internal coordinate contributions to each normal mode (Table 2). The experimental spectrum is dominated by the lowest frequency mode, ν8 (369 cm-1), which is approximately four times more intense than the next strongest mode, ν2 (1576 cm-1). It can be seen from Figure 4, however, that the transform scaling factor |φ(ωL) - φ(ωL-Ωk)|2 will suppress the intensity of lower frequency modes relative to the higher frequency modes: this results from cancellation between the φ(ωL) and φ(ωL-Ωk) terms as the scattered frequency approaches the incident frequency.28 Thus, the strong enhancement of ν8 observed in the experimental spectrum implies a large dimensionless shift (∆) along this mode (to counter suppression by the transform scaling factor). Our analysis shows that the large shift along ν8 follows from the eigenvector phasing. Contributions from each internal coordinate displacement in the excited state reinforce when projected onto the ν8 normal coordinate, giving rise to a significant dimensionless shift along this mode. As can be seen from Table 2, the largest inverse eigenvector elements for ν8 arise from CRCm stretching and pyrrole ring (ip 1) and NCRCm bending motions. The positive sign and large magnitude of both the eigenvector inverse elements for these internal coordinates and their associated excited-state displacements (Table 1) cause their respective contributions to constructively add up to produce a high dimensionless shift along ν8. The importance of eigenvector phasing is further emphasized when examining the enhancement pattern of high-frequency vibrations. An important result in the calculated RR spectrum is that the obserVed disparity of the ν2 and ν3 intensities is reproduced. Although these modes both contain comparable contributions from CRCm and CβCβ stretching coordinates, the contributions are in phase for ν2 and out of phase for ν3, causing a cancellation of intensity in the latter case (as was earlier suggested11). Thus, in spite of large excited-state expansion along CRCm and CβCβ stretching coordinates, their contributions to the displacement along ν3 are canceled by their phasing and hence there is little net origin shift along this mode. If the mixing between these coordinates is altered Via kinematic effects of substituents, ν3 gains intensity: for example, isotope shifts establish that ν2 and ν3 for octaalkylporphyrins consist of nearly pure CβCβ and CRCm stretching motions, respectively.40,41 As expected from our model, ν2 and ν3 display comparable enhancement in the Soret RR spectra of nickel(II) octaalkylporphyrins.12 Consideration of internal coordinate phasing is of particular relevance with regard to the Soret RR spectra of heme proteins. RR spectra of the CO complex of myoglobin (MbCO) show weak activation of ν3, and a transform analysis has determined the major contribution to the intensity to arise from a nonCondon mechanism with only a weak Condon amplitude
Rush et al.
Figure 6. Comparison of the experimental RR spectrum of NiP at 406.7 nm with the calculated spectra using bond changes only. (A) The experimental spectrum of 1 mM NiP in CS2 taken with 406.7 nm excitation. The CS2 peaks are labeled with an *. (B) The calculated spectrum using bond changes only as dictated by the INDO MO calculation. (C) The calculated spectrum using bond changes only as dictated by the screened potential MO calculation. All spectra are normalized to the 1576 cm-1 ν2 peak.
present.8 Our results suggest that phasing of the internal coordinate contributions may be responsible for the small origin shift along ν3. Although both the INDO and SPMO calculations reasonably reproduce the relatiVe intensities of the totally symmetric modes, the intensity of the ν8 overtone (724 cm-1) is overestimated in the INDO calculation and underestimated in the SPMO calculation (Figure 5). These discrepancies imply that the INDO bond displacements are too large and the SPMO bond displacements are too small, because the fundamental:overtone ratio is a measure of the absolute value of ∆. As can be seen in eqs 4 3N-6 L -1∆Re)]), the funand 5 (where ∆k ) [(xΩk/5.8065)(∑j)1 kj j 2 damental depends only on ∆k , while the overtone depends on ∆k4;3 therefore
∆k ≈
( ( )( 4
I0f2 k
|φ(ωL) - φ(ωL - Ωk)|2
))
I0f1 |φ(ωL) - 2φ(ωL - Ωk) + φ(ωL - 2Ωk)|2 k
1/2
(13)
This formula was used to estimate the factor by which the calculated bond displacements should be scaled: 0.81 ( 10% for the INDO determined geometric changes, and 2.4 ( 10% for the SPMO determined geometric changes. These factors bring the scaled values for the two calculations into good agreement (Table 1). The new relative intensities of the overtone produced by these changes are shown as the dotted lines in Figure 5. The uncertainties in the scale factors are
Raman Intensities of Metalloporphyrins and Heme Proteins
J. Phys. Chem., Vol. 100, No. 29, 1996 12083
Figure 8. Activation of nontotally-symmetric modes by a Jahn-Teller mechanism: potential surfaces are viewed along symmetric (Qs, left) and nontotally symmetric (Qns, right) nuclear coordinates. The excited state (E) minimum is displaced along the symmetric coordinate by ∆s relative to the ground state (G). Lowering of the excited-state symmetry (dashed curves) introduces displacements ∆ns along the non-totally symmetric coordinates which are absent if D4h symmetry is retained (solid curve).
Figure 7. Comparison of the experimental RR spectrum of NiP at 406.7 nm with the calculated spectra using the transform and frequency squared scaling. (A) The experimental spectrum of 1 mM NiP in CS2 taken with 406.7 nm excitation. The CS2 peaks are labeled with an *. (B) The calculated spectrum using full geometry changes as dictated by the INDO MO calculation and the transform scaling. (C) The calculated spectrum using full geometry changes as dictated by the INDO MO calculation and frequency squared scaling. All spectra are normalized to the 1576 cm-1 ν2 peak.
associated with possible small contributions from ν7 and ν16 to the 2ν8 band profile. Results can now be compared using the approximations described in section III. First, we examine the treatment which considers changes in bond lengths only, eq 8 (Figure 6). Qualitative agreement with the experimental spectrum is obtained in the 1000-1700 cm-1 region where the modes are primarily composed of bond stretching motions; however, correspondence is poorer at lower frequencies where bending motions contribute more. The activation of ν7 (which now dominates the predicted 2ν8 intensity at 724 cm-1), is overestimated: a large enhancement of this mode is not experimentally observed. This approximation also underestimates the intensity of ν8 relative to the higher frequency modes. When angle coordinates are included, the sign of angle changes together with the phasing of bending and stretching motions for ν7 and ν8 is responsible for suppression of ν7 intensity and increased enhancement of ν8. This brings the calculated spectra into closer agreement with the experimental spectra. Next we consider the approximation leading to eq 9 where the intensities are scaled by the square of the vibrational frequencies (Ωk2) rather than the transform values. As mentioned in section II, previous time-domain treatments have shown that this approximation is only valid in the limit of shorttime dynamics. It is nevertheless often used to simulate spectra due to its computational simplicity. The resulting intensities can seriously be in error as illustrated in Figure 7, where Ωk2 scaling is seen to greatly underestimate the intensity of the
lowest frequency mode ν8 when the spectra are normalized to ν2. In general, the parabolic dependence of Ωk2 will introduce a scaling error unless the frequency range of the bands being compared is small. C. Enhancement of Jahn-Teller Active Modes. Although the enhancement of non-totally symmetric modes in RR spectra is generally ascribed to a B-term (Herzberg-Teller) mechanism, contributions from an A-term mechanism can arise in the case of an excited-state Jahn-Teller (JT) distortion. This lowers the symmetry of the excited state and introduces origin shifts along the JT-active modes (Figure 8). The extent of JT distortion in the S1 and S2 singlet excited states of porphyrins has been related to the form of the electronic wave functions:21 it has been demonstrated that the intrastate (JT) coupling term vanishes in the case of strong configuration interaction (CI), corresponding to the case where the coefficients c1 ) c2 ) 2-1/2 in eqs 11 and 12. However, the magnitude of the coupling term increases as the degree of CI is reduced and the coefficients c1 and c2 deviate from equality. A comparison of the Soret RR and absorption spectra of a series of metalloporphyrins has correlated the enhancement of nontotally symmetric modes with increasing strength of the Q0 absorption band (which reflects reduction in the extent of CI), consistent with the prediction from the four-orbital model.44 To determine whether the moderate activation of non-totally symmetric modes (for example: ν10, ν11, and ν13) observed in the Soret RR spectra of NiP can be accounted for solely by an A-term mechanism, we have attempted to model their enhancement by lowering the symmetry of the resonant S2 state. The degenerate S2 structure was distorted toward the geometry expected for a nondegenerate state consisting of the x (or y) component alone, and the RR spectra were calculated for several intermediate structures using the scaled INDO geometric changes. Figure 9 shows the spectrum calculated with a 75% Jahn-Teller distorted structure (i.e., distorted half-way between the symmetric and nondegenerate x states). As expected, the totally symmetric modes remain insensitive to this distortion and scaling of the geometric parameters; their symmetric eigenvectors average any x-y inequivalence and the same relative intensities are obtained as for the D4h S2 structure. However, the scaled geometry changes predict a smaller ν8 overtone, and the distortion to a D2h structure activates modes of b1g symmetry, namely modes ν10, ν11, ν12, ν13, ν15, ν16, and ν17. Of these modes, ν10, ν11, and ν13 are the only ones which are clearly resolved experimentally. The modes ν12, ν15, ν16,
12084 J. Phys. Chem., Vol. 100, No. 29, 1996
Rush et al. substituent-, metal-, and ligand-dependent effects on resonance Raman intensities of metalloporphyrins. The potential application of this model to heme proteins is of considerable interest and made possible by recent advances in the development of force fieds for naturally occurring heme groups. This will provide a means to quantitatively analyze variations in heme RR intensities in order to follow electronic and structural interactions which are occurring within the protein environment. Acknowledgment. The authors are grateful to Mark A. Thompson for providing a version of ARGUS version 2.0 prior to its official release, which was modified to compute excitedstate bond orders; and Pawel M. Kozlowski and Peter Pulay for supplying the results of their SQM/DFT calculation on NiP prior to its release in print. T.R. III is the recipient of a NIH postdoctoral fellowship (F32 HL09322-01). This work was supported by NIH grant GM 33576. References and Notes
Figure 9. Comparison of the experimental RR spectrum of NiP at 406.7 nm with the calculated spectra using a 75% Jahn-Teller distorted excited state structure. (A) The experimental spectrum of 1 mM NiP in CS2 taken with 406.7 nm excitation. The CS2 peaks are labeled with an *. (B) The calculated spectrum using full geometry changes of a 75% Jahn-Teller distorted excited state as dictated by the INDO MO calculation.
and ν17 are predicted to be present but obscured by the enhancement of the stronger symmetric modes as shown in Figure 9. Although the enhancement of the above-mentioned b1g modes are predicted to occur, the calculated intensities relative to the totally symmetric modes are not accurately reproduced even for a large distortion. We infer that accurate modeling of non-totally symmetric mode enhancement requires consideration of both Condon and non-Condon mechanisms. The transform approach can be extended to include non-Condon coupling,7,8,28 and respective contributions from each mechanism can be determined by simulating the RR excitation profiles7,28 for the modes of interest. V. Conclusions The present study provides a basis for understanding the electronic and vibrational factors which govern the Soretresonant enhancement of Raman-active vibrations of metalloporphyrins. The excellent agreement between observed and calculated RR spectra confirms the reliability of the groundstate force field, and demonstrates that a four-orbital description for the electronic wave function of the S2 excited state is sufficient to approximate its structural properties. Although the measurement and simulation of RR excitation profiles is required for accurate determination of geometry changes and to confirm the validity of the implicit assumptions, the results from this study provide valuable insight into sources of the observed intensity for each vibrational mode with regard to the groundstate normal mode structure and excited-state electronic properties. This approach is currently being extended to examine
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