Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Physical Chemistry C

Monte Carlo Explicitly Correlated Second-Order Many-Body Green’s Function Calculations of Semiconductor Band Gaps

J. César Cruz, So Hirata

Published Sep 28, 2024• 2 citations
Publisher

Abstract

A systematically converging series of ab initio, postdensity functional, size-consistent, electron-correlated approximations is desired for predictive computing of electronic band structures of insulating, semiconducting, and metallic solids. A series that meets all of these desiderata (except the applicability to metals) is ab initio many-body Green’s function theory based on Gaussian-type orbital (GTO) basis sets. Here, its leading-order approximation, the second-order Green’s function (GF2) method in the diagonal and frequency-independent approximations with the aug-cc-pVDZ basis set, is applied to the fundamental band gaps of three semiconductors (diamond, silicon, and silicon carbide in the zincblende structure) using cluster models. Corrections are made to the basis set incompleteness errors by the explicit correlation (F12) ansatz (GF2-F12) for the valence band edges. The crystals are modeled as surface-passivated clusters of increasing sizes, whose wave functions are expanded by up to 2709 GTO basis functions. Immense computational costs of these calculations are overcome by the highly scalable stochastic algorithm of the Monte Carlo GF2-F12 method, whose operation cost per state increases only as a cubic power of system size, which has a tiny memory footprint and easily achieves near-perfect parallel efficiency on thousands of CPUs or on hundreds of GPUs. The correlated, F12-corrected highest-occupied and lowest-unoccupied molecular orbital energy (HOMO–LUMO) gap is 5.78 ± 0.07 eV for C 87 H 76 as compared with the experimental value of the fundamental (indirect) band gap of bulk diamond at 5.48 eV. The correlated, F12-corrected HOMO–LUMO gaps for Si 75 H 76 and Si 32 C 43 H 76 are 2.56 ± 0.15 and 3.50 ± 0.12 eV, respectively, which are expected to decrease further with increasing cluster sizes. The experimental fundamental (indirect) band gaps of bulk silicon and silicon carbide are 1.17 and 2.42 eV, respectively.

DOI
10.1021/acs.jpcc.4c03848

Related papers

Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy

Alexander E. Doran, So Hirata

We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O( n 4 ) and O( n 5 ) of molecular size ( n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O( n 5 ) and O( n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.

Journal of Chemical Theory and Computation•Oct 3, 2019•15 citations

Quantum Chemistry in Solid-State Simulations: Gaussian Basis Sets Development and ab-initio Green's Function Based Realistic System Applications

Yanbing Zhou

Understanding emergent many-body phenomena in correlated materials remains one of the grandest challenges in solid-state quantum chemistry calculations. The individuality of electrons diminishes as correlation increases, which leads to the emergence of new features in the system. Seeking to address such problems accurately requires the aid of non-perturbative many-body techniques as well as the corresponding high-quality basis sets. In periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, and an elemental basis set independent of the physical properties of materials may lead to significant simulation inaccuracies. In fact, with the rapid progress of quantum chemistry methods in condensed-phase simulations, the need for a library of reliable Gaussian basis sets explicitly designed for periodic calculations has become urgent. In the meantime, the development of parameter-free, systematic and reliable quantum chemistry methods for simultaneously treating weak and strong correlations in periodic systems has never stopped. Their ability to treat realistic solid materials needs to be tested, and the obtained results can serve as benchmarks for empirical parameter-guided solid-state calculations. In this thesis, we present numerical studies of various solid-state systems, emphasizing the design and optimization of Gaussian basis sets and assessing the band structure properties of the periodic solid systems. The work begins by introducing the theoretical framework of the electronic structure theory and presenting the fundamentals of a fully self-consistent GW approximation and the self-energy embedding theory (SEET) in Chapters 2 and 3. It continues with a description of the Gaussian basis set optimization scheme we devised for solid-state quantum chemistry calculations. The scheme we present is designed to avoid a lack of material specificity within a given basis set by simultaneously minimizing the total energy of the system and optimizing the band energies when compared to the reference plane wave calculation while accounting for the overlap matrix condition number. We compare the quality of the Gaussian basis sets generated via our method against the existing basis sets. And the optimization scheme is tested to yield improved results. We finally present a quantitative study of electronic properties of realistic materials with the treatment of Green's function based weakly correlated self-consistent GW approximation and the strongly correlated self-energy embedding theory. In the transition metal oxide BiVO3, we found the inaccurate illustration of orbitals including t_{2g} in transition metal Vanadium by DFT. Our calculation shows a systematic trend of DFT failure in illustrating both insulating and metallic solutions, demonstrating the dangers of using DFT when the experimental data is scarce.

•Jan 1, 2022•0 citations

Ab Initio Many-Body Theory for Solids Based on the Green's Function Formalism

Chia-Nan Yeh

Electronic structure of materials from first principles is essential for one-to-one comparisons between theories and experiments. However, the many-body effects resulting from electron correlation and their interplay with other degrees of freedom pose a great challenge to design an ab initio approximate method with good accuracy and efficiency at the same time. In this thesis, we present our recent developments of ab initio many-body theory for solids based on the finite-temperature Green’s function formalism. We begin by reviewing the basic concepts and definitions of realistic electronic Hamiltonians (Ch. 2) as well as Green’s functions at finite temperature (Ch. 3). Overviews of self-consistent many-body perturbation theories and quantum embedding theories are then presented in Chs. 4 and 5 as complementary materials for the following chapters. In Ch. 6, we present algorithms and implementation details for the finite-temperature self-consistent GW (scGW) method in Gaussian-type orbitals for solids. Our implementation is based on the finite-temperature Green’s function formalism in which all equations are solved on the imaginary axis, without resorting to analytical continuation during the self-consistency. No quasiparticle approximation is employed and all matrix elements of self-energies are explicitly evaluated. We show agreement with other, differently formulated finite-temperature scGW implementations. By migrating computationally intensive calculations to GPUs, we obtain scalable results on large supercomputers with nearly optimal performance. Inclusion of relativistic effects in scGW is discussed in Ch. 7. Our formulation of relativistic scGW is based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and their interplay with electron correlation without adjustable parameters. We examine the effect of the X2C1e by comparison to the established four-component formalism and reach excellent agreement. The simplicity of X2C1e enables construction of higher order theories, such as embedding theories, on top of perturbative calculations. In Ch. 8, we use self-energy embedding theory (SEET) to study the spectra of the prototypical undistorted cubic perovskites SrVO3 and SrMnO3. In the strongly correlated metallic SrVO3, we find that the usual attribution of the satellite peaks at -1.8 eV to Hund or Hubbard physics in the t2g orbitals is inconsistent with our calculations. In the strongly correlated insulator SrMnO3 we recover insulating behavior due to a feedback effect between the strongly correlated orbitals and the weakly correlated environment. Our calculation shows a systematic convergence of spectral features as the space of strongly correlated orbitals is enlarged, paving the way to a systematic parameter-free study of correlated perovskites. Lastly, in Ch. 9, we apply the Green’s function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems in strongly correlated solids within the framework of SEET. We describe the implementation details of our Green’s function coupled cluster (GFCC) solver, investigate its performance, and highlight potential advantages and problems on examples of impurities created for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems.

•Jan 1, 2022•0 citations

Periodic Coupled-Cluster Green’s Function for Photoemission Spectra of Realistic Solids

Katelyn Laughon, Jason M. Yu, Tianyu Zhu

We present an efficient implementation of the coupled-cluster Green’s function (CCGF) method for simulating photoemission spectra of periodic systems. We formulate the periodic CCGF approach with Brillouin zone sampling in the Gaussian basis at the coupled-cluster singles and doubles (CCSD) level. To enable CCGF calculations of realistic solids, we propose an active-space self-energy correction scheme by combining CCGF with the cheaper many-body perturbation theory ( GW ) and implement the model order reduction (MOR) frequency interpolation technique. We find that the active-space self-energy correction and MOR techniques significantly reduce the computational cost of CCGF while maintaining the high accuracy. We apply the developed CCGF approaches to compute spectral properties and band structure of silicon (Si) and zinc oxide (ZnO) crystals using triple-ζ Gaussian basis sets and medium-size k -point sampling and find good agreement with experimental measurements.

arXiv:2208.07478•The Journal of Physical Chemistry Letters•Sep 26, 2022•15 citations

Excited state calculations in solids by auxiliary-field quantum Monte Carlo

Fengjie Ma, Shiwei Zhang, Henry Krakauer

We present an approach for ab initio many-body calculations of excited states in solids. Using auxiliary-field quantum Monte Carlo, we introduce an orthogonalization constraint with virtual orbitals to prevent collapse of the stochastic Slater determinants in the imaginary-time propagation. Trial wave functions from density-functional calculations are used for the constraints. Detailed band structures can be calculated. Results for standard semiconductors are in good agreement with experiments; comparisons are also made with GW calculations and the connections and differences are discussed. For the challenging ZnO wurtzite structure, we obtain a fundamental band gap of 3.26(16) eV, consistent with experiments.

arXiv:1211.4635•New Journal of Physics•Sep 13, 2013•52 citations

Stochastic Self-Consistent Second-Order Green’s Function Method for Correlation Energies of Large Electronic Systems

Daniel Neuhauser, Roi Baer, Dominika Zgid

The second-order Matsubara Green's function method (GF2) is a robust temperature-dependent quantum chemistry approach, extending beyond the random-phase approximation. However, until now the scope of GF2 applications was quite limited as they require computer resources that rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: estimating of the self-energy from the previous step's Green's function and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size, while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2), which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains with up to 1000 electrons, showing that the approach is numerically stable, efficient, and accurate. The stochastic errors are very small, on the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling; hence we obtain an extremely fast stochastic MP2 (sMP2) method as a byproduct. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.

arXiv:1707.08296•Journal of Chemical Theory and Computation•Sep 29, 2017•54 citations
A systematically converging series of ab initio, postdensity functional, size-consistent, electron-correlated approximations is desired for predictive computing of electronic band structures of insulating, semiconducting, and metallic solids. A series that meets all of these desiderata (except the applicability to metals) is ab initio many-body Green’s function theory based on Gaussian-type orbital (GTO) basis sets. Here, its leading-order approximation, the second-order Green’s function (GF2) method in the diagonal and frequency-independent approximations with the aug-cc-pVDZ basis set, is applied to the fundamental band gaps of three semiconductors (diamond, silicon, and silicon carbide in the zincblende structure) using cluster models. Corrections are made to the basis set incompleteness errors by the explicit correlation (F12) ansatz (GF2-F12) for the valence band edges. The crystals are modeled as surface-passivated clusters of increasing sizes, whose wave functions are expanded by up to 2709 GTO basis functions. Immense computational costs of these calculations are overcome by the highly scalable stochastic algorithm of the Monte Carlo GF2-F12 method, whose operation cost per state increases only as a cubic power of system size, which has a tiny memory footprint and easily achieves near-perfect parallel efficiency on thousands of CPUs or on hundreds of GPUs. The correlated, F12-corrected highest-occupied and lowest-unoccupied molecular orbital energy (HOMO–LUMO) gap is 5.78 ± 0.07 eV for C 87 H 76 as compared with the experimental value of the fundamental (indirect) band gap of bulk diamond at 5.48 eV. The correlated, F12-corrected HOMO–LUMO gaps for Si 75 H 76 and Si 32 C 43 H 76 are 2.56 ± 0.15 and 3.50 ± 0.12 eV, respectively, which are expected to decrease further with increasing cluster sizes. The experimental fundamental (indirect) band gaps of bulk silicon and silicon carbide are 1.17 and 2.42 eV, respectively.
We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O( n 4 ) and O( n 5 ) of molecular size ( n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O( n 5 ) and O( n 6 ) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C 60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. They display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.
Understanding emergent many-body phenomena in correlated materials remains one of the grandest challenges in solid-state quantum chemistry calculations. The individuality of electrons diminishes as correlation increases, which leads to the emergence of new features in the system. Seeking to address such problems accurately requires the aid of non-perturbative many-body techniques as well as the corresponding high-quality basis sets. In periodic systems, a given element may be present in different spatial arrangements displaying vastly different physical and chemical properties, and an elemental basis set independent of the physical properties of materials may lead to significant simulation inaccuracies. In fact, with the rapid progress of quantum chemistry methods in condensed-phase simulations, the need for a library of reliable Gaussian basis sets explicitly designed for periodic calculations has become urgent. In the meantime, the development of parameter-free, systematic and reliable quantum chemistry methods for simultaneously treating weak and strong correlations in periodic systems has never stopped. Their ability to treat realistic solid materials needs to be tested, and the obtained results can serve as benchmarks for empirical parameter-guided solid-state calculations. In this thesis, we present numerical studies of various solid-state systems, emphasizing the design and optimization of Gaussian basis sets and assessing the band structure properties of the periodic solid systems. The work begins by introducing the theoretical framework of the electronic structure theory and presenting the fundamentals of a fully self-consistent GW approximation and the self-energy embedding theory (SEET) in Chapters 2 and 3. It continues with a description of the Gaussian basis set optimization scheme we devised for solid-state quantum chemistry calculations. The scheme we present is designed to avoid a lack of material specificity within a given basis set by simultaneously minimizing the total energy of the system and optimizing the band energies when compared to the reference plane wave calculation while accounting for the overlap matrix condition number. We compare the quality of the Gaussian basis sets generated via our method against the existing basis sets. And the optimization scheme is tested to yield improved results. We finally present a quantitative study of electronic properties of realistic materials with the treatment of Green's function based weakly correlated self-consistent GW approximation and the strongly correlated self-energy embedding theory. In the transition metal oxide BiVO3, we found the inaccurate illustration of orbitals including t_{2g} in transition metal Vanadium by DFT. Our calculation shows a systematic trend of DFT failure in illustrating both insulating and metallic solutions, demonstrating the dangers of using DFT when the experimental data is scarce.
Electronic structure of materials from first principles is essential for one-to-one comparisons between theories and experiments. However, the many-body effects resulting from electron correlation and their interplay with other degrees of freedom pose a great challenge to design an ab initio approximate method with good accuracy and efficiency at the same time. In this thesis, we present our recent developments of ab initio many-body theory for solids based on the finite-temperature Green’s function formalism. We begin by reviewing the basic concepts and definitions of realistic electronic Hamiltonians (Ch. 2) as well as Green’s functions at finite temperature (Ch. 3). Overviews of self-consistent many-body perturbation theories and quantum embedding theories are then presented in Chs. 4 and 5 as complementary materials for the following chapters. In Ch. 6, we present algorithms and implementation details for the finite-temperature self-consistent GW (scGW) method in Gaussian-type orbitals for solids. Our implementation is based on the finite-temperature Green’s function formalism in which all equations are solved on the imaginary axis, without resorting to analytical continuation during the self-consistency. No quasiparticle approximation is employed and all matrix elements of self-energies are explicitly evaluated. We show agreement with other, differently formulated finite-temperature scGW implementations. By migrating computationally intensive calculations to GPUs, we obtain scalable results on large supercomputers with nearly optimal performance. Inclusion of relativistic effects in scGW is discussed in Ch. 7. Our formulation of relativistic scGW is based on the exact two-component formalism with one-electron approximation (X2C1e) and non-relativistic Coulomb interactions. Our theory allows us to study scalar relativistic effects, spin-orbit coupling, and their interplay with electron correlation without adjustable parameters. We examine the effect of the X2C1e by comparison to the established four-component formalism and reach excellent agreement. The simplicity of X2C1e enables construction of higher order theories, such as embedding theories, on top of perturbative calculations. In Ch. 8, we use self-energy embedding theory (SEET) to study the spectra of the prototypical undistorted cubic perovskites SrVO3 and SrMnO3. In the strongly correlated metallic SrVO3, we find that the usual attribution of the satellite peaks at -1.8 eV to Hund or Hubbard physics in the t2g orbitals is inconsistent with our calculations. In the strongly correlated insulator SrMnO3 we recover insulating behavior due to a feedback effect between the strongly correlated orbitals and the weakly correlated environment. Our calculation shows a systematic convergence of spectral features as the space of strongly correlated orbitals is enlarged, paving the way to a systematic parameter-free study of correlated perovskites. Lastly, in Ch. 9, we apply the Green’s function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems in strongly correlated solids within the framework of SEET. We describe the implementation details of our Green’s function coupled cluster (GFCC) solver, investigate its performance, and highlight potential advantages and problems on examples of impurities created for antiferromagnetic MnO and paramagnetic SrMnO3. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable by existing accurate impurity solvers such as exact diagonalization. However, when correlations become strong, the singles and doubles approximation used in GFCC could lead to instabilities in searching for the particle number present in impurity problems.
We present an efficient implementation of the coupled-cluster Green’s function (CCGF) method for simulating photoemission spectra of periodic systems. We formulate the periodic CCGF approach with Brillouin zone sampling in the Gaussian basis at the coupled-cluster singles and doubles (CCSD) level. To enable CCGF calculations of realistic solids, we propose an active-space self-energy correction scheme by combining CCGF with the cheaper many-body perturbation theory ( GW ) and implement the model order reduction (MOR) frequency interpolation technique. We find that the active-space self-energy correction and MOR techniques significantly reduce the computational cost of CCGF while maintaining the high accuracy. We apply the developed CCGF approaches to compute spectral properties and band structure of silicon (Si) and zinc oxide (ZnO) crystals using triple-ζ Gaussian basis sets and medium-size k -point sampling and find good agreement with experimental measurements.
We present an approach for ab initio many-body calculations of excited states in solids. Using auxiliary-field quantum Monte Carlo, we introduce an orthogonalization constraint with virtual orbitals to prevent collapse of the stochastic Slater determinants in the imaginary-time propagation. Trial wave functions from density-functional calculations are used for the constraints. Detailed band structures can be calculated. Results for standard semiconductors are in good agreement with experiments; comparisons are also made with GW calculations and the connections and differences are discussed. For the challenging ZnO wurtzite structure, we obtain a fundamental band gap of 3.26(16) eV, consistent with experiments.
The second-order Matsubara Green's function method (GF2) is a robust temperature-dependent quantum chemistry approach, extending beyond the random-phase approximation. However, until now the scope of GF2 applications was quite limited as they require computer resources that rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: estimating of the self-energy from the previous step's Green's function and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size, while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2), which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains with up to 1000 electrons, showing that the approach is numerically stable, efficient, and accurate. The stochastic errors are very small, on the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling; hence we obtain an extremely fast stochastic MP2 (sMP2) method as a byproduct. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures.