J . Phys. Chem. 1990, 94, 4420-4425
4420
Determination of Excited-State Potential Energy Surfaces In the Franc&-Condon Region from Electronic Absorption Spectra Soo-Y. Lee Department of Chemistry. National University of Singapore, Kent Ridge, S.051I Singapore (Received: July 6 , 1989; I n Final Form: November 30, 1989)
A fast and efficient method using the first and second moments of the electronic absorption spectrum is described to determine
the harmonic fit to the excited-state potential about the ground-state equilibrium configuration, i.e., the Franck-Condon region. It is an improvement over the conventional harmonic Franck-Condon analysis in the following ways. (a) It uses the minimum information to determine the harmonic fit to the excited-state potential in the Franck-Condon region. (b) It is easy to use, since it is not a fitting procedure. (c) It gives a definite sign to the shift in bond length. (d) It provides one bound to the actual shift in bond length. (e) It is computationally more efficient and gives accurate results. Examples of some diatomics and a polyatomic with separable modes are given to illustrate the method.
1. Introduction Basically, two approaches have been used to determine excited-state potential surfaces for molecules from their electronic absorption spectra. The first approach is by Franck-Condon analysis, which is well-described in the work of Herzberg.l*2 This is a time-independent method that makes use of the eigenvalues and eigenfunctions for harmonic or Morse potentials. In this method, the initial (usually ground) vibrational state is assumed known, and the parameters for the excited-state potential are adjusted until the calculated Franck-Condon factors give a good fit of the experimental absorption spectrum. The harmonic or Morse potential so derived is often interpreted as an expansion of the excited-state potential about its minimum. This interpretation has its difficulty as pointed out by von Niessen et aL3 It is also often the case that getting both line spacings and intensities to match between the calculated and observed spectrum is not achievable. Moreover, when the harmonic approximation is made, symmetry prevents discriminating between bond lengthening and bond shortening accompanying the change in electronic state; arguments based on the electronic structure of the excited state have to be used. Other versions of FranckCondon analysis such as the semiclassical Rydberg-Klein-Rees for diatomics4 and basis set methods5 do not have to assume harmonic or Morse potentials. However, they do require that sufficient spectral lines be observable before the actual potential can be determined. Unfortunately, many molecules of interest do not give sufficient spectral data for these methods to be applicable. The second, more recent approach is the time-frame viewpoint developed for various spectroscopies by Heller and co-workers.6 Heller7**showed that the electronic absorption profile I(w) is given by the Fourier transform of an autocorrelation function C(t) which is the overlap integral between two nuclear wave functions C(t) = ($I+(t))
(1.1)
Here 16) = plxg,)is the nuclear wave function prepared on the excited electronic state by a Franck-Condon transition and is given by a product of the initial vibrational wave function Ixgi) in the ( I ) Herzberg, G . In Molecular Spectra and Molecular Structure I . Spectra of Diatomic Molecules; Van Nostrand: New York, 1966. (2) Herzberg, G. Molecular Spectra and Molecular Structure II. Spectra and Electronic Structure of Polyatomic Molecules; Van Nostrand: New York, 1966. (3) von Niessen, W.; Cederbaum, L. S.; Domcke, W.; Schirmer, J. In Computational Methods in Chemistry; Bargon, J., Ed.; Plenum Press: New York, 1980; pp 65-102. (4) Zare, R. N . J . Chem. Phys. 1964, 40, 1934. ( 5 ) Lill, J. V.; Parker, G. A.; Light, J. C. Chem. Phys. Lett. 1982,89,483. Heather, R . W.; Light, J. C. J . Chem. Phys. 1983, 79, 147. Kanfer. S.; Shapiro, M . J . Phys. Chem. 1984, 88, 3964. (6) Heller, E. J. Ace. Chem. Res. 1981, 14, 368. (7) Heller, E. J . J . Chem. Phys. 1978, 68, 2066. (8) Heller. E J J C h m Phys. 1978, 68, 3891
0022-3654/90/2094-4420$02.50/0
ground electronic state with the transition dipole moment k . The other wavepacket, I$(t)) = exp(-iH,t/h)lr#J), is simply the evolution of the nonstationary wavepacket 16) on the excited-state surface which has vibrational Hamiltonian He, and this gives the vibrational dynamics that take place following the Franck-Condon transition. At short times, the wavepacket I d ( t ) ) samples the Franck-Condon region of the excited-state potential, and the short time autocorrelation function gives the gross features of the absorption spectrum such as the mean energy and the width. At longer times, the return overlaps in C(t) builds up the vibrational structures in the absorption spectrum. Assuming the Condon approximation ( p = constant) as well as harmonic ground- and excited-state potentials, an analytic expression has been derived for C(t).93'0 By further assuming equal frequencies for groundand excited-state oscillators, Rusk9 showed that at a quarter period the magnitude of the autocorrelation function gives the very simple expression
IC(T=j/,)l = exp(-A2/2)
( 1 .2)
where A is the (dimensionless) shift in equilibrium geometry between the two electronic states. This forms the basis of Ruscic's method to determine the shift in bond length as follows. (a) The inverse Fourier transform of the electronic absorption profile is taken to give an experimental C(t). (b) The period of the excited-state oscillator is determined from the time interval between recurrences in IC(t)l. (c) The value of IC(t)l at a quarter period is used in eq 1.2 to determine A. (d) The sign of the shift in bond length is determined by graphical analysis where the experimental IC(t)l is compared with a harmonic IC(f)l near t = 0; the excited-state harmonic oscillator used to calculate the harmonic IC(t)l has frequency determined in step b and a shift in the potential minimum determined in step c. Despite the severity and arbitrariness in some of Ruscic's assumptions, the method yielded remarkably good results for the shift in the equilibrium bond length of some diatomic molecule ions: and this has been explained.I0 It has further been shown by Lee et a1.I0 that the derivatives in the magnitude and phase of C(r) at t = 0 could be used to determine the harmonic fit to the excited-state potential in the Franck-Condon region, and this could also be used to obtain a value for A. The importance of such information lies in the fact that theoretical calculations of potential energy surfaces are largely confined to the Franck-Condon region, and the method therefore permits a direct comparison of theory and experiment. In this paper we look at a third method using the moments of the spectrum. This method actually preceded the time-frame approach, but so far it has only been used for a qualitative understanding of electronic absorption spectra rather than to deduce parameters for the excited-state potentials. Lax" has given the (9) Ruscic, B. J . Chem. Phys. 1986, 85, 3776. (IO) Lee. Soo-Y.; Lim, S. K . J . Chem. Phys. 1988, 88. 3417.
0 1990 American Chemical Society
Excited-State Potential Energy Surfaces
The Journal of Physical Chemistry, Vol. 94, No. 1I , 1990 4421
first two moments in terms of the expectation values of the difference between excited- and ground-state potentials. More recently, CoalsonI2 evaluated the third moment which provides a measure of the skewness of the potential. The original intent of this method is the understanding of diffuse spectra in terms of the potential energy surfaces, where only limited experimental information is available. Here we take a different view: we seek limited information about the potential energy surfaces from the electronic absorption or emission spectra. The fact that the sum rules are presented in the energy frame makes it more easily accessible to the experimentalists. In section 2, the time-frame viewpoint is used to derive and evaluate the first two spectral moments explicitly in terms of the ground- and excited-state harmonic oscillator parameters. The spectral moments are related to the derivatives of the autocorrelation function evaluated at t = 0 when the wavepacket I4(t)) is sampling the Franck-Condon region, and hence the excited-state harmonic oscillator deduced by the moment method is an approximation to the actual excited-state surface in the Franck-Condon region. In section 3 the moment method is applied to some diatomics which have previously been studied by using the autocorrelation function app r o a ~ h ~and * ' a~ polyatomic with separable normal modes which has been studied by Franck-Condon a n a l y ~ i s . ' ~
where we have defined (2.1 1) Assuming the Condon approximation ( p = constant), the ratio of the first moment to the zeroth moment yields
= (ih)Ol)(O)/C(O) = (Xgol(He - EcdIXgO)
and the ratio of the second moment to the zeroth moment yields (( h 8 ) ~ ) --
2. Theory Assuming the Born-oppenheimer approximation and a constant damping function (Le., Dirac delta function line shape), Heller7-* showed that the photoabsorption line shape is given by I ( w ) = Kwa(w) (2.1) Here K is a constant, w is the light frequency, and g(w), defined as the absorption profile, is given by the Fourier transform a(@) = (1 / 2 r ) S m -m d t eiEf/h(41e-iHer/h 14)
where the transform energy E = hw
+ E,
(2.2)
and the autocorrelation function C(t) is defined to be c(t) = (4le-i(&-Ea)t/hI4 )
(2.7)
Taking the inverse Fourier transform of eq 2 5 , the autocorrelation function can be determined from the absorption profile (2.8)
Defining the nth moment of the absorption profile by 1:dw
8"u(w)
(2.9)
it can then be shown by differentiating eq 2.8 n times with respect to t and setting t = 0 that (11) Lax, M. J . Chem. Phys. 1952, 20, 1752.
(h8)2
((halo) ( ( h 8 ) o ) 2
1 Imll dqo)
P(0)
- (ih)2-
0) +
= (Xgol(He - Eeo)21Xgo)- (Xgol(Hc - Ea)lXgo)2 (2.14) Returning to the autocorrelation function C ( t ) ,it can be expressed in terms of its modulus IC(t)l and phase cp(t) as
C ( t ) = IC(t)l exp(icpW1
(2.15)
From eq 2.7, C*(t) = C(-t), and hence IC(t)l is an even function. Moreover, C(0) = (414) is real, which implies that cp(0) = 0. It is then straightforward to show that
( i h ) d ' ) ( O ) / C ( O= ) -h (d$)) -
(2.16) r=o
(2.6)
(8")
(2.13)
The zeroth moment gives the area under the absorption profile. The mean energy of the absorption profile is given by the first moment, eq 2.12, but for the width of the absorption profile it is necessary to evaluate the mean square energy
(2.5)
where the transform energy 8 here is defined as ha = hw + Ego- E d
C(r) = S *-m d w e-'&'U(W)
I(w)/w
= (xgoI(He - Eeo)21Xgo)
and
a(@) = (1 / 2 r ) S -m m d teiGT ( t )
( h o ) 2 ( I (/a) ~)
= (i h ) 2 d 2 )0)( / C(0)
(2.3)
ground electronic state with eigenvalue EBoand eigenfunction IxBo). The transition dipole moment is denoted by p, and the vibrational Hamiltonian of the excited state is He. Taking energies relative to the u'= 0 level with eigenvalue Ed in the excited state, we can write eq 2.2 as
1:dw
JIdw
((hay) ----
14) = P I X ~ O ) (2.4) The molecule is assumed to be initially in the u" = 0 level of the
(2.12)
(12) Coalson, R. D. J . Chem. Phys. 1984, 81, 2891. (13) Tsubi, M.; Hirakawa, A. Y.; Hoshino, T.; Ishiguro, T.; Kimura, K.; Katsumata, S. J . Mo/.Spectrosc. 1976, 63, 80.
and
Comparing these with eqs 2.12 and 2.14, we deduce that the method of Lee et a1.,I0 which uses the slope of cp(t) and the curvature of IC(t)l as t 0 to determine the harmonic approximation to the excited-state potential in the Franck-Condon region, is equivalent to using the mean energy (first moment) and width (second moment) of the absorption profile. It is much easier to evaluate the moments of the absorption profile rather than to carry out an inverse Fourier transform to find the autocorrelation function C(t) followed by an evaluation of its derivatives as t 0. In order to evaluate the matrix elements in eqs 2.12 and 2.14, let the ground- and excited-state potentials be harmonic with vibrational Hamiltonians given by
-
-
(2.18)
4422
Lee
The Journal of Physical Chemistry, Vol. 94, No. I I , 1990
and
I' r
He = h w
1 w,2
-p2 + - -(Q2
A)2
2wg2
J
+ E,
'
i
(2.19)
ye
Here p (=-i d/dQ) is the momentum operator; Q, the dimensionless coordinate, is related to the difference of the bond length R from the ground-state equilibrium R, by (2.20)
Q = (mw,/h)'/'(R - R,)
with reduced mass m; E,, E, are the energies at the equilibrium configurations, and w,, we are the angular frequencies of the ground- and excited-state harmonic oscillators, respectively. Finally, A is the dimensionless shift of the minima of the two surfaces A = ( m w , / h ) 1 / 2 ( R-, R,)
= (mw,/h)1/2S
(2.21)
The parameters for these two harmonic oscillators and their relation to the anharmonic ground- and excited-state potentials are shown in Figure I . We consider the molecule to be initially in the u" = 0 vibrational state of the ground electronic surface x ~ o ( Q )= r-'l4exp(-Q2/2) Using the following equations for the operators on the harmonic oscillator wave function Ixn) PlXn) = ~
i
(2.22) and Q acting
[ ( + nI)li21Xn+l) - niizl~n-l)l (2.23)
1
Qlxn) = ~ [ (+ nI)l'21Xn+,) + n 1 / 2 1 ~ n - l ) l
(2.24)
it is straightforward to derive from eqs 2.12 and 2.14 the mean energy and mean square energy of the absorption profile in terms of the parameters of the ground- and excited-state harmonic oscillators with the following results
Figure 1. Harmonic oscillator approximation (solid curve) to the ground-state Ve and excited-state V, potentials in the Franck-Condon region. Ground-state oscillator has frequency w,, energy of E, a t minimum, and energy of Ego at the d' = 0 level with vibrational wave function xgo. Excited-state oscillator has frequency we, energy E, at minimum, energy of Ed at the u ' = 0 level, and a shift in the minimum relative to the ground state by A. Absorption energy is denoted by hw.
and
Here, we have defined the parameters f and a as
f = ( w c / w g - I ) / ( w e / w g + 1)
(2.27)
a = y2(l + A A 2
(2.28)
and The first and second moments can be calculated from the experimental absorption profile; then, if w, is known, eqs 2.25 and 2.26 can be used to solve for the unknown parameters we and A of the excited-state oscillator. The sign of A is decided by comparing we with the spectroscopic frequency w, corresponding to the harmonic expansion of the excited-state potential about its minimum. The frequency w, can be determined by a BirgeSponer plot or approximately from the vibronic spectrum as h w , E(u'=l) - E(u'=O) (2.29)
=
In the case of bond lengthening, A > 0, the excited-state anharmonic potential about the ground-state minimum Q = 0 is a hard wall and we > w,. This is shown in Figure 2. It is also clear that A < 4;thus, a lower bound to the actual shift in bond length 4 is given by the moment method. On the other hand, for bond shortening, A < 0, it is the soft wall of the excited-state potential that is reached at Q = 0, and we < w,. Moreover, -A > -4 and an upper bound to the actual decrease in bond length is obtained in this case. This is shown in Figure 3. The figures also suggest
INTEAWUCLEAR DISTANCE
-
Figure 2. Bond lengthening in the excited state. Hard wall of the excited-state potential is reached at the configuration corresponding to the ground-state minimum. Harmonic frequency we in the FranckCondon region is greater than the harmonic frequency w, about the excited-state minimum. Shift in bond length estimated by the moment method is denoted by A, and the actual shift is denoted by 4, with A
4. that a better estimate of the shift in bond length can be obtained by using a mean frequency = (we 0,)/2 for the excited-state oscillator and assuming that such an oscillator gives the correct
+
The Journal of Physical Chemistry, Vol. 94, No. 11, I990 4423
Excited-State Potential Energy Surfaces
TABLE I: Absorption Energies Ed and Peak Intensities I,, for Selected States of HzrNZyNO,0, and ZnAr
I
II
species H2+(X2Zgt)
U'
0 1 2 3 4 5 6 7 8 9
IO II 12 13 14 15 0
I
IKI+RHucLEAR DISTANCE
2 3 4 5 6 0 1 2 3 4 5 0 1 2 3 4 6 7 8 9
-
Figure 3. Bond shortening in the excited state. Soft wall of the excited-state potential is reached at the configuration corresponding to the ground-state minimum. Harmonic frequency we in the Franck-Condon region is less than the harmonic frequency w, about the excited-state minimum. Shift in bond length estimated by the moment method is denoted by -A, and the actual shift is denoted by -A,, with A > 4.
mean energy for the absorption profile. This implies solving eq 2.25 with i3 in place of we to obtain a refined shift A in bond length. The example below will show that A is very close to As.
3. Application to Diatomic and Separable Polyatomic Spectra The moment method is applied to some diatomic photoelectron spectra for the following species and transitions: H2+XzZg+ H2 X'Z +,N2+A2H, N2 XIZg+,NO+ XIZ+ NO X2H, and 0 2 ' X%, O2 X3Z;. The first two cases involve bond lengthening, and the next two cases involve bond shortening. The last three cases have been studied by using the comparatively more tedious autocorrelation function method9J0 We have also included a recent example of laser-induced fluorescence spectrum of ZnAr X'Zo+) transition.18 van der Waals molecules for the (C'II, Here the change in bond length is very large (about 1.2 A), and we are really testing the limits of the theory. The observed energies E,, and relative peak intensities Id as a function of the excited-state vibrational quantum number u'for the various species are collected in Table I. For such well-resolved vibronic spectra which are treated as discrete, the zeroth, first, and second moments of the absorption profile are simply given by
- -
-
-
-
(14) Gardner, J. L.; Samson, J. A. R. J . Electron Spectrosc. Relat. Phenom. 1976,8, 123. ( I S ) Gardner, J. L.; Samson, J. A. R. J . Chem. Phys. 1974, 60, 3711. (16) Edqvist, 0.; Asbrink, L.; Lindholm, E. Z . Narurforsch.. A 1971, 26, 1407. Edqvist, 0.; Lindholm, E.;Selin, L. E.; Sjogren, H.; Asbrink, L. Ark. Fys. 1970, 40, 439. (17) Edqvist, 0.; Lindholm, E.; Selin, L. E.; Asbrink, L. Phys. Scr. 1970, I , 25. Nicholls, R. W. J . Phys. B 1968, I , 1192. (18) Wallace, 1.; Bennett, R.R.; Breckenridge, W. H. Chem. Phys. Lert. 1988, 153, 127. (19) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955; Chapter 4.
IO 11 12 13
Elf
Id
ref
15.4254 15.6971 15.9530 16.1936 16.4 194 16.6308 16.8281 17.0113 17.1808 17.3365 17.4783 17.6061 17.7 199 17.8197 17.9055 17.9773 16.70 16.93 17.16 17.38 17.61 17.83 18.04 9.25 9.54 9.83 10.11 10.40 10.67 12.071 12.305 12.533 12.756 12.976 5.760937 5.766162 5.770975 5.775460 5.779542 5.783299 5.786701 5.789694
0.448 0.922 1.000 0.969 0.713 0.572 0.423 0.308 0.208 0.132 0.089 0.058 0.039 0.024 0.015 0.005 0.871 1.000 0.763 0.435 0.191 0.07 1 0.025 0.51 1.oo 0.91 0.49 0.17 0.025 0.43 1 1.000 0.923 0.446 0.138 0.067 0.213 0.461 0.8 18 1.ooo 0.694 0.310 0.117
14
15
16
17
18
TABLE II: Excited-State Parameters Calculated Using the First and Second Moments of the Absorption Profile" we,
Wsr
cm-'
cm-l
6, A 8, A 6,, A H2+(X2Zgt) 4395.24 3588.3 2191 0.266 0.327 0.318 0.0795 0.0772 2358.57 2234.6 1855 0.0729 N2'(A2n,) NOt(XIZt) 1904.20 1871.3 2339 -0.0981 -0.0871 -0.0875 -0.0957 -0.0911 02'(X2ng) 1580.19 1409.5 1887 -0.112
species
cm-l
ZnAr(C'II,)
20.5
10.6
61 -3.30
-1.00
-1.21
the ground-state oscillator frequency about ground-state minimum. we is the excited-state oscillator frequency about ground-state minimum. us is the excited-state oscillator frequency about excitedstate minimum. 6 is the shift in the bond length estimated by the moment method. 6 is the refined shift in bond length using D = (we + w,)/2 in place of we in eq 2.25. 6, is the shift in bond length by rotational analysis. wg is
These moments are used on the left-hand side of eqs 2.25 and 2.26; then using the known ground oscillator frequency w,, the equations are solved for o,and A by the Newton-Raphson method. The shift in bond length 6 is related to the dimensionless variable A by eq 2.21. Table I1 summarizes the results for the ground-state oscillator frequency og,the excited-state oscillator frequency we in the Franck-Condon region, the excited-state oscillator frequency us(=[E,,, - E O , = O ] / about h) the potential minimum, the harmonic shift 6, the refined shift 8 as discussed above, and the literature shift 6, typically obtained by analysis of rotational spectra. For typical shifts in bond length of about f0.1 A, the moment method gives results to beter than 5% of those obtained from hard-to-get rotational spectra. Even in the case of the ZnAr
4424
Lee
The Journal of Physical Chemistry, Vol. 94, No. 11, 1990
TABLE 111: Vibrational Structure of the 230-nm Band of Thioformamide assignment
energy,
0,'
43020 43337 43637 43783 44072
I (740 cm-I) 0 1.00 0 0.21 0 0.04 1 0.71 1 0.18
44514
0.68
cm-l
44813
0.18
1; 1;
45259
0.36
)
45998 46730
0.21 0.07
0 ' 1
u;
L','
(3 17 cm-I) 0
( I500
1
2 0 1
0 0 0 0 0 0 1 0
0 1
2 2
0
I for
1.oo
0.21 0.04 0.7 1 0.18 0.275 0.405 0.287 0.073 0.111 0.099 0.07
van der Waals complex where the shift in bond length is very large, the simple moment method gave relatively good results. The sign of the shift is easily determined by comparing we with us-it is positive when we > ws and negative when we < us. For a polyatomic example with separable modes, we chose the ultraviolet absorption spectrum of thioformamide gas, HCSNH2. The 230-nm band corresponding to the ? T * ~(nN-ncs) ~ transition has well-resolved vibrational structures whose energies, peak intensities, and vibrational assignments were determined by Tsuboi et aI.l3 and summarized in the first three columns of Table 111. The intensity assignments for overlapping lines presented in the last column were calculated as follows. Assuming that the modes are separable, the intensity 1(0,0,2) was determined from the following relation
-
1(0,0,2) --I(O,O,O) --
I( 1,0,2)
I( 1,O,O)
(3.4)
From the known values for the other intensities, we calculated 1(0,0,2) = 0.099. The intensity of the 45998-cm-' line is 0.21; therefore, by the difference of intensities, I(2,0,1) = 0.1 11. Next, the sum of intensities 1(2,0,0)
+ I(O,O,l)
= 0.68
(3.5)
and product of intensities I(2,0,0)I(0,0,1) = I(2,0,l) = 0.111
(3.6)
were solved to give 1(2,0,0) = 0.275 and I(O,O,l)= 0.405. With these results, we calculated I(l,O,l) using the relation (3.7) giving I(l,O,l) = 0.287. It then followed from the intensity of the 45 259-cm-l line that 1(3,0,0) = 0.073. We need not bother about decomposing the 44 8 13-cm-' line as we do not need it for the moment calculation. The intensities corresponding to the C = S stretch, LNCS deformation, and C-N stretch which were used for the moment calculations are summarized in Table IV, and the results are given in Table V. The reduced m a w s of the three motions which are required in eq 2.20 are calculated from the diagonal elements of the C matrixI9 G(C=S) = j ~ c+ ps
(3.8)
G(LNCS) = pCN2c(N
+ pCS2& + b C N 2 + pCS2 - 2PCN PCS cos a)PC (3.9) G(C-N) = Pc -t PN
Cl'
(3.10)
where ccX is the reciprocal mass, p is the reciprocal bond length, and a is the N C S bond angle, all in the ground electronic state. The reduced mass is given by G-I. The literature values for the shift in bond length 6, were those calculated by Tsuboi et a1.I3 using the harmonic Franck-Condon analysis. By symmetry, the harmonic Franck-Condon analysis can only give the magnitude
0 I 2 3
0 0 0 0
0 0 0
0 I
0 0 0
0 0 0
2
I
cm-1
U,f
U,'
cm-I) calcn
1 1
2
TABLE IV: Energies and Peak Intensities for Various Vibrational Modes of Thioformamide Used in the Moment Calculations assignment energy,
C=S Stretch 0 43 020 0 43 783 0 44514 0 45 259
1 .oo 0.7 1 0.275 0.073
LNCS Deformation 0 43 020 0 43 337 0 43 637
1.oo
0.21 0.04
Stretch
C-N 0
1 .oo
43 020 44514 45 998
1
2
0.405 0.099
TABLE V: Excited-State Parameters for Tbioformamide Calculated Using the First and Second Moments of the Absorption Profile Corresponding to the Transition x*c-s (nN-rg.s)(l~b
-
transn
w.
w.
w,
6
8
6,
C=S C-N LNCS
843 1443 439
771 1415 367
740 1500 317
0.083 -0.055 4.1
0.083 -0.053 4.3
0.08 -0.06 -4
a wg
imum.
is the ground-state oscillator frequency about ground-state minwe is the excited-state oscillator frequency about ground-state
minimum. osis the excited-state oscillator frequency about excitedstate minimum. 6 is the shift in bond length estimated by the moment method. 8 is the refined shift in bond length or angle using i3 = (we + w,)/2 in place of o, in eq 2.25. 6, is the values obtained by Tsuboi et al.I3 using harmonic Franck-Condon analysis. b w values are given in cm-'. 6 values are given in A and deg. of the shift in bond length or bond angle. The sign of the shift was determined by Tsuboi et al. based on the contribution of the +N=C-Sstructure to the excited electronic state. N o such assumption is required in the moment method. There is good agreement in the sign and amount of shift in the potential minima for the C=S and C-N stretch, but there is a disagreement in the sign of the shift in the NCS bond angle. The moment method predicts that the NCS bond angle should increase by 4.3O in the excited state, but Tsuboi et al. argue that it should decrease by 4'. The argument used by Tsuboi et al. is based on the comparison between the gas-phase and condensed-phase structures of thioformamide where there is contribution from the +N=C-Sstructure in the condensed phase. However, it should be noted that the structures in such a comparison are in the electronic ground state for both phases, whereas for the spectrum considered here we should be comparing the excited-state structure with the ground-state one. 4. Conclusion
While it is true that we should use as much information as possible to determine as fully as possible the excited-state potential energy surface, nevertheless, if we seek a harmonic approximation to the excited-state surface, then clearly there are only two parameters to determine (or three if one includes the potential minimum) and just two pieces of information would suffice. We have presented a method using the minimum information-the first and second spectral moments of the absorption profile-to determine the harmonic approximation to the excited-state potential in the Franck-Condon region. This method takes advantage of the time-dependent view of how an absorption spectrum is built up as a consequence of the vibrational dynamics on the excited-state surface, which a fitting procedure such as FranckCondon analysis does not. Franck-Condon analysis using a harmonic model cannot give the sign change for the bond length shift; either an anharmonic model or qualitative knowledge about the difference between the ground- and excited-state electronic structure is necessary. But the sum rule method with just a harmonic model is sufficient to give an unambiguous sign change
J . Phys. Chem. 1990, 94, 4425-4431 for the bond length shift, and this is clearly easier to use. The current sum rule method gives one bound to the excited-state bond length, and this is useful as a consistency check when we proceed to higher order sum rules for anharmonic models. Solving the two simultaneous equations (2.25) and (2.26) for the frequency and shift of the harmonic model is computationally more efficient than a Franck-Condon analysis to fit the absorption spectrum.
4425
Finally, the energy-frame sum rule approach is also more direct and efficient than the time-frame autocorrelation function approach, although both approaches are equivalent.
Acknowledgment. The author gratefully acknowledges receipt of a research grant RP111/82 from the National University of Singapore.
Time-Resolved Electron Paramagnetic Resonance Studies on the Lowest Excited Triplet States of Aliphatic Carbonyl Compounds Keisuke Tominaga, Seigo Yamauchi,t and Noboru Hirota* Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan (Received: August 14, 1989; In Final Form: January 5, 1990)
The time-resolved electron paramagnetic resonance (TREPR) technique is applied to study the properties of the lowest excited triplet states of aliphatic carbonyls such as acyclic and cyclic ketones, aldehydes, and the methyl derivatives of cyclohexanone. The zero-field splittings (zfs) and the populating rates obtained from the analysis of the TREPR spectra are found to depend considerably on the individual molecule, particularly alkyl groups attached to the carbonyl group. From the analysis of the zfs, the distortion angles in the pyramidally distorted triplet states are estimated, for example, 40' and 32' for acetone and isopropionaldehyde,respectively. The populating rates are also analyzed to examine the triplet'geometry. A good correlation is found between the angles obtained from the analyses of the zfs and those from the populating rates. A brief discussion is given on the reactivities of the triplet states such as hydrogen abstraction and a-cleavage reactions in relation to the observed triplet geometries.
1. Introduction The photochemical and spectroscopic properties of the excited states of aliphatic carbonyl compounds (alkanones) have been a topic of much interest in view of their structures and reactivities,I4 The excited states of alkanones are expected to have distorted structures, being very different from those of the ground states5v6 This structural change is believed to play an important role in photochemical reactions. The photophysical and photochemical properties of alkanones have been investigated extensively in the past three decade^.^-^^ A number of studies have been made on the excited singlet states to clarify the electronic, vibrational, and rotational structure^.^-^^ In particular, recent studies using supersonic jet spectroscopy have revealed details of the potential energy surfaces of the excited singlet states of simple alkanones such as a ~ e t o n e . ~ -As ' ~ for the lowest excited triplet (TI) s t a t e ~ , ~a~ few - ~ l alkanones, such as cy~lopentanone,~~ cyclohexanone,282-indan0ne?~and acetone,3O were examined by the optically detected magnetic resonance (ODMR) technique, and the magnitudes of the zero-field splittings (zfs) and the decay properties of the spin sublevels were obtained. There are also many theoretical investigations on the radiative and nonradiative decay p r o c e ~ s e s . ~ lDavidson -~~ et al. used a b initio calculations to show that the zfs for the distorted TI states of formaldehyde are smaller than the planar However, studies of the triplet properties of alkanones are rather limited compared with those of aromatic carbonyls despite the fact that the simple alkanones are the most fundamental of the carbonyl molecules. Consequently, there is limited information about their zfs and dynamic properties. For example, although the spacings among the triplet sublevels have been obtained from the ODMR studies, there often remain ambiguities about the exact level schemes. We do not have sufficient data on the zfs of TI alkanones to compare with the predictions of the theoretical calculations. As for reactivity, cyclic alkanones are known to undergo a-cleavage reactions in the T,states. This reactivity is known to be sensitive
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Present address: Chemical Research Institute of Non-Aqueous Solutions, Tohoku University. Katahira 2-1-1, Sendai 980, Japan.
0022-3654190/2094-4425$02.50/0
to the ring size and methyl substitution,2J5 but the exact reason for this is unclear. Therefore, it is desirable to make a detailed
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0 1990 American Chemical Society
