Chemical Reaction Topology Theory (II)

Define the wave function on the reaction path
σ:0,1→R (R is the atomic configuration space) as
ψ(σ, r) (t, r)=ψ (σ(t), r), t ∈0, 1 (r is the electronic coordinates)
It is proved that the homotopy claases ψ_σ of the reaction path wavefunctions (functional of reaction pathes) (RPW) are the same as the reaction path homotopy classes, namely, ifσ₁∩σ₂=ω, then ψσ₁ ∩σ₂=ω, here ψ_σ is defined as
ψ (σ,r), σ∈σ
Homotopy expansion of the reaction propagator is derivedK\left(R_2, t_2 ; R_1, t_1\right)=\sum_t \in \Pi_1\left(F^-(A), R_1\right) \int_0^1 \mathrm~d u e^i S\leftt \sigma_M\righthere Π₁(F^-(A),R₁) is the foundamental group of F^-(A). And the RPW can be written according to the initial state
ψ_σ=∫dR₁∫_0σ¹due^isσ_Mψ(R₁, t₁)
As an example, we have studied a moving paticale on a circle. The homotopy expanaion is

(Z is the foundamental group of a circle) the integration in the summation is the contribution or the particle rotating along the circle for n times to K(v₀, 1; v₀, 0).