In molecule, since nuclei are much heavier than electrons, they are considered as classical particle, this is semi-quantum molecular dynamics. The electronic states in molecule are time-dependent superpositions of all the eigenstates, and the electron distribution is changing with the time, this is dynamical evolution. At the same time, the nuclear motion makes the base vectors of the eigenstates continuously varying, and it indirectly induces additional evolution, it is kinematical evolution.

Recent years, the semi-quantum molecular dynamics, especially the Ehrenfest dynamics, have got further progress,^[1–4] where the dynamical evolution can be expressed analytically, and its calculation is transparent. However, the kinematical one depends on complicated numerical calculation,^[5–7] which causes some confusions. It is needed to establish an analytic formula to express the kinematical evolution, then all the results become definite. This paper will provide one approach based on the connection in the geometry to derive such an expression.

A molecule consists of N nuclei and M electrons, the coordinates of nuclei are {R} = (R₁ ⋯ R_α ⋯ R_3N) and that of electrons are x = (x₁ ⋯ x_β ⋯ x_3M). R determines the configuration and shape of the molecule.

The electronic Hamiltonian is

The time-dependent Schrödinger equation is

Actually Eq. (4) is the solution of Schrödinger Eq. (3). Here , which depends only on R, is the component of the electronic state ψ^k(x, t) in the Hilbert space. In other words, is the electronic state in φ_n representation. When molecular configuration R changes slowly, the base vectors φ_m(x | R) follow to vary, and the electronic state evolves.

At the beginning t = t₀, R = R₀, if an electron stays in the eigen-function φ_l(x | R₀), i.e.

Nuclei obey the Newtonian equation

By using the generalized Hellmann-Feynman Theorem,^[8] Eq. (6) can be simplified to

By solving the combined equations of Schrödinger Eq. (3) and Newtonian Eq. (7), both the electronic states C_m(R) and nuclear motion R(t) can be obtained. Since nuclear motion is much slower than that of electrons, a practical method to solve these combined equations is the iteration approach step by step, each step takes a short time interval Δt. In solving the Schrödinger Eq. (3), within Δt, R can be fixed and Eq. (3) becomes

For each step, during the interval Δt, the configuration is R, and ψ(x, t) is expanded in Eq. (4). At the end of that step, the time is t′ = t + Δt, the configuration becomes R′ = R + ΔR, and ψ^k(x, t + Δt) should be expanded by φ_n(x | R′),

From Eq. (11), once the electronic state at t is known, the state at t+Δt can be obtained.

Thus, the evolution of the electronic state is demonstrated.

In each step, the starting time t can be taken as t₀. Then Eq. (11) becomes

This equation shows that the evolution of the electronic state or ψ^k(x, t) contains two parts: The first part comes from Schrödinger equation, it is the dynamical evolution caused by the phase change. The second part ⟨φ_l(x | R′) |φ_n(x|R)⟩ comes from the transformation of base vector φ_l(x | R), it is the kinematic evolution.

The rest problem is how to calculate the matrix ⟨φ_l(x|R + ΔR)|φ_n(x|R)⟩. A primary method is numerical calculation, and such method needs a lot of computer work. As usual, heavy numerical calculation may cause uncertainty, one is energy shift. If we can derive an analytic expression for the kinematic evolution ⟨φ_l(x|R′)|φ_m(x|R)⟩, it is desirable. This goal can be achieved by using the connection of fiber bundle. The details are presented in the following.

The change from configuration R_i to R_j is a deformation operator D(j, i), which includes unit element D(i, i)=I, inverse element D⁻¹(i, j) = D(j, i), multiplication D(k, j) × D(j, i) = D(k, i), associativity of products [D(l, k) × D(k, j)] × D(j, i) = D(l, k) × [D(k, j) × D(j, i)]. Here, the multiplication has a restriction: two operators can multiply only they are connected. With such multiplication, the deformation operators form a special group, and the deformation group D(i, j) can be represented in terms of matrix, based on the Hamiltonian (1).

In base space R, at any point, a linear Hilbert space with the base vectors φ_n(x|R) can be attached, it is a fiber bundle.

During the deformation from R to R′, the bases are transformed from φ_n(x|R) to φ_n(x|R′). Since φ_n(x|R) are complete set, φ_n(x|R′) can be expanded in terms of φ_n(x|R),

From Eq. (13), we can get

Very close to R(R₁,R₂,…,R_α,…,R_3N), the coordinates of R′ are (R₁,R₂,…, R_α + Δ R_α,…,R_3N). i.e. R′ = R + ΔR_α,

In geometry, this rate demonstrates the orientation correlation between neighboring base vectors of fiber bundles. Here, in the molecule, it is the connection of deformation. In base space R, is a vector with the component index α. In the Hilbert space of fiber bundle, is a tensor with the indexes (n, m).

By the way, in algebra, is the infinitesimal generator of the representation D_nl(R′, R).

By using the perturbation theory, we can get

By using , from Eq. (17), we can get

Finally, by solving combined equations (7) and (23), the evolutions of electronic state ψ(x, t) and the nuclear configuration R(t) can be determined.

Polymers are chain-like molecules. Since they are one-dimensional systems, the polymers have prominent self-trapping, which is a typical deformation effect. The self-trapping produces many novel phenomena in polymers, such as photoinduced spin-flipping of charge carries and charge-flipping of spin carries. The Ehrenfest dynamics has been used to study these processes.^[6–7] But these works have a problem, that their numerical calculations are very heavy, even getting some uncertainty. The connection of fibre bundle can be used to reduce the numerical calculation and make results transparent.

Molecule is a physical model of the fiber bundle, which can be used to build analytic expression for molecular dynamics and simplify numerical calculation.