** 7B3G NP-D8.K 825 8B3G NP 08 K 495 5B3G NP 08 K 1036 6B3G NP-D8.K**

**3.4.6 Vibronic Activity of the Totally Symmetric Vibrations of Naphthalene**

**In allowed transitions, vibronic mixing by totally symmetric**
**vibrations is a demonstrated effect in, for example, azulene [Lacey,**

**9**

**McCoy and Ross 1973] and phenanthrene [Craig and Small 196Ä], but there **
**is a need for a wider appreciation of its incidence.**

**Our vibronic calculations have accordingly been extended to the**
**a modes of naphthalene, but some further theory is first required. ** **The **

**8**

**exposition below, while embodying the methods formulated by Craig and **
**Small, is concerned with vibronic effects on total intensities rather **
**than with the intensities of individual progression members and the **

**argument follows a different course, resulting in a differently expressed **
**conclusion, to that of Craig and Small.**

**We assume that the ground and excited electronic states have **
**geometries related by a displacement 6 along a totally symmetric **

**coordinate Q. ** **For such coordinates there is no obvious choice for the **
**origin of 0, and so for definiteness we adopt the following definition. **
**Let the electronic transition moment for the transition between the two**

states be a = a(Q) — the same notation as in Sec. 3.1.3, though for
simplicity we ignore state labels. We also write *a.Q = a(Qfr)* for the
transition moment at the geometry of the upper state iind aQ = a(Qg) for -
that at the geometry of the lower. Over the range of Q of practical
interest ve assume that a varies linearly with Q, with slope

m 3 a 3a iq-j Qo 3qJ Qtf (3.4.1)

Now define the origin of Q as that of linearly extrapolated geometry for which a = 0. Fig. 3.4d illustrates this, and defines new coordinates

Q" = Q - Q Ö (3.4.2)

Q ’ = Q - Q o = Q - Qo ” 5 (3.4.3) where

6 = Q ’ - Q" . (3.4.4)

Fig. 3.4d: Transition moment a as a function of totally symmetric coordinates Q.

Note that since the sense of Q is arbitrary, we may always choose it such that the relative magnitude of the transition moments in the two equilibrium geometries is expressed through 6, which is positive if ctg >a", and negative if a' < a " . Also defined in the figure is the mean transition moment

a 0 = ^(aj+a'o') = m Q 0 , (3.4.5) where

Q0 « 3i(Qq - Qq) - = Q ’ -%<5 . (3.4.6) For transitions v ’ - On and O ’ ~ v" (primes are added to keep track of the origins of the. respective vibrational wavefunctions), the vibronic transition moments are respectively

<v'|a|0"> = m<v ’ IQ10" > - (3.4.7) ( v"I a I 0 T > - m(v"IQI 0 ’ ) . (3.4.8)

For coordinates Q, originating at the rest position of an harmonic oscillator

Q 10> = (2ß)”^|l> , (3.A.9)

where

3 = 47r2vy/h . (3.4.10)

So, using (3.4.7), the coordinate QM and in coordinate Q*, we get

(3.4.3) the

< v ’ | a | 0M ) - ct^v’ |0n )+ (23”)"’^ ra<v* |l"> (3.4.11) ( v"|a|0 1 ) = aj<v"|0'>+ (23’)"^ m<vn |l’> (3.4.12)

Squaring, summing over v ’ or v" as appropriate, effecting closure and noting that (0[0> = < 1 j1> = 1 and ( 0 11> = 0 gives the total intensities

(omitting constants) as

Iabs = (a,J)2 + m 2/2ß" (3.4.13) b l u e r = («5)2 + m z/2B’ • (3.4.14) These equations may be recast with the aid of (3.4.5) and (3.4.6), to yield

Iab° = («ü+ ^ 2 ö 2 + m 2 /23M) - ^ 0 (3.4.15) Ifluor = ( S o + W ö Z + H i ^ a ^ + m ö ä o • (3.4.15) For the present purpose we assume 3' - 3” - 3 (whence the term m6a0 , produces the interference) and introduce

a* = 2 2 :n 0 # = (23)“'1 m (3.4.17) zpt

I

**Now aK , in a calculation, is the change of transition moment**
**evaluated at a displacement 2**

**‘zpt** **from Qjj (or Q^)** **If**/ 0** is the ratio**
**of the first to the zeroth progression members in a Franck-Condon**

**progression. ** **Where vibronic effects are suspected, the ratio should be**
**determined (from a spectrum) using a progression built on a false origin**

**(where vibronic effects due to other modes enter only in second order).**

**Using (3.4.17) and (3.4.13), v/e obtain**

**"abs**
**f luor**
**a g j l + (X**
**1/0i* 1 )**
*( k\*
**a***

_{— ~ (x }

_{y2}

_{y2}

**I****6 I**** U l/0;**

**a***

**0**

*****

**0**

**-**

**(3.4.19)**

**Calculations on naphthalene are now given in Table 3.4f. ** **The**
**most active vibration is 5a^; ** **the total induced intensity is 20% of the**
**intrinsic intensity.**

**Table 3.4f**

**Vibronically induced transition moments (in /\) due to totally**

**symmetric vibrations, in naphthalene. ** **The calculated**

**value of q q is 0.1185. ** **Frequencies in cm""1 .**

**Mode**

**V calc**

**a**

**Vobs** **l«*lb** **103x(a*)2** **|a*/a0 |**

**1** **3065** **3058** **0.0148** **0.2193** **0.1249**
**2** **3041** **3025** **0.0045** **0.0205** **0.0382**
**3** **1603** **1579** **0.0242** **0.5856** **0.2042**
**4** **1468** **1465** **0.0124** **0.1538** **0.1046**
**5** **1418** **1380** **0.0291** **0.8491** **0.2458**
**6** _{1160}**1148** **0.0047** **0.0217** **0.0393**
**7** **1014** **1021** **0.0169** **0.2856** **0.1426**
**8** _{720}_{765}_{0.0172}_{0.2972}_{0.1454}**9** **484** **514** **0.0200** **0.4000** **0.1687**
**Total** **2.8329 = 0.2016 c^'2**
**a ** **From Kydd [1969],**
**b ** **a*= ** *\a (2-h***Qzpc)-«(Q0)|.**

**Mode 5a ** **also appears to he the most: active vibration in the**
**8**

**[Stockburger 1973] one finds I ^ ^ = 0 . 6 , from a p r o g r e s s i o n built on the **
**strong false origin due to 8 b _ .**

' ■ ' O

**Since we do not know w h e t h e r 6 is posit i v e or negative (to do'**
**so, it w o u l d be necessary to compare p r o g r e s s i o n lengths in absorption**
**and emission, as illustrated by Craig and Small [1968]), we use odj for**
**and from (3.4.19) find for the p r o g r e s s i o n in 5a ** **built on the origin**

**8**
**abs**
**f luor**
**fluojr**
**\ b s**
**1 .097 + 0 . 3 80**
**1.097 - 0.380** **-1-1.48 .**

**We conclude that vibronic activity due to m ode 5a^ is likely to **
**be d e t e ctable in naphthalene. ** **Unfortunately, the observed span of the **
**abso r p t i o n s p e ctrum is short (it is soon o v e rlaid by a stronger **

**transition) and most of the intensity derives from false origins. ** **The **

**f l uorescence is clearer, but the origin is normally self absorbed and its **
**intensity not easily measured. ** **Hence naph t h a l e n e is not the molecule we **
**w o u l d choose to p u rsue these issues.**

**In Sec. 3.5 we investigate another, much more spectacular case **
**of vib r o n i c interaction by totally symmetric interactions, in azulene, **
**but there it appears that <5~0, w h ence I ^** **q = 0. ** **The interesting**

**transition moment interference effects are therefore not significant in **
**that case.**

**3.5 ** **VIBRONIC COUPLING IN AZULENE **

**3.5,1 ** **Introduction**

**This section treats two transitions in azulene. ** **Most attention **

**is given to the 3500 X transition to the second excited state, assigned **
**Aj -A i, wdierein most of the intensity derives from a particularly strong **

**i nteraction w i t h a still higher Aj state; ** **the a^ m ode m a i n l y responsible **
**has frequency 1580 cm 1 (ground state value) but there are undoubtedly **
**other v i b r o n i c origins, notably one near 1050 cm” 1 .**

**Sec. 3.5.5 deals w i t h the Bj-eAj (M-polarized) transition at **
**7000 **

### X,

**w h i c h has v i b r o nically induced L - p o larized intensity, recently**

**d o c u mented by Small and K u s serow [1974].**

**In treating the 3500 X system, we note that Lacey, McCoy and **

**Ross [1973] concluded that Franck-Condon active ** **v ibrations were not **