On the perturbation description of radiationless transitions

Volume II, number

1

1

CHEMICAL PHYSICS LETTERS

May 1972

ON THE PERTURBATION DESCRIPTION OF RADIATIONLESS TRANSITIONS* G. ORLANDi Division

ofChernistty,

Xational

** and W. SIEBRAND

Research Couwd of Canada, Uftawa, Ontario Kl.4 OR6, Carlada

Received

A nonempirical criterion is derived for determining description of radiationless transitions.

There have been a number of recent arguments [l-12] about the proper perturbation description of radiationless transitions. In this description the molecule is initially prepared in an approximate molecular eigenstate In> which subsequently relaxes towards an exact eigenstate by mixing with a dense manifold of approximate eigenstates I(m)>. The initial state results from the action of an external perturbation, usually the radiation Geld, which interacts selectively with the exact eigenstates of the molecular hamiltonian 3cfi, so as to produce an approximate eigenstate which can be represented as the exact eigenstate of a truncated molecular hamiltonian J& = JC, - V. The signal monitored is usually emission from In,, where again the approximate eigenstate is used to account for the selectivity of the interaction with the radiation field. This selectivity is usually represented accurately enough by the assumption that all the transition dipole moment governing the decay of In) and I {nr}) is concentrated in the single state In>. Then the relaxation of Irt> towards the compound state In, (m)> competes with radiative decay of In) and is called a radiationless transition. If both decay processes are exponential, their rate constants k, and k, are determined by the phenomenoiogical equations

* Issued as NRCC NO. 12456. ** NRCC Visiting Scientist 1970-71. Present and permanent address: Laboratorio di Fotochimica e Radiazioni d’Alta Energia de1 CNR, Bologna, It&.

1 February

1972

the representation

apprcpriate

k = kL + kg ;

to a first-order

d = krKkr + ks) >

perturbation

(1)

where ic and @ are the observed decay rate constant and quantum yield. In this description the non-radiative rate constant is given to first order by the Golden Rule kg = (2n/fi)p,,,

lonl WI’

)

(2)

where pm is the density of states in I {m);. Thus to calculate kq through (2) it is essential to select the correct zeroth-order representation In>, since a wrong choice for 1)~)implies also a wrong choice for V and hence an erroneous matrix element. Although this ambiguity has been discussed extensively in the literature [I---12] , no consensus seems to have emerged as yet. Among the representations that have been discussed are crude [ 1,2] and adiabatic [3,8,12-IS] Born-Oppenheimer (BO) states, states based on uncorrelated electrons [4] and purespin BO states [5,6,8]. Higher-order terms have been included [6, IO, 121 and an empirical criterion has been proposed [8] for the selection of In> based on the nature of the observed signaL In the present note we present a non-empirical criterion which is simpler to apply and more powerful than the empirical one. The following analysis concerns only transitions observed through signals that decay exponentially in time. These transitions are singled out firstly because they are by far the most cdmmcn from an experimental point of view, and secondly because they and only they can be described by means of eq. (2) and 19

Volume 14, number

CHEMICAL PHYSICS LETTERS

1

thus suffer from the ambiguity associated with the choice of reprevlntation. A non-exponeniial transition wili always require a more extensive analysis. We shall show that the fundamental assumption of exponential decay leads to constraints which unequivocally determine the best first-order representation of In> and I(m)). First we show that it’ is not enough to extend the expansion, of which eq. (2) is the first term, to infinite order. Formally this is achieved by replacing the operator V in (1) by the Ievel shift operator [ 157 R=V+VP(E

II -iQ’PV

3

(3)

where x,,

=xg

tpvp;

P= 1 -~,z>Ozl.

(41

k, still depends on the initial choice of IH), j(m)} and V. Physicafly this is to be expected since the states 1,~)and 1(ml) are selected through their interaction with the radiation field [! l] . Hence k, should be described in terms of the hamiltonian tJC,+M,

(9

where ;ifp denotes the radiation field and M its interaction with the moieculc, here represented by the dipole operator. This leads to a new Perturbation formulation where Jco=3c;

Golden Rule k= k, + kq = (2n/@(+“J2

‘ps]6’q~2) ,

(111

where pr and pq are the densities of the states J(mr;x)) and I {mq;O)), respectively. Thus a neces-

that jC’,l 4 IC,l and ICs], so that we can set Cp = 0. This implies that only the state [tr>in the manifold of compound states IFI,(mq)) carries transition dipole strength to Iowerenergy states I{m,}>. The matrix elements Cr and Cg in (7) and (S) are of infinite-order form. To reduce them to first-order form, we carry cut a partial diagonalization of the sary condition

for (11)

to hold

is

matrix [ 161 _The higher-order terms in (7) a::d (8) mix lm> with other moiecular eigenstates, namely states with an energy E =AEnI and this mixing

hamiltonian

It is readily seen that with this replacement

sf=K,

1 May 1972

can be in~o~orated in the description of In-r>.However, complete diagonalization is impossible since the radiation field used to generate and monitor the signal dis~~rin~i~latesbetween states viith E - E,, where the vafue and range ofk’,;, is determined by the wavelength and wavelength range of the radiation field. The most direct way to remove-the intermediate states with E f E,, is to select X given by (IO) as the new zeroth-order hamiltonian. However under typical experimental conditions configuration interaction due to M is completely negligible, so that we can partition the total hamiltonian in a simpler way as folIOWS

tJcF

;

3c’=M+

v.

(6) x=p+ffc”’

The relevant eigenstates of X? are In;@, Inr,;~) and Imq;O), where K repreants a photon of a given energy and polarization, and 0 represents the vacuum state. The matrix elements of the level-shift operator R connecting these states are [IS] c, = ~~~~~:~~l~~t~‘~(~~~ -z)-lP
-jr)-‘PX’ln;O)

; ;

;

iP=jc,

+3qt

+PVP;

i?=M+Lff’P+FVO+OVQ,

Ii3

(71

where 0 = 1 -P = Jn;O) (n;Ol. This procedure reduces (7) and (8) to

03)

C, =

<{i+K)Ihf+hfP(~,

cq

<{%,;o)i

=

-jr)-lP(fif+v)iTl;o)

V1_flfP(E,,

;

-~~-lP(~~t~~,~;~}.

(13)

If we neglect processes involving more than one photon we finalfy get =

xc +FIT, +PfV+M)P.

(101

The interactions (7)-(g) lead to exponential decay if we can represent the rate constant through the ;20 ..,_ ,:

,;

,.;.

.’ .

C, = C{i;;I’;~)fii+ [CIE,, i+n

-~j)-l~li;O)O.;U]]Vlm;~~

;

Volume

14, number 1

CHEMICAL

PHYSICS

The new molecular zeroth-order.states /j%i)> are eigenstates of Z* gnm . by (121, whereas the old states i(m)> were chosen as eigenstates of X0 given by (6). The two zeroth-order hamiltonians jc”o and X0 differ by a term PVP which depends on In>and thus on the nature of the excitation and observation processes. All actual calculations available thus far [ I- 141 are based on matrix elements C$ = ~(z~z,;O}JVJn;O~ instead of on (14). In this sense they are 211approximations and their relative merits must be judged on the basis of their quantitative accuracy. Generally speaking the best representation in terms of eigenfunctions of X0 is that which yields the smallest higher-order matrix eiements. Since PUP varies with in>, there is no general prescription for finding this representation: it all depends on the system at hand. However, some general rules of practical interest can be given if we restrict ourselves to radiationless transitions involving the electronic ground state and the lower excited electronic stxtes. Let I(r71]) be the electronic ground state and 1(m,)> the lowest vibrational levels of this state. This assumption implies that we h3ve used a BO-type approxinlation to separate the electronic and vibrational parts of the wavefunction, in keeping with the nature of optical selection rules. Let us now partially diagonal&e the hamiltoni~ matrix and consider the transformed states I{Gr}> which are eigenstates of @f = 3& + PVP. This hamiltonian differs from the exact molecular hamiltonian JX’,f only to the extent that Pdiffers from the unit operator in its effect on a particular state. Since typically l(mr)} has a much lower energy than In>, P tends to approach the unit operator for these states, so that the states I(%r]) will be close to exact moIecular eigenstates. Consider now the states I (m,) > degenerate with In). In this case the dominant interaction of I{mq)) wih often be the interaction with &, so that P will approach the null operator. Then JCt* approaches jc$ for this state and I{%,}, can be approximated by /(mqJ>. However the condition that for exponential decay to occur all transition dipole moment to lower states I(%,}) must be concentrated in a single state in>, implies that <(~z~;O)iMl(%r;~)) =: 0, where Im,) approaches an eigenstate of ;rC&and I,&;,)an eigenstate of SC,. Hence the repre~ntation to be chosen for lm>and In>should not only concentrate all transitions dipole moment in a single level, but should simuttaneously provide an

LETTERS

1 May

1972

accurate description of the molecular ground state. In general this criterion is easier to handle than the requirement that higher-order matrix elements should be minimized [ 10,123. To illustrate this and to contribute to the solution of a we&known problem, we consider the internal conversion from the lowest excited singlet state St to the ground state So of a typical polyatomic molecule. In particular we wish to compare the adisbatic BO representation trt”> = @,,(G Q)~~,CQ~ f

fj3

with the crude BO representation In5 = ~~~(~,~~~A~(Q~ I)

(16)

where q and Q denote the sets of electronic and nuclear coordinates, respectiveIy, and Qo is a suitabfy chosen nuclertr equilibrium configuration. For many molecules both representations are equally satisfactory in separating the state [?z>with transition dipofe moment to I( “r)> from the state I(~to}) without such transition dipole moment. Since @,,(q. Qo) contains fewer v3~at~on3~ parameters than +,(q,Q), it should be subject to stronger configuration interaction with states of different energy. However, it is difficult CO turn this into a quantitative argument, since there are so many of these interactions, often of opposite sign. On the other h3nd, it is immediately obvious that @,,r(q, Q) is a better approximation to the exact efectronic ground state than +,,(q, Qo)_ Hence our critericn immediately indicates the superiority of the adiabatic owx the crude BO representation for typical S 1v+ So transitions. it is clear, however, that the argument breaks down if I(nz,)> is not the ground state but an excited st3te close to and strongly coupled with other excited states, Let us now consider triplet-to-ground state (T1--+ So) intersystem crossing in such a molecule, generated through direct optical excitation of the triplet manifold. To make use of the spin selection rule as we11 as the electronic-vibrational selection ruie, we start with a pure-spin BO basis I??),Inr). The single emitting state is the triplet state; to concentrate ah transition dipole moment to So states in this state, we must treat spin-orbit coupling between degenerate T, and So ievels as a perturbation. Since there is no vibronic coupling between pure-spin state of different multiplicity, the nuclear kinetic-energy operator can 21

Volutie

14, number

CHEMICAL PHYSICS LETTERS

1

be incorporated in the zeroth-order hamiltonian X0. This yields a rtew representation I??), I?%),which contains full vibronic configuration interaction, as well as partial spin-orbit coupling, namely with all higher excited states. In a planar aromatic molecule this would include full nu spin-orbit coupling which is the dominant source of multiplicity mixing [6]. It follows that I(%,)) is indeed an essentially exact moIecufar eigenstate in this description. The radiationless transition matrix element is simply (($)PC~* 13, where 3cs0 is the spin--orbit coupling operator. 3y expanding this matrix eIement in the pure-spin BO set we obtdn higher-order matrix elements in addition to ({m,)IJCsoIn), as discussed in detail elsewhere [6, 173. These examp!es show that OUI method gives a simple and unique answer to the problem of finding the best first-order description for a radiaiionless transition to the ground state. The argument is readily extended to radiationless transitions not involving the ground

state

but competing

with

a radiative

transition

state, e.g., S,-+ T, competing with S, + S, fluorescence. For a detailed discussion of this process we refer to ref. [17]. to the ground

We are indebted to Victor Lawetz for helpful comments. G.O. thanks the Consiglio Nazionale delle Recerche of Italy for the award of a fellowship.

1 May 1972

References [l] D.&l. Burland and G.W. Robinson, Proc. NatL Acad, Sci. 66 (1970) 2.57. 121 B. Sharfand R. Silbey, Chem. Phys. Lettzrs 4 (1969) 423; 5 (1970) 314; 9 (1971) 125. (31 K.F. Freed and J. Jortner, J. Chem. Phys. 50 (1969) 2916. [4] B. Sharf, Chem. Phys Letters 6 (1970) 364. [S] B.R. Henry and W. Siebrand, J. Chit-n. Phys (1970) 33; R. Lefebvre, J. Chim. Phys. (1970) 38. (61 W. Siebrand, Chem. Phys. Letters 6 (1970) 192; B.R. Henry and W. Siebrand, J. Chem. Phys. 54 (1971) 1072. [ 7 J G. Orlandi and W. Siebrand, Chem. Phys. Letters 8 (19Ti) 473. [S] W. Siebrand,Chem. Phys. Letters Y (1971j 157. [9] R. tefebvre, Chem. Phys Letters 8 (1971) 306. [IO] K.F. Freed rend WM. Gelbart, Chem. Phys. Letters 10 (1971) 187. [ll] W. Rhodes, Chem. Phys. Letters 11 (1971) 179. [ 121 A. Nitzan and 3. Jortner, Chem. Phys Letters ll(l971) 458. [ 131 S.H. Lin, I. Chem. Phys. 44 (1966) 3759. ! 141 hf. Bixon and J. Jortner. J. Chem. Phys 48 (1968) 715. [ 151 bf. GoldbeGer and K. Watson, Collision theory (Wiley, New York, 1964) ch. 8. [ 161 U. Fsno, Phys Rev. 124 (1961) 1866. (171 V. Lawetz, G. Orlandi md W. Siebrand, J. Chem. Phys., to be published.