Adiabatic Connection Research Articles

Potential Energy Surfaces without Unphysical Discontinuities: The Coulomb Hole Plus Screened Exchange Approach.

In this work, we show the advantages of using the Coulomb hole plus screened exchange (COHSEX) approach in the calculation of potential energy surfaces (PES). In particular, we demonstrate that, unlike perturbative GW and partial self-consistent GW approaches, such as eigenvalue self-consistent GW and quasi-particle (QP) self-consistent GW, the COHSEX approach yields smooth PES without irregularities and discontinuities. Moreover, we show that the ground-state PES obtained from the Bethe-Salpeter equation (BSE), within the adiabatic connection fluctuation dissipation theorem, built with QP energies obtained from perturbative COHSEX on top of Hartree-Fock (BSE@COHSEX@HF) yield very accurate results for diatomic molecules close to their equilibrium distance. When self-consistent COHSEX QP energies and orbitals are used to build the BSE equation, the results become independent of the starting point. We show that self-consistency worsens the total energies but improves the equilibrium distances with respect to BSE@COHSEX@HF. This is mainly due to the changes in the screening inside the BSE.

Adiabatic connection in spin-current density functional theory

The spin-current density functional theory (SCDFT), when formulated in a basis of Pauli spinors, provides a proper theoretical framework for the study of materials in an arbitrarily oriented external magnetic field and/or upon inclusion of spin-dependent relativistic effects, such as spin-orbit coupling. The SCDFT is formulated in terms of the particle-number density $n$, the Cartesian components of the magnetization ${m}_{x},$ ${m}_{y},$ and ${m}_{z}$, the orbital-current density $\mathbit{j}$, and the three spin-current densities ${\mathbit{J}}^{x},$ ${\mathbit{J}}^{y},$ and ${\mathbit{J}}^{z}$, where each of these density variables depends on specific blocks of the density matrix. Exchange-correlation (xc) functionals within the SCDFT should therefore depend on all of these eight fundamental density variables: ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z},\mathbit{j},{\mathbit{J}}^{x},{\mathbit{J}}^{y},{\mathbit{J}}^{z}]$, which makes their parametrization a formidable task. Here, we formulate the adiabatic connection of the SCDFT for a treatment of exact Fock exchange in the theory. We show how the inclusion of a fraction of Fock exchange in standard functionals of the (spin) DFT (either in their collinear or noncollinear versions: ${F}_{xc}[n],$ ${F}_{xc}[n,{m}_{z}]$ and ${F}_{xc}[n,{m}_{x},{m}_{y},{m}_{z}]$) allows for the two-electron potential to depend on all those blocks of the density matrix that correspond to the eight density variables of the SCDFT, in a sensible and yet practical way. In particular, in the local-density and generalized-gradient approximations of the SCDFT, the treatment of the current densities solely from the Fock exchange term is formally justified by the short-range behavior of the exchange hole. We discuss that the adiabatic coupling strength parameter modulates the two-electron coupling of the orbital- and spin-current densities with the particle-number density and magnetization. Formal considerations are complemented by numerical tests on a periodic model system in the presence of spin-orbit coupling and in the absence of an external magnetic field.

Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms.

We study in detail the first three leading terms of the large coupling-strength limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory. We first focus on the H atom, both in the spin-polarized and the spin-unpolarized cases, reporting numerical and analytical results. In particular, we derive an asymptotic equation that turns out to have simple analytical solutions for certain channels. The asymptotic H atom solution for the spin-unpolarized case is then shown to be variationally optimal for the many-electron spin-restricted closed-shell case, providing expressions for the large coupling-strength density functionals up to the third leading order. We also analyze the H2 molecule and the uniform electron gas.

Generalizing Double-Hybrid Density Functionals: Impact of Higher-Order Perturbation Terms.

Connections between the Görling–Levy (GL) perturbation theory and the parameters of double-hybrid (DH) density functional are established via adiabatic connection formalism. Moreover, we present a more general DH density functional theory, where the higher-order perturbation terms beyond the second-order GL2 one, such as GL3 and GL4, also contribute. It is shown that a class of DH functionals including previously proposed ones can be formed using the present construction. Based on the proposed formalism, we assess the performance of higher-order DH and long-range corrected DH formed on the Perdew–Burke–Ernzerhof (PBE) semilocal functional and second-order GL2 correlation energy. The underlying construction of DH functionals based on the generalized many-body perturbation approaches is physically appealing in terms of the development of the non-local forms using more accurate and sophisticated semilocal functionals.

Extended Koopmans' theorem in the adiabatic connection formalism: Applied to doubly hybrid density functionals.

A rigorous framework that combines the extended Koopmans' theorem (EKT) with the adiabatic connection (AC) formalism of density functional theory is developed here, namely, EKT-AC, to calculate the vertical ionization potentials (IPs) of molecular systems. When applied to the doubly hybrid density functional approximations (DH-DFAs), the EKT-DH approach is established for the B2PLYP-type DHs with one-parameter and two-parameters, as well as the XYG3-type DHs. Based on EKT-DH, an approximation of the KT type is introduced, leading to the KT-DH approach. The IP-condition that the calculated vertical IPs with EKT-DH or KT-DH are to reproduce the experimental IPs closely is applied to investigate the commonly used DH-DFAs for such a purpose and is utilized as a principle for DH-DFA developments. Considering the systematic improvements, as well as its numeric stability, we recommend the KT-B2GPPLYP approach as a pragmatic way for vertical IP calculations.

Search for correlation-induced adiabatic paths between distinct topological insulators

Correlations in topological states of matter provide a rich phenomenology, including a reduction in the topological classification of the interacting system compared to its non-interacting counterpart. This happens when two phases that are topologically distinct on the non-interacting level become adiabatically connected once interactions are included. We use a quantum Monte Carlo method to study such a reduction. We consider a 2D charge-conserving analog of the Levin-Gu superconductor whose classification is reduced from $\mathbb{Z}$ to $\mathbb{Z}_4$. We may expect any symmetry-preserving interaction that leads to a symmetric gapped ground state at strong coupling, and consequently a gapped symmetric surface, to be sufficient for such reduction. Here, we provide a counter example by considering an interaction which (i) leads to a symmetric gapped ground state at sufficient strength and (ii) does not allow for any adiabatic path connecting the trivial phase to the topological phase with $w=4$. The latter is established by numerically mapping the phase diagram as a function of the interaction strength and a parameter tuning the topological invariant. Instead of the adiabatic connection, the system exhibits an extended region of spontaneous symmetry breaking separating the topological sectors. Frustration reduces the size of this long-range ordered region until it gives way to a first order phase transition. Within the investigated range of parameters, there is no adiabatic path deforming the formerly distinct free fermion states into each other. We conclude that an interaction which trivializes the surface of a gapped topological phase is necessary but not sufficient to establish an adiabatic path within the reduced classification. In other words, the class of interactions which trivializes the surface is different from the class which establishes an adiabatic connection in the bulk.

Reduced Density Matrix-Driven Complete Active Apace Self-Consistent Field Corrected for Dynamic Correlation from the Adiabatic Connection.

The recently proposed multireference adiabatic connection (AC) formalism [Pernal, Phys. Rev. Lett. 120, 013001 (2018)] is applied to recover dynamic electron correlation effects lacking in variational two-electron reduced density matrix (v2RDM)-driven complete active space self-consistent field theory (CASSCF). The AC approach is validated by computing potential energy curves for the dissociation of molecular nitrogen and the symmetric double dissociation of H2O while enforcing two sets of approximate N-representability conditions in the underlying v2RDM-driven CASSCF calculations (either two-particle or two-particle plus partial three-particle conditions). The AC yields smaller absolute errors than second-order N-electron perturbation theory (NEVPT2) at all molecular geometries for both sets of the N-representability conditions considered. The efficacy of the approach for thermochemistry is also assessed for a set of 31 small-molecule reactions. When imposing partial three-particle N-representability conditions, mean and maximum unsigned errors in reaction energies from the AC are superior to those from NEVPT2.

MAP: An MP2 Accuracy Predictor for Weak Interactions from Adiabatic Connection Theory.

Second-order Møller-Plesset perturbation theory (MP2) approximates the exact Hartree-Fock (HF) adiabatic connection (AC) curve by a straight line. Thus, by using the deviation of the exact curve from the linear behavior, we construct an indicator for the accuracy of MP2. We then use an interpolation along the HF AC to transform the exact form of our indicator into a highly practical MP2 accuracy predictor (MAP) that comes at a negligible additional computational cost. We show that this indicator is already applicable to systems that dissociate into fragments with a nondegenerate ground state, and we illustrate its usefulness by applying it to the S22 and S66 datasets.

Many‐body response of benzene at monolayer MoS2: Van der Waals interactions and spectral broadening

AbstractModels of surface enhancement of molecular electronic response properties are challenging for two reasons: (a) molecule‐surface interactions require a simultaneous solution of the molecular and the surface dynamic response (a daunting task), and (b) when solving for the electronic structure of the combined molecule + surface system, it is not trivial to single out the particular physical effects responsible for enhancement. To tackle this problem, in this work, we apply a formally exact decomposition of the system's response function into subsystem contributions by using subsystem density functional theory (DFT), which grants access to dynamic polarizabilities and optical spectra. In order to access information about the interactions between the subsystems, we extend a previously developed subsystem‐based adiabatic connection fluctuation‐dissipation theorem of DFT to separate the additive from the nonadditive correlation energy and identify the nonadditive correlation as the van der Waals interactions. As an example, we choose benzene adsorbed on monolayer MoS2. We isolate the contributions to benzene's dynamic response arising from the interaction with the surface, and for the first time, we evaluate the enhancements to the effectiveness of C6 coefficients as a function of benzene‐MoS2 distance and adsorption site. We also quantify the spectral broadening of the benzene's electronic excited states due to their interaction with the surface. We find that the broadening has a similar decay law with the molecule‐surface distance as the leading van der Waals interactions (ie, R−6) and that the surface enhancement of dispersion interactions between benzene molecules is less than 5% but is still large enough (0.5 kcal/mol) to likely play a role in the prediction of interface morphologies.

Long-range-corrected multiconfiguration density functional with the on-top pair density.

We propose a multiconfiguration density functional combining a short-range density functional approximation with a novel long-range correction for dynamic correlation effects. The correction is derived from the adiabatic connection formalism so that the resulting functional requires access only to one- and two-electron reduced density matrices of the system. In practice, the functional is formulated for wavefunctions of the complete active space (CAS) type and the short-range density functional part is made dependent on the on-top pair density via auxiliary spin densities. The latter allows for reducing the self-interaction and the static correlation errors without breaking the spin symmetry. We study the properties and the performance of the non-self-consistent variant of the method, termed lrAC0-postCAS. Numerical demonstration on a set of dissociation energy curves and excitation energies shows that lrAC0-postCAS provides accuracy comparable with more computationally expensive ab initio rivals.

Exchange-Correlation Energy Densities and Response Potentials: Connection between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond.

We analyze in depth two widely used definitions (from the theory of conditional probability amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn–Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg–Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.

Divergence of Many-Body Perturbation Theory for Noncovalent Interactions of Large Molecules.

Prompted by recent reports of large errors in noncovalent interaction (NI) energies obtained from many-body perturbation theory (MBPT), we compare the performance of second-order Mo̷ller-Plesset MBPT (MP2), spin-scaled MP2, dispersion-corrected semilocal density functional approximations (DFAs), and post-Kohn-Sham random phase approximation (RPA) for predicting binding energies of supramolecular complexes contained in the S66, L7, and S30L benchmarks. All binding energies are extrapolated to the basis set limit, corrected for basis set superposition errors, and compared to reference results of the domain-based local pair-natural orbital coupled-cluster (DLPNO-CCSD(T)) or better quality. Our results confirm that MP2 severely overestimates binding energies of large complexes, producing relative errors of over 100% for several benchmark compounds. RPA relative errors consistently range between 5 and 10%, significantly less than reported previously using smaller basis sets, whereas spin-scaled MP2 methods show limitations similar to MP2, albeit less pronounced, and empirically dispersion-corrected DFAs perform almost as well as RPA. Regression analysis reveals a systematic increase of relative MP2 binding energy errors with the system size at a rate of approximately 0.1% per valence electron, whereas the RPA and dispersion-corrected DFA relative errors are virtually independent of the system size. These observations are corroborated by a comparison of computed rotational constants of organic molecules to gas-phase spectroscopy data contained in the ROT34 benchmark. To analyze these results, an asymptotic adiabatic connection symmetry-adapted perturbation theory (AC-SAPT) is developed, which uses monomers at full coupling, whose ground-state density is constrained to the ground-state density of the complex. Using the fluctuation-dissipation theorem, we obtain a nonperturbative "screened second-order" expression for the dispersion energy in terms of monomer quantities, which is exact for non-overlapping subsystems and free of induction terms; a first-order RPA-like approximation to the Hartree, exchange, and correlation kernel recovers the macroscopic Lifshitz limit. The AC-SAPT expansion of the interaction energy is obtained from Taylor expansion of the coupling strength integrand. Explicit expressions for the convergence radius of the AC-SAPT series are derived within RPA and MBPT and numerically evaluated. While the AC-SAPT expansion is always convergent for nondegenerate monomers when RPA is used, it is found to spuriously diverge for second-order MBPT, except for the smallest and least polarizable monomers. The divergence of the AC-SAPT series for MBPT is numerically confirmed within RPA; prior numerical results on the convergence of the SAPT expansion for MBPT methods are revisited and support this conclusion once sufficiently high orders are included. The cause of the failure of MBPT methods for NIs of large systems is missing or incomplete "electrodynamic" screening of the Coulomb interaction due to induced particle-hole pairs between electrons in different monomers, leaving the effective interaction too strong for AC-SAPT to converge. Hence, MBPT cannot be considered reliable for quantitative predictions of NIs, even in moderately polarizable molecules with a few tens of atoms. The failure to accurately account for electrodynamic polarization makes MBPT qualitatively unsuitable for applications such as NIs of nanostructures, macromolecules, and soft materials; more robust nonperturbative approaches such as RPA or coupled cluster methods should be used instead whenever possible.

Grating Lobe-Free Beam Steering through Optical Phase Array Using Phase-Compensated Two Index-Mismatched Silicon Wires-Based Emitters

The beam formation can be treated as the diffraction pattern. A 1-D light detection and ranging beam steering could be derived through a phase shifter array using Rayleigh–Sommerfeld Diffraction, which is then utilized to demonstrate grating lobe-free beam steering from the optical phase array emitter with half-wavelength pitches. The half-wavelength pitch cannot demonstrate beam formation without any evanescent coupling blocking between emitters. Here, two index-mismatched silicon wires in the emitter array are proposed by the optical phase compensation through waveguide width adjustment, to avoid the complex and addressable thermal control on the phase shifters. Moreover, the same output optical waveguide mode needs to be further considered to demonstrate the grating lobe-free beam steering. In order to get the adiabatic connection between two different pitches between the phase shifter and emitter, an optical path equalization will also be applied.

Semiempirical van der Waals method for two-dimensional materials with incorporated dielectric functions

A density functional theory based semiempirical van der Waals (vdW) method with dielectric functions being incorporated is developed for two-dimensional materials. The coefficients of interatomic pairwise potentials are derived from atomic polarizabilities obtained via a Clausius-Mossotti relation dedicated for layered crystals. The method not only can efficiently describe the dispersion energy for a range of planar graphene-like materials at nearly the same accuracy as the adiabatic connection fluctuation-dissipation theorem, but also rationalizes experimentally measured relative interfacial strengths of heterostructures and interlayer registry of hexagonal boron nitride that have plagued other vdW methods.

Asymptotic Behavior of the Exchange‐Correlation Energy Density and the Kohn‐Sham Potential in Density Functional Theory: Exact Results and Strategy for Approximations

AbstractDensity functional theory (DFT) is nowadays the leading theoretical framework for quantum description of materials from first principles. Being an exact theory on one hand and computationally efficient on the other hand, DFT allows to address large and complex many‐electron systems and accurately predict their properties. The predictive power of DFT critically depends though on an accurate approximation to the generally unknown exchange‐correlation (xc) energy functional. Approximations can be constructed from first principles by satisfying known properties of the exact functional. In this work I review two such exact properties: the asymptotic behavior of the xc energy density per particle and the asymptotic behavior of the Kohn‐Sham potential, in finite many‐electron systems. The derivation of the asymptotic forms for both quantities is reviewed, employing the concepts of the adiabatic connection and of the xc hole with relation to the first quantity and the exact electron factorization approach for the second one. Furthermore, it is shown that the correct asymptotic behavior of one of the aforementioned quantities does not guarantee a correct behavior of the other. These quantities are related via the xc hole response function, which is defined, examined and its exact exchange part is analytically derived. The extent to which existing xc approximations satisfy the named exact properties is reviewed and the relationship between correct asymptotics and freedom from one‐electron self‐interaction in DFT is discussed. Finally, a strategy for development of advanced approximations for exchange and correlation with a correct asymptotic behavior is suggested.

An adiabatic connection for doubly-occupied configuration interaction wave functions.

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.

Ensemble Green's function theory for interacting electrons with degenerate ground states

An ensemble Green's function formalism, based on the von Neumann density matrix approach, to calculate one-electron excitation spectra of a many-electron system with degenerate ground states is proposed. A set of iterative equations for the ensemble Green's function and self-energy is derived and a simplest approximation corresponding to an ensemble $GW$ approximation is naturally obtained. The derivation is based on the Schwinger functional derivative technique and does not assume any adiabatic connection between a noninteracting and an interacting ground state.

Adiabatic connection in density functional theory in two-dimensions: A semi-analytic wavefunction based study for two-electron atomic systems.

This work focuses on studying the adiabatic-connection in density functional theory in two dimensions. It employs a recently developed accurate form of wavefunction for two-electron systems. The explicit semianalytic form of the wavefunction makes it possible to calculate ground state wavefunctions, energies, densities, and the resulting properties for the scaled Coulomb interaction between the electrons at fixed density accurately. The results so obtained for the correlation energies are then used as the reference values for studying the performance of two-dimensional correlation energy functionals.

Many-Body Quantum Dynamics by the Reduced Density Matrix Based on Time-Dependent Density-Functional Theory.

We evaluate the density matrix of an arbitrary quantum mechanical system in terms of the quantities pertinent to the solution of the time-dependent density functional theory (TDDFT) problem. Our theory utilizes the adiabatic connection perturbation method of Görling and Levy, from which the expansion of the many-body density matrix in powers of the coupling constant λ naturally arises. We then find the reduced density matrix ρ_{λ}(r,r^{'},t), which, by construction, has the λ independent diagonal elements ρ_{λ}(r,r,t)=n(r,t), n(r,t) being the particle density. The off-diagonal elements of ρ_{λ}(r,r^{'},t) contribute importantly to the processes unaccessible via the density, directly or by the use of the known TDDFT functionals. Of those, we consider the momentum-resolved photoemission, doing this to the first order in λ, i.e., on the level of the exact exchange theory. In illustrative calculations of photoemission from the quasi-2D electron gas and isolated atoms, we find quantitatively strong and conceptually far-reaching differences with the independent-particle Fermi's golden rule formula.

Capturing the Dynamic Correlation for Arbitrary Spin-Symmetry CASSCF Reference with Adiabatic Connection Approaches: Insights into the Electronic Structure of the Tetramethyleneethane Diradical.

The recently proposed approach to multireference dynamic correlation energy based on the adiabatic connection (AC) is extended to an arbitrary spin symmetry of the reference state. We show that both the spin-free AC approach and its computationally inexpensive approximation, AC0, when combined with a complete active space wave function, constitute viable alternatives to the perturbation-based and density-functional-based multiconfiguration methods. In particular, the AC0 approach, thanks to its favorable scaling with the system size and the size of the active space, allows for treating larger systems than its perturbation-based counterparts while maintaining comparable accuracy. We show the method's robustness on illustrative chemical systems, including the elusive tetramethyleneethane (TME) diradical, potential energy surfaces of which present a challenge to most computational approaches. For the latter system, AC0 outperforms other methods, staying in close agreement with the full configuration interaction quantum Monte Carlo benchmark. A careful analysis of the contributions to the correlation energy of TME's lowest singlet and triplet states reveals the subtle interplay of the dynamic and static correlation as the key to understanding the shape of the diradical's potential energy surfaces.