J. Phys. Chem. 1981, 85,3409-3415
it should be noted that the use of a local dielectric function becomes somewhat questionable if the atomic point charges constituting the molecule are close to the boundary of the sphere. Then the image attraction to the sphere becomes very large and an overestimate of the size of the electrostatic correction will occur. While this effect can be corrected for in the theory by taking into account the coupling of the charges to the spherical surface polariton modes, such a calculation would probably not improve the accuracy of the theory very much due to the other afore-
3409
mentioned uncertainties. The present theory will obviously work best when the charge distribution is localized near the center of the molecule and when the spherical cavity has a large radius. In conclusion, the electrostatic effect correction together with hydrogen-bonding considerations proves useful to explain trends in amine basicities. As a next endeavor one must try to improve eq 1 still further by introducing a shape factor for series 2 instead of placing them in a spherical cavity.
Temperature Dependence of Absorption Spectra for Orbitally Allowed Electronic Transitions of Polyatomic Molecules Ernest Grunwald Department of Chemistty, Brandeis Universi@, Waltham, Massachusetts 02254 (Received: May 27, 198 1)
A statistical approach is presented for dealing with the temperature dependence of UV/visible molecular electronic absorption bands in cases of extensive and unresolved fine structure. The model is based on the Born-Oppenheimer approximation and assumes that the many vibrational modes of a polyatomic molecule all make independent contributions. Explicit expressions are obtained for the mean (mi)and the variance (ut) of a modewise distribution as functions of the mean excitation number (ni)of the respective mode. For the overall absorption band m = Emi and u2 = Cut. Practical methods are suggested for applying the results.
With the availability of pulsed high-power infrared lasers, gas temperatures above 1000 K are readily reached, and such temperatures, as well as chemical changes that result, can often be monitored by UV/visible absorption spectroscopy.lV2 Because of this, and because of current interest in high-temperature combustion and environmental chemistry,14there is renewed interest in the effects of high temperatures on the widths, shapes, and maxima of molecular electronic absorption bands. The basic principles for such an analysis are wellknown: but rigorous application is possible only when the electronic absorption band shows resolvable fine structure. Many electronic absorption bands, indeed most bands of medium-sized and large molecules, are experimentally indistinguishable from continua. For such bands, detailed analysis is precluded, yet one can still use theory to predict the form of the mathematical relationships between bandwidth, band maximum, and temperature. Actual application of the resulting equations wili then involve some empirical curve fitting and adjustment of parameters. This partly theoretical approach is preferable to pure empiricism because there is at least some basic understanding, and the range of validity of the relationships can be defined. A notably successful example of this approach is the treatment, by Sulzer and Wieland, of temperature effects on diatomic absorption bands leading to dissociative electronic state^.^,^ In the present work we shall consider the temperature dependence of apparently continuous electronic absorption (1)Garcia, D.;Grunwald, E. J. Am. Chem. SOC.1980,102,6407-11. (2)Grunwald, E.;Lonzetta, C. M.; Popok, S. J.Am. Chem. SOC.1979, 101,5062-4. (3)Herzberg, G.In ‘‘MolecularSpectra and Molecular Structure”; Van Nostrand Princeton, NJ, 1966;Vol. 3, pp 128-82. (4)Sulzer, P.; Wieland, K. Helo. Phys. Acta 1952,25,653-76. (5)Passchier, A. A.; Christian, J. D.; Gregory, N. W. J.Phys. Chem. 1967, 71,937-42.
bands of medium-sized and large molecules. Consideration will be limited to electronic transitions that are allowed by orbital selection rules in which both upper and lower states are bound states. The theoretical model will comprise well-known approximations, similar in kind to those adopted by Sulzer and Wieland.4 For orbitally allowed electronic transitions to bound electronic states, basic theory predicts extensive fine structure due to concurrent vibrational and rotational energy change^.^ For small molecules the density of such fine-structure lines is often low enough to permit resolution and analysis. However, both density and absolute number increase very rapidly with the number of molecular vibrational modes, especially at elevated temperatures, and even for medium-sized molecules the number of such lines which must be included in a detailed analysis at, e.g., 1000 K, may be said to be astronomical. Fortunately, as the number of vibrational modes increases, the probability laws applicable to large numbers begin to apply, and in particular the central limit theorem and its corollaries6 lead to a manageably simple and predictable phenomenology.
Theoretical Model In the following, unprimed symbols will denote properties of the ground electronic state and primed symbols those of the excited electronic state. Rotational energy changes associated with an electronic transition make a relatively small contribution to absorption bandwidth for polyatomic molecules and will be neglected. At elevated temperatures, thermal vibrational excitation is significant, and the electronic ground state must be treated as a mixture of discrete vibrational subspecies. Let s denote the number of vibrational modes. Let ni denote the vibrational quantum number and vi the frequency of (6) Feller, W. “An Introduction to Probability Theory and Its Applications”, 3rd ed.; Wiley: New York, 1968;Vol. 1, pp 253-6,266-70.
0022-3654/81/2085-3409$01.25/00 198 1 American Chemical Society
3410
the ith normal mode. A given vibrational subspecies is then characterized by the set nl, n2,..., n, of its vibrational quantum numbers. Let f i = exp(-hvi/kT). At thermal equilibrium at temperature T, the relative population p of the subspecies nl, n2, ..., n, is then given by eq 1,
= pl(n1) Pz(n2)
p,(n,) (2) adopting the harmonic oscillator approximation, which includes treating the modes as independent. 1 - fi = 1/ (partition function) for the ith mode. In eq 2 (1- fi)fi"i is written simply as pi(ni)and denotes the probability that a molecule chosen at random will have ni quanta of vibrational energy in the ith mode, irrespective of the state of excitation of the other modes. Let (n]denote the complete set nl, n2,..., n, for a given subspecies in the ground electronic state, and let ( n )denote a similar set n;, n2/,..., n,' in the excited electronic state. If one neglects molecular rotation, the transition moment Rev(n,n?for the absorption process (e,n) {e',n')is then given by eq 3, where $ev denotes total wave function
-
Rev(n,n? = S$ev'(n?* M$ev(n) d7
(3)
(electronic X vibrational), M electric dipole vector, and * the complex on jug ate.^ To obtain a working relationship, $ev and $e,' are separated into electronic and vibrational wave functions according to the Born-Oppenheimer approximation3?' ($ev - xexv;$e,' = xJx,'). Furthermore, xe and x:, which in principle depend on the nuclear coordinates, are approximated by xe(0)and x,'(O) for the nuclear coordinates in the ground-state potential minimum. The resulting eq 4 Rev(n,n? =
SXJ(O)* MeXe(0) dre Jx,'(~? xv(n) dTv
(4) = [C(n,n?l (5) is the Condon approximation (for further discussion, see ref 7), whose usefulness is limited to orbitally allowed transitions. Equation 4 is conveniently rewritten in the form of eq 5, where R,(O) denotes the purely electronic transition moment (independent of (n,n'))and C(n,n? is the vibrational overlap integral. The absolute extinction Y(n,n?and the relative extinction G(n,n?associated with eq 5 are then given by eq 6 and 7, where p ( n ) accounts for Y(n,n? = IRe(0)12 p ( n ) C(n,n?' (6)
G(n,n? = A n ) (7) the relative ground-state population. The corresponding absorption wavenumber v(n,n? is given by eq 8, where v(0,O) is the wavenumber of the 0,O transition. 5
v(n,ii? = v(0,O)
+ i=c1 (n[v[ - npi)
(8)
In the preceding, the 2s vibrational quantum numbers of the set (n,n)were considered fixed, and eq 7 and 8 therefore represent a specific point in the 3,G plane. This point is part of a discrete distribution whose totality, for all physically significant sets (n,n),constitutes the desired electronic absorption band. Modewise Representation In eq 8, v is a sum of modewise contributions, plus a constant term v(0,O). ~~~
Grunwald
The Journal of Physical Chemistty, Vol. 85, No. 23, 1981
~
(7) Azumi, T.; Matsuzaki, K. Photochem. Photobiol. 1977,25,315-26.
In eq 6 and 7, IRe(0)l2is a constant factor, and p ( n ) can be separated into modewise factors according to eq 2. We now wish to consider the conditions under which C(n,nq2 can be separated into modewise factors. Let Q1, Q2,..., Q, and Q{, Q2/, ..., Q,' represent respectively the mass-weighted normal displacement coordinates of the ground and excited electronic states. The vibrational wave functions representing the respective set (n) and (n')can then be separated into modewise factors. x
V
~
= xi,nl(Qi)
)
x,'(n? = x'i,n/(Qi')
x2,nz(Q2) ~'2,n;(Q2/)
X~,~,(QJ (9) X',,~;(Q,')
(10)
As a first step toward a modewise representation of the overlap integral, one characterizes each Qi and Qf in terms of a specific internal vibrational coordinate (or combination of such coordinates) whose displacement makes the dominant contribution, and one numbers the modes so that analogous motions are given the same subscripts in both ground and excited electronic states. For instance, if Qlo is nominally an X-Y stretch in the electronic ground state, then Q1d is the analogous nominal X-Y stretch of the same symmetry in the upper state. However, a correlation be-, tween the Q and Q'values can be established in this way only if the topological structure of the molecule remains intact on electronic excitation. This is to say, none of the ground-state bonds break and no new bonds form in the excited state, although of course the bond energies and equilibrium molecular geometry may change. Having established this correlation, one next assumes that each Q[ depends only on the correlated Qi. That is, any mixing of different vibrations of the same symmetry in going from the ground to the excited electronic state is neglected. In a rigorous analysis this assumption would not be a c ~ e p t a b l e . ~ J But ~ J ~for the present limited purpose of obtaining the temperature dependence of continuous absorption bands, the assumption seems adequate and may indeed be unavoidable. If Q[ depends only on Qi, physical considerations require a linear relationship, eq 11,where the parameter 6qi asQ[ = Qi - 6qi
(11)
sumes a specific value for each modes8 A brief review of physical considerations seems appropriate at this point. Each normal displacement coordinate Qi is a specific normalized linear combination of mass-weighted Cartesian displacement coordinates ( a x , = x , - xO,,; 6y, = y, - yo,; etc.; the x,y,z's are mass-weighted) including in principle all atoms in the m o l e ~ u l e .If~ one writes Qi = q i- qo,i,it follows that qo,i is the same linear combination (as Qi) of mass-weighted Cartesian equilibrium coordinates. In principle Q[ is a different linear combination (from Qi)of Cartesian displacement coordinates. However, because &[ and Qi are dominated by displacements of the same internal coordinate or comprise similar blends of such (8) Duschinsky, F. Acta Physicochim. URSS 1937,7,551-9.
(9)Wilson, E.B.;Decius, J. C.; Cross, P. C. "Molecular Vibrations"; McGraw-Hill: New York, 1955;pp 11-33, 290. (10) Manneback, C. Physica 1951,17,1001-10. Natl. Bur. (11)Shimanouchi, T.Natl. Stand. Ref. Data Ser. (US'., Stand.) 1972,No.39,p 139. (12)McDiarmid, R J. Chem. Phys. 1976,64, 514-21. (13)Squillacote, M. E.;Sheridan, R. S.; Chapman, 0. L.; Anet, F. A. L. J. Am. Chem. SOC.1979,101,3657-9. (14)Coe, D. S.;Steinfeld, J. I. Chem. Phys. Lett. 1980, 76,485-9. (15)Warschel, A.; Karplus, M. Chem. Phys. Lett. 1972,17, 7-14. (16)Lasaga, A. C.; Aerni, R. J.; Karplus, M. J. Chem. Phys. 1980,73, 5230-43.
The Journal of Physical Chemistty, Vol. 85, No. 23, 1981 3411
Temperature Dependence of Absorption Spectra
displacements, one assumes that the particular linear combination which characterizes Q/ will also characterize Qi. Thus Q[ = q i - qo,[, where qi is the same variable as in Qi = qi - qo,i. Equation 11 is obtained on eliminating q; and letting 6qi = qb,; - qo,i. Because eq 11 is of the same form as the relationship for the change in internuclear distance on electronic excitation for a diatomic molecule, we shall refer to eq 11as the model of diatomic analogy. However, for polyatomic molecules the properties of 6qi and of 6qi/qoiare intuitively less obvious. Symmetry Propertie~.~JThe qo,is share, with the equilibrium molecular geometry, the property that they must remain unchanged under all symmetry operations. At the same time each qoi belongs to the symmetry species of its respective vibrational mode. These two requirements can be satisfied simultaneously for nonsymmetric vibrational modes only if qo,i = 0. Thus qo,iis different from zero only for vibrations which belong to the totally symmetric species of the molecular point group. Delocalization. To the extent that normal mode displacements are delocalized, 6qi may be different from zero for several vibrational modes even if the given electronic transition is narrowly localized in the molecule. Magnitude. If the upper state is a bound state, rules relating bond distance to bond order make it unlikely that 6qi/qo,ifor nominal stretching vibrations will be greater than -0.1. For modes associated with other internal coordinates, 6qi/qo,ican be more variable.
Spectral Density According to the model of eq 11, Q: depends on Qi only. Thus on introducing eq 9-11, and letting drv = dQl dQ2 ... dQ,, the vibratinal overlap integral is given by eq 12. C(n,n? =
xl,nl(QddQ1 5 ~ ’ 2 , ~ z ) ( Q -2 dQ2 ... lx’s,n:(Qs - 6q,) xs,n8(Qs) dQ, (12)
l x ’ l , n / ( Q 1-
6q2) xz,nz(Q2)
According to eq 12 C(n,n? has been separated into a product of modewise overlap integrals. Let Ci(ni,n[)= sx’i,n!xi,nidQi, and introduce eq 2 and 7. The relative extinction is then given by eq 13. In view of eq 11, [CiS
G(n,n? =
JJPi(ni) [Ci(ni,ni’)12 1=1
(13)
(ni,n[)]’is a modewise Franck-Condon factor. It will be recalled that eq 13 applies to a specific transition (e,n) (e’+? of a specific subspecies (n)of the electronic ground state. To obtain the overall spectral distribution, we begin by showing that eq 13 implies sets of independent modewise distributions. For definiteness, let n2,n;, ..., n,,n,‘ be fixed and allow nl and nl’ to vary independently. This will generate a subset {S,) of G and a corresponding subset (S,’) of n.
-
IS,)=
fi
[cl(n,,n1’)1211 Pi(ni) [ci(ni,n[)12 i=2
(14)
+
{ S i )= nJn,f{(n,’nl’)- n1nJ)) 8(0,0)
+ i=2 c (n[n[ S
- nini)
(15) Jn ,.(...I) denotes that the set includes all physically significant elements obtained by varying n, and n,’ independently. In eq 15, nl’nl’ - nlnl is the physical variable (wavenumber) whose overall distribution we wish to obtain. In eq 14, pl(n1) [C1(n1,n,’)l2is the correspondingprobability. To demonstrate that it is normalized, one adduces prop-
erties of Franck-Condon factors (eq 16)1° and of Boltzm
C Cl(nl,n
