Self-consistent Second-order Perturbation Theory Research Articles

Green/WeakCoupling: Implementation of fully self-consistent finite-temperature many-body perturbation theory for molecules and solids

The accurate ab-initio simulation of molecules and periodic solids with diagrammatic perturbation theory is an important task in quantum chemistry, condensed matter physics, and materials science. In this article, we present the WeakCoupling module of the open-source software package Green, which implements fully self-consistent diagrammatic weak coupling simulations, capable of dealing with real materials in the finite-temperature formalism. The code is licensed under the permissive MIT license. We provide self-consistent GW (scGW) and self-consistent second-order Green's function perturbation theory (GF2) solvers, analysis tools, and post-processing methods. This paper summarises the theoretical methods implemented and provides background, tutorials and practical instructions for running simulations. Program summaryProgram Title:Green/WeakCouplingCPC Library link to program files:https://doi.org/10.17632/2ysyhzww6t.1Developer's repository link:https://github.com/Green-Phys/green-mbptProgramming language:C++17, CUDA, PythonLicensing provisions: MIT LicenseExternal routines/libraries:MPI >= 3.0, BLAS, Eigen >= 3.4.0, cmake >= 3.18, cuBLAS.Nature of problem: The simulation of periodic solids and molecules using diagrammatic perturbation theorySolution method: We present an open-source implementation of the fully self-consistent finite-temperature many-body perturbation theory formalism at the GW and second-order perturbation theory level.

A Theoretical Study on the Excited States of the KCCH Radical

Based on the C2v symmetry, the low-lying electronic states of the KCCH radical was studied in the complete active pace self-consistent field (CASSCF) and multi-configurational second-order perturbation theory (CASPT2) methods. For the low-lying electronic states of KCCH, the calculated harmonic vibrational modes are in agreement with theoretical reports. Moreover, the electron transitions of KCCH, from X1Σ+ to 21Σ+, 11Π, 21Π, 11Σ-, 13Σ+, 13Π, 23Π and 13Σ-, are predicted at 5.43, 5.59, 7.87, 5.50, 5.15, 5.73, 5.53 and 6.82 eV, respectively. The CASSCF/CASPT2 potential energy curves (PECs) were calculated for K+-loss dissociation from the 13Σ+, 23Σ+,13Π ,21Σ, 11Π and 21Π states. Analyses of electron transition and dissociation curves show the ionic properties in the metal–ligand bond for KCCH. The dissociation energy of the KCCH (X1Σ+)→CCH-(11Σ)+K+(S0) is 4.423 eV at the CASSCF/CASPT2 level.

Effect of propagator renormalization on the band gap of insulating solids

We present momentum-resolved spectral functions and band gaps from bare and self-consistent second-order perturbation theory for insulating periodic solids. We establish that, for systems with large gap sizes, both bare and self-consistent perturbation theory yield reasonable gaps. However, smaller gap sizes require a self-consistent adjustment of the propagator. In contrast to results obtained within a quasiparticle formalism used on top of bare second-order perturbation theory, no unphysical behavior of the band gap is observed. Our implementation of a fully self-consistent, $\mathrm{\ensuremath{\Phi}}$-derivable, and thermodynamically consistent finite-temperature diagrammatic perturbation theory forms a framework on which embedding theories such as the dynamical mean-field theory and self-energy embedding theories can be implemented.

A Theoretical Probe for Structures, Metal–Metal Bonding, and Electronic Spectra of Paramagnetic Tetrapyrrolic RuII Complex

Dimeric complexes (RuIIPz)2 have been investigated using density functional theory (DFT), where Pz is a porphyrazine ligand that features a 16-atom, 18-π-electron cyclic polyene aromatic skeleton. Structural optimizations in various configurations and spin states indicate that (RuPz)2 favours a Pz–Pz staggered conformer over an eclipsed one; the paramagnetic triplet state with the staggered configuration is found as the global ground state. This agrees with experimental magnetic results of (RuOEPor)2 (OEPor = octaethylporphyrin) and (RuPc)2 (Pc = phthalocyanine). The Ru–Ru bond length was optimized to be 2.38 Å, close to the experimental bond length of 2.40–2.41 Å. The Ru2 doubly bonded nature has been evidenced by the Ru–Ru stretching vibrational frequency of 202 cm–1, bond energy of 30.7 kcal mol–1, and electronic arrangement of σ2π4(nonbonding-δ)4(π*)2. Further confirmation was obtained from high-level wave function theory calculations (complete active space self-consistent field and n-electron valence state second-order perturbation theory). Associated with the solvation of the explicit pyridine accounting for the first coordination sphere and the implicit continuum model for the long-range interaction, the electronic spectra of tetrapyrrolic ruthenium complex were calculated at the time-dependent DFT level.

Vibronic structure of the 2Πu ground electronic state of dicyanoacetylene cation revisited by PFI-ZEKE photoelectron spectroscopy and ab initio calculations

The X+2Πu ← X1Σ+g transition of dicyanoacetylene has been recorded for the first time using pulsed-field-ionisation zero-kinetic-energy photoelectron spectroscopy. The analysis of the photoelectron spectrum allowed an accurate determination of the adiabatic ionisation potential of C4N2 (Ei, ad./hc = 95,479 ± 2 cm−1) and a description of the vibrational structure of the electronic ground state of the cation which is affected by the Renner–Teller effect and the spin–orbit interaction. The spin–orbit coupling constant was measured as −52 ± 2 cm−1. These results are supported by ab initio calculations performed at the complete active space self-consistent field and second-order perturbation theory levels of theory, with extrapolation to the complete basis set limit.

Dynamical instabilities and transient short-range order in the fermionic Hubbard model

We study the dynamics of magnetic correlations in the half-filled fermionic Hubbard model following a fast ramp of the repulsive interaction. We use Schwinger-Keldysh self-consistent second-order perturbation theory to investigate the evolution of single-particle Green's functions and solve the non-equilibrium Bethe-Salpeter equation to study the dynamics of magnetic correlations. This approach gives us new insights into the interplay between single-particle relaxation dynamics and the growth of antiferromagnetic correlations. Depending on the ramping time and the final value of the interaction, we find different dynamical behavior which we illustrate using a dynamical phase diagram. Of particular interest is the emergence of a transient short-range ordered regime characterized by the strong initial growth of antiferromagnetic correlations followed by a decay of correlations upon thermalization. The discussed phenomena can be probed in experiments with ultracold atoms in optical lattices.

Long-lived nonequilibrium states in the Hubbard model with an electric field

We study the single-band Hubbard model in the presence of a large spatially uniform electric field out of equilibrium. Using the Keldysh nonequilibrium formalism, we solve the problem using perturbation theory in the Coulomb interaction $U$. We present numerical results for the charge current, the total energy of the system, and the double occupancy on an infinite-dimensional hypercubic lattice with nearest-neighbor hopping. The system is isolated from an external bath and is in the paramagnetic state. We show that an electric field pulse can drive the system to a steady nonequilibrium state, which does not evolve into a thermal state. We compare results obtained within second-order perturbation theory (SOPT), self-consistent SOPT, and iterated perturbation theory (IPT). We also discuss the importance of initial conditions for a system which is not coupled to an external bath.

Self-Consistent Second-Order Perturbation Theory of Correlation Effect in bcc Iron at Finite Temperatures

The self-consistent second-order perturbation theory is applied to a realistic multiband Hubbard model of bcc iron with realistic local Coulomb interaction terms, and both the single-electron Green's function and the dynamical transverse susceptibility are calculated while preserving the spin rotational symmetry. The overall bandwidth reduction is obtained and is in agreement with experiments and previous theoretical results, but no significant additional mass enhancement near the Fermi surface is found. The spin wave mode is found to evolve in the dynamical spin fluctuation spectrum, whose frequency vanishes correctly in the long wavelength limit. The spin stiffness constant \(D\) is found to be in good agreement with other theoretical results and is smaller than the experimental value.

Dynamical density and spin response functions of two-dimensional correlated fermion systems: Self-consistent second-order perturbation theory

Dynamical density and spin responses of a two-dimensional fermion system are studied with the self-consistent second-order perturbation theory using a model Hamiltonian. Special attention is paid to the density response at a large wave vector, where it was reported that a sharp mode develops below the particle-hole continuum in two-dimensional liquid ${}^{3}$He. We find that at a large wave vector, a peak in the dynamical density structure factor shifts to lower energy as interaction sets in, but it is not sharp enough to represent a well-defined mode because of the strong damping effect of the quasiparticles. We point out that it is necessary to treat the large damping effect properly in explaining the propagating mode found in two-dimensional liquid ${}^{3}$He.

Correlation effect in bcc iron at finite temperatures

Self-consistent second-order perturbation theory is applied to a multi-band Hubbard model to study the static and the dynamical properties of iron at finite temperatures. Reasonable values of the Curie temperature and the Curie constant are obtained for a realistic band structure and realistic values of the local Coulomb interaction. The dynamical transverse susceptibility in the ferromagnetic phase is calculated, and a spin wave mode vanishing in the long-wavelength limit is correctly obtained. The spin stiffness constant is found to be enhanced compared with that obtained using the random phase approximation.

A CAS study on S-loss and O-loss dissociation mechanisms of the SO2+ ion in the C, D, and E states

In the present work, we mainly study dissociation of the C (2) B(1) , D(2) A(1) , and E(2) B(2) states of the SO(2) (+) ion using the complete active-space self-consistent field (CASSCF) and multiconfiguration second-order perturbation theory (CASPT2) methods. We first performed CASPT2 potential energy curve (PEC) calculations for S- and O-loss dissociation from the X, A, B, C, D, and E primarily ionization states and many quartet states. For studying S-loss predissociation of the C, D, and E states by the quartet states to the first, second, and third S-loss dissociation limits, the CASSCF minimum energy crossing point (MECP) calculations for the doublet/quartet state pairs were performed, and then the CASPT2 energies and CASSCF spin-orbit couplings were calculated at the MECPs. Our calculations predict eight S-loss predissociation processes (via MECPs and transition states) for the C, D, and E states and the energetics for these processes are reported. This study indicates that the C and D states can adiabatically dissociate to the first O-loss dissociation limit. Our calculations (PEC and MECP) predict a predissociation process for the E state to the first O-loss limit. Our calculations also predict that the E(2) B(2) state could dissociate to the first S- and O-loss limits via the A(2) B(2) ← E(2) B(2) transition. On the basis of the 13 predicted processes, we discussed the S- and O-loss dissociation mechanisms of the C, D, and E states proposed in the previous experimental studies.

Electronic states of the C6H5CN+ ion studied using multiconfiguration wave functions

Electronic states of the C6H5CN+ ion have been studied within C2v symmetry using the complete active space self-consistent field (CASSCF) and multiconfiguration second-order perturbation theory (CASPT2) methods in conjunction with an atomic natural orbital basis set. Adiabatic excitation energies (T 0), vertical excitation energies (T v), and relative energies () at the ground-state geometry of the C6H5CN molecule were calculated for eight electronic states of the C6H5CN+ ion. The CASPT2//CASSCF T 0 and CASPT2 T v and calculations all indicate that the 1 2B1, 1 2A2, 2 2B1, 1 2B2, 1 2A1, 2 2A1, 2 2B2, and 3 2B1 states are the eight lowest-lying states of C6H5CN+. In conjunction with the MS-CASPT2 T 0 and oscillator strength f calculations, we assigned the X, A, B, C, D, E, F, and G states of the C6H5CN+ ion to 1 2B1, 1 2A2, 1 2B2, 2 2B1, 1 2A1, 2 2A1, 2 2B2, and 3 2B1, respectively.

A CASPT2//CASSCF study of the quartet excited state $$\tilde{a}^{4}A^{\prime\prime}$$ of the HCNN radical

Complete active space self-consistent field and second-order multiconfigurational perturbation theory methods have been performed to investigate the quartet excited state \({\tilde{a}}^{4}{A^{\prime\prime}}\) potential energy surface of HCNN radical. Two located minima with respective cis and trans structures could easily dissociate to CH \(({\tilde{a}}^{4}\Sigma^{-})\) and \(N_{2} ({\tilde{X}}^{1}\Sigma_{\rm g}^{+})\) products with similar barrier of about 16.0 kcal/mol. In addition, four minimum energy crossing points on a surface of intersection between \({\tilde{a}}^{4}A^{\prime\prime}\) and X (\(X={\tilde{X}}^{2}A^{\prime\prime}\) and \({\tilde{A}}^{2}A^{\prime}\)) states are located near to the minima. However, the intersystem crossing \({\tilde{a}}^{4}A^{\prime\prime} \rightarrow X\) is weak due to the vanishingly small spin–orbit interactions. It further indicates that the direct dissociation on the \({\tilde{a}}^{4}{A^{\prime\prime}}\) state is more favored. This information combined with the comparison with isoelectronic HCCO provides an indirect support to the recent experimental proposal of photodissociation mechanism of HCNN.

X, A, B, C, and D States of the C6H5F+ Ion Studied Using Multiconfiguration Wave Functions

Electronic states of the C6H5F+ ion have been studied within C2v symmetry by using the complete active space self-consistent field (CASSCF) and multiconfiguration second-order perturbation theory (CASPT2) methods in conjunction with an atomic natural orbital basis. Vertical excitation energies (Tv) and relative energies (Tv') at the ground-state geometry of the C6H5F molecule were calculated for 12 states. For the five lowest-lying states, 1(2)B1, 1(2)A2, 2(2)B1, 1(2)B2, and 1(2)A1, geometries and vibrational frequencies were calculated at the CASSCF level, and adiabatic excitation energies (T0) and potential energy curves (PEC) for F-loss dissociations were calculated at the CASPT2//CASSCF level. On the basis of the CASPT2 T0 calculations, we assign the X, A, B, C, and D states of the ion to 1(2)B1, 1(2)A2, 2(2)B1, 1(2)B2, and 1(2)A1, respectively, which supports the suggested assignment of the B state to (2)(2)B1 by Anand et al. based on their experiments. Our CASPT2 Tv and Tv' calculations and our MRCI T0, Tv, and Tv' calculations all indicate that the 2(2)B1 state of C6H5F+ lies below 1(2)B2. By checking the relative energies of the asymptote products and checking the fragmental geometries and the charge and spin density populations in the asymptote products along the CASPT2//CASSCF PECs, we conclude that the 1(2)B1, 1(2)B2, and 1(2)A1 states of C6H5F+ correlate with C6H5+ (1(1)A1) + F (2P) (the first dissociation limit). The energy increases monotonically along the 1(2)B1 PEC, and there are barriers and minima along the 1(2)B2 and 1(2)A1 PECs. The predicted appearance potential value for C6H5+ (1(1)A1) is very close to the average of the experimental values. Our CASPT2//CASSCF PEC calculations have led to the conclusion that the 1(2)A2 state of C6H5F+ correlates with the third dissociation limit of C6H5+ (1(1)A2) + F (2P), and a preliminary discussion is presented.

Dissociation reactions of the vinyl radical in the state

Abstract In the present theoretical work we have explored mechanisms of dissociation reactions of the vinyl radical in the A 2 A ″ state (C2H3 ( A 2 A ″ )) and examined possible pathways for nonadiabatic dissociation of C2H3 ( A 2 A ″ ) into C2H2 ( X 1 Σ g + ). In the calculations we used the complete active space self-consistent field (CASSCF) and multiconfiguration second-order perturbation theory (CASPT2) methods in conjunction with the cc-pVDZ and cc-pVTZ basis sets. Mechanisms for the following three dissociation channels of C2H3 in the A 2 A ″ state were explored: (1) C2H3 ( A 2 A ″ ) → C2H2 (trans, 3 A u ) + H, (2) C2H3 ( A 2 A ″ ) → C2H2 (cis, 3 A 2 ) + H, and (3) C2H3 ( A 2 A ″ ) → H2CC ( 3 A 2 ) + H. The CASSCF and CASPT2 potential energy curve calculations for the C2H3 ( A 2 A ″ ) dissociation channels (1)–(3) indicate that there is neither transition state nor intermediate for each of the channels. At the CASPT2//CASSCF/cc-pVTZ level, the dissociation energies for channels (1)–(3) are predicted to be 84.3, 91.1, and 86.9 kcal/mol, respectively. For a recently observed nonadiabatic dissociation of C2H3 ( A 2 A ″ ) into C2H2 ( X 1 Σ g + ) + H [J. Chem. Phys. 111 (1999) 3783], two previously suggested internal conversion (IC) pathways were examined based on our CASSCF and CASPT2 calculations. Our preliminary CASSCF and CASPT2 calculations indicate that the assumed IC pathway via the twisted C2H3 ( A 2 A ′ ) structure might be feasible. The CASSCF/cc-pVTZ geometry optimization and frequency analysis calculations were performed for the four C2v bridge structures in the 2 B 2 , 2 A 2 , 2 B 1 , and 2 A 1 states along the pathways of the 1 2 A ′ ( X 2 A ′ ), 1 2 A ″ ( A 2 A ″ ), 2 2 A ″ , and 2 2 A ′ states of C2H3, respectively, and the CASPT2//CASSCF/cc-pVTZ energetic results indicate that the assumed IC pathway, via a C2v ( 2 A 2 ) structure and then 2 A 2 / 2 A 1 surface crossing, be not feasible since at their excitation wavelengths (327.4 and 366.2 nm) the C2v ( 2 A 2 ) structure could not be accessed.

Orbital orderings and excitations in ferromagnetic metallic manganites

Orbital orderings in ferromagnetic manganites are studied. We study a model which describes the ferromagnetic metallic manganites at absolute zero by the self-consistent second order perturbation theory (SCSOPT). The orbital wave excitation of the effective Hamiltonian at half-filling in the strong correlation limit is also studied.

Correlation effect of the doped Kondo insulator described on the basis of the periodic Anderson model

We study the periodic Anderson model away from half filling in the framework of self-consistent second-order perturbation theory. The carrier concentration dependence of the density of states spectrum, spin and charge susceptibilities, and magnetization process is systematically investigated. We find that the coefficient $\ensuremath{\gamma}$ of the linear-in-$T$ specific heat and the spin susceptibility ${\ensuremath{\chi}}_{\mathrm{s}}(T)$ at $T=0$ decrease as the carrier concentration decreases near half filling, while $\ensuremath{\gamma}$ and ${\ensuremath{\chi}}_{\mathrm{s}}(T=0)$ themselves are strongly enhanced by the Coulomb interaction. In the temperature dependence of the spin susceptibility and the magnetic-field dependence of the magnetization, it is found that the characteristic energy that corresponds to the hybridization gap appearing at half filling is observed even away from half filling, and another characteristic energy appears in the system with finite doping. Although there are apparently two distinguishable energy scales away from half filling, it is shown that they are related by the Fermi liquid parameter and both are nothing but energy scales that characterize quasiparticles constructed in the renormalized hybridization band.

Theory of Kondo insulators under strong magnetic field

The Kondo insulators are band insulators with strong electron correlation. Their properties are uncovered especially by application of a strong magnetic field. Their electronic structures are described by the dynamical mean-field theory or the self-consistent second-order perturbation theory under the field, and a mechanism for the field-induced insulator-to-metal transition recently discovered for YbB 12 is clarified. Special interest will be placed on the first-order transition found experimentally, the mechanism for which is analyzed by investigating the competition between the Zeeman and the Kondo effects on the energy gap, and by calculating the energy versus magnetization curves. Magnetic anisotropy will be discussed in terms of a single-ion model under crystalline field.

Effect of crystalline electric field splitting in heavy Fermion systems

The excitation spectra in the multi-band periodic Anderson model with the crystalline electric field (CEF) are investigated within the self-consistent second-order perturbation theory. It is shown that the k -dependence of the hybridization, V( k) , reflecting the symmetry of the lowest CEF level, is essential to explain the low-energy properties, even if the CEF splitting cannot be observed by the neutron scattering, such as the Kondo semiconductor, CeNiSn.

Many-body effects on the optical conductivity of the correlated band insulators

The two-band Hubbard model with the density of states obtained from the band calculation for FeSi is investigated. The temperature dependence of the density of states and the optical conductivity are calculated with self-consistent second-order perturbation theory. It is found that the calculated optical conductivity reproduces the experimental result that the gap in the optical conductivity is filled up at a lower temperature than the gap size. It is also found that the peak at the gap edge shifts to the lower-frequency region due to the correlation effect, as is seen in the experiments.