9956
J. Phys. Chem. 1993,97, 9956-9968
Vibronic Coupling in Ni(I1) Porphine Derived from Resonant Raman Excitation Profiles E. Unger, U. Bobinger, W. Dreybrodt, and R. Schweitzer-Stenner' Institute of Experimental Physics, University of Bremen, 28359 Bremen, Germany Received: March 24, 1993; In Final Form: June 25, 1993' We have measured theresonant Raman excitation profiles of Ni(I1) porphine in a C S 2 solution for 15 fundamental modes: the AI, modes YZ, u4, Y6, and US; the B1, modes Y ~ O ,~ 1 1 Y13, , Y ~ S ,and Y17; the '428 modes ~ 1 9 YZI, , ~ 2 2 and , Yz6; and the B28 modes ~ 2 and 9 ~ 3 4 . Several Raman lines were found which are indicative of the existence of a second conformer. Three of them correspond to the modes Y Z , u4, and Y19. The data cover the resonant regions of the Q and the Q, bands and the preresonant region of the 8 band. The depolarization ratio of all lines investigated was found to be independent of the excitation wavelength, in accordance with expectations of a molecule with D4h symmetry. The excitation profiles were first analyzed by use of a theory which describes intra- and interstate coupling in terms of a time-independent perturbation theory (Shelnutt, J. A. J . Chem. Phys. 1981, 74, 6644-6657) in the framework of Gouterman's four-orbital model. This approach accounts for the excitation profiles of all ,428 modes and most of the AI, modes, but it fails for all B l , modes and the Bz, mode ~ 3 4 . In a second step, we considered multimode contributions to the scattering process and additional intrastate coupling for the B1, modes within the Q and 8 states. This yields satisfactory fits to all Raman excitation profiles (REPs), but rather large deviations from the four-orbital model must be assumed. In a third attempt, we took into account possible contributions from a scattering process which involves Bl,-type pseudo JahnTeller coupling between the porphyrin ground state and a state created by an electron transfer from the d,(Ni) orbital to the e, orbital of the porphyrin macrocycle. Thus, it was possible to describe all REPS without relaxation of the restrictions imposed by the four-orbital model. The coupling parameters thus derived are consistent with the absorption spectrum.
Introduction Metalloporphyrinsplay an important role in various biological molecules, where they are embedded into a protein matrix of low symmetry. Such molecules are hemoglobin, which acts as transport molecule in blood; myoglobin, which stores and releases oxygen in the muscles;' and various cytochromes, which are involved in the electron-transfer process of the respiratory chain. Therefore, understanding the properties of metalloporphyrinsis motivated by biophysical and biochemical interest. Since metalloporphyrins have a relatively simple electronic structure2 with two dipole-allowed transitions in the region of visible light, resonance Raman scattering is a powerful tool to obtain information about the properties of the corresponding excited states and their response to changes in the environment of the *-system of the porphyrin ~keleton.~-~ Such changes can be obtained by replacing either the peripheral substituents (PS) or the central atom. Both can reduce the symmetry of the porphyrin from the ideal D4h. Polarized Raman measurements are a suitable method for detecting the symmetry of the investigated molecule. The depolarization ratio (DPR) is recorded as a function of the excitation wavelength. Its dispersion (DPD) yields information about the symmetry of the molecule. In high symmetries such as 04h, 0 4 , Du, Du, or Cb,all Raman lines for theoretical reasons ~how~depolarization ratios (DPRs) independent of the frequency of the exciting light. However, this is not the case for porphyrins in heme proteins and for some Ni(I1) porphyrin derivatives in solution,- because the actual symmetry is reduced by its environment and the PS below the symmetries listed above. This has stimulateda variety of theoretical investigations,aimed mostly at understanding DPDs and the properties of the Raman excitation profiles (REPs).'v4 Most of these theoretical approaches consider strong vibronic coupling but are too complex to be used in a fitting procedure5to the experimental data. Therefore, an alternative approach was developed in our laboratory9 using the time-dependent perturbationtheory formulated by Loudon.10 This *Abstract published in Aduunce ACS Abstrucrs. September 1, 1993.
U
R/
R'
Figure 1. Structure of Ni(I1) porphine (R = R' = H).
approach is valid only in the weak coupling limit, but it yields analytical expressionsfor the REPs and DPDs of the fundamental modes which are suitable for a fitting procedure, thus allowing one toextract thevibroniccouplingparametersand their symmetry properties.5 This theory was used to study the behavior of Fez+protoporphyrin IX, the heme group in hemoglobin, myoglobin, or cytochromec. In these proteins, the DPD of the prominent Raman lines depends on the properties of the solvent, e.g., the pH and ionic strength of an aqueous solution.".l* Using the above theoretical expressions,it was possible to extract parameterswhich are related to symmetry-classified distortions by lowering the symmetry of the heme.9 Alterations of these distortions, caused by allosteric interactions, could then successfully be related to changes of the biological function.5 In view of these results, it is desirable to reinvestigate isolated porphyrins to discriminate between the symmetry-loweringeffects of the peripheral substituents and the protein environment. To this end, we have measured REPS and DPDs of a variety of Ni(11) porphyrins in our laboratory such as nickel octaethylporphine13 (Figure 1 with R = ethyl, R' = H), nickel tetraphenylporphine ('Figure 1 with R = H, R' = phenyl), and nickel octaalkyltetraphenylp~rphine~~ (Figure 1 with R = alkyl, R' =
0022-3654/93/2097-9956$04.00/00 1993 American Chemical Society
Vibronic Coupling in Ni(I1) Porphine
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9957
phenyl). The investigation of the DPDs shows that some of these are nonplanar. Ruffled structures were also found in a series of investigations by Shelnutt and co-workers.lS18 To compare all these data, one needs a reference system with planar symmetry and negligible influence of the peripheral substituents on the macrocycle. Ni(I1) porphine (Nip) (Figure 1) is the simplest compound of this type. Amazingly, it has never been studied with respect to its REPs, in spite of respective theoretical model calculations performed by Shelnutt’ on the basis of Gouterman’s four-orbital model in strict D4h symmetry. Therefore, we have investigated the polarized REPs of the highest frequency modes of NIP in the preresonant region of the 3band and in resonance with the Q bands. To obtain thevibronic coupling parameters, we used a modification of Shelnutt’s approach3 based on the assumption that the matrix elements of the transition between the al, and az,, ground-state orbitals and the first unoccupied orbitals e, are different. From this, we obtain analytical expressions which are used for a fit to the experimental data. In the ideal D4h symmetry, the Raman spectra arise from lines corresponding to vibrational modes which transform like the irreducible representations AI,, Az,, Bl,, and B2,.19 For all the Ais, Azg,and Bls lines, we find good agreement to the data in the frame of the four-orbital model. The REPs of the B1,, however, can only be understood by this model if the interaction between the r-electrons of the macrocycle and the d-electrons of Ni2+ is taken into account. “ry There are a variety of theoretical derivations of the Raman tensor of porphyrins. A general formulation, which is also valid in the case of strong vibronic coupling, has been suggested by Shelnutt3 and also by Zgierski and Pawlikow~ki.~The authors, in a time-independent theory, calculated the vibronic states of the molecule. The Raman tensor is derived by inserting these wave functions into the Kramers-Heisenberg-Dirac (KHD) dispersion formula. This approach has been especially used for porphyrins in an environment of low symmetry. Unfortunately,these theories are not suitable for fit procedures to the experimentally observed DPDs and REPs. Therefore, Shelnutt3 proposed a simpler formulation of the absorption coefficient and the Raman scattering cross section for highly symmetric porphyrins in the weak coupling limit, which yields analytical expressions for both quantities. In the following, we use this approximation with some modifications. The vibronic Hamiltonian of the molecule is given by
H = H,+ fit= f i e ( i , O ) + fin(&) + Bl(i,e)
(1)
&(F,O) is the electronic Hamiltonian at fixed nuclear positions in the equilibrium configuration. i re resents the electronic coordinates. The nuclear Hamiltonian &&) with the vector of normal coordinates QIcontains the effective potential of the electronic ground state Ig). In the harmonic approximation, fin@) is separated in the basis -of the normal coordinates. Therefore, its eigenfunctions are Ivs) = I~~,)lu~,)...Iv~~>,where u l . . . v ~are the occupation quantum numbers for the N normal vibrations. The frequency of a normal mode i in the ground state is termed Qf. The solution of the SchrBdinger equation for 7f0is the prod_uct of the pure electronic state 11) and the nuclear wave function (v,):
a
err
&‘(F,G)
couples the electronic and nuclear states. In a linear
approximation, it is substituted by Z,(d&’/dQ,) Q,. In the weak coupling limit, the eigenfunction 11,;) of hsc can be calculated in a perturbation series:
11,;) =
II), Im),In), and (p) denote the electronic basis functions. ,);I
and),;I are nuclear wave functions, as is All diagonal terms with vanishing energy denominators are excluded in the sums of eq 3. f,),
Equation 3 exhibits a close analogy to the commonly used adiabatic Hertzberg-Teller expansion. It must be emphasized, however, that it is entirely different.20 Equation 3 represents a nonadiabatic expansion in terms of the crude Born-Oppenheimer adiabatic vibronic states Il$,), which fully contains the HertzbergTeller approach. This can be seen by replacing all energy differences E t ) - E:; of the vibronic states by E$) i.e., the energy difference of the electronic states. Then by use of the closure property in the summation over ,; i, and,; the HertzbergTeller expansion of 11,;) emerges in all orders. Therefore, eq 3 is not a nonadiabatic correction to the Hertzberg-Teller formulation but an entirely different approach. As will be shown in the Discussion section, it contains also contributions from the Dushinsky rotation.21J0
E3,
Equation 3 is not valid if there are modes i j with accidentally degenerate frequencies or frequenciesso close to each other that the coupling energy is not weak compared to their energy distance. Then one has to choose another basis set with linear combinations of ll)l&) and 1 l ) Z j ) . This procedure is described in Appendix A. (;,IQ&,) vanishes unless the vibrational quantum numbers of mode i, vi and f i i , differ by 1 (vi = pi f 1) and all other quantum numbers are equal.
To derive the selection rules of (ml(a&/aQ,)(l) for a Ramanallowed mode i, the electronic states must be considered in detail. The electronic wave functions II), Im), ...for porphyrins in D4h, symmetry are given by the four-orbital model of Gouterman.22 Thegroundstatelg) isan Al,symmetricsingletstatecharacterized by two highest occupied orbitals (HOMOS) lazy) and la,,) with a small energy distance expressed by the parameter d = E.- E,,,. The lowest unoccupied orbital (LUMO)is a degenerate lesry). The dipole-allowed transitions l a d +lesxy) and lesxy) yield the four lowest excited configurations la2,eoxY) and (aluesry),respectively. The transition moments are called R1 for k) laluepxy)and R2 for k) la2,eox,,,). Because all these excited configurationsexhibit E,symmetry and their energies lie close to each other, there is a strong configuration interaction, mixing the two-electron orbitals.
-
-
-
Unger et al.
9958 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 E b
-$!
818:
4.0
E
0
E *&
8
- u, sin 2u
4,'
E E 3.0 'I.
a,'
a,cos2u
2.0
- a, sin 2u
0
1.0
a2g:
0.0
Flpw 2. Absorption spectrum measured in the range from the Q band to the B band. The experimental conditions arc explained in the text.
L p
-b, sin2u
wavenumber in 1000cm" '1s:
In the two-electron picture, the two lowest excited states exhibit E,,(xy) symmetry and are designated as Q and B. They are given by
b2i
b,
008
2u
-b2sin2u b2-2~
1
0 0 a, ca 2v ul' + 4,sin 2u 0 0 11,' 0 , sin 2v a, ca 2u
[" z2 7 "1
01 c a 2 u
B
8 n 0
0
+
0
0
a 2 0
o
0 6 , sin 2u 0
b,
b, sin 2u
0
-b1
0
-b, sin 2~
COS~U
o
CQS~U
0
-b2sin2u 0 0 b2ca2v bZcos2~ 0 0 b, sin 2u
0
1 1
-b, COS 2~
b2 COS 2~
0
b2 sin 2v
0
(7) aI, a2, bl, and b2 are defined as the Herzberg-Teller coupling matrix elements for states with u = 0 (5050 states). a1' is the respective Franck-Condon matrix element. To calculate the absorptioncoefficient e of the transitions from the ground state to the Gouterman states, we use Fermi's golden rule:
The unmixing parameter u is given by u
= 1/2 arctan(6/2A)
(5) where A is the matrix element of configuration interaction between the states )azue,,) and la1,eWJ). The dipole matrix elements MQand Me for the transition into the Q and B states are
lg,a) is the wave function of the ground state and 11,;) an
electronically and vibrationally excited state, where rr; is the corresponding half-width. P denotes the electronic dipole operator. w is the frequency of the absorbed light. Equation 8 accountsfor the dampingeffect by substituting a Lorentzfunction normalized with respect to its area for the energy-conserving &distribution. Ed = 0 is defined as the zero energy. Er; is the energy of ll,;), which is a correction of the energy of ll)[;t). It can be expressed by the excited-state vibrational frequencies
4')
nf:
El; =: El
+ x h Q f ( u ,+ 1/2)
(9)
I
The differential Raman cross section for a mode i is calculated from the KHD formula;*O Since R1n R222 and u 20 is much more extended than that for vz1 and ~ 2 2 . Thus, the reliable part of the data suggests that the DPRs of the Azg lines are not dispersive. Some interesting features of the observed REPs should be mentioned. Shelnutt's theory predicts that all modes should show a Q, intensity stronger than the Q intensity. The REPs displayed in Figure 9 reveal, however, that this is not the case. If one regards the areas of the resonance peaks, it can be shown that
Vibronic Coupling in Ni(I1) Porphine 17
1,9
.
21
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9961
21
m ~.
2.0Ki 0.0
3.0
0.8
0.4 0.2
0.2
0.0
20 4J
0.0 1 .o
0.0 0
*
,
4
I
~
I .o
0-
0.0 1.5 1 .o
0.5
0.0
20
r
17
19
21
23
I
4.0
5.0 2.0
2.5
--
Figure 7. DPDn and REPsof Az, modes. The Raman frequencies are listed in Table 111. For further explanation, scc Figure 6.
n:
Figure 8. DPDs and REPSof Bg modes. The Raman frequencies Q: are listed in Table IV. For further explanation, see Figure 6.
all BI, modm exhibit more intensity in the Q resonance than in the Q, resonance. The same observation was earlier made on Ni(1I) octaethylporphyrin and ferrocytochrome c.13927
0.0 17
19
21
23
0.0 17
19
21
23
Figure 9. DPDs and REPs of B1, modes. The Raman frequencies i# are listed in Table V. For further explanation, see Figure 6.
Comparison of the REPs of v22, V26, and V19 in Figure 7 shows that the half-width rQ8, of the Q, resonance is generally not equal to the width rQ= rpa of the Q resonance. This is not surprising because the relaxation of the Q state can differ if a vibrational mode is excited. Therefore, the vibrational states have different lifetimes. To account for this, we have used mode-specific values of rQl,. The REPs of the AI, modes V6 and YZ exhibit a third maximum at the high-energy side of the Q, position, which is clearly beyond experimental errors. This cannot be described by the first-order approach. It indicates multimode contributions as predicted by earlier theories.9.28 This will be discussed in detail below. First-Order Fit Procedure: The simplest approach to fit the REPS is to use only the part of eq 3 which is linear in the vibronic coupling parameters. The following procedure was employed to obtain the fits. The electronic parameters EQ,Es, rQ,Fa, MQ, and M Bwere obtainedfrom the absorption spectrumby the energy positions of the Q and 3 band, their half-widths, and their integrated areas.
9962
Unger et al.
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
TABLE I: Electronic Parameters Common for All Modes’ EQ 18410 cm-1 ~~~~~~~~
TABLE III: Parameters of A% Modes
~~~
25000 cm-I 200 cm-1 500 cm-I 0.43 X Cm 2.30 X lO-Z9 Cm -1.3’ 0.88 0.48
Ea
rQ
ra MQ Ma V
NQlMS NalMa
fY np
I
I1 I11
rsr
I
a2
0 For definition, see text. Accuracy: A E p = AEa = 30 cm-I, 10% for all other quantities.
I1 111 I I1 I11
rQR
TABLE Ik Parameters (Defined k the Text) of A,. Modes. v6
v9
994 1065 900 1040 I1 880 1020 111 860 1040 rgn 2.4 3.0 83 zil I 213 77 I1 140 101 111 173 77 -34 ai’ I -37 I1 68 -40 I11 83 200 rQR I 230 200 I1 250 200 I11 230
n:
I
v4*
v4
1367 1250 1270 1270 3.2 85 81 95 -37 -38
1376 1300 1300 1300 3.2 159 159 190 -58 -55 -61 270 270 270
-40
280 280 280
v2*
v2
1564 1572 1530 1530 1550 1550 1550 1550 4.0 4.0 52 122 59 136 63 144 -46 -99 -44 -95 -52 -110 270 270 270 270 270 270
a Units in cm-I. I: calculated with first-order wave functions. 11: with third-order wave functions containingfree QQ and 2333 parameters. 111: with eg-egcoupling to nickel. Accuracy: An: = 1 cm-I, An? = 30 cm-1, ArQR = 30 cm-1, vibronic coupling matrix elements 10%.
The unmixing parameter v was obtained by fits to the REPs. For the first-order fits, only the two leading terms of the Raman tensor in eq B4 were used. Close inspection of these terms shows that the intrastate coupling parameter mainly determines the Raman intensity in the 8-and &-band region, because Ms is much larger than MQ. The interstate parameter determines the Raman intensity in the Q- and Q,-band region. For each mode i, the ratio of thesecoupling parameters can therefore be evaluated from the REP by comparison of the Q and Q, resonance intensity with the preresonance intensity of the 3 band. According to eq 7, this ratio is tan 2v for the B1, and Bz, modes. Thus, having determined all electronic parameters we have fitted the REPs of all vibrational modes. For each mode, the excited-state vibrational frequency QF was used as a free parameter and was adjusted to match the resonance position of the REP at EQ+ haQ. Since our data do not cover the 8,region, we assumed Q,8 = Q,d for simplicity. The half-width rQ8, was also used as a line-specific parameter to reproduce the shape of the Q, resonance. The fit to the REPs yields vibronic coupling parameters al, a’l, a2,61, and 62 in units related to the internal standard. To scale them to absolute units, we have utilized the absorption spectrum of NiP shown in Figure 2. The 4, absorption band was fitted by the second and fourth term of eq B2 using the coupling parameters above scaled by a common factor. It is interesting to note that from the knowledge of the REPs in arbitrary units a reliable determination of the coupling parameters in absolute units is possible, if the absorption spectrum is known. This experimentally is an important fact, since determination of the Raman cross section in absolute units is not an easy task. Results of First-Order Fits. The best fits to the data are displayed as solid lines in Figures 6-9. The fit to the Q and Q, bands of the absorption spectrum yields the long dashed line in Figure 14. Table I lists the electronic parameters used for all fits (NQ and Ns are defined in the subsection Extension of the Electronic State Model). The values MQ = 0.43 x 10-29 Cm =
v22
v21
V26
VI9
v19*
1003 950 950 950 2.4 118 91 106 170 290 290
1137 1070 1110 1100 2.6 112 90 109 190 270 270
1315 1180 1220 1210 2.2 497 400 497 240 280 270
1603 1470 1520 1500 6.2 618 513 612 380 340 330
1618 1480 1510 1510 6.2 337 281 339 380 350 330
TABLE W. Parameters of
I I1 I11 rOR
I I1
82
111
I
rQR
11 111
TABLE V
Modes v34
V29
1193 1090 1120 1120 6.4 149 -70 74 170 200 200
1354 1210 1260 1260 3.2 227 -188 233 190 170 190
Parameters of BI. Modes v1s
v17
1002 940 940 940 3.4 204 133 81
1058 1020 lo00 1020 7.O 212 146 58
VI3
v11
v10 ~
fY
QlP
I
I1 111
i-? 81’”’’
bK PQR
I I1 I11 I I1 111 I I1
I11
-
6010 6 170 170 170
-
8610 8 170 170 170
1184 1100 1110 1100 2.6 157 146 11
-
50/150 7 170 270 170
~~
1502 1380 1380 1380 6.8 358 227 88
1648 1580 1580 1610 7.6 493 390 207
15510 9 250 250 250
lOa/O
-
-
9 270 200 200
1.3D and Ms = 2.30 X Cm = 7.0D can be compared with the values for oxyhemoglobinz9(MQ= 0.6 D, Ms = 7.2D).The deviation of MQ is not surprising, because MQ is sensitive to changes in v. Using eq 6, the configuration dipole transition matrix elements RI and Rz were calculated as R1 = 1.36 X Cm = 4.1 D and RZ= 1.90 X lez9 Cm = 5.8 D. This contradicts earlier studies,4*5 where R1and RZwere assumed to be identical. Tables 11-V (values labeled by I) give the vibronic parameters for the fits to the AI,, Azo, and B2# REPs. The vibronic coupling parameters (51,&’, (52, 62, and 61 are given in energies (cm-1) and are written as (51 = a1 ( 61Q&ab). Figures 6-8 show that satisfactory fits are obtained for almost all AI, and All lines and for the B2 mode ~ 2 9 . The REPS of all B1g modes investigated and the B2, mode v34, however, could not be r e p r o d u d (see Figures 8 and 9). Moreover, the REP of the AI# mode V6 depicts another considerable strong maximum at 20 000 cm-l, which is not accounted for by the fits (Figure 6). In the following, we try to rationalize these discrepancies. Multimode Fitting Procedure. The appearance of the Q b resonance (Le., a resonance of a state with two excited vibrations) in the REP of the V6 mode is somewhat surprising, because corresponding maxima in the REPs of the remaining modes are weak or not detectable. On the other hand, the coupling parameters derived from the first-order fits are strong compared to those found in other porphyrins.13.2’ This holds in particular for the interstate parameters a2 of V19 and V26. Thus, one may
Vibronic Coupling in Ni(I1) Porphine
The Journal of Physical Chemistry, Vo1.97, No.39, 1993 9963
:i 2.0
17 19 21 23 Rgure 10. REP of the V.5 mode. The solid line results from fitting procedure 111. The dotted line is calculated by neglecting the vibronic coupling parameters of all other modes. The fit depicted as the dashed line takes into account the contribution of the v19 mode.
expect that these modes provide multimode contributions to the scattering tensor, which must be described by higher order terms of the perturbation theory. Therefore, we used the full third-order wave function of eq 3 to describe the contributions of other vibrations to the scattering amplitude of the particular Raman-active vibration. This yields the cross section given in eq B3. The fits were performed by an iterative procedure. The parameters obtained from the first-order fit to all Raman lines investigated were inserted into eq B3. The REPS were then refitted, varying only the parameters of the considered Raman mode. This yields a new set of parameters which are again scaled to absolute units by fitting the Q, absorption band. These steps were repeated until convergence was achieved. Resultsof Multimode Fits. Figure 10visualizes the multimode contribution to the REPSof the V.5 mode. The dotted line displays the first-order contribution, which solely involves the Raman vibration. It exhibits a strong Q resonance intensity and a much weaker resonance enhancement at EQ ha$. The dashed line emerges if the coupling parameters of the v19 mode are inserted in the higher order terms of eq B4. This gives rise to an additional resonance at EQ ha:,, which causes an upshift of the effective Q, resonance position. Additionally, a small shoulder shows up at the Q2, resonance, Le., at E@ + ha$ ha:,. Finally, the solid line displays the result of the multimode fit including all modes as described above. The superposition of the multimode contribution from all modes causes a stronger Q, intensity and yields a sound reproduction of the observed Q h resonance. It turned out that all the REPS described well by the firstorder theory are also fitted by the multimode procedure. In addition, the multimode approximation yields good fits to the REPs of V6 and v34. The REPS of the B1, modes, however, still cannot be fitted. This discrepancy may be indicative of a failure of the four-orbital model. Extension of the Electronic-State Model. As a consequence, we relaxed its restrictions by introducing new intrastate coupling parameters identical to those employed in earlier studies.4 For each BI, mode, a matrix M is added to the vibronic coupling matrix given by eq 7.
+
F i ?,/
0.3 17 0.2 0.1
0.3
19
* . C, 4
*a,.
....’.
.._......,.
0.4 0.2
.
17
19
21
23
0.4
0.2
0.1
0.0 4.0
0.2 0.0
0.8
+
+
rb,’
0
0
0
1
61’ and bl” describe additional intrastate coupling within the Q and B states, which for some reason operates only for B1, modes. These parameters may be attributed to two-electron matrix elements30 or to interactions of the Q and B states with lower or higher energy states of the central nickel and the porphyrin macrocycle.31
2.0
0.4
0.0
0.0
Figure 11. DPDs and REPs of A,, modes (same data as in Figure). The solid lines result from fitting procedure 11; the dashed lines are calculated by procedure 111. If procedures I1 and I11 yield nearly identical fits, only the fits of I1 are given.
We now refitted the REPsof all B1, modes using themultimode procedure with 61’ and 61’’ as additional free parameters. The solid lines in Figures 11-14 show that this provides a satisfactory reproduction of the REPs. The Q2, resonance of the REP of V6 at 20 000 cm-l is slightly underestimated, but qualitatively the fits account for the third-order contributions displayed by some of the REPs. The corresponding fit to the Q, absorption band yields the solid line in Figure 14. The values labeled by I1 in Tables 11-V are the coupling parameters emerging from this procedure. They are discussed at the end of this section. Large intrastate bl’ parameters, however, are necessary to reproducethe strong Q resonance of B1, modes. The thus-obtained fits to the REPS of the v11 and ~ 1 modes 3 (solid lines in Figure 14) exhibit slight third-order resonances which are not shown by the data. Moreover,it is not yet understood why these parameters should only exist for BI, BEPs.
Unger et al.
9964 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 17
19
21
23
u
I7 I
0.2
0.2
0.0
20
17
19
21
0.0
23
r--l
17 17
19
21
19
21
23
23
1
" I
-.-
I
I
8.0
4.0
2.0
0.0 17
I9
17
I9
~
2oj 0
21
23
21
2?
.
17
I
Q)
I
19
21
20 4
,
17
2?
1
aD
, , , ,I?,
, , z:,,
, Z?, ,
,
17
, , , *l?l*,. 2 ( l , , ,
2?.,
,
I
I ..
6 4.0 * o / , j y g * , f , 2I .o. 0 [ , j ( g , , f r 2.0
l.5u ;;k,i 0.0
17
19
21
0.0
23
17
19
21
23
Figure 12. DPDs and REPs of A28 modes. For further explanation, scc Figure 11.
-.-
I
-'-
I
I
I .o
0.5
a .
0.0 17
19
21
23
0.0
17
19
21
23
Figure 13. DPDs and REPSof B a modes. For further explanation, scc
Figure 11.
Another explanation is that the B1g modes may gain Raman intensityfrom another process. The ground state may vibronically couple to a low-lying B1, symmetric state lblB). Such a state is
Figure 14. DPDs and REPs of B1, modes. For further explanation, see Figure 11.
predicted by INDO calculations32 at an energy about 13 OOO cm-I above the ground state. It results from a transfer of one electron from the highest occupied to the lowest unoccupied nickel orbital, Le., from the AI, symmetric 3d9 to the B1, symmetric 3d9-9. Since the porphyrin ground state transforms like Alg, this yields vibronic coupling only for B1, modes. Considerationof this process in eq B3 requiresthat an additional term pw(u, u = x, y, z)is added to the Raman tensor b,, namely:
Vibronic Coupling in Ni(I1) Porphine
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9965
0.5
--- fitdataI
xxx x
E0 t 0.4
EE
1 0.3
.E
+
(0
8E
0.2
0
g a
= (gl@);l,/dQOlbl,) (b(Q&) as fit parameters. Since &Is is unknown, it was adjusted to an arbitrary value of 4000 cm-1. This yields the dotted lines in Figures 11-14, which are shown only when they deviate from the solid line fits. The short dashed line in Figure 14 represents the corresponding fit to the Q, absorption band. The respective parameters are listed in Tables 11-V and are labeled by 111. The values of NQ,Ne,and bK = (gl(dfZe/dQ,)lble)(GlQ&) listed in Tables I and V wereobtained by normalizing the dipole matrix elements due to N Q ~ Ns2 = Ma2. Comparisonof the Multimode Fitting Procedures I, II, and ItI. The parameters for the AI, modes derived by the fitting prooedures I, 11, and I11 are given in Table 11. Comparison of I, 11, and I11 shows that the excited-state frequencies QQ, Franck-Condon parameters &’,and Q, resonance half-widths l’Q,are only slightly different. This also holds for the Herzberg-Teller parameters 81 of all modes except the V6 mode. This is not surprising because procedure I neglects multimode contributions, which strongly influence the Q and Q, resonance intensities. For the Azgmodes, Table I11 shows that the Herzberg-Teller parameters 82 resulting from procedures I, 11, and I11 exhibit major differences. This results from the large interstate coupling of u19 and Y26, which provides strong third-order contributions to the Q, resonance intensities of all Bl,, B2,, and A?, modes. The Al, vibrations are less affected for symmetry reasons. Comparison is more difficult for the BI, parameters listed in Table V because I, 11, and I11 are based on different parameters. The fits in Figure 9 show that procedure I cannot reproduce the REPs of the Blgmodes. Thus, the parameter values cannot be regarded as reliable. In 11, the ratio of the Q and Q, resonance intensity is determined by the interference between intra- and interstate coupling contributions. This is constructive for the Q and destructive for the Q, resonance, if both parameters exhibit the same sign. In procedure 111,the terms containing bKdescribe constructive interference. It is therefore apparent that procedure I11 yields lower QB parameters 61 than procedure 11. As a consequence, all other vibronic coupling parameters are larger in procedure I11 than in 11, because smaller 61 values give rise to a larger scaling factor in the fit of the Q, absorption band.
0.1
0
0.0
1’s
$0
wavenumber in 1000cm” Figure 15. Absorption spectrum of NiP in the Q- and Qv-bandregion. The data (same as in Figure 2) and the fits resulting from procedures I, 11, and I11 are shown and are assigned in the figure.
the energy denominator of the Q-resonant term is smaller than (Ebls + ha:) in the Qv-resonant term. Therefore, Pw provides larger contributions to the Raman intensity at the Q than at the Q, resonance. This qualitatively shows that by introducing 81 into eq B4 the large Q intensity exhibited by the B1, REPs can be accounted for without violating the four-orbital model. The contribution of P becomes less effective if E b l s is too large. For the above-mentioned Ebl = 13 000cm-l, unreasonably large coupling parameters (gl(dh’/dQ,)blg) must be assumed to reproduce the data. We therefore wonder whether there might be other states of B1, symmetry with lower energy. Indeed, such a Ibl,) state can be found by the following considerations. We have investigated the Ni configurations with lower energy. Two of the 3d orbitals have E, symmetry. The INDO calculations show that these 3d, orbitals ( a = x, y ) are approximately 10 000 cm-1 lower in energy than the e, symmetric LUMOs of the porphyrin A system. In the ground-state configuration, these 3d orbitals are occupied by four electrons. If one electron is transferred to a LUMO,this gives rise to four excited singlet configurations. These new configurations, denoted le&fil,a2,,) and le&’&1,,aZu), transform like AI, + Bl,, and le&&lUa2,,) and lefl’&1,,a~,,) transform like A2, Bw The notation lefl’gyalUa2,,) means that one electron is taken from the nickel orbital e’, to the LUMO e., The fourfold degeneracy of these states is removed by configurational interaction. The former configurations split into an Al, and a B1, symmetric state:
+
bI8)= 1/fi(Iewe’ual,,a2u)
- Ie,e’walua2u))
(14)
Because the matrix element C’of the repulsiveelectronicCoulomb interaction between the two configurations is positive, the energy E b l g of Ibl,) is downshifted by C’compared to the energy of 3d,,. E.1, is upshifted by C’. The remaining configurations split into the corresponding states la2,) and (b2,). From the two states lalo) and Ibl,), only Ibl,) contributes significantly to the Raman tensor due to its lower energy. For the states la2,) and (bz,), the vibroniccoupling elements are small, because due to their electronic configuration they exhibit only a small overlap to the ground state wave function. It is therefore justified to take into account only the Ibl,) state. In a multimode fitting procedure, we introduced the products NQbKandNsbK of the dipole matrix elements NQ= (bl,lP,IQ,), NB = (b@JBB,), and the vibronic coupling matrix element bK
Discussion Contributions to the Scattering Amplitude. In an attempt to fit the REPs of 18 Raman lines, we have utilized a formalism for the Raman cross section which is based on the expansion of the electronic wave functions of the Q and B states in terms of higher vibronic states by a nonadiabatic perturbation approach earlier proposed by Shelnutt.’ In first order, we takeintoaccount vibronic coupling between and within the Q and B manifold. Thecoupling parameters are expressed in accordance with the restrictions imposed by Gouterman’s four-orbital model.’ In a first extension of Shelnutt’s model, the half-widths F d , rQ&, Flea, and Fa, and the frequency of the Raman vibrations in the ground and the excited states (Of, Qp) are allowed to be different. Only a part of the REPs can then be fitted by this model. Therefore, we extended the perturbation approach further by considering multimode contributions to the scattering process by adding perturbation terms of higher order. No additional parameters are then needed, because the same coupling matrix elements are used for the first- and third-order wave functions. In our earlier work, we calculated the multimode Raman tensor by a time-dependent approach (TDR).9 The TDR was applied to NiOEP13 and to ferrocytochrome c.27 Comparison with the approach used in this work shows that the TDR contains doubleresonant terms in fifth order, where one state is involved twice.
9966 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
Similar contributionsare also present in all higher orders leading to multiresonancedenominators. This is a disadvantagebecause the small energy denominators cause the series to converge more slowly. However, by using many-body field methods it is possible to summarize the series of multiresonanceterms to infinite order (comparison of the two approaches will be the subject of further studies). The result shows that the time-independentexpression eq B3 is a more reasonable approximation.
Unger et al. Using Tables 11-V, we obtain values of Q for all Ais, BI,, and Bz, modes less than 0.06, mostly about 0.02. Only for coupling of the Azg modes V26 and v19 is a larger value Q = 0.27 obtained. The contribution @ ;' of Dushinsky rotation to the Raman tensor for mode 1 in the weak coupling limit has been estimated by Siebrand and Zgierskizl as
The multimode approach enabled us to describe the QzV resonances between 20 000 and 21 000 cm-l, which were not accounted for by the first-order approach. The Q resonance intensities of all the Blglines, however, could not be reproduced. Hence, in a third extension of the theory we considered two different contributions: (1) a relaxation of the four-orbital model and (2) vibronic coupling between the ground state and a chargetransfer state. Both attempts yield excellent fits to the data. Some especially interesting fit parameters are the intrastate coupling matrix elements (see eq 7) and the excited-state frequencies s2F. These parameters provide information about the excited-state potential for the nuclear motion, which is not easily availablefrom other experimentalmethods. This potential is characterized by the position of its minimum, which is shifted by SQ, with respect to the ground-state potential and by its force constant, the change of which is reflected by fl?. SQ, can arise either by Franck-Condon shift for AI, vibrations or by JahnTeller effect for BI, and Bz, modes. It is related to the intrastate matrix elements in eq 733 by
(Q.l(ak'/aQ,)IQp) is given by eq 7 in connection with Tables 11-V. This yields, for instance, for the Al, modes
a1 and izl'
are listed in Table 11.
The excited-state frequencies $2: are generally lower than the corresponding frequencies il: in the ground state, indicating weaker force constants for nuclear vibrations in the Q state. The difference is especially large for the modes V6, v4*, V26r 119, v19*, ~ 2 9 and , vI1. An apparent regularity is that these are modes with relatively large interstate coupling parameters. The modes with large SQ, are V6, v4, v4*, v2, and v2*. Because of the small unmking parameter v, 'Q, is generally small for B1, and Bzg modes. One may further ask the question, whether contributions assigned to Dushinsky coupling21.z0are also taken into account. Vibroniccouplingbetween two electronicstates Im) and In) leads to a rotation of the excited states' normal coordinatesof the same symmetry representation with respect to the corresponding set of normal coordinates in the ground state.z0 In the case of two normal coordinates Q1 and Qz, the rotation angle 4 is given by21
ahr
allr
4( m , a e l ~(nbe," ) )
E, and E, are the energies of the electronic states Im) and In), and Of and Slf are the frequencies of the above modes.
Thus, Dushinsky rotation yields Raman intensity of mode 1 in the resonance positions hw = EQ,Jand hw = E Q ,,+ ~ ti@. The above term is in third order of the linear coupling aH'/aQ. Such terms are also contained in our approach (cf. eq B4). We thus conclude that at least in the weak coupling limit our approach contains the effects of Dushinsky rotation. Siebrand and Zgierski have shown that the Dushinsky effect gives also contributions by excitation of two vibrational quanta, giving rise to resonance at the position hw = E Q , + ~ hflf + h@. Such terms are smaller than the terms in eq 18 by a factor (Of - @)/(nf + Q?). This is corroborated in the excitation profiles of the v19 and V26 modes. In spite of comparably large Dushinsky rotation, no resonance in this region is detected. Conformers of NIP. To our surprise, the structural sensitive marker modes VZ, v4, VIO, v19, v29, and v34 appear twice in the spectrum. The two lines of each 'doublet" differ in terms of their intensity. For each "doublet", the REP of the weaker line can be scaled onto that of the stronger one. Moreover, the halfwidths of the corresponding lines are identical in the limit of accuracy (see Tables 11-V). One may explain this finding by assuming that a part of the NiP molecules forms dimers. This can be ruled out because the measured intensities do not depend on the concentration of the sample. Therefore, we assign the above pairs to different conformers C and C* (see Figures 3-5). One may suspect that at least one of the above conformers is nonplanar. Coexistence of planar and nonplanar conformations has recently been established for NiOEP. Czernuscewicz et al.34 and Alden et a1.16 found that its v10 band is rather broad and can be decomposed into two different sublines. The subline at higher energy was assigned to the planar conformation. Highly resolved spectra of NiOEP observed in our laboratory have revealed that other core size marker bands u3 and u19 also exhibit this type of heterogenity.35 Results from recent Raman experimentson Ni(I1) octaalkyltetraphenylporphyrins suggest that the frequencies of the core size marker lines decrease with increasing nonplanarity of the ma~rocycle.~sIn a series of planar octaalkylporphyrins with different metals and different alkylsl5 and in a series of planar protoporphyrins,36 however, these lines are upshifted with increasing core size. Our data on NiP cannot be incorporated into one of these pictures. While v34*, v19*, and VIO* are upshifted with respect to their positions in C, vz, v4, and vz9 appear at the lower frequency
Vibronic Coupling in Ni(1I) Porphine side (Figures 3-5). An interesting regularity, however, should be mentioned. The smaller subline of the modes with a strong contribution of C,-C, stretch is upshifted, whereas the C,-C, and Cp-C, stretching modes show the opposite shift. At present, our data do not allow us to infer the structure of C and C*. This is the subject of further investigation.
Conclusions We have shown that NiP is a suitable model substance to test the Gouterman four-orbital model. The fit parameters to the REPS and DPDs can be determined without ambiguity. In summary, the following statements can be made: The DPRs show no significant deviation from the expected constant values in D4h symmetry. The REPs of all Als, Azs, and Bzg modes can be fitted by a theory, which considers interactions in the framework of Gouterman's four-orbital model and multimode contributions to the Raman tensor. To rationalize the REPSof the B1, modes, however, either additional intrastate coupling or pseudo Jahn-Teller coupling between the ground state and a BI,-type excited state must be taken into account. The fits to the REPs are consistent with the assumption that the vibronic coupling matrix elements are small compared to the energy distance E ~ E Q Strong . coupling only occurs for modes with a small energy distance and relatively large Q.23 coupling parameters such as v10 and v19. We account for this by an extra diagonalization of the respective energy matrix. The unmixing parameter u is small (i.e., 1.3'), showing that the electronic states IQ)and la)result from a nearly 5050 mixing of the lowest excited configurations. This leads to very small Jahn-Teller terms. Another consequence is that the extinction in the Q band is predominantly determined by the transition dipole moment R1 Rz of the Gouterman 5050 state Q O . ~ Franck-Condon terms are generally smaller than the Q-B matrix elements. This implies that the Q, absorption band arises from the coupling to the .23 state rather than to the Q state. Qpis generally smaller than Q:. This indicates a weaker force field in the excited state.
The Journal of Physical Chemistry, Vol. 97, No. 39, 1993 9967
Appendix B. Absorption Coefficient and Raman Crow section In the Theory section, expressionsfor the absorptioncoefficient (eq 8) and for the differential Raman cross section duf/dQ (q 10) were gi_venin terms of the total vibrational molecular wave functionsIl,u). To obtain an expression taking into account thirdorder contributions of the wave functions given by eq 3, this equation has to be inserted into eq 8 to obtain c and into eq 10 to obtain the Raman cross section. The absorption is then described by: e
+
\
+
Acknowledgment. We thank Dipl. Phys. Walter Jentzen and Dipl. Phys. Gerasimos Karvounis for their helpful discussions and for reading the manuscript.
with
Et' = E, + x h Q f ( v f+ 1/2) I
and
EI; = E,
+ ChOf(v, + 1/2) I
Appendix A. Degenerate Perturbation Metbad In the Theory section, it was stated that eq 3 is only valid in the weak coupling limit. Suppose two normal modes i j exist with frequencies 0: and :Q close to each other. The weak coupling limit is only valid if the vibronic interaction energy C of the states IQ&) and IQ,&,) is small compared to h(QfQ:). Otherwise, the degenerated perturbation method must be applied. This method redefines the basis functions using linear combinations of IQ,)l&) and (Qe)lZ,), which only have diagonalvibronic coupling matrix elements. These linear combinations are the eigenvectors of the matrix
Equation 3 is then used with the new basis set.
All terms with a final state of two or more excited vibrations are neglected in eq B2. For convenience, the index g in the harmonic oscillator functions is omitted. It must be emphasized that for each given absorption band all terms in eq B2 are added coherently. However, there is no interference between different absorptionbands. This is because each absorption band correspondsto a fiied value of 1 and v and because the sum over 1 and v is taken outside the square of eq B2. The Raman cross section is given by:
where eq B4 is given in Chart I. The sums run over the Gouterman states defined in q 4 and exclude all terms containing a zero energy denominator.
Unger et al.
9968 The Journal of Physical Chemistry, Vol. 97, No. 39, 1993
CHART I
+
5
L
'3
(En,- h~ - ir,*)(E#)- E$)(E#) - E$)(&
-.
+
L '3 (EI;~) - EE~)(E);O)- E$)(E/; - ho - irl;)(@')
-E$;)
ahf
permutations of indices in -and Q:j i j
aQ
ahJ
permutations of indices in -and Q: j i j
aQ
ahJ
j j i , ijj
-
permutations of indices in -and Q: j i j
aQ
References and Notes (1) Antonini, E.; Brunori, M. Hemoglobin and myoglobin in their reactionr wirh ligands;North Holland Publishing: Amsterdam-London, 1971. (2) Adar, F.In ThePorphyrinr;Dolphin, D., Ed.; Academic Press: New York, 1978;Vol. 3a, pp 167-209. (3) Shelnutt, J. A. J. Chem. Phys. 1981,74,6644-6657. Pawlikowski, M. Chem. Phys. 1982,65,335-367. (4) Zgicrski, M.Z.; (5) Schweitzer-Stenner, R. Q.Rev. Biophys. 1989,22, 381497. (6) CoUms, S.W.; Champion, P. M.; Fitchen, D. B. Chem. Phys. Lett.
1976,40,416420. (7) Debois, A.; Lutz, M.;Banerjee, R. Biochemistry 1978,18, 15101518. (8) Oertling, W. A,; Salehi, A.; Chung, Y. C.; Leroi, G. E.;Chang, C. K.; Babcock, G. T.J. Phys. Chem. 1987,91,5887-5898. (9) Schweitzer-Stenner. R.; Dreybrodt, W. J. Raman Spectrosc. 1985, 16, 111-123. (10) Loudon, R. Quantum theory of light; Claredon Press: Oxford, 1979. (1 1) Wedekind, D.; Schweitzer-Stenner, R.; Dreybrodt, W. Biochim. Bioohvs. Acta 1985.830..~~ 224-232. ili) Schweitzer-Stenner, R.; Wedekind, D.; Dreybrodt, W. Biophys. J. ~
0) - E@)) pa,
~
1986,49,1077-1088.
(13) Bobinger, U.;Schweitzer-Stenner. R.; Dreybrodt, W. J.Phys. Chem.
1991,95,7625-7635. (14) Stichternath, A.; Schweitzer-Stenner, R.; Dreybrodt, W.; Mak, R. S.W.;Li, X.Y.;Sparks. L. D.; Shelnutt, J. A.; Medforth, C. J.; Smith. K. M. J. Phys. Chem. 1993,97,3701-3708. (15) Shelnutt,J.A.;Medforth,C. J.;Baker,M.D.;Barkiga,K.M.;Smith, K. M. J. Am. Chem. Soc. 1991.40774087. (16) Alden, R. G.;Crawford;B. A.; Doolen, R.; Ondrias, M. R.; Shelnutt, J. A. J. Am. Chem. Soc. 1989,111,2070-2072. (17) Shelnutt, J. A.; Majumdcr, S. A.; Sparks, L. D.; Hobb, J. D.; Medforth, C. J.; Senge, M. 0.;Smith, K.M.; Miura, M.; Luo, L.; Quirke, J. M. E. J. Raman Specrrosc. 1992,23,523-529.
j j i , ijj
+
-
j j i , ijj
(18) Shelnutt, J. A,; Ortiz, V. J. Phys. Chem. 1985,89,47334739. (19) Spiro, T.G.; Strekas, T.C. J. Am. Chem. Soc. 1974,96,338-345. (20) Fischer, G. Vibronic coupling, Academic Press: London, 1984. (21) Siebrandt, W.; Zgienki, M. Z . Chem. Phys. b i t . 1979,62,3-8. (22) Gouterman, M. J. Chem. Phys. 1959,30, 1139-1161. (23) Lacy, W. B.; Rowlen, K. L.; Hams, J. M. Appl. Spectrosc. 1991,45, 1598-1603. (24) Stichtemath, A. Diploma Thesis, Bremen, 1989. (25) Schrader, B.; Mcycr, W. RamanlIR-Atlas organ. Verbindungen; Verlag Chemic: Weinheim, 1975. (26) Li, X . Y.; Czernuszewicz, S.;Kincaid, J. R.; Su, Y. 0.; Spiro, T.G. J. Phys. Chem. 1990,94,3147. (27) Schweitzer-Stenner. R.;Bobinger, U.; Dreybrodt, W. J. Raman spectrosc. 1991,22, 65-78. (28) Shelnutt, J. A.; O'Shea, D. C. J. Chem. Phys. 1978,69,5361-5374. (29) Hsu, M.C.; Woody, R. W. J. Am. Chem. Soc. 1971,93,3515-3525. (30) Curtis, W. C.; Weigang, 0. E. J. Chem. Phys. 1975,63,2135-2143. (31) Weiss, C.; Kobayashi, H.; Goutennan, M. J. Mol. Spccrrosc. 1965, 16,415450. (32) Courtney, S.H.; Jedju, T.M.; Friedman,J. M.; Rothberg, L.; Alden, R. G.; Park, M. S.;Ondnas, M. R. J. Opt. Soc. Am. 1990, 7, 1610-1614. (33) Garouo, M.; Galluui, F. J. Chem. Phys. 1975,64,1720-1723. (34) Czcmuszcwicz, R. S.;Li, X. Y.; Spiro, T. G. J. Am. Chem. Soc. 1989,111, 7024-7031. (35) Jentzen, W.; Dreybrodt, W.; Schweitzer-Stenner, R. Blophys.J. 1993, 64,A155. (36) Choi. S.;Spiro, T.G.; Langry, K.C.; Smith, K. M.; Budd. D. L.; La Mar, G. N. J. Am. Chem. Soc. 1982,104,43454351.
