On the theory of two-photon n → π* transitions

IS March 1973

CHEh;lICAL PHYSICS LETTERS

Volume 19, number 2

ON THE

THEORY

n + n* TRANSITIONS

OF TWO-PHOTON Robert

A. HARRIS

Department of Chemistrq’, University of CaIifoinia, Berkeley, Califomia 94720. USA Received 11 August 1972. Revised.manuscript received 21 December

It is shown that two-photon spectroscopy uniquely determines tion amplitude is the sum of two distinct parts.

In solution, n * II* transitions are often suppressed by IT+ X* transitions. In this note we point out that in the case of linear conjugated molecules two-photon spectroscopy provides a unique method of determintransition amplitudes. In addition, n + rr* ingn-tn* transitions may also be obtained. These n-electron states are eigenstates of a different n-electron harniltonian than the normal n-electron hamiltonian. Because of the nonlinear nature of two-photon spectroscopy three separate experiments can be carried out by varying the polarization of the incident beams [I]. Two such cross sections in the dipole approximation are

1972

n -+ z * transitions

in line3r molecuies. The transk

a~p?F_n++ti/l_t

/.l =

-

!j>

i and z^are the unit vet tors, a is a matrix element bt-tween !ocalized n-electron orbit& and b is a matrix element between a nonbonding orbitaland a localized II orbital. In general a 3 b. pm* is the cpera:or which generates n --f 5~* transitions, and ,u,_+ is the operator which generates pi * IT* transitions. Substitute expression (5) into eqs. (I) and (2). The resulting cross sections look like 61

= a4/T1 I2 +b417’212

+n2b22Re

T, T: ,

_ad 6 ) = tz2b2113~2.

and

(2) IO), In), If> are the ground, virtual intermediate and final states of the molecule respectively, and Ai(wlw2) are defined

A,:(wlW2)=(E~l-t2wl)-’ f(E~,-hp, where E. +fiwl

+fiw;=Ef

.

The transition moment operator for a linear molecule ,+-I

nonbonding

rnagnitudk

is reduced

spectroscopy

intensity,

(3)

(7)

T,, T2 and B are the appropriate second-order transition amplitudes. In ordinary spectroscopy in a region of overlapping transitions the absorption coefficient is a2 f b2. There 6 1 need not teach anything new. However 6, is all or none. It uniquely gives n + ~1* transitions. Its order of photon

by

(6)

this effect

from

ordinary

?T+ 7~* twa-

by 62/~2_ With increasing should

laser

be observable.

We shall formally evaIuate quantity B for the case of one nonbonding site [2]. If we use the assumptions of ref. [2], -all reference to nonbonding eIectrons may be removed. The resulting amplitude B takes the form,

sites takes the form 201 ‘,

_, Volume. 19, number

2

CHEhlICAL

PHYSICS LETTERS

15 Ear&

1973

B:/ a judicious choice of frequencies it may be possible to isolate one of the two terms in B. This will more or less occur when one of the frequencies approaches 3 given excitaiion frequency such as E, + Z

C& creates a reelectron in a L&din orbital in the nonbonding site. The energy factors are A;{o&

= (E”f z - Aw$-l

- (E,.f. a: - rrwJl, (9)

an’d A;{wlw2)

= (Er- liw

$-I

- (q-t?up

.

(10)

The ground state Ig> is the ground state ofH(Z), the n-ele&%n h~~toni~ with two nonbondi~~el~c-tfons. Also, H(2)lZ) =

ep ,

(11)

where

5=U42E+F;. The eig&aIue

(12) equation forH( 1) is

H(l)124 = (fVt. Z)lY) .

(13)

We see that the amplitude 3 is the 5u.q of two different “paths”. One path leads through intermediate -r-eIectron states which are ground-state n-electron

states. The other path travels through n:electron states which are solution ofJ’Y(l). If orz may isolate the fatter term, then two-photon spectroscopy is the spectroscopy of n + 1~*transitions of H[ 1) with each transition weighted by the “Franck-Condon” factor WC& lg>:The former pathway is n -+ R* spectroscopy ofH(2) with each transition weighted by the “Franck -Condon” ampIi~de~ G.J$‘& If).

or eI. A simple analoeJI of the above effect may be found in tha case of two electronic states, each with a single mode of vibration. The two pathways correspond to (a) vibrational transitions in the ground electronic s&es folIowed by vibronic transitions to the excited electronic state, and (b) the reverse. There are two effects which we have negIected which m;ly spoil our theory. If the equiIib~u~1 position of the final state is bent, then even in the Franck -Condon limit 7r-+ n* transitions may occur with a 7 component. The ~pIitude of this effect depends upon the detailed relationship of the various finalstate-intermediate-state and intermediate-statei~it~~-state ~ra~ck-London factors. The second effect is due to bending vibrations either in the z^ory^ directions. The cross section in this latter case wil! be the sJrn of the cross section considered in this note and a Franck-Condon-forbidden cross section due to the vibronic effect. In strongIy linear mo!ecuies iike those constructed from pofy-acetylenes these effects shou:ld be small. I s4ish to thank my friend W.M. M&lain pleasurable discussions.

for many

References [if P.R. Monson and W.&f. McClain, J, Chem, Phys. 53 (1970) 29. [2] R.A. Harris and L.&l. Falicov, J. Chem. Phys. 55 (1971) 2931.