and excited-state potential energies as functions of molecular

H. E. HOLLOWAY, R. V. NAUMAN, AND J. H. WHARTON

4474

The Electronic Structure and Spectra of 2-Phenylnaphthalene. Ground- and Excited-State Potential Energies as Functions of Molecular Conformation1 by Homer E. Holloway, Robert V. Nauman, and James H. Wharton Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803 (Received April $0,1968)

The potential energy as a function of molecular conformation of the ground electronic state of 2-phenylnaphthalene is considered to be the sum of the steric repulsion energy between nonbonded hydrogens and the T delocalization energy. The computation of the steric repulsion energy as a function of molecular conformation has been based on a model which includes both the van der Waals repulsion energy and the molecular deformation energy. The delocalization energy has been computed as a function of conformation by semiempirical SCF-MO methods. For the ground electronic state the equilibrium conformation is predicted to be that with an angle of 30” between ring planes. The theoretical electronic spectrum has been computed by the semiempirical SCE-MO-CI method as a function of molecular conformation. The theoretical spectrum for the 30” conformation is found t b be in excellent agreement with the experimental spectrum. The potential energy as a function of molecular conformation of the excited electronic states has been estimated from the steric repulsion energy developed for the ground-state and the delocalization energies computed from the results of the configuration interaction calculations. A planar equilibrium conformation is predicted for the lowest excited state. The computations predict that thermal equilibration of the lowest excited state should give rise to a Stokes shift between the absorption and fluorescence spectrum of 920 cm-l. This agrees favorably with the experimentalvalue of 1290 cm-I. The extent of inhomogeneous broadening of the absorption spectrum as a function of temperature predicted from the ground- and excited-state potential functions has been found to be in good agreement with experimental observations. Finally, the dramatic temperature effect observed for the p band of 2‘-methyl-2-phenylnaphthalene is qualitatively explained in terms of the potential functions of 2-phenylnaphthalene, I n the preceding paper the spectra of the 2-phenylnaphthalenes have been analyzed in some detaiL2 Considerable information about the conformations and the energy effects associated with changes in conformation was deduced from comparisons of the experimental spectra. I n addition, there were observed significant temperature effects which could be interpreted only qualitatively by consideration of the experimental data alone. Here we describe our efforts to compute the energy effects associated with changes in conformation of 2phenylnaphthalene and to interpret quantitatively the temperature effects observed in the absorption spectrum. Even a semiquantative agreement between experiment and theory requires rather accurate computations of the change in ground-state energy with conformation and of the change in excited-state energies with conformation. Since u-7r separability is probably not a good approximation for twisted composite polynuclear aromatics and since the u framework is important to the steric effects in the untwisted conformations, it would seem that a full (T-T computation is necessary. However, the equilibrium conformation of 2-phenylnaphthalene approaches the planar conformation and hence the a-T interaction makes the greatest contribution to the delocalization” energy in this range of conformation. In addition, the u-u interaction can be included as an empirical, nonbonded hydrogen repulsion in which both ((

The Journal of Physical Chemistry

the van der Waals repulsion and the molecular distortion are included in the steric repulsion energy. Thus in the case of 2-phenylnaphthalene the latter approach is expected to give better results than those that can currently be obtained from a full u-7r computation.

Methods Electronic Energy States. As a guide to the interpretation of the electronic spectrum of 2-phenylnaphthalene, a computation based upon a configuration interaction model of the Pariser-Parr-Pople typea using SCF-MO’s as basis functions has been carried out. Since the computation consisted of a straightforward application of the method, only the empirical parameters and basic assumptions are recapitulated. The input data were based on the structure shown in Figure 1 in which sp2 hybridization is assumed and all bond lengths are taken to be 1.40 A, except the interannular bond which is assumed to be 1.48 A. The latter distance is based on the fact that the central bond in planar biphenyl has been measured to be 1.50 i f 4 and the interannular bond in 2-phenylnaphthdene should be slightly shorter. (1) Adapted from the Ph.D. dissertation of H. E. Holloway, Louisiana State University, 1967. (2) H. E. Holloway, R. V. Nauman, and J. H. Wharton, J . Phus. Chem., 72, 4468 (1968). (3) R. G . Parr, “Quantum Theory of Molecular Electronic Structure,” W. A. Benjamin, Inc., New York, N. Y., 1963. (4) A. Hargreaves and S. H. Riser, Acta Crystallogr., 15, 366 (1962).

THEELECTRONIC STRUCTURE AND SPECTRA OB 2-PHENYLNAPHTHALENE

Figure 1. Geometry of 2-phenylnaphthalene distances between nonbonded hydrogens).

(dl

and da

thalene. The merits of this assumption will be discussed as the magnitudes of the various interaction terms are evaluated. Thus the analysis is in fact that for biphenyl. A general theory for the quantitative evaluation of the steric repulsive energy of planar biphenyls has been given by Westheimer and MayerlS and has been applied to a calculation of rates of racemization by We~theimer.’~In this method the total steric repulsion energy is given by

E, The empirical parameters were puvO = -2.39 eV for adjacent carbons in the ring skeletons, puVO = 0 for nonadjacent carbons, pz,ll = [LS2,11(1 - Sz,11)/S0(1 So)] /3’J cos e for the bonded carbons between rings,5 ycc = 11.08 eV, and Y~~ = 14.397/(1.300 Rrv) eV/ii.6 Calculations were made for the 0, 30, 45, 90” conformations. I n the configuration interaction model for each conformation 25 singly excited configurations (!Pj-l!P&) and the ground-state configuration were used. The computations were carried out with an IBM 7040 digital computer using a program written by Bloor and Gilson.’ Steric Effects. Numerous authors have considered the steric effect in overcrowded molecules, especially in the case of biphenyl. Adrian8has applied the “perfectpairing” approximation of valence bond theory to a rigid molecular model. Goodwin and Morton-Blakeg developed a simple nonbonded H-H potential function by empirically fitting a function consisting of the Huckel molecular orbital resonance stabilization energy combined with an assumed form for the steric repulsion energy to the experimentally measured equilibrium conformation; again, a rigid model was assumed. Golebriewski and Parczewskilo developed a method for calculating the most stable conformation by minimizing the combined delocalization energy and the steric repulsion energy. The steric repulsion energy included the van der Waals repulsion energy and the molecular strain energy, This division of the steric repulsion energy into two contributions appears to offer a better rationale than does that based on the rigid molecule; however, these authors did not develop a potential function relating the steric repulsion to the molecular conformation. I n addition, the authors used the hardsphere model for van der Waals repulsions developed by Coulson and Senetll; this model was later criticized by Coulson and Haigh.12 In the paper by Coulson and Haigh an extensive evaluation of steric interaction between nonbonded hydrogens has been made, and 14 H-H van der Waals potential functions have been examined. These authors conclude that atoms are “softer” than they had been assumed to be. I n the evaluation of the steric effect in 2-phenylnaphthalene, it is assumed that the steric potential function of biphenyl will closely approach that of 2-phenylnaph-

+

4475

= ‘/zCatqt2 i

+ u ( d i ) + U(dz)

(1)

where a, is the force constant associated with the displacement qt along the normal coordinate and U ( d ) is the van der Waals repulsion between nonbonded hydrogens separated by the distance dl or dz. The distances dl and d2 are given by the equations dl

dio

+ Cbtqt i

(2)

i

I n these equations dlo and d20 are the separations between the nonbonded hydrogen pairs in the undistorted planar molecule. In the case of biphenyl, dlo and dzo are equal. The constants bB and b,‘ are geometric factors which relate the increase in dl and dz to the magnitude of the displacements. The minimum value of E , with respect to all possible variations in the normal coordinates that influence dl and dz is found by solving the set of equations given by

The van der Waals potential function used was that given by R!tulleri5 and is 2818.1 U ( d ) = -(d/adB

+ 63,335e-2.645d’aQkcal/mol

(4)

in which d is expressed in angstroms and a. = 0.529 A. This equation was selected because (1) it was derived t o (5) Reference 3, p 100. (6) N. Mataga and K. Nishimoto, Z. Phys. Chem. (Frankfurt), 13, 140 (1967). (7) J. E. Bloor and B. R. Gilson, “Closed-Shell SCF-LCAO-MO,” Quantum Chemistry Program Exchange Q C P E 7 1.1, Indiana University, Bloomington, Ind., 1966. (8) F. J. Adrian, J . Chem. Phys., 28, 608 (1958). (9) T. H. Goodwin and D. A. Morton-Blake, Theor. Chim. Acta, 1, 458 (1963). (10) A. Golebriewski and A. Parczewski, ibid., 7, 171 (1967). (11) C. A. Coulson and 8. Senet, J . Chem. SOC.,1813 (1955). (12) C. A. Coulson and C. W. Haigh, Tetrahedron, 19, 527 (1963). (13) F. H. Westheimer and J. E. Mayer, J . Chem. Phys., 14, 733 (1946). (14) F. H. Westheimer, ibid., 15, 252 (1947). (15) A. Muller, Proc. Roy. SOC.,A154, 624 (1936).

Volume 72,Number 18 December 1968


4476 agree with the observed splittings in the infrared spectra that arise from the intermolecular force fields caused by overcrowding and (2) it appears to be one of the potential functions that can be considered to be consistent with the experimental results in the analysis of Coulson and Haigh. The vibrational force constants used are given in Table I. With the exception of the C-C interannular stretch, the force constants are those used by Westheimer in the case of 2,2'-dibromo-4,4'-dicarboxybiphenyl. The force constant corresponding to the interannular stretch was calculated by means of an assumed linear relation between bond order and the stretching force constant. Carbon-carbon single and double bonds and benzene bonds were used as reference points.

Table I1 : Geometric Factors, bi, ba' Vibration

bi

Interannular stretch C-C-H bend

1.00 1.08 x 10-8 cm/radian -0.50

C-H stretch Benzene rings 6a 8a (6a 8a)* 12

+ +

1.2-5 2.55 1.60

bi'

1.30 5.00 1.60

Table I11 : Contributions to the Steric Repulsion Energy of Planar Biphenyl Displacement Vibration

Table I : Values of the Force Constants, ai Vibration

Interannular stretch C-C-H bend C-H stretch Benzene ring 1" 6+8 (6 8 ) * 12 14 19

+

ai

6 . 1 X lo6 dyn/cm 0.86 X 10-11 dyne cm/radiana 5 . 0 X lo6 dyn/cm

45.9 X l o 5 dyn/cm 13.7 X lo5 dyn/cm 450 X 106 dyn/cm 46.2 X lo6 dyn/cm 138 X 106 dyn/cm 74.6 X lo6 dyn/cm

(Pi)

A. Model Including Ring Distortions Interannular stretch 0.022 R C-C-H bend 0.084 Ladian C-H stretch 0.004 A (6a sa) 0.008 d (ea 8a)* 0.002 R 12 0.004 d 2U(d) (d = 2.04 A)

+ +

0.21 1.76 0.02 0.12 0.28 0.09 3.00 __

5.48

The notation corresponds to that of Westheimer's normalcoordinate analysis.

B. Model Neglecting Ring Distortions Interannular stretch 0.0235 C-C-H bend 0.0901 radian 2U(d) (d = 2.018 A)

0.24 2.01 3.30 __

5.55

In the case of planar undeformed biphenyl the bond lengths have been taken to be the following: C-H, 1.08 A; ring C-C, 1.40 A; and interannular C-C, 1.48 A. The bond angles were taken to be 120'. The choice of 1.48 A for the interannular bond length was based upon the bond distance measured for biphenyl in the vapor16 where steric repulsion is absent. The distortion of the molecule is assumed to be sufficiently small that second- and higher order terms relating the H-H distance to the magnitude of the vibration in question can be neglected. Only three of the ring vibration normal-coordinate force constants (6 8, 6 8*, and 12) and the corresponding geometric factors have been used. Analysis shows that the remainder of the force constants are so large or that the geometric factors are so small that the rings are essentially undistorted in these modes and thus the distortion energy is negligible. The geometric factors b, and b,' are given in Table 11. Table I11 gives the displacements, the energy associated with the displacements, and the total steric repulsion energy for planar biphenyl. The data show that the C-H stretch can be neglected and that the strain energy involving ring distortions is quite small (0.49 kcal/ mol) as expected.

+


+

In view of the small contributions of the ring distortions to the change in distance between nonbonded hydrogens and to the interaction energy, the calculation was repeated and the ring distortions and C-H stretch were neglected. These results are given in Table 111. Neglect of the ring distortions severely alters the contribution of the C-C-H bend to the total energy but does not increase the total interaction energy significantly. Thus, because of the stiffness of the benzene rings, their distortions can be neglected in considering the total steric repulsion energy; however, the error in the individual contributions is apt to be quite large. The predicted 0.0235-A stretch of the C-C interannular bond combined with the assumed 1.48 A bond length for undeformed biphenyl gives 1.5035 A for the interannular bond length in the planar molecule, a result that is in excellent agreeyent with e ~ p e r i m e n t . ~I n addition, the predicted 2.08-A nonbonded hydrogen distance is in fair agreement with the 1.97-A distance estimated from experiment." (16) A. Almenningen and 0. Bastiansen, Kgl. Norslce Videnslcab. Selskabs SkriJter, 4 (1958). (17) See ref 12, p 643.

THEELECTRONIC STRUCTURE AND SPECTRA O F 2-PHENYLNAPHTHALENE With the assumption that ring distortions can be neglected, biphenyl serves as a good model for 2-phenylnaphthalene, provided that the assumption that d10 = &o is justifiable in the case of 2-phenylnaphthalene and that d for 2-phenylnaphthalene and d for biphenyl are equal. Analysis of the bond orders of planar 2-phenylnaphthalene indicates that dlo E 1.79 A and dto E 1.81 8,while do for biphenyl was 1.80 8. Thus a slight bending (-0.003 radian) of the C-C-C angles a t the interannular bond would tend to make these distances equal and would involve very little strain energy (0.001 kcal/ mol). Thus the steric repulsion energy of planar biphenyl should be applicable to 2-phenylnaphthalene. If one considers only the interannular stretch and the C-C-H bends to be modes to relieve the van der Waals interaction, then calculation of the total repulsive energy as a function of the angle between planes is considerably reduced in complexity. The calculation involves fixing the molecule a t various angles, calculating the corresponding dowhich is given by do = (12.50 - 9.24

COS

e)'/*

(5)

and minimizing the repulsive energy. It is assumed that the hydrogens are displaced from each other along a line joining their undisplaced positions. The distortion involves both the in-plane and out-of-plane bending constants. The out-of-plane bending constant is 0.39 X dyn cm/radian2.18 The computation was made for the 0, 10, 20, 30, 40, and 60" conformations. The resulting repulsive potential function is given by the upper curve (E,) in Figure 2. Resonance Energy. Several semiempirical methods commonly used t o calculate increased resonance stabilization due to twist have been examined. Most of these methods include as the major contribution the function

in which p,, is the K bond order, @,, is the resonance integral, and 0 is the angle between planes. Bond orders have been calculated by a variety of methods, while firs is generally selected to agree with the thermochemical data and an experimental picture of relatively free rotation. If the attitudes previously applied to biphenyl are applied to 2-phenylnaphthalene, the potential function obtained for the ground state indicates the planar conformation t o be the most stable. I n addition, the potential function shows very little energy variation. Thus there is predicted a broad molecular distribution which when combined with the transition energies calculated as a function of e suggests that the electronic absorption bands should be extremely broad. Both the planar ground-state conformation and extremely broad bands are in complete discord with the experimental results. Without adjusting ,8 to an unrealistically small value and changing @ between ground

4477

Figure 2. Potential energy as a function of conformation for the ground electronic state (Es, steric repulsion energy; E,, delocalization energy; ET, total energy).

and excited states, the empirical methods developed for biphenyl are not applicable to 2-phenylnapththalene. It seems more nearly consistent to use the same model to predict resonance stabilization energies that has been used to calculate state energies. Pople'g has successfully used the SCF method to calculate ground-state resonance energies without making a significant adjustment of @. Hence, the energy of delocalization across the interannular bond gained by untwisting the molecule is given by

El(e) is the energy eigenvalue of the ith SCF-A40 a t a given 8. The function obtained is shown as the lower curve (E,) in Figure 2. A second-order correction to this function results from the variation in interannular bond length with the variation in angle between planes. This effect requires that the resonance stabilization plus steric repulsion should be minimized in one step. However, our @ us. bond length function is essentially invariant to the 0.02 A bond stretch predicted by the steric interaction. For t h e j t h excited state the energy of delocalization across the interannular bond gained by untwisting the molecule is given by

in which cq(0) is the transition energy for the e conformation calculated from the configuration interaction model. (18) R.C.Lord and D. H. Andrews, J. Phys. Chern., 41, 149 (1937). (19) J. A. Pople, Trans. Faraday SOC.,49, 1375 (1953); A. Briokstock and J. A. Pople, ibid., 50, 901 (1954). volume 72,Number 18 December 1068

4478

Results and Discussion Ground-State Potential Function. The ground-state potential function (E,) is shown in Figure 2 and is the sum of E,@) and EB(0). The minimum energy is pre-

dicted at t9 = 30". In the range -40" < 8 > 40" the function is relatively flat but is predicted to rise rather sharply above 40". These results are consistent with the intuitive prediction that the equilibrium conformation of 2-phenylnaphthalene should exhibit a greater tendency toward planarity than does biphenyl; the equilibrium angle in biphenyl has been determined to be 42".16 In addition the results are consistent with the analysis of the electronic absorption spectra of 2-phenylnaphthalene. The absorption spectra indicate that the equilibrium conformation in the ground state is angular; spectra of molecules in rigid hydrocarbon glasses are being compared with computational predictions for molecules in the gas phase. The minimal portion of the potential function is so broad that one must work at low temperatures in order to observe sharper spectra that result from a narrow distribution of molecules; low vapor pressure with the resulting need for long path lengths that lead to appreciable scattering prohibifs the observation of vapor spectra at low temperatures. Consequently, the undesirable but necessary alternative of observing the spectra of 2-phenylnaphthalene in solution was adopted. The predicted molecular distributions at 300 and 77"K, in which classical Boltzmann statistics are assumed, are shown in Figure 3. At room temperature a substantial population of the planar conformation is indicated, while a t 77°K essentially no population of the planar conformation is indicated. Even a t high temperature the molecular distribution is essentially confined to the region of =k4Oo. However, in the computation of the steric repulsion energy and the delocalization energy U--a separability was assumed, and in the case of molecules like 2-phenylnaphthalene U-T separability may be a rather poor approximation for the 90" conformation. Thus at high angles the model assuming separate U-u (steric repulsion function) interaction and separate -a--a (delocalization function) must be a poor approximation. On the other hand, as 0 approaches zero, u-T separability is a much better approximation. Consequently, the contour of the potential function is considered to be a good approximation below 8 =

45". Electronic Energy States. Figure 4 is a graphical representation of the bonding MO's of the subsystems and shows the mixing of the localized &lo'sthat occurs m the molecule approaches the planar conformation. The solid line implies that a great deal of subsystem character (>go%) is retained by the indicated MO's in the planar composite system. Dotted lines imply very little contribution, while the dashed lines imply extensive delocalization. As indicated, the major effect arises from the mixing of the B1 orbital of benzene with The Journal of Physical Chemistry

H. E. HOLLOWAY, R. V. NAUMAN, AND J. H.WHARTON

L

1

8 Figure 3. Molecular distribution functions for the ground electronic state a t 77 and 300°K.

90DEG.

0-DEG.

90 DEG.

Figure 4. A correlation diagram for the molecular orbital energy levels of planar and twisted 2-phenylnaphthalene. The solid lines indicate the subsystem MO's that make the greatest contribution (greater than goo/,) to the delocalized MO's while the dotted lines indicate the minor contributors. The dashed lines indicate that the subsystem MO's make essentially equal contributions to the delocalized MO'g. Configurational excitations are denoted by arrows 1-4. Excitations 1 and 3 show no charge-transfer characteristics, while excitations 2 and 4 show partial charge-transfer characteristics.

the B1, orbital of naphthalene. The MO's of 2-phenylnaphthalene can be classified as delocalized ( 5 and 7), partially localized (1, 2, 3, 4, and S), and localized (6).

THEELECTRONIC STRUCTURE AND SPECTRA OF 8-PHENYLNAPHTHALENE

4479

Table IV : Transition Energies and Transition Moments

Banda

Wave function

a P

P P'

... ... ...

... .

I

.

... Clar's notation.

-!3O0A E ~

4.26 4.50 5.74 6.44 6.10 5.97 6.27 6.19 7.12 7.00 AE in electron volts.

- 4 5 ' 7

M

0.0 0.80 2.24 1.11 0.0 0.0 0.05 0.35 0.0 0.0

AE

4.16 4.40 5.17 5.95 6.04 6.07 6.29 6.52 6.73 6.94

M

0.Q 0.75 2.00 0.77 0.75 1.07 0.49 0.58 1.06 0.51

Conformations 730'-'0AE M AE

4.11 4.35 5.03 5.85 6.00 6.06 6.29 6.57 6.69 6.93

' Experimental values from

Thus three types of nonpolar configurational excited states and three types of polar configurational excited states (charge transfer of varying extents) are included in the configuration interaction model. Consideration of only first-order configurational interaction indicates that the a and P transitions involve polar excited states (delocalized to partially localized), while the p transition involves a nonpolar excited state (partially localized to partially localized). Higher excited states include a great deal of charge-transfer character, giving rise to extensive charge-transfer mixing in the configuration interaction calculation. The transition energies and transition moments obtained from the configuration interaction computation are given in Table IV along with the experimental results. For all conformations with e less than 45",the calculated spectrum is in satisfactory agreement with the experimental results. The calculation for the 30' conformation agrees best with the experimental results ; however, this agreement must be considered to be fortuitous because of the limited accuracy of the computation. It is significant that the calculated and measured spectra are in agreement in the range of 0 predicted t o be pertinent by the ground-state potential function. According to the configuration interaction model, the energy of the a and p transitions is not extremely sensitive to the angular conformation, while the energy of the P transition is predicted to be very sensitive to the conformation. This is interpretable in terms of mixing the and charge-transfer excited states (\Ee-W9 - \kg-'\kll Qb-l\kg - \kS-l\klz) with the state corresponding to the P transition. The experimental spectrum is compared with the calculated spectrum in Figure 5. Only those transitions which are computed to have nonzero transition moments (except the a! transition) are included in the calculated spectrum. As observed, the calculated spectrum is in excellent agreement with the experimental spectrum for the a,p , and P transitions. The measured

Xm.,

0.0 0.80 2.02 0.69 0.76 1.16 0.47 0.42 1.08 0.52

4.00 4.29 4.93 5.77 5.97 6.05 6.29 6.92 6.67 6.59

M

0.0 0.90 2.02 0.61 0.84 1.18 0.45 0.54 1.11 0.17

-ExptlcAE

4.21 4.33 4.98 5.85

... ... ... ...

... ...

e

-4

x 102 1.25 x 104 5.7 x 104 4 . 1 x 104

... ... ... ... ... ...

of the room-temperature-solution spectrum.

0

30

Figure 5. Comparison of the theoretical spectrum of the 30' conformation with the experimental spectrum. Calculated transitions (solid lines and short dashed line) are those of Table IV in the order of increasing energy. Lengths of solid lines are proportional to the transition moment squared, while the (Y transition is dashed to indicate a transition moment of zero. The tall vertical dashed lines show the calculated upper and lower limits for the energy of the p transition (90 and 0' conformations, respectively).

intensity of the P' transition is considerably higher than that predicted by the transition moment of the \kl-'\kl0 transition. This indicates that a better interpretation of the band labeled P' is the superposition of the \kl-Wl0, QO-l\ke - \kg-lQ~l, and \k,s"\E9 - "ks-'\klz transitions. The sum of the squares of the corresponding transition moments gives a good account of the intensity observed for the P' band. The tall vertical dashed lines in Figure 5 indicate the range of energies calculated for the p transition as the conformation is changed from 90 to 0". The calculations indicate that no conformation has other allowed transitions in this range. Stokes Shift. The potential functions for the ground state and first three excited states are shown in Figure 6. The excited-state potential functions are the sum of the delocalization energy (eq 8) computed for each excited state and the steric repulsion energy calculated for the ground state. Use of the ground-state steric repulsion Volume 74,Number 13 December 1968


4480

/

-24.0

I

-40t

I

/

/

I

-8.0

-20

0

20

40

60

80

8 Figure 6. Potential energy as a function of conformation of the ground electronic state and the excited states associated with the CY, p, and p transitions.

energy for the excited states assumes that the geometry does not change upon excitation and that the vibrational force constants pertinent to the steric repulsion do not vary between ground and excited state. The calculated bond orders for the excited states indicate that the geometry change involving the distance between nonbonded hydrogens is small; the interannular bond shortens but the adjacent ring bonds lengthen. The interannular force constant should increase by 1015%, while the C-C-H bending force constant should not change significantly. A 15% increase in the interannular force constant increases the calculated steric repulsion energy by less than 0.1 kcal/mol and this would tend to be canceled by a lowering of the ringdistortion force constants which were neglected in the simplified model. I n spite of the numerous approximations involved, the potential functions still include the major factors involved and hence should provide a basis for at least a qualitative interpretation of the experimental data. Examination of the potential function for the ground state shows that the low-temperature absorption should involve molecules distributed about the 30" conformation (see Figure 4 for the distribution a t 77°K). In the case of the lowest excited state the most stable conformation is predicted to be planar. Thus, if thermal equilibration takes place during the lifetime of the excited state, emission should involve molecules distributed about the planar conformation. The calculated gap between absorption and emission is 930 cm-', which compares favorably with the experimental value of 1290 cm-1.2 The data are also consistent with the angular ground state and the planar lowest excited state The Journal of Physical Chemistry

that were indicated in the interpretation of the experimental studies of 2-phenylnaphthalene. Inhomogeneous Broadening. The major contributions to the increased band broadening observed for 2-phenylnaphthalene compared with that of a more rigid molecule such as naphthalene are (1) the change in vibrational frequency (ring stretch for example) with change in conformation, (2) the vibrational progressions associated with the low-frequency torsional oscillation (0 -t 1, 0 -t 2, etc., excitations), and (3) changes in population of the low-frequency torsional levels. The latter may also be interpreted to be a shift in the transition energy with conformation when the conformation is determined by the intersection of the torsional vibrational level with the potential function. The nearconstant 1400-cm-' ring stretching frequency observed from 2-phenylnaphthalene, 2'-methyl-2-phenylnaphthalene, and other derivatives indicates that the stretching frequencies are not sufficiently sensitive to the conformation to account for the broadening observed. The extent of band broadening due to the vibrational progressions involving the torsional oscillations is difficult to estimate; however, it is small compared with the extent of band broadening associated with changes in population of these levels. Since the population of the torsional levels is temperature dependent, the temperature effect on the electronic absorption spectrum provides a sensitive test of the potential functions of Figure 6. Our discussion assumes that the population can be treated as a classical distribution; this assumption is probably valid for 2-phenylnapththalene in solution. At 77°K the vast majority of the molecules are predicted to be distributed between the 15 and 35" conformations. If one assumes that a given vibrational band of the p electronic band is a Dirac function, the superposition of Dirac functions for all conformations in the interval would form a total band width of approximately 300-400 cm-'; the total band width corresponds to the variation in transition energy in the interval 15" < B > 35". This predicted broadening is insufficient to obliterate the 1400-cm-' ring stretch frequency, and the structure is observed a t 77°K. At room temperature the molecules are distributed between the -40 and 40" conformation. Superposition of the spectra of all the conformations in this interval would form a band 1000-1200-cm-' total width. This is just sufficient to blend vibrational peaks separated by 1400 cm-l. Again, the experimental spectrum agrees with the theoretical predictions. I n the 77-300°K temperature range the extent of band broadening is essentially the same for the a, p , and @ transitions. That this should be the case can be seen by observing the potential functions for the excited states. The excited-state potential functions are nearly parallel in the range of the ground-state conformations populated at room temperature and below. However, at higher angles (>45") the potential func-

THEELECTRONIC STRUCTURE AND

tion for the excited state associated with the /3 transition diverges rapidly from the ground-state potential function. At a temperature sufficiently high to populate the high-angle ground-state conformations substantially, the /3 transition should broaden extensively. This broadening should be observed as a collapse of the maximum at 40,000 cm-' and an intensity increase in the region from 40,000 to 48,000 cm-l. Theoretically, the /3 band should have a 10,000-cm-' maximum half-width when a 3000 cm-l natural half-width is included. Attempts have been made to observe the predicted broadening of the /3 band. The solution absorption spectra have been measured at temperatures up to 100". These spectra show a slight (5%) loss of intensity in the maximum at 40,000 cm-1 and a slight broadening toward higher energies. These observations support the indicated sharp rise in the ground-state potential function above 40" but from the data we are unable to estimate the exact magnitude. Attempts to measure high-temperature vapor spectra have not been successful because of the low vapor pressure of 2-phenylnaphthalene and its instability above 150". The spectrum of 2'-methyl-2-phenylnaphthalene confirms the expected broadening of the /3 transition in the substituted 2-phenylnaphthalenes. At 77°K the spectrum is very similar to that of 2-phenylnaphthalene; this indicates a narrow distribution of molecules. At room temperature the intensity of the /3 band has collapsed and the band is much broader than that of 2phenylnaphthalene. These observations can be interpreted in terms of an equilibrium configuration at a larger angle in the case of 2'-methyl-2-phenylnaphthalene. The distribution is "centered" in the region (-60") where a small variation in conformation gives rise to a large variation in /3 transition energy. Thus, a distribution width comparable with the width predicted a t room temperature in the case of 2-phenylnaphthalene but occurring in the range 55" < 6 > 125" gives rise to a very broad /3 band. We are currently conducting an extensive study of the band shapes of 2-phenylnaphthalene and its derivatives. From fitting the experimental band contours G ( v , T) to the relation

G(v, T) = EN$@,T)g,(v, e> i

4481

SPECTRA OF 2-PHENYLNAPHTHALENE

(9)

in which Nr(O, 7)is a population factor and gt(e) is the band shape function, we expect to obtain experimental transition energies as a function of conformation and the contour of the ground-state potential function. Sensitivity of Results to Parameters. I n the semiempirical analysis our assumptions have been justified on the basis of experimental data taken from very similar molecular systems. However, there remains the possibility that the results are quite sensitive to the assumed parameters and that the agreement obtained is accidental.

The theoretical results show the greatest sensitivity to the resonance integral between rings and to the assumed coordinates in the analysis of the steric repulsion energy. In the latter the interannular bond length is by far the most critical coordinate. Thus the mating of the two rings is the critical factor. The form for the resonance integral given in the hiIethods section has been found by us to give the best agreement with experimental spectra for other molecules involving long bonds. This function does not vary sufficiently in the range of bond lengths from 1.45 to 1.52 A to change the results of our computation. I n order to observe any effect from varying the resonance integral, we must break away from this function. I n Table V the results Table V : Sensitivity of the Results to the Parameters. Variation of Resonance Integral 8,

eV 2.39 2.15 1.91

-Conformation, degGround Excited stateb etatea

0 0

-18 30 -38

-10

Stokes shift,,

om -'

Temp effect

-400 930 -1200

Very small Small Large

Equilibrium ground-state conformation. cited-state conformation.

Equilibrium ex-

of the analysis using /3 = 2.39 cos 19 and 1.91 cos 0 are compared with the previous analysis (P = 2.15 cos e). The steric repulsion energy used was that given in Figure 2. The effect of varying /3 has been determined from the approximationz0 dE

=

2pz,nd/3COS O

(10) A spot check using the SCF computations shows this to be a valid approximation when used in conjunction with the previous results for each conformation. Increasing the resonance integral from 2.15 to 2.39 eV does not significantly alter the main results of the analysis. The calculated gap between absorption and emission is smaller because the predicted groundand excited-state conformations more closely approach one another. It is significant that for a planar groundstate prediction the value of /3 must be 2.60 eV, considerably higher than the resonance integral within the rings. Decreasing the resonance integral to 1.91 eV gives an acceptable solution to all the experimental observations except for the predicted temperature effect in the absorption spectrum. I n this case a shallow ground-state potential function combined with the predicted 38" equilibrium conformation indicates that the /3 transition should be quite broad at room temperature. This was not the case. However, the lower resonance integral gives a better account of the gap (20) C. A. Coulson and H. C. Longuet-Higgins, Proc. Roy. Soc., A191, 39 (1947).

Volume 7.9, Number 18 December 1968


4482

bonded hydrogen distances at very little cost in energy. Further analysis shows that the steric repulsion curves for the different interannular bond lengths rapidly converge as the angle between rings increases. The implication is that within reasonable limits the solutions to the steric contribution do not critically depend upon the assumed coordinates.

3.0

x

?

0

2 .o

Uv

LL

Table VI: Sensitivity of the Results to the Parameters. Variation of Interannular Bond Length 1.o

Undeformed bond length,

A

1.0

2.0

3.0

4.0

1.46 1.48 1.50

q, Ax102 Figure 7. Graphical solutions for the displacement along the interannular bond length corresponding to minimum steric repulsion energy fur bond lengths of 1.46, 1.48, and 1.50 in undeformed planar biphenyl.

between absorption and fluorescence than do the other resonance integrals. I n addition the predicted groundand excited-state conformations are acceptable. This analysis indicates that the theoretical results are not ultrasensitive to the value of the resonance integral between rings and that the best over-all results would be 2.00 eV. probably obtained with /3 The sensitivity of the steric repulsion computation to changes in assumed coordinates has been investigated by varying the interannular bond distance for the undeformed molecule from 1.46 to 1.50 A. The positions q1e corresponding to the minimum energies are found by solving the system of equations defined by eq 3. For the stretching of the interannular bond length this equation takes the form

-

for the planar molecule. Graphically, the solution is shown in Figure 7 using interannular bond lengths of 1.46, 1.48, and 1.50 8 for the undeformed molecule. Inspection shows that the qre and Fr(qte)which enter into the energy are not very sensitive to the assumed interannular bond length. I n Table VI the results of the total steric repulsion calculation are given for the planar molecule as a function of bond length. The results show that slight relaxations of the stretching and bending modes give rise to essentially constant non-


Equil bond length,

H-H

ii

8,

Steric energy*

1.4848 1.5035 1.5223

2,0101 2.0181 2.0269

6.18 5.50 5.03

Nonbonded hydrogen distance for the planar molecule. energy for the planar conformation in kilocalories per mole. a

* Steric

Conclusions The agreement between the experimental observations and theoretical predictions in the case of 2-phenylnaphthalene is extremely gratifying. This is especially true since our parameters have been taken from similar molecular systems. It seems possible within the framework of the computation to obtain a much better fit between experiment and theory by empirical methods. Perhaps this will be a worthwhile endeavor when more experimental data become available. The initial success in the theoretical analysis of 2-phenylnaphthalene indicates that a combined theoretical-experimental attack on the spectra of the 1- and 2'-substituted 2-phenylnaphthalenes will provide considerable insight into the nature of intramolecular interactions. Thus far, 12 derivatives of 2-phenylnaphthalene have been synthesized and are currently being studied. Acknowledgment. We wish to thank Dr. Jimmie McDonald for measuring the vapor spectrum of 2-phenylnaphthalene. H. E. H. gratefully acknowledges the support of the National Science Foundation in the form of two appointments to Summer Research Participation Programs for College Teachers and a Science Faculty Fellowship. We are especially grateful to one of the referees who made comments that showed deep understanding and suggested methods for improving and clarifying the discussion.

and excited-state potential energies as functions of molecular

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