We present a low-complexity algorithm to calculate the correlation energy of periodic systems in second-order M\o ller-Plesset perturbation theory (MP2). In contrast to previous approximation-free MP2 codes, our implementation possesses a quartic scaling, $\mathcal O (N^4)$, with respect to the system size $N$ and offers an almost ideal parallelization efficiency. The general issue that the correlation energy converges slowly with the number of basis functions is eased by an internal basis set extrapolation. The key concept to reduce the scaling is to eliminate all summations over virtual orbitals which can be elegantly achieved in the Laplace transformed MP2 (LTMP2) formulation using plane wave basis sets and Fast Fourier transforms. Analogously, this approach could allow to calculate second order screened exchange (SOSEX) as well as particle-hole ladder diagrams with a similar low complexity. Hence, the presented method can be considered as a step towards systematically improved correlation energies.
Using first principles only, we calculate the melting point of MgO, also called periclase or magnesia. The random phase approximation (RPA) is used to include the exact exchange as well as local and nonlocal many-body correlation terms, in order to provide high accuracy. Using the free energy method, we obtain the melting temperature directly from the internal energies calculated with DFT. The free energy differences between the ensembles generated by the molecular dynamics simulations are calculated with thermodynamic integration or thermodynamic perturbation theory. The predicted melting temperature is ${T}_{m}^{\text{RPA}}=3043\ifmmode\pm\else\textpm\fi{}86\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and the values obtained with the PBE and SCAN functionals are ${T}_{m}^{\text{PBE}}=2747\ifmmode\pm\else\textpm\fi{}59\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and ${T}_{m}^{\text{SCAN}}=3032\ifmmode\pm\else\textpm\fi{}53\phantom{\rule{0.28em}{0ex}}\mathrm{K}$.
We present first-principles calculations for the fcc noble gas solids Ne, Ar, and Kr applying the adiabatic connection fluctuation-dissipation theorem (ACFDT) to evaluate the correlation energy. The ACFDT allows us to describe long-range correlation effects including London dispersion or van der Waals interaction on top of conventional density functional theory calculations. Even within the random phase approximation, the typical $1∕{V}^{2}$ volume dependence for the cohesive energy of the noble gas solids is reproduced, and equilibrium cohesive energies and lattice constants are improved compared to density functional theory calculations. Furthermore, we present atomization energies for ${\mathrm{H}}_{2}$, ${\mathrm{N}}_{2}$, and ${\mathrm{O}}_{2}$ within the same post-density-functional-theory framework, finding an excellent agreement with previously published data.
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The random phase approximation (RPA) systematically overestimates the magnitude of the correlation energy and generally underestimates cohesive energies. This originates in part from the complete lack of exchange terms, which would otherwise cancel Pauli exclusion principle violating (EPV) contributions. The uncanceled EPV contributions also manifest themselves in form of an unphysical negative pair density of spin-parallel electrons close to electron-electron coalescence. We follow considerations of many-body perturbation theory to propose an exchange correction that corrects the largest set of EPV contributions while having the lowest possible computational complexity. The proposed method exchanges adjacent particle/hole pairs in the RPA diagrams, considerably improving the pair density of spin-parallel electrons close to coalescence in the uniform electron gas (UEG). The accuracy of the correlation energy is comparable to other variants of Second Order Screened Exchange (SOSEX) corrections although it is slightly more accurate for the spin-polarized UEG. Its computational complexity scales as $\mathcal O(N^5)$ or $\mathcal O(N^4)$ in orbital space or real space, respectively. Its memory requirement scales as $\mathcal O(N^2)$.
We present an embedding approach to treat local electron correlation effects in periodic environments. In a single consistent framework, our plane wave based scheme embeds a local high-level correlation calculation [here, Coupled Cluster (CC) theory], employing localized orbitals, into a low-level correlation calculation [here, the direct Random Phase Approximation (RPA)]. This choice allows for an accurate and efficient treatment of long-range dispersion effects. Accelerated convergence with respect to the local fragment size can be observed if the low-level and high-level long-range dispersions are quantitatively similar, as is the case for CC in RPA. To demonstrate the capabilities of the introduced embedding approach, we calculate adsorption energies of molecules on a surface and in a chabazite crystal cage, as well as the formation energy of a lattice impurity in a solid at the level of highly accurate many-electron perturbation theories. The absorption energy of a methane molecule in a zeolite chabazite is converged with an error well below 20 meV at the CC level. As our largest periodic benchmark system, we apply our scheme to the adsorption of a water molecule on titania in a supercell containing more than 1000 electrons.
The construction of accurate pseudopotentials with good convergence properties for the first-row and transition elements is discussed. We show that by combining an improved description of the pseudowavefunction inside the cut-off radius with the concept of ultrasoft pseudopotentials introduced by Vanderbilt optimal compromise between transferability and plane-wave convergence can be achieved. With the new pseudopotentials, basis sets with no more than 75-100 plane waves per atom are sufficient to reproduce the results obtained with the most accurate norm-conserving pseudopotentials.
We illustrate how ab initio numerical simulation methods can be used to check and improve structure models for i(Al–Pd–Mn). By focusing our study on the optimization of a small approximant, we obtain a number of general structural and compositional rules simple enough to be applicable to the quasicrystalline structure itself.
The melting properties of aluminum are calculated from first-principles molecular-dynamics simulations using density-functional theory in the local-density approximation. We calculate a melting temperature of 890 K at zero pressure, to be compared to the experimental value of 933 K. An elaborate discussion of the techniques employed is presented. The solid- and liquid-state free energies are obtained via coupling constant integration. The respective reference systems are the quasiharmonic crystal and the Lennard-Jones fluid. Good quality of the Brillouin zone sampling is shown to be crucial. The strategy followed is expected to be applicable to a wide range of liquid metals.