Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation

Abstract

Density Functional Theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-density-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addition to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's density-based exchangecorrelation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of O(r 3 ), yields substantial improvements over traditional DFT in the description of static correlation in chemical structures and processes such as singlet biradicals and bond dissociations.

Comparison of Density-Matrix Corrections to Density Functional Theory

“…Consider the energy functional for DFT E normalD normalF normalT [ ρ ] = T normals [ ρ ] + V [ ρ ] + F normalx normalc [ ρ ] where ρ is the one-electron density, T s [ρ] is the non-interacting kinetic energy functionalthe kinetic energy from the single Slater determinant that yields the density ρ, V [ρ] is the sum of the one-electron (external) potential and the Coulomb potential, and F xc [ρ] is the exchange–correlation functional. As in ref , we convert DFT into a 1-RDMFT by replacing the non-interacting kinetic energy by the full kinetic energy and adding a 1-RDM-based correction functional C [ 1 D ] E normalR normalD normalM normalF normalT false[ 1 D false] = E normalD normalF normalT + normalT false[ 1 D false] + C false[ 1 D false] where E DFT+T [ 1 D ] is defined as E normalD normalF normalT + normalT false[ 1 D false] = E normalD normalF normalT [ ρ ] + false( T false[ 1 D false] − T s false[ ρ false] …”

“…Note that assumption (ii) is an approximation for the exact C [ 1 D ] functional. Using these assumptions, we previously obtained the following form in ref E normalR normalD normalM normalF normalT false[ 1 D false] = E normalD normalF normalT + normalT false[ 1 D false] − normalT normalr [ false( W 1 false( 1 D − D 2 1 false) ] where 1 W is an arbitrary positive semidefinite weight matrix. By taking 1 W to be a weighted identity matrix w 1 I , we produce the final form of our correction E normalR normalD normalM normalF normalT false[ 1 D false] = E normalD normalF normalT + normalT false[ 1 D false] + w false( normalT normalr false[ 1 D 2 − D 2 1 false] ) If the 1-RDM is idempotent, we note that this correction vanishes, and if the 1-RDM is not idempotent, it is nonzero and serves to remove the double counting of the correlated kinetic energy and to account for static correlation that is missing from traditional DFT.…”

“…Another area of research which focuses on fractional occupations for describing static correlation is presented by one-electron reduced density matrix functional theory (1-RDMFT). − These approaches utilize Gilbert’s theorem to express the ground-state energy as a functional of the one-electron reduced density matrix (1-RDM) rather than the wavefunction Ψ(12... N ) where D 1 false( 1 ; 1̅ false) = ∫ Ψ false( 12 ... N false) Ψ * false( 1̅ 2 ... N false) d 2 ... d N with each roman number representing the spatial and spin coordinates of an electron. Using the 1-RDM as the fundamental variable allows for the description of static correlation through fractional occupation numbers. − Natural orbital functional theory presents a related approach which uses the natural orbitals and their occupation numbers, which may be obtained from the 1-RDM, to reconstruct the 2-RDM subject to N -representability conditions. ,− Taking inspiration from these methods, we previously transformed KS-DFT into a 1-RDMFT to retain the favorable computational scaling of KS-DFT while enabling the description of static correlation. , …”

Explicit-by-Implicit Treatment of Natural Orbital Occupations Using First- and Second-Order Optimization Algorithms: A Comparative Study

“…In addition, unlike the canonical orbitals, the optimization of the natural orbitals can reflect the strong correlation effect through fractional occupations. Different approximations for have been developed from different philosophies and strategies during the past few decades. − Here, to evaluate the performances of different methods and algorithms on the natural occupation optimizations, the power functionals are utilized, which take the form, With eq , a series of 1-RDM functionals can be generated by adjusting the power ω. The unique feature of the power functionals is that the occupation optimizations present different difficulties as the parameter ω changes.…”

Static Electron Correlation in Anharmonic Molecular Vibrations: A Hybrid TAO-DFT Study

“…Kohn–Sham density functional theory (KS-DFT) makes use of noninteracting auxiliary orbitals that are described by a single Slater determinant when constructing the one-electron probability density and therefore suffers from this kind of error. Most forms of KS-DFT have been found to perform poorly for systems that have are known to have “multireference” character, and attempts have been made to overcome this limitation. − Thermally-assisted-occupation density functional theory (TAO-DFT) enables the calculation of static correlation within DFT through the use of fractional orbital occupations maintained with a fictitious temperature, θ. ,− The complexity of this method scales similarly to KS-DFT when increasing the number of electrons in the system, and yet it has been shown to give a similar accuracy to computationally more expensive wave function based methods − which scale very rapidly with increasing numbers of electrons. − …”