Orbital-free density functional theory

Key takeaways

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

Reference excerpt

In computational chemistry, orbital-free density functional theory (OFDFT) is a quantum mechanical approach to electronic structure determination which is based on functionals of the electronic density. It is most closely related to the Thomas–Fermi model. Orbital-free density functional theory is, at present, less accurate than Kohn–Sham density functional theory models, but it has the advantage of being fast, so that it can be applied to large systems.

Kinetic energy of electrons: an orbital-dependent functional The Hohenberg–Kohn theorems guarantee that, for a system of atoms, there exists a functional of the electron density that yields the total energy. Minimization of this functional with respect to the density gives the ground-state density from which all of the system's properties can be obtained. Although the Hohenberg–Kohn theorems tell us that such a functional exists, they do not give us guidance on how to find it. In practice, the density functional is known exactly except for two terms. These are the electronic kinetic energy and the exchange–correlation energy. The lack of the true exchange–correlation functional is a well known problem in DFT, and there exists a huge variety of approaches to approximate this crucial component. In general, there is no known form for the interacting kinetic energy in terms of electron density. In practice, instead of deriving approximations for interacting kinetic energy, much effort was devoted to deriving approximations for non-interacting (Kohn–Sham) kinetic energy, which is defined as (in atomic units)

T S [ { ϕ i } ] = ∑ i = 1 N ⟨ ϕ i | − 1 2 ∇ 2 | ϕ i ⟩ , {\displaystyle T_{S}[\{\phi _{i}\}]=\sum _{i=1}^{N}\langle \phi _{i}|-{\frac {1}{2}}\nabla ^{2}|\phi _{i}\rangle ,}

where | ϕ i ⟩ {\displaystyle |\phi _{i}\rangle } is the i-th Kohn–Sham orbital. The summation is performed over all the occupied Kohn–Sham orbitals.

Thomas-Fermi (TF) kinetic energy One of the first attempts to do this (even before the formulation of the Hohenberg–Kohn theorem) was the Thomas–Fermi model (1927), which wrote the kinetic energy as

T TF [ n ] = 3 10 ( 3 π 2 ) 2 3 ⏟ C T F ∫ [ n ( r ) ] 5 3 d 3 r . {\displaystyle T_{\text{TF}}[n]=\underbrace {{\frac {3}{10}}(3\pi ^{2})^{\frac {2}{3}}} _{C_{TF}}\int [n(\mathbf {r} )]^{\frac {5}{3}}\,d^{3}r.}

This expression is based on the homogeneous electron gas (HEG) and a Local Density Approximation (LDA), thus, is not very accurate for most physical systems. By formulating Kohn–Sham kinetic energy in terms of electron density, one avoids diagonalizing the Kohn–Sham Hamiltonian for solving for the Kohn–Sham orbitals, therefore saving the computational cost. Since no Kohn–Sham orbital is involved in orbital-free density functional theory, one only needs to minimize the system's energy with respect to the electron density. An important bound for the TF kinetic energy is the Lieb-Thirring inequality.

Von Weizsäcker (vW) kinetic energy A notable historical improvement of the Thomas-Fermi model is the von Weizsäcker (vW) kinetic energy (1935), which is exactly the kinetic energy for noninteracting bosons and can be regarded as a Generalized Gradient approximation (GGA).

T vW [ n ] = 1 8 ∫ ∇ n ( r ) ⋅ ∇ n ( r ) n ( r ) d 3 r = ∫ n ( r ) ( − 1 2 Δ ) n ( r ) d 3 r {\displaystyle T_{\text{vW}}[n]={\frac {1}{8}}\int {\frac {\nabla n(\mathbf {r} )\cdot \nabla n(\mathbf {r} )}{n(\mathbf {r} )}}d^{3}r=\int {\sqrt {n(\mathbf {r} )}}(-{\frac {1}{2}}\Delta ){\sqrt {n(\mathbf {r} )}}d^{3}r}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Orbital-free density functional theory

Start with the simplest possible case. Write down what Orbital-free density functional theory claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Orbital-free density functional theory 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 Orbital-free density functional theory 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 Orbital-free density functional theory

In research: Orbital-free density functional theory appears in mathematics 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 Orbital-free density functional theory 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: Orbital-free density functional theory is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational chemistry, Density functional theory, Quantum chemistry, so understanding it makes those chapters shorter.
In everyday life: Look for Orbital-free density functional theory 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 Orbital-free density functional theory in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Orbital-free density functional theory 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 Orbital-free density functional theory out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Orbital-free density functional theory in simple terms?

In computational chemistry, orbital-free density functional theory (OFDFT) is a quantum mechanical approach to electronic structure determination which is based on functionals of the electronic density. It is most closely related to the Thomas–Fermi model.

Why does Orbital-free density functional theory matter?

Because it connects several mathematics 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 Orbital-free density functional theory?

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 Orbital-free density functional theory.