1062
SOTES
tribution function of the acceptor molecules, which is not a plausible procedure. Roznian6 indeed applies the proper method, which involves some extended calculations. Remarkably enough, he arrives a t Forster’s final results. Galanin hiniself5J treats the problem by a different approach, averaging already in the differential equation (1). The resulting formula is identical with that of Forster. The question now arises: how is it that Forster’s method, which is open to the above criticism, still yields the correct result? It is the purpose of this communication to clarify this question. Following Galanin’s a p p r o a ~ hthe , ~ decay with time of the number of excited donor molecules n(t) is given by
(say, by some additional mechanism) the integration of the corresponding new equation ( 7 ) would have yielded an integral, not necessarily equal to that obtained by Forster’s approach. I t thus seems, that the proper result obtained by Forster nevertheless does not justify his statistical reasoning. Acknowledgment. The author wishes to thank Dr.
A. Weinreb for helpful discussions. (6) I. M. Rozrnan, Opt. i Spektroakopiya, 4, 536 (1958). (7) M. D. Galanin, Zh. Ekaperim. i Teor. Fiz., 28, 485 (1955).
The Temperature Dependence of the cis-trans Photoisomerization of Azo Compounds : Theoretical Considerations
where f(R) = 1/To(Ro/R)6and N(R,t) is the number of acceptor niolecules whose distance from an excited donor molecule at time t is R. It is assumed that an acceptor molecule, once excited, can exert no further quenching eftict on a donor molecule. Therefore, N(R,t) decays exponentially with time with a rate determined by f(R).
N ( R , ~= ) Noe-f(R)l Substituting (6) in ( 5 ) he obtains
k?d!dt
=
--{:
+ No
lm
e-“”‘f(R)w(R)dR) n(t) (7)
In integrating this equation, one may change the order of integration with respect to the variables R and t. The result is
This relation developed by Galanin, however, contains an integral which is identical with that of Flirster (eq. 3), and which, after proper approximations, leads to an identical result. The fact that Galanin’s correct approach coincides niatheniatically with Forster’s incorrect approach is a “coincidence,” due to the assumption that the acceptor molecules decay exponentially with the coefficient f(R). This causes f(R) to appear in eq. 7 both as a fartor and as an exponent, and this again causes the result upon integration to be equal to that of Forster. Had we assunled N(R,t) to decrease at a differerit rate, The J o l ~ f ’ 7 of d Physacul Chrmaatry
by David R. Kearns Department of Chemistry, University of California, Riverside, California (Received January 18. 1964)
Recent of the temperature and wave length sensitivity of the cis e trans photoisomerization of aromatic azo compounds have revealed that (i) trans, and, &, for the quantum yields, &, for the cis the trans cis photoisomerizations decrease with decreasing temperature, indicating that an activation energy is required for photoisomerization; (ii) the activation energy for the cis + trans transformation is smaller than for the trans + cis transformation; is less than unity a t all tem(iii) the sum of 4c and peratures studied; and (iv) excitation of an azo molecule to its lowest (n-r*) singlet state results in a greater photoisomerization quantum yield than does excitation to the higher lying (r-r*) state. In an attempt to understand some of these observations we have applied molecular orbital theory (with overlap included) to a calculation of the dependence of the ground state and of the excited (n-r*) and (r-r*) state energies of simple aliphatic azo compounds on the angle of rotation about the N-N bond axis. In this note we present the results of these calculations and show how they offer a possible theoretical explana-
-
-
(1) P. B. Birnbaurn and D. W. G. Syles, Trans. Faraday ~ o c . 50, , 1192 (1954). (2) G . Zirnmerman, L. Chow, and E. Paik, J . A m . Chem. SOC.,8 0 , 3525 (1955).
(3) E. Fischer, ibid., 82, 3249 (1960). (4) J. Malkin and E. Fischer. J . Phya. Chem., 66, 2482 (1962).
NOTES
tion of the experimental observations on aromatic azo compounds. According to MO theory, the two nitrogen 2P, atomic orbitals pea and psb of a planar aliphatic azo molecule can be combined to form a set of molecular orbitals A* = (Pz, f P Z b ) / d 2 ( 1 f SO,,) with corresponding energies er* = ( a , f popp)/(l f Sopp) where SO,, =: (Pz,(i)lPzb(i)) = 0.22, a, = (Pza(i)/ X(i)lPz8(i)? = 0, bopp = (PEa(i)lX(i)lPEb(i)?= -20 kK.,and X(i) is some effective one-electron Hamiltonian. Values for these integrals, as well as those given below, were obtained previously from spectral s t ~ d i e s . ~ -In ~ a similar fashion, the nitrogen sp2 hybridized lone-pair atomic orbitals, na and nb, may be combined to _ form molecular orbitals N* = (n,, f ~ _ n b ) / d ( l f Sonn)2with corresponding energies enk = cy, f Ponn/(l f Sonn), where Sonn= (n.(i)/nb(i)) = 0.11, an = (n,,(i)lX(i)/na(i)) = -8 kK.,and Bonn = (na(i)lX(i)1nb(i)? = -6 kK. For the planar isomers the n and pz and the K* and A& orbitals are orthogonal. For nonplanar isomers, however, the n and p orbitals on adjacent centers are not orthogonal, and consequently the X+’ and A_‘ orbitals and the N-‘ and A+’ orbitals will overlap, although N+’ and N-’ remain orthogonal to A+’ and A_‘, respectively. (Primes are used here to indicate that the n and p orbitals on one center have been rotated by some angle e about the N-N axis.) If the normalized molecular orbitals N+’, N-’, A+ ‘, and A_‘ are chosen as the basis functions for the calculation, the problem of determining orbital energies is reduced to one of solving the following two 2 X 2 secular determinants
1063
another. A value of -10 kK. was assigned to pnp on the basis of overlap considerations. Orbital energies, e’, were obtained by solving the secular determinants for various angles 0 and these results are shown in Figure la. The angular dependence of the energies of the ground state and the excited (n-r*) and ( P A * ) states were obtained by summing the appropriate orbital energies, and these results are presented in Figure l b . As the calculations were based on a oneelectron approximation, excited singlet and triplet states of the same configuration are not distinguished. This can easily be rectified for the (n-r*) states by introducing the singlet-triplet splitting factor, which according to AS110 theory will be determined primarily by one-center exchange integrals involving n and the p orbitals on the same center.’ By the use of the wave functions which we obtain from our calculations for the planar and perpendicular isomers and by evaluation of the appropriate integrals, the singlet-triplet splitting was found to change from 5 l o 16.6 kK. in going from a planar to the perpendicular isomer. Inclusion of this factor in the calculation of the state energies leads to the results given by the dotted curves in Figure l b . Since we would like to use the above results to discuss the photoisomerization of the aromatic azo compounds, it is necessary to consider how the aromatic substituents will modify our initial results. I n the first place, as a result of the greater electron delocalizatioii in the aromatic than in the aliphatic azo compounds, aromatic substituents will reduce the antibonding character of the x* orbital involved in the n + x* transition, without a corresponding reduction in the ground-(Don,
+
- SOnpe) sin e
d(1 SO,, cos e)(i -
sonn cos e)
- PonnCOS e 1 - SOnn COS e
=
0.
an
and d ( 1 - SO,,COS an
1
where Pnp = (na(i)/X(i)iPYb(i))= -10 kK., and SOnp = (na(i)lPyb(i))= 0.18. In writing these expressions we made use of the fact, that the overlap between n orbitals on adjacent centers, and naturally between 11 and p orbitals, is primarily of the Atype, the u cont,ributions to the overlap cancelling one
+
POnn COS
0
COS
e
+ sonn
e)(i + #ann COS e)
=o
-
state bond order. This alone would increase the energies of all the excited states of the perpendicular ( 5 ) S. F Mason, J . Chem. Soe., 1246 (1957); 493 (1962). (6) D. It. Kearns and .\I A. . El Bayoumi, J . (‘hem 1’hy8., 38, 1508 (1963). (7) L. Goodman and It. W. Harrcll, ihid., 3 0 , 11:31 (1959)
XOTES
1064
I
I
-
I
I
I
20
Y
X
u)
g IO (3 a
W
z w o -I 4
+
N-
2?,
a 0-10 -20
TI i.
la
50
c
40
Y X
u)
g 30
$2W
z
W W l-
20
a
+ VI IO
0
ANGLE OF ROTATION Figure 1. ( a ) The variation of the orbital energies with angle of rotation about the N-N bond for a simple, aliphatic azo molecule. The N, and ?T* designations are appropriate only for the planar isomers. ( b ) The angular dependence of the ground state and the lowest (n-r*) and ( ?T-?T*)states of a simple aliphatic azo molecule: solid line, oneelectron approximation : dashed line, one-electron results corrected for the effect of the singlet-triplet splitting.
isomer relative to those for the planar isomers. Secondly, as a result of steric interactions between the aromatic groups, trans isomers will in general be more stable than cis isomers. This effect is easily incorporated into our results by raising the energy of all of the electronic states of the cis isomer by some amount AEsterlc with respect to the trans isomer. If we incorporate these two modifications into the results obtained for the aliphatic azo compounds, then they should be at least qualitatively useful for discussioii of thc photoisonierization of the aromatic azo compounds. 13:xatiiiiiation of the dotted curves of Figure l b indivat es that for a iiiolccule in it s lowest (n-r*) singlet
?'/IC
. r n w n n l of I'hysical Chemisfru
state there will be an energy barrier to rotation from the planar to the perpendicular configuration. This barrier is due partly to the angular dependence of the orbital energies and primarily to the angular dependence of the singlet,-triplet splitting. If a planar isomer undergoes intersystem crossing from the lowest (n-r*) singlet state to the lowest (n-r*) triplet state, which is also predicted to be the lowest excited state, we observe that no barrier to rotation from the planar to the perpendicular configuration is expected. As the curves in Figure l b for the (r-r*) states indicate, a rather large activation energy is required for a molecule to be converted from a planar to the perpendicular configuration. Since the singlet-triplet splitting of the (r-r*) states is large for both the planar and the perpendicular isomers, inclusion of this factor in the calculations would not qualitatively alter the orbital results. On the basis of the above considerations we conclude that, when an azo molecule is in its lowest (r-r*) singlet or triplet state or lowest (n-r*) singlet state, an activation energy will be required for isomerization. No activation energy should be required when the molecule is in its lowest (n-r*) triplet state. These results would appear to offer an explanation for the fact that the photoisomerization of the aromatic azo compounds is temperature dependent. If the effect of steric interaction is included in our results, we note that there will necessarily be a lowering of the (n-r*) excited state barrier for the cis + perpendicular conversion without any effect on the reverse transitions. I n qualitative argeement with this expectation it is found that the quantum yield for the cis + trans photoisomerization of azobenzene is almost constant from room temperature to -180", whereas the quantum yield for the trans + cis photoisomerization decreases by a factor of 2 over this same temperature range. Similar results were obtained with other azo compounds.* Continuing with the assumption that our calculations are qualitatively applicable to aromatic azo compounds, we predict that the excitation process which is most efficient in populating the (n-r*) triplet state will be the most efficient one for promoting isomerization. Excitation directly to the (n-r*) singlet state should be more efficient in this respect than excitation to the lowest (r-r*) singlet state, which would allow more opportunity for radiationless transitions to occur which are competitive with formation of the (n-r*) triplet state. Experimentally, excitation in the n + r* band does result in higher quantuni yields for the photoisomerization than does excitation in the r --t r* band.4 This would not necessarily
NOTES
1065
have been the case if isomerization could occur in the (PX*) singlet state. We conclude that the results of our MO calculation provide a possible explanation for a number of the features of the photoisomerization of azo aromatics including the origin of the excited state barrier to rotation about the N-N bond and the wave length sensitivity. Finally, by way of justification of our use of an MO method we note that (i) when an MO calculation, including overlap, is carried out for ethylene the results are qualitatively quite similar to those obtained by including the effects of electron-electron interaction explicitly8; and (ii) recent RIO calculations, including overlap, on the ground-state energies of a number of aliphatic and olefinic systems have been found to be quite reliable for predicting the stability of various isomeric configprations. While these successes with other systems do not guarantee the success of our calculations on azo molecules, they are certainly encouraging.
Acknowledgment. The support of the U. S. Public Health Service (Grant No. GM-10499) is most gratefully acknowledged. (8) R. G. Parr and B. L. Crawford, J . Chem. Phys., 16, 626 (1948). (9) R. Hoffman, ibid., 39, 1397 (1963).
Note on the Thermodynamics of Formation of Dolomite
by F. Halla Martinstrasse 88, Vienna 1 1 0 , Austria
(Received July 2, 1964)
For the free energy of formation of dolomite (D) from its solid components calcite (C) and magnesite (M) Stout and Robie’ recently gave the value AGO = - 2700 cal., starting from the dissociation pressures of D and 11,as determined by Graf and Goldsmith2 and Marc and Simek,3 and from their own entropy values. This AGO value differs considerably from earlier values between - 500 and - 1000 cal. obtained in the solubility measurements of Halla.4 Stout and Robie saw the reason for the divergence in the lack of reversibility in the solubility experiments involving two solid residues and in the erroneous use of the first power of the mean activity coefficient instead of the third. However, these well-founded objections
have already been considered in a recent paper of Halla5 (and Van Tassel) which has been overlooked by Stout and Robie and which gave AGO = -1720 f 280 cal. Stout and Robie also made a correctional recalculation of Halla’s data using the solubilities mD = 3.25 X and m M = 13.6 X low3of Yanat’eva6 and found AGO = -2700 cal., in agreement with their first value. Between their data and Halla’s - 1720 cal. a considerable difference still exists. It is explained by the fact that the mD value employed by Stout and Robie is given in ref. too low compared with mD = 4.5 x 5 and repeatedly confirmed. It has to be mentioned here that -1720 f 280 cal. already represents an average including Kramer’s’ value. That it coincides fairly well with the value derived directly from the pH determination is no independent confirmation but shows only the correctness of the formula employed in the evaluation of the latter. Another reason for backing our value lies in the fact that, for the enthalpy derived with the aid of Stout and Robie’s entropy value, AS0z98 = 0.81 f 0.29, one obtains AHoZg8 = -1720 - 240 = -1960 f 370 cal., which agrees with the value -1600 given by Roth and Muller-Mangolds within the error limits. This would seem to plead for our value,g but, on the other hand, no serious reason may be found which would invalidate Stout and Robie’s result. The only explanation for the divergence may be sought, at the moment, in a difference of the dispersity and, consequently, of the surface energy of the different samples of magnesite employed in the investigations. Irregularities which may be connected with the problem in question have already been observed. lo A redetermination of the enthalpies of the substances involved had to include those of the magnesite gel of Kraubath, Styria, and those of the well-crystallized magnesite of high purity, obtained by hydrothermal decomposition of solutions of Mg(HCO& (1) J. W. Stout and R. A. Robie, J . Phys. Chem., 67, 2248 (1963). (2) D. L. Graf and J. R. Goldsmith, Geochim. Cosmochim. Acta, 7 , 109 (1955).
(3) R. Marc and A. Simek, 2 . anorg. Chem., 82, 17 (1913). (4) F. Halla, 2. physik. Chem. (Frankfurt), 17, 368 (1958); 21, 349 (1959); 25, 267 (1960). (5) F. Halla, Sedimentology, 1, 191 (1962). (6) 0. K. Yanat’eva, Zh. Neorgan. Khim., 1, 1475 (1956). (7) J. R. Kramer, J . Sediment. Petrol., 2 9 , 465 (1959). (8) W. A. Roth and D. Muller-Mangold. “ Landolt-Bornsteins physikalisch-chemische Tabellen,” 5th Ed.. Vol. IIIc, p. 2762. (9) The attempt made in ref. 5 to calculate AGO on the same lines as Stout and Robie proved to be fallacious because the mixing energy was not considered, (io) F. Halla and R. Van Tassel, Radez Rundschau, 42 (19641.
Volume 69, Kumber 9
March 1966
