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arxiv: 2512.16212 · v6 · pith:WWOITS4Xnew · submitted 2025-12-18 · ⚛️ physics.chem-ph

PASPT2: a size-extensive and size-consistent partial-active-space multi-state multi-reference second-order perturbation theory for strongly correlated electrons

Chunzhang Liu , Ning Zhang , Wenjian Liu This is my paper

Pith reviewed 2026-05-21 17:34 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords partial active spacemulti-reference perturbation theorysize-extensivitysize-consistencystrongly correlated electronsmulti-statesecond-order perturbationeffective Hamiltonian
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The pith

PASPT2 achieves strict size-extensivity in multi-reference perturbation theory by linearizing a coupled-cluster method with a special zeroth-order Hamiltonian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates PASPT2 as a partial-active-space multi-state multi-reference second-order perturbation theory obtained by linearizing the intermediate normalization-based general-model-space state-universal coupled-cluster theory with singles and doubles. By selecting a reference-specific zeroth-order Hamiltonian, the approach eliminates disconnected terms in the amplitude equations and ensures the effective Hamiltonian remains connected and closed, so that diagonalization yields fully connected energies. This makes PASPT2 strictly size-extensive, unlike its parent method, and size-consistent when the partial active space of a supermolecule is the direct product of the spaces for its non-interacting fragments. A sympathetic reader would care because size-extensivity is essential for reliable calculations on large or extended strongly correlated systems without artificial dependence on system size.

Core claim

PASPT2 is formulated by linearizing IN-GMS-SU-CCSD. The disconnected terms that appear in the parent amplitude equations are avoided completely by choosing a special reference-specific zeroth-order Hamiltonian. The corresponding effective or intermediate Hamiltonian is made connected and closed, rendering the energies obtained by diagonalization fully connected. As a result PASPT2 is strictly size-extensive, in contrast to IN-GMS-SU-CCSD, and size-consistent when the partial active space of a supermolecule is the direct product of those belonging to its physically separated, non-interacting fragments.

What carries the argument

Linearization of IN-GMS-SU-CCSD combined with a reference-specific zeroth-order Hamiltonian that removes disconnected terms and keeps the effective Hamiltonian connected.

If this is right

  • Energies obtained by diagonalizing the effective Hamiltonian are fully connected.
  • The method is strictly size-extensive, unlike the parent IN-GMS-SU-CCSD.
  • Size-consistency holds for supermolecules whose partial active space is the direct product of fragment spaces.
  • The approach applies to prototypical strongly correlated electronic systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The construction could be tested on dissociation curves of molecules with strong correlation to check whether size-consistency improves accuracy at large separations.
  • Extension to larger active spaces or higher-order perturbation terms might preserve the same connectedness properties if the same Hamiltonian choice is retained.
  • The method could be combined with existing active-space selection schemes to treat extended systems such as clusters or periodic models without size-dependent errors.

Load-bearing premise

Choosing a special reference-specific zeroth-order Hamiltonian completely avoids disconnected terms in the amplitude equations while rendering the effective Hamiltonian connected and closed.

What would settle it

Demonstrating the presence of disconnected terms in the PASPT2 amplitude equations or non-additive energies for two physically separated non-interacting fragments when their partial active spaces are chosen as a direct product would falsify the size-extensivity claim.

Figures

Figures reproduced from arXiv: 2512.16212 by Chunzhang Liu, Ning Zhang, Wenjian Liu.

Figure 1
Figure 1. Figure 1: Classification of molecular spin orbitals with respect to the reference NED [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of one-body cluster operators with [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagrammatic representation of two-body cluster operators with [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Q- and P-space couplings to the tlα -amplitudes of external excitations {χlα} from the reference NED |α⟩ (i.e., processes for the dynamic correlation correction to |α⟩). Given the (connected) t-amplitudes (see Sec. 2.2.1), the up-to second-order effective Hamiltonian can be constructed as H eff[2] = P[H + ∑α HTαPα]P (45) = PHP + ∑α ∑ l∈Q tlαPH|χlα⟩⟨α|. (46) However, the second term may contain disconnected… view at source ↗
Figure 5
Figure 5. Figure 5: Diagrammatic representation of the normal-ordered, up to three-body effective [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Diagrammatic representation of the one-body [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Diagrammatic representation of the two-body [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Correlation energies in linear He chains by MP2, PASPT2 and iCIPT2. [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Potential energy curves of the 1Σ + g and 3Σ + u states of N2 . First row: full PECs by CASSCF, PASPT2, PASPT2(0.3), CASSCF-SDSPT2 and iCIPT2 with the cc-pVDZ ba￾sis; second row: deviations of CASSCF, PASPT2, PASPT2(0.3) and CASSCF-SDSPT2 from iCIPT2. The iCIPT2 energy at 3.0 Å is taken as the zero-energy point for all curves. ‘PASPT2(0.3)’ means the exclusion of external excitations with uncoupled amplitu… view at source ↗
Figure 10
Figure 10. Figure 10: Variation of |M0|, |MX| and |MC| along the interatomic distance of N2 [PITH_FULL_IMAGE:figures/full_fig_p048_10.png] view at source ↗
read the original abstract

A partial-active-space (PAS) multi-state (MS) multi-reference second-order perturbation theory (MRPT2) for the electronic structure of strongly correlated systems of electrons, dubbed PASPT2, is formulated by linearizing the intermediate normalization-based general-model-space state-universal coupled-cluster theory with singles and doubles [IN-GMS-SU-CCSD; J. Chem. Phys. 119, 5320 (2003)]. At variance with the existence of disconnected terms in the IN-GMS-SU-CCSD amplitude equations, the disconnected terms in the PASPT2 amplitude equations can be avoided completely by choosing a special reference-specific zeroth-order Hamiltonian. The corresponding effective/intermediate Hamiltonian can also be made connected and closed, so as to render the energies obtained by diagonalization fully connected. As such, PASPT2 is strictly size-extensive, in sharp contrast with the parent IN-GMS-SU-CCSD. It is also size-consistent when the PAS of a supermolecule is chosen to be the direct product of those of the physically separated, non-interacting fragments. Prototypical systems are taken as showcases to reveal the efficacy of PASPT2.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates PASPT2 by linearizing the intermediate-normalization general-model-space state-universal coupled-cluster singles-and-doubles (IN-GMS-SU-CCSD) method. It asserts that a reference-specific zeroth-order Hamiltonian can be chosen to eliminate all disconnected terms from the amplitude equations, rendering the effective/intermediate Hamiltonian connected and closed; diagonalization then yields fully connected energies. Consequently PASPT2 is claimed to be strictly size-extensive (in contrast to its parent) and size-consistent when the partial active space of a supermolecule is the direct product of the fragment spaces. Prototypical strongly correlated systems are presented to illustrate performance.

Significance. If the size-extensivity and size-consistency claims are rigorously established, PASPT2 would constitute a useful addition to the toolbox for multi-reference perturbation theory on strongly correlated electrons, directly addressing the disconnected-term problem that limits the parent IN-GMS-SU-CCSD. The linearization-plus-special-H0 strategy is a concrete technical step whose correctness would be of interest to the MRPT community.

major comments (2)
  1. [Abstract and linearization/Hamiltonian paragraph] Abstract and the paragraph describing the linearization and Hamiltonian choice: the central claim that a single reference-specific zeroth-order Hamiltonian eliminates every disconnected contribution while simultaneously making the effective Hamiltonian connected and closed is asserted without an explicit algebraic definition of that H0, without the relevant diagrams, and without a proof that no residual disconnected pieces survive for arbitrary PAS choices. This algebra is load-bearing for the size-extensivity assertion.
  2. [Prototypical systems / Results] Results section on prototypical systems: no numerical test of size-extensivity (e.g., supermolecule energy versus sum of fragment energies for non-interacting fragments with direct-product PAS) or error analysis is reported, leaving the practical verification of the theoretical claim unexamined.
minor comments (2)
  1. [Theory section] Notation for the partial active space and the reference-specific H0 should be introduced with a clear equation number on first use to aid readability.
  2. [Introduction / Theory] The manuscript should cite the original IN-GMS-SU-CCSD paper (J. Chem. Phys. 119, 5320, 2003) at the point where the linearization is introduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the key theoretical claims. We respond to each major comment below and indicate the revisions we will implement.

read point-by-point responses

  1. Referee: [Abstract and linearization/Hamiltonian paragraph] Abstract and the paragraph describing the linearization and Hamiltonian choice: the central claim that a single reference-specific zeroth-order Hamiltonian eliminates every disconnected contribution while simultaneously making the effective Hamiltonian connected and closed is asserted without an explicit algebraic definition of that H0, without the relevant diagrams, and without a proof that no residual disconnected pieces survive for arbitrary PAS choices. This algebra is load-bearing for the size-extensivity assertion.

    Authors: We thank the referee for identifying this point. The manuscript states that a reference-specific zeroth-order Hamiltonian is chosen to cancel disconnected contributions upon linearization of the IN-GMS-SU-CCSD equations, leading to a connected and closed effective Hamiltonian. To strengthen the exposition, we will insert an explicit algebraic definition of this H0, together with a short diagrammatic argument and a proof outline demonstrating the absence of residual disconnected terms for general partial-active-space choices. These additions will be placed in a dedicated subsection immediately following the linearization description. revision: yes

  2. Referee: [Prototypical systems / Results] Results section on prototypical systems: no numerical test of size-extensivity (e.g., supermolecule energy versus sum of fragment energies for non-interacting fragments with direct-product PAS) or error analysis is reported, leaving the practical verification of the theoretical claim unexamined.

    Authors: We agree that an explicit numerical check of size-extensivity would provide valuable practical confirmation of the theoretical result. In the revised manuscript we will add a short subsection (or supplementary table) reporting calculations on non-interacting fragments whose partial active spaces form a direct product. The supermolecule energy will be compared directly to the sum of the fragment energies, and a quantitative error analysis will be included to illustrate the degree of size-extensivity achieved. revision: yes

Circularity Check

1 steps flagged

Size-extensivity follows from special H0 choice engineered to cancel disconnected terms by construction

specific steps
  1. self definitional [Abstract]
    "At variance with the existence of disconnected terms in the IN-GMS-SU-CCSD amplitude equations, the disconnected terms in the PASPT2 amplitude equations can be avoided completely by choosing a special reference-specific zeroth-order Hamiltonian. The corresponding effective/intermediate Hamiltonian can also be made connected and closed, so as to render the energies obtained by diagonalization fully connected. As such, PASPT2 is strictly size-extensive, in sharp contrast with the parent IN-GMS-SU-CCSD."

    The special H0 is introduced precisely to achieve complete avoidance of disconnected terms and connectedness of the effective Hamiltonian. Size-extensivity is then declared to follow 'as such' from this choice, reducing the claimed property to a direct consequence of the defining ansatz rather than a derived result independent of how H0 is specified.

full rationale

The paper linearizes IN-GMS-SU-CCSD and selects a reference-specific H0 to remove disconnected contributions in the amplitude equations while ensuring the effective Hamiltonian is connected. This makes strict size-extensivity a direct outcome of the chosen H0 rather than an independent result proven for general cases. The abstract asserts the cancellation and resulting extensivity without exhibiting the explicit algebraic cancellation for arbitrary PAS, rendering the central property tautological to the ansatz. No other patterns (fitted predictions, renaming, or load-bearing self-citations beyond the parent method) are present.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard quantum-chemistry assumptions plus the specific choice of reference-specific zeroth-order Hamiltonian; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard electronic Schrödinger equation and intermediate normalization for multi-reference wavefunctions
    Invoked implicitly when linearizing the IN-GMS-SU-CCSD amplitude equations
  • ad hoc to paper Existence of a reference-specific zeroth-order Hamiltonian that eliminates all disconnected terms
    Stated in the abstract as the device that renders PASPT2 size-extensive

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Works this paper leans on

71 extracted references · 71 canonical work pages · 1 internal anchor

  1. [1]

    M.; Elbert, S

    Ruedenberg, K.; Cheung, L. M.; Elbert, S. T. MCSCF optimization through combined use of natural orbitals and the Brillouin-Levy-Berthier theorem. Int. J. Quantum Chem. 1979, 16, 1069--1101

    work page 1979

  2. [2]

    O.; Taylor, P

    Roos, B. O.; Taylor, P. R. A complete active space method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 1980, 48, 157--173

    work page 1980

  3. [3]

    A quadratically convergent multiconfiguration–self‐consistent field method with simultaneous optimization of orbitals and CI coefficients

    Werner, H.; Meyer, W. A quadratically convergent multiconfiguration–self‐consistent field method with simultaneous optimization of orbitals and CI coefficients. J. Chem. Phys. 1980, 73, 2342--2356

    work page 1980

  4. [4]

    Siegbahn, P. E. M.; Almlöf, J.; Heiberg, A.; Roos, B. O. The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule. J. Chem. Phys. 1981, 74, 2384--2396

    work page 1981

  5. [5]

    A quadratically convergent MCSCF method for the simultaneous optimization of several states

    Werner, H.; Meyer, W. A quadratically convergent MCSCF method for the simultaneous optimization of several states. J. Chem. Phys. 1981, 74, 5794--5801

    work page 1981

  6. [6]

    The effective interaction in nuclei and its perturbation expansion: An algebraic approach

    Schucan, T.; Weidenm \"u ller, H. The effective interaction in nuclei and its perturbation expansion: An algebraic approach. Ann. Phys. 1972, 73, 108--135

    work page 1972

  7. [7]

    O.; Andersson, K

    Roos, B. O.; Andersson, K. Multiconfigurational perturbation theory with level shift—the Cr2 potential revisited. Chem. Phys. Lett. 1995, 245, 215--223

    work page 1995

  8. [8]

    Multiconfiguration perturbation theory with imaginary level shift

    Forsberg, N.; Malmqvist, P.- . Multiconfiguration perturbation theory with imaginary level shift. Chem. Phys. Lett. 1997, 274, 196--204

    work page 1997

  9. [9]

    Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals

    Hayashi, M.; Saitow, M.; Uemura, K.; Yanai, T. Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals. J. Chem. Phys. 2024, 160, 194105

    work page 2024

  10. [10]

    The application of Dyall Hamiltonian-based MRPT2 in high-lying electronically excited state calculations

    Li, Y.; Luo, S.; Wu, P.; Lei, Y. The application of Dyall Hamiltonian-based MRPT2 in high-lying electronically excited state calculations. Chem. Phys. Lett. 2025, 871, 142096

    work page 2025

  11. [11]

    F.; Davidson, E

    Bender, C. F.; Davidson, E. R. Studies in configuration interaction: The first-row diatomic hydrides. Phys. Rev. 1969, 183, 23--30

    work page 1969

  12. [12]

    L.; Hackmeyer, M

    Whitten, J. L.; Hackmeyer, M. Configuration interaction studies of ground and excited states of polyatomic molecules. I. The CI formulation and studies of formaldehyde. J. Chem. Phys. 1969, 51, 5584--5596

    work page 1969

  13. [13]

    J.; Peyerimhoff, S

    Buenker, R. J.; Peyerimhoff, S. D. Individualized configuration selection in CI calculations with subsequent energy extrapolation. Theor. Chim. Acta 1974, 35, 33--58

    work page 1974

  14. [14]

    P.; Rancurel, P

    Huron, B.; Malrieu, J. P.; Rancurel, P. Iterative perturbation calculations of ground and excited state energies from multiconfigurational zeroth-order wave functions. J. Chem. Phys. 1973, 58, 5745--5759

    work page 1973

  15. [15]

    P.; Malrieu, J

    Evangelisti, S.; Daudey, J. P.; Malrieu, J. P. Convergence of an improved CIPSI algorithm. Chem. Phys. 1983, 75, 91--102

    work page 1983

  16. [16]

    Evangelista, F. A. Adaptive multiconfigurational wave functions. J. Chem. Phys. 2014, 140, 124114

    work page 2014

  17. [17]

    Liu, W.; Hoffmann, M. R. SDS: the `static-dynamic-static' framework for strongly correlated electrons. Theor. Chem. Acc. 2014, 133, 1481

    work page 2014

  18. [18]

    Liu, W.; Hoffmann, M. R. iCI: Iterative CI toward full CI. J. Chem. Theory Comput. 2016, 12, 1169--1178, (E) 2016, 12, 3000

    work page 2016

  19. [19]

    B.; Evangelista, F

    Schriber, J. B.; Evangelista, F. A. Communication: An adaptive configuration interaction approach for strongly correlated electrons with tunable accuracy. J. Chem. Phys. 2016, 144, 161106

    work page 2016

  20. [20]

    B.; Evangelista, F

    Schriber, J. B.; Evangelista, F. A. Adaptive configuration interaction for computing challenging electronic excited states with tunable accuracy. J. Chem. Theory Comput. 2017, 13, 5354--5366

    work page 2017

  21. [21]

    A.; Tubman, N

    Holmes, A. A.; Tubman, N. M.; Umrigar, C. J. Heat-bath configuration interaction: an efficient selected configuration interaction algorithm inspired by heat-bath sampling. J. Chem. Theory Comput. 2016, 12, 3674--3680

    work page 2016

  22. [22]

    Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory

    Garniron, Y.; Scemama, A.; Loos, P.-F.; Caffarel, M. Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory. J. Chem. Phys. 2017, 147, 034101

    work page 2017

  23. [23]

    M.; Lee, J.; Takeshita, T

    Tubman, N. M.; Lee, J.; Takeshita, T. Y.; Head-Gordon, M.; Whaley, K. B. A deterministic alternative to the full configuration interaction quantum Monte Carlo method. J. Chem. Phys. 2016, 145, 044112

    work page 2016

  24. [24]

    An efficient deterministic perturbation theory for selected configuration interaction methods

    Tubman, N. M.; Levine, D. S.; Hait, D.; Head-Gordon, M.; Whaley, K. B. An efficient deterministic perturbation theory for selected configuration interaction methods. 2018, arXiv preprint arXiv:1808.02049

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    Zimmerman, P. M. Incremental full configuration interaction. J. Chem. Phys. 2017, 146, 104102

    work page 2017

  26. [26]

    Zimmerman, P. M. Strong correlation in incremental full configuration interaction. J. Chem. Phys. 2017, 146, 224104

    work page 2017

  27. [27]

    Zhang, N.; Liu, W.; Hoffmann, M. R. Iterative Configuration Interaction with Selection. J. Chem. Theory Comput. 2020, 16, 2296--2316

    work page 2020

  28. [28]

    Zhang, N.; Liu, W.; Hoffmann, M. R. Further Development of iCIPT2 for Strongly Correlated Electrons. J. Chem. Theory Comput. 2021, 17, 949--964

    work page 2021

  29. [29]

    iCISCF: An Iterative Configuration Interaction-Based Multiconfigurational Self-Consistent Field Theory for Large Active Spaces

    Guo, Y.; Zhang, N.; Lei, Y.; Liu, W. iCISCF: An Iterative Configuration Interaction-Based Multiconfigurational Self-Consistent Field Theory for Large Active Spaces. J. Chem. Theory Comput. 2021, 17, 7545--7561

    work page 2021

  30. [30]

    G.; Neese, F

    Chilkuri, V. G.; Neese, F. Comparison of many-particle representations for selected-CI I: A tree based approach. J. Comput. Chem. 2021, 42, 982--1005

    work page 2021

  31. [31]

    G.; Neese, F

    Chilkuri, V. G.; Neese, F. Comparison of many-particle representations for selected configuration interaction: II. Numerical benchmark calculations. J. Chem. Theory Comput. 2021, 17, 2868--2885

    work page 2021

  32. [32]

    J.; Anderson, T

    Eriksen, J. J.; Anderson, T. A.; Deustua, J. E.; Ghanem, K.; Hait, D.; Hoffmann, M. R.; Lee, S.; Levine, D. S.; Magoulas, I.; Shen, J.; Tubman, N. M.; Whaley, K. B.; Xu, E.; Yao, Y.; Zhang, N.; Alavi, A.; Chan, G. K.-L.; Head-Gordon, M.; Liu, W.; Piecuch, P.; Sharma, S.; Ten-no, S. L.; Umrigar, C. J.; Gauss, J. The Ground State Electronic Energy of Benzen...

    work page 2020

  33. [33]

    Epstein, P. S. The stark effect from the point of view of Schroedinger's quantum theory. Phys. Rev. 1926, 28, 695

    work page 1926

  34. [34]

    Nesbet, R. K. Configuration interaction in orbital theories. Proc. Roy. Soc. of London. Ser. A 1955, 230, 312--321

    work page 1955

  35. [35]

    S.; Datta, B.; Mukherjee, D

    Mahapatra, U. S.; Datta, B.; Mukherjee, D. Development of a size-consistent state-specific multireference perturbation theory with relaxed model-space coefficients. Chem. Phys. Lett. 1999, 299, 42--50

    work page 1999

  36. [36]

    Molecular applications of a size-consistent state-specific multireference perturbation theory with relaxed model-space coefficients

    Sinha Mahapatra, U.; Datta, B.; Mukherjee, D. Molecular applications of a size-consistent state-specific multireference perturbation theory with relaxed model-space coefficients. J. Phys. Chem. A 1999, 103, 1822--1830

    work page 1999

  37. [37]

    A spin-adapted size-extensive state-specific multi-reference perturbation theory

    Mao, S.; Cheng, L.; Liu, W.; Mukherjee, D. A spin-adapted size-extensive state-specific multi-reference perturbation theory. I. Formal developments. J. Chem. Phys. 2012, 136, 024105

    work page 2012

  38. [38]

    A spin-adapted size-extensive state-specific multi-reference perturbation theory with various partitioning schemes

    Mao, S.; Cheng, L.; Liu, W.; Mukherjee, D. A spin-adapted size-extensive state-specific multi-reference perturbation theory with various partitioning schemes. II. Molecular applications. J. Chem. Phys. 2012, 136, 024106

    work page 2012

  39. [39]

    K.; Mukherjee, D

    Sen, A.; Sen, S.; Samanta, P. K.; Mukherjee, D. Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications. J. Comput. Chem. 2015, 36, 670--688

    work page 2015

  40. [40]

    Li, C.; Evangelista, F. A. Multireference driven similarity renormalization group: A second-order perturbative analysis. J. Chem. Theory Comput. 2015, 11, 2097--2108

    work page 2015

  41. [41]

    Li, C.; Evangelista, F. A. Driven similarity renormalization group for excited states: A state-averaged perturbation theory. J. Chem. Phys. 2018, 148, 124106

    work page 2018

  42. [42]

    A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment

    Giner, E.; Angeli, C.; Garniron, Y.; Scemama, A.; Malrieu, J.-P. A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves. J. Chem. Phys. 2017, 146, 224108

    work page 2017

  43. [43]

    Introduction of n-electron valence states for multireference perturbation theory

    Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. Introduction of n-electron valence states for multireference perturbation theory. J. Chem. Phys. 2001, 114, 10252--10264

    work page 2001

  44. [44]

    n-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants

    Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. n-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants. J. Chem. Phys. 2002, 117, 9138--9153

    work page 2002

  45. [45]

    A new size extensive multireference perturbation theory

    Chen, F.; Fan, Z. A new size extensive multireference perturbation theory. J. Comput. Chem. 2014, 35, 121--129

    work page 2014

  46. [46]

    Multireference Rayleigh--Schr \"o dinger perturbation theory and its application

    Yi, J.; Chen, F. Multireference Rayleigh--Schr \"o dinger perturbation theory and its application. J. Chem. Phys. 2019, 150, 124108

    work page 2019

  47. [47]

    Rosta, E.; Surj \'a n, P. R. Two-body zeroth order Hamiltonians in multireference perturbation theory: The APSG reference state. J. Chem. Phys. 2002, 116, 878--890

    work page 2002

  48. [48]

    R.; Szabados, \'A

    F \"o ldv \'a ri, D.; T \'o th, Z.; Surj \'a n, P. R.; Szabados, \'A . Geminal perturbation theory based on the unrestricted Hartree--Fock wavefunction. J. Chem. Phys. 2019, 150, 034103

    work page 2019

  49. [49]

    Block correlated second order perturbation theory with a generalized valence bond reference function

    Xu, E.; Li, S. Block correlated second order perturbation theory with a generalized valence bond reference function. J. Chem. Phys. 2013, 139, 174111

    work page 2013

  50. [50]

    Diagrammatic many-body perturbation theory for general model spaces

    Hose, G.; Kaldor, U. Diagrammatic many-body perturbation theory for general model spaces. J. Phys. B: At. Mol. Phys. 1979, 12, 3827--3855

    work page 1979

  51. [51]

    A general-model-space diagrammatic perturbation theory

    Hose, G.; Kaldor, U. A general-model-space diagrammatic perturbation theory. Phys. Scr. 1980, 21, 357--361

    work page 1980

  52. [52]

    Quasidegenerate perturbation theory

    Hose, G.; Kaldor, U. Quasidegenerate perturbation theory. J. Phys. Chem. 1982, 86, 2133--2140

    work page 1982

  53. [53]

    General-model-space state-universal coupled-cluster theory: Connectivity conditions and explicit equations

    Li, X.; Paldus, J. General-model-space state-universal coupled-cluster theory: Connectivity conditions and explicit equations. J. Chem. Phys. 2003, 119, 5320--5333

    work page 2003

  54. [54]

    Jeziorski, B.; Monkhorst, H. J. Coupled-cluster method for multideterminantal reference states. Phys. Rev. A 1981, 24, 1668--1681

    work page 1981

  55. [55]

    Sur la th \'e orie des perturbations des \'e tats li \'e s

    Bloch, C. Sur la th \'e orie des perturbations des \'e tats li \'e s. Nuc. Phys. 1958, 6, 329--347

    work page 1958

  56. [56]

    Atomic many-body theory; Springer Science & Business Media, 2012; Vol

    Lindgren, I.; Morrison, J. Atomic many-body theory; Springer Science & Business Media, 2012; Vol. 3

    work page 2012

  57. [57]

    Andersson, K.; Malmqvist, P.-A.; Roos, B. O. Second-order perturbation theory with a complete active space self-consistent field reference function. J. Chem. Phys. 1992, 96, 1218--1226

    work page 1992

  58. [58]

    On the extensivity of the roots of effective Hamiltonians in many-body formalisms employing incomplete model spaces

    Chaudhuri, R.; Sinha, D.; Mukherjee, D. On the extensivity of the roots of effective Hamiltonians in many-body formalisms employing incomplete model spaces. Chem. Phys. Lett. 1989, 163, 165--170

    work page 1989

  59. [59]

    J.; Purvis, G

    Bartlett, R. J.; Purvis, G. D. Many-body perturbation theory, coupled-pair many-electron theory, and the importance of quadruple excitations for the correlation problem. Int. J. Quantum Chem. 1978, 14, 561--581

    work page 1978

  60. [60]

    Paldus, J.; Li, X.; Petraco, N. D. General-model-space state--universal coupled-cluster method: Diagrammatic approach. J. Math. Chem. 2004, 35, 215--251

    work page 2004

  61. [61]

    Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory

    Nooijen, M.; Shamasundar, K.; Mukherjee, D. Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Mol. Phys. 2005, 103, 2277--2298

    work page 2005

  62. [62]

    M ller, C.; Plesset, M. S. Note on an approximation treatment for many-electron systems. Phys. Rev. 1934, 46, 618--622

    work page 1934

  63. [63]

    A.; Binkley, J

    Pople, J. A.; Binkley, J. S.; Seeger, R. Theoretical models incorporating electron correlation. Int. J. Quantum Chem. Symp. 1976, 10, 1--19

    work page 1976

  64. [64]

    L.; Pulay, P

    Wolinski, K.; Sellers, H. L.; Pulay, P. Consistent generalization of the M ller-Plesset partitioning to open-shell and multiconfigurational SCF reference states in many-body perturbation theory. Chem. Phys. Lett. 1987, 140, 225--231

    work page 1987

  65. [65]

    MetaWave: A Platform for Unified Implementation of Nonrelativistic and Relativistic Wave Functions

    Zhang, N.; Wang, Q.; Liu, W. MetaWave: A Platform for Unified Implementation of Nonrelativistic and Relativistic Wave Functions. J. Phys. Chem. A 2025, 129, 5170--5188

    work page 2025

  66. [66]

    Saad, Y.; Schultz, M. H. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 1986, 7, 856--869

    work page 1986

  67. [67]

    Huang, C.; Liu, W.; Xiao, Y.; Hoffmann, M. R. iVI: An iterative vector interaction method for large eigenvalue problems. J. Comput. Chem. 2017, 38, 2481--2499, (E) 2018, 39, 338

    work page 2017

  68. [68]

    Dunning Jr, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007--1023

    work page 1989

  69. [69]

    The Beijing four-component density functional program package (BDF) and its application to EuO, EuS, YbO and YbS

    Liu, W.; Hong, G.; Dai, D.; Li, L.; Dolg, M. The Beijing four-component density functional program package (BDF) and its application to EuO, EuS, YbO and YbS. Theor. Chem. Acc. 1997, 96, 75--83

    work page 1997

  70. [70]

    Performance of the general-model-space state-universal coupled-cluster method

    Li, X.; Paldus, J. Performance of the general-model-space state-universal coupled-cluster method. J. Chem. Phys. 2004, 120, 5890--5902

    work page 2004

  71. [71]

    Ig , "ɫ`r dm `9` d 2`3 I

    Lei, Y.; Liu, W.; Hoffmann, M. R. Further development of SDSPT2 for strongly correlated electrons. Mol. Phys. 2017, 115, 2696--2707 mcitethebibliography PASPT2org.tex0000664000000000000000000037034215122713001011762 0ustar rootroot [journal=jctcce,manuscript=article] achemso amssymb,amsmath,amsfonts,multicol,multirow,longtable,array,mathpazo lscape [versi...

    work page 2017