We develop relativistic short-range exchange energy functionals for four-component relativistic range-separated density-functional theory using a Dirac-Coulomb Hamiltonian in the no-pair approximation. We show how to improve the short-range local-density approximation exchange functional for large range-separation parameters by using the on-top exchange pair density as a new variable. We also develop a relativistic short-range generalized-gradient approximation exchange functional which further increases the accuracy for small range-separation parameters. Tests on the helium, beryllium, neon, and argon isoelectronic series up to high nuclear charges show that this latter functional gives exchange energies with a maximal relative percentage error of 3 %. The development of this exchange functional represents a step forward for the application of four-component relativistic range-separated density-functional theory to chemical compounds with heavy elements.
We lay out the extension of range-separated density-functional theory to a four-component relativistic frame-work using a Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. This formalism combines a wave-function method for the long-range part of the electron-electron interaction with a density(-current) functionalfor the short-range part of the interaction. We construct for this formalism a short-range exchange local-densityapproximation based on calculations on a relativistic homogeneous electron gas with a modified Coulomb-Breitelectron-electron interaction. More specifically, we…
Using an end-to-end differentiable implementation of the Kohn-Sham self-consistent field equations, we obtain an accurate neural network-based exchange and correlation (XC) functional of the electronic density. The functional is optimized using information on both energy and density while exact constraints are enforced through an appropriate neural network architecture. We evaluate our model against different families of XC approximations and show that, for non-empirical functionals, there is a strong linear correlation between energy and density errors. We use this correlation to define a nov…
We train a neural network as the universal exchange-correlation functional of density-functional theory that simultaneously reproduces both the exact exchange-correlation energy and potential. This functional is extremely non-local, but retains the computational scaling of traditional local or semi-local approximations. It therefore holds the promise of solving some of the delocalization problems that plague density-functional theory, while maintaining the computational efficiency that characterizes the Kohn-Sham equations. Furthermore, by using automatic differentiation, a capability present …
In this chapter, we provide a review of ground-state Kohn-Sham density-functional theory of electronic systems and some of its extensions, we present exact expressions and constraints for the exchange and correlation density functionals, and we discuss the main families of approximations for the exchange-correlation energy: semilocal approximations, single-determinant hybrid approximations, multideterminant hybrid approximations, dispersion-corrected approximations, as well as orbital-dependent exchange-correlation density functionals. The chapter aims at providing both a consistent bird's-eye…
University of Oxford（牛津大学）；STFC Rutherford Appleton Laboratory（科学技术设施委员会卢瑟福·阿普尔顿实验室）
Improving the predictive capability of molecular properties in ab initio simulations is essential for advanced material discovery. Despite recent progress making use of machine learning, utilizing deep neural networks to improve quantum chemistry modelling remains severely limited by the scarcity and heterogeneity of appropriate experimental data. Here we show how training a neural network to replace the exchange-correlation functional within a fully-differentiable three-dimensional Kohn-Sham density functional theory (DFT) framework can greatly improve simulation accuracy. Using only eight ex…
Density functional theory (DFT) is one of the primary approaches to get a solution to the many-body Schrodinger equation. The essential part of the DFT theory is the exchange-correlation (XC) functional, which can not be obtained in analytical form. Accordingly, the accuracy improvement of the DFT is mainly based on the development of XC functional approximations. Commonly, they are built upon analytic solutions in low- and high-density limits and result from quantum Monte Carlo or post-Hartree-Fock numerical calculations. However, there is no universal functional form to incorporate these dat…
Ab initio molecular dynamics (AIMD) simulations using hybrid density functionals and plane waves are of great interest owing to the accuracy of this approach in treating condensed matter systems. On the other hand, such AIMD calculations are not routinely carried out since the computational cost involved in applying the Hartree Fock exchange operator is very high. In this work, we make use of a strategy that combines adaptively compressed exchange operator formulation and multiple time step integration to significantly reduce the computational cost of these simulations. We demonstrate the effi…
We present a new class of non-adiabatic approximations in time-dependent density functional theory derived from an exact expression for the time-dependent exchange-correlation potential. The approximations reproduce dynamical step and peak features in the exact potential that are missing in adiabatic approximations. Central to this approach is an approximation for the one-body reduced density-matrix as a functional of the Kohn-Sham density-matrix, and we demonstrate two such examples.
Gross--Oliveira--Kohn (GOK) ensemble density-functional theory (GOK-DFT) is a time-\textit{independent} extension of density-functional theory (DFT) which allows to compute excited-state energies via the derivatives of the ensemble energy with respect to the ensemble weights. Contrary to the time-dependent version of DFT (TD-DFT), double excitations can be easily computed within GOK-DFT. However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous ensemble derivative contribution to the exc…
Machine learning techniques have received growing attention as an alternative strategy for developing general-purpose density functional approximations, augmenting the historically successful approach of human designed functionals derived to obey mathematical constraints known for the exact exchange-correlation functional. More recently efforts have been made to reconcile the two techniques, integrating machine learning and exact-constraint satisfaction. We continue this integrated approach, designing a deep neural network that exploits the exact constraint and appropriate norm philosophy to d…
A generalized exchange-correlation functional: the Neural-Networks approach
A Neural-Networks-based approach is proposed to construct a new type of exchange-correlation functional for density functional theory . It is applied to improve B3LYP functional by taking into account of high-order contributions to the exchange-correlation functional. The improved B3LYP functional is based on a neural network whose structure and synaptic weights are determined from 116 known experimental atomization energies, ionization potentials, proton affinities or total atomic energies which were used by Becke in his pioneer work on the hybrid functionals [J. Chem. Phys. ${\bf 98}$ , 5648…
The Breit correction, the finite-light-speed correction for the Coulomb interaction of the electron-electron interaction in $ O \left( 1/ c^2 \right) $, is introduced to density functional theory (DFT) based on the non-relativistic reduction with the local density approximation. Using this newly developed relativistic DFT, it is found that the possible outer-most electron of lawrencium atom is the $ p $ orbital instead of the $ d $ orbital, which is consistent with the previous calculations based on wave-function theory. A possible explanation of the anomalous behavior of its first ionization …
A relativistic density-functional theory based on a Fock-space effective quantum-electrodynamics (QED) Hamiltonian using the Coulomb or Coulomb-Breit two-particle interaction is developed. This effective QED theory properly includes the effects of vacuum polarization through the creation of electron-positron pairs but does not include explicitly the photon degrees of freedom. It is thus a more tractable alternative to full QED for atomic and molecular calculations. Using the constrained-search formalism, a Kohn-Sham scheme is formulated in a quite similar way to non-relativistic density-functi…
Exchange interactions are a manifestation of the quantum mechanical nature of the electrons and play a key role in predicting the properties of materials from first principles. In density functional theory (DFT), a widely used approximation to the exchange energy combines fractions of density-based and Hartree-Fock (exact) exchange. This so-called hybrid DFT scheme is accurate in many materials, for reasons that are not fully understood. Here we show that a 1/4 fraction of exact exchange plus a 3/4 fraction of density-based exchange is compatible with a correct quantum mechanical treatment of …
As a new approach to efficiently describe correlation effects in the relativistic quantum world we propose to consider reduced density matrix functional theory, where the key quantity is the first-order reduced density matrix (1-RDM). In this work, we first introduce the theoretical foundations to extend the applicability of this theory to the relativistic domain. Then, using the so-called no-pair (np) approximation, we arrive at an approximate treatment of the relativistic effects by focusing on electronic wavefunctions and neglecting explicit contributions from positrons. Within the np appro…
Water on BN doped benzene: A hard test for exchange-correlation functionals and the impact of exact exchange on weak binding
密度泛函理论（DFT）对弱相互作用复合物的研究主要集中在范德华色散力的重要性上，而对交换作用的研究则少之又少。在这里，通过利用水和硼和氮掺杂的苯衍生物（1，2-氮杂硼）之间的微妙的结合，我们展示了如何精确的交换可以改变复杂的结合构象。基准值已经计算了三个方向的水单体的1，2-氮杂硼杂环从明确相关的量子化学方法，我们也使用扩散量子蒙特卡罗。对于一个流行的DFT交换相关泛函的主机，我们表明，缺乏确切的交换导致错误的最低能量方向的水对1，2-氮杂硼杂环。因此，我们建议，高比例的确切交换和相关的电子结构的改善可能需要掺杂表面上的物理吸附位点和复杂的有机分子的准确预测。同时，为了预测正确的绝对相互作用能，需要通过色散包含泛函来增强对交换的准确描述，并且某些非局部范德华泛函（optB 88-和optB 86 b-vdW）对于绝对相互作用能表现得非常好。通过与水在苯和环硼氮烷（B $_3$ N $_3$ H $_6$）上的比较，我们表明这些结果可能对水与掺杂石墨烯表面的相互作用具有影响，并提出了调节相互作用能的可能方式。
An alternative type of approximation for the exchange and correlation functional in density functional theory is proposed. This approximation depends on a variable $u$ that is able to detect inhomogeneities in the electron density $ρ$ without using derivatives of $ρ$. Instead, $u$ depends on the orbital energies which can also be used to measure how a system differs from the homogeneous electron gas. Starting from the functional of Perdew, Burke, and Ernzerhof (PBE) [Phys. Rev. Lett. 77, 3865 (1996)], a functional depending on $u$ is constructed. Tests on the lattice constant, bulk modulus, an…
Van-der-Waals exchange-correlation functionals and their high pressure and warm dense matter applications
Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR)（德累斯顿-罗森多夫亥姆霍兹中心）；Michigan State University（密歇根州立大学）；University of Iowa（爱荷华大学）；Center for Advanced Systems Understanding (CASUS)（高级系统理解中心）
