Mathematical formalization inventory¶
This inventory covers all 45 method sections in the current method map, the 19 sections of the feature matrix, and selected explicit dispatch registries. The 198 variant labels include algorithms and backends; they are not a count of independent theories or proofs.
The RHF pilot and the UHF, density-fitting and MP2 extensions prove selected real finite-basis identities. Each coverage entry distinguishes checked claims from outstanding work. Method maturity, empirical accuracy and production status remain those of the method map. A family row does not imply coverage of every spin, boundary condition, derivative, functional or parameter set.
The machine-readable source is methods.json. Run python formal/check_inventory.py to detect new or removed catalog sections, dispatch selectors, missing source paths, or stale generated text. This coverage check tracks these declared catalogs; it is not an exhaustive analysis of every executable branch or option combination.
Mean-field methods¶
M01 Hartree-Fock: RHF, UHF, ROHF¶
Variants: RHF; UHF; ROHF.
Proof obligations: Metric density projectors; electron count; spin occupations; Fock/energy consistency; stationarity.
Coverage: RHF core pilot; selected UHF algebraic claims and OH implementation/reference checks passed; ROHF unproved.
Implementation entry points: cpp/src/rhf.cpp, cpp/src/uhf.cpp, python/vibeqc/rohf.py.
Existing method references: Guest, M. F. and Saunders, V. R. (1974). *On methods for converging open-shell Hartree-Fock wave-functions.* Molecular Physics, **28**(3), 819–828. doi:[10.1080/00268977400102171](https://doi.org/10.1080/00268977400102171), Roothaan, Clemens C. J. (1960). *Self-Consistent Field Theory for Open Shells of Electronic Systems.* Reviews of Modern Physics, **32**(2), 179–185. doi:[10.1103/RevModPhys.32.179](https://doi.org/10.1103/RevModPhys.32.179).
M02 Kohn-Sham density-functional theory: RKS, UKS, ROKS¶
Variants: RKS; UKS; ROKS; LDA; GGA; meta-GGA; global hybrids; range-separated hybrids; VV10; custom functional mixtures.
Proof obligations: Density-to-grid map; XC energy/potential derivatives; spin scaling; exchange mixing; quadrature.
Coverage: Inventoried.
Implementation entry points: cpp/src/rks.cpp, cpp/src/uks.cpp, python/vibeqc/roks.py.
M03 SKALA-1.1 neural exchange-correlation functional¶
Variants: SKALA-1.1.
Proof obligations: Full-grid provider contract; density features; network expression; differentiability and supported properties.
Coverage: Inventoried; empirical accuracy remains external.
Implementation entry points: python/vibeqc/skala.py.
M04 Dispersion corrections: D3(BJ), D4, gCP and the 3c composites¶
Variants: D3(BJ); D4; gCP; HF-3c; PBEh-3c; B97-3c; B3LYP-3c; r2SCAN-3c; wB97X-3c; HSE-3c.
Proof obligations: Pair/triple counting; damping; parameter provenance; component accounting; energy gradients.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/dispersion.py, cpp/src/dispersion.cpp, python/vibeqc/composites.py.
Existing method references: Caldeweyher, Eike, Ehlert, Sebastian, et al. (2019). *A generally applicable atomic-charge dependent London dispersion correction.* Journal of Chemical Physics, **150**(15), 154122. doi:[10.1063/1.5090222](https://doi.org/10.1063/1.5090222), Grimme, Stefan, Ehrlich, Stephan, and Goerigk, Lars (2011). *Effect of the damping function in dispersion corrected density functional theory.* Journal of Computational Chemistry, **32**(7), 1456–1465. doi:[10.1002/jcc.21759](https://doi.org/10.1002/jcc.21759).
M05 DFT+U¶
Variants: DFT+U.
Proof obligations: Projector metric; occupation eigenvalues; Dudarev trace and derivative; per-spin versus total density.
Coverage: Inventoried.
Implementation entry points: cpp/src/dft_plus_u.cpp, python/vibeqc/dft_plus_u.py.
Existing method references: Dudarev, S. L., Botton, G. A., et al. (1998). *Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study.* Physical Review B, **57**(3), 1505–1509. doi:[10.1103/PhysRevB.57.1505](https://doi.org/10.1103/PhysRevB.57.1505).
M06 Density fitting: RI-J, RIJK and RIJCOSX¶
Variants: RI-J; RIJK; RIJCOSX; auxiliary-basis generation.
Proof obligations: Coulomb-metric least squares; metric truncation; fitting error; separate COSX quadrature error.
Coverage: Coulomb-metric factor identities and public/resident fitting contractions checked; projection and approximation-error theorems pending.
Implementation entry points: cpp/src/df.cpp, cpp/src/jk_builder.cpp, cpp/src/cosx.cpp.
Existing method references: Eichkorn, Karin, Treutler, Oliver, et al. (1995). *Auxiliary basis sets to approximate Coulomb potentials.* Chemical Physics Letters, **240**(4), 283–290. doi:[10.1016/0009-2614(95)00621-A](https://doi.org/10.1016/0009-2614(95)00621-A), Neese, Frank, Wennmohs, Frank, et al. (2009). *Efficient, approximate and parallel Hartree-Fock and hybrid DFT calculations. A ‘chain-of-spheres’ algorithm for the Hartree-Fock exchange.* Chemical Physics, **356**(1–3), 98–109. doi:[10.1016/j.chemphys.2008.10.036](https://doi.org/10.1016/j.chemphys.2008.10.036).
M07 Effective core potentials¶
Variants: ECP; inline ECP primitives; spin-orbit ECP route gates.
Proof obligations: Effective electron counts; nonlocal operator; angular projectors; scalar versus spin-orbit scope.
Coverage: Inventoried.
Implementation entry points: cpp/src/ecp.cpp.
M08 Implicit solvation: CPCM, COSMO and COSMO-RS¶
Variants: CPCM; COSMO; COSMO-RS.
Proof obligations: Boundary linear system; dielectric factors; surface charge/energy sign; self-consistent coupling; residual chemical potentials.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/solvation/cpcm.py, python/vibeqc/solvation/cosmors/cosmospace.py.
M09 Getting the SCF to converge¶
Variants: DIIS; EDIIS; ADIIS; KDIIS; damping; level shift; SOSCF; Newton; TRAH; quadratic SCF; OpenTrustRegion; initial guesses; finite-temperature occupations; frontier ensembles.
Proof obligations: Residual definitions; constrained extrapolation; occupation trace; assumptions for convergence and stability; root selection.
Coverage: Inventoried; no general convergence claim.
Implementation entry points: cpp/src/rhf.cpp, cpp/src/diis.cpp, cpp/include/vibeqc/orbital_scf.hpp.
Existing method references: Hu, Xiangqian and Yang, Weitao (2010). *Accelerating self-consistent field convergence with the augmented Roothaan-Hall energy function.* Journal of Chemical Physics, **132**(5), 054109. doi:[10.1063/1.3304922](https://doi.org/10.1063/1.3304922), Kudin, Konstantin N., Scuseria, Gustavo E., and Cancès, Eric (2002). *A black-box self-consistent field convergence algorithm: One step closer.* Journal of Chemical Physics, **116**(19), 8255–8261. doi:[10.1063/1.1470195](https://doi.org/10.1063/1.1470195), Pulay, Péter (1980). *Convergence acceleration of iterative sequences. The case of SCF iteration.* Chemical Physics Letters, **73**(2), 393–398. doi:[10.1016/0009-2614(80)80396-4](https://doi.org/10.1016/0009-2614(80)80396-4), Saunders, V. R. and Hillier, I. H. (1973). *A “Level-Shifting” method for converging closed shell Hartree-Fock wave functions.* International Journal of Quantum Chemistry, **7**(4), 699–705. doi:[10.1002/qua.560070407](https://doi.org/10.1002/qua.560070407), Seeger, Rolf and Pople, John A. (1977). *Self-consistent molecular orbital methods. XVIII. Constraints and stability in Hartree-Fock theory.* Journal of Chemical Physics, **66**(7), 3045–3050. doi:[10.1063/1.434318](https://doi.org/10.1063/1.434318).
Single-reference correlation¶
M10 Moller-Plesset perturbation theory: MP2, RI-MP2, SCS/SOS-MP2, UMP2¶
Variants: MP2; RI-MP2; SCS-MP2; SOS-MP2; UMP2; ROHF-MP2.
Proof obligations: Spin sums; occupied/virtual index domains; denominators; frozen-core dressing; canonical/RI equivalence limits.
Coverage: Canonical restricted MP2 scalar identities and water implementation/reference checks passed; full finite sums and perturbation derivation pending.
Implementation entry points: cpp/src/mp2.cpp, cpp/src/ump2.cpp, python/vibeqc/rohf.py.
Existing method references: Feyereisen, Martin, Fitzgerald, George, and Komornicki, Andrew (1993). *Use of approximate integrals in ab initio theory. An application in MP2 energy calculations.* Chemical Physics Letters, **208**(5), 359–363. doi:[10.1016/0009-2614(93)87156-W](https://doi.org/10.1016/0009-2614(93)87156-W), Grimme, Stefan (2003). *Improved second-order Møller-Plesset perturbation theory by separate scaling of parallel- and antiparallel-spin pair correlation energies.* Journal of Chemical Physics, **118**(20), 9095–9102. doi:[10.1063/1.1569242](https://doi.org/10.1063/1.1569242), Jung, Yousung, Lochan, Rohini C., et al. (2004). *Scaled opposite-spin second order Møller-Plesset correlation energy: An economical electronic structure method.* Journal of Chemical Physics, **121**(20), 9793–9802. doi:[10.1063/1.1809602](https://doi.org/10.1063/1.1809602), Møller, Christian and Plesset, Milton S. (1934). *Note on an Approximation Treatment for Many-Electron Systems.* Physical Review, **46**(7), 618–622. doi:[10.1103/PhysRev.46.618](https://doi.org/10.1103/PhysRev.46.618).
M11 Double-hybrid functionals: B2PLYP, DSD-PBEP86, revDSD-PBEP86-D4¶
Variants: B2PLYP; DSD-PBEP86; revDSD-PBEP86-D4; PWPB95; custom double hybrids.
Proof obligations: Energy component weights; same/opposite-spin separation; orbitals and frozen-core convention; dispersion accounting.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/runner.py.
Existing method references: Grimme, Stefan (2006). *Semiempirical hybrid density functional with perturbative second-order correlation.* Journal of Chemical Physics, **124**(3), 034108. doi:[10.1063/1.2148954](https://doi.org/10.1063/1.2148954).
M12 Coupled cluster: CCSD, CCSD(T) and the open-shell variants¶
Variants: CCSD; CCSD(T); UCCSD; UCCSD(T); ROHF-CCSD; ROHF-CCSD(T).
Proof obligations: Anticommutators; normal ordering; BCH termination assumptions; projected residuals; spin adaptation; triples prefactors.
Coverage: Inventoried; symbolic codegen is not a proof.
Implementation entry points: cpp/src/ccsd.cpp, cpp/src/ccsd_triples.cpp, cpp/src/uccsd.cpp.
Existing method references: Čížek, Jiří (1966). *On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods.* Journal of Chemical Physics, **45**(11), 4256–4266. doi:[10.1063/1.1727484](https://doi.org/10.1063/1.1727484), Purvis, George D. III and Bartlett, Rodney J. (1982). *A full coupled-cluster singles and doubles model: The inclusion of disconnected triples.* Journal of Chemical Physics, **76**(4), 1910–1918. doi:[10.1063/1.443164](https://doi.org/10.1063/1.443164), Raghavachari, Krishnan, Trucks, Gary W., et al. (1989). *A fifth-order perturbation comparison of electron correlation theories.* Chemical Physics Letters, **157**(6), 479–483. doi:[10.1016/S0009-2614(89)87395-6](https://doi.org/10.1016/S0009-2614(89)87395-6).
M13 The wider coupled-cluster and coupled-pair family: CC2, CC3, CCSDT, QCISD(T), CEPA(n)¶
Variants: CC2; CC3; CCSDT; CCD; LCCD; LCCSD; BCCD; BCCD(T); QCISD; QCISD(T); CEPA(0); CEPA(1); CEPA(2); CEPA(3).
Proof obligations: Variant-specific retained diagrams; perturbative order; Brueckner orbital condition; limiting reductions.
Coverage: Inventoried.
Implementation entry points: cpp/src/ccsd.cpp, python/vibeqc/runner.py.
Existing method references: Christiansen, Ove, Koch, Henrik, and Jørgensen, Poul (1995). *The second-order approximate coupled cluster singles and doubles model CC2.* Chemical Physics Letters, **243**(5–6), 409–418. doi:[10.1016/0009-2614(95)00841-Q](https://doi.org/10.1016/0009-2614(95)00841-Q), Handy, Nicholas C., Pople, John A., et al. (1989). *Size-consistent Brueckner theory limited to double substitutions.* Chemical Physics Letters, **164**(2–3), 185–192. doi:[10.1016/0009-2614(89)85013-4](https://doi.org/10.1016/0009-2614(89)85013-4), Koch, Henrik, Christiansen, Ove, et al. (1997). *The CC3 model: An iterative coupled cluster approach including connected triples.* Journal of Chemical Physics, **106**(5), 1808–1818. doi:[10.1063/1.473322](https://doi.org/10.1063/1.473322), Meyer, Wilfried (1973). *PNO-CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane.* Journal of Chemical Physics, **58**(3), 1017–1035. doi:[10.1063/1.1679283](https://doi.org/10.1063/1.1679283), Noga, Jozef and Bartlett, Rodney J. (1987). *The full CCSDT model for molecular electronic structure.* Journal of Chemical Physics, **86**(12), 7041–7050. doi:[10.1063/1.452353](https://doi.org/10.1063/1.452353), Pople, John A., Head-Gordon, Martin, and Raghavachari, Krishnan (1987). *Quadratic configuration interaction. A general technique for determining electron correlation energies.* Journal of Chemical Physics, **87**(10), 5968–5975. doi:[10.1063/1.453520](https://doi.org/10.1063/1.453520).
M14 Local correlation: DLPNO-MP2, DLPNO-CCSD(T), DLPNO-(T1) and the open-shell solvers¶
Variants: DLPNO-MP2; DLPNO-CCSD; DLPNO-CCSD(T); DLPNO-(T1); open-shell local MP2 and CCSD(T); periodic local correlation.
Proof obligations: Pair domains; nonorthogonal local metrics; pair counting; PNO truncation; canonical limit and screened-term error.
Coverage: Inventoried.
Implementation entry points: cpp/src/dlpno_mp2.cpp, cpp/src/dlpno_uccsd.cpp, python/vibeqc/dlpno/mp2.py.
Existing method references: Guo, Yang, Riplinger, Christoph, et al. (2018). *Communication: An improved linear scaling perturbative triples correction for the domain based local pair-natural orbital based singles and doubles coupled cluster method [DLPNO-CCSD(T)].* Journal of Chemical Physics, **148**(1), 011101. doi:[10.1063/1.5011798](https://doi.org/10.1063/1.5011798), Pinski, Peter, Riplinger, Christoph, et al. (2015). *Sparse maps—A systematic infrastructure for reduced-scaling electronic structure methods. I. An efficient and simple linear scaling local MP2 method that uses an intermediate basis of pair natural orbitals.* Journal of Chemical Physics, **143**(3), 034108. doi:[10.1063/1.4926879](https://doi.org/10.1063/1.4926879), Riplinger, Christoph and Neese, Frank (2013). *An efficient and near linear scaling pair natural orbital based local coupled cluster method.* Journal of Chemical Physics, **138**(3), 034106. doi:[10.1063/1.4773581](https://doi.org/10.1063/1.4773581), Saitow, Masaaki, Becker, Ute, et al. (2017). *A new near-linear scaling, efficient and accurate, open-shell domain-based local pair natural orbital coupled cluster singles and doubles theory.* Journal of Chemical Physics, **146**(16), 164105. doi:[10.1063/1.4981521](https://doi.org/10.1063/1.4981521).
M15 Configuration interaction with singles and doubles: CISD¶
Variants: CISD.
Proof obligations: Determinant excitation phases; Slater-Condon matrix elements; Hermiticity; variational subspace inclusion.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/solvers/_cisd.py.
Existing method references: Pople, J. A., Seeger, R., and Krishnan, R. (1977). *Variational configuration interaction methods and comparison with perturbation theory.* International Journal of Quantum Chemistry, **12**(S11), 149–163. doi:[10.1002/qua.560120820](https://doi.org/10.1002/qua.560120820).
Multireference and exact methods¶
M16 CASCI and CASSCF¶
Variants: CASCI; CASSCF.
Proof obligations: Active/inactive partition; core dressing; determinant Hamiltonian; orbital stationarity; state averaging.
Coverage: Inventoried.
Implementation entry points: cpp/src/casci.cpp, python/vibeqc/solvers/_casscf.py.
Existing method references: Roos, Björn O., Taylor, Peter R., and Siegbahn, Per E. M. (1980). *A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach.* Chemical Physics, **48**(2), 157–173. doi:[10.1016/0301-0104(80)80045-0](https://doi.org/10.1016/0301-0104(80)80045-0).
M17 CASPT2, NEVPT2 and MRCI¶
Variants: CASPT2; NEVPT2; MRCI.
Proof obligations: Zeroth-order Hamiltonian; contraction metric; denominators and shifts; excitation manifold; variant assumptions.
Coverage: Inventoried.
Implementation entry points: cpp/src/nevpt2_ops.cpp, python/vibeqc/solvers.
Existing method references: Andersson, Kerstin, Malmqvist, Per-Åke, et al. (1990). *Second-order perturbation theory with a CASSCF reference function.* Journal of Physical Chemistry, **94**(14), 5483–5488. doi:[10.1021/j100377a012](https://doi.org/10.1021/j100377a012), Andersson, Kerstin, Malmqvist, Per-Åke, and Roos, Björn O. (1992). *Second-order perturbation theory with a complete active space self-consistent field reference function.* Journal of Chemical Physics, **96**(2), 1218–1226. doi:[10.1063/1.462209](https://doi.org/10.1063/1.462209), Angeli, Celestino, Cimiraglia, Renzo, et al. (2001). *Introduction of n-electron valence states for multireference perturbation theory.* Journal of Chemical Physics, **114**(23), 10252–10264. doi:[10.1063/1.1361246](https://doi.org/10.1063/1.1361246), Werner, Hans-Joachim and Knowles, Peter J. (1988). *An efficient internally contracted multiconfiguration-reference configuration interaction method.* Journal of Chemical Physics, **89**(9), 5803–5814. doi:[10.1063/1.455556](https://doi.org/10.1063/1.455556).
Excited states and response properties¶
M19 Linear-response TDDFT: Casida, Tamm-Dancoff, CIS¶
Variants: TDDFT; Casida response; TDA; CIS.
Proof obligations: Response A/B blocks; spin factors; excitation metric; stability assumptions; oscillator strength sum rules.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/tddft.py.
Existing method references: Martin, Richard L. (2003). *Natural transition orbitals.* Journal of Chemical Physics, **118**(11), 4775–4777. doi:[10.1063/1.1558471](https://doi.org/10.1063/1.1558471).
M20 Green’s-function ionisation energies: GF2 and OVGF¶
Variants: GF2; OVGF.
Proof obligations: Self-energy diagrams; frequency denominators; pole residues; spin and analytic-continuation conventions.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/solvers.
Existing method references: Cederbaum, L. S. (1975). *One-body Green’s function for atoms and molecules: theory and application.* Journal of Physics B: Atomic and Molecular Physics, **8**(2), 290–303. doi:[10.1088/0022-3700/8/2/018](https://doi.org/10.1088/0022-3700/8/2/018), von Niessen, W., Schirmer, J., and Cederbaum, L. S. (1984). *Computational methods for the one-particle Green’s function.* Computer Physics Reports, **1**(2), 57–125. doi:[10.1016/0167-7977(84)90002-9](https://doi.org/10.1016/0167-7977(84)90002-9).
M21 Static polarizability and analytic Hessians: coupled-perturbed SCF¶
Variants: CPHF; CPKS; static polarizability; analytic RHF/UHF/RKS/UKS Hessians; finite-difference Hessians.
Proof obligations: Orbital response equation; full perturbation derivative; Pulay terms; Hessian symmetry; translational modes.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/cphf.py, cpp/src/hessian_integrals.cpp.
Existing method references: Pople, J. A., Krishnan, R., et al. (1979). *Derivative studies in Hartree-Fock and Møller-Plesset theories.* International Journal of Quantum Chemistry, **16**(S13), 225–241. doi:[10.1002/qua.560160825](https://doi.org/10.1002/qua.560160825).
Semiempirical methods¶
M22 MSINDO¶
Variants: MSINDO.
Proof obligations: Retained integrals; empirical parameters; core repulsion; spin occupations; analytical energy derivatives.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/semiempirical/methods/msindo.py.
Existing method references: Ahlswede, Bernd and Jug, Karl (1999). *Consistent modifications of SINDO1: I. Approximations and parameters.* Journal of Computational Chemistry, **20**(6), 563–571. doi:[10.1002/(SICI)1096-987X(19990430)20:6<563::AID-JCC1>3.0.CO;2-2](https://doi.org/10.1002/(SICI)1096-987X(19990430)20:6<563::AID-JCC1>3.0.CO;2-2).
M23 Density-functional tight binding: DFTB0 and SCC-DFTB¶
Variants: DFTB0; SCC-DFTB; spin-polarized SCC-DFTB.
Proof obligations: Slater-Koster interpolation; charge SCC functional; overlap populations; gamma kernels; repulsive terms.
Coverage: Inventoried.
Implementation entry points: cpp/src/semiempirical/dftb0.cpp, python/vibeqc/semiempirical/dftb0.py.
Existing method references: Elstner, M., Porezag, D., et al. (1998). *Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties.* Physical Review B, **58**, 7260. doi:[10.1103/PhysRevB.58.7260](https://doi.org/10.1103/PhysRevB.58.7260), Porezag, D., Frauenheim, Th., et al. (1995). *Construction of tight-binding-like potentials on the basis of density-functional theory: Application to carbon.* Physical Review B, **51**, 12947. doi:[10.1103/PhysRevB.51.12947](https://doi.org/10.1103/PhysRevB.51.12947).
M24 GFN2-xTB¶
Variants: GFN2-xTB; molecular and periodic GFN2 variants.
Proof obligations: Anisotropic multipoles; charge response; dispersion derivatives; cutoffs and periodic image convention.
Coverage: Inventoried.
Implementation entry points: cpp/src/semiempirical/methods/xtb, python/vibeqc/semiempirical/methods/gfn2_dispersion.py.
Existing method references: Bannwarth, Christoph, Ehlert, Sebastian, and Grimme, Stefan (2019). *GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions.* Journal of Chemical Theory and Computation, **15**(3), 1652. doi:[10.1021/acs.jctc.8b01176](https://doi.org/10.1021/acs.jctc.8b01176).
M25 NDDO methods: PM6, OM1, OM2, OM3 and the PM7 gate¶
Variants: PM6; OM1; OM2; OM3; PM7 gate.
Proof obligations: NDDO integral selection; orthogonalization corrections; parameters; energy/gradient consistency; explicit missing terms.
Coverage: Inventoried; gated routes remain gated.
Implementation entry points: cpp/src/semiempirical/methods, python/vibeqc/semiempirical.
Existing method references: Scholten, Mirjam (2003). *Semiempirische Verfahren mit Orthogonalisierungskorrekturen: Die OM3 Methode.*, Stewart, James J. P. (2007). *Optimization of parameters for semiempirical methods V: Modification of NDDO approximations and application to 70 elements.* Journal of Molecular Modeling, **13**, 1173–1213. doi:[10.1007/s00894-007-0233-4](https://doi.org/10.1007/s00894-007-0233-4), Weber, Wolfgang and Thiel, Walter (2000). *Orthogonalization corrections for semiempirical methods.* Theoretical Chemistry Accounts, **103**, 495–506. doi:[10.1007/s002149900083](https://doi.org/10.1007/s002149900083).
M26 SECCM: the semiempirical cyclic cluster model¶
Variants: SECCM-DFTB0; SECCM-SCC-DFTB; SECCM-GFN2; SECCM-PM6; SECCM-OMx; MSINDO cyclic models.
Proof obligations: Finite group/topology; images and pair domains; per-cell normalization; electrostatic boundary; ensemble occupations.
Coverage: Inventoried; experimental.
Implementation entry points: cpp/src/semiempirical/seccm, python/vibeqc/semiempirical/seccm.
Existing method references: Bredow, Thomas, Geudtner, Gerald, and Jug, Karl (2001). *Development of the cyclic cluster approach for ionic systems.* Journal of Computational Chemistry, **22**(1), 89–101. doi:[10.1002/1096-987X(20010115)22:1<89::AID-JCC9>3.0.CO;2-7](https://doi.org/10.1002/1096-987X(20010115)22:1<89::AID-JCC9>3.0.CO;2-7).
Machine-learned interatomic potentials¶
M27 MACE¶
Variants: MACE; MACE-MP; MACE-OFF; cyclic MACE construction A.
Proof obligations: Graph periodic images; equivariance; cutoff continuity; force and cell derivatives; model/checkpoint identity.
Coverage: Inventoried; empirical accuracy remains external.
Implementation entry points: python/vibeqc/mlip.
Existing method references: Batatia, Ilyes, Kovács, Dávid Péter, et al. (2022). *MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields.* Advances in Neural Information Processing Systems, **35**, 11423–11436. doi:[10.52202/068431-0830](https://doi.org/10.52202/068431-0830).
Periodic systems¶
M28 Bloch’s theorem and k-point sampling¶
Variants: Gamma and multi-k RHF/UHF/RKS/UKS/ROHF; symmetry-reduced k meshes.
Proof obligations: Bloch phase convention; conjugation; reciprocal mesh weights; electron count; time reversal and symmetry action.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/periodic_scf_accelerators.py, cpp/src/periodic_scf.cpp.
Existing method references: Monkhorst, Hendrik J. and Pack, James D. (1976). *Special points for Brillouin-zone integrations.* Physical Review B, **13**(12), 5188–5192. doi:[10.1103/PhysRevB.13.5188](https://doi.org/10.1103/PhysRevB.13.5188).
M29 The periodic Coulomb problem and Ewald summation¶
Variants: 3D Ewald; 2D slab electrostatics; 1D electrostatics; truncated Coulomb.
Proof obligations: Neutrality; self/background/surface terms; zero mode; convergent split and boundary-dependent limits.
Coverage: Inventoried.
Implementation entry points: cpp/src/ewald.cpp, python/vibeqc/periodic.
Existing method references: Ewald, Paul Peter (1921). *Die Berechnung optischer und elektrostatischer Gitterpotentiale.* Annalen der Physik, **369**(3), 253-287. doi:[10.1002/andp.19213690304](https://doi.org/10.1002/andp.19213690304), Parry, D. E. (1975). *The electrostatic potential in the surface region of an ionic crystal.* Surface Science, **49**, 433–440. doi:[10.1016/0039-6028(75)90362-3](https://doi.org/10.1016/0039-6028(75)90362-3).
M30 GDF: Gaussian density fitting, with RSGDF and MDF¶
Variants: GDF; RSGDF; MDF.
Proof obligations: Momentum conservation; complex auxiliary metric; near/far split; G=0 terms; fitting/truncation errors.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/periodic_gdf_route.py, cpp/src/periodic_gdf_short_range.cpp.
Existing method references: Gygi, F. and Baldereschi, A. (1986). *Self-consistent Hartree-Fock and screened-exchange calculations in solids: Application to silicon.* Physical Review B, **34**, 4405–4408. doi:[10.1103/PhysRevB.34.4405](https://doi.org/10.1103/PhysRevB.34.4405), Sun, Qiming, Berkelbach, Timothy C., et al. (2017). *Gaussian and plane-wave mixed density fitting for periodic systems.* Journal of Chemical Physics, **147**(16), 164119. doi:[10.1063/1.4998644](https://doi.org/10.1063/1.4998644), Ye, Hong-Zhou and Berkelbach, Timothy C. (2021). *Fast periodic Gaussian density fitting by range separation.* Journal of Chemical Physics, **154**(13), 131104. doi:[10.1063/5.0046617](https://doi.org/10.1063/5.0046617).
M31 GPW and GAPW: the plane-wave route¶
Variants: GPW; GAPW.
Proof obligations: FFT normalization; Poisson kernel; smooth/augmentation decomposition; grid aliasing and compensation charges.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/periodic_gapw_augment.py, python/vibeqc/periodic_gapw_grid.py.
Existing method references: Blöchl, P. E. (1994). *Projector augmented-wave method.* Physical Review B, **50**(24), 17953–17979. doi:[10.1103/PhysRevB.50.17953](https://doi.org/10.1103/PhysRevB.50.17953), VandeVondele, Joost and Hutter, Jürg (2003). *An efficient orbital transformation method for electronic structure calculations.* Journal of Chemical Physics, **118**(10), 4365–4369. doi:[10.1063/1.1543154](https://doi.org/10.1063/1.1543154).
M32 BIPOLE: the exact Ewald J split¶
Variants: BIPOLE.
Proof obligations: Near/far domain partition; multipole normalization; Ewald identity; duplicate/self exclusion; truncation error.
Coverage: Inventoried.
Implementation entry points: cpp/src/bipole_ewald_gram.cpp, cpp/src/bipole_contractor.cpp, python/vibeqc/pbc_bipole.py.
Existing method references: Saunders, V. R., Freyria-Fava, C., et al. (1992). *On the electrostatic potential in crystalline systems where the charge density is expanded in Gaussian functions.* Molecular Physics, **77**, 629–665. doi:[10.1080/00268979200102671](https://doi.org/10.1080/00268979200102671).
M33 Periodic RIJCOSX and the 2-D slab route¶
Variants: Periodic RIJCOSX; 2D slab J/K.
Proof obligations: Boundary-specific exchange and Coulomb kernels; quadrature; mesh symmetry and energy consistency.
Coverage: Inventoried.
Implementation entry points: cpp/src/periodic_cosx_jk_builder.cpp, python/vibeqc/periodic.
Existing method references: de Leeuw, S. W. and Perram, J. W. (1979). *Electrostatic lattice sums for semi-infinite lattices.* Molecular Physics, **37**(4), 1313–1322. doi:[10.1080/00268977900100951](https://doi.org/10.1080/00268977900100951), Parry, D. E. (1975). *The electrostatic potential in the surface region of an ionic crystal.* Surface Science, **49**, 433–440. doi:[10.1016/0039-6028(75)90362-3](https://doi.org/10.1016/0039-6028(75)90362-3).
M34 Electronic structure of solids: bands, DOS, COOP/COHP, bond orders, QTAIM¶
Variants: Bands; DOS; COOP; COHP; periodic bond orders; periodic QTAIM.
Proof obligations: Bloch normalization; energy/reference conventions; population metrics; integration and partition sum rules.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/qtaim.py, python/vibeqc/periodic.
Existing method references: Bader, R. F. W. (1985). *Atoms in molecules.* Accounts of Chemical Research, **18**(1), 9–15. doi:[10.1021/ar00109a003](https://doi.org/10.1021/ar00109a003), Dronskowski, Richard and Blöchl, Peter E. (1993). *Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations.* Journal of Physical Chemistry, **97**(33), 8617–8624. doi:[10.1021/j100135a014](https://doi.org/10.1021/j100135a014), Hughbanks, Timothy and Hoffmann, Roald (1983). *Chains of trans-edge-sharing molybdenum octahedra: metal-metal bonding in extended systems.* Journal of the American Chemical Society, **105**(11), 3528–3537. doi:[10.1021/ja00349a027](https://doi.org/10.1021/ja00349a027).
M35 Phonons, stress and variable cells¶
Variants: Phonons; cell stress; elastic response; variable-cell optimization.
Proof obligations: Mass weighting; force-constant symmetries; acoustic sum rule; strain/cell derivative and stress sign.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/basis_optimization/phonons.py, python/vibeqc/periodic.
M36 Surface embedding¶
Variants: Green-function surface embedding.
Proof obligations: Schur complements; self-energy boundaries; layer indexing; causal Green functions and matching conditions.
Coverage: Inventoried; experimental.
Implementation entry points: python/vibeqc/periodic_embedding.
Existing method references: Inglesfield, J. E. (1981). *A method of embedding.* Journal of Physics C: Solid State Physics, **14**(26), 3795-3806. doi:[10.1088/0022-3719/14/26/015](https://doi.org/10.1088/0022-3719/14/26/015).
M37 AICCM: the ab initio cyclic cluster model¶
Variants: Gamma-CCM; chi-CCM; AICCM; cyclic HF/DFT/MP2/CC variants.
Proof obligations: Finite translation group; characters; sewing phases; cyclic domains; reductions and per-cell energy.
Coverage: Inventoried; experimental.
Implementation entry points: python/vibeqc/periodic/ccm, python/vibeqc/periodic/chi.
Existing method references: Peintinger, Michael F. and Bredow, Thomas (2014). *The cyclic cluster model at Hartree-Fock level.* Journal of Computational Chemistry, **35**(11), 839–846. doi:[10.1002/jcc.23550](https://doi.org/10.1002/jcc.23550).
Workflows on the energy surface¶
M38 Geometry optimisation, relaxed scans and intrinsic reaction coordinates¶
Variants: Geometry optimization; relaxed scans; IRC; internal coordinates; BFGS; L-BFGS; trust region.
Proof obligations: Coordinate Jacobians; energy gradient; Hessian update; step acceptance; constrained stationarity.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/geomopt, python/vibeqc/irc.py.
Existing method references: Pulay, P. (1969). *Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I. Theory.* Molecular Physics, **17**(2), 197–204. doi:[10.1080/00268976900100941](https://doi.org/10.1080/00268976900100941).
M39 Transition states: nudged elastic band, dimer and eigenvector following¶
Variants: NEB; climbing-image NEB; dimer; eigenvector following.
Proof obligations: Projected forces; tangent normalization; saddle index; translation/rotation treatment; convergence assumptions.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/neb.py, python/vibeqc/geomopt.
M40 Vibrational frequencies and thermochemistry¶
Variants: Vibrational analysis; IR/Raman intensities; harmonic thermochemistry; quasi-RRHO.
Proof obligations: Mass-weighted Hessian; unit conversions; zero modes; partition functions; standard-state and low-frequency conventions.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/thermo.py, python/vibeqc/runner.py.
M41 Molecular dynamics, metadynamics and conical intersections¶
Variants: Molecular dynamics; thermostats; metadynamics; conical intersections.
Proof obligations: Integrator map; timestep error; thermostat distribution; bias derivatives; state crossings and branch conditions.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/md.py, python/vibeqc/runner.py.
M42 Basis-set optimisation¶
Variants: Basis-set optimization; periodic basis optimization; counterpoise objectives.
Proof obligations: Exponent positivity; contraction normalization; linear-dependence handling; objective/gradient consistency.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/basis_optimization.
Analysis of the wavefunction¶
M43 Charges, bond orders and localised orbitals¶
Variants: Mulliken; Lowdin; Hirshfeld; CM5; IAO; IBO; Boys; Pipek-Mezey; bond orders; QTAIM.
Proof obligations: Metric populations; partition sum rules; localization invariance; charge conservation; real-space integration.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/iao.py, python/vibeqc/iao_population.py, python/vibeqc/qtaim.py.
Existing method references: Hirshfeld, F. L. (1977). *Bonded-atom fragments for describing molecular charge densities.* Theoretica Chimica Acta, **44**(2), 129–138. doi:[10.1007/BF00549096](https://doi.org/10.1007/BF00549096), Knizia, Gerald (2013). *Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts.* Journal of Chemical Theory and Computation, **9**(11), 4834–4843. doi:[10.1021/ct400687b](https://doi.org/10.1021/ct400687b), Mayer, I. (1983). *Charge, bond order and valence in the ab initio SCF theory.* Chemical Physics Letters, **97**(3), 270–274. doi:[10.1016/0009-2614(83)80005-0](https://doi.org/10.1016/0009-2614(83)80005-0), Wiberg, Kenneth B. (1968). *Application of the Pople-Santry-Segal CNDO method to the cyclopropylcarbinyl and cyclobutyl cation and to bicyclobutane.* Tetrahedron, **24**(3), 1083–1096. doi:[10.1016/0040-4020(68)88057-3](https://doi.org/10.1016/0040-4020(68)88057-3).
M44 Energy decomposition and orbital entanglement¶
Variants: Energy decomposition; orbital entanglement.
Proof obligations: Partition definitions; energy component sum; reduced density spectra; entropy normalization and domains.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/eda.py, python/vibeqc/solvers.
Existing method references: Legeza, Ors and Solyom, Jeno (2003). *Optimizing the density-matrix renormalization group method using quantum information entropy.* Physical Review B, **68**(19), 195116. doi:[10.1103/PhysRevB.68.195116](https://doi.org/10.1103/PhysRevB.68.195116), Morokuma, Keiji (1971). *Molecular orbital studies of hydrogen bonds. III. C=O…H-O hydrogen bond in H2CO…H2O and H2CO…2H2O.* Journal of Chemical Physics, **55**(3), 1236–1244. doi:[10.1063/1.1676210](https://doi.org/10.1063/1.1676210), Su, Peifeng and Li, Hui (2009). *Energy decomposition analysis of covalent bonds and intermolecular interactions.* Journal of Chemical Physics, **131**(1), 014102. doi:[10.1063/1.3159673](https://doi.org/10.1063/1.3159673).
M45 Natural orbitals, natural transition orbitals and volumetric data¶
Variants: Natural orbitals; natural transition orbitals; cube grids; density/orbital volumes.
Proof obligations: Metric eigenproblems; SVD invariances; occupancy sum; grid interpolation and spatial normalization.
Coverage: Inventoried.
Implementation entry points: python/vibeqc/runner.py, python/vibeqc/output.
Feature matrix coverage¶
Feature section |
Inventory records |
|---|---|
Molecular SCF |
M01, M02, M21 |
Wavefunction methods (v0.9.0-v0.12.0) |
M16, M17, M18 |
Post-HF correlation methods |
M10, M11, M12, M13, M14, M15 |
Density fitting + RIJCOSX (v0.8.0) |
M06 |
Periodic SCF (native GDF / GPW / BIPOLE / Γ-CCM / χ-CCM) |
M28, M29, M30, M31, M32, M33, M37 |
Semiempirical methods + MLIP (v0.11.0-v0.12.0) |
M22, M23, M24, M25, M26, M27 |
Effective core potentials |
M07 |
SCF convergence aids |
M09 |
XC functionals (v0.8.0 expanded) |
M02, M03, M11 |
Ewald summation (Phase 12e, mostly shipped) |
M29, M32 |
External data fetcher ( |
Interchange and orchestration |
Job queue ( |
Interchange and orchestration |
Properties |
M19, M20, M21, M34, M35, M40, M43, M44, M45, Additional response and analysis |
Solvation (CPCM / COSMO v0.9.0, COSMO-RS v0.15.158) |
M08 |
Input / output |
Interchange and orchestration |
Basis sets |
M42, Integrals and basis |
Crystal / lattice infrastructure |
M28, M29, M35, M36, M37 |
Validation |
all |
Known issues |
all |
Dispatch coverage¶
The following named registries are checked without importing or executing the calculation engine. Aliases and gated selectors remain explicit.
_CITYPE_SUPPORTED:bccd,bccd(t),cc2,cc3,ccd,ccsd,ccsd(t),ccsdt,cepa(0),cepa(1),cepa(2),cepa(3),cepa0,cepa1,cepa2,cepa3,cisd,lccd,lccsd,qcisd,qcisd(t)._CITYPE_UNIMPLEMENTED:cepa,cepa(n),cepan._CC_VARIANT_METHODS:cc2,ccd,cepa(0),cepa(1),cepa(2),cepa(3),lccd,lccsd,qcisd,qcisd(t)._BCCD_METHODS:bccd,bccd(t)._DOUBLE_HYBRID_METHODS:b2plyp,dsd-pbep86,pwpb95,revdsd-pbep86._WB97X_D_METHODS:wb97x-d,wb97xd.
Expansion order¶
Extend the RHF core through UHF and density fitting, then MP2. Reuse those tensor and metric definitions for coupled cluster and periodic methods. Derivative and response proofs must name every energy contribution they differentiate. Empirical models need both an exact model specification and separate accuracy evidence.