arXiv subjects

So Hirata

Publications and source records attributed to So Hirata.

At least 19 recordsLinked to original sources

arXiv2026

Degenerate coupled-cluster theory

A size-extensive, converging, black-box, ab initio coupled-cluster ($\Delta$CC) ansatz is introduced that computes the energies and wave functions of states from any degenerate or nondegenerate Slater-determinant references with any numbers of $\alpha$- and $\beta$-spin electrons, any patterns of orbital occupancy, any spin multiplicities, and any spatial symmetries. For a nondegenerate reference, it reduces to the single-reference coupled-cluster ansatz. For a degenerate multireference, it is a natural coupled-cluster extension of degenerate Moeller-Plesset perturbation ($\Delta$MP) theory. For ionized and electron-attached references, it is a coupled-cluster Green's function, although the present theory is convergent toward the full-configuration-interaction (FCI) limits, while Feynman-Dyson many-body Green's function (MBGF) theory generally is not. Its single-excitation instance is a projection Hartree-Fock theory as per the Thouless theorem, which may be useful for core ionizations, high-spin states, and possibly electron affinities. A new multireference coupled-cluster theory for a general model space is also developed. This quasidegenerate coupled-cluster (QCC) theory is exactly converging, but not black-box, and intended for strong correlation. Determinant-based, general-order algorithms of $\Delta$CC and QCC theories are implemented and compared with configuration-interaction (CI) and equation-of-motion coupled-cluster (EOM-CC) theories through octuple excitations and with $\Delta$MP and MBGF theories up to the nineteenth order. An algebraic, optimal-scaling algorithm of $\Delta$CC theory is computer-synthesized at the levels of single excitations ($\Delta$CCS) and of single and double excitations ($\Delta$CCSD). The order of performance is: QCC $\approx$ $\Delta$CC > EOM-CC > CI at the same order or QCC $\approx$ $\Delta$CC > $\Delta$MP > MBGF at the same cost scaling.

physics.chem-ph↗

arXiv2025

MO-HEOM: Extending Hierarchical Equations of Motion to Molecular Orbital Space

Studies of quantum thermal effects on molecular excitation dynamics have often relied on oversimplified models, such as energy eigenstates or low-dimensional potentials, which fail to capture the complexity of real chemical systems. In reality, molecules are spatially extended and embedded in anisotropic environments, where molecular orbitals (MOs) play a central role in determining quantum behavior. To advance beyond these limitations, we propose a three-dimensional rotationally invariant system-bath (3D-RISB) model within the MO framework, with explicit inclusion of intramolecular vibrational motion. From this MO foundation, we derive numerically ``exact'' hierarchical equations of motion (MO-HEOM). As a demonstration, we analyze hydrogen molecules and hydrogen molecular ions with vibrational degrees of freedom, revealing their linear absorption spectra.

physics.chem-ph↗

arXiv2024

Thermal quasi-particle theory

The widely used thermal Hartree-Fock (HF) theory is generalized to include the effect of electron correlation while maintaining its quasi-independent-particle framework. An electron-correlated internal energy (or grand potential) is postulated in consultation with the second-order finite-temperature many-body perturbation theory (MBPT), which then dictates the corresponding thermal orbital (quasi-particle) energies in such a way that all fundamental thermodynamic relations are obeyed. The associated density matrix is of a one-electron type, whose diagonal elements take the form of the Fermi-Dirac distribution functions, when the grand potential is minimized. The formulas for the entropy and chemical potential are unchanged from those of Fermi-Dirac or thermal HF theory. The theory thus constitutes a finite-temperature extension of the second-order Dyson self-energy of one-particle many-body Green's function theory and can be viewed as a second-order, diagonal, frequency-independent, thermal inverse Dyson equation. At low temperature, the theory approaches finite-temperature MBPT of the same order, but it may outperform the latter at intermediate temperature by including additional electron-correlation effects through orbital energies. A physical meaning of these thermal orbital energies is proposed (encompassing that of thermal HF orbital energies, which has been elusive) as a finite-temperature version of Janak's theorem.

cond-mat.stat-mech↗

arXiv2024

Graph-theoretical estimates of the diameters of the Rubik's Cube groups

A strict lower bound for the diameter of a symmetric graph is proposed, which is calculable with the order $n$ and other local parameters of the graph such as the degree $k\,(\geq 3)$, even girth $g\,(\geq 4)$, and number of $g$-cycles traversing a vertex, which are easily determined by inspecting a small portion of the graph (unless the girth is large). It is applied to the symmetric Cayley graphs of some Rubik's Cube groups of various sizes and metrics, yielding slightly tighter lower bounds of the diameters than those for random $k$-regular graphs proposed by Bollob\'{a}s and de la Vega. They range from 60% to 77% of the correct diameters of large-$n$ graphs.

math.CO↗

arXiv2024

Thermal mean-field theories

Several closely related ab initio thermal mean-field theories for fermions, both well-established and new ones, are compared with one another at the formalism level and numerically. The theories considered are Fermi-Dirac theory, thermal Hartree-Fock (HF) theory, two modifications of the thermal single-determinant approximation of Kaplan and Argyres, and first-order finite-temperature many-body perturbation theory based on zero-temperature or thermal HF reference. The thermal full-configuration-interaction theory is used as the benchmark.

cond-mat.stat-mech↗

arXiv2024

Probabilistic estimates of the diameters of the Rubik's Cube groups

The diameter of the Cayley graph of the Rubik's Cube group is the fewest number of turns needed to solve the Cube from the hardest initial configuration. For the 2$\times$2$\times$2 Cube, the diameter is 11 in the half-turn metric, 14 in the quarter-turn metric, 19 in the semi-quarter-turn metric, and 10 in the bi-quarter-turn metric. For the 3$\times$3$\times$3 Cube, the diameter was determined by Rokicki et al. to be 20 in the half-turn metric and 26 in the quarter-turn metric. This study shows that a modified version of the coupon collector's problem in probability theory can predict the diameters correctly for both 2$\times$2$\times$2 and 3$\times$3$\times$3 Cubes insofar as the quarter-turn metric is adopted. In the half-turn metric, the diameters are overestimated by one and two, respectively, for the 2$\times$2$\times$2 and 3$\times$3$\times$3 Cubes, whereas for the 2$\times$2$\times$2 Cube in the semi-quarter-turn and bi-quarter-turn metrics, they are overestimated by two and underestimated by one, respectively. Invoking the same probabilistic logic, the diameters of the 4$\times$4$\times$4 and 5$\times$5$\times$5 Cubes are predicted to be 48 (41) and 68 (58) in the quarter-turn (half-turn) metric, whose precise determinations are far beyond reach of classical supercomputing. The probabilistically estimated diameter is shown to obey the approximate formula of $\ln N / \ln r + \ln N / r$, where $N$ is the number of configurations and $r$ is the branching ratio.

cs.DM↗

arXiv2023

Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators

Using a general-order ab initio many-body Green's function method, we numerically illustrate several pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron Feynman propagators. (i) The perturbation expansion of the frequency-dependent self-energy is not convergent at the exact self-energy in many frequency domains. (ii) An odd-perturbation-order self-energy has a qualitatively wrong shape and, as a result, many roots of the corresponding Dyson equation are nonphysical in that the poles may be complex or residues can exceed unity or be negative. (iii) A higher even-order self-energy consists of vertical lines at many frequencies, predicting numerous phantom poles with zero residues. (iv) Infinite partial resummations of diagrams by vertex or edge renormalization tend to exacerbate these pathologies. (v) The nonconvergence is caused by the nonanalyticity of the rational-function form of the exact Green's function at many frequencies, where the radius of convergence of its Taylor expansion is zero. This is consistent with the fact that (vi) Pad\'{e} approximants (power-series expansions of a rational function) can largely restore the correct shape and poles of the Green's function. Nevertheless, not only does the nonconvergence render higher-order Feynman-Dyson diagrammatic perturbation theory useless for many lower-lying ionization or higher-lying electron-attachment states, but it also calls into question the validity of its combined use with the ans\"{a}tze requiring the knowledge of all poles and residues. Such ans\"{a}tze include the Galitskii-Migdal identity, the self-consistent Green's function methods, and some models of the algebraic diagrammatic construction.

quant-ph↗

arXiv2023

Finite-temperature many-body perturbation theory for vibrations: Recursions, algebraic reduction, second-quantized reduction, diagrammatic rules, linked-diagram theorem, finite-temperature self-consistent field, and general-order algorithm

A unified theory is presented for finite-temperature many-body perturbation expansions of the anharmonic vibrational contributions to thermodynamic functions: the free energy, internal energy, and entropy. The theory is diagrammatically size-consistent at any order, as ensured by the linked-diagram theorem proved here, and thus applicable to molecular gases and solids on an equal footing. It is also a basis-set-free formalism, just like its underlying Bose-Einstein theory, capable of summing anharmonic effects over an infinite number of states analytically. It is formulated by the Rayleigh-Schrodinger-style recursions, generating sum-over-states formulas for the perturbation series, which unambiguously converges at the finite-temperature vibrational full-configuration-interaction limits. Two strategies are introduced to reducing these sum-over-states formulas into compact sum-over-modes analytical formulas. One is a purely algebraic method that factorizes each many-mode thermal average into a product of one-mode thermal averages, which are then evaluated by the thermal Born-Huang rules. Canonical forms of these rules are proposed, dramatically expediting the reduction process. The other is finite-temperature normal-ordered second quantization, which is fully developed in this study, including a proof of thermal Wick's theorem and the derivation of a normal-ordered vibrational Hamiltonian at finite temperature. The latter naturally defines a finite-temperature extension of size-extensive vibrational self-consistent field theory. These reduced formulas can be represented graphically as Feynman diagrams with resolvent lines, which include anomalous and renormalization diagrams. Two order-by-order and one general-order algorithms of computing these perturbation corrections are implemented and applied up to the eighth order. The results show no signs of Kohn-Luttinger-type nonconvergence.

cond-mat.stat-mech↗

arXiv2022

Nonvanishing quadrature derivatives in the analytical gradients of density functional energies in crystals and helices

It is shown that the quadrature derivatives in some analytical gradients of energies evaluated with a multi-centre radial-angular grid do not vanish even in the limit of an infinitely dense grid, causing severe errors when neglected. The gradients in question are those with respect to a lattice constant of a crystal or to the helical angle of a chain with screw axis symmetry. This is in contrast with the quadrature derivatives in atomic gradients, which can be made arbitrarily small by grid extension. The disparate behaviour is traced to whether the grid points depend on the coordinate with respect to which the derivative of energy is taken. Whereas the nonvanishing quadrature derivative in the lattice-constant gradient is identified as the surface integral arising from an expanding integration domain, the analytical origin of the nonvanishing quadrature derivative in the helical-angle gradient remains unknown.

cond-mat.mtrl-sci↗

arXiv2022

Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method

A second-order many-body perturbation correction to the relativistic Dirac-Hartree-Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-CDF (cumulative distribution function) method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vast improvement over the quintic scaling of the conventional, deterministic second-order many-body perturbation method. The algorithm is also easily and efficiently parallelized with demonstrated 92% strong scalability going from 64 to 4096 processors for a fixed job size.

quant-ph↗

arXiv2022

General solution to the Kohn-Luttinger nonconvergence problem

A simple, but general solution is proposed for the Kohn-Luttinger problem, i.e., the nonconvergence of the finite-temperature many-body perturbation theory with its zero-temperature counterpart as temperature is lowered to zero under some circumstances. How this nonconvergence can be avoided by altering the reference wave function is illustrated numerically by using up to the fifth order of the perturbation theory.

cond-mat.stat-mech↗

arXiv2021

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles

A comprehensive and detailed account is presented for the finite-temperature many-body perturbation theory for electrons that expands in power series all thermodynamic functions on an equal footing. Algebraic recursions in the style of the Rayleigh-Schrödinger perturbation theory are derived for the grand potential, chemical potential, internal energy, and entropy in the grand canonical ensemble and for the Helmholtz energy, internal energy, and entropy in the canonical ensemble, leading to their sum-over-states analytical formulas at any arbitrary order. For the grand canonical ensemble, these sum-over-states formulas are systematically transformed to sum-over-orbitals reduced analytical formulas by the quantum-field-theoretical techniques of normal-ordered second quantization and Feynman diagrams extended to finite temperature. It is found that the perturbation corrections to energies entering the recursions have to be treated as a nondiagonal matrix, whose off-diagonal elements are generally nonzero within a subspace spanned by degenerate Slater determinants. They give rise to a unique set of linked diagrams -- renormalization diagrams -- whose resolvent lines are displaced downwards, which are distinct from the well-known anomalous diagrams of which one or more resolvent lines are erased. A linked-diagram theorem is introduced that proves the size-consistency of the finite-temperature many-body perturbation theory at any order. General-order algorithms implementing the recursions establish the convergence of the perturbation series towards the finite-temperature full-configuration-interaction limit unless the series diverges. Normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree--Fock and first-order many-body perturbation theories.

cond-mat.stat-mech↗

arXiv2021

Low-temperature breakdown of many-body perturbation theory for thermodynamics

It is shown analytically and numerically that the finite-temperature many-body perturbation theory in the grand canonical ensemble has zero radius of convergence at zero temperature when the energy ordering or degree of degeneracy for the ground state changes with the perturbation strength. When the degeneracy of the reference state is partially or fully lifted at the first-order Hirschfelder-Certain degenerate perturbation theory, the grand potential and internal energy diverge as $T \to 0$. Contrary to earlier suggestions of renormalizability by the chemical potential $μ$, this nonconvergence, first suspected by W. Kohn and J. M. Luttinger, is caused by the nonanalytic nature of the Boltzmann factor $e^{-E/k_\text{B}T}$ at $T=0$, also plaguing the canonical ensemble, which does not involve $μ$. The finding reveals a fundamental flaw in perturbation theory, which is deeply rooted in the mathematical limitation of power-series expansions and is unlikely to be removed within its framework.

physics.chem-ph↗

arXiv2020

Finite-temperature many-body perturbation theory in the grand canonical ensemble

A finite-temperature many-body perturbation theory is presented that expands in power series the electronic grand potential, chemical potential, internal energy, and entropy on an equal footing. Sum-over-states and sum-over-orbitals analytical formulas for the second-order perturbation corrections to these thermodynamic properties are obtained in a time-independent, nondiagrammatic, algebraic derivation, relying on the sum rules of the Hirschfelder-Certain degenerate perturbation energies in a degenerate subspace as well as nine algebraic identities for zeroth-order thermal averages of one- through four-indexed quantities and of products thereof. They reproduce numerically exactly the benchmark data obtained as the numerical derivatives of the thermal-full-configuration-interaction results for a wide range of temperature.

physics.chem-ph↗

arXiv2019

Finite-Temperature Many-Body Perturbation Theory in the Canonical Ensemble

Benchmark data are presented for the zeroth- through third-order many-body perturbation corrections to the electronic Helmholtz energy, internal energy, and entropy in the canonical ensemble in a wide range of temperature. They are determined as numerical $λ$-derivatives of the respective quantities computed by thermal full configuration interaction with a perturbation-scaled Hamiltonian, $\hat{H}=\hat{H}_0+λ\hat{V}$. Sum-over-states analytical formulas for up to the third-order corrections to these properties are also derived as analytical $λ$-derivatives. These formulas, which are verified by exact numerical agreement with the benchmark data, are given in terms of the Hirschfelder-Certain degenerate perturbation energies and should be valid for both degenerate and nondegenerate reference states at any temperature down to zero. The results in the canonical ensemble are compared with the same in the grand canonical ensemble.

physics.chem-ph↗

arXiv2019

Scalable approximation of Green's function for estimation of anharmonic energy corrections

A method based on separated integration to estimate anharmonic corrections to energy and vibration of molecules in a second-order diagrammatic vibrational many-body Green's function formalism has already been presented. A severe bottleneck in extending this approach to bigger molecules is that the storage of the Green's function scales exponentially with the number of atoms in the molecule. In this article, we present a method that overcomes this limitation by approximating the Green's function in the Hierarchical Tucker tensor format. We illustrate that the storage cost is linear in dimension and hence one can obtain accurate representations of the Green's function for a molecule of any size. Application of this method to estimate the second-order correction to energy of molecules illustrates the advantage of this approach.

physics.chem-ph↗

arXiv2019

Numerical Evidence Invalidating Finite-Temperature Many-Body Perturbation Theory

Low-order perturbation corrections to the electronic grand potential, internal energy, chemical potential, and entropy of a gas of noninteracting, identical molecules at a nonzero temperature are determined numerically as the $λ$-derivatives of the respective quantity calculated exactly (by thermal full configuration interaction) with a perturbation-scaled Hamiltonian, $\hat{H}_0 + λ\hat{V}$. The data thus obtained from the core definition of any perturbation theory serve as a benchmark against which analytical formulas can be validated. The first- and second-order corrections from finite-temperature many-body perturbation theory disagree with these benchmark data. This is because the theory neglects the variation of chemical potential with $λ$, thereby failing to converge at the exact, full-interaction ($λ=1$) limit, unless the exact chemical potential is known in advance. The renormalized finite-temperature perturbation theory [S. Hirata and X. He, J. Chem. Phys., 138, 204112 (2013)] is also found to be incorrect.

physics.chem-ph↗

arXiv2019

Converging finite-temperature many-body perturbation theory that conserves the average number of electrons

A finite-temperature perturbation theory for the grand canonical ensemble is introduced that expands chemical potential in a perturbation series and conserves the average number of electrons, ensuring charge neutrality of the system at each perturbation order. Two classes of (sum-over-state and reduced) analytical formulas are obtained in a straightforward, algebraic, time-independent derivation for the first-order corrections to chemical potential, grand potential, and internal energy, with the aid of several identities of the Boltzmann sums also introduced in this study. These formulas are numerically verified against benchmark data from thermal full configuration interaction. In the zero-temperature limit, the finite-temperature perturbation theory reduces analytically to and is consistent with the Møller-Plesset perturbation theory, but only for a nondegenerate ground state. For a degenerate ground state, its correct zero-temperature limit is not the Møller-Plesset perturbation theory, but the Hirschfelder-Certain degenerate perturbation theory.

physics.chem-ph↗