Finite-temperature many-body perturbation theory for anharmonic 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, i.e., the free energy, internal energy, and entropy. The theory is diagrammatically size-consistent at any order, as ensured by the linked-diagram theorem proved in this study, 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-Schrödinger-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 reduce 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.

ω3 fatty acid metabolite, 12-hydroxyeicosapentaenoic acid, alleviates contact hypersensitivity by downregulation of CXCL1 and CXCL2 gene expression in keratinocytes via retinoid X receptor α.

ω3 fatty acids show potent bioactivities via conversion into lipid mediators; therefore, metabolism of dietary lipids is a critical determinant in the properties of ω3 fatty acids in the control of allergic inflammatory diseases. However, metabolic progression of ω3 fatty acids in the skin and their roles in the regulation of skin inflammation remains to be clarified. In this study, we found that 12-hydroxyeicosapentaenoic acid (12-HEPE), which is a 12-lipoxygenase metabolite of eicosapentaenoic acid, was the prominent metabolite accumulated in the skin of mice fed ω3 fatty acid-rich linseed oil. Consistently, the gene expression levels of Alox12 and Alox12b, which encode proteins involved in the generation of 12-HEPE, were much higher in the skin than in the other tissues (eg, gut). We also found that the topical application of 12-HEPE inhibited the inflammation associated with contact hypersensitivity by inhibiting neutrophil infiltration into the skin. In human keratinocytes in vitro, 12-HEPE inhibited the expression of two genes encoding neutrophil chemoattractants, CXCL1 and CXCL2, via retinoid X receptor α. Together, the present results demonstrate that the metabolic progression of dietary ω3 fatty acids differs in different organs, and identify 12-HEPE as the dominant ω3 fatty acid metabolite in the skin.

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-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 92% strong scalability going from 64 to 4096 processors.

Monte Carlo MP2-F12 for Noncovalent Interactions: The C60 Dimer.

A scalable stochastic algorithm is presented that can evaluate explicitly correlated (F12) second-order many-body perturbation (MP2) energies of weak, noncovalent, intermolecular interactions. It first transforms the formulas of the MP2 and F12 energy differences into a short sum of high-dimensional integrals of Green's functions in real space and imaginary time. These integrals are then evaluated by the Monte Carlo method augmented by parallel execution, redundant-walker convergence acceleration, direct-sampling autocorrelation elimination, and control-variate error reduction. By sharing electron-pair walkers across the supermolecule and its subsystems spanned by the joint basis set, the statistical uncertainty is reduced by one to 2 orders of magnitude in the MP2 binding energy corrected for the basis-set incompleteness and superposition errors. The method predicts the MP2-F12/aug-cc-pVDZ binding energy of 19.1 ± 4.0 kcal mol-1 for the C60 dimer at the center distance of 9.748 Å.

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, and 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 upward, 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 toward the finite-temperature full-configuration-interaction limit unless the series diverges. The normal-ordered Hamiltonian at finite temperature sheds light on the relationship between the finite-temperature Hartree-Fock and first-order many-body perturbation theories.

Stochastic evaluation of fourth-order many-body perturbation energies.

A scalable, stochastic algorithm evaluating the fourth-order many-body perturbation (MP4) correction to energy is proposed. Three hundred Goldstone diagrams representing the MP4 correction are computer generated and then converted into algebraic formulas expressed in terms of Green's functions in real space and imaginary time. They are evaluated by the direct (i.e., non-Markov, non-Metropolis) Monte Carlo (MC) integration accelerated by the redundant-walker and control-variate algorithms. The resulting MC-MP4 method is efficiently parallelized and is shown to display O(n5.3) size-dependence of cost, which is nearly two ranks lower than the O(n7) dependence of the deterministic MP4 algorithm. It evaluates the MP4/aug-cc-pVDZ energy for benzene, naphthalene, phenanthrene, and corannulene with the statistical uncertainty of 10 mEh (1.1% of the total basis-set correlation energy), 38 mEh (2.6%), 110 mEh (5.5%), and 280 mEh (9.0%), respectively, after about 109 MC steps.

Maternal ω3 docosapentaenoic acid inhibits infant allergic dermatitis through TRAIL-expressing plasmacytoid dendritic cells in mice.

BACKGROUND: Maternal dietary exposures are considered to influence the development of infant allergies through changes in the composition of breast milk. Cohort studies have shown that ω3 polyunsaturated fatty acids (PUFAs) in breast milk may have a beneficial effect on the preventing of allergies in infants; however, the underlying mechanisms remain to be investigated. We investigated how the maternal intake of dietary ω3 PUFAs affects fatty acid profiles in the breast milk and their pups and reduced the incidence of allergic diseases in the pups. METHODS: Contact hypersensitivity (CHS) induced by 2,4-dinitrofluorobenzene (DNFB) and fluorescein isothiocyanate was applied to the skin in pups reared by mother maintained with diets mainly containing ω3 or ω6 PUFAs. Skin inflammation, immune cell populations, and expression levels of immunomodulatory molecules in pups and/or human cell line were investigated by using flow cytometric, immunohistologic, and quantitative RT-PCR analyses. ω3 PUFA metabolites in breast milk and infant's serum were evaluated by lipidomics analysis using LC-MS/MS. RESULTS: We show that maternal intake of linseed oil, containing abundant ω3 α-linolenic acid, resulted in the increased levels of ω3 docosapentaenoic acid (DPA) and its 14-lipoxygenation products in the breast milk of mouse dams; these metabolites increased the expression of TNF-related apoptosis-inducing ligand (TRAIL) on plasmacytoid dendritic cells (pDCs) in their pups and thus inhibited infant CHS. Indeed, the administration of DPA-derived 14-lipoxygenation products to mouse pups ameliorated their DNFB CHS. CONCLUSION: These findings suggest that an inhibitory mechanism in infant skin allergy is induced through maternal metabolism of dietary ω3 PUFAs in mice.

Development of a bivalent food poisoning vaccine: augmented antigenicity of the C-terminus of Clostridium perfringens enterotoxin by fusion with the B subunit of Escherichia coli Shiga toxin 2.

Food poisonings caused by Clostridium perfringens and Shiga toxin (Stx)-producing Escherichia coli (STEC) occur frequently worldwide; however, no vaccine is currently available. Therefore, we aimed to develop a bivalent vaccine against C. perfringens and STEC infections. Although it has been considered that the C-terminal region of C. perfringens enterotoxin (C-CPE) could be a good vaccine antigen to block the binding to its receptor, it was insufficient for induction of a protective immune response because of the low antigenicity. However, the fusion of C-CPE with Stx2 B subunit (Stx2B) augmented the antigenicity of C-CPE without affecting the antigenicity of Stx2B. Indeed, high levels of C-CPE-specific neutralizing IgG were found in the serum of mice immunized with the fusion protein Stx2B-C-CPE. Additionally, comparable and substantial levels of Stx2B-specific neutralizing IgG were induced in mice receiving Stx2B-C-CPE or Stx2B alone. These antibody responses against C-CPE and Stx2B lasted for at least 48 weeks, which were sufficient for protective immunity in vitro and in vivo, indicating that Stx2B-C-CPE could induce long-term protective immunity. As an underlying mechanism, ex vivo stimulation with Stx2B, but not with C-CPE, induced cytokine production from splenic T cells collected from mice immunized with Stx2B-C-CPE, suggesting that Stx2B-specific, but not C-CPE-specific, T cells were induced by the immunization with Stx2B-C-CPE and plausibly promoted immunoglobulin class switching of both Stx2B- and C-CPE-specific B cells from IgM to IgG. These findings collectively indicate that Stx2B-C-CPE is a T-cell-antigen-supplement-type bivalent vaccine, which could be an efficient against C. perfringens and STEC infections.

Convergence acceleration of Monte Carlo many-body perturbation methods by using many control variates.

The use of many control variates is proposed as a method to accelerate the second- and third-order Monte Carlo (MC) many-body perturbation (MC-MP2 and MC-MP3) calculations. A control variate is an exactly integrable function that is strongly correlated or anti-correlated with the target function to be integrated by the MC method. Evaluating both integrals and their covariances in the same MC run, one can effect a mutual cancellation of the statistical uncertainties and biases in the MC integrations, thereby accelerating its convergence considerably. Six and thirty-six control variates, whose integrals are known a priori, are generated for MC-MP2 and MC-MP3, respectively, by systematically replacing one or more two-electron-integral vertices of certain configurations by zero-valued overlap-integral vertices in their Goldstone diagrams. The variances and covariances of these control variates are computed at a marginal cost, enhancing the overall efficiency of the MC-MP2 and MC-MP3 calculations by a factor of up to 14 and 20, respectively.

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

A finite-temperature many-body perturbation theory is presented, which 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 the zeroth-order thermal averages of one- through four-indexed quantities and 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 temperatures.

Convergence acceleration of Monte Carlo many-body perturbation methods by direct sampling.

In the Monte Carlo many-body perturbation (MC-MP) method, the conventional correlation-correction formula, which is a long sum of products of low-dimensional integrals, is first recast into a short sum of high-dimensional integrals over electron-pair and imaginary-time coordinates. These high-dimensional integrals are then evaluated by the Monte Carlo method with random coordinates generated by the Metropolis-Hasting algorithm according to a suitable distribution. The latter algorithm, while advantageous in its ability to sample nearly any distribution, introduces autocorrelation in sampled coordinates, which, in turn, increases the statistical uncertainty of the integrals and thus the computational cost. It also involves wasteful rejected moves and an initial "burn-in" step as well as displays hysteresis. Here, an algorithm is proposed that directly produces a random sequence of electron-pair coordinates for the same distribution used in the MC-MP method, which is free from autocorrelation, rejected moves, a burn-in step, or hysteresis. This direct-sampling algorithm is shown to accelerate second- and third-order Monte Carlo many-body perturbation calculations by up to 222% and 38%, respectively.

Dietary coconut oil ameliorates skin contact hypersensitivity through mead acid production in mice.

Coconut oil is used as a dietary oil worldwide, and its healthy effects are recognized by the fact that coconut oil is easy to digest, helps in weight management, increases healthy cholesterol, and provides instant energy. Although topical application of coconut oil is known to reduce skin infection and inflammation, whether dietary coconut oil has any role in decreasing skin inflammation is unknown. In this study, we showed the impact of dietary coconut oil in allergic skin inflammation by using a mouse model of contact hypersensitivity (CHS). Mice maintained on coconut oil showed amelioration of skin inflammation and increased levels of cis-5, 8, 11-eicosatrienoic acid (mead acid) in serum. Intraperitoneal injection of mead acid inhibited CHS and reduced the number of neutrophils infiltrating to the skin. Detailed mechanistic studies unveiled that mead acid inhibited the directional migration of neutrophils by inhibiting the filamentous actin polymerization and leukotriene B4 production required for secondary recruitment of neutrophils. Our findings provide valuable insights into the preventive roles of coconut oil and mead acid against skin inflammation, thereby offering attractive therapeutic possibilities.

Immunological association of inducible bronchus-associated lymphoid tissue organogenesis in Ag85B-rHPIV2 vaccine-induced anti-tuberculosis mucosal immune responses in mice.

We previously reported that Ag85B-expressing human parainfluenza type 2 virus (Ag85B-rHPIV2) was effective as a nasal vaccine against tuberculosis in mice; however, the mechanism by which it induces an immune response remains to be investigated. In the present study, we found that organogenesis of inducible bronchus-associated lymphoid tissue (iBALT) played a role in the induction of antigen-specific T cells and IgA antibody responses in the lung of mice intra-nasally administered Ag85B-rHPIV2. We found that expression of Ag85B was dispensable for the development of iBALT, suggesting that HPIV2 acted as an iBALT-inducing vector. When iBALT organogenesis was disrupted in Ag85B-rHPIV2-immunized mice, either by neutralization of the lymphotoxin pathway or depletion of CD11b+ cells, Ag85B-specific immune responses (i.e. IFN γ-producing T cells and IgA antibody) were diminished in the lung. Furthermore, we found that immunization with Ag85B-rHPIV2 induced neutrophil and eosinophil infiltration temporally after the immunization in the lung. Thus, our results show that iBALT organogenesis contributes to the induction of antigen-specific immune responses by Ag85B-rHPIV2 and that Ag85B-rHPIV2 provokes its immune responses without inducing long-lasting inflammation.

The 17,18-epoxyeicosatetraenoic acid-G protein-coupled receptor 40 axis ameliorates contact hypersensitivity by inhibiting neutrophil mobility in mice and cynomolgus macaques.

BACKGROUND: Metabolites of eicosapentaenoic acid exert various physiologic actions. 17,18-Epoxyeicosatetraenoic acid (17,18-EpETE) is a recently identified new class of antiallergic and anti-inflammatory lipid metabolite of eicosapentaenoic acid, but its effects on skin inflammation and the underlying mechanisms remain to be investigated. OBJECTIVE: We evaluated the effectiveness of 17,18-EpETE for control of contact hypersensitivity in mice and cynomolgus macaques. We further sought to reveal underlying mechanisms by identifying the responsible receptor and cellular target of 17,18-EpETE. METHODS: Contact hypersensitivity was induced by topical application of 2,4-dinitrofluorobenzene. Skin inflammation and immune cell populations were analyzed by using flow cytometric, immunohistologic, and quantitative RT-PCR analyses. Neutrophil mobility was examined by means of imaging analysis in vivo and neutrophil culture in vitro. The receptor for 17,18-EpETE was identified by using the TGF-α shedding assay, and the receptor's involvement in the anti-inflammatory effects of 17,18-EpETE was examined by using KO mice and specific inhibitor treatment. RESULTS: We found that preventive or therapeutic treatment with 17,18-EpETE ameliorated contact hypersensitivity by inhibiting neutrophil mobility in mice and cynomolgus macaques. 17,18-EpETE was recognized by G protein-coupled receptor (GPR) 40 (also known as free fatty acid receptor 1) and inhibited chemoattractant-induced Rac activation and pseudopod formation in neutrophils. Indeed, the antiallergic inflammatory effect of 17,18-EpETE was abolished in the absence or inhibition of GPR40. CONCLUSION: 17,18-EpETE inhibits neutrophil mobility through GPR40 activation, which is a potential therapeutic target to control allergic inflammatory diseases.

RAQET: Large-scale two-component relativistic quantum chemistry program package.

The Relativistic And Quantum Electronic Theory (RAQET) program is a new software package, which is designed for large-scale two-component relativistic quantum chemical (QC) calculations. The package includes several efficient schemes and algorithms for calculations involving large molecules which contain heavy elements in accurate relativistic formalisms. These calculations can be carried out in terms of the two-component relativistic Hamiltonian, wavefunction theory, density functional theory, core potential scheme, and evaluation of electron repulsion integrals. Furthermore, several techniques, which have frequently been used in non-relativistic QC calculations, have been customized for relativistic calculations. This article introduces the brief theories and capabilities of RAQET with several calculation examples. © 2018 Wiley Periodicals, Inc.

Similarity-transformed equation-of-motion vibrational coupled-cluster theory.

A similarity-transformed equation-of-motion vibrational coupled-cluster (STEOM-XVCC) method is introduced as a one-mode theory with an effective vibrational Hamiltonian, which is similarity transformed twice so that its lower-order operators are dressed with higher-order anharmonic effects. The first transformation uses an exponential excitation operator, defining the equation-of-motion vibrational coupled-cluster (EOM-XVCC) method, and the second uses an exponential excitation-deexcitation operator. From diagonalization of this doubly similarity-transformed Hamiltonian in the small one-mode excitation space, the method simultaneously computes accurate anharmonic vibrational frequencies of all fundamentals, which have unique significance in vibrational analyses. We establish a diagrammatic method of deriving the working equations of STEOM-XVCC and prove their connectedness and thus size-consistency as well as the exact equality of its frequencies with the corresponding roots of EOM-XVCC. We furthermore elucidate the similarities and differences between electronic and vibrational STEOM methods and between STEOM-XVCC and vibrational many-body Green's function theory based on the Dyson equation, which is also an anharmonic one-mode theory. The latter comparison inspires three approximate STEOM-XVCC methods utilizing the common approximations made in the Dyson equation: the diagonal approximation, a perturbative expansion of the Dyson self-energy, and the frequency-independent approximation. The STEOM-XVCC method including up to the simultaneous four-mode excitation operator in a quartic force field and its three approximate variants are formulated and implemented in computer codes with the aid of computer algebra, and they are applied to small test cases with varied degrees of anharmonicity.

Monte Carlo explicitly correlated many-body Green's function theory.

A highly scalable stochastic algorithm is proposed and implemented for computing the basis-set-incompleteness correction to the diagonal, frequency-independent self-energy of the second-order many-body Green's function (GF2) theory within the explicitly correlated (F12) formalism. The 6-, 9-, 12-, and 15-dimensional integrals comprising the F12 correction are directly evaluated by the Monte Carlo method using appropriate weight functions for importance sampling. The method is naturally and easily parallelized, involves minimal memory space and no disk I/O, and can use virtually any mathematical form of a correlation factor. Its computational cost to correct all ionization energies (IEs) is observed to increase as the fourth power of system size, as opposed to the fifth power in the case of the deterministic counterparts. The GF2 calculations and their F12 corrections for the first IEs of C60 and C70 were executed on 128 graphical processing units (GF2) and 896 central processing units (F12), respectively, to reach the results with statistical errors of 0.04 eV or less. They showed that the basis-set-incompleteness (from aug-cc-pVDZ) accounts for only 50%-60% of the deviations from experiments, suggesting the significance of higher-order perturbation corrections.

Computerized implementation of higher-order electron-correlation methods and their linear-scaling divide-and-conquer extensions.

We have implemented a linear-scaling divide-and-conquer (DC)-based higher-order coupled-cluster (CC) and Møller-Plesset perturbation theories (MPPT) as well as their combinations automatically by means of the tensor contraction engine, which is a computerized symbolic algebra system. The DC-based energy expressions of the standard CC and MPPT methods and the CC methods augmented with a perturbation correction were proposed for up to high excitation orders [e.g., CCSDTQ, MP4, and CCSD(2)TQ ]. The numerical assessment for hydrogen halide chains, polyene chains, and first coordination sphere (C1) model of photoactive yellow protein has revealed that the DC-based correlation methods provide reliable correlation energies with significantly less computational cost than that of the conventional implementations. © 2017 Wiley Periodicals, Inc.

Perspective: Explicitly correlated electronic structure theory for complex systems.

The explicitly correlated approach is one of the most important breakthroughs in ab initio electronic structure theory, providing arguably the most compact, accurate, and efficient ansatz for describing the correlated motion of electrons. Since Hylleraas first used an explicitly correlated wave function for the He atom in 1929, numerous attempts have been made to tackle the significant challenges involved in constructing practical explicitly correlated methods that are applicable to larger systems. These include identifying suitable mathematical forms of a correlated wave function and an efficient evaluation of many-electron integrals. R12 theory, which employs the resolution of the identity approximation, emerged in 1985, followed by the introduction of novel correlation factors and wave function ansätze, leading to the establishment of F12 theory in the 2000s. Rapid progress in recent years has significantly extended the application range of explicitly correlated theory, offering the potential of an accurate wave-function treatment of complex systems such as photosystems and semiconductors. This perspective surveys explicitly correlated electronic structure theory, with an emphasis on recent stochastic and deterministic approaches that hold significant promise for applications to large and complex systems including solids.

One-particle many-body Green's function theory: Algebraic recursive definitions, linked-diagram theorem, irreducible-diagram theorem, and general-order algorithms.

A thorough analytical and numerical characterization of the whole perturbation series of one-particle many-body Green's function (MBGF) theory is presented in a pedagogical manner. Three distinct but equivalent algebraic (first-quantized) recursive definitions of the perturbation series of the Green's function are derived, which can be combined with the well-known recursion for the self-energy. Six general-order algorithms of MBGF are developed, each implementing one of the three recursions, the ΔMPn method (where n is the perturbation order) [S. Hirata et al., J. Chem. Theory Comput. 11, 1595 (2015)], the automatic generation and interpretation of diagrams, or the numerical differentiation of the exact Green's function with a perturbation-scaled Hamiltonian. They all display the identical, nondivergent perturbation series except ΔMPn, which agrees with MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 but converges at the full-configuration-interaction (FCI) limit at n=∞ (unless it diverges). Numerical data of the perturbation series are presented for Koopmans and non-Koopmans states to quantify the rate of convergence towards the FCI limit and the impact of the diagonal, frequency-independent, or ΔMPn approximation. The diagrammatic linkedness and thus size-consistency of the one-particle Green's function and self-energy are demonstrated at any perturbation order on the basis of the algebraic recursions in an entirely time-independent (frequency-domain) framework. The trimming of external lines in a one-particle Green's function to expose a self-energy diagram and the removal of reducible diagrams are also justified mathematically using the factorization theorem of Frantz and Mills. Equivalence of ΔMPn and MBGF in the diagonal and frequency-independent approximations at 1≤n≤3 is algebraically proven, also ascribing the differences at n = 4 to the so-called semi-reducible and linked-disconnected diagrams.