Green's functions for separable potentials

Summary The convergence properties of approximate one-particle Green's functions derived from separable potential expansions are examined. A convergence criterion for the basis set to be used in a Löwdin-type inner projection is established and illustrated by calculations of matrix elements of the Coulomb Green's function based on separable potentials of finite rank. It is then suggested that a separable potential ansatz for a many-electron wave function may be introduced into Frenkel's time-dependent variation principle in order to obtain approximate response functions with the continuum of one-particle states explicitly included. A preliminary outline of the formalism at the time-dependent Hartree-Fock level is presented.     

An alternative multipolar expansion for intermolecular potential functions

We have derived a new multipolar expansion for intermolecular potential-energy functions with applications in molecular physics, theoretical chemistry, and mathematical physics. The new formulation employs a separation of radial and angular terms with a simple index structure that leads to computational efficiency and ease of physical interpretation. For the case of the Coulomb interaction, we compare the present formulation with two conventional multipole expansions: the Cartesian tensor and the irreducible spherical tensor expansions. The new formalism leads to efficient numerical algorithms that are useful for general applications beyond intermolecular potentials. In addition to the electrostatic Coulomb interaction, we illustrate the formalism with applications to special function theory and a bipolar expansion involved in potential theory.     

Calculation of optical rotation with time-periodic magnetic-field-dependent basis functions in approximate time-dependent density-functional theory

We report the implementation of a method for the calculation of optical rotation. This method is based on the time-dependent density-functional theory and utilizes time-periodic magnetic-field-dependent basis functions. The calculations are based on a density fit. It is demonstrated that additional terms in the analytical expression appearing from derivatives of the approximated Coulomb potential are necessary to provide the gauge-origin independence of the results within a given numerical accuracy. Contributions from these terms also restore the symmetry between the electric and magnetic perturbations in the optical rotation tensor.     

Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2

A new approximation for post-Hartree-Fock (HF) methods is presented applying tensor decomposition techniques in the canonical product tensor format. In this ansatz, multidimensional tensors like integrals or wavefunction parameters are processed as an expansion in one-dimensional representing vectors. This approach has the potential to decrease the computational effort and the storage requirements of conventional algorithms drastically while allowing for rigorous truncation and error estimation. For post-HF ab initio methods, for example, storage is reduced to O(d·R·n) with d being the number of dimensions of the full tensor, R being the expansion length (rank) of the tensor decomposition, and n being the number of entries in each dimension (i.e., the orbital index). If all tensors are expressed in the canonical format, the computational effort for any subsequent tensor contraction can be reduced to O(R(2)·n). We discuss details of the implementation, especially the decomposition of the two-electron integrals, the AO-MO transformation, the M?ller-Plesset perturbation theory (MP2) energy expression and the perspective for coupled cluster methods. An algorithm for rank reduction is presented that parallelizes trivially. For a set of representative examples, the scaling of the decomposition rank with system and basis set size is found to be O(N(1.8)) for the AO integrals, O(N(1.4)) for the MO integrals, and O(N(1.2)) for the MP2 t(2)-amplitudes (N denotes a measure of system size) if the upper bound of the error in the l(2)-norm is chosen as ε = 10(-2). This leads to an error in the MP2 energy in the order of mHartree.     

Freeon tensor product states and the unitary group formulation of the many-electron problem

The freeon tensor product basis provides a rapid method for the evaluation of matrix elements in the unitary group formulation of quantum chemistry. The method employs fast transformations between the Gel'fand and freeon tensor product basis.     

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518–2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097–3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn–Sham matrix formation. © 2019 Wiley Periodicals, Inc.     

Fast and accurate Coulomb calculation with Gaussian functions

Coulomb interaction is one of the major time-consuming components in a density functional theory (DFT) calculation. In the last decade, dramatic progresses have been made to improve the efficiency of Coulomb calculation, including continuous fast multipole method (CFMM) and J-engine method, all developed first inside Q-Chem. The most recent development is the advent of Fourier transform Coulomb method developed by Fusti-Molnar and Pulay, and an improved version of the method has been recently implemented in Q-Chem. It replaces the least efficient part of the previous Coulomb methods with an accurate numerical integration scheme that scales in O(N2) instead of O(N4) with the basis size. The result is a much smaller slope in the linear scaling with respect to the molecular size and we will demonstrate through a series of benchmark calculations that it speeds up the calculation of Coulomb energy by several folds over the efficient existing code, i.e., the combination of CFMM and J-engine, without loss of accuracy. Furthermore, we will show that it is complementary to the latter and together the three methods offer the best performance for Coulomb part of DFT calculations, making the DFT calculations affordable for very large systems involving thousands of basis functions.     

A self-consistent coulomb bath model using density fitting

A self-consistent Coulomb bath model is presented to provide an accurate and efficient way of performing calculations for interfragment electrostatic and polarization interactions. In this method, a condensed-phase system is partitioned into molecular fragment blocks. Each fragment is embedded in the Coulomb bath due to other fragments. Importantly, the present Coulomb bath is represented using a density fitting method in which the electron densities of molecular fragments are fitted using an atom-centered auxiliary basis set of Gaussian type. The Coulomb bath is incorporated into an effective Hamiltonian for each fragment, with which the electron density is optimized through an iterative double self-consistent field (DSCF) procedure to realize the mutual many-body polarization effects. In this work, the accuracy of interfragment interaction energies enumerated using the Coulomb bath is tested, showing a good agreement with the exact results from an energy decomposition analysis. The qualitative features of many-body polarization effects are visualized by electron density difference plots. It is also shown that the present DSCF method can yield fast and robust convergence with near-linear scaling in performance with increase in system size.     

Irreducible tensor operator methods in intermediate-field coupling

The algebra of irreducible tensor operators is developed in the intermediate-field coupling case. The Wigner-Eckart theorem is formulated for a simple irreducible tensor operator as well as for the Kronecker and scalar products of these operators. The expressions required for the calculation of Coulomb repulsion, crystal field splitting, spin-orbit interaction, and Zeeman effect are given in detail. Recent applications to various problems in spectroscopy and magnetism of transition metal compounds are referred to.     

Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials

We present auxilliary basis sets for the atoms H to At – excluding the Lanthanides – optimized for an efficient treatment of molecular electronic Coulomb interactions. For atoms beyond Kr our approach is based on effective core potentials to describe core electrons. The approximate representation of the electron density in terms of the auxilliary basis has virtually no effect on computed structures and affects the energy by less than 10⁻⁴ a.u. per atom. Efficiency is demonstrated in applications for molecules with up to 300 atoms and 2500 basis functions. Received: 17 December 1996 / Accepted: 8 May 1997     

Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Mo̸ller-Plesset perturbation theory

Many approximations have been developed to help deal with the O(N(4)) growth of the electron repulsion integral (ERI) tensor, where N is the number of one-electron basis functions used to represent the electronic wavefunction. Of these, the density fitting (DF) approximation is currently the most widely used despite the fact that it is often incapable of altering the underlying scaling of computational effort with respect to molecular size. We present a method for exploiting sparsity in three-center overlap integrals through tensor decomposition to obtain a low-rank approximation to density fitting (tensor hypercontraction density fitting or THC-DF). This new approximation reduces the 4th-order ERI tensor to a product of five matrices, simultaneously reducing the storage requirement as well as increasing the flexibility to regroup terms and reduce scaling behavior. As an example, we demonstrate such a scaling reduction for second- and third-order perturbation theory (MP2 and MP3), showing that both can be carried out in O(N(4)) operations. This should be compared to the usual scaling behavior of O(N(5)) and O(N(6)) for MP2 and MP3, respectively. The THC-DF technique can also be applied to other methods in electronic structure theory, such as coupled-cluster and configuration interaction, promising significant gains in computational efficiency and storage reduction.     

Poisson-transformed density fitting in relativistic four-component Dirac-Kohn-Sham theory

We present recent developments in the implementation of the density fitting approach for the Coulomb interaction within the four-component formulation of relativistic density functional theory [Belpassi et al., J. Chem. Phys. 124, 124104 (2006)]. In particular, we make use of the Poisson equation to generate suitable auxiliary basis sets and simplify the electron repulsion integrals [Manby and Knowles, Phys. Rev. Lett. 87, 163001 (2001)]. We propose a particularly simple and efficient method for the generation of accurate Poisson auxiliary basis sets, based on already available standard Coulomb fitting sets. Just as is found in the nonrelativistic case, we show that the number of standard auxiliary fitting functions that need to be added to the Poisson-generated functions in order to achieve a fitting accuracy equal or, in some cases, better than that of the standard procedure is remarkably small. The efficiency of the present implementation is demonstrated in a detailed study of the spectroscopic properties and energetics of several gold containing systems, including the Au dimer and the CsAu molecule. The extraction reaction of a H(2)O molecule from a Au(H(2)O)(9) (+) cluster is also calculated as an example of mixed heavy-light-atom molecular systems. The scaling behavior of the algorithm implemented is illustrated for some closed shell gold clusters up to Au(5) (+). The increased sparsity of the Coulomb matrices involved in the Poisson fitting is identified, as are potential computational applications and the use of the Poisson fitting for the relativistic exchange-correlation problem.     

Metal-Free Photocatalytic CO2 Reduction to CH4 and H2O2 under Non-sacrificial Ambient Conditions

Photocatalytic CO₂ reduction to CH₄ requires photosensitizers and sacrificial agents to provide sufficient electrons and protons through metal-based photocatalysts, and the separation of CH₄ from by-product O₂ has poor applications. Herein, we successfully synthesize a metal-free photocatalyst of a novel electron-acceptor 4,5,9,10-pyrenetetrone (PT), to our best knowledge, this is the first time that metal-free catalyst achieves non-sacrificial photocatalytic CO₂ to CH₄ and easily separable H₂O₂. This photocatalyst offers CH₄ product of 10.6 μmol ⋅ g⁻¹ ⋅ h⁻¹ under non-sacrificial ambient conditions (room temperature, and only water), which is two orders of magnitude higher than that of the reported metal-free photocatalysts. Comprehensive in situ characterizations and calculations reveal a multi-step reaction mechanism, in which the long-lived oxygen-centered radical in the excited PT provides as a site for CO₂ activation, resulting in a stabilized cyclic carbonate intermediate with a lower formation energy. This key intermediate is thermodynamically crucial for the subsequent reduction to CH₄ product with the electronic selectivity of up to 90 %. The work provides fresh insights on the economic viability of photocatalytic CO₂ reduction to easily separable CH₄ in non-sacrificial and metal-free conditions.     

Calculation of origin-independent optical rotation tensor components in approximate time-dependent density functional theory

We outline an implementation of the origin-independent optical rotation tensor, which includes electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability. The method is based on approximate time-dependent density functional theory. We utilize time-periodic magnetic-field-dependent basis functions as well as a modified velocity-gauge formulation of dynamic polarizability tensors in order to obtain a gauge-origin independence. To ensure gauge-origin independence of the results within a given numerical accuracy, density fit coefficient derivatives are employed. A damping constant has been introduced into the linear response equations to treat both resonance and nonresonance regions of optical activity. We present calculations for trans-2,3-dimethyloxirane and derivatives thereof as well as calculations for androst-4,17-dien-3-one. In the Appendix, we derive the equivalence between the common-gauge origin and gauge-including atomic orbitals formulations for the optical rotation tensor in time-dependent DFT.     

On robust density fitting in molecules and extended systems

Two theorems for robust density fitting are proved: A. For a given electron density, the best robust density fitting approximation to the Coulomb electron repulsion energy results from the unconstrained Coulomb metric fit, B. For infinite periodic systems, the necessary condition for correct long-range behavior of any robust density fitting approximation of the Coulomb electron repulsion energy is the exact reproduction of the number of electrons.     

Electron correlation effects in the Fe dimer

The potential energy surface of the Fe dimer is investigated on the basis of density functional theory in the generalized gradient approximation (GGA). Electron correlation effects are taken into account explicitly within the GGA+U approach. We find a value of 2.20 eV for the Coulomb repulsion parameter U to describe the Fe dimer best, yielding a 9 Sigma(g)- ground state with an interatomic separation of 2.143 A. Agreement of the associated vibrational frequency, binding energy, ionization potential, and electron affinity with experimental data as well as corresponding results calculated within a high-level ab initio approach is improved significantly compared to conventional GGA. The effect of U on calculated geometric and magnetic properties of larger Fe clusters is discussed.     

Long‐range analysis of density fitting in extended systems

Density fitting scheme is analyzed for the Coulomb problem in extended systems from the correctness of long‐range behavior point of view. We show that for the correct cancellation of divergent long‐range Coulomb terms it is crucial for the density fitting scheme to reproduce the overlap matrix exactly. It is demonstrated that from all possible fitting metric choices the Coulomb metric is the only one which inherently preserves the overlap matrix for infinite systems with translational periodicity. Moreover, we show that by a small additional effort any non‐Coulomb metric fit can be made overlap‐preserving as well. The problem is analyzed for both ordinary and Poisson basis set choices. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008