Radiationless transitions in alternant hydrocarbons

VobJme

8. number

CHEMICAL PHYSICS LETTERS

1

RADIATIONLESS

TRANSITIONS .

IN ALTERNANT

HYDROCARBONS

U. SOMMJZR

lnstitut fiir i%ysi.kalisclte Chemie der UniuersitZt Stuttgati, Stuttgart. Germany

Received 6 November

-.

1970

A generat selection rule for radiationless transitions in alternant hydrocarbons. analogous to ffiarfor radiative transitions, is derived. The basic expressions for the evaluation of the electronic part of the kinetic interaction terms within the framework of the semiempirical SCF MO CI scheme are given.

The quantum mechanical treatment of radiationless transitions requires the evaluation of matrix eleApplying the Herzberg-Tether expansion [l] and ments arising from the kinetic interaction operator T at m first order the electronic part of these maa straightforward perturbation treatment it is found tl?-. trix elements is governed by expressions of the tyhe 12.31

where 9k,(Q) are adiabatic electronic wavefunctions and He(Q) is the molecular nuclei. Since a/aQ, only acts on the nuclear coordinates one obtains

aHe

hamiitonian for fixed

I

(2)

aQY Ed,- is the column vector for the pth nucleus constructed from the inverse ‘X-l forming nuclear into normal coordinates at the nuclear configuration Q = 0.

of the matrix CKtrans-

(3) is independent of electronic

coordinates

and (4) I_

.is a one-electron operator. If the a-electron approximation is invoked, W(Q) does not contain the u-electronic coordinates ad the operator pV only refers to s-electrons. The o-electrons are summarily accounted for by replacing Zcr by the effective core charges pPff. In the SCF MO CI scheme for closed shell systems the molecuIar waveftmctions are set up as superpositions of singly and multiply excited electronic configurations which are antisymmetrized products of molecular spin orbitals [4]. The ground state can thus be ‘Written as Iso> =.ab+w

b,(2))

l.5~39. . . k+wwm

For singly excited singlet c&figurations

. . ._

_I-

-.

:-

,.,.

:.,.-

--. _-

,,,

. _.

‘:

..‘,

,.

-

:.-,.

,,

.-. _-.. ._-_-

:_.__

.. .

._

113. -.

._

Volume 8, number 1,

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/-.._

CHEMICAL PHYkCS h”+ERS .

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_.

1 January 1971

.-- ..

(5) where‘ the summation extencls overii! ground state are given by

-Intrdduction of LCAO orbit4s

.

orbitals occupied in the ground .state. Interactions ,with the

(with real coefficients)

and ZDb~apprtiimation

leads to

From these relations one can derive a simple selection rule for radiationless transitions in alternant hydrocarbons [6] analogous to that for radiative transitions. Denoting paired orbitals by U, t(‘, V, o’, etc. the following

types of matrix element between (+) or.(-)

(UU’ *+/\m47; * wu’ >,

(II

2w’I~cv f

: cI.1,

(UL” f

WV

(uu’ f ua’l;uu’ i

of -configurations arise:

U+zu#v;

u+t#u:

VI’),

vu*> ;.

Due to the p&ring propert)

Furthermore;

superpositions

of t( and U’ etc. one has

in the expression

the sum

=qff 25 c2 sv

=

\ 1

*=a

for alteknant hydrocarbons. Since, for reasons of consistency, charges must be treated on the same basis as the two-electron ivy = -2 K&) s=a

the co&ombik interactions of core two-centre repulsion terms one obtains

@&),_

(10)

With the help of-relations (5) to (10) the matrix elements (i) to (III) can be expanded. &d it is readily seen that they-vanish identically in all cases where both superposition signs are eqtial. Conversely, ifthe signs ve.opposite, the kkatrix elements reduce to. . (I)-

?(vill

-?2+&,l v$(f$

00 (III)

-5

_-2 {{zqjvJ&).-

.6 tne.ca& _-. : ,._ Y,.:‘ _:.:’ :

v$u(g),; :

; _‘ _; &4(i)Q4(i))j;

_

,. i :-,: ‘

’

of .~int~qkti&s -_‘.I; .&piththe ground state, the IS&X .. :. ., ,. _:. :. : ,_:. ._.., .”

: :‘

- ..

elekents

~_.

,_

.,_;:’ .

-1. ~ .

\

.-

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Volume 8. number 1

CHEMICAL-PHYSICS LETTERS

L-Jenuarg

19il

(11)

vanish identically for the (-) superposition

since,

on account of (7),

(12) So therefore

behaves like a (-) superposition

in accordance

with the usual assignment.

Thus, within the present approximations, radiationless transitions symbol are first order forbidden. As a consequence of the present considerations, the vectors

xj.lv = -e2(v(i)

IFI

2

between states of tpe same

(W -)

(13)

v(i)>

cli must be evaluated from the same semi-empirical the SCF MO scheme. In particular

analytical function that is used for the yWLvintegrals

in

(14) A more complete description of the mathematical formalism necessary trix elements (1) and related expressions will be given in a future paper.

for the evaluation of the ma-

The author wishes to thank Professor Th. Fdrster for many helpful and stimulating discussions_ is indebted to the Gorres-Gesellschaft who supported this work by a research grant.

He

REFERENCES [l] G. Hexberg and E. Teller, 2 Phys. Chem. 21 (1933) 410. [2] J. Jortner, S. A. Rice and R. M. Hochstrasser, in: Advances in photochemistry. eds. 6. N. Pitts et al. (Interscience, New York,

1969).

[3] U.Sommer. Thesis, Universitlt Stuttgart (1969). [4] C.C. J.Roothaan. Rev.Mod. Phys. 23 (1951) 69. i5J E. U. Condon and G. H. Shortley, The theory of atomic spectra (Cambridge Univ. Press.

[S] H. C. Longuet-Higgins. J. Chem. Phys. 18 (1950) 265.

, .-_

London. 1964).