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Wu et al. J. Mater. Inf. 2025, 5, 14 https://dx.doi.org/10.20517/jmi.2024.77 Page 5 of 15

although studies have shown that high coverage of adsorbates can trigger surface reconstruction and
significantly change the surface’s catalytic properties [29,34,37] . For potential adsorbate desorption issues after
geometric optimization, we can avoid them by introducing an empirical distance threshold in the initial
guess modeling before optimization, effectively reducing the computational cost of unreasonable
configuration guesses. In our studied systems, the differences in adsorption energy between adsorption
configurations primarily depend on the adsorption sites, so enumerating the combination space of
adsorption sites on the surface conveniently yields our target configuration guess space, which can be
quickly achieved for a finite-sized slab.

The enumeration of adsorption configuration guess space is conducted on eight Cu surfaces, including
(100), (110), (111), (210), (221), (310), (311), and (322), which are considered to be worth exploring by
researchers [38-42] . As shown in Figure 2, the enumeration process is divided into three steps: surface site
search, adsorption configuration enumeration, and deduplication of equivalent configuration guesses. In the
site search part, we define the collection of Cu atoms on the surface that can bond with adsorbates and their
surrounding environment as a site. Therefore, the types of sites depend on the size of the Cu atom
collection and its local environment. Here, we abstract sites and their local environments into graphs using
graph theory methods and determine the uniqueness of sites by judging whether the site graphs are
[25]
isomorphic . Using graph theory methods, we conveniently identified 68 unique sites from the eight Cu
surfaces and used a combination letter naming method to distinguish different site types, for example, the
“Bb” site type, where “B” indicates the site has a coordination number of n = 2, and “b” indicates it is the
second graph structure among sites with the same coordination number, and so on. This purely geometric
definition of sites does not restrict the morphology of the sites, reducing the damage that biased
understanding may cause in constructing a complete configuration guess space [25,36] . Its generality and
scalability make it convenient to apply to the construction of configuration guesses in other high coverage
systems [43-45] . The graph-theoretical site comparison method allows for a simple and rapid quantitative
description of differences between sites.

After detecting all adsorption sites on the Cu surfaces, we used a method of filling CO molecules on the
clean Cu surface with distance limitations to achieve the enumeration of configuration guesses, with the
main requirements being: CO adsorbs monodentately on the Cu surface with only C contacting Cu atoms;
the orientation of CO molecules is set to the vector sum of the direction vector from its coordinating Cu
atom to the C atom; the filling distance limitation requires that the interatomic distance between different
CO molecules must not be less than 2.3 Å. Consequently, we obtained approximately 44 million initial
adsorption configuration guesses, with the number of configurations enumerated on different index faces
increasing with CO coverage before decreasing [Supplementary Table 1]. Based on the site type naming
method, we named the adsorption configuration according to the combination types of sites they belong to,
for example, “Aa2Ab1Da1Dc2” site combination type, indicating 2, 1, 1, 2 CO molecules adsorb on “Aa”,
“Ab”, “Da”, “Dc” types of sites, respectively. Moreover, we defined a simplified site combination type, such
as the simplified type for “Aa2Ab1Da1Dc2” being “AD”. Subsequent configuration data sampling
calculations will be based on these two types of site combinations. Notably, the previous enumeration
process did not consider the intrinsic symmetry of different Cu surfaces; for example, the Cu(111) surface
corresponds to the p3m1 plane group [Supplementary Tables 3 and 4] [46-48] . The existence of such planar
symmetry results in a large number of duplicate structures in the enumerated configuration guesses. As
shown in Supplementary Figure 1, two adsorption configurations A and B on Cu(111) with two CO
molecules that do not correspond to the same atomic coordinates may actually be equivalent structures.
Through symmetry operations, we obtained approximately 7 million independent initial adsorption
configuration guesses from the 44 million, significantly narrowing the target configuration space range