Orbital-free density functional theory

The Hohenberg–Kohn theorems{{cite journal |last=Hohenberg |first=P. |author2=Kohn, W. |title=Inhomogeneous Electron Gas |journal=Physical Review |year=1964 |volume=136 |issue=3B |pages=B864–B871 |doi=10.1103/PhysRev.136.B864 |doi-access=free |bibcode = 1964PhRv..136..864H }} 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)

: $T\_S[\backslash \{\backslash phi\_\{i\}\backslash \}]\; =\; \backslash sum\_\{i=1\}^\{N\}\; \backslash langle\backslash phi\_i|-\backslash frac\{1\}\{2\}\backslash nabla^2|\backslash phi\_i\backslash rangle,$

where $|\backslash phi\_i\backslash rangle$ is the i-th Kohn–Sham orbital. The summation is performed over all the occupied Kohn–Sham orbitals.

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{{cite book |last=Ligneres |first=Vincent L. |title=Handbook of Materials Modeling |year=2005 |publisher=Springer Netherlands |pages=137–148 |author2=Emily A. Carter |editor=Syndey Yip |chapter=An Introduction to Orbital Free Density Functional Theory}}{{cite journal|author1=Thomas, L. H.|title=The Calculation of Atomic Fields|journal=Math. Cambridge Philos. Soc.|volume=23|pages=542–548|year=1927|issue=5 |doi=10.1017/S0305004100011683|bibcode=1927PCPS...23..542T }}{{cite journal|author1=Fermi, E.|title=Un Metodo Statistico per la Determinanzione di alcune Prioprieta dell'Atomo|journal=Rend. Accad. Naz. Lincei|volume=6|pages=602|year=1927|doi=10.1007/BF01351576}}

: $T\_\backslash text\{TF\}[n]\; =\; \backslash underbrace\{\backslash frac\{3\}\{10\}\; (3\backslash pi^2)^\{\backslash frac\{2\}\{3\}\}\}\_\{C\_\{TF\}\}\; \backslash int\; [n(\backslash mathbf\; r)]^\{\backslash frac\{5\}\{3\}\}\; \backslash ,d^3r.$

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.

A notable historical improvement of the Thomas-Fermi model is the von Weizsäcker (vW) kinetic energy (1935),{{cite journal|title=Zur Theorie der Kernmassen|journal=Zeitschrift für Physik|year=1935|first=C. F. v.|last=Weizsäcker|volume=96|issue=7–8|pages=431–458|doi=10.1007/BF01337700|bibcode = 1935ZPhy...96..431W |s2cid=118231854 }} which is exactly the kinetic energy for noninteracting bosons and can be regarded as a Generalized Gradient approximation (GGA).

: $T\_\{\backslash text\{vW\}\}[n]=\backslash frac\{1\}\{8\}\backslash int\; \backslash frac\{\backslash nabla\; n(\backslash mathbf\; r)\backslash cdot\backslash nabla\; n(\backslash mathbf\; r)\}\{n(\backslash mathbf\; r)\}d^\{3\}r=\backslash int\backslash sqrt\{n(\backslash mathbf\; r)\}(-\backslash frac\{1\}\{2\}\backslash Delta)\backslash sqrt\{n(\backslash mathbf\; r)\}d^\{3\}r$

A conceptually really important quantity in OFDFT is the Pauli kinetic energy.{{cite journal|author1=Bartolotti, L. J.|author2=Acharya, P. K.|title=On the functional derivative of the kinetic energy density functional|journal=Journal of Chemical Physics|volume=77|pages=4576–4585|year=1982|issue=9 |doi=10.1063/1.444409}}{{cite journal|author1=Levy, M.|author2=Ou-Yang, H.|title=Exact Properties of the Pauli potential for the square root of the electron density and the kinetic energy density functional|journal=Physical Review A|volume=38|pages=625|year=1988|doi=10.1103/PhysRevA.38.625}}{{cite journal|author1=Smiga, S.|author2=Siecinska, S.|author3=Fabiana, E.|title=Methods to generate reference total Pauli and kinetic potentials|journal=Physical Review B|volume=101|pages=165144|year=2020|doi=10.1103/PhysRevB.101.165144|arxiv=2005.03526 }}{{cite journal|author1=Holas, A.|author2=March, N. H.|title=Construction of the Pauli potential, Pauli energy, and effective potential from the electron density|journal=Physical Review A|volume=44|pages=5521–5536|year=1991|issue=9 |doi=10.1103/PhysRevA.44.5521|bibcode=1991PhRvA..44.5521H }} As the Kohn-Sham correlation energy links the real system of interacting electrons to the artificial Kohn-Sham (KS) system of noninteracting electrons, the Pauli kinetic energy links the KS system to the fictitious system noninteracting model bosons. In the same way as the KS interacting energy it is highy KS-orbital dependent and must be in practice approximated.

: $T\_\{\backslash text\{P\}\}[n]\backslash equiv\; T\_\{\backslash text\{S\}\}[n]-T\_\{\backslash text\{vW\}\}[n]$

The term Pauli comes from the fact, that the functional is related to the Pauli exclusion principle. $T\_\{\backslash text\{P\}\}[n]=0$ for an electron number of $N\backslash leq\; 2$.

The exchange energy in orbital-free density functional theory (OFDFT) is the Dirac exchange{{cite journal|author1=Dirac, P. A. M.|title=Note on Exchange Phenomena in the Thomas Atom|journal=Proc. Cambridge Philos. Soc.|volume=26|pages=376–385|year=1930|issue=3 |doi=10.1017/S0305004100016108|bibcode=1930PCPS...26..376D }} as a Local Density Approximation (LDA) (1930)

It is related to the homogeneous electron gas (HEG).

: $E\_\{x\}^\{\backslash text\{LDA\}\}[n]=-\backslash underbrace\{\backslash frac\{3\}\{4\}\backslash bigg(\backslash frac\{3\}\{\backslash pi\}\backslash bigg)^\{1/3\}\}\_\{C\_\{x\}\}\backslash int[n(\backslash mathbf\; r)]^\{\backslash frac\{4\}\{3\}\}d^\{3\}r$

An important bound for the LDA exchange energy is the Lieb-Oxford inequality.

State of the art kinetic energy density functionals for orbital-free density functional theory and still subject to ongoing research are the so called nonlocal (NL) kinetic energy density functionals such as e.g. the Huang-Carter (HC) functional {{cite journal|author1=Huang, C.|author2=Carter, E.|title=Nonlocal orbital-free kinetic energy density functional for semiconductors|journal=Physical Review B|volume=81|pages=045206|year=2010|issue=4 |doi=10.1103/PhysRevB.81.045206|bibcode=2010PhRvB..81d5206H }}{{cite journal|author1=Shao, X.|author2=Mi, W.|author3=Pavanello, M.|title=Revised Huang-Carter nonlocal kinetic energy functional for semiconductors and their surfaces|journal=Physical Review B|volume=104|pages=044118|year=2021|issue=4 |doi=10.1103/PhysRevB.104.045118|bibcode=2021PhRvB.104d5118S }} (2010), the Mi-Genova-Pavenello (MGP) functional {{cite journal|author1=Mi, W.|author2=Genova, A.|author3=Pavanello, M.|title=Nonlocal kinetic energy functionals by functional integration|journal=Journal of Chemical Physics|volume=148|pages=184107|year=2018|issue=18 |doi=10.1063/1.5023926|pmid=29764156 |arxiv=1704.08943 |bibcode=2018JChPh.148r4107M }} (2018) or the Wang-Teter (WT) functional {{cite journal|author1=Wang, L.-W.|author2=Teter, M. P.|title=Kinetic-energy functional of the electron density|journal=Physical Review B|volume=45|pages=13196–13220|year=1992|issue=23 |doi=10.1103/PhysRevB.45.13196|pmid=10001400 |bibcode=1992PhRvB..4513196W }} (1992). They admit the general form

$T\_\{\backslash text\{NL\}\}[n]=C\_\{\backslash text\{NL\}\}\backslash iint\; d^\{3\}rd^\{3\}r\text{'}n(\backslash mathbf\; r)^\{\backslash alpha\}K[n](r,r\text{'})n(\backslash mathbf\; r\text{'})^\{\backslash beta\}$

where $\backslash alpha$ and $\backslash beta$ are arbitrary fractional exponents, $K[n](r,r\text{'})$ is a nonlocal KEDF kernel and $C\_\{\backslash text\{NL\}\}$ some constant.

The analogue to the Kohn-Sham (KS) equations (1965) in Orbital-free density functional theory (OFDFT) is the Levy-Perdew-Sahni (LPS) equation {{cite journal |author1=Levy, M |author2=Perdew, J. P.|author3=Sahni, V.|title=Exact differential equation for the density and ionization energy of a many-particle system|journal=Physical Review A|year=1984 |volume=30|issue=5 |pages=2745–2748 |doi=10.1103/PhysRevA.30.2745|bibcode=1984PhRvA..30.2745L }}{{cite journal |author1=Lehtomäki J.|author2=Makkonen, I.|author3=Caro, M. A.|author4=Harju, A.|author5=Lopez-Avecedo, O.|title=Orbital-free density functional theory implementation with the projector augmented-wave method|journal=Journal of Chemical Physics|volume=141|pages=234102|year=2014| issue=23 |doi=10.1063/1.4903450| pmid=25527914 |arxiv=1408.4701v2| bibcode=2014JChPh.141w4102L }} (1984), an effectively bosonic Schrödinger equation

: $\backslash bigg(-\backslash frac\{1\}\{2\}\backslash Delta+v\_\{S\}(\backslash mathbf\; r)+v\_\{P\}(\backslash mathbf\; r)\backslash bigg)\backslash sqrt\{n(\backslash mathbf\; r)\}=\backslash mu\backslash sqrt\{n(\backslash mathbf\; r)\}$

where $v\_\{S\}(\backslash mathbf\; r)$ is the Kohn-Sham (KS) potential, $v\_\{P\}(\backslash mathbf\; r)$ the Pauli potential, $\backslash mu$ the highest occupied KS orbital and $\backslash sqrt\{n(\backslash mathbf\; r)\}$ the square root of the density. One big benefit of the LPS equation being so intimately related to the KS equations is that an existing KS code can be easily modified in an OF code with ejecting all orbitals except for one in the Self-Consistent-Field (SCF) cycle.

Starting from Euler-Lagrange equation of density functional theory $\backslash frac\{\backslash delta\; T\_\{S\}[n]\}\{\backslash delta\; n(\backslash mathbf\; r)\}+v\_\{S\}(\backslash mathbf\; r)=\backslash mu$, simultaneously adding and subtracting the von Weizsäcker potential, i.e. the functional derivative of the vW functional and acknowleding the definition of the Pauli kinetic energy, while the functional derivative of the Pauli kinetic energy with respect to the density is the Pauli potential $\backslash underbrace\{\backslash frac\{\backslash delta\; T\_\{vW\}[n]\}\{\backslash delta\; n(\backslash mathbf\; r)\}\}\_\{v\_\{vW\}(\backslash mathbf\; r)\}+v\_\{S\}(\backslash mathbf\; r)+\backslash underbrace\{\backslash frac\{\backslash delta\; T\_\{P\}[n]\}\{\backslash delta\; n(\backslash mathbf\; r)\}\}\_\{v\_\{P\}(\backslash mathbf\; r)\}=\backslash mu$. Expanding the functional derivative via chain rule $\backslash underbrace\{\backslash frac\{\backslash delta\backslash sqrt\{n(\backslash mathbf\; r)\}\}\{\backslash delta\; n(\backslash mathbf\; r)\}\}\_\{1/\backslash sqrt\{n(\backslash mathbf\; r)\}\}\backslash underbrace\{\backslash frac\{\backslash delta\}\{\backslash delta\backslash sqrt\{n(\backslash mathbf\; r)\}\}\backslash int\backslash sqrt\{n(\backslash mathbf\; r)\}(-\backslash frac\{1\}\{2\}\backslash Delta)\backslash sqrt\{n(\backslash mathbf\; r)\}d^\{3\}r\}\_\{-\backslash frac\{1\}\{2\}\backslash Delta\backslash sqrt\{n(\backslash mathbf\; r)\}\}+v\_\{S\}(\backslash mathbf\; r)+v\_\{P\}(\backslash mathbf\; r)=\backslash mu$ and as a last step multiplying both sides by the square root of the density $\backslash sqrt\{n(\backslash mathbf\; r)\}$ yields the LPS equation.

With the linear transformation $\backslash sqrt\{n(\backslash mathbf\; r)\}\backslash mapsto\backslash frac\{1\}\{\backslash sqrt\{N\}\}\backslash phi\_\{B\}(\backslash mathbf\; r)$ and by defining the bosonic potential as $v\_\{B\}(\backslash mathbf\; r)\backslash equiv\; v\_\{S\}(\backslash mathbf\; r)+v\_\{P\}(\backslash mathbf\; r)$ the LPS equation evolves to the bosonic Schrödinger equation

$\backslash bigg(-\backslash frac\{1\}\{2\}\backslash Delta+v\_\{B\}(\backslash mathbf\; r)\backslash bigg)\backslash phi\_\{B\}(\backslash mathbf\; r)=\backslash mu\backslash phi\_\{B\}(\backslash mathbf\; r)$.

Note that the normalization constraint in Bra–ket notation $\backslash langle\backslash phi\_\{B\}|\backslash phi\_\{B\}\backslash rangle=1$ holds, since $N[n]=\backslash int\; n(\backslash mathbf\; r)d^\{3\}r$.

Quite recently also a time-dependent version of OFDFT has been developed.{{cite journal|author1=Jiang, K.|author2=Pavanello, M.|title=Time-dependent orbital-free density functional theory: Background and Pauli kernel approximations|journal=Physical Review B|volume=103|pages= 245102|year=2021|issue=24 |doi=10.1103/PhysRevB.103.245102|arxiv=2102.06174 |bibcode=2021PhRvB.103x5102J }} It is also implemented in DFTpy.

As a free open-source software package for OFDFT DFTpy has been developed by the Pavanello Group.{{Cite web |title=Pavanello Research Group / DFTpy · GitLab |url=https://gitlab.com/pavanello-research-group/dftpy |access-date=2025-04-24 |website=GitLab |language=en}}{{Cite web |title=DFTpy: Density Functional Theory with Python — DFTpy 2.0.0 documentation |url=http://dftpy.rutgers.edu/ |archive-url=http://web.archive.org/web/20241024062324/http://dftpy.rutgers.edu/ |archive-date=2024-10-24 |access-date=2025-04-24 |website=dftpy.rutgers.edu |language=en}} It was launched in 2020.{{cite journal|author1=Shao, X.|author2=Jiang, K.|author3=Mi, W.|author4=Genova, A|author5=Pavanello, M.|title=DFTpy: An efficient and object-oriented platform for orbital-free DFT simulations|journal=WIREs Comput. Mol. Sci.|volume=11|issue=1|pages=1482|year=2021|doi=10.1002/wcms.1482|arxiv=2002.02985 }} The most recent version is 2.1.1.

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