inorganic chemistry

where cj represents the full set of eigenfunctions with the
associated eigenvaluesEj of the unperturbed system Hamilto-
nianH 0 , when the coefficientscjare time dependent. The values
ofcjobtained in the solution of this set of equations are related to
the probability of finding the system in any particular state at
any later time. It is not generally possible to find exact solutions,
and the time-dependent perturbation theory is usually employed
to obtain approximate solutions.
Thetransitionprobabilityper unittimegivenby thetime-depen-
dentperturbationtheory,thatFerminamedGoldenRuleinviewof
its prevalence in radiationless transitions, has the form

kGR¼

2 p

ℏ

jjVfi^2 rf E

ðÞ 0

i



(4)

whererf(Ei(0)) represents the density of final unperturbed states
at the energy of the initial stationary state, andVfiis the matrix
element of the perturbation between the initial and final unper-
turbed states

Vfi¼

ð

cfVcidt (5)

and the integration is over all the space, represented byt. Equa-
tion (4) is valid provided that the final states form a quasi-contin-
uum of states over an energy rangedEin the neighborhood of
Ei(0)and for values oftthat satisfy the relation

t

ℏ

dE

(6)

Theseconditionsareusuallyverifiedforpicosecondandnanosecond
transitions occurring in largemolecules.For example, thedensityof
vibrational states of the ground electronic state of anthracene at the
energy of its first excited singlet state (ES1¼75.5 kcal/mol) is in the
range of 10^11 – 1017 states/(cal mol^1 )( 48 ).
Using the Born–Oppenheimer approximation and assuming
that only the electronic distribution is perturbed,Eq. (5) can be
written as

Vfi¼

ð

ffVfidte

ð

uv 0 uvdtn (7)

where the first factor measures the extension of the electronic
redistribution induced by the perturbation from the initial state
i to the initial state f

204 LUIS G. ARNAUT