Superposition Of Atomic Potentials Research Articles

Accelerating the Convergence of Self-Consistent Field Calculations Using the Many-Body Expansion.

The balance between cost-effective and sufficiently accurate methods represents the proverbial "promised land" for quantum chemistry calculations. The burden thus falls upon theoretical and computational chemists to provide such alternatives to mitigate the issues that arise from the employ of finite computing resources. In this paper, we attempt to demonstrate the importance of the quality of the initial guess for the self-consistent field (SCF) calculation when considering cost reduction techniques. We broach this challenge by using the many body expansion (MBE) to yield high quality density matrices (DMs) which, in turn, are applied as an SCF initial guess. The MBE-DM approaches combined with purification schemes and distance-based cutoff schemes can serve as initial guesses to reduce the SCF cycles necessary for convergence or derive energy directly through one Fock build. To this end, four unique types of clusters including water clusters, fluoride anion water clusters, sodium cation water clusters, and ammonium-bisulfate salt clusters have been used to test the performance of MBE-DM where its truncation at three-body expansion, MBE(3)-DM, shows vast improvement for those four clusters with reductions in the number of SCF cycles up to 40% as compared with the traditional superposition of atomic densities (SAD) guess. Other types of typical initial guesses, superposition of atomic potentials (SAP) and basis set projection (BSP), perform much worse than MBE-DM and SAD. In addition, the MBE-DM shows consistency across an array of fragment types irrespective of charges, size, level of theory, and basis set selection. Through MBE(3)-DM with the distance cutoff and the average purification scheme, the energy can be obtained directly with a mere 3.2 mH of the mean absolute deviation (MAD) for (H2O)N=6-55 which is at least 73 times better than the energy prediction using the typical initial guesses (SAD, SAP, and BSP). The corresponding MAD per monomer is only 0.14 mH which reaches the threshold of the "dynamical accuracy". The promising results of the methods outlined in this paper not only indicate two direct routes for computational cost reduction but also lay the possible foundation for composite techniques (i.e., ab initio sampling) that make best use of near converged values as their starting point.

Efficient implementation of the superposition of atomic potentials initial guess for electronic structure calculations in Gaussian basis sets.

The superposition of atomic potentials (SAP) approach has recently been shown to be a simple and efficient way to initialize electronic structure calculations [S. Lehtola, J. Chem. Theory Comput. 15, 1593-1604 (2019)]. Here, we study the differences between effective potentials from fully numerical density functional and optimized effective potential calculations for fixed configurations. We find that the differences are small, overall, and choose exchange-only potentials at the local density approximation level of theory computed on top of Hartree-Fock densities as a good compromise. The differences between potentials arising from different atomic configurations are also found to be small at this level of theory. Furthermore, we discuss the efficient Gaussian-basis implementation of SAP via error function fits to fully numerical atomic radial potentials. The guess obtained from the fitted potentials can be easily implemented in any Gaussian-basis quantum chemistry code in terms of two-electron integrals. Fits covering the whole periodic table from H to Og are reported for non-relativistic as well as fully relativistic four-component calculations that have been carried out with fully numerical approaches.

Fully numerical calculations on atoms with fractional occupations and range-separated exchange functionals

A recently developed finite element approach for fully numerical atomic structure calculations [S. Lehtola, Int. J. Quantum Chem. 119, e25945 (2019)] is extended to the description of atoms with spherically symmetric densities via fractionally occupied orbitals. Specialized versions of Hartree-Fock as well as local density and generalized gradient approximation density functionals are developed, allowing extremely rapid calculations at the basis set limit on the ground and low-lying excited states even for heavy atoms. The implementation of range-separation based on the Yukawa or complementary error function (erfc) kernels is also described, allowing complete basis set benchmarks of modern range-separated hybrid functionals with either integer or fractional occupation numbers. Finally, computation of atomic effective potentials at the local density or generalized gradient approximation levels for the superposition of atomic potentials (SAP) approach [S. Lehtola, J. Chem. Theory Comput. 15, 1593 (2019)] that has been shown to be a simple and efficient way to initialize electronic structure calculations is described. The present numerical approach is shown to afford beyond microhartree accuracy with a small number of numerical basis functions, and to reproduce literature results for the ground states of atoms and their cations for $1 \leq Z \leq 86 $. Our results indicate that the literature values deviate by up to 10 {\mu}Eh from the complete basis set limit. The numerical scheme for the erfc kernel is shown to work by comparison to results from large Gaussian basis set calculations from the literature. Spin-restricted ground states are reported for Hartree-Fock and Hartree-Fock-Slater calculations with fractional occupations for $1 \leq Z \leq 118$.

Assessment of Initial Guesses for Self-Consistent Field Calculations. Superposition of Atomic Potentials: Simple yet Efficient.

Electronic structure calculations, such as in the Hartree–Fock or Kohn–Sham density functional approach, require an initial guess for the molecular orbitals. The quality of the initial guess has a significant impact on the speed of convergence of the self-consistent field (SCF) procedure. Popular choices for the initial guess include the one-electron guess from the core Hamiltonian, the extended Hückel method, and the superposition of atomic densities (SAD). Here, we discuss alternative guesses obtained from the superposition of atomic potentials (SAP), which is easily implementable even in real-space calculations. We also discuss a variant of SAD which produces guess orbitals by purification of the density matrix that could also be used in real-space calculations, as well as a parameter-free variant of the extended Hückel method, which resembles the SAP method and is easy to implement on top of existing SAD infrastructure. The performance of the core Hamiltonian, the SAD, and the SAP guesses as well as the extended Hückel variant is assessed in nonrelativistic calculations on a data set of 259 molecules ranging from the first to the fourth periods by projecting the guess orbitals onto precomputed, converged SCF solutions in single- to triple-ζ basis sets. It is shown that the proposed SAP guess is the best guess on average. The extended Hückel guess offers a good alternative, with less scatter in accuracy.

Application of the layer Korringa-Kohn-Rostoker method to the calculation of near-edge structure in x-ray-absorption and electron-energy-loss spectroscopy

Green's-function methods are frequently used in the calculation of both the extended and the near-edge structures observed in x-ray-absorption and electron-energy-loss spectroscopies. To date, calculations based upon these methods have tended to be based upon a superposition of atomic potentials used to represent the crystal potential, with no attempt to calculate the self-consistent electronic potential. Many features in the near-edge region relate to charge redistribution and therefore are only approximately described by non-self-consistent electronic potentials. In this paper we show that the layer Korringa-Kohn-Rostoker method can be used in the same way as conventional Green's-function theories for near-edge structure, with the added advantage that the self-consistent ground-state charge is used. Spectra calculated in this manner, and compared with those obtained from other Green's-function methods, demonstrate that self-consistency is necessary to show certain features such as molecular orbital splitting in ${\mathrm{TiO}}_{2}$ (rutile).

BAND STRUCTURE OF VACANCIES AND DISLOCATIONS IN DIAMOND

The results of a first principles calculation of the energy levels and bands of ideal vacancies as well as partial dislocations in diamond are described. The potential is constructed by the superposition of atomic potentials, together with a local exchange potential. The recursion method, generalised to deal with overlap integrals, is used to compute the energy levels.

Band-structure effects in field-emission energy distributions in tungsten

A calculation of the total energy distribution of field-emitted electrons from the (100) and (110) planes of tungsten is presented. The potential inside the metal is represented by a superposition of atomic potentials in the muffin-tin approximation as evaluated by Mattheiss. The crystal is terminated abruptly at the surface and the potential barrier outside the surface is represented by a straight line. Using the mathematical apparatus developed by low-energy-electron-diffraction theorists, the band structure of tungsten is reproduced and the tunneling probability calculated by matching the wave function outside the metal to a superposition of Bloch waves, including a number of evanescent waves, inside the metal. The total energy distribution from the (100) plane exhibits a sharp peak at 0.29 eV below the Fermi level which suggests that the experimentally observed hump at about the same energy is due to band- structure effects. It is shown that this peak in the total-energy-distribution curve is due to electron states associated with certain sections of the constant- energy surface and it is argued that spin-orbit interaction is not likely to remove this peak. The calculated distribution from the (110) plane does not show any marked deviation from the free-electron behavior in agreement withmore » the experimental observations. For the (100) plane the ratio of the exact total energy distribution to its corresponding value evaluated in the free-electron approximation for the (100) plane is considerably smaller than unity away from the peak in the total-energy-distribution curve. For the (110) plane it is of the order of unity.« less

Band-structure effects in field-emission energy distributions in tungsten

A calculation of the total energy distribution of field-emitted electrons from the (100) and (110) planes of tungsten is presented. The potential inside the metal is represented by a superposition of atomic potentials in the muffin-tin approximation as evaluated by Mattheiss. The crystal is terminated abruptly at the surface and the potential barrier outside the surface is represented by a straight line. Using the mathematical apparatus developed by low-energy-electron-diffraction theorists we reproduce the band structure of tungsten and calculate the tunneling probability by matching the wave function outside the metal to a superposition of Bloch waves, including a number of evanescent waves, inside the metal. The total energy distribution from the (100) plane exhibits a sharp peak at 0.29 eV below the Fermi level which suggests that the experimentally observed hump at about the same energy is due to band-structure effects. We show that this peak in the total-energy-distribution curve is due to electron states associated with certain sections of the constant-energy surface and we argue that spin-orbit interaction is not likely to remove this peak. The calculated distribution from the (110) plane does not show any marked deviation from the free-electron behavior in agreement with the experimental observations. For the (100) plane the ratio of the exact total energy distribution to its corresponding value evaluated in the free-electron approximation for the (100) plane is considerably smaller than unity away from the peak in the total-energy-distribution curve. For the (110) plane it is of the order of unity.

Crystal Potentials in Energy-Band Calculations of Noble Metals

The prevalent prescriptions for constructing crystal potentials are used to determine the energy bands for silver. The Kohn-Rostoker method is applied to determine the energy eigenvalues at the symmetry points $\ensuremath{\Gamma}$, $X$, and $L$ and along the $\ensuremath{\Delta}$, $\ensuremath{\Lambda}$, and $\ensuremath{\Sigma}$ axes. General conclusions with respect to the suitability of a given potential to noble metals are drawn from a comparison of the important energy-level separations and Fermi-surface parameters of copper (from available calculations) and silver (this calculation) with experiment. It is concluded that the most reasonable representation of the crystal potential for noble metals is obtained by a superposition of atomic potentials, with a proper choice of electronic configuration.

Energy Bands for Solid Argon

Energy bands have been calculated for solid argon using the augmented plane-wave method. The crystal potential used in this calculation has been approximated by a superposition of atomic potentials and involves the Slater free-electron exchange approximation. In order to estimate the error introduced by this approximation, a quantitative comparison has been made between the Hartree-Fock and free-electron exchange potentials for atomic argon. The results indicate that the free-electron exchange approximation is most accurate for the less tightly bound valence electrons, though errors of the order of 0.1 Ry remain. The present energy bands are found to be in qualitative agreement with the earlier results of Knox and Bassani, though there are important quantitative differences. The present calculation yields a band gap of 13.3 eV, which is in fair agreement with the experimentally observed value of 14.3 eV.