Adiabatic semi-empirical parametric method for computing electronic-vibrational spectra of complex molecules part 1. Polyenes and diphenylpolyenes

Journal of

MOLECULAR STRUCTURE ELSEVIER

Journal of Molecular Structure 407 (1997) 177-198

Adiabatic semi-empirical parametric method for computing electronic-vibrational spectra of complex molecules Part 1. Polyenes and diphenylpolyenes V.I. Baranov, L.A. Gribov*, V.O. Djenjer, D.Yu. Z e l e n t ' s o v Vernadsky Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, Kosygin Str. 19, 117975 Moscow, Russia

Received 5 August 1996; accepted 4 October 1996

Abstract

A parametric semi-empirical method for the calculation of the vibrational structure of the electronic spectrum and the determination of the parameters of the molecular excited state potential surface has been developed. The method is based on the adiabatic molecular model and is unique for all sets of parameters of the excited states (first and second derivatives of the matrix of coulombic and resonant one-electron integrals with respect to the internal coordinates). Simplified analytical expressions for the changes in the molecular potential surfaces on excitation, which account only for the first-order terms, are obtained. It is shown that the parameters possess distinct local properties and may be transferred in a homologous series of molecules. The number of most significant parameters, sufficient to describe the molecular model adequately and to obtain satisfactory quantitative results, is very small. Calculations of geometry changes and vibronic spectra for some polyene and diphenylpolyene molecules using only two parameters show good quantitative agreement with experimental data. It is possible to create a special data bank of molecular fragments for vibronic spectroscopy with relatively small structural groups (e.g. H>O = for polyenes and related compounds) and to use it to compute the excited state properties of complex molecules and their vibronic spectra employing the suggested parametric method. © 1997 Elsevier Science B.V. Keywords: Diphenylpolyene; Excited state property; Polyene; Vibronic spectra calculation

I. Introduction

Based on the theory of electronic-vibrational spectra, the spectral method for the investigation of the structure and physical and chemical properties of complex polyatomic molecules in the excited electronic state is one of the most promising. An analysis of the vibrational structure of the absorption and fluorescence spectra provides more complete, high-quality * Corresponding author. Fax: 00 7 095 938 2054.

data on the electronically excited states of complex molecules than the much used quantum chemical investigations of pure electronic states and transitions of polyatomics. Considerable progress has been made in experimental methods in recent years and the possibility exists for the recording of high-resolution fine structure spectra. On the one hand, the available methods (such as Shpolskii spectra and free jet spectra) have been refined and, on the other, new methods have appeared (e.g. zero kinetic energy photoelectron

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V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

spectroscopy). Experimental spectra with a resolution of about 1 cm -I and the bandwidth of a single vibronic line of the same order are not unique. This fact essentially enhances the use of theoretical methods in vibronic spectroscopy and calls for their further development. Two approaches, starting from adiabatic and nonadiabatic molecular models, may be used for the development of a theory. Earlier [ 1], we justified the idea of constructing the non-adiabatic parametric theory of molecular spectra based on the direct solution of the total variational problem. The fundamental feature of such an approach is the lack of the concept of a molecular potential surface (PS) itself, whereas the vibrational structure of the spectrum is specified by the vibronic coupling matrix elements of the variational problem. Thus these matrix elements can be taken as the molecular model parameters [2]. In contrast, the adiabatic approach has its origin in the concept of molecular PSs in the ground and excited electronic states, and their difference determines the vibrational structure of the electronic spectrum. Much success has been achieved in this line of investigations and reasonably accurate and efficient methods for computing all the matrix elements have been devised [3-5]. Hence, the basic challenge of adiabatic theory lies in the determination of changes in the PS parameters (geometry and force field) on going from the ground state to the excited state. Leaving out the consideration of the isomeric and conformational intramolecular rearrangements, we can state from the experimental data that these changes are reasonably small (in the region of several per cent) for large-sized complex polyatomics. To find them, three approaches may be used. The first method involves the direct ab initio quantum chemical calculation of the PS parameters. However, this does not provide the required accuracy (primarily due to errors in the adiabatic approximation [6]) and also involves empirical parameters (in particular, scaling factors [7]) even for the ground state calculations. Moreover, owing to the size of the molecules of interest, it is impractical to perform the extensive geometry optimization in the excited electronic states; therefore there is a need to use approximate semi-empirical methods, which also are not sufficiently accurate and efficient for computing excited

PSs of complex molecules. The best method (QCFF/ PI method and its modifications [8]) has errors comparable with the changes in geometry and force constants on excitation. The second approach, which involves the calculation of PS displacements along the normal coordinate axes via the corresponding matrix elements of vibronic coupling [9], has its main disadvantage in the fact that there is no way to determine the molecular model parameters of the excited states in the internal coordinates. The third approach uses semi-empirical correlations between the bond order, bond length and force constant (BOLF relationship) for the bonds [3,10] and the technique of atomic rehybridization by excitation for the valence angles [11]. This approach makes it possible to obtain theoretical spectra which not only qualitatively conform to the experimental results, but also quantitatively reproduce the main spectral regularities (see, for example, Refs. [3,5,11,12]). By this means, it is possible to interpret the spectra as well as to adjust the molecular model parameters through the solution of the inverse spectral problem [13]. Nevertheless, no wholly satisfactory theory within the framework of the adiabatic approximation is yet available, and its further development is required, above all in the choice of an efficient parameter set. The performance of reasonably accurate, large-scale, fast and, more importantly, predictive calculations of the vibronic spectra is of fundamental importance for the spectral investigation of complex polyatomics. As shown in Refs. [4,14], this can be done only with the help of a parametric approach based on the "data bank" idea and a physically correct molecular model and parameter set. The latter, as well as the method as a whole, must have a semi-empirical nature (in particular, because of significant errors of the adiabatic model itself [61). The possibility of devising a parametric semiempirical theory of electronic-vibrational spectra based on the adiabatic molecular model and a special physically meaningful set of parameters has been discussed elsewhere [ 15]. This approach has its origin in the description of the molecular PS with the help of two parameter sets, i.e. the first and second derivatives of the coulombic and resonant one-electron integrals, on the basis of hybrid atomic orbitals (HAOs) with respect to the internal coordinates. The

V.L Baranovet al./Journalof MolecularStructure407 (1997) 177-198 first derivatives determine the changes in molecular geometry and the second the changes in the force constants. These parameters are identical for the different electronic states, including the ground state. Their peculiarities were analysed qualitatively in Ref. [2]. Within the framework of such a model approach, analytical expressions for the changes in the PS on excitation may be simplified to the major terms, diagonal in the HAO basis. In addition, by accounting for the structure of the electronic density matrix rearrangements, the local properties of the parameters and their differences in magnitude, it can be determined that the number of these parameters is sufficiently small. The transferability of these parameters in a series of related compounds and the optimal semi-empirical choice of their values for a given molecular fragment can provide a satisfactory description of the molecular spectral properties in the excited adiabatic electronic states. These points are examined in this paper in detail. On the basis of concrete calculations (for polyenes and diphenylpolyenes), the quantitative analysis of the parameters is performed. The possibility of the prediction of the vibrational structure of the electronic spectrum is also discussed.

is necessary to use another set of "hidden" parameters, which offer the following fundamental properties: 1. clear physical meaning and close relation with PS parameters; 2. dependence on peculiarities of small-sized local atomic groups of molecules ("locality") which makes it possible to isolate typical structural groups in a molecule (molecular fragment); 3. transferability in a homologous series of related compounds containing typical fragments; 4. difference in magnitude and the ability to choose a moderate number of the most significant parameters for a given molecular fragment; 5. independence from small changes in the electronic density distribution, which allows the use of a unified parameter set for different excited electronic states. To construct a consistent parametric semi-empirical theory of electronic-vibrational spectra, we use the method given in Ref. [16] for the determination of the excited state PS parameters and some ideas on molecular model parametrization from Refs. [2,15]. In the harmonic approximation, the PS of the nth electronic state takes the form 1 - OAn QO E" =E~ +hnQ° + ~a

2. Adiabatic parametric approach In the adiabatic approximation, molecular models in combining electronic states are described by the PS parameters, namely the location of the PS minimum (equilibrium geometry, s) and its curvature (force constants, u). This parameter system is reasonably convenient for constructing a parametric theory of IR spectra because it has clear physical meaning and transferability in a series of related molecules or molecular fragments. However, the vibrational structure of the electronic spectrum is defined by the changes in the PS parameters on excitation (As, Au) which are sufficiently small for complex molecules (of the order of 5%). Consequently, the requirement on their transferability is two orders of magnitude greater than that for the parameters of the ground state (s °, u°). In addition, the values of these parameters are significantly different for various excited states. Thus, in the parametric theory of vibronic spectra, it

179

(1)

where Q0 are the normal coordinates (NCs) of the molecular ground state and E~ is the energy of the nth state at the equilibrium geometry (Q0 = 0). For the ground state, we have

1 ,.0AQ0 E" =E~0+ ~O.°A°a°=E°+ ~Q

(2)

where the diagonal matrix of the squared normal frequencies A in the ground state can be found by the standard technique for computing IR molecular vibrations [4,17]. According to the Hellman-Feynman theorem, the generalized force aN conjugated to the NC Q~ (taken with the opposite sign) will look like

c?En f . c?Hc aN= ~ - - / O j ~,odV Q~

(3)

or in matrix form

a.=Sp(t~ OHc'~ OQOj

(4)

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V.L Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

where P" is the electronic density matrix in the nth state, He =H+SVnn, H is a matrix of coulombic and resonant one-electron integrals, S is a matrix of AO overlap integrals and V~n=~,a.b(ZaZb/rab ) is a nuclear potential energy. Analogously, the matrix of the second derivatives can be defined as follows

.

/

O2He "~

l

/ae"O.e

OP"OH~'~ (5)

With Eq. (1) and the well-known relation between the normal and internal coordinates for the ground state qO =kOqOo, it is easy to determine the changes in molecular geometry on excitation [16]

As n = s n - s ° = - L° (A n) - Ja n

(6)

When deriving Eq. (6) we assume that the system of internal coordinates itself does not change on excitation but only its origin location. In the case of the small structural changes studied here, this assumption always holds. For the ground state, we have a ° = 0, and so eqn (6) can be rearranged as follows A s n= - L q0( A n ) - I A a n

(7)

where

Aan=a"-a°=Sp(AP

" OHe'~ :L°qSp(Ap" OHe'~

oooj

qO} (8)

and A/~ = / ~ - P ° is a matrix of electronic density change. The force constants in the excited electronic state u~l can be easily found when it is considered that the linear term in Eq. (1) (expressed as a function of the internal coordinates) must vanish 1 -n-O-ngO

E n =Enin + ~ q L e a l~pq

n

n

. 0 --n~'0

u =%,a %

(9)

(10)

Then, for the force constant changes, we have the expression 02He

Au"k t = oa P [/ A r" ~ } \

\

oqkoql /

1 I" OAP n OHe OAP n OHe'~ + ~ Sp ~ ~ OqOt t- Oq~ ~q~ j

(11)

The changes in the molecular parameters, as a rule, are small in comparison with their equilibrium values; therefore we can restrict our consideration to the firstorder terms in Eq. (7), replacing A n by A ° and dropping the term proportional to the product AAnAa ". By this, the force constants (as well as the vibrational frequencies) are considered to be equal to their values in the ground state when computing the geometry change. The error in the determination of As ~ is certainly less than 10% for such an approximation. Then, with Eq. (2) and Eq. (8), we have

As n = _ LOqA- I Aa n = _ LqA

LqSp AP n c~q0}

(12)

In turn, neglecting the derivatives of the density matrix change, we can write, for the force constants, the following expression correct to the first-order terms)

\

Oq~Oq~,l

(13)

Within the framework of this approximation, based on the small PS change on excitation, the relationship between the change in the electronic density matrix and the change in the geometry and force constants is linear. This result agrees well with the familiar BOLF relationships [3,10], which are also close to linear over a sufficiently wide range of density matrix change. This confirms the correctness of using such an approximation in the parametric semi-empirical theory of electronic-vibrational spectra. It should be noted that all the necessary quantities on the right-hand side of Eq. (12) and Eq. (13) can be calculated with the help of exact ab initio quantum chemical methods (or semi-empirical methods). However, as mentioned above, this is not to say that the PS parameters defined by this strategy will allow the spectral experiment to be described with the required accuracy, because the adiabatic model itself has significant errors [6]. Thus, instead of exact matrix elements, it is necessary to use empirical parameters, which are not close to the exact values; moreover, they must differ essentially from those values. For example, currently available ab initio force constants involve the use of empirical factors different from unity (of the order of 0.7-0.9 [7]) even for the ground electronic state.

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198 Reference to Eq. (12) and Eq. (13) shows that the changes in the molecular PS parameters As~, Au~; are functions not only of the density matrix change but also of the first (OHe/Oq °) and second (02He/(Oq°Oq°)) derivatives of the He matrix with respect to the internal coordinates at the equilibrium geometry. These are precisely the quantities which are the desired "hidden" parameters of the theory satisfying all the necessary requirements.

constants, which is well known and much used in the theory of IR spectra. In an effort to elucidate the parameter properties in more detail and to obtain their numerical values, we have performed calculations of polyene molecules (butadiene, hexatriene, octatetraene) with the repetitive structural group H>C= with sp2-type HAOs for the carbon atom and a Is-type AO for the hydrogen atom. Previously devised methods and programs for computing matrix elements and electronic-vibrational spectra [3,18] have been used. These calculations confirm previous qualitative estimations as a whole. The parameters can be divided into groups of a given type depending on the a- or 7r-character of the HAO and the kind of internal coordinate (OHts/Oq~), where index t designates the type of HAO, k is the kind of internal coordinate (b, bond; va, valence angle, etc.) and r, s and i are the serial numbers of HAOs and internal coordinates respectively (see Fig. 1). The parameters have distinct local peculiarities: those immediately related to the given fragment tt>C = have the greatest values and all the others are one order of magnitude (or even more) less. For example, for hexatriene, (OH~/Oqbi)=3(OH~/Oqb)= 7(OH~/Oqb)=zO(OH~/Oqb). Hence it follows that we can separate out the group H>C= with its parameter system for the polyene chains. The values of parameters of the same type vary slightly both for the different fragments in the molecule and for the different molecules in the series. These variations are less than 50% for the maximal case. For instance, parameter OHr'r,laqVia for the different groups is equal to 0.2 for butadiene, 0.2 and 0.3 for hexatriene and 0.2, 0.3

3. Peculiarities of the parameters With the use of HAOs, parameters OHe/OqOk, 02He/(OqOOq °) are defined by the structural features of the local molecular fragment and independent of the electronic density distribution. Thus this parameter set is unique for different excited electronic states and accounts for the fragmentary structure of polyatomic molecules. Even qualitative analysis [2,15] shows that these parameters must offer all the properties mentioned above that are necessary for semi-empirical theory. In particular, their locality is a consequence of familiar local properties of coulombic and resonant integrals which find wide application in modem semi-empirical theories of electronic spectra and structures of polyatomics (see, for instance, Ref. [4]). After differentiation with respect to the internal coordinates, this property must become even more distinct. Because long-range interactions caused by electronic density redistribution are absent in these parameters, their transferability in a series of related molecules must also be better (in any case, not worse) than that of force

xr= x o"

~-.

x Tr

.4-- "-~"~'-J

x=

~-d J./"~,~,.)-,

s ~ i /

q Vb k \ x~_, \ ~

181

I

r'

qbm / . ~ , , .

Fig. 1. HAOs and internal coordinatesof molecularfragment.

ab

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V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198 o

b

and 0.4 for octatetraene, and parameter OHrJOqi is equal to 0.4, 0.5; 0.4, 0.5, 0.6; and 0.5, 0.7, 0.8, 0.8 respectively. Thus we can remark on the transferability of these parameters and construct complex molecular models containing polyene chains with the help of the fragmentary approach bonding the separate fragments together. The number of parameters is rather small because, even for the given fragment, they vary in magnitude by more than two orders. Thus we can separate out the most significant parameters while neglecting the others. These parameters for polyenes are listed below (average values are given in parentheses): r b OH~,/Oqb (2.0), OHrr/Oqi J (0.8), OH~,/Oqbi (0.5),

OH~s/Oq b (0.3), OHr~/Oq va (0.3), OHrrr/OqiV,?i,, (-0.15). The changes in the molecular PS parameters As, Au depend not only on the values of these derivatives but also on the density matrix change. The peculiarities of AP" for the 7r-Tr* transitions in conjugated polyatomics (in particular, for polyenes) lead to the fact that the parameters OH~JOqbi and 02Hr~/(Oqbi) 2 make the major contribution to As, Au (more than three orders of magnitude greater than that of the other parameters). Therefore the number of parameters describing the lowest excited states for r-electron

molecular systems is very small. To a first approximation, for polyene molecules, we can account only for two of them, namely OHrJOqi '~ b which defines the change in molecular geometry and 02H~s/(Oqb) 2 which defines the change in the force constants.

4. Results

and discussion

To check the efficiency of the parametric theory of the vibronic spectra developed here, to determine the possibility of performing predictive calculations and to evaluate the numerical values of the parameters used, it is necessary to perform calculations of the molecular structure in the excited states and the vibrational structure of the electronic spectra for different series of molecular compounds and to compare them with experimental and other theoretical results. Molecules with a well-defined vibrational structure of the electronic spectra, such as polyenes and their derivatives (in particular, diphenylpolyenes), which have been intensively studied by theoretical and experimental methods, are ideal for the analysis of the suggested method.

Table I Changes in bond lengths (Al,, A) and valence angles (Aa,, rad ~of butadiene, hexatriene and octatetraene on excitationa nb

Butadiene

Hexatriene

Octatetraene

-0.091 (-0.060) 0.088 (0.098)

-0.064 (-0.040) 0.050 (0.053) 0.095 (0.072)

-0.048 (-0.043) 0.031 (0.034) 0.064 (0.073) -0.072 (-0.065)

0.028 -0.021 -0.007 -0.034 -0.029 0.063

0.013 -0.011 -0.003 -0.022 -0.015 0.037 0.063 -0.036 -0.027

0.007 -0.007 -0.001 -0.016 -0.009 0.025 0.035 -0.017 -0.017 -0.028 -0.005 0.033

Bonds 1

2 3 4 Angles 1

2 3 4 5 6 7 8 9 10 I1 12

a The values obtained with the BOLF relationship are given in parentheses. bFor internal coordinate numbering, refer to Figs. 2-4.

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177 198

4.1. Short polyenes: butadiene, hexatriene and octatetraene For these molecules, changes in the geometry and force constants have been calculated for the transition to the first excited singlet state of ~Bu symmetry within the framework of this simple parametric adiabatic model. Because there are no first-hand experimental data on the molecular structure in the excited states, the results are compared with the estimations of these values using the BOLF relationship obtained previously [ 12,13]. Any parametric method is semi-empirical in nature, so that it calls for the adjustment of the parameter set with regard to the experimental data. This adjustment accounts for the errors caused by inaccurate quantum chemical calculations of the matrix elements as well as the adiabatic model as a whole. Such correction of parameters for the I'I>c= group has been performed in this case. The results of the calculations with the parameter bx2 = 0.1 values OH~s/Oqb=O.055 a.u. and 0 2Ha-r¢,(0 qi) a.u. are listed in Table 1 and Table 2. It is evident from Table 1 that the use of one parameter common for all the groups in all the molecules makes it possible to obtain quantitatively satisfactory values of the geometry changes. The appropriate sign of the bond length changes and their numerical values are obtained (variation averages 15%; the maximal value is 35% for hexatriene (HT); variations of the same order also occur when solving inverse vibronic problems [13]), and the same is true for the force constants. The parametric approach also allows the calculation of changes in the valence angles (as well as any other kind of internal coordinate), which have been considered as being equal to zero in previous

183

calculations. These values are sufficiently less (in relative units) than the bond length changes (on average, 20%). This fact is in general agreement with the data obtained when solving inverse problems for HT and octatetraene (OT) (15%, see Ref. [13]). For most angles, we also have good quantitative agreement with the results of the inverse problem solution. Agreement between the calculated and experimental spectra serves as a major criterion for the correctness of the devised approach and the chosen parameter set, as well as for adequate consistency between the actual and model structures of the molecules. For the most part, this is due to the lack of direct experimental methods for the determination of the molecular geometry in the excited states and the essential errors of quantum chemical evaluations. We have carried out calculations of the absorption spectra (So ~ $1) of butadiene (BD), HT and OT molecular models. The results depicted in Figs. 2 - 4 show satisfactory qualitative and quantitative agreement with the experimental data. In all instances, prominent vibrational bands seen in the experiment are well reproduced. Their locations (vibrational frequencies) and intensities are predicted properly: the errors in the frequencies are less than 50 cm -1 (200 cm -l for low-resolution spectrum of BD with F W H M - 6 0 0 cm-J), and those in the relative intensities are less than 30% (for BD and HT, 10%). Not only are the most intense lines reproduced in the spectra, but also the rather weak vibrational structure is observed (such as the low-frequency bands at approximately 200 and 300 cm -j and the weak 1100 and 1300 cm -1 bands near the moderately intense 1200 cm -~ band of OT, see Fig. 4). An important point is that successive refinements of the molecular model (namely, accounting first only for the changes in bond lengths and then for the

Table 2 Changes in the force constants (Au.., × 10 6 cm 2) of butadiene, hexatriene and octatetraene on excitation a nb Bonds 1 2 3 4

Butadiene

Hexatriene

Octatetraene

0.702 (0.590) -0.962 (-1.070)

0.492 (0.370) -0.482 (-0.530) -0.906 (-0.950)

0.334 -0.298 -0.642 0.514

a The values obtained with the BOLF relationship are given in parentheses. b For internal coordinate numbering, refer to Figs. 2-4.

(0.250) (-0.340) (-0.720) (0.380)

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V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

1,o-

1/'?'

0,5-

I

1200

J

1

!

2400

Wavenumbers (era - I ) Fig. 2. Experimental (1) [19] and calculated (2) absorption spectra of butadiene.

changes in the force constants and valence angles) lead to an improvement in the agreement between the theoretical and experimental spectra for all three molecules. This is clearly seen in Fig. 3, where we present the results of such calculations for HT. The most prominent feature here is the correct redistribution of intensity between the vibrational bands at approximately 1200 and 1600 cm -I. We emphasize that no individual parameter fitting has been performed when computing these spectra.

Only two non-zero parameters equal for all the molecules have been used. Even such a crude "first approximation model" allows the calculation of not only the low structure spectrum of BD but also the sufficiently high-resolution spectra of HT and OT.

4.2. Decapentaene We have attempted to calculate and interpret the fine structure spectrum of the S0(1 lAg) ---* $2(11Bu)

ro-~ i

o.5J, iI • o

0

o

tO00

2doo

\ °,

~.

3Joo

Fig. 3. Experimental (I) [20] and calculated (with (3) and without (2) accounting for the valence angle changes) absorption spectra of hexatriene.

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

185

1.0-

I /\) 0,5-

l

~,0-

l

I

i

O, 5-

0

.if

2

.

J,

2(Joo

'

-4dO0

k,6 '

Wavenumbers (era- I ) Fig, 4. Experimental(I) 121] and calculated (2) absorptionspectra of octatetraene. transition for the next molecule in the polyene series: decapentaene CIoHI2 (DP). In doing so, we have rested upon the experience gained when computing BD, HT and OT and used all the previously devised methods, such as the fragmentary approach, the set of "hidden" parameters and the fast and accurate algorithms for matrix element calculations. It is worth noting that no additional approximations or fittings have been used, because the investigation of the possible performance of predictive calculations starting from the available data bank of molecular fragments and parameters is among the major problems to be solved in this work. The vibronic spectra of the first five polyenes have been experimentally studied in Ref. [22]. It has been shown that all the spectra are similar and may be described with the help of two frequencies (us = 1234 cm -I and v~ = 1626 cm -l in the case of DP) which correspond to the single C - C and double C=C stretching vibrations respectively. However, in Ref. [23], there is another experimental spectrum of DP, which differs drastically not only from that in Ref. [22] but also from the typical spectra of both shorter [20,21,24] and longer [25,26] polyenes. Strong anharmonicity and vibrations with a total quantum

number equal to five are present in this spectrum, whereas they are missing from the other spectra. These peculiarities call for special investigation and can be elucidated only through the theoretical calculations and analysis performed in this work. The DP model in the ground state has been constructed with the fragmentary approach by "gluing" the fragments of shorter polyenes together and transferring the force field parameters. The role of terminal groups decreases with increasing chain length and the values of the parameters for interior groups H>C = tend to the equal limiting value. Therefore the values of the bond lengths and force constants at the junction of two fragments have been taken to be equal to those of the neighbouring interior groups. Further quantum chemical calculations of the electron density distribution and evaluation of these parameters using the BOLF relationship give analogous results. Experimental data [27] and other theoretical estimations [28-30] also support this conclusion. For example, in Ref. [28], the following values were given: 1.347 A for the terminal C=C bonds; 1.361 and 1.363 ,~ for the interior C=C bonds; 1.464 and 1.459 ,& for the interior C - C bonds (the difference in length for bonds of the same kind is less than 0.4%).

186

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

Table 3 Bond lengths (l, ,~) of decapentaene in the ground (So) and excited (S ~, $2) states" Bondb

So

Sz

$2 [29]

S~ [28]

CI=C2 C2-C3 C3=C4 C4-C5 C5=C6

1.337 1.458 1.368 1.458 1.368

1.357 (20) 1.429 (-29) 1.421 (53) 1.404 (-54) 1.435 (67)

1.354 (19) 1.444 (-31 ) 1.413 (72) 1.411 (-62) 1.413 (72)

1.381 (34) 1.411 (-53) 1.440 (79) 1.396 (-63) 1.429 (56)

"The changes in bond lengths (,5l, x 10 ~ ,~) are given in parentheses. b Atoms are numbered sequentially from the beginning of the chain. In our model, the~, are considered to be equal to 1.340, 1.350 and 1.455 A respectively. The calculation of the IR spectrum in the ground state shows very good quantitative agreement of the vibrational frequencies with the experimental data, so that further refinement of the model parameters by solving the inverse problem has not been performed. To determine the PS properties of DP in the excited $2 state, we used the parametric semi-empirical method discussed above, with the parameters obtained from the calculations of the shorter polyenes (BD, HT and OT) without any correction of their values. By this means, we tested the transferability of the parameters and the possibility of carrying out predictive calculations of the spectral and structural properties for longer molecules of this series. The results are shown in Table 3. It is of interest to compare our data with those of other theoretical calculations (also given in Table 3) I and experimental estimations. As can be seen, the results agree well as a whole, having identical signs for the changes in bond lengths and similar ratios of different values. The average difference is 0.01 ~, (20% of the length changes and less than 1% of the lengths themselves). Considering the accuracy of quantum chemical methods, this result may be thought of as wholly satisfactoryo. Experimental evaluations [22] ( ~ c = c = 0.085 A and ARc_c = -0.081 A; changes in all bonds of the same kind are considered to be equal) are also of the same order. Their slight overestimation is probably attributable to the neglect of the changes in the valence angles. The fundamental peculiarity of the new method is

its ability to provide a means of computing changes in internal coordinates of any kind, including the angles. The changes in the valence angles of DP (see Table 4) are comparatively small so that, with regard to the forms of the normal vibrations, we might expect that the deformation vibrations are not active in the spectrum, and the experiments confirm this conclusion. Nevertheless, consideration of the valence angle changes becomes very important for computing the intensities of the active valence vibrational modes (see the analogous results for the shorter polyenes). Some regular trends are observed for the geometry change in polyenes on excitation. They are well known for the bonds: single C - C bonds decrease and double C = C bonds increase, with the average

Angleb

S0

$2

$1 [28]

CI-C2-C3 C2-C3-C4 C3-C4-C5 C4-C5-C6 HI-CI-H2 C2-C1-H1 C2-CI-H2 CI-C2-H3 C3-C2-H3 C2-C3-H4 C4-C3-H4 C3-C4-H5 C5-C4-H5 C4-C5-H6

122.4 122.4 122.4 122.4 120.4 119.8 19.8 19.8 17.8 17.8 19.8 19.8 17.8 17.8

122.1 (-0.3) 121.2 (-1.2) 121.4 (-1.0) 121.0 (-I.4) 120.7 (0.3) 19.7 (-0.1) 19.6 (-0.2)

121.6 (-0.7) 121.7 (-0.1) 121.8 (-0.1)

We emphasize that the accuracy of the available quantum chemical methods is still low when computing such large molecules, and the aim of this comparison is to display qualitative agreementof the results obtained by different approaches.

aThe changes in the valence angles (As, deg) are given in parentheses. b Atoms are numbered sequentially from the beginning of the chain.

Table 4 Valence angles (o~, deg) of decapentaene in the ground (So) and excited (S ~, $2) states~

l 18.6 (-3.2)

19.2 (-0.6)

18.7 (0.9) 19.8 (2,0) 19.0 (-0.8) 18.6 (-1.2) 12O.0 (2.2) 120,4 (2.6)

187

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

increase in terminal C=C bonds being 30%-50% less. The absolute values of As decrease monotonically with increasing chain length, tending to limiting values for an infinite chain (polymer). Trends can also be determined for the valence angles. First, it is clearly seen that the C - C - C angles always tend to decrease, so that the molecule shrinks. Secondly, the terminal H - C - H angles increase sharply asymmetrically and, consequently, the C - C - H angles change so that those located on one side of the molecule bend in the same direction, with opposite directions on opposite sides of the molecule. By this means, the asymmetry of the molecule decreases, since the greater C C - H angles decrease and the smaller ones increase. The maximum values occur for the central atomic groups and they rapidly decrease towards the ends of the chain. Thirdly, the absolute values of the angle changes decrease monotonically with increasing chain length. Starting from these general regularities, we can now predict qualitatively the geometry change on excitation for any molecule containing polyene fragments. The calculation of the S~ ---* $2 transition vibrational structure for DP has been carried out with the help of the approach described elsewhere [5]. The result is depicted in Fig. 5. As for the shorter polyenes, accounting for angle changes is essential in this case because they introduce important corrections to the relative intensities of the observed bands. In

particular, they are responsible for the correct ratio between the relative intensities of the two main bands (single- and double-bond stretching modes) with calculated frequencies of 1249 and 1613 cm <. Comparison of the calculated spectrum with the experimental spectrum (see Fig. 5) shows very good agreement for both the vibrational frequencies (errors are less than 20 cm <) and relative intensities (5%-10%). Hence, the molecular structure in the excited state is described with a high degree of accuracy by the molecular model provided. This is especially important because we have used the minimum possible number of parameters for the excited state model

1

2

-360o

6

Wa.venumbers (cm-t) Fig. 5. Experimental(1) [22] and calculated (2) absorptionspectra of decapemaene.

1

1

L

t

!

I

I

1

1

. !

t

l

Wavenumbers (x 10 ~ cm-') Fig. 6. Experimental (l) [23] and calculated (2) absorption spectra of decapentacne.

188

V.1. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

Table 5 Vibronic band positions (u, c m - i ), spacings (Al,, cm -I) and anharmonicity (anh = Au - Au~,k., cm -i ) for all-trans-decapentaene n

~,~

Assignment ~

Au ~

anh

Assignment

Au

anh

1 2 3 4 5 6 7 8 9 10 11 12 13

30390 31660 32040 33260 33570 34790 35190 36020 36410 36830 37620 38040 38490

Origin u~ ~d ~ + ud 2ua ~ + 2~j 3~ 2(v~ + ~ ) ~ + 3Ud 4vd 2 ~ + 3~d U~ + 4~d 5Ua

1270 1650 2870 3180 4400 4800 5630 6020 6440 7230 7650 8100

50 120 170 150 190 200 160 260 220 150

Origin ~ ua 2u~ ~ + ~d 2ud 2P~ + ud U, + 2~d 3Ud

1220 1620 2450 2840 3260 4050 4470 4920

l0 0 20 -10 -10 60

a Ref. [231.

and have not performed any fitting of the parameters, i.e. the calculation is not only qualitative but also quantitative prediction. Shown in Fig. 6 is another absorption spectrum of DP obtained in an argon matrix at 20 K [23]. It should be mentioned that its interpretation is a considerable challenge because of the intense background and low resolution, but it is immediately obvious that this spectrum does not have a structure typical of linear conjugated polyenes [19-22,24-26]. A more detailed analysis displays the following peculiarities. First, the vibrational frequencies at 1270 and 1650 cm -I are slightly overestimated when compared with other experimental data [22]. Secondly, and more importantly, the essential anharmonicity reported but unexplained in Ref. [23] is present in the spectrum (on average, 150-200 cm -I for all the bands, see Table 5), whereas it is completely absent in the spectra of both the shorter [20,21,24] and longer [25,26] polyenes (the anharmonicity is, as a rule, less than 15-20 cm-~; maximal value is about 50 cm J for the 3~'d band). Finally, some typical bands (such as 2u0 are absent in the spectrum, so that it does not have a characteristic appearance; instead, vibrations with a total quantum number of five are observed. (In other available experiments, there are no overtones with a total quantum number greater than four.) In this context, some doubts are cast on the treatment of this spectrum, in particular, on the assignment of the 30 390 cm -~ band to the 0 - 0 transition. It can be seen that, if we take the 33 570 cm -j band (3180 cm -~

distant from the first) as the origin, and consider the part of the spectrum starting from it, all the problems noted above are solved at once. The frequencies become equal to 1220 and 1620 cm -~ respectively, and form equidistant progressions with an anharmonicity of less than 20 cm -j (except for the 3pd band, equal to 60 cm -~) and a total quantum number up to three (see Table 5). Their relative intensities are also in much better agreement with those from Ref. [22] and the calculated data. On the other hand, in this case, the assignment of the three intense bands in the region 30 0 0 0 - 3 2 000 cm -~, which separately are typical of the linear polyene structure, still remains unclear. The following explanation can be suggested. The spectrum in Ref. [23] presents a superposition of two spectra from different origins having similar vibrational structure, but different energies of the electronic 0 - 0 transition, To check this assumption, a theoretical spectrum involving two spectra similar to that in Fig. 5 (with the second shifted to the red by 3180 cm -j and multiplied by a common fitting factor of 0.6) is presented in Fig. 6. A very good agreement is observed between this "summarized" spectrum and the experimental spectrum. Consequently, the assumption made is justified, but the question of the nature of the two origins remains unresolved. This cannot be caused by the formation of D P - A r complexes, because the energy shift in this case is tens (maximum, hundreds) of reciprocal centimetres (see, for example, the data for tetracene [31]). It may

189

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

possibly be caused by the presence of different isomers in the sample, although in this case the shift is also about 100 cm -1 [20,21]. The most plausible explanation is that DP loses its C2h symmetry because of structural distortion in the Ar matrix and the lowest excited state S~ also becomes optically allowed. Therefore we observe absorption in both the S I and $2 states with approximately equal probabilities. However, in this case, the difficulty is in the fact that the S~ state in DP lies much lower than $2 (approximately 25 000 cm -1 [32]) and, as far as we know, there are no other electronic states in this energy region (25 0 0 0 - 3 5 000 cm-I). Thus this problem needs further experimental and theoretical investigation, in particular, accurate ab initio calculations of the low-lying states of DP, its isomers and its complexes with Ar,

~6SI

4.3. D e c a t e t r a e n e ISO00

Another attempt to carry out predictive calculations was performed with all-trans-2,4,6,8-decatetraene (DT) (or dimethyl-substituted OT). As with DP, the molecular model was constructed from fragments: two methyl groups were attached to the OT base and the force field parameters at the junctions of the fragments were fitted to attain better agreement with the experimental IR frequencies of the ground state. Thereafter, electronic density distributions in the ground So and first optically allowed S 2(lBu) electronic states were calculated with the CNDO\S method and the changes in the geometry and force field were obtained with the help of the parametric approach described above. The results listed in Table 6 show one interesting feature different from all the molecules investigated previously: the change in the terminal single C - C

34J~0

3?0OO

l~xcit~tion Energy (cm -l) Fig. 7. Experimental (1) [24] and calculated (2) fluorescenceexcitation spectra of jet-cooled decatetraene. bond length is equal to zero (or negligibly small). This means that, for the lowest singlet states, the excitation of electron density is localized within the 7rsystem of the polyene chain and does not affect the terminal substituted groups. As will be seen later, an analogous result is valid for diphenyl-substituted polyenes. The calculation of the S 0 "--' S z absorption spectrum and its comparison with the experimental spectrum taken from Ref. [24] reveal other interesting features not seen in the experiment (see Fig. 7 and Fig. 8; for all the calculated bands, FWHM = 23 cm -j is taken to

Table 6 Bond lengths (/, ,~) and valence angles (ct, deg) of decatetraene in the ground (So) and excited ($2) statesa Bondb

S0

S2

Angleb

S0

S2

C1 -C2 C2=C3 C3-C4 C4=C5 C5-C6

1.511 1.350 1.455 1.350 1.455

1.511 (0) 1.386 (33) 1.412 (-43) 1.410 (60) 1.393 (-62)

CI-C2=C3 C2=C3-C4 C3-C4=C5 C4=C5-C6

122.4 122.4 122.4 122.4

121.75 (-0.65) 122.04 (-0.36) 121.13 (-1.27) 121.38 (-1.02)

"The changes in bond lengths (AI, × 10 -3 ,~) and valence angles (Aa, deg) are given in parentheses. b Atoms are numbered sequentially from the beginning of the chain.

V.I. Baranov et aL/Journal of Molecular Structure 407 (1997) 177-198

190

fi

It tl

t ,

{

I

;

//A t"t

36000

/1

16250

1

~\

36100

Excita,tion Energy (cm -z) Fig. 8. Data points: measured excitation spectrum of jet-cooled decatetraene from Ref. [24]. The full line is the best fit with four Iorentzian functions [24] and the broken line is our calculation.

be equal to the average experimental FWHM of the 0 - 0 and 132 cm -l bands). In the low-energy region (0-1000 cm-l), all the experimentally observed bands are present in the calculation. As in the experiment, the most intense band in this region is at 139 (132) 2 cm -~. Its calculated relative intensity is approximately twofold less than the experimental value, which may be considered as sufficiently good agreement because we used the fragmentary approach and did not solve the inverse problem (therefore the vibrational forms and all the other quantities are rather poorly reproducible for such low-frequency vibrations). The same is also true for the other low-frequency bands at 254 (273) and 404 (405) cm -~, which are present in the calculated spectrum but have underestimated intensities. The calculation shows that more vibronic bands of low intensity occur in this region (at 583 and 944 cm -j) but, because of the strong background, they cannot be reliably distinguished in the experimental spectrum. In the 1000-2000 cm -~ region, the situation is more complicated. The calculation shows that the intense peaks at 1230 and 1651 cm -j, which were assigned to totally symmetric carbon-carbon single- and doublebond stretches in Ref. [24], have in fact a complex structure non-resolved in the experiment: the 1651 cm -l peak consists of two bands at 1641 and

2 Experimental values are given in parentheses.

1662 cm -j with a maximum at 1643 cm -~ and the 1230 cm -1 peak consists of four bands at 1222, 1239, 1260 and 1289 cm -~ with a maximum at 1236 cm -J. Fig. 7 demonstrates that the band at 1289 cm -1 is responsible for the right shoulder of the 1230 cm -I peak, which is clearly seen in the experimental spectrum, but was not fitted by the contour in Ref. [24] because only one lorentzian function was used. The presence of several vibronic lines also explains the broadening of the 1230 and 1651 cm -1 peaks (experimental FWHM equal to 43 and 40 cm -~ respectively) in comparison with the 0 - 0 and 132 cm -I bands (FWHM of 22 and 24 cm-J). With regard to the relative intensity, the 1651 cm -~ peak is in excellent agreement with the experimental peak: 0.54 and 0.52 respectively. For the 1230 cm -z peak, there is some disagreement in the experimental data (see Fig. 7 and Fig. 8): in Fig. 7, an experimental ratio of the 1230/1651 cm -~ peak intensities of approximately 0.62 is obtained which agrees well with the calculated value of 0.6; however, in Fig. 8, this ratio is equal to 0.4. Nevertheless, even in this case, the error is less than 15% from the intensity of the 0 - 0 transition. The intensities of the combination bands at 1365 and 1784 cm -~ are underestimated because of the underestimation of the 132 cm -~ band intensity. As a whole, the calculated spectrum is in very good agreement with the experimental data in terms of the frequencies and relative intensities.

4.4. Long polyenes The last application of the suggested method for the polyenes involves the calculation of the excited state structure and spectra of the optically allowed S 0(11A g) --* $2(1 IBu) transition for all-trans-l,3,5,7,9,11,13tetradecaheptaene and its dimethyl-substituted derivative (2,4,6,10,12,14-hexadecaheptaene). Again, the molecular models were constructed by "gluing" shorter polyene fragments together and the parameters obtained for the first four short polyenes were used without any fitting. The values of the geometry changes are presented in Table 7 and the spectra are depicted in Fig. 9 and Fig. 10. As can be seen from the figures, apart from a slight underestimation of the intensity of the C - C stretching vibration mode at approximately 1250 cm -t, the calculated spectra are in very good agreement with the experimental curves,

V.L Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

191

Table 7 Bond lengths (l, A) and valence angles (a, deg) of tetradecaheptaene and hexadecaheptaene in the ground (So) and excited (S 2) states a Bond b Tetradecaheptaene C1=C2 C2-C3 C3=C4 C4-C5 C5=C6 C6-C7 C7~C8 Hexadecaheptaene C 1-C2 C2=C3 C3-C4 C4-=C5 C5-C6 C6=C7 C7-C8 C8=C9

So

S2

1.340 1.455 1.350 1.455 1.350 1.455 1.350

1.351 1.437 1.378 1,477 1,393 1,404 1.400

1.500 1.350 1.455 1.350 1.455 1.350 1.455 1.350

1.500 1.362 1.436 1.378 1.417 1,393 1.405 1.398

Angle b

So

S2

(I 1) (-18) (28) (-38) (43) (-51) (50)

C1~C2-C3 C2-C3=C4 C3=C4-C5 C4-C5=C6 C5=C6-C7 C6-C7=C8

122.4 122.4 122.4 122.4 122.4 122.4

122.26 121.79 121.97 121.52 121.65 121.47

(-0.14) (-0.61) (-0,43) (-0,88) (-0.75) (-0.93)

(0) (12) (-19) (28) (-38) (43) (-50) (48)

C 1-C2=C3 C2=C3-C4 C3-C4=C5 C4=C5-C6 C5-C6=C7 C6=C7-C8 C7-C8=C9

122.4 122.4 122.4 122.4 122.4 122,4 122.4

122.19 122.23 121.79 121.96 121.55 121,66 121,49

(-0.21) (-0.17) (-0.61) (-0.44) (-0.85) (-0.74) (-0,91)

The changes in bond lengths (Al, × 10 -3 A) and valence angles (Ate, deg) are given in parentheses. b Atoms are numbered sequentially from the beginning of the chain.

which are typical of polyenes. Unfortunately, the available experimental spectra are of medium resolution, and the fine effects in these spectra cannot be discussed; however, it is worth noting that the results obtained confirm the high predictive ability of the method devised and the transferability of the parameters in the polyene series.

4.5. Diphenylpolyenes The calculations performed show that the suggested parameter set describes the spectral properties and structures of the polyenes in the excited states adequately. In this context, the following factors are of considerable interest: 1. the possibility of calculating the vibronic spectra of more complicated molecules within the framework of this parametric approach; 2. the transferability of the chosen parameter set to molecules that do not belong among the polyenes but have similar structure (contain polyene fragments); 3. the extent of universality of the polyene fragment H~,,C=. Thus we performed calculations of the vibronic spectra of some diphenylpolyenes (DPPs):

diphenylbutadiene (DPB), diphenylhexatriene (DPH), diphenyloctatetraene (DPO) and 1,6-di(4'methylphenyl)- 1,3,5-hexatriene (DMPH). DPP models in the ground electronic state were constructed by attaching phenyl rings to the appropriate polyene chain. In deciding on such a model, we accounted for the following facts. Firstly, the geometries of the polyene and phenyl parts differ very slightly from those of separate polyene and benzene molecules. Secondly, quantum chemical calculations show that the electronic density of the fragments remains unchained on connection, i.e. the r-system of the polyene does not conjugate with the r-systems of the phenyl rings. Therefore DPP molecules have three 7r-electron systems, which are weakly coupled. Thirdly, the vibrational modes of the fragments remain independent of one another after connection and the matrix of the vibrational forms has an approximate block-diagonal structure. Therefore it can be stated that the additive attachment of the fragments is quite valid in this case. When constructing DPP molecular models in excited electronic states, we used the parameters OH~/Oqb=O.055 a.u. and 82H~J(dqb) 2 = 0.1 a.u. (obtained previously for the pure polyenes) for polyene chains without any correction. The polyene group H>C= is very similar to the relevant group of the

192

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

I

300

400 W~velength

500

(nrn)

Fig. 10. Experimental (a) [26] and calculated (b) spectra of tetradecaheptaene in 77 K EPA.

-,

,~

300

i

350

A

4(30

i

450

Wavelength (rim) Fig. 9. Experimental(1) [26] and calculated (2) absorptionspectra of hexadecaheptaene. phenyl ring, whereas the values of 3He/Oq ° and 02He] (Oq°) 2 are primarily determined by the geometry of the fragment. The difference is only in the bond order: 0.3 (C-C) and 0.9 ( C : C ) for polyenes and 0.67 for benzene. This structural similarity allows us to transfer the polyene parameter set to the phenyl rings as a first approximation. Moreover, for the So -'-* S~.2 transitions, the change in the electronic density distribution of the polyene fragment is one order of magnitude more than that of the terminal substituted groups (see Refs. [11,33] and discussion about DT above), and the vibrations of the polyene C - C and C=C bonds are still the most active in the spectra of DPPs. The errors caused by this transfer of parameters from polyene to phenyl fragments would thus be expected to be negligibly small and to affect the

spectra only slightly. We carried out calculations with these parameters, with parameters taken from acene molecules [34] and without any consideration of the phenyl rings; these calculations confirm the possibility of such an approach. The results of the calculations are presented in Table 8 and Table 9. Since direct experimental data on these quantities are not available, we also present the values obtained using the BOLF relationship and inverse problem solution. It is evident that the changes in the polyene fragments are essentially greater (5-10 times) than the changes in the phenyl rings. This supports the assumption that the phenyl fragments add little to the electronic-vibrational spectra of DPPs. As with the polyene series, the changes in the bond lengths and force constants increase on moving away from the terminal phenyl rings towards the centre of the polyene chain. Consequently, the changes in the central part of the polyene fragment contribute significantly to the formation of the vibrational structure of the electronic transition. On going from shorter to longer molecules, a peculiar effect of saturation occurs. The changes in

V.I. Baranov et al./Journal of Molecular Structure 407 (I 997) 177-198

193

Table 8 Changes in bond lengths (Al,,, x 10 -3 A) and valence angles (Aa,, × 10 -3 rad) of some diphenylpolyenes on excitation a nb

Bonds 1 2 3 4 5 6 12 14 16 18 20 Angles 1-12 6-12 I-6 1-7 1-2 2-7 2-8 2-3 3-8 3-9 3-4 4-9 4 - 10 4-5 5-10 5-11 5-6 6-11 12-13 12-14 13-14 14-15 15-16 14-16 16-17 16-18 17-18 18-19 19-20 18-20

DPB

DPH, DMPH

DPO

19 (a), 25 (b) ~ , -7 7,8 9, 12 -7. -8 16, 20 -40, -40, -38 (c) 51, 62, 68 -59, -53, - 5 4

12 (a), 10 (b) -5, -5 5,5 6,5 -5, -5 11, 10 -29, -25 40, 34 -50, - 3 0 60, 50

10 (a), 12 (b) --4, -3 4,3 5,5 -4, -3 9, 10 -23, -21 31, 37 -43, -38 45, 49 -53, -45

4, -10 8, 10 -12, 0 -10 1 9 2 6 -8 2 -2 0 -8 5 3 7 3 -10 19,4 4, 0 -23, -4 -20, -17 38, 4 -18, 13

4 5 -9 -7 1 6 1 4 -5 1 -I 0 -6 3 3 5 2 -7 13 5 -18 -17 30 -13 42 -22 -20

3 4 -7 -5 I 4 I 3 -4 1 -1 0 -5 3 2 4 I -5 9 5 -14 -14 24 -10 25 -10 -15 -20 23 -3

Results of calculation using the parametric method (a), BOLF relationship [10] (b) and solution of the inverse problem [13] (c). b For the bond numbering, refer to Fig. 11.

the bond lengths evidently decrease (more distinctly in the polyene part than in the phenyl rings) as the size of the molecule increases. The average bond length changes are equal to 0.050 --- 0.008 A for DPB, 0.045 +- 0.012 A for DPH and 0.039 +-- 0.011 ,~

for DPO. A similar saturation is also common to the angular coordinates. Hence, it may be assumed that, in the infinite chain limit, there comes a point when the addition of new I'I>C= groups will have no effect on the geometry change on excitation.

194

V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

Table 9 Changes in the force constants(Au,,, × 10+cm-2) of some diphenylpolyeneson excitation" nb

DPB

DPH, DMPH

DPO

1

-20 (-19)

-I 3 (-14)

-10 (-9)

2 3 4 5 6 12 14 16

6 (5) -8 (-6) -10 (-9) 7 (6) -17 (-16) 31 (23) -54 (-61) 44 (31)

4 (5) -5 (-4) -7 (-7) 5 (4) -12 (-12) 22 (20) -40 (-53) 38 (35)

3 (3) -4 (-3) -5 (-4) 4 (3) -9 (-8) 17 (10) -31 (-37) 30 (22)

-54 (-65)

-43 (-48)

Bonds

18

20

36 (26)

"The values obtainedwith the BOLF relationshipare given in parentheses. bFor the bond numbering,refer to Fig. 11. This correlates well with the small PS changes in complex molecules and is the reason why the use of PS properties of combining electronic states (such as the location of minima and curvature) faces severe problems when computing the vibrational structure of the electronic spectra of complex polyatomics. Compatibility between the calculated molecular structures and the actual structures is supported by the results of the electronic-vibrational spectrum calculations. The calculated absorption spectra of DPB, I

DPH, DMPH and DPO (and for the last two molecules, the fluorescence spectra as well) are depicted in Figs. 11-14. Good qualitative agreement is seen: all the major vibrational bands are present in the spectra. Frequency errors are less than 50 cm -j (except for the 3000 cm -~ band of the low structure spectrum of DMPH with F W H M - 6 5 0 cm-I). The low-frequency vibrations, which may significantly affect the appearance of the spectrum, are well reproduced. The calculated relative intensities are slightly

....

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V.I. Ba ranov et al./Journal of Molecular Structure 407 (1997) 177-198

]

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underestimated, but this may be caused not only by the errors in the molecular models but also by the low quality of the experimental data. For example, in the experimental spectrum of DPO (see Fig. 14), the intensity of the 1200 cm -l band ( C - C stretch) is greater than that of the 1640 cm q band (C=C stretch). This differs drastically from the intensity ratio of similar bands in the spectra of related molecules (Figs. 11-13) as well as in the structurally similar polyene molecules (Figs. 2-5). Moreover, an opposite ratio between the intensities of the 1200 and 1600 cm -~ bands can be seen in the fluorescence spectrum of DPO. The latter is more correct from our standpoint (note also that the Dushinsky effect is small for this molecule). Similar conclusions may be drawn about the ratio of the 0 - 0 and 0-1 band intensities of DPH. In the experimental spectrum of this molecule, the presence of the strong background hinders comparison with the calculated spectrum. We also carried out estimations of the influence of various molecular fragments and their parameters on the resulting spectrum. In Fig. 11 and Fig. 12, the full lines show the spectra calculated without accounting for angle changes. Even in this case good qualitative agreement with experiment is observed with underestimated intensities of the 1640 cm -J band and its

overtones. The difference is rather significant (approximately 80%) and, by accounting for the changes in angles, a better agreement with the experimental spectra is obtained. Thus it is important that the devised parametric approach permits changes of any kind of internal coordinate to be calculated, as opposed to the previously used semi-empirical BOLF relationship. In Fig. 13 and Fig. 14, the full lines show the spectral curves calculated for the molecular models with frozen phenyl rings, i.e. when only changes in the polyene chain were taken into account. The differences from the results for the total model (chain line in Fig. 13 and Fig. 14) are minimal. This confirms once again the above conclusion about the small phenyl ring contribution to the DPP vibronic spectra. It should be stressed that the solution of the inverse vibronic problem and individual parameter fitting to improve the agreement between the theoretical and experimental spectra were not performed when computing these electronic-vibrational spectra of DPPs. Only two parameters transferred from the polyenes (identical for all the molecules and their H,C= groups) were used, and this crude approximation allows us to reproduce the location and intensity of the major vibronic bands of the vibronic spectra of

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V.I. Baranov et al./Journal of Molecular Structure 407 (1997) 177-198

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W ~ v e n u m b e r s ( c m -1 ) Fig. 14. Emission (a) and absorption (b) spectra of diphenyloctatetraene. Experimental curve (- - - ) [38] and curves calculated with ( - • - ) and without ( - - ) taking into account changes in the phenyl rings.

DPPs. Therefore the concept of the "locality" and transferability of the parameters (see above) is correct and can be successfully used for the investigation and, more importantly, prediction of the structure of complex molecular compounds in their excited electronic states and the electronic-vibrational spectra. 5. Conclusions From the results of the calculations and their analysis, we can conclude that it is possible to predict the electronic-vibrational spectra of complex molecules and their structure in the excited electronic states using the semi-empirical parametric approach described herein. The distinctive feature of this method is the choice of a radically new system of parameters which describe the adiabatic molecular model. These parameters are defined as the first and second derivatives of the matrix of coulombic and resonant one-electron integrals on the basis of HAOs with respect to the internal coordinates. They possess all the properties required to serve as parameters of the semi-empirical theory of vibronic spectra, namely "locality", transferability in a homologous series of molecules, independence from small electronic density change, ability to be ranked by their values and a small number of significant parameters sufficient for the adequate description of molecular models.

This makes it possible to create a special data bank of molecular fragments with the aim of using it in the construction of new molecular models. Relatively small structural groups (for example, of the form H>C= for polyenes and diphenylpolyenes) may be taken as molecular fragments. Of special importance is the fact that the results obtained with this method are stable with regard to small variations in the parameters. Moreover, the method provides the ability to perform fast and accurate predictive calculations of molecular structures and spectra which are reliable and suitable for sophisticated analysis. If need be, these results may be adjusted using the solution of the inverse vibronic problem. The results of calculations of acene and azine [34] molecules using this method confirm the above conclusions.

Acknowledgements The authors gratefully acknowledge partial financial support of this investigation by the Russian Foundation of Fundamental Investigations, Grant No. 95-03-08808.

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198

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