Prediction of the Lowest-Energy Non-Koopmans Doublet States and the First Ionization Potentials of Polycyclic Aromatic Systems from Their Triplet-State Energies
M U D A S S I R M. HUSAIN, ZAHID H. KHAN,* and EDWIN HASELBACH Department of Physics, Jamia Millia Islamia (Central University), Jamia Nagar, New Delhi-110025, India (M.M.H., Z.H.K.J; Inter-University Centre for Astronomy and Astrophysics (IUCAA), Post Bag 4, Ganeshkhind, Pune-411007, India (Z.H.K.); and Institut de Chimie Physiqe de l'Universitd Fribourg, Pdrolles, CH-1700 Fribourg, Switzerland (E.H.)
This paper presents some new relations connecting the triplet-state en- ergy of an alternant aromatic hydrocarbon with that of its excited doublet state as well as its first ionization potential. The energy relation between the doublet and triplet states is basically a simplified version of the SDT relation but the predictive power of the proposed correlation is consid- erably better in spite of the drastic assumptions made in the model. Incorporation of molecular size in the doublet-triplet correlation has no appreciable effect on the predictions, but its inclusion in the ionization potential vs. triplet-state energy relation leads to a significant improve- ment in the results. It is explained on the basis of our new finding that the Coulomb term appearing in the expression for the ionization potential behaves in the same manner as the molecular size, which thus accounts for electron interaction in the model. The proposed correlations provide a simple but powerful means to estimate the first ionization potentials and excited doublet-state energies of polycyclic aromatic hydrocarbons directly from the knowledge of their triplet-state energies.
In the early eighties, Haselbach and co-workers L,2 gave very interesting relations for alternant aromatic hydro- carbons connecting their excited singlet (S), doublet (D) and triplet (T) states that originate from the one-electron highest occupied molecular orbital to lowest unoccupied molecular orbital (HOMO ~ LUMO) excitation. Guided by these so-called "SDT relations", Khan et al. 3 suggested a simple correlation between the excited singlet and dou- blet states. In this paper, our approach has been extended further to cover the triplet state as well, thus giving a new relation between the excited doublet state of the cation which is of n o n - K o o p m a n s type 4 and the triplet state of the parent hydrocarbon. In addition, another correlation is proposed between the first ionization potential (IP0 of a molecule and its triplet-state energy. The effect of in- corporation of "molecular size" in the proposed model formulations is also examined.
This work, apart from being academic in nature, is also of considerable practical importance. For instance, in- formation about the spectroscopic data such as ionization potentials for polycyclic aromatic hydrocarbons (PAHs) and electronic doublet states of their monopositive ions is of immense interest in astrophysics since there is strong evidence for such molecular species being found in the interstellar matter, both in their neutral 5,6 and in ionized 7-1°
Received 20 October 1994; accepted 21 January 1995. * Author to whom correspondence should be sent.
forms. It has been estimated that interstellar PAHs may contain a maximum of ~ 100-200 carbon atoms, 5 where- as in the laboratory the number of carbon atoms per PAH is considerably less. In view of this consideration, the proposed relations may be used to furnish information about IPs and doublet-state energies for such polyaro- matic systems for which laboratory data are currently not available. Apart from this benefit, information about doublet-state energies of PAH cations can be very useful in the understanding of fluorescence quenching mecha- nisms involving the electron-acceptor quenching agents where the PAH species produced may be the radical cat- ion. In recent years, Acree and co-workers 11-13 have car- fled out a detailed investigation of fluorescence quenching of a variety of PAHs and have particularly examined the role of nitromethane as a fluorescence quenching agent for discriminating between alternant and nonalternant hydrocarbons.
THEORETICAL CONSIDERATIONS
Doublet vs. Triplet State. Consider the simple Hiickel molecular orbital (MO) model for an alternant hydro- carbon as depicted in Fig. l a. Assuming that the MOs remain unaltered on ionization of the molecule ("frozen core" approximation) and neglecting electron interaction (El) and configuration interaction (CI) in the first instance, the model cannot distinguish between the singlet, doublet, and triplet states arising from HOMO ~ LUMO exci- tation. However, with the use of the electron interaction in the model, the excited singlet (Sx), doublet (Dx), and triplet (Tx) state energies can be expressed as
E(Sx) = ~ , - Em -- Jmn + 2Km, , (1)
E(Dx) = ~. - ~,~ + Jmm - 2Jmn + Km., (2)
E(Tx) = ~, - tim - - Jmn (3)
where ~n and 6 m are energies of the nth and mth MOs, and Jm, and Kin, represent the Coulomb and exchange terms, respectively. The above energy states with reference to the ground singlet state So are shown in Fig. lb. In the ZDO approximation, which holds good for alternant sys- tems, Jmm = Jm," Substituting this expression in Eq. 2, the latter can be rewritten in the form
without e[ectTon interaction with etectron in teract ion
(a) (b) F~o. 1. (a) A Hiickel molecular orbital (MO) diagram depicting the highest occupied MO (HOMO) and the lowest unoccupied MO (LUMO) for an alternant hydrocarbon together with its first ionization potential, IP~. (b) Ground (So), excited singlet (S~), and triplet (T~) states of an aromatic hydrocarbon (M) and the doublet (Dx) state of its monopositive ion (M + ). K,,, represents the exchange term.
With the abbreviations S, D, and T for E(S,), E(Dx), and E ( T , ) , respectively, Eq. 5 assumes the form
D = V2(S + T). (6)
With the inclusion of CI in the model, the above relation is transformed to
D = (S. T) '/'. (7)
Equations 6 and 7 were first given by Haselbach and co- workers, 1,2 and these are commonly known as SDT re- lations.
To derive our own formulation, we now combine Eqs. 3 and 4 to obtain the relation
E ( D , ) : E ( T , ) + Kin,, (8)
which shows that the doublet and triplet states are sep- arated by the exchange term Kin,. Thus, to compute the doublet-state energies from the above relation, one must separately evaluate the exchange term for each aromatic hydrocarbon.
Ionization Potential vs. Triplet State. From the simple MO model in Fig. la, it is obvious that the first ionization potential can be correlated with the HOMO --~ LUMO excitation energy (~, - ~m) according to the following ex- pression:
IPl = - a + V2(~. - E m ) (9)
where a is the HiJckel Coulomb parameter. In the above expression, % -Em is simply the triplet-state energy E(T~) provided that the model does not include electron inter- action. With this assumption, we can write
IPl = - a + lhE(Tx), (without El). (10)
However, if we include electron interaction in the model, ~, - em in Eq. 9 shall have to be replaced by E(Tx) + Jm, in accordance with relation 3. In such a case, Eq. 9 as- sumes the form
2
f3Z W Z
/(-" f #]# \ I o n i z a t i o n ~r"" ""~.
Potent ia l "'If'
q -~ ~DoubLet
, /2 , , ~ , ,',, - / s at. ~---~,
: \ ' , , / / i ! : i r r p e i ',, / ' , : ~ i ~f" ~ / : i : ', : i : 1 st&re i ~¢ ', i
AROMATIC HYDROCARBONS
FIG. 2. Variation of the observed doublet (Dx) and triplet (T,) state energies and the first IPs for different polycyclic aromatic systems.
IP, = - a + I/2E(T,) + 1/~l .... (with El). (l 1)
Computation of the ionization potential thus requires information about the Hfickel parameter a as well as information about the Coulomb term Jm,, which have to be computed for each case. Such a complex approach is naturally very unpractical, and an alternative solution may have to be found for a better applicability of the correlations. For this effort, let us first examine the prob- lem from the viewpoint of observed spectroscopic data on ionization potentials and excited-state energies.
T H E EXPERIMENTAL SCENARIO
A plot of the observed energies for the doublet states, ~4-18 the triplet states,19 and the first ionization potentials 2o for 23 polyaromatic systems is shown in Fig. 2. The corre- sponding experimental data are given in Table I, covering a variety of hydrocarbons. A close examination of the figure reveals a nearly uniform pattern in the variation of the energies for all the three cases. This observation suggests that these spectroscopic quantities may be cor- related with some simple formulas.
RESULTS AND DISCUSSION
E(Dx) vs. E ( T 3 Correlat ion. From the experimental data for the doublet and triplet states plotted in Fig. 2, it ap- pears that the exchange term in Eq. 8 should be a constant. However, its computation for the hydrocarbons under investigation reveals that Kin, has a wide fluctuation be- tween 0.33 and 0.98 eV) This problem can, therefore, be solved by using the parameterization technique in which the exchange term Kin, and the coefficient of E(Tx), which is unity, are replaced by the parameters a and b, respec- tively. Equation 8 may thus be written in the generalized form:
E(Dx) = a + bE(Tx). (12)
The parameters a and b in the above relation are deter- mined by the least-squares fit of the experimental data. For 23 data points, this approach yields to the regression:
APPLIED SPECTROSCOPY 853
TABLE I. Energies (in eV) for the observed triplet states (T~), the excited doublet states (D~), and the first ionization potentials (IP0 of polycyclic aromatic systems along with the calculated values of the latter two. NR represents the number of benzene rings in a molecule, and J~. is the Coulomb integral computed using an SCF procedure.
Observed Calculated Observed Calculated System energies energies energies energies
no. Aromatic hydrocarbons N R J,,,, E(T~)" E(DO" E ( D O ~ IP~ I ~ IP~
Calculated from the relation E(D~) °"' = 0.58 + 0.89E(T0 °b'. d See Ref. 20.
Calculated from the relation IP'~ ~ = 5.85 + 0.70E(T.) °b~. f Calculated from the relation IP? ~ = 6.85 + 0.60E(T~) °~ - 0.10NR.
E(Dx) ~' = 0.58 + 0.89 E(Tx) °b~,
S E [E(Dx)] = 0.18 eV. (13)
The doublet-state energies computed with the above re- lation are given in Table I, and a plot of observed and calculated results is shown in Fig. 3. It is found that the SDT relation (6) has a root mean square error (RMSE) of 0.43 eV, which improves to 0.35 eV when the CI is also considered in the model (see Table II). In contrast, the relatively small RMSE value of 0.17 eV as obtained from Eq. 13 is very impressive. This result occurs despite the fact that one requires only the triplet-state energy to calculate the energy of the doublet state; whereas, for the
TABLE II. Comparison of the root mean square errors (RMSEs) for different correlations for the excited doublet-state energies E(D,), triplet state energies E(T~), and the first ionization potentials IP 1. E~
Relation RMSE (eV) W
Doublet state: E(Dx) ~ = 0.58 + 0.89E(Tx) . . . . E(D,)- ' = 0.5{E(Sx) ob~ + E(T~)oh~} b E ( D , ) - ' = {E(S,)o"s.E(T,)ob~} °.5c
Triplet state: E(T~) . . . . . 4.73 + 0.94IP~ b'" E ( T x ) cat = -9.65 + 1.05IP? bs + 0.15NR"
. Relations proposed in this work. b The SDT relation without configuration interaction (Eq. 6). c The SDT relation with configuration interaction (Eq. 7).
computation of the same quantity, the SDT relation re- quires additional information about the singlet-state en- ergy as well. The decrease in the slope of the regression line from unity can be considered to be due to noninclu-
3.5
E(~)cal:o.s8 ÷0.89 E(Tx)°bs /
SE [EfDx) ] = O.IBeV . /
3,0 • / .
O 0 • •
2.5
2.0 •
1.5
1.0 I I I i I 1.0 1.5 2.0 2.5 3.0 3.5
E (Dx) °bs (eV)
Fio. 3. A plot of observed lowest-energy non-Koopmans doublet states for different PAHs over those calculated from the relation E ( D , ) ~ = 0.58 + 0.89E(Tx) obs.
854 Volume 49, Number 6, 1995
8.5
Iplcal = 5 . 8 5 + 0 . 7 0 E(Tx) °bs
SE(IP1) = 0 . 1 9 e V
8 . 0
> 7 . 5
• e • •
t •
7 . 0
6 . 5 , t t , i i 6 . 5 7 . 0 7 .5 8 . 0 8 .5
Ipl°bs(eV)
FiG. 4. Observed first ionization potentials for PAHs as compared with those calculated from the expression IN"' = 5.85 + 0.70E(T0 °b~.
sion o f CI in the formula, which may appreciably affect the energy levels.
IPi vs. E(Tx) Correlation. To derive a relation between the first ionization potentials and the triplet states, we proceed in a similar manner and replace - a and 1/2 in Eq. 10 by empirical parameters a' and b', respectively, thus t ransforming it to the form
IP~ = a' + b 'E(Tx) . (14)
The values o f the parameters are again determined by the least-squares fit method, which results in the following regression:
I P ? ' = 5.85 + 0.70 E(Tx) ob~,
SE(IP0 = 0.19 eV, (15)
with an RMSE value o f 0.18 eV. A comparison of ob- served and calculated IP~ values is shown in Fig. 4, and the results of calculations are presented in Table I. The large shift in the slope o f Eq. 15 from the expected value o f 0.5 is not surprising since it neglects both the electron interaction and the configuration interaction.
The Effect of Molecular Size. As ment ioned earlier, the incorporat ion of electron interaction in Eq. 11 requires evaluat ion o f the Coulomb term Jm, for each molecular system separately. Using a self-consistent field (SCF) pro- cedure, we have computed the Jm, values for all the ar- omat ic hydrocarbons under investigation, which vary be- tween 2.96 and 4.53 eV. Since this approach is quite cumbersome, we looked for the replacement o f J,, , by a more convenient term, viz., "molecular size". A plot of the number of benzene rings (NR) in molecules over their Jm, values is shown in Fig. 5, which is simply described by the linear relation
Jmn = 4.54 - 0.19 NR,
SE (J,,,) = 0.15 eV. (16)
J m n
(eV)
N R FIG. 5. A plot of the number of benzene rings (NR) in molecules over the energies of the coulomb term J,,. as computed by a SCF procedure. The best linear fit is given by the relation, J,,,, = 4.54 - 0.19NR.
This observat ion suggests that Jmn in Eq. 11 can be re- placed by NR as the third parameter , resulting in the fol- lowing general expression:
IP? ~= a" + b"E(Tx) °bs + c"NR. (17)
F ro m the least-squares fit o f the experimental data, the following regression is obtained:
IP~ t = 6.58 + 0.60 E(Tx) obs - 0.10 NR,
SE (IP,) = 0.08 eV. (18)
The calculated values for the first IPs are given in Table I and a plot o f the observed IPs over those o f the c o r n -
v
~r H
8.5
IP1¢°1 = 6..58 • 0.60 E (Tx)°bs_ 0.10NR /
8.0 sz(z ~,} = o.oe ev
. / 75 /
7.0
6.5 ~ I I I 6.5 7.0 7.5 B.O 8.5
[ p1 °bs (eV)
Flo. 6. Observed first ionization potentials for PAHs as compared with those calculated from the expression It~ ~ = 6.58 + 0.60E(Tx) °~' - 0.10NR, where NR represents the number of benzene rings in a mol- ecule.
A P P L I E D S P E C T R O S C O P Y 8 5 5
puted ones is shown in Fig. 6. The effect of molecular size can clearly be seen in the above relation, where the coefficient of E(Tx) has improved from the previous value of 0.7 to 0.6, which is closer to 0.5, as expected from Eq. 11. The present results show a remarkable improvement over those obtained from Eq. 15, which did not include electron interaction. Thus, the molecular size, in effect, serves as a substitute for the Coulomb term Jm, in Eq. 11. This observation is further supported from our cal- culations, which reveal that the incorporation of molec- ular size in the doublet vs. triplet state correlation (which already considers electron interaction) does not lead to any significant improvement in the results.
To get a better feeling about the proposed regressions for the excited doublet-state energies and the first ioniza- tion potentials, the root mean square errors for the var- ious relations are presented in Table II and compared with those obtained from the SDT relations.
E(T,,) vs. IP~ Correlation. It may often be desirable to get information about the triplet-state energies from the known experimental data for the first IPs. To achieve this, we have derived regressions for E(Tx) using two- and three-parameter fits of the relevant experimental data us- ing the same approach as adopted in the earlier sections of this paper. The regressions obtained are as follows:
The crucial role of molecular size in such correlations becomes once more obvious from the fact that the stan- dard error in Eq. 20 is reduced by one-half with the in- clusion of Ng in the regression.
C O N C L U S I O N
The E(Dx) vs. E(Tx) relation connecting the excited doublet and triplet states of polycyclic aromatic systems has the following special features: (1) it is the simplest possible relation connecting the two states which enables one to predict the energy of the doublet state of an alter- nant hydrocarbon merely if its triplet-state energy is
known; (2) in contrast to SDT relations, it has a better predictive power despite its drastic assumptions such as the frozen core approximation and neglect of configura- tion interaction. Furthermore, it has been demonstrated that the triplet-state energy can furnish information about the first ionization potentials of PAHs with a fairly good accuracy, especially when the relation also incorporates molecular size. This observation is based on our finding that the molecular size plays the same role as the Coulomb term Jm,, and, in effect, it describes the electron inter- action in the model.
ACKNOWLEDGMENTS
Z. H. Khan gratefully acknowledges the Department of Science & Technology, New Delhi, for a research grant, and M. M. Husain is thankful to the above agency for a Junior Research Fellowship. E. Has- elbach expresses his gratefulness to Schweizerischer Nationalfonds zur Frrderung der wissenschaftlichen Forschung.
1. P. Forster, R. Gschwind, E. Haselbach, U. Klemm, and J. Wirz, Nov. J. Chim. 4, 365 (1980).
2. E. Haselbach, U. Klemm, R. Gschwind. T. Bally, L. Chassot, and S. Nitsche, Helv. Chim. Acta 65, 2464 (1982).
3. Z. H. Khan, M. M. Husain, and E. Haselbach, Appl. Spectrosc. 47, 2140 (1993).
4. Z. H. Khan, Can. J. Appl. Spectrosc. 37, 82 (1992). 5. L.J. Allamandola, A. G. G. M. Tielens, andJ. R. Barker, Astrophys.
J. Suppl. Ser. 71, 733 (1989). 6. J. L. Puget and A. Lrger, Ann. Rev. Astron. Astrophys. 27, 161
(1989). 7. A. Lrger and J. L. Puget, Astron. Astrophys. 137, L5 (1984). 8. L.J. Allamandola, A. G. G. M. Tielens, and J. R. Barker, Astrophys.
J. 290, L25 (1985). 9. O. Pafisel, G. Berthier, and Y. Ellinger, Astron. Astrophys. 266,
L1 (1992). 10. F. Salama and L. J. Allamandola, J. Chem. Soc. Farad. Trans. 89,
2277 (1993). 11. S. A. Tucker, W. E. Acree, Jr., B. P. Cho, R. G. Harvey, and J. C.
Fetzer, Appl. Spectrosc. 45, 1699 (1991). 12. V. L. Amszi, Y. Cordero, B. Smith, S. A. Tucker, W. E. Acree, Jr.,
C. Yang, E. Abu-Shaqara, and R. G. Harvey, Appl. Spectrosc. 46, 1156 (1992).
13. S. A. Tucker, H. Darmodjo, W. E. Acree, Jr., J. C. Fetzer, and M. Zander, Appl. Spectrosc. 46, 1260 (1992).
14. Z. H. Khan, Z. Naturforsch. 39a, 668 (1984). 15. Z. H. Khan, Z. Naturforsch. 42a, 91 (1987). 16. Z. H. Khan, Spectrochim. Acta 44A, 313 (1988). 17. Z. H. Khan, Spectrochim. Acta 45A, 253 (1989). 18. Z. H. Khan, Acta Phys. Polon. A82, 937 (1992). 19. J. B. Birks, Photophysics of Aromatic Molecules (Wiley-Intersci-
ence, New York, 1970), pp. 182-183. 20. P. Eilfeld and W. Sehmidt, J. Electron Spectrosc. Rel. Phenom. 24,
