The computation of vibrational spectra of diatomic molecules through the exact diagonalization of algebraically determined matrixes based on powers of Morse coordinates is made substantially more efficient by choosing a properly adapted quantum-mechanical basis, specifically tuned to the molecular potential. A substantial improvement is achieved while still retaining the full advantage of the simplicity and numerical light-weightedness of an algebraic approach. In the scheme we propose, the basis is parameterized by two quantities which can be adjusted to best suit the molecular potential through a simple minimization procedure.
Using finite basis sets, it is shown how to construct a local Hamiltonian, such that one of its infinitely many degenerate eigenfunctions is the ground state full configuration interaction (FCI) wave function in that basis set. Formally, the local potential of this Hamiltonian is the optimized effective potential and the exact Kohn-Sham potential at the same time, because the FCI wave function yields the exact ground-state density and energy. It is not the aim of this paper to provide a new algorithm for obtaining FCI wave functions. A new potential is the goal.
Molecule-optimized basis sets, based on approximate natural orbitals, are developed for accelerating the convergence of quantum calculations with strongly correlated (multi-referenced) electrons. We use a low-cost approximate solution of the anti-Hermitian contracted Schr{ö}dinger equation (ACSE) for the one- and two-electron reduced density matrices (RDMs) to generate an approximate set of natural orbitals for strongly correlated quantum systems. The natural-orbital basis set is truncated to generate a molecule-optimized basis set whose rank matches that of a standard correlation-consistent b…
We show that the finite-basis optimized effective potential (OEP) equations exhibit previously unknown singular behavior.Imposing continuity, we derive new well-behaved finite-basis-set OEP equations that determine OEP for any orbital and any large enough potential basis sets and which adopt an analytic solution via matrix-inversion.
Vibrational spectra can be computed without storing full-dimensional vectors by using low-rank sum-of-products (SOP) basis functions. We introduce symmetry constraints in the SOP basis functions to make it possible to separately calculate states in different symmetry subgroups. This is done using a power method to compute eigenvalues and an alternating least squares method to optimize basis functions. Owing to the fact that the power method favours the convergence of the lowest states, one must be careful not to exclude basis functions of some symmetries. Exploiting symmetry facilitates making…
We investigate the use of optimized correlation consistent gaussian basis sets for the study of insulating solids with auxiliary-field quantum Monte Carlo (AFQMC). The exponents of the basis set are optimized through the minimization of the second order Møller--Plesset perturbation theory (MP2) energy in a small unit cell of the solid. We compare against other alternative basis sets proposed in the literature, namely calculations in the Kohn--Sham basis and in the natural orbitals of an MP2 calculation. We find that our optimized basis sets accelerate the convergence of the AFQMC correlation e…
We present, within Kohn-Sham Density Functional Theory calculations, a quantitative method to identify and assess the partitioning of a large quantum mechanical system into fragments. We then show how within this framework simple generalizations of other well-known population analyses can be used to extract, from first principles, reliable electrostatic multipoles for the identified fragments. Our approach reduces arbitrariness in the fragmentation procedure, and enables the possibility to assess, quantitatively, whether the corresponding fragment multipoles can be interpreted as observable qu…
Arguments showing that exchange-only optimized effective potential (xOEP) methods, with finite basis sets, cannot in general yield the Hartree-Fock (HF) ground state energy, but a higher one, are given. While the orbital products of a complete basis are linearly dependent, the HF ground state energy can only be obtained via a basis set xOEP scheme in the special case that all products of occupied and unoccupied orbitals emerging from the employed orbital basis set are linearly independent from each other. In this case, however, exchange potentials leading to the HF ground state energy exhibit …
We revisit the collocation method of Manzhos and Carrington (J. Chem. Phys. 145, 224110, 2016) in which a distributed localized (e.g., Gaussian) basis is used to set up a generalized eigenvalue problem to compute the eigenenergies and eigenfunctions of a molecular vibrational Hamiltonian. Although the resulting linear algebra problem involves full matrices, the method provides a number of important advantages. Namely: (i) it is very simple both conceptually and numerically, (ii) it can be formulated using any set of internal molecular coordinates, (iii) it is flexible with respect to the choic…
We present an approach for generating local numerical basis sets of improving accuracy for first-principles nanoplasmonics simulations within time-dependent density functional theory. The method is demonstrated for copper, silver, and gold nanoparticles that are of experimental interest but computationally demanding due to the semi-core d-electrons that affect their plasmonic response. The basis sets are constructed by augmenting numerical atomic orbital basis sets by truncated Gaussian-type orbitals generated by the completeness-optimization scheme, which is applied to the photoabsorption spe…
We describe an accurate and scalable implementation for the computation of molecular nuclear magnetic resonance shieldings, J-couplings, and magnetizabilities within nonrelativistic semilocal density functional theory, based on numeric atom-centered orbital (NAO) basis sets. We compare the convergence to the basis set limit for two established types of NAO basis sets, called NAO-VCC-nZ and FHI-aims-09, to several established Gaussian-type basis sets. The basis set limit is reached faster for the NAO basis sets than for standard correlation consistent Gaussian-type basis sets (cc-pVnZ, aug-cc-p…
Firstly, a sequential symmetry adaptation procedure is derived for semidirect product groups. Then, this sequential symmetry adaptation procedure is used in the development of new method named Monomer Basis Representation (MBR) for calculating the vibration-rotation-tunneling (VRT) spectra of molecular clusters. The method is based on generation of optimized bases for each monomer in the cluster as a linear combination of some primitive basis functions and then using the sequential symmetry adaptation procedure for generating a small symmetry adapted basis for the solution of the full problem.…
In this work, we developed and showcased the occ-RI-K algorithm to compute the exact exchange contribution in density functional calculations of solids near the basis set limit. Within the gaussian planewave (GPW) density fitting, our algorithm achieves a 1-2 orders of magnitude speedup compared to conventional GPW algorithms. Since our algorithm is well-suited for simulations with large basis sets, we applied it to 12 hybrid density functionals with pseudopotentials and a large uncontracted basis set to assess their performance on band gaps of 25 simple solids near the basis set limit. The la…
The exchange-only optimized effective potential method is implemented with the use of Slater-type basis functions, seeking for an alternative to the standard methods of solution with some computational advantages. This procedure has been tested in a small group of closed shell atoms and diatomic molecules, for which numerical solutions are available. The results obtained with this implementation have been compared to the exact numerical solutions and to the results obtained when the optimized effective equations are solved using the Gaussian-type basis sets. This Slater-type basis approach lea…
Given a partition of a large system into an active quantum mechanical (QM) region and its environment, we present a simple way of embedding the QM region into an effective electrostatic potential representing the environment. This potential is generated by partitioning the environment into well defined fragments, and assigning each one a set of electrostatic multipoles, which can then be used to build up the electrostatic potential. We show that, providing the fragments and the projection scheme for the multipoles are chosen properly, this leads to an effective electrostatic embedding of the a…
We present a classically solvable model that leads to optimized low-depth quantum circuits leveraging separable pair approximations. The obtained circuits are well suited as a baseline circuit for emerging quantum hardware and can, in the long term, provide significantly improved initial states for quantum algorithms. The associated wavefunctions can be represented with linear memory requirement which allows classical optimization of the circuits and naturally defines a minimum benchmark for quantum algorithms. In this work, we employ directly determined pair-natural orbitals within a basis-se…
A general procedure for the optimization of atomic density-fitting basis functions is designed with the balance between accuracy and numerical stability in mind. Given one-electron wavefunctions and energies, weights are assigned to the product densities, modeling their contribution to the exchange and second-order correlation energy, and a simple weighted error measure is minimized. Generally-contracted Gaussian auxiliary basis sets are optimized to match the wavefunction basis sets [D. N. Laikov, Theor. Chem. Acc. 138, 40 (2019)] for all 102 elements in a scalar-relativistic approximation [D…
Using the optimized effective potential method in conjunction with the semi-analytical approximation due to Krieger, Li and Iafrate, we have performed fully self-consistent exact exchange-only density-functional calculations for diatomic molecules with a fully numerical basis-set-free molecular code. The results are very similar to the ones obtained with the Hartree Fock approach. Furthermore we present results for ground states of positive atomic ions including correlation contributions in the approximation of Colle and Salvetti. It is found that the scheme performs significantly better than …
We devise an efficient practical method for computing the Kohn-Sham exchange-correlation potential corresponding to a Hartree-Fock electron density. This potential is almost indistinguishable from the exact-exchange optimized effective potential (OEP) and, when used as an approximation to the OEP, is vastly better than all existing models. Using our method one can obtain unambiguous, nearly exact OEPs for any finite one-electron basis set at the same low cost as the Krieger-Li-Iafrate and Becke-Johnson potentials. For all practical purposes, this solves the long-standing problem of black-box c…
