A new formalism was recently proposed to improve random phase approximation (RPA) correlation energies by including approximate exchange effects [B. Mussard et al., J. Chem. Theory Comput. 12, 2191 (2016)]. Within this framework, by keeping only the electron-hole contributions to the exchange kernel, two approximations can be obtained: An adiabatic connection analog of the second order screened exchange (AC-SOSEX) and an approximate electron-hole time-dependent Hartree-Fock (eh-TDHF). Here we show how this formalism is suitable for an efficient implementation within the plane-wave basis set. The response functions involved in the AC-SOSEX and eh-TDHF equations can indeed be compactly represented by an auxiliary basis set obtained from the diagonalization of an approximate dielectric matrix. Additionally, the explicit calculation of unoccupied states can be avoided by using density functional perturbation theory techniques and the matrix elements of dynamical response functions can be efficiently computed by applying the Lanczos algorithm. As shown by several applications to reaction energies and weakly bound dimers, the inclusion of the electron-hole kernel significantly improves the accuracy of ground-state correlation energies with respect to RPA and semi-local functionals.

We derive a mean-field model that is based on a two-component Pauli-like equation and incorporates quantum, spin, and relativistic effects up to second order in 1/c. Using a Lagrangian approach, we obtain the self-consistent charge and current densities that act as sources in the Maxwell equations. A physical interpretation is provided for the second-order corrections to the sources. The Maxwell equations are also expanded to the same order. The resulting self-consistent model constitutes a suitable semi-relativistic approximation to the full Dirac-Maxwell equations.

Within a formalism based on dielectric matrices, the electron-hole time-dependent Hartree-Fock (eh-TDHF) and the adiabatic connection second-order screened exchange (AC-SOSEX) are promising approximations to improve ground-state correlation energies by including exchange effects beyond the random phase approximation (RPA). We introduce here an algorithm based on a Gram-Schmidt orthogonalization (GSO) procedure that significantly reduce the number of matrix elements to be computed to evaluate the response functions that enter in the formulation of these two methods. By considering the A24 test set, we show that this approach does not lead to a significant loss of accuracy and can be effectively applied to compute the small interaction energies involved in weakly bound dimers. Importantly, the GSO method significantly extends the applicability of the eh-TDHF and AC-SOSEX to large systems. This is shown by considering the S22 test set, which includes dimers with up to one hundred valence electrons requiring hundreds of thousands of plane-waves in the basis set. By comparing our results to coupled-cluster benchmark values, we show that the inclusion of exchange effects beyond the RPA significantly improves the accuracy, with mean absolute errors that decrease by almost 40% for the A24 test set and by almost 50% for the S22 test set. This approach based on dielectric matrices is particularly suited for plane-wave implementations and might be used in the future to improve the description of the correlation energy in solid state applications.