GW approximation

Key takeaways

GW approximation belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect GW approximation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of GW approximation from memory before moving on to harder problems.

Reference excerpt

The GW approximation is a method used to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's function G and the screened Coulomb interaction W

Σ = i G W − G W G W G + ⋯ {\displaystyle \Sigma =iGW-GWGWG+\cdots }

can be truncated after the first term:

Σ ≈ i G W {\displaystyle \Sigma \approx iGW}

In other words, the self-energy is expanded in a formal Taylor series in powers of the screened interaction W and the lowest order term is kept in the expansion in GW approximation.

Theory The above formulae are schematic in nature and show the overall idea of the approximation. More precisely, if we label an electron coordinate with its position, spin, and time and bundle all three into a composite index (the numbers 1, 2, etc.), we have

Σ ( 1 , 2 ) = i G ( 1 , 2 ) W ( 1 + , 2 ) − ∫ d 3 ∫ d 4 G ( 1 , 3 ) G ( 3 , 4 ) G ( 4 , 2 ) W ( 1 , 4 ) W ( 3 , 2 ) + . . . {\displaystyle \Sigma (1,2)=iG(1,2)W(1^{+},2)-\int d3\int d4\,G(1,3)G(3,4)G(4,2)W(1,4)W(3,2)+...}

where the "+" superscript means the time index is shifted forward by an infinitesimal amount. The GW approximation is then

Σ ( 1 , 2 ) ≈ i G ( 1 , 2 ) W ( 1 + , 2 ) {\displaystyle \Sigma (1,2)\approx iG(1,2)W(1^{+},2)}

To put this in context, if one replaces W by the bare Coulomb interaction (i.e. the usual 1/r interaction), one generates the standard perturbative series for the self-energy found in most many-body textbooks. The GW approximation with W replaced by the bare Coulomb yields nothing other than the Hartree–Fock exchange potential (self-energy). Therefore, loosely speaking, the GW approximation represents a type of dynamically screened Hartree–Fock self-energy. In a solid state system, the series for the self-energy in terms of W should converge much faster than the traditional series in the bare Coulomb interaction. This is because the screening of the medium reduces the effective strength of the Coulomb interaction: for example, if one places an electron at some position in a material and asks what the potential is at some other position in the material, the value is smaller than given by the bare Coulomb interaction (inverse distance between the points) because the other electrons in the medium polarize (move or distort their electronic states) so as to screen the electric field. Therefore, W is a smaller quantity than the bare Coulomb interaction so that a series in W should have higher hopes of converging quickly. To see the more rapid convergence, we can consider the simplest example involving the homogeneous or uniform electron gas which is characterized by an electron density or equivalently the average electron-electron separation or Wigner–Seitz radius r s {\displaystyle r_{s}} . (We only present a scaling argument and will not compute numerical prefactors that are order unity.) Here are the key steps:

The kinetic energy of an electron scales as 1 / r s 2 {\displaystyle 1/r_{s}^{2}}

The average electron-electron repulsion from the bare (unscreened) Coulomb interaction scales as 1 / r s {\displaystyle 1/r_{s}} (simply the inverse of the typical separation) The electron gas dielectric function in the simplest Thomas–Fermi screening model for a wave vector q {\displaystyle q} is

ϵ ( q ) = 1 + λ 2 / q 2 {\displaystyle \epsilon (q)=1+\lambda ^{2}/q^{2}}

where λ {\displaystyle \lambda } is the screening wave number that scales as r s − 1 / 2 {\displaystyle r_{s}^{-1/2}}

Typical wave vectors q {\displaystyle q} scale as 1 / r s {\displaystyle 1/r_{s}} (again typical inverse separation) Hence a typical screening value is ϵ ∼ 1 + r s {\displaystyle \epsilon \sim 1+r_{s}}

The screened Coulomb interaction is W ( q ) = V ( q ) / ϵ ( q ) {\displaystyle W(q)=V(q)/\epsilon (q)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with GW approximation

Start with the simplest possible case. Write down what GW approximation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to GW approximation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about GW approximation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of GW approximation

In research: GW approximation appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses GW approximation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: GW approximation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Quantum field theory, so understanding it makes those chapters shorter.
In everyday life: Look for GW approximation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study GW approximation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what GW approximation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain GW approximation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is GW approximation in simple terms?

The GW approximation is a method used to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the self-energy Σ in terms of the single particle Green's function G and the screened Coulomb interaction W Σ = i G W − G W G W G + ⋯ {\displaystyle \Sig…

Why does GW approximation matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study GW approximation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on GW approximation.