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Dynamical quantum mechanical phenomena in nature are governed by Time-Dependent Schrödinger Equation (TDSE), which plays the same role as the Newton’s second law in classical mechanics

in mathematical sense,

There exist too many approach to solve TDSE such as time-dependent perturbation theory, Green function and evolution operator technique.

One may find the conduction phenomenon in a molecular wire a time-dependent process, i.e., the electrons are transported from one end to the other end. Therefore, TDSE can be employed to simulate this process, utilizing evolution of the system’s wave function from its initial state to final state:

One may find the conduction phenomenon in a molecular wire a time-dependent process, i.e., the electrons are transported from one end to the other end. Therefore, TDSE can be employed to simulate this process, utilizing evolution of the system’s wave function from its initial state to final state:

For infinitesimal time interval, evolution operator is defined as

In practice, matrix representation of evolution operator can be used in some finite basis set, e.g., atom-centered Gaussian type orbitals. The below equation describes superposition of the time-dependent wave function over eigenstates of the time-indepndent Hamiltonian, i.e., stationary states of the unperturbed Hamiltonian in the absence of applied voltage:

Using time-dependent perturbation theory, it can be shown that the coefficients vary with time as:

Once evolution operator is determined, the system’s wave function at any time can be computed and all physical properties will be available. On the other hand, the time dependence of observable A can be directly computed by Heisenberg equation: