Perturbation theory

Perturbation theory is an essential tool in quantum chemistry, allowing us to perform approximate calculations in systems where the Hamiltonian is too complicated for exact solutions. At its core, perturbation theory provides a systematic way to find an approximate solution to a problem by starting from an exact solution to a similar, simpler problem. This approach is incredibly valuable in quantum chemistry, where many-body interactions often lead to complex systems that lack closed-form solutions.

Introduction to perturbation theory

In quantum mechanics, we often deal with systems described by the Hamiltonian H, which is the sum of two terms:

H = H0 + λH′
    

Here, H0 is the Hamiltonian of a simple, unperturbed system for which we can solve the Schrödinger equation exactly:

h0 ψn(0) = en(0) ψn(0)
    

The term λH′ denotes the perturbation, which is a small correction to the simple system. The parameter λ is often assumed to be a small number, and in many cases, we can find the solution as a power series in λ:

En = En(0) + λEn(1) + λ² En(2) + ...
    

Similarly, the wave function ψ_n can also be expanded as follows:

ψn = ψn(0) + λψn(1) + λ²ψn(2) + ...
    

First-order perturbation theory

First-order perturbation theory is the simplest approximation. It focuses on finding the first corrections to the energy and wave functions introduced by the perturbation. We substitute the expansion into the Schrödinger equation and collect terms by powers of λ to solve iteratively.

Hψn = En ψn
    

This yields a series of equations based on different powers of λ. For λ0, we have the original, unaffected equation. For λ1, the energy correction is given by:

En(1) = ⟨ψn(0) |H′|ψn(0) ⟩
    

This result indicates that the first-order correction to the energy is the expected value of the disturbance with respect to the unperturbed wave function.

Second-order perturbation theory

Second-order perturbation theory provides a more accurate estimate by including higher-order terms. Second-order corrections to the energy En(2) can be obtained from terms of order λ²:

En(2) = ∑m≠n |⟨ψm(0) |H′|ψn(0) ⟩|² / (En(0) − Em(0) )
    

where m represents the other states of the system that are unaffected. This expression calculates how the different states contribute to the energy improvement, taking into account their respective energy differences.

Example: Stark effect

The Stark effect describes the splitting and shifting of spectral lines of atoms and molecules due to the presence of an external electric field. Let us consider the hydrogen atom in an electric field as an example.

The unaffected Hamiltonian H0 of the hydrogen atom is given by:

H0 =-ħ²/2m∇²-e²/r
    

The disturbance H′ caused by the external electric field along z axis can be represented as:

H′ = efz
    

where F is the field strength. Applying first-order perturbation theory, the ground state corrections vanish, because the electric dipole elements for the hydrogen ground state under z are zero. Thus, we need second-order corrections to observe the transitions.

An example in quantum chemistry: the helium atom

The helium atom with two electrons is a perfect example where perturbation theory proves useful. The unperturbed Hamiltonian of the helium atom H0 (ignoring electron repulsion) is:

H0 = -ħ²/2m (∇²1 + ∇²2) - Ze²/r1 - Ze²/r2
    

The perturbative Hamiltonian H′ is the electron-electron repulsion:

H′ = E²/|R1 - R2|
    

Turbulence theory helps to calculate the energy corrections to the ground state of helium, and gives better results than ignoring interactions.

Mathematical representation

The perturbation method involves careful mathematical expansion and substituting the solution into the Schrödinger equation. Consider a system where the total Hamiltonian expressed with perturbations is:

H = H0 + εH′
    

Here, ε serves as a bookkeeping parameter, which initially allows the perturbation to probe successively smaller units.

In the perturbed problem the eigenvalue equation becomes:

(H0 + εH′) (ψn(0) + εψ(1) + …) = (En(0) + εEn(1) + …) (ψn(0) + εψ(1) + …)
    

By balancing independent equations at comparable orders of ε, we can find the wave-function and energy solutions sequentially.

Visual example of perturbation theory

₀ Ψ = Ψ₀ + λΨ₁

In this simple visual representation, Ψ0 represents an initial unaffected wave function, while Ψ = Ψ0 + λΨ1 represents the affected wave function.

Benefits and limitations

Perturbation theory offers notable advantages:

• Analytical simplicity in estimating energy levels and wave functions.
• Useful for understanding the impact of small changes on complex systems.
• Applicable to various branches including quantum field theory and solid state physics.

However, it has its limitations:

• This is valid only if the disturbance is minor; major disturbances may invalidate its use.
• Convergence is not guaranteed in strongly interacting systems.
• This may fail in degenerate states or when the energy denominator approaches zero.

Summary

Perturbation theory is invaluable for solving complex systems without closed-form solutions in quantum chemistry. It repeatedly refines solutions, considering only minor changes to the original, easily manageable problems. Despite its limitations under strong perturbations, it serves as a cornerstone technique providing insights into physical systems ranging from the atomic to the molecular levels.

Comments