Jefferson E. Bates

Electron correlation methods based on the random phase approximation

In the past decade, the random phase approximation (RPA) has emerged as a promising post-Kohn–Sham method to treat electron correlation in molecules, surfaces, and solids. In this review, we explain how RPA arises naturally as a zero-order approximation from the adiabatic connection and the fluctuation-dissipation theorem in a density functional context. This is contrasted to RPA with exchange (RPAX) in a post-Hartree–Fock context. In both methods, RPA and RPAX, the correlation energy may be expressed as a sum over zero-point energies of harmonic oscillators representing collective electronic excitations, consistent with the physical picture originally proposed by Bohm and Pines. The extra factor 1/2 in the RPAX case is rigorously derived. Approaches beyond RPA are briefly summarized. We also review computational strategies implementing RPA. The combination of auxiliary expansions and imaginary frequency integration methods has lead to recent progress in this field, making RPA calculations affordable for systems with over 100 atoms. Finally, we summarize benchmark applications of RPA to various molecular and solid-state properties, including relative energies of conformers, reaction energies involving weak and covalent interactions, diatomic potential energy curves, ionization potentials and electron affinities, surface adsorption energies, bulk cohesive energies and lattice constants. RPA barrier heights for an extended benchmark set are presented. RPA is an order of magnitude more accurate than semi-local functionals such as B3LYP for non-covalent interactions rivaling the best empirically parametrized methods. Larger but systematic errors are observed for processes that do not conserve the number of electron pairs, such as atomization and ionization.

Communication: Random phase approximation renormalized many-body perturbation theory

We derive a renormalized many-body perturbation theory (MBPT) starting from the random phase approximation (RPA). This RPA-renormalized perturbation theory extends the scope of single-reference MBPT methods to small-gap systems without significantly increasing the computational cost. The leading correction to RPA, termed the approximate exchange kernel (AXK), substantially improves upon RPA atomization energies and ionization potentials without affecting other properties such as barrier heights where RPA is already accurate. Thus, AXK is more balanced than second-order screened exchange [A. Grüneis et al., J. Chem. Phys. 131, 154115 (2009)], which tends to overcorrect RPA for systems with stronger static correlation. Similarly, AXK avoids the divergence of second-order Møller-Plesset (MP2) theory for small gap systems and delivers a much more consistent performance than MP2 across the periodic table at comparable cost. RPA+AXK thus is an accurate, non-empirical, and robust tool to assess and improve semi-local density functional theory for a wide range of systems previously inaccessible to first-principles electronic structure calculations.

Harnessing the meta-generalized gradient approximation for time-dependent density functional theory

Density functionals within the meta-generalized gradient approximation (MGGA) are widely used for ground-state electronic structure calculations. However, the gauge variance of the kinetic energy density τ confounds applications of MGGAs to time-dependent systems, excited states, magnetic properties, and states with strong spin-orbit coupling. Becke and Tao used the paramagnetic current density to construct a gauge invariant generalized kinetic energy density τ. We show that τ(W)≤τ, where τ(W) is the von Weizsäcker kinetic energy density of a one-electron system. Thus, replacing τ by τ leads to current-dependent MGGAs (cMGGAs) that are not only gauge invariant but also restore the accuracy of MGGAs in iso-orbital regions for time-dependent and current-carrying states. The current dependence of cMGGAs produces a vector exchange-correlation (XC) potential in the time-dependent adiabatic Kohn-Sham (KS) equations. While MGGA response properties of current-free ground states become manifestly gauge-variant to second order, linear response properties are affected by a new XC kernel appearing in the cMGGA magnetic orbital rotation Hessian. This kernel reflects the first-order coupling of KS orbitals due to changes in the paramagnetic current density and has apparently been ignored in previous MGGA response implementations. Inclusion of the current dependence increases total computation times by less than 50%. Benchmark applications to 109 adiabatic excitation energies using the Tao-Perdew-Staroverov-Scuseria (TPSS) MGGA and its hybrid version TPSSh show that cMGGA excitation energies are slightly lower than the MGGA ones on average, but exhibit fewer outliers. Similarly, the optical rotations of 13 small organic molecules show a small but systematic improvement upon inclusion of the magnetic XC kernel. We conclude that cMGGAs should replace MGGAs in all applications involving time-dependent or current-carrying states.

Synthesis of the (N2)3− Radical from Y2+ and Its Protonolysis Reactivity To Form (N2H2)2− via the Y[N(SiMe3)2]3/KC8 Reduction System

Examination of the Y[N(SiMe(3))(2)](3)/KC(8) reduction system that allowed isolation of the (N(2))(3-) radical has led to the first evidence of Y(2+) in solution. The deep-blue solutions obtained from Y[N(SiMe(3))(2)](3) and KC(8) in THF at -35 °C under argon have EPR spectra containing a doublet at g(iso) = 1.976 with a 110 G hyperfine coupling constant. The solutions react with N(2) to generate (N(2))(2-) and (N(2))(3-) complexes {[(Me(3)Si)(2)N](2)(THF)Y}(2)(μ-η(2):η(2)-N(2)) (1) and {[(Me(3)Si)(2)N](2)(THF)Y}(2)(μ-η(2):η(2)-N(2))[K(THF)(6)] (2), respectively, and demonstrate that the Y[N(SiMe(3))(2)](3)/KC(8) reaction can proceed through an Y(2+) intermediate. The reactivity of (N(2))(3-) radical with proton sources was probed for the first time for comparison with the (N(2))(2-) and (N(2))(4-) chemistry. Complex 2 reacts with [Et(3)NH][BPh(4)] to form {[(Me(3)Si)(2)N](2)(THF)Y}(2)(μ-N(2)H(2)), the first lanthanide (N(2)H(2))(2-) complex derived from dinitrogen, as well as 1 as a byproduct, consistent with radical disproportionation reactivity.

Analytical First-Order Molecular Properties and Forces within the Adiabatic Connection Random Phase Approximation.

The random phase approximation (RPA) is an increasingly popular method for computing molecular ground-state correlation energies within the adiabatic connection fluctuation-dissipation theorem framework of density functional theory. We present an efficient analytical implementation of first-order RPA molecular properties and nuclear forces using the resolution-of-the-identity (RI) approximation and imaginary frequency integration. The centerpiece of our approach is a variational RPA energy Lagrangian invariant under unitary transformations of occupied and virtual reference orbitals, respectively. Its construction requires the solution of a single coupled-perturbed Kohn-Sham equation independent of the number of perturbations. Energy gradients with respect to nuclear displacements and other first-order properties such as one-particle densities or dipole moments are obtained from partial derivatives of the Lagrangian. Our RPA energy gradient implementation exhibits the same [Formula: see text] scaling with system size N as a single-point RPA energy calculation. In typical applications, the cost for computing the entire gradient vector with respect to nuclear displacements is ∼5 times that of a single-point RPA energy calculation. Derivatives of the quadrature nodes and weights used for frequency integration are essential for RPA gradients with an accuracy consistent with RPA energies and can be included in our approach. The quality of RPA equilibrium structures is assessed by comparison to accurate theoretical and experimental data for covalent main group compounds, weakly bonded dimers, and transition metal complexes. RPA outperforms semilocal functionals as well as second-order Møller-Plesset (MP2) theory, which fails badly for the transition metal compounds. Dipole moments of polarizable molecules and weakly bound dimers show a similar trend. RPA harmonic vibrational frequencies are nearly of coupled cluster singles, doubles, and perturbative triples quality for a set of main group compounds. Compared to the ring-coupled cluster based implementation of Rekkedal et al. [J. Chem. Phys. 2013, 139, 081101.], our method scales better by two powers of N and supports a semilocal Kohn-Sham reference. The latter is essential for the good performance of RPA in small-gap systems.

Isolation of (CO)1– and (CO2)1– Radical Complexes of Rare Earths via Ln(NR2)3/K Reduction and [K2(18-crown-6)2]2+ Oligomerization

Deep-blue solutions of Y(2+) formed from Y(NR(2))(3) (R = SiMe(3)) and excess potassium in the presence of 18-crown-6 at -45 °C under vacuum in diethyl ether react with CO at -78 °C to form colorless crystals of the (CO)(1-) radical complex, {[(R(2)N)(3)Y(μ-CO)(2)][K(2)(18-crown-6)(2)]}(n), 1. The polymeric structure contains trigonal bipyramidal [(R(2)N)(3)Y(μ-CO)(2)](2-) units with axial (CO)(1-) ligands linked by [K(2)(18-crown-6)(2)](2+) dications. Byproducts such as the ynediolate, [(R(2)N)(3)Y](2)(μ-OC≡CO){[K(18-crown-6)](2)(18-crown-6)}, 2, in which two (CO)(1-) anions are coupled to form (OC≡CO)(2-), and the insertion/rearrangement product, {(R(2)N)(2)Y[OC(═CH(2))Si(Me(2))NSiMe(3)]}[K(18-crown-6)], 3, are common in these reactions that give variable results depending on the specific reaction conditions. The CO reduction in the presence of THF forms a solvated variant of 2, the ynediolate [(R(2)N)(3)Y](2)(μ-OC≡CO)[K(18-crown-6)(THF)(2)](2), 2a. CO(2) reacts analogously with Y(2+) to form the (CO(2))(1-) radical complex, {[(R(2)N)(3)Y(μ-CO(2))(2)][K(2)(18-crown-6)(2)]}(n), 4, that has a structure similar to that of 1. Analogous (CO)(1-) and (OC≡CO)(2-) complexes of lutetium were isolated using Lu(NR(2))(3)/K/18-crown-6: {[(R(2)N)(3)Lu(μ-CO)(2)][K(2)(18-crown-6)(2)]}(n), 5, [(R(2)N)(3)Lu](2)(μ-OC≡CO){[K(18-crown-6)](2)(18-crown-6)}, 6, and [(R(2)N)(3)Lu](2)(μ-OC≡CO)[K(18-crown-6)(Et(2)O)(2)](2), 6a.

Dinitrogen Reduction via Photochemical Activation of Heteroleptic Tris(cyclopentadienyl) Rare-Earth Complexes

Dinitrogen can be reduced by photochemical activation of the Ln(3+) mixed-ligand tris(cyclopentadienyl) rare-earth complexes (η(5)-C5Me5)(3-x)(C5Me4H)(x)Ln (Ln = Y, Lu, Dy; x = 1, 2). [(C5Me4R)2Ln]2(μ-η(2):η(2)-N2) products (R = H, Me) are formed in reactions in which N2 is reduced to (N═N)(2-) and (C5Me4H)(-) is oxidized to (C5Me4H)2. Density functional theory indicates that this unusual example of rare-earth photochemistry can be rationalized by absorptions involving the (η(3)-C5Me4H)(-) ligands.

Synthesis, Structure, and Physical Properties for a Series of Trigonal Bipyramidal MII–Cl Complexes with Intramolecular Hydrogen Bonds

A series of transition metal chloro complexes with the tetradentate tripodal tris(2-amino-oxazoline)amine ligand (TAO) have been synthesized and characterized. X-Ray structural analyses of these compounds demonstrate the formation of the mononuclear complexes [M(II)(TAO)(Cl)](+), where M(II) = Cr, Mn, Fe, Co, Ni, Cu and Zn. These complexes exhibit distorted trigonal-bipyramidal geometry, coordinating the metal through an apical tertiary amine, three equatorial imino nitrogen atoms, and an axial chloride anion. All the complexes possess an intramolecular hydrogen-bonding (H-bonding) network within the cavity occupied by the metal-bound chloride ion. The metal-chloride bond distances are atypically long, which is attributed to the effects of the H-bonding network. Nuclear magnetic resonance (NMR) spectroscopy of the Zn complex suggests that the solid-state structures are representative of that observed in solution, and that the H-bonding interactions persist as well. Additionally, density functional theory (DFT) calculations were carried out to probe the electronic structures of the complexes.

Fully relativistic complete active space self-consistent field for large molecules: Quasi-second-order minimax optimization

We develop an efficient algorithm for four-component complete active space self-consistent field (CASSCF) methods on the basis of the Dirac equation that takes into account spin-orbit and other relativistic effects self-consistently. Orbitals are optimized using a trust-region quasi-Newton method with Hessian updates so that energies are minimized with respect to rotations among electronic orbitals and maximized with respect to rotations between electronic and positronic orbitals. Utilizing density fitting and parallel computation, we demonstrate that Dirac-Coulomb CASSCF calculations can be routinely performed on systems with 100 atoms and a few heavy-elements. The convergence behavior and wall times for octachloridodirhenate(III) and a tungsten methylidene complex are presented. In addition, the excitation energies of octachloridodirhenate(III) are reported using a state-averaged variant.

Varying the Lewis Base Coordination of the Y2N2 Core in the Reduced Dinitrogen Complexes {[(Me3Si)2N]2(L)Y}2(μ-η2:η2-N2) (L = Benzonitrile, Pyridines, Triphenylphosphine Oxide, and Trimethylamine N-Oxide)

The effect of the neutral donor ligand, L, on the Ln(2)N(2) core in the (N═N)(2-) complexes, [A(2)(L)Ln](2)(μ-η(2):η(2)-N(2)) (Ln = Sc, Y, lanthanide; A = monoanion; L = neutral ligand), is unknown since all of the crystallographically characterized examples were obtained with L = tetrahydrofuran (THF). To explore variation in L, displacement reactions between {[(Me(3)Si)(2)N](2)(THF)Y}(2)(μ-η(2):η(2)-N(2)), 1, and benzonitrile, pyridine (py), 4-dimethylaminopyridine (DMAP), triphenylphosphine oxide, and trimethylamine N-oxide were investigated. THF is displaced by all of these ligands to form {[(Me(3)Si)(2)N](2)(L)Y}(2)(μ-η(2):η(2)-N(2)) complexes (L = PhCN, 2; py, 3; DMAP, 4; Ph(3)PO, 5; Me(3)NO, 6) that were fully characterized by analytical, spectroscopic, density functional theory, and X-ray crystallographic methods. The crystal structures of the Y(2)N(2) cores in 2-5 are similar to that in 1 with N-N bond distances between 1.255(3) Å and 1.274(3) Å, but X-ray analysis of the N-N distance in 6 shows it to be shorter: 1.198(3) Å.