# 6: Adiabatic Approximation

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- 107250

In quantum mechanics, the adiabatic approximation refers to those solutions to the Schrödinger equation that make use of a time-scale separation between fast and slow degrees of freedom, and use this to find approximate solutions as product states in the fast and slow degrees of freedom. Perhaps the most fundamental and commonly used version is the Born–Oppenheimer (BO) approximation, which underlies much of how we conceive of molecular electronic structure and is the basis of potential energy surfaces. The BO approximation assumes that the motion of electrons is much faster than nuclei due to their large difference in mass, and therefore electrons adapt very rapidly to any changes in nuclear geometry. That is, the electrons “adiabatically follow” the nuclei. As a result, we can solve for the electronic state of a molecule for fixed nuclear configurations. Gradually stepping nuclear configurations and solving for the energy leads to a potential energy surface, or adiabatic state. Much of our descriptions of chemical reaction dynamics is presented in terms of propagation on these potential energy surfaces. The barriers on these surfaces are how we describe the rates of chemical reactions and transition state. The trajectories along these surfaces are used to describe mechanism.

More generally, the adiabatic approximation can be applied in other contexts in which there is a time-scale separation between fast and slow degrees of freedom. For instance, in the study of vibrational dynamics when the bond vibrations of molecules occur much faster than the intermolecular motions of a liquid or solid. It is also generally implicit in a separation of the Hamiltonian into a system and a bath, a method we will often use to solve condensed matter problems. As widely used as the adiabatic approximation is, there are times when it breaks down, and it is important to understand when this approximation is valid, and the consequences of when it is not. This will be particularly important for describing time-dependent quantum mechanical processes involving transitions between potential energy sources.

- 6.1: Born–Oppenheimer Approximation
- Exact solutions using the molecular Hamiltonian are intractable for most problems of interest, so we turn to simplifying approximations. The BO approximation is motivated by noting that the nuclei are far more massive than an electron. When the distances separating particles is not unusually small, the kinetic energy of the nuclei is small relative to the other terms in the Hamiltonian. This means that the electrons move and adapt rapidly—adiabatically—in response to shifting nuclear positions.

- 6.2: Nonadiabatic Effects
- Even without the BO approximation, we note that the nuclear-electronic product states form a complete basis in which to express the total vibronic wavefunction. The is referred to as the coupled channel Hamiltonian can be constructed with terms that describe deviations from the BO approximation and are referred to as nonadiabatic terms. These depend on the spatial gradient of the wavefunction in the region of interest, and act to couple adiabatic Born–Oppenheimer states.

- 6.3: Diabatic and Adiabatic States
- Although the Born–Oppenheimer surfaces are the most straightforward and commonly calculated, they may not be the most chemically meaningful states.

- 6.4: Adiabatic and Nonadiabatic Dynamics
- The BO approximation never explicitly addresses electronic or nuclear dynamics, but neglecting the nuclear kinetic energy to obtain potential energy surfaces has implicit dynamical consequences.

- 6.5: Landau–Zener Transition Probability
- The adiabatic approximation has significant limitations in the vicinity of curve crossings. This phenomenon is better described through transitions between diabatic surfaces. The Landau–Zener expression gives the transition probabilities as a result of propagating through the crossing between diabatic surfaces.