Ground-state Correlation Energy Research Articles

Dielectric Matrix Formulation of Correlation Energies in the Random Phase Approximation: Inclusion of Exchange Effects.

Starting from the general expression for the ground state correlation energy in the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, it is shown that the dielectric matrix formulation, which is usually applied to calculate the direct random phase approximation (dRPA) correlation energy, can be used for alternative RPA expressions including exchange effects. Within this famework, the ACFDT analog of the second order screened exchange (SOSEX) approximation leads to a logarithmic formula for the correlation energy similar to the direct RPA expression. Alternatively, the contribution of the exchange can be included in the kernel used to evaluate the response functions. In this case, the use of an approximate kernel is crucial to simplify the formalism and to obtain a correlation energy in logarithmic form. Technical details of the implementation of these methods are discussed, and it is shown that one can take advantage of density fitting or Cholesky decomposition techniques to improve the computational efficiency; a discussion on the numerical quadrature made on the frequency variable is also provided. A series of test calculations on atomic correlation energies and molecular reaction energies shows that exchange effects are instrumental for improvement over direct RPA results.

Ground state octupole correlation energy with effective forces

The ground state octupole correlation energy is computed with the D1M variant of the Gogny force in different theoretical frameworks and analyzed in detail. First I consider the correlation energy gained at the mean field level by breaking reflection symmetry. Next I consider the energy gain coming from symmetry (parity) restoration and finally I analyze the ground state correlation energy after configuration mixing with axially symmetric octupole states. The impact of the latter on theoretical binding energies indicates that octupole correlations do not affect in a significant way the trend and systematic of binding energies and therefore can not improve the performance of theoretical models in this respect. In particular, the too-large ‘shell gaps’ predicted by self-consistent mean field models and relevant in astrophysics scenarios are not altered by the octupole correlations.

Restricted second random phase approximations and Tamm-Dancoff approximations for electronic excitation energy calculations.

In this article, we develop systematically second random phase approximations (RPA) and Tamm-Dancoff approximations (TDA) of particle-hole and particle-particle channels for calculating molecular excitation energies. The second particle-hole RPA/TDA can capture double excitations missed by the particle-hole RPA/TDA and time-dependent density-functional theory (TDDFT), while the second particle-particle RPA/TDA recovers non-highest-occupied-molecular-orbital excitations missed by the particle-particle RPA/TDA. With proper orbital restrictions, these restricted second RPAs and TDAs have a formal scaling of only O(N(4)). The restricted versions of second RPAs and TDAs are tested with various small molecules to show some positive results. Data suggest that the restricted second particle-hole TDA (r2ph-TDA) has the best overall performance with a correlation coefficient similar to TDDFT, but with a larger negative bias. The negative bias of the r2ph-TDA may be induced by the unaccounted ground state correlation energy to be investigated further. Overall, the r2ph-TDA is recommended to study systems with both single and some low-lying double excitations with a moderate accuracy. Some expressions on excited state property evaluations, such as ⟨Ŝ(2)⟩ are also developed and tested.

Scaling in the correlation energies of atomic ions

We show through numerical investigations that the ground-state correlation energies of atomic ions follow an unexpectedly simple scaling relation, ${E}_{c}\ensuremath{\approx}{Z}^{4/3}{f}_{c}(Z/N)$, where $N$ is the number of electrons, $Z$ is the atomic number, and ${f}_{c}$ is a universal function, for which an analytic expression with a one-parameter fit can be provided. The relation agrees well with several sets of correlation energies obtained from different methods for atomic ions with $N=2,...,18$ and $Z=2,...,28$. Moreover, our relation gives a good agreement with neutral atoms up to $N\ensuremath{\approx}90$. Our main result is readily applicable to estimating correlation energies of heavy elements, for which there are no available data in the literature. The simplicity of the relation may also have implications in the development of correlation functionals within density-functional theory.

Linear-response time-dependent density-functional theory with pairing fields.

Recent development in particle-particle random phase approximation (pp-RPA) broadens the perspective on ground state correlation energies [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A 88, 030501 (2013), Y. Yang, H. van Aggelen, S. N. Steinmann, D. Peng, and W. Yang, J. Chem. Phys. 139, 174110 (2013); D. Peng, S. N. Steinmann, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 104112 (2013)] and N ± 2 excitation energies [Y. Yang, H. van Aggelen, and W. Yang, J. Chem. Phys. 139, 224105 (2013)]. So far Hartree-Fock and approximated density-functional orbitals have been utilized to evaluate the pp-RPA equation. In this paper, to further explore the fundamentals and the potential use of pairing matrix dependent functionals, we present the linear-response time-dependent density-functional theory with pairing fields with both adiabatic and frequency-dependent kernels. This theory is related to the density-functional theory and time-dependent density-functional theory for superconductors, but is applied to normal non-superconducting systems for our purpose. Due to the lack of the proof of the one-to-one mapping between the pairing matrix and the pairing field for time-dependent systems, the linear-response theory is established based on the representability assumption of the pairing matrix. The linear response theory justifies the use of approximated density-functionals in the pp-RPA equation. This work sets the fundamentals for future density-functional development to enhance the description of ground state correlation energies and N ± 2 excitation energies.

Influence of the exchange-correlation potential in methods based on time-dependent density-functional theory

Time-dependent density-functional methods are used to compute excitation energies and, via the adiabatic-connection fluctuation-dissipation theorem, ground-state correlation energies of atoms, ions, and the H${}_{2}$ molecule at various bond lengths. Various exchange-correlation potentials ${v}_{xc}$ and exchange-correlation kernels ${f}_{xc}$ are tested. Accurate exchange-correlation potentials are found to be essential for getting accurate energies. Methods employing in the Kohn-Sham self-consistency process the exact local Kohn-Sham exchange potential while neglecting completely the correlation potential lead to better excitation and correlation energies than methods with exchange-correlation potentials within the local density approximation or the generalized gradient approximation. Taking into account the exact exchange-correlation potential and thus the exact Kohn-Sham potential further improves excitation and correlation energies.

Description of electronic excited states using electron correlation operator

The electron correlation energy in a chemical system is defined as a difference between the energy of an exact energy for a given Hamiltonian, and a mean-field, or single determinant, approximation to it. A promising way to model electron correlation is through the expectation value of a linear two-electron operator for the Kohn-Sham single determinant wavefunction. For practical reasons, it is desirable for such an operator to be universal, i.e., independent of the positions and types of nuclei in a molecule. The correlation operator models the effect of electron correlation on the interaction energy in a electron pair. We choose an operator expanded in a small number of Gaussians as a model for electron correlation, and test it by computing atomic and molecular adiabatic excited states. The computations are performed within the Δ Self-Consistent Field (ΔSCF) formalism, and are compared to the time-dependent density functional theory model with popular density functionals. The simplest form of the correlation operator contains only one parameter derived from the helium atom ground state correlation energy. The correlation operator approach significantly outperforms other methods in computation of atomic excitation energies. The accuracy of molecular excitation energies computed with the correlation operator is limited by the shortcomings of the ΔSCF methodology in describing excited states.

AN ATTEMPT AT A RESONATING MEAN-FIELD THEORETICAL DESCRIPTION OF THERMAL BEHAVIOR OF TWO-GAP SUPERCONDUCTIVITY

The resonating mean-field theory (Res-MFT) has been applied and shown to effectively describe two-gap superconductivity (SC). Particularly at T = 0 using a suitable chemical potential, the two-gap SC in MgB2 has been well described by the Res-Hartree-Bogoliubov theory (Res-HBT). The Res-HB ground state generated with HB wavefunctions almost exhausts the ground-state correlation energy in all the correlation regimes. In this paper we make an attempt at a Res-MF theoretical description of thermal behavior of the two-gap SC. In an equal energy-gap case we find a new formula leading to a higher critical temperature Tc than the Tc of the usual HB formula.

Fluctuating parts of nuclear ground-state correlation energies

Background: Heavy atomic nuclei are often described using the Hartree-Fock-Bogoliubov (HFB) method. In principle, this approach takes into account Pauli effects and pairing correlations while other correlation effects are mimicked through the use of effective density-dependent interactions. Purpose: Investigate the influence of higher order correlation effects on nuclear binding energies using Skyrme's effective interaction. Methods: A cut-off in relative momenta is introduced in order to remove ultraviolet divergences caused by the zero-range character of the interaction. Corrections to binding energies are then calculated using the quasiparticle-random-phase approximation (QRPA) and second order many-body perturbation theory (MBPT2). Result: Contributions to the correlation energies are evaluated for several isotopic chains and an attempt is made to disentangle which parts give rise to fluctuations that may be difficult to incorporate on the HFB level. The dependence of the results on the cut-off is also investigated. Conclusions: The improved interaction allows explicit summations of perturbation series which is useful for the description of some nuclear observables. However, refits of the interaction parameters are needed to obtain more quantitative results.

Treatment of the like-particle pairing of quartets

The ground state correlations induced by a general pairing Hamiltonian in a finite system of like fermions are described in terms of four-body correlated structures (quartets). These are real superpositions of the products of two pairs of particles in time-reversed states. Quartets are determined variationally through an iterative sequence of diagonalizations of the Hamiltonian in restricted model spaces and are, in principle, all distinct from one another. The ground state is represented as a product of quartets to which, depending on the number of particles (supposed to be even, in any case), an extra collective pair is added. The extra pair is also determined variationally. In the case of pairing in a spherically symmetric mean field, both the quartets and the extra pair (if any) are characterized by a total angular momentum J = 0. Realistic applications of the quartet formalism are carried out for Sn isotopes with valence neutrons in the 50–82 neutron shell. The exact ground state correlation energies, occupation numbers and pair transfer matrix elements are reproduced to a very high degree of precision. The formalism also lends itself to a straightforward and accurate description of the lowest seniority 0 and 2 excited states of the pairing Hamiltonian. A simplified representation of the ground state as a product of identical quartets is eventually discussed and found to improve considerably upon the more traditional particle-number projected BCS approach.

Microscopic description of quadrupole-octupole coupling in Sm and Gd isotopes with the Gogny energy density functional

The interplay between the collective dynamics of the quadrupole and octupole deformation degree of freedom is discussed in a series of Sm and Gd isotopes both at the mean-field level and beyond, including parity symmetry restoration and configuration mixing. Physical properties such as negative-parity excitation energies and $E1$ and $E3$ transition probabilities are discussed and compared to experimental data. Other relevant intrinsic quantities such as dipole moments, ground-state quadrupole moments or correlation energies associated with symmetry restoration and configuration mixing are discussed. For the considered isotopes, the quadrupole-octupole coupling is found to be weak and most of the properties of negative-parity states can be described in terms of the octupole degree of freedom alone.

Assessment of correlation energies based on the random-phase approximation

The random-phase approximation to the ground state correlation energy (RPA) in combination with exact exchange (EX) has brought the Kohn–Sham (KS) density functional theory one step closer towards a universal, ‘general purpose first-principles method’. In an effort to systematically assess the influence of several correlation energy contributions beyond RPA, this paper presents dissociation energies of small molecules and solids, activation energies for hydrogen transfer and non-hydrogen transfer reactions, as well as reaction energies for a number of common test sets. We benchmark EX + RPA and several flavors of energy functionals going beyond it: second-order screened exchange (SOSEX), single-excitation (SE) corrections, renormalized single-excitation (rSE) corrections and their combinations. Both the SE correction and the SOSEX contribution to the correlation energy significantly improve on the notorious tendency of EX + RPA to underbind. Surprisingly, activation energies obtained using EX + RPA based on a KS reference alone are remarkably accurate. RPA + SOSEX + rSE provides an equal level of accuracy for reaction as well as activation energies and overall gives the most balanced performance, because of which it can be applied to a wide range of systems and chemical reactions.

GROUND-STATE ENERGY OF THE ELECTRON GAS WITH THE MODIFIED COULOMB POTENTIAL 1/rp

In this paper, due to the effect of positively-charged screening holes, Coulomb potential energy 1/r is modified to be 1/rp, which is assumed to deviate slightly from the former. Using many-body perturbation theory, we obtain a simple analytic representation of the ground-state energy and correlation energy for a uniform electron gas. Our results agree with those obtained by the numerical and semi-analytic methods at low-density limit. Higher ground-state energies at high-density limit are calculated from our model. High order r expansion terms are found at high-density region. A curve of transition density versus p is drawn via the Misawa spin-scaling relation, which is in consistent with Perdew's study at low-density limit.

Correlation energy of two electrons in a ball

We study the ground-state correlation energy E(c) of two electrons of opposite spin confined within a D-dimensional ball (D > or = 2) of radius R. In the high-density regime, we report accurate results for the exact and restricted Hartree-Fock energy, using a Hylleraas-type expansion for the former and a simple polynomial basis set for the latter. By investigating the exact limiting correlation energy E(c)(0) = lim(R-->0)E(c) for various values of D, we test our recent conjecture [P.-F. Loos and P. M. W. Gill, J. Chem. Phys. 131, 241101 (2009)] that in the large-D limit, E(c)(0) approximately -delta(2)/8 for any spherically symmetric confining external potential, where delta = 1/(D-1).

Benchmarking GW against exact diagonalization for semiempirical models

We calculate groundstate total energies and single-particle excitation energies of seven pi conjugated molecules described with the semi-empirical Pariser-Parr-Pople (PPP) model using self-consistent many-body perturbation theory at the GW level and exact diagonalization. For the total energies GW captures around 65% of the groundstate correlation energy. The lowest lying excitations are overscreened by GW leading to an underestimation of electron affinities and ionization potentials by approximately 0.15 eV corresponding to 2.5%. One-shot G_0W_0 calculations starting from Hartree-Fock reduce the screening and improve the low-lying excitation energies. The effect of the GW self-energy on the molecular excitation energies is shown to be similar to the inclusion of final state relaxations in Hartree-Fock theory. We discuss the break down of the GW approximation in systems with short range interactions (Hubbard models) where correlation effects dominate over screening/relaxation effects. Finally we illustrate the important role of the derivative discontinuity of the true exchange-correlation functional by computing the exact Kohn-Sham levels of benzene.

The ground state correlation energy of the random phase approximation from a ring coupled cluster doubles approach

We present an analytic proof demonstrating the equivalence between the random phase approximation (RPA) to the ground state correlation energy and a ring-diagram simplification of the coupled cluster doubles (CCD) equations. In the CCD framework, the RPA equations can be solved in O(N(4)) computational effort, where N is proportional to the number of basis functions.

Multireference many-electron correlation energies from two-electron reduced density matrices computed by solving the anti-Hermitian contracted Schrödinger equation

Two-electron reduced density matrices (2-RDMs) have recently been directly calculated by solving the anti-Hermitian contracted Schr\"odinger equation (ACSE) to obtain $95$--$100\phantom{\rule{0.3em}{0ex}}%$ of the ground-state correlation energy of atoms and molecules with the accuracy increasing with the size of the one-electron basis set [Mazziotti, Phys. Rev. Lett. 97, 143002 (2006).] In this paper, the ACSE method is extended to treat strong multireference correlation effects that are often important at nonequilibrium molecular geometries. While previous ACSE calculations have employed an initial 2-RDM from the Hartree-Fock method, we initialize the solution of the ACSE with a 2-RDM guess from a multiconfiguration self-consistent field calculation. Applications are made to multireference correlation in the potential energy surfaces of the molecules $\mathrm{HF}$, ${\mathrm{H}}_{2}\mathrm{O}$, and ${\mathrm{C}}_{2}$ in polarized valence double-zeta basis sets while ${\mathrm{N}}_{2}$ is treated in polarized valence double- and triple-zeta basis sets. Accurate ground-state energies and 1-RDM occupation numbers are obtained at both equilibrium and nonequilibrium geometries where the energies are within a few millihartrees of those from full configuration interaction.

Two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation: Enhanced energies and properties with larger basis sets

Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.

Extensions of r12 corrections to CC2-R12 for excited states

As known since about two decades, R12 methods, which include terms linear in the interelectronic distance r(12) in the wave function, improve substantially the basis set convergence of the ground state correlation energy. In a previous study, however, it was found that the same approach does not give a similar systematic improvement if applied to excited states in the framework of coupled cluster response theory. In the present work, we examine the reason for this behavior and show that the inclusion of additional orbitals in the construction of the r(12) pair functions leads to an enhanced basis set convergence (and thus a balanced description) also for the excited states.

Coulomb repulsive correlation in systems with radial confinement: quantum dots and the Overhauser model in an external magnetic field

The ground-state correlation energy of an electron–negative ion system in a spherical quantum dot (QD) with parabolic confinement in an external homogeneous magnetic field is investigated. Both cases of finite and infinite potential barrier on the QD surface are considered. The theoretical analysis is carried out using a variational approach. The ground state energy of a two-electron system in a QD with harmonic-oscillator potential in a magnetic field is also determined as a function of the strength of the magnetic field and parabolic confinement. The influence of the finite potential barrier on the correlation energy of the two-electron system is also discussed. A first investigation of electron–electron correlation in a two-dimensional (2D) system with the screened Coulomb potential proposed by Overhauser but in the presence of a perpendicular magnetic field is shown to be another important application of the proposed variational wave functions.