Second-order Perturbation Theory Correction Research Articles

Survival of the most transferable at the top of Jacob’s ladder: Defining and testing the ωB97M(2) double hybrid density functional

A meta-generalized gradient approximation, range-separated double hybrid (DH) density functional with VV10 non-local correlation is presented. The final 14-parameter functional form is determined by screening trillions of candidate fits through a combination of best subset selection, forward stepwise selection, and random sample consensus (RANSAC) outlier detection. The MGCDB84 database of 4986 data points is employed in this work, containing a training set of 870 data points, a validation set of 2964 data points, and a test set of 1152 data points. Following an xDH approach, orbitals from the ωB97M-V density functional are used to compute the second-order perturbation theory correction. The resulting functional, ωB97M(2), is benchmarked against a variety of leading double hybrid density functionals, including B2PLYP-D3(BJ), B2GPPLYP-D3(BJ), ωB97X-2(TQZ), XYG3, PTPSS-D3(0), XYGJ-OS, DSD-PBEP86-D3(BJ), and DSD-PBEPBE-D3(BJ). Encouragingly, the overall performance of ωB97M(2) on nearly 5000 data points clearly surpasses that of all of the tested density functionals. As a Rung 5 density functional, ωB97M(2) completes our family of combinatorially optimized functionals, complementing B97M-V on Rung 3, and ωB97X-V and ωB97M-V on Rung 4. The results suggest that ωB97M(2) has the potential to serve as a powerful predictive tool for accurate and efficient electronic structure calculations of main-group chemistry.

Benchmark of Dynamic Electron Correlation Models for Seniority-Zero Wave Functions and Their Application to Thermochemistry.

Wave functions restricted to electron-pair states are promising models to describe static/nondynamic electron correlation effects encountered, for instance, in bond-dissociation processes and transition-metal and actinide chemistry. To reach spectroscopic accuracy, however, the missing dynamic electron correlation effects that cannot be described by electron-pair states need to be included a posteriori. In this Article, we extend the previously presented perturbation theory models with an Antisymmetric Product of 1-reference orbital Geminal (AP1roG) reference function that allows us to describe both static/nondynamic and dynamic electron correlation effects. Specifically, our perturbation theory models combine a diagonal and off-diagonal zero-order Hamiltonian, a single-reference and multireference dual state, and different excitation operators used to construct the projection manifold. We benchmark all proposed models as well as an a posteriori Linearized Coupled Cluster correction on top of AP1roG against CR-CC(2,3) reference data for reaction energies of several closed-shell molecules that are extrapolated to the basis set limit. Moreover, we test the performance of our new methods for multiple bond breaking processes in the homonuclear N2, C2, and F2 dimers as well as the heteronuclear BN, CO, and CN+ dimers against MRCI-SD, MRCI-SD+Q, and CR-CC(2,3) reference data. Our numerical results indicate that the best performance is obtained from a Linearized Coupled Cluster correction as well as second-order perturbation theory corrections employing a diagonal and off-diagonal zero-order Hamiltonian and a single-determinant dual state. These dynamic corrections on top of AP1roG provide substantial improvements for binding energies and spectroscopic properties obtained with the AP1roG approach, while allowing us to approach chemical accuracy for reaction energies involving closed-shell species.

Variational grand-canonical electronic structure of Li+Li at ~104 K with second-order perturbation theory corrections

An ab initio variational grand-canonical electronic structure mean-field method, based on the Gibbs–Peierls–Bogoliubov minimum principle for the Gibbs free energy, is applied to the di-lithium (Li+Li) system at temperatures around T ≈ 104 K and electronic chemical potential of μ ≈ −0.1E h . The method is an extension of the Hartree–Fock approach to finite temperatures. We first study the Li2 molecule at a frozen inter-nuclear distance of R = 3 A as a function of temperature. The mean-field electronic structure changes smoothly as temperature increases, up to 104 K, where a sharp spontaneous spin-polarization emerges as the variational mean-field solution. Further increase in the temperature extinguishes this polarization. We analyze the mean-field behavior using a correlated single-site Hubbard model and show it arises from an attempt of the mean-field to mimic the polarization of the spin–spin correlation function of the exact solution. Next, we keep constant the temperature at 104 K and examine the electronic structure as a function of inter-nuclear distance R. At R = 3.7 A, a crossing between two free energy states occurs: One state is “spin-unpolarized” (becomes lower in energy when R > 3.7 A), while the other is “spin polarized”. This crossing causes near-discontinuous jumps in calculated properties of the system and is associated with using the noninteracting electron character of our mean-field approach. Such problems will likely plague FT-DFT calculations as well. We use second-order perturbation theory (PT2) to study effects of electron correlation on the potential of mean force between the two colliding Li atoms. We find that PT2 correlation free energy at ~104 K is larger than at 0 K and tends to restore the spin-polarized state as the lowest free energy solution.

Analytical nuclear gradients of the explicitly correlated Møller–Plesset second-order energy

The analytical computation of nuclear gradients has been derived and implemented for the explicitly correlated second-order Møller–Plesset method (MP2-F12). The implementation has been accomplished in the TURBOMOLE program package for ansatz MP2-F12/2*A. A Slater-type geminal expanded in six Gaussian geminals (STG-6G), a complementary auxiliary basis set (CABS), and robust density fitting approximations are used. In addition, a second-order perturbation theory correction for single excitations into the complementary auxiliary basis set (CABS singles) is included to reduce the Hartree–Fock error. Smooth convergence towards the basis set limit is observed for a selection of molecules. For computations on dimers of weakly interacting molecules in small basis sets, explicitly correlated second-order Møller–Plesset theory outperforms conventional second-order Møller–Plesset theory because basis set superposition errors are largely avoided at the MP2-F12/2*A level.

Comparing periodic-orbit theory to perturbation theory in the asymmetric infinite square well

An infinite square well with a discontinuous step is one of the simplest systems to exhibit non-Newtonian ray-splitting periodic orbits. This system is analyzed using both time-independent perturbation theory (PT) and periodic-orbit theory and the approximate formulas for the energy eigenvalues derived from these two approaches are compared. The periodic orbits of the system can be divided into classes according to how many times they reflect from the potential step. Different classes of orbits contribute to different orders of PT. The dominant term in the second-order PT correction is due to non-Newtonian orbits that reflect from the step exactly once. In the limit in which PT converges, the periodic-orbit theory results agree with those of PT, but outside of this limit the periodic-orbit theory gives much more accurate results for energies above the potential step.

Analytic Calculation of First-order Molecular Properties at the Explicitly-correlated Second-order Møller-Plesset Level

Abstract Formulae are derived and implemented for the analytic calculation of first-order molecular properties at the level of explicitly-correlated second-order Møller-Plesset perturbation theory. In this theory, which is denoted as MP2-F12 theory, Slater-type geminals are used to expand the first-order wave function. A second-order perturbation theory correction for single excitations into a complementary auxliary basis set is also included. At the MP2-F12 level, it seems sufficient to restrict the analytic calculation of energy derivatives to the level of standard approximation A of MP2-F12 theory and to assume the extended Brillouin condition to hold. Smooth and rapid convergence towards the basis-set limit is observed for the dipole moments of a selection of small closed- and open-shell molecules when calculated at the RI-MP2-F12/2A*[T+V] + CABS singles level in augmented correlation-consistent polarized valence double-, triple-, and quadruple-zeta basis sets that have been optimized especially for use in MP2-F12 theory.

Phase diagram of magnetic structures in Fe/Cr/Fe films with nanoscale geometric inhomogeneities of interfaces

Results of an analysis of magnetic structures (mutual orientations of the iron-layer magnetizations) in three-layered Fe/Cr/Fe films formed due to the exchange interactions inside the layers of iron and chromium and on the Fe/Cr interfaces are given. The analysis was performed under the following assumptions: (1) the iron layers are ferromagnetic, and the magnetic structure of chromium layers has the form of transverse linearly polarized spin-density waves; (2) the iron-chromium exchange interaction is considered as a weak disturbance as compared to the exchange interactions inside the iron and chromium layers; therefore, they were considered as first-order and second-order perturbation-theory corrections; (3) the relative value of these corrections depends on the degree of roughness of the Fe/Cr interfaces; it was assumed that the roughness has a nanometer scale. With such approximations, the mutual orientation of the magnetizations of iron layers can be parallel, antiparallel, or noncollinear. The limits of stability of these structures depending on the thickness of the chromium interlayer and on the ratio of the first-to-second-order corrections have been determined. The main attention was paid to the conditions of the loss of stability with respect to changes in the phase of the spin-density wave, which can result in jumps in the curves of magnetization and magnetization reversal of the Fe/Cr films.

Infrared Spectra and Ab Initio Calculations for the F-−(CH4)n (n = 1−8) Anion Clusters

Infrared spectra of mass-selected F- -(CH4)n (n = 1-8) clusters are recorded in the CH stretching region (2500-3100 cm-1). Spectra for the n = 1-3 clusters are interpreted with the aid of ab initio calculations at the MP2/6-311++G(2df 2p) level, which suggest that the CH4 ligands bind to F- by equivalent, linear hydrogen bonds. Anharmonic frequencies for CH4 and F--CH4 are determined using the vibrational self-consistent field method with second-order perturbation theory correction. The n = 1 complex is predicted to have a C3v structure with a single CH group hydrogen bonded to F-. Its spectrum exhibits a parallel band associated with a stretching vibration of the hydrogen-bonded CH group that is red-shifted by 380 cm-1 from the nu1 band of free CH4 and a perpendicular band associated with the asymmetric stretching motion of the nonbonded CH groups, slightly red-shifted from the nu3 band of free CH4. As n increases, additional vibrational bands appear as a result of Fermi resonances between the hydrogen-bonded CH stretching vibrational mode and the 2nu4 overtone and nu2+nu4 combination levels of the methane solvent molecules. For clusters with n < or = 8, it appears that the CH4 molecules are accommodated in the first solvation shell, each being attached to the F- anion by equivalent hydrogen bonds.

First Singlet (n,π*) Excited State of Hydrogen-Bonded Complexes between Water and Pyrimidine

Hydrogen bonds from water to excited-state formaldehyde and from water to excited-state pyridine have been shown to display novel motifs to traditional hydrogen bonds involving ground states, with, in particular for H2O:pyridine, strong interactions involving the electron-rich pi cloud dominating the (n,pi) excited state. We investigate H2O:pyrimidine and various dihydrated species and reveal another motif, one in which the hydrogen bonding can dramatically alter the electronic structure of the excited state. Such effects are rare for ground-state interactions for which hydrogen bonding usually acts to merely perturb the electronic structure of the participating molecules. It arises as the (n,pi*) excitation of isolated pyrimidine is delocalized over both nitrogens but asymmetric hydrogen bonding causes it to localize on just the noninteracting atom. As a result, the excited-state hydrogen bond in H2O:pyrimidine is suprisingly very similar to the ground-state structure. These results lead to an improved understanding of the spectroscopy of pyrimidine in liquid water, and to the prediction that stable excited-state hydrogen bonds in H2O:pyrimidine should be observable, despite failure of experiments to actually do so. They also provide a simple model for the intricate control over primary charge separation in photosynthesis exerted by hydrogen bonding, and for solvent-induced electron localization in symmetric mixed-valence complexes. All conclusions are based on strong parallels found between the results of calculations performed using density-functional theory (DFT) and time-dependent DFT (TDDFT), complete-active-space self-consistent-field (CASSCF) with second-order perturbation-theory correction (CASPT2) theory, and equation-of-motion coupled cluster (EOM-CCSD) theory, calculations that are verified through detailed comparison of computed properties with experimental data for both the isolated molecules and the ground-state hydrogen bond.

The First Singlet (n,π*) and (π,π*) Excited States of the Hydrogen-Bonded Complex between Water and Pyridine

The excited-state hydrogen bonding between a pyridine molecule and a water molecule has been investigated by a series of theoretical methods including direct and time-dependent density functional theory (DFT and TD-DFT), complete-active-space self-consistent-field (CASSCF) with second-order perturbation-theory correction (CASPT2), and equation-of-motion coupled-cluster (EOM-CCSD). All calculations indicate that the water:pyridine complex on the ground state has strong hydrogen-bonding with binding enthalpies ranging from 4.5 to 5.9 kcal mol-1 after basis set superposition error, zero-point, and thermal correction, with the water molecule lying perpendicularly to the pyridyl plane (total Cs symmetry for the complex). This is in reasonable agreement with experiment and also with previous DFT and MP2 (second-order Møller−Plesset perturbation theory) calculations with large basis sets. Similar results are obtained for hydrogen bonding to the lowest (π,π*) excited state, S2 (1B2). However, for this complex in its first (n-π*) state, S1 (1B1), pyridine is found to adopt a boat configuration of only Cs symmetry with the water above the pyridyl plane. Both the EOM-CCSD and CASPT2 calculations indicate that reasonably strong hydrogen-bonding occurs to pyridine in the (n-π*) state, with the calculated bond enthalpies ranging from 4.0 to 4.5 kcal mol-1. Hence, we find that excited-state hydrogen bonding to azines remains important, but that it has a different motif from the usual linear hydrogen bonding found in ground-state systems. For the (n,π*) excited state, the hydrogen bonding is to the electron-enhanced π cloud of the aromatic ring. A new, much more complex picture is presented for hydrogen bonding in azines which is qualitatively consistent with observed spectroscopic data.

Tetramethylene: A CASPT2 study

The potential energy surface (PES) of the 1,4-tetramethylene biradical has been reinvestigated at the complete-active-space self-consistent field (CASSCF), complete active space with second-order perturbation theory correction and multi-reference configuration interaction (MRCI) levels of theory. The choice of basis set, the size inconsistency of the MRCI wavefunction and the selection of the active space of the CASSCF wavefunction each have a considerable influence on the shape of the PES. In particular, the relative energy of the so-called GF and TF structures depends significantly on the level of theory employed. The results of the study suggest that the experimentally observed intermediate is more likely to be due to entropic trapping rather than the presence of a transition state.

Dipole-Bound Electron Attachment to Uracil−Water Complexes. Theoretical ab Initio Study

Ab initio calculations performed in this work found positive electron affinities for all three possible doubly H-bonded complexes of the uracil molecule with a single water molecule. In all cases the excess electron is bound by the the dipole field of the complex. No conventional stable “valence” anionic states were found with the theoretical procedure used in this work (SCF + second-order perturbation theory corrections for the electron correlation effects). The attachment of the excess electron lowers the relative energy differences between the three complexes, making their coexistence more probable. Structural changes in the uracil−water complex upon attachment of an electron were also found. The anion's equilibrium geometry had noticeably shortened hydrogen-bond lengths and a shifted orientation of the water molecule with respect to the uracil molecule compared to the neutral system.

Second-order perturbation theory corrections to effective Fermi resonance coupling constants

Second-order perturbation theory corrections to effective Fermi resonance coupling constants have been derived analytically within the framework of the traditional rectilinear normal coordinate approach to the analysis of anharmonic molecular vibrations. These corrections have been evaluated for a range of methyl halide molecules (including deuterated isotopomers) and are found to rationalise qualitatively discrepancies between the empirical effective Fermi resonance parameters and theoretical normal coordinate cubic force constants.