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This work provides a rigorous analysis of Bain-type cuboidal lattice transformations, which connect the face-centered cubic (fcc), mean-centered cubic (mcc), body-centered cubic (bcc), and axially centered cubic (acc) lattices. Our study incorporates a general (n, m) Lennard-Jones (LJ) two-body potential and a long-range repulsive Axilrod\u2013Teller\u2013Muto (ATM) three-body potential. The two-body lattice sums and their meromorphic continuations are evaluated to full precision using super-exponentially convergent series expansions. Furthermore, we introduce a novel approach to computing three-body lattice sums by converting the multi-dimensional sum into an integral involving products of Epstein zeta functions. This enables us to evaluate three-body lattice sums and their meromorphic continuations to machine precision within minutes on a standard laptop. Using our computational framework, we analyze the stability of cuboidal lattice phases relative to the close-packed fcc structure along a Bain transformation path for varying ATM coupling strengths. We analytically demonstrate that the ATM cohesive energy exhibits an extremum at the bcc phase and show numerically that it corresponds to a minimum for repulsive three-body forces along the Bain path. Our results indicate that strong repulsive three-body interactions can destabilize the fcc phase and render bcc energetically favorable for soft LJ potentials. However, even in this scenario, the bcc phase remains susceptible to further cuboidal distortions. These results suggest that the stability of the bcc phase is, besides vibrational, temperature, and pressure effects, strongly influenced by higher than two-body forces. Because of the wrong short-range behavior of the triple\u2013dipole ATM model, the LJ potential is limited to exponents n &amp;gt; 9 for the repulsive wall, otherwise one observes distortion into a set of linear chains collapsing to the origin.<\/jats:p>","DOI":"10.1063\/5.0276677","type":"journal-article","created":{"date-parts":[[2025,9,2]],"date-time":"2025-09-02T10:08:26Z","timestamp":1756807706000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Exact lattice summations for Lennard-Jones potentials coupled to a three-body Axilrod\u2013Teller\u2013Muto term applied to cuboidal phase transitions"],"prefix":"10.1063","volume":"163","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9586-1003","authenticated-orcid":false,"given":"Andres","family":"Robles-Navarro","sequence":"first","affiliation":[{"name":"Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland 1 , Private Bag 102904, 0745 Auckland,","place":["New Zealand"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5103-0400","authenticated-orcid":false,"given":"Shaun","family":"Cooper","sequence":"additional","affiliation":[{"name":"School of Natural and Computational Sciences, Massey University Albany 2 , Private Bag 102904, Auckland 0745,","place":["New Zealand"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4004-713X","authenticated-orcid":false,"given":"Andreas A.","family":"Buchheit","sequence":"additional","affiliation":[{"name":"Saarland University, Department of Mathematics 3 , P.O. Box 15 11 50, D-66041 Saarbr\u00fccken,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-3323-3455","authenticated-orcid":false,"given":"Jonathan K.","family":"Busse","sequence":"additional","affiliation":[{"name":"Saarland University, Department of Mathematics 3 , P.O. Box 15 11 50, D-66041 Saarbr\u00fccken,","place":["Germany"]},{"name":"German Aerospace Center (DLR), Institute of Software Technology, High-Performance Computing Department 4 , Linder H\u00f6he, Cologne 51147,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4118-3048","authenticated-orcid":false,"given":"Antony","family":"Burrows","sequence":"additional","affiliation":[{"name":"Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland 1 , Private Bag 102904, 0745 Auckland,","place":["New Zealand"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1259-147X","authenticated-orcid":false,"given":"Odile","family":"Smits","sequence":"additional","affiliation":[{"name":"The School of Mathematics and Physics, University of Queensland 5 , Brisbane QLD 4072,","place":["Australia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4845-686X","authenticated-orcid":false,"given":"Peter","family":"Schwerdtfeger","sequence":"additional","affiliation":[{"name":"Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland 1 , Private Bag 102904, 0745 Auckland,","place":["New Zealand"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,9,2]]},"reference":[{"key":"2025090206082458800_c1","volume-title":"Phase Transitions and Critical Phenomena","year":"1971"},{"key":"2025090206082458800_c2","doi-asserted-by":"publisher","first-page":"C4-3","DOI":"10.1051\/jphyscol:1982401","article-title":"Landau theory and martensitic phase transitions","volume":"43","year":"1982","journal-title":"J.\u00a0Phys. 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