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Lie group A group G which has a compati- value. Statistically, it represents the mean life
ble manifold structure such that both the product · expectancy of an excited species. In a reacting
: G × G → G and the inversion i : G → G : system in which the decrease in concentration
g → g −1 are differentiable maps. of a particular chemical species is governed by
Left translations L : G → G defined by a first-order rate law, it is equal to the recipro-
g
L (h) = g · h are diffeomorphisms.If v ∈ T G cal of the sum of the (pseudo)unimolecular rate
e
g
is a tangent vector in the unit e ∈ G we can constants of all processes which cause the decay.
define a vector field λ (g) = T L (v) which is When the term is used for processes which are
g
e
v
left-invariant, i.e., TL λ (h) = λ (g · h).Any not first order, the lifetime depends on the initial
g v
v
left-invariant vector field on G is obtained in this concentration of the species, or of a quencher,
way. The set X (G) of left-invariant vector fields and should be called apparent lifetime instead.
L
form a Lie subalgebra of the Lie algebra of all
limiting current The limiting value of a
vector fields X(G). The Lie algebra X (G) is faradaic current that is approached as the rate of
L
isomorphic to T G as a vector space, so that it is the charge-transfer process is increased by vary-
e
finite dimensional (dim(X (G)) = dim T G = ing the potential. It is independent of the applied
L
e
dim G). It is called the Lie algebra of G.
potential over a finite range, and is usually evalu-
Analogously, right-translations R : G → G
g ated by subtracting the appropriate residual cur-
defined by R (h) = h · g are diffeomorphisms.
g rent from the measured total current. A limiting
If v ∈ T G is a tangent vector we can define
e current may have the character of an adsorption,
a vector field ρ (g) = T R (v) which is right- catalytic, diffusion,or kinetic current, and may
g
v
e
invariant, i.e., TR ρ (h) = ρ (h · g).Any include a migration current.
v
g v
right-invariant vector field on G is obtained in
this way. The set X (G) of right-invariant vector line formula A two-dimensional represen-
R
fields form a Lie subalgebra of the Lie algebra tation of molecular entities in which atoms are
of vector fields X(G). The Lie algebra X (G) shown joined by lines representing single or
R
is canonically isomorphic to X (G) as a vec- multiple bonds, without any indication or impli-
L
tor space, so that it is also finite dimensional cation concerning the spatial direction of bonds.
(dim(X (G)) = dim G). Clearly, if the group linear A vector space is often called a linear
R
G is Abelian, then X (G) and X (G) coincide. space. A map T : V → W from a linear space V
R
L
into a linear space W is called linear or a linear
Lie-Poisson bracket Let g be a Lie algebra,
transformation or a linear operator if T(x + y)
g its dual space and <, >: g × g → R
∗
∗
= T(x) + T(y) and T(αx) = αT (x) for all
the natural pairing, <µ, ξ >= µ(ξ). The Lie-
x, y ∈ V , α ∈ C.
∗
Poisson bracket of any F, G : g → R is defined
by linear chain A chain with no branch points
intermediate between the boundary units.

δF δG
{F, G} (µ) =±/µ, , 0
± linear functional A linear scalar valued
δµ δµ
function f : V → C on a vector space V . Some-
where δF ∈ g is the functional derivative of F at times continuity of f is also assumed.
δµ
µ defined by
linear group (of a vector space V ) The
1 δF group of automorphisms of V (which is a group
lim [F(µ + tν) − F(µ)] =/ν, 0. with respect to composition). It is denoted by
t→0 t δµ
GL(V ).
With the Lie-Poisson bracket, g is a Poisson Also the matrix group of all invertible (finite)
∗
manifold. matrices. It can be noncanonically identified
m
with GL(R ), and it is denoted by GL(m, R).
lifetime (mean lifetime), τ τ τ The lifetime of If dim(V ) = m, then there exists a group iso-
a chemical species which decays in a first-order morphism between GL(V ) and GL(m, R) which
process is the time needed for a concentration is induced by the choice of a basis of V ; both
of this species to decrease to 1/e of its original GL(V ) and GL(m, R) are Lie groups.
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