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We introduce new functionals for the Kohn–Sham correlation energy that are based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem and are named σ-functionals. Like in the well-established direct random phase approximation (dRPA), σ-functionals require as input exclusively eigenvalues σ of the frequency-dependent KS response function. In the new functionals, functions of σ replace the σ-dependent dRPA expression in the coupling-constant and frequency integrations contained in the ACFD theorem. We optimize σ-functionals with the help of reference sets for atomization, reaction, transition state, and non-covalent interaction energies. The optimized functionals are to be used in a post-self-consistent way using orbitals and eigenvalues from conventional Kohn–Sham calculations employing the exchange–correlation functional of Perdew, Burke, and Ernzerhof. The accuracy of the presented approach is much higher than that of dRPA methods and is comparable to that of high-level wave function methods. Reaction and transition state energies from σ-functionals exhibit accuracies close to 1 kcal/mol and thus approach chemical accuracy. For the 10 966 reactions of the W4-11RE reference set, the mean absolute deviation is 1.25 kcal/mol compared to 3.21 kcal/mol in the dRPA case. Non-covalent binding energies are accurate to a few tenths of a kcal/mol. The presented approach is highly efficient, and the post-self-consistent calculation of the total energy requires less computational time than a density-functional calculation with a hybrid functional and thus can be easily carried out routinely. σ-Functionals can be implemented in any existing dRPA code with negligible programming effort. [ABSTRACT FROM AUTHOR]

Analytic energy gradients with respect to nuclear coordinates are derived and implemented for the self-consistent direct random phase approximation (sc-dRPA) method. In contrast to the more common non-self-consistent dRPA methods, the sc-dRPA method does not require a choice for the approach to generate the Kohn–Sham orbitals and eigenvalues serving as input for the dRPA correlation functional. The fact that the sc-dRPA total energy is variational facilitates the calculation of analytic gradients. The analytic gradients are tested against numerical ones and then used to calculate equilibrium geometries and vibrational frequencies for various molecules including weakly bonded dimers and transition metal compounds. The sc-dRPA method can compete in accuracy with Møller–Plesset perturbation theory of second order and with conventional density-functional methods within the generalized gradient approximation or of hybrid type. Indeed, sc-dRPA geometries and vibrational frequencies are most accurate in many cases. Moreover, the sc-dRPA method is robust in the sense that it is applicable to all considered molecules, whereas conventional density-functional methods are not applicable to dispersion bonded dimers, and Møller–Plesset perturbation theory of second order erroneously predicts a number of molecules to be unbound and yields completely wrong vibrational frequencies in some cases. The coupled cluster singles doubles methods yield geometries and vibrational frequencies of a quality that is inferior to that of the other considered methods. [ABSTRACT FROM AUTHOR]

Analytic energy gradients with respect to nuclear coordinates for an exact exchange-only (EXX) Kohn–Sham method are presented. In the underlying EXX method, the exact exchange potential is obtained as the electrostatic potential of an exchange charge density, which is determined via the optimized effective potential method. Parts of the presented calculation of analytic EXX energy gradients can be reused for analytic energy gradients in self-consistent Kohn–Sham methods treating correlation via the adiabatic-connection fluctuation–dissipation theorem, e.g., methods relying on the random phase approximation. A version of the analytic EXX energy gradients that uses density-fitting is shown to be highly efficient. The accuracy of the analytic energy gradients is tested by comparison with numerically calculated gradients. [ABSTRACT FROM AUTHOR]

A fully self-consistent hybrid dRPA (direct random phase approximation) method, named sc-H[γ]dRPA, is presented with γ = 1/3. The exchange potential of the new method contains a fraction γ of nonlocal Hartree-Fock-like exchange besides the exact local Kohn-Sham (KS) exchange potential. The sc-H[γ]dRPA method, in contrast to a straightforward self-consistent dRPA method within the KS formalism, does not suffer from convergence problems for systems with small eigenvalue gaps. Moreover, the sc-H[γ]dRPA method yields distinctively more accurate reaction, isomerization, and transition state energies than other dRPA approaches, e.g., the frequently used non-self-consistent dRPA method using orbitals and eigenvalues from a KS calculation with the exchange-correlation potential of Perdew, Burke, and Ernzerhof (PBE). The sc-H[γ]dRPA method outperforms second-order Møller-Plesset perturbation theory and coupled cluster singles doubles methods while exhibiting a more favorable scaling of computational costs with system size. A value of γ = 1/3 is shown to be a good choice also for a dRPA@PBE[γ] method, which is a non-self-consistent dRPA method using orbitals and eigenvalues from the hybrid PBE0 method with an admixture of γ = 1/3 of exact exchange instead of the 25% of the PBE0 functional. The dRPA@PBE[γ] method yields reaction, isomerization, and transition state energies that are as good as the sc-H[γ]dRPA ones but is computationally simpler and more efficient because it does not require the self-consistent construction of the dRPA correlation potential. The direct sc-H[γ]dRPA, on the other hand, in contrast to all standard density-functional methods, yields qualitatively correct correlation potentials. [ABSTRACT FROM AUTHOR]