J . Phys. Chem. 1987, 91, 5885-5887 'E
I
1
1500
loo0
I
500
RAMAN SHIFT (cm-'
Figure 7. (A) Raman spectrum of hydroquinone radical cation obtained by the method illustrated in Figure 6. The spectrum is an average of -6000 experiments. (B) Spectrum of p-benzosemiquinoneradical anion excited at 430 nm (of ref 8) given for comparison.
the spectrum of the radical cation after subtraction of the pbenzosemiquinone contribution from the observed recording. Since the radical cation is present to only -20%, this spectrum is relatively weak and noisy so that except for the strong line at 1644 cm-l and a modestly intense line at 1172 cm-I the remaining CO features are very weak. We note, in particular, that the stretching mode, which is observed with fair intensity at 1435 cm-' in the spectrum of p-benzosemiquinone anion and at 15 1 1 cm-l in that of p-benzosemiquinone, is not readily apparent in Figure 6A. The very weak feature at 1426 cm-' in the difference spectrum is attributed to this mode. The spectrum of the radical anion is given in Figure 7B for comparison. It is seen that the lines of the radical cation at 1644, 1172, and 474 cm-I correspond
5885
closely to lines of the radical anion which have been definitively They are assigned to the vga, vga, and v~~modes of the similarly assigned in the present case. There is also a very weak feature at 1535 cm-I which is tentatively assigned to the 6(OH) mode. One notes in Figure 7 that the vga band of the radical cation is somewhat broader than that of the radical anion. This difference very likely reflects a slight difference between the cis and trans configurations of the OH groups such as has been attributed to similar broadening observed in the case of the 1,4-dimethoxybenzene radical ~ a t i o n . ~ ~ , ~ ~ It is clear from the spectra of Figure 7 that the structural features of the radical cation and anion are very similar. This aspect is also manifest in the ESR parameters which show the unpaired spin populations on the ring to be about the same. This similarity is expected since these radicals are isoelectronic with both having nine T electron systems and of almost similar symm e t r ~ . ~ The ' principal difference in the Raman spectra is in the lack of appreciable resonance enhancement of the mode in the radical cation. Apparently, the nature of the C O bonds changes relatively little on electronic excitation of the radical cation. The p-benzosemiquinone radical is, of course, also isoelectronic but is of lower symmetry2?and therefore manifests a larger number of lines in the resonance Raman spectrum. In particular, the visa CH bending motion, which is symmetry-forbidden in the radical cation and anion, becomes very apparent in the neutral radical. The lower symmetry of p-benzosemiquinone also results in an increase in the double-bond character of the C O bond and a corresponding increase in the v~~ frequency. Registry No. HOC6H,0., 3225-30-7; hydroquinone radical cation, 34507-04-5; D,, 7782-39-0. (25) Hester, R. E. In Advances in Infrared and Raman Spciroscopy; Wiley: New York, 1976;Vol. 4, p 317. (26) Ernstbrunner, E.;Girling, R. B.; Grossman, W. E. L.; Hester, R. E. J . Chem. Soc., Perkin Trans. 2 1978, 177. (27) Thep-benzosemiquinone anion radical belongs to the DIhpoint group (ref 5). The trans isomer of hydroquinone radical cation has c,h symmetry, but the Raman spectrum shows that its electronic and vibrational states can still be classified on the basis of the D2hsymmetry. The p-benzosemiquinone radical spectrum is consistent with the approximate C, point group symmetry.
WKB Edge-Effect Correctlons to Numerical Wave Functions for Asymmetric Potentials P. ZdmeEnik, P. Simon,* and L. Valko Department of Physical Chemistry, Faculty of Chemical Technology, Slovak Technical University, (3-812 37 Bratislava. Czechoslovakia (Received: April 27, 1987)
A correction to a previously reported (Le Roy, Sprague, and Williams) Wentzel-Kramers-Brillouin (WKB) "edge-effect" corrections to the phase and amplitude of numerical free particle wave functions is derived and tested. In tunneling calculations for asymmetric potential barriers, the present approach gives results coinciding with exact ones.
1. Introduction For most one-dimensional potential energy barriers, the quantum mechanical tunneling probability must be calculated numerically, since exact closed form solutions to the corresponding Schrodinger equation are not known. A variety of numerical procedures for performing such calculations have been and it has been ~ h o w n that ~ . ~the accuracy achieved depends on (1) Belford, G. B.; Kuppermann, A,; Phipps, T. E. Phys. Reu. 1962, 128, 524.
(2) Le Roy, R. J.; Quickert, K. A,; Le Roy, D. J. Trans. Faraday SOC. 1970, 66, 2997. (3) Le Roy, R. J.; Sprague, E. D.; Williams, F. J . Phys. Chem. 1972, 76, 546.
(4) Truhlar, D. G.; Kuppermann, A. J . Am. Chem. Soc. 1971, 93, 1840. (5) Wyatt, R. E. J . Chem. Phys. 1969, 51, 3489.
0022-3654/87/2091-5885$01.50/0
the integration mesh and on how far the integration is required to propagate into the asymptotic regions as the potential smoothly approaches its asymptotes. In order to accelerate the convergence of such calculations, Le Roy et aL3 proposed the use of a first-order WKB correction to the phase and amplitude associated with the numerical wave function at the boundaries of the numerical integration interval. The present communication points out that while their phase and amplitude correction expressions3are correct for symmetric potential barriers for which the potential approaches zero at the asymptotes, they are not valid in general. A correct generalized phase and amplitude correction expression is presented, and its utility is demonstrated by trial calculations for Eckart6 barriers. ( 6 ) Eckart, C. Phys. Reu. 1930, 35, 1303
0 1987 American Chemical Society
5886 The Journal of Physical Chemistry, Vol. 91, No. 23, 1987
TABLE I: Transmission Probabilities for Eckart’s Function Calculated by Closed Formulae and by Numerical Integration ( V , = 41.86 kJ/mol, JI = 1.45 A, m = 0.48339 emu) A V = -25.1 kJ/mol 1.0694 X 10” 5.3899 X 10“ lo4 1.1572 X 3.6737 X 7.0543 X 0.5374 0.9345 0.9933
0.10 0.25 0.40 0.60 0.80 1.oo 1.20 1.40
1.0694 5.3899 1.1572 3.6737 7.0543 0.5374 0.9345 0.9934
0.10 0.25 0.40 0.60 0.80 1.oo 1.20 1.40
2.6365 X 5.5799 x 10-7 2.6469 X 1.6624 X lo-’ 5.1844 X lo-* 0.5424 0.9503 0.9959
0.80 1.oo 1.2 1.4
A V = +25.1 kJ/mol 1.7491 X 1.7491 X 0.5519 0.5519 0.9751 0.9751 0.9988 0.9988
X X X X X
AV= 0 2.6365 X 5.5799 x 10-7 2.6469 X 1.6624 X 5.1844 X 0.5424 0.9503 0.9959
1.6229 X 6.7206 X 10” 2.3067 X 3.7308 X lo-’ 9.7750 X lo-’ 0.7443 1.0195
ZiimeEnIk et al. obtain a given level of accuracy. Le Roy et aL3 addressed the latter question by proposing the use of WKB corrections to the phase and amplitude of the wave functions at the ends of the range of numerical integration, y+ and y-. They pointed out that inclusion of these corrections transforms the wave function of eq 2 into the form
+b)= [ai/a(j)] l P [ ~ , * ” e ~ [ ~ ~ y + 6 *+C ~,*“e-i[ai~y+6iCv)l] v)l (4) where Ari” and A,*” are the actual asymptotic amplitudes, and the phase corrections 6&) are defined as
2.6365 X 5.5799 x 10-7 2.6469 X 1.6624 X 5.1844 X 0.5424 0.9503 0.9959
However, their final expressions for these phase corrections (eq A8 and subsequent expressions in ref 3) are valid only for potential barriers which approach zero at both asymptotes. In general, the integrands of eq 5 and 6 may be expanded as ai -
cub’) = ai
- (aiz- B [ V b ’ )- V ( k t . ~ ) ] ’ / ~ ()7 )
5.3162 X 0.4326 0.6915 0.7480
Closed formulae. Our results. Original program.
2. The WKB Phase and Amplitude Corrections Following ref 2 and 3, the Schrodinger equation governing the one-dimensional motion of particles of mass m impinging on a potential barrier V(x) may be written in the dimensionless form2v3
wherey = x / a , E = E/Vo,V b ) = V(x)/Vo,B = 2mV&/h2, and the (in principle arbitrary) energy and length scaling factors V, and a are set equal to the characteristic height and width parameters for the barrier in question. In the barrier penetration + m and problem, the potential approaches asymptotes as x -m, and in these limits the wave function becomes a linear combination of plane waves
-
+b)= Aretmu+ Aie-IaQ
where the approximation of truncating the binomial expansion in the third line of this equation clearly becomes increasingly accurate with increasing lyl. The final result here has a rather different form than the corresponding integrand of eq A8 of ref 3, ( 4 2 )[ V ( j ? / E ] . While the two become identical for the case of a symmetric barrier for which F ( f m ) = 0, eq A8 of ref 3 is not correct for the more general case of asymmetric potential barriers.* In conclusion, therefore, the correct general expressions for the WKB edge-effect phase corrections are
(2)
where A, and Ai are the amplitudes of plane waves moving to the right (increasing y ) and left (decreasing y ) , respectively, and ai = ( B [ E- v(*a)])1/2. Barrier transmission probabilities are then obtained by comparing the asymptotic amplitudes A, and A , to the left and right of the barrier.’-5 For most smooth barriers, the wave function behavior of eq 2 is only achieved in the limits y f m where the potential becomes perfectly flat. In practical numerical calculations, however, it is necessary to truncate the range of integration by assuming that eq 2 becomes exact at some finite (reduced) distances y - and y + lying, respectively, to the left and right of the barrier maximum. While this truncation introduces some error, it was pointed out in ref 2 that its magnitude could be systematically controlled by defining the cutoff distances y , and y- by the requirement that for some specified value of Z (say Z = the WKB convergence criterion’
and the corresponding amplitude corrections are given as A * b ) = [ai/.Cv)1”2
(10)
+
t[.b)l-*d a b ) / d y l 5 z
(3)
be satisfied for all y Iy- and y 1 y+. Use of the criterion of eq 3 increases computational efficiency by providing a way of assuring that some consistent level of accuracy is achieved. However, it does not in itself reduce the integration interval b-,y+] which must be spanned in order to (7) See, e&: Merzbacher, E. Quantum Mechanics, Wiley: New York, 1961; Chapter 7.
3. Test of the WKB Edge-Effect Correction To illustrate the utility of the present procedure, it has been used to calculate barrier transmission probabilities for various asymmetric Eckart potentials, one of the few cases for which the exact quantal barrier permeability is known in closed f ~ r m . ~The .~ barriers used all had a height (relative to the asymptote V(-m) = 0) of V, = 41.86 kJ/mol, and a characteristic width ~ a r a m e t e r ~ . ~ of a = 1.45 A, while the barrier height asymmetry AVwas in turn set equal to -25.1,0.0, and +25.1 kJ/mol. The range of numerical integration b-,y+] was then determined by using eq 3 with Z = 10-5.
For a range of reduced energies E , Table I compares the exact quantal tunneling probabilities (column 2) with values calculated numerically by using either the present edge-effect correction (8) R. J. Le Roy has expressed agreement with this conclusion (private communication, 1986). (9) Bell, R. P. The Tunnel Effect in Chemistry; Chapman and Hall: London, 1980.
J. Phys. Chem. 1987,91, 5887-5898 (column 3) or the phase correction of ref 3 (column 4). It is evident that the phase correction expression of ref 3 is valid for the case AV = 0, but not when AV # 0. In conclusion, therefore, we have confirmed that use of the type of WKB edge-effect correction originally proposed by Le Roy et
5887
ala3can greatly accelerate the convergence of numerical tunneling probability calculations for potentials with long-range tails. However, for asymmetric barriers, the expression for the WKB phase correction given in ref 3 is in error and should be replaced by the present more general eq 8 and 9.
Vibratlonal, Electronic, and Structural Properties of Cobalt, Copper, and Zinc Octaethylporphyrln ?r Cation Radicals W. Anthony Oertling, Asaad Salehi, Young C. Chung? George E. Leroi, Chi K. Chang,* and Gerald T. Babcock* Department of Chemistry and Michigan State University Shared Laser Laboratory, Michigan State University, East Lansing, Michigan 48824-1322 (Received: May 26, 1987)
Optical and resonance Raman (RR) spectroscopic characterization of the oxidation products of several metallwtaethylporphyrins has been carried out. One-electron oxidation of the macrocycle yields a series of divalent metal substituted octaethylporphyrin a cation radicals, M"OEP'+C104- (M = Ni, Co, Cu, and Zn).The porphyrin core vibrational frequencies above 1450 cm-I of these complexes are described by a linear function of center to pyrrole nitrogen distances. A comparison of these structural correlations and of Raman depolarization ratios with those of the parent MOEP compounds is used to establish vibrational mode assignments for the cations. The agreement between the correlation parameters of the MOEP and MOEP'+C104suggests similar potential energy distributions in the normal modes of both species. We find that the frequenciesof the stretching modes with predominantly CbCb character increase, whereas those with C,C, and C,N character decrease in the cation radical relative to the neutral metalloporphyrin. Similar trends in Soret band maxima for the a cation radicals and their parent compounds reflect changes in the relative energy of the aZu(a)orbital. These structural correlations seem to be essentially insensitive to 2A2uvs 2Aluradical designation. With the vibrational mode correlations as a guide to evaluation of porphyrin core geometry, we have carried out a detailed analysis of the oxidation products of Co"0EP and we suggest structures for the two-electron-oxidized species Co"'OEP'+2C104- and Co"'OEP'+2Br-. Differences in the high-frequency vibrations of these two compounds are interpreted in terms of expansion or possible ruffling of the porphyrin core in the latter relative to the former compound. RR excitation in the 600-680-nm region of the CdnOEP+2Br- absorptions shows a lack of anomalously polarized scattering and produces spectra similar to those obtained with near-UV excitation. This suggests the absence of strong Herzberg-Teller coupling between the excited electronic states of this u cation radical.
Introduction Oxidized states of metalloporphyrins (MP) participate in the redox chemistry of a variety of biological structures including light-harvesting photosynthetic systems and heme proteins. Oxidation of the M P may occur at the central metal, at the porphyrin ligand, or a t both locations. The latter, doubly oxidized metalloporphyrin-structures have been implicated in the catalytic cycles of heme peroxidases, P450 monooxygenases, and catalases.' Because of the extensive A system, oxidation at the porphyrin yields a delocalized A cation radial; indeed, in the case of the special pair bacteriochlorophyll a dimer in the photosynthetic bacterial reaction center, radical character may be shared between two porphyrin rings.2 Characterization of metalloporphyrin A cation radicals is thus of interest because of their widespread Occurrence in nature and because of the often unusual chemistry in which they participate. Both chemical and electrochemical preparations of a variety of metallooctaethylporphyrin (MOEP) and metallotetraphenylporphyrin (MTPP) A cation radicals have been reported.M These oxidized complexes have been characterized principally by optical absorption,' EPR? MCD? NMR,losll and IRIZspectroscopies and X-ray cry~tallography.'~ Although resonance Raman (RR) spectroscopy has been applied to a few metalloporphyrin A cation radical (MP'+) systems, these studies are limited in scope for various reasons. Attempts to measure R R scattering from the oxoferrylporphyrin A cation radical of horseradish peroxidase and compound I (HRP-I) are complicated by photochemi~try,'~.'~ special precautions are required to obtain its spectrum.I6 Analysis of the R R scattering from the bacteriochlorophyll cation radical Present address: Department of Chemistry, Northeastern University,
Boston, MA 021 15.
is obscured by the complexity of the parent system." The work of Yamaguchi et a1.'* on MTPP'+ complexes is not strictly relevant (1) (a) Frew, J. E.; Jones, P. In Advances in Inorganic and Bioinorganic Mechanism; Academic: New York, 1984; Vol. 3, pp 175-215. (b) Hewson, W. D.; Hager, L: P. In The Porphyrins; Dolphin, D., Ed.; Academic: New York, 1979; Vol. 4, pp 295-332. (2) (a) Norris, J. R.;Sheer, H.; Katz, J. J. in The Porphyrins; Dolphin, D., Ed.; Academic: New York, 1979; Vol. 4, pp 159-195. (b) Lubitz, W.; Lendzian, F.; Plato, M.; Mabius, K.; Triinkle, E. Springer Ser. Chem. Phys. 1985, 42, 164-173.
(3) Fuhrhop, J. H. Srrucr. Bonding (Berlin) 1974, 18, 1-67. (4) Dolphin, D.; Mulijiani, Z.; Rousseau, K.; Borg, D. C.; Fajer, J.; Felton, R. H. Ann. N.Y. Acad. Sci. 1973, 206, 177-200. (5) Wolberg, A.; Manassen, J. J. Am. Chem. SOC.1970,92,2982-2991. (6) Phillippi, M. A.; Goff, H. M. J . Am. Chem. SOC. 1982, 104, 6026-6034. (7) Carnieri, N.; Harriman, A. Znorg. Chim. Acta 1982, 62, 103-107. (8) (a) Fajer, J.; Davis, M. S. in The Porphyrins; Dolphin, D., Ed.; Academic: New York, 1979; Vol. 4, pp 197-256. (b) Fajer, J.; Borg, D. C.; Forman, A.; Felton, R. H.; Vegh, L.; Dolphin, D. Ann. N.Y. Acad. Sci. 1973, 206, 349-364. (9) (a) Browett, W. R.; Stillman, M . J. Znorg. Chim. Acta 1981,49,69-77. (b) Browett, W. R.;Stillman, M. J. Biochim. Biophys. Acta 1981,660, 1-7. (10) (a) Godziela, G.M.; Goff, H. M. J. Am. Chem. SOC.1986, 108, 2237-2243. (b) Goff, H. M.;Phillippi, M. A. J. Am. Chem. SOC.1983, 105, 7567-757 1. (1 1) Morishima, I.; Takamuki, Y.; Shiro, Y. J. Am. Chem. Soc. 1984, 106, 7666-7672. (12) Shimomura, E. T.; Phillippi, M. A.; Goff, H. M. Scholz, W. F.; Reed, C. A. J. Am. Chem. SOC.1981, 103, 6778-6780. (13) (a) Barkigia, K. M.; Spaulding, L. D.; Fajer, J. Znorg. Chem. 1983, Bertrand, J. A.; Felton, R. 22, 349-351. (b) Spaulding, L. D.; Eller, P. G.; H. J. Am. Chem. SOC.1974, 96, 982-987. (14) Nadezhdin, A. D.; Dunford, H. B. Phorochem. Photobiol. 1979,29, 899-903. (15) (a) Teraoka, J.; Ogura, T.; Kitagawa, T. J . Am. Chem. SOC.1982, 104, 7354-7356. (b) Van Wort, H. E.; Zimmer, J. J. Am. Chem. SOC.1985, 107, 3319-3381.
0022-3654/87/2091-5887$01.50/0 0 1987 American Chemical Society
