Coulomb integrals, i.e., matrix elements of bare or screened Coulomb interaction between one-electron orbitals, are fundamental objects in many approaches developed to tackle the challenging problem of calculating the electronic structure of strongly correlated materials. In this paper, Coulomb integrals are analyzed by considering both the point-group symmetry of the site occupied by the atom in the crystal or molecule and the permutation symmetries of the orbitals in the integrals. In particular, the case where one-electron orbitals form the basis of a general (i.e., a real, complex, or pseudoreal) irreducible representation is considered. Explicit formulas are provided to calculate all integrals of the interaction tensor in terms of a minimum set of independent ones. The effect of a symmetry breaking is also investigated by describing Coulomb integrals of a group in terms of those of one of its subgroups. We developed the specific example of $\mathrm{O}(3)$ as the larger group which can therefore be used to quantify the deviation of a specific system from the spherical symmetry. Possible applications of the presented framework include the calculation of solid-state and molecular spectroscopies via multiplet techniques, dynamical mean-field theory, or the $GW$ approximation.
Coulomb integrals, i.e., matrix elements of bare or screened Coulomb interaction between one-electron orbitals, are fundamental objects in many approaches developed to tackle the challenging problem of calculating the electronic structure of strongly correlated materials. In this paper, Coulomb integrals are analyzed by considering both the point-group symmetry of the site occupied by the atom in the crystal or molecule and the permutation symmetries of the orbitals in the integrals. In particular, the case where one-electron orbitals form the basis of a general (i.e., a real, complex, or pseudoreal) irreducible representation is considered. Explicit formulas are provided to calculate all integrals of the interaction tensor in terms of a minimum set of independent ones. The effect of a symmetry breaking is also investigated by describing Coulomb integrals of a group in terms of those of one of its subgroups. We developed the specific example of O(3) as the larger group which can therefore be used to quantify the deviation of a specific system from the spherical symmetry. Possible applications of the presented framework include the calculation of solid-state and molecular spectroscopies via multiplet techniques, dynamical mean-field theory, or the GW approximation.
An implementation of the single-site density matrix renormalization group (DMRG) approach for the quantum-chemical application to calculate approximate as well as exact electronic ground-state energies and wave functions of atoms and molecules is presented.The algorithm is formulated in terms of the variational and sizeconsistent class of wave functions, matrix product states, and their operator analogues, matrix product operators.A procedure to construct quantum-chemical Hamiltonian operators with long-range Coulomb interactions as matrix product operators is developed.The many-electron Hamiltonian operators are represented as sums and products of one-electron operators.For the efficient matrix product operator construction of quantum-chemical Hamiltonians, different basis spaces including discrete position and momentum space as well as molecular and atomic orbitals are investigated.Discrete position space yields the best matrix product operator size scaling, but introduces artificial long-range couplings in the kinetic energy term of the Hamiltonian which are a consequence of the DMRG algorithm in terms of matrix product states.The huge amount of discrete space points needed for an accurate description poses a problem because of the electron-electron interaction.Therefore, a momentum space or Fourier series expansion of the electron-electron interaction is shown which makes it possible to compress its matrix product operator representation.DMRG applied to quantum-chemical Hamiltonians in matrix product operator form can serve as means to directly (without mean-field treatment in advance) investigate quantum-chemical problems and find the analytically exact ground-state solution of small atoms and molecules.The next basis space analyzed in detail is the atomic orbital basis.The matrix product operator construction and the direct application of the DMRG method in this basis is examined.A physical compression scheme based on a singular value decomposition of the electron-electron interaction in the atomic orbital basis is demonstrated.Furthermore, DMRG calculations in the atomic orbital basis for certain systems where mean-field theory yields qualitatively wrong results and where higher-order approximative correction methods like truncated configuration interaction and coupled cluster cannot fully capture strong electron correlation effects are compared to results obtained in the molecular orbital basis.The comparison shows that atomic orbitals describe the physical situation well and thus lead to a tremendous reduction of the matrix product state dimension and the computational effort.i
A point group is a set of spatial symmetry operations in molecular systems and is an indispensable tool for analyzing molecular orbitals and spectroscopy experiments in chemistry. Several quantum algorithms to exploit this symmetry have been proposed, but practical implementations of point-group symmetry operations and the detailed symmetry analysis of realistic many-electron wavefunctions are still missing. In this work, we propose an ancilla-free hybrid method to analyze point-group symmetries of many-electron states, which works for both abelian and non-abelian groups. For a given wavefunction, our method calculates the projection weights of point-group irreducible representations by applying orbital rotations derived from the eigenvectors of the representation matrices, making it applicable to arbitrary basis functions. The usefulness of our approach is demonstrated through numerical simulations of benzene and ferrocene molecules. Furthermore, we perform a hardware demonstration of the weight calculation of the ground state and the first excited state of benzene in D2h symmetry, using up to 32 qubits of IBM's ibm_kawasaki device. By combining a tensor-network based encoding scheme and error mitigation techniques, we find the weights of irreducible representations for both states are faithfully reproduced within a few percent error. Our results suggest that the proposed method serves as a practical tool for analyzing symmetry properties of many-electron wavefunctions in realistic material simulations on near-term and early fault-tolerant quantum computers.
The calculation of the many-electron (screened) Coulomb and exchange integrals is a very common task to perform in modern electronic structure theory. While an analytical treatment of the complete integrals is too complicated to be attempted directly, essential insight can be gained by focusing on the most significant contributions, as determined by a physical criterion. The monopole term of the screened Coulomb interaction is particularly important, since it defines the Hubbard interaction U among a set of localized electrons, which is an essential parameter for effective models of strongly correlated systems. Here, we derive an analytical solution for the matrix elements of the screened Coulomb interaction on a plane-wave basis, which is routinely used in the most common methods for electronic structure calculations. Screening is treated using a Yukawa potential, which is suitable to describe the valence electrons in the Fermi level region. For the solution of the integrals, the plane waves and the Yukawa potential are first expanded in Bessel functions of first and second kind. Then, by means of the lower and upper incomplete gamma functions, we are able to obtain closed-form integrals of a series that remains convergent for realistic parameters. Our exact solution for the radial integrals of the monopole term can find usage in plane-wave codes for electronic structure calculations, both as an output tool as well as within the computational cycle, as e.g., for many-body extensions of density-functional theory. Considering the importance of Bessel functions in solid state physics and electronic structure theory, it is also easy to foresee that our solution to the various integrals across this work may become useful to several other problems in the field. • Analytical method for matrix elements of the screened Coulomb interaction in plane wave. • Analysis of the monopole term, significant as the Hubbard U in strongly correlated systems. • Closed-form solutions achieved for both long-range and short-range components of integrals. • Method generalizable to evaluate higher-order terms. • Anticipated applicability to plane-wave codes for physical analysis and hybrid functional routines.
Developing accurate and computationally efficient methods to calculate the electronic structure and total energy of correlated-electron materials has been a very challenging task in condensed matter physics and materials science. Recently, we have developed a correlation matrix renormalization (CMR) method which does not assume any empirical Coulomb interaction U parameters and does not have double counting problems in the ground-state total energy calculation. The CMR method has been demonstrated to be accurate in describing both the bonding and bond breaking behaviors of molecules. In this study, we extend the CMR method to the treatment of electron correlations in periodic solid systems. Using a linear hydrogen chain as a benchmark system, we show that the results from the CMR method compare very well with those obtained recently by accurate quantum Monte Carlo (QMC) calculations. We also study the equation of states of three-dimensional crystalline phases of atomic hydrogen. We show that the results from the CMR method agree much better with the available QMC data in comparison with those from density functional theory and Hartree-Fock calculations.
Strong correlations arising from interactions between particles are responsible for many phenomena in condensed matter physics, from superconductivity to heavy fermions. An accurate theoretical description of such complex systems requires to solve so-called many-body problems, which are inherently difficult to tackle due to the exponential growth of the underlying Fock space with the size of the system. Usually, single-particle descriptions or perturbation theory at first orders fail to provide satisfactory results in correlated regimes, and pertinent Ansätze or cutting edge numerical methods are required to solvethese problems.Among the wide variety of strongly correlated electronic systems, quantum impurity systems are of central interest. These systems, where a few localized degrees of freedom that undergo strong interactions hybridize with a larger system of free particles, have been widely studied for decades. Beyond their relevance to study dilute dynamic impurities in metals, quantum dots, or more complex nanostructures, they are exploited in a broader class of problems through dynamical mean field theory (DMFT), that is designed for fully interacting lattice problems. Despite the effort devoted to studying of these problems,some questions remain open and new approaches are required to address them.In this thesis, quantum impurity problems are tackled with the use of natural orbitals, a representation rooted in the framework of quantum chemistry. The numerical renormalization group, the standard non-perturbative method to treat these systems, is employed to demonstrate the hierarchical structure of correlations unveiled by natural orbitals. An ansatz for the wavefunction of the ground state is then proposed, separating explicitely a small set of correlated orbitals to a large uncorrelated one, leaving the problem to a few-body complexity. This simplification brought new insights for simulating quantumimpurity problems, and a new algorithm is developed to generate an optimized set of correlated orbitals independently of existing methods, going beyond their usual limitations.In the first place, the algorithm is used to compute the screening cloud in large one-dimensional chains and in a realistic 2D square lattice coupled to a localized adatom. This observable is of great interest to understand the screening mechanism of the magnetic moment of the impurity that plays a central role in transport properties, for unidimensional systems or in bulk metals experiencing RKKY interactions. Spatial correlations were found to be highly non-symmetric, breaking the s-wave picture. In one direction, corre-lations vanish way faster than the usual isotropic correlation length, while on the other one they spread further than this same expected length. This anisotropy suggests that impurities in bulk systems may interact differently depending on their relative positions with respect to the lattice.Taking further advantage of the efficiency of this new algorithm allowed us to perform a statistical study on large systems to investigate correlations around a quantum impurity in a disordered host, averaging over a large amount of realizations, which was not possible with previous approaches. The question motivating this study was the fate of the correlation cloud spreading into the disordered electronic bath. Although spatial correlations are expected to vanish when the disorder increases, long range correlationssurvive to some configurations of the potential, inducing a bimodal structure of probability distribution functions of relevant scales.Finally, these disordered problems are investigated in the non-interacting situation with random matrix theory, which reproduced qualitatively the results obtained before. The joint distribution of one-particle orbitals energies and amplitudes is calculated, which allows to compute any disordered averaged local correlation functions, and calculations in the large-N limit are presented.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.