Wave-function-based correlation treatments of bulk metals have reached the point where a multireference (MR) version of the method of increments, using the approximately size-extensive MR averaged coupled-pair functional (MRACPF), recovers about 97% of the experimental correlation contribution to the cohesive energies of the face-centered cubic (fcc) metals Ca and Sr. However, that treatment explicitly omits the long-range charge-fluctuation correlation that screens conduction electrons in a metal — a part that can be approximated by the random-phase approximation (RPA) — and it rests on MR correlation methods that are not rigorously size-extensive, with residual truncation and basis-set errors of roughly ±3% and 1–2%. The open problem is to complete the incremental correlation treatment of bulk metals: formulate and compute the long-range charge-fluctuation (RPA-type) contribution in a double-counting-free combination with the short-range multireference incremental sums, and quantify its impact on the incremental sums and on cohesive energies for Ca and Sr.

Background

The cohesive energy of a solid — the energy gained by assembling atoms into the bulk crystal — can be decomposed in wave-function-based quantum chemistry into a Hartree–Fock (HF) part and a correlation part. The method of increments makes the correlation part computable for extended systems: the occupied orbitals of a periodic HF calculation are localized (in practice one works with finite embedded clusters cut from the crystal), partitioned into groups centered on atoms or bonds, and the correlation energy is expanded as a sum of one-body increments (each group correlated alone), two-body increments (pairs correlated together), and so on. The expansion converges when increments decay fast enough with distance and order, which holds for insulators and many weakly bonded solids but is delicate in metals, where electronic states are delocalized and the conduction electrons screen each other. For metals the incremental scheme therefore works with a well-localizable embedded model system that approaches the real metal only gradually across the levels of the expansion.

A second difficulty is the choice of correlation method for each increment. It must be size-extensive — the energy must scale correctly with the number of independent subsystems — because the incremental sum is only meaningful for additive pieces. Single-reference coupled cluster with perturbative triples, CCSD(T), the workhorse for insulators, fails for fcc calcium and strontium: their valence s,p bands hybridize with low-lying empty d bands near the Fermi energy (the energy separating occupied from empty states), so several electronic configurations contribute with comparable weight. The proper zero-order description is then multireference, typically a complete-active-space self-consistent-field (CASSCF) wave function — a multiconfigurational reference in which a small set of near-degenerate orbitals is optimized while all electron configurations within that active space are included — on top of which dynamic correlation is added by a multireference correlation method. Available MR correlation methods — multireference configuration interaction (MRCI) and the averaged coupled-pair functional (ACPF) of Gdanitz and Ahlrichs — are only approximately size-extensive. Voloshina and Paulus (2014) used MRACPF, validating it against size-corrected MRCI, and showed for Ca and Sr that CCSD and CCSD(T) break down (perturbative triples even diverge) while the MR incremental sum through three-body increments already gives a balanced description, recovering about 97% of the experimental correlation contribution to cohesion. They estimated ±3% residual error from truncating the incremental expansion and 1–2% each from basis-set incompleteness at the correlated and HF levels.

What the 2014 treatment deliberately left out is the long-range piece. In a metal, electronic correlation separates into a short-range part, expressible through local increments, and a long-range charge-fluctuation part, governed by screening of the conduction electrons, which is naturally described by the random-phase approximation (RPA) — a sum of ring diagrams equivalent to screening the Coulomb interaction by the density response of the electron gas, an idea going back to Heilingbrunner and Stollhof's ab initio correlation calculation for metallic lithium (1993). Combining a local short-range treatment with an RPA long-range part without double counting either regime is a nontrivial and unsolved step for the incremental scheme. Meanwhile, an adjacent, methodologically distinct line has advanced for simple (single-reference) metals: plane-wave periodic coupled-cluster theory, where second-order perturbation theory suffers an infrared divergence in zero-gap systems, cured by modified perturbative triples (Masios et al., 2023) or nonperturbative triples (Neufeld and Berkelbach, 2023); local coupled cluster embedded in an RPA description of the periodic environment (Schäfer et al., 2021); and converged coupled-cluster ground states for metals exploiting the weak coupling between long-range and short-range correlation (Schäfer, 2025). These developments show the long-range/short-range separation is physically controllable in metals, but they are single-reference, plane-wave constructions and do not address multireference metals treated through localized incremental expansions.

Problem Statement

Complete the wave-function-based incremental correlation treatment of bulk metals left unfinished by the first multireference treatment of fcc Ca and Sr (Voloshina and Paulus, 2014). Specifically: (1) formulate a double-counting-free combination in which the short-range correlation is collected by the (multireference) method of increments and the long-range charge-fluctuation/screening correlation is treated by a random-phase-approximation-type approach, so that the two parts add to the total correlation contribution to the cohesive energy of a metal; and (2) compute this combination for at least the paradigmatic multireference metals Ca and Sr, and quantify the impact of the long-range part on the incremental sums (by order and by distance class) and on the final cohesive energies.

References

  1. Elena Voloshina and Beate Paulus, 'First Multireference Correlation Treatment of Bulk Metals', J. Chem. Theory Comput. 10, 1698-1706 (2014). DOI: 10.1021/ct401040t; preprint arXiv:1402.6514. https://doi.org/10.1021/ct401040t
  2. Andrea Heilingbrunner and Gernot Stollhof, 'Ab initio correlation calculation for metallic lithium', J. Chem. Phys. 99, 6799-6809 (1993). DOI: 10.1063/1.465823. https://doi.org/10.1063/1.465823
  3. B. Paulus, 'The method of increments - a wavefunction-based ab initio correlation method for solids', Phys. Rep. 428, 1-52 (2006). DOI: 10.1016/j.physrep.2006.01.003. https://doi.org/10.1016/j.physrep.2006.01.003
  4. Elena Voloshina and Beate Paulus, 'Wave-function-based ab initio method for metals: Application of the incremental scheme to magnesium', Phys. Rev. B 75, 245117 (2007). DOI: 10.1103/PhysRevB.75.245117. https://doi.org/10.1103/PhysRevB.75.245117
  5. Daniel Koch, E. Fertitta and Beate Paulus, 'Calculation of the static and dynamical correlation energy of pseudo-one-dimensional beryllium systems via a many-body expansion', J. Chem. Phys. 145, 024104 (2016). DOI: 10.1063/1.4955317. https://doi.org/10.1063/1.4955317
  6. E. Fertitta, D. Koch, B. Paulus, G. Barcza and Ö. Legeza, 'Towards a multiconfigurational method of increments', Mol. Phys. 116, 1471-1482 (2018). DOI: 10.1080/00268976.2018.1444208. https://doi.org/10.1080/00268976.2018.1444208
  7. Hermann Stoll, 'Toward a wavefunction-based treatment of strong electron correlation in extended systems by means of incremental methods', J. Chem. Phys. 151, 044104 (2019). DOI: 10.1063/1.5109860. https://doi.org/10.1063/1.5109860
  8. K. G. Steenbergen, J.-M. Mewes, L. F. Pašteka, H. W. Gäggeler, G. Kresse, E. Pahl and P. Schwerdtfeger, 'The cohesive energy of superheavy element copernicium determined from accurate relativistic coupled-cluster theory', Phys. Chem. Chem. Phys. 19, 32286-32295 (2017). DOI: 10.1039/C7CP07203A. https://doi.org/10.1039/C7CP07203A
  9. Inge Røeggen and Bin Gao, 'On the origin of bonding in metals: lithium as a case study', Mol. Phys. 120, e2117662 (2022). DOI: 10.1080/00268976.2022.2117662. https://doi.org/10.1080/00268976.2022.2117662
  10. Tobias Schäfer, Florian Libisch, Georg Kresse and Andreas Grüneis, 'Local embedding of coupled cluster theory into the random phase approximation using plane waves', J. Chem. Phys. 154, 011101 (2021). DOI: 10.1063/5.0036363. https://doi.org/10.1063/5.0036363
  11. Nikolaos Masios, Andreas Irmler, Tobias Schäfer and Andreas Grüneis, 'Averting the Infrared Catastrophe in the Gold Standard of Quantum Chemistry', Phys. Rev. Lett. 131, 186401 (2023). DOI: 10.1103/PhysRevLett.131.186401. https://doi.org/10.1103/PhysRevLett.131.186401
  12. Verena A. Neufeld and Timothy C. Berkelbach, 'Highly Accurate Electronic Structure of Metallic Solids from Coupled-Cluster Theory with Nonperturbative Triple Excitations', Phys. Rev. Lett. 131, 186402 (2023). DOI: 10.1103/PhysRevLett.131.186402. https://doi.org/10.1103/PhysRevLett.131.186402
  13. James M. Callahan, Malte F. Lange and Timothy C. Berkelbach, 'Dynamical correlation energy of metals in large basis sets from downfolding and composite approaches', J. Chem. Phys. 154, 211105 (2021). DOI: 10.1063/5.0049890. https://doi.org/10.1063/5.0049890
  14. Tobias Schäfer, 'Ground States for Metals from Converged Coupled Cluster Calculations', J. Phys. Chem. Lett. 16, 17-23 (2025). DOI: 10.1021/acs.jpclett.4c03134. https://doi.org/10.1021/acs.jpclett.4c03134
  15. Ruiheng Song, Xiliang Gong, Aamy Bakry and Hong-Zhou Ye, 'Random phase approximation-based local natural orbital coupled cluster theory', J. Chem. Phys. 164, 214114 (2026). DOI: 10.1063/5.0320612. https://doi.org/10.1063/5.0320612