*2.3. Energetics*

In this part, we focus on the relative stability of *pc*, *al*, and *wc* conformers for both pristine calixarenes and complexes. The stability order can be established by comparing total energies, which are sums of the DFT+D electronic energy calculated at minimum and the nuclear contribution, given by zero-point vibrational energy (ZPVE). In practice, since these total energies have large absolute values and are similar to each other, it is the most useful to find the lowest conformer and to report relative differences of energies for the two remaining conformers. It should be emphasized that the energetic order of complexes of different calixarene conformers with the same amino acid, which results from this procedure, does not always correspond to the order of the interaction energies since a difference in conformers' energies contains additional deformation contributions for the interacting molecules. In particular, the calixarene's deformation may vary substantially for different complexes because of the large flexibility of calixarenes.

The selected complexes in their optimal geometries have been studied with the help of several methods in order to perform a detailed analysis of their energetics with special attention paid to energy partitioning. A method of choice for such a study is symmetry-adapted perturbation theory (SAPT) [44,88], which allows us to obtain not only intermolecular interaction energies but also their components, such as electrostatic, induction, dispersion, and their exchange counterparts, with a well-defined physical interpretation,

$$E\_{\rm int} = E\_{\rm elst}^{(1)} + E\_{\rm exch}^{(1)} + E\_{\rm ind}^{(2)} + E\_{\rm exch-ind}^{(2)} + E\_{\rm disp}^{(2)} + E\_{\rm exch-disp}^{(2)} + \delta E\_{\rm HF}.\tag{1}$$

In this study, we utilized the simplest SAPT variant, denoted as SAPT(HF) or alternatively as SAPT0 [89] SAPT0 is also a method of choice for further F-SAPT analysis.

Supermolecular interaction energies were obtained on the MP2 and Spin-Component-Scaled MP2 (SCS-MP2) [90] levels of theory with the jun-cc-pVDZ basis set [91–93]. Since the DF implementation of these methods was used, the corresponding default auxiliary basis sets [94–96] were applied in these calculations, which were performed with the Psi4 program [97]. In order to estimate the accuracy of the interaction energies with respect to the basis set, we performed the Complete-Basis-Set (CBS) extrapolation [98] for the correlation part of the interaction energy for one selected complex. For all supermolecular calculations, the Boys–Bernardi counterpoise correction was employed to avoid the basis-set superposition error [99].

Additionally, we studied the interaction energies with the help of a Symmetrized Systematic Molecular Fragmentation on the third level [49] with interaction energies between fragments calculated with the MP2 method (SSMF3-MP2) method. The SSMF arises from the SMF by Annihilation (SMFA) [100], and differs from the original one by treatment of branched molecules [49]. In the SMFA, one first defines molecular units, which are composed either of a single nonhydrogen atom, or atoms connected by multiple bonds, both with attached hydrogen atoms. Then, a molecule is fragmented into overlapping units, where broken single bonds are saturated with hydrogen atoms. Therefore, the molecule is represented as a sum of overlapping fragments with positive and negative weights, and the energy is calculated as a sum of the energies of overlapping fragments corrected with a sum of interaction energies of non-overlapping units [100–102], i.e., if the fragmentation of a molecule M into fragments Gi with weights wi is performed, the total energy of M is expressed as

$$\rm E(M) = \sum\_{\rm i} w\_{\rm i} E(G\_{\rm i}) + E\_{\rm nb\prime} \tag{2}$$

where E(X) is the energy of the molecule *X* and Enb is the (usually small) interaction energy of the non-overlapping parts. The size of fragments is given in terms of the number of units, and as a rule, one obtains fragments of size *n* and *n* + 1 for the *n*th level of fragmentation. In the SMFA approach, the fragments which incorporate branches are bigger and can be formed of *n* + 2 or more units, which makes the calculation more expensive computationally. In our recent publication [49] we proposed an alternative treatment of branching points, which keeps an immediate symmetry around branches (therefore the name) and–more importantly–preserves the number of units at cost of introducing fractional weights. Numerical tests show that the third fragmentation level (i.e., SSMF3) has sufficient accuracy for practical applications. We also studied the applicability of the SSMF3 approach for intermolecular interaction energies and found that the computationally expensive long-range correlation energy contribution to the interaction energy is well reproduced on the SSMF3 level [50].

The SMF approach may be also utilized to obtain the interaction energies. If one molecule (M) is large enough to undergo fragmentation, while the second one (A) is small and will be left unfragmented, then the interaction energy between M and A can be obtained by utilizing a *method* to calculate the interaction energy for each pair Gi with A, and then by applying the formula,

$$\mathrm{E}\_{\mathrm{int}}^{\mathrm{method}}(\mathrm{MA}) = \sum\_{\mathrm{i}} \mathrm{w\_{i}} \mathrm{E}\_{\mathrm{int}}^{\mathrm{method}}(\mathrm{G\_{i}A}). \tag{3}$$

This is exactly the case of complexes with calixarenes, where the amino acids are left unfragmented, while the calixarene host is fragmented at the SSMF3 level [50]. Altogether 24 such fragments are generated for the CX case: twelve 6-methyl-2,2 -methanediyl-diphenol, six 2,2 -methanediyl-di-phenol, and six 2,6-dimethylphenol molecules, so 24 (short) calculations of the interaction energies should be performed instead of one long calculation. The timing gain may be significant, especially for high-level electron-correlated theories, because of the steep scaling of such methods with respect to the molecular size, and since the calculations of fragments can be performed in parallel.

The SSMF3 partitioning has been performed by a homemade C++ program written by one of us (E.M.).

### *2.4. Molecular Properties*

For pristine calix[6]arene, in addition to the total energy calculations, we performed computations of selected first-order and second-order properties (electric dipole moments and static dipole polarizabilities) on the HF and MP2 levels. For these properties, we applied Equation (2) with energies replaced by the corresponding property and neglecting the nonbonding contribution. The HF and MP2 properties for SSMF3 fragments were obtained analytically with Molpro [103] (all electrons were correlated in the MP2 method because of the limitations of the MP2 polarizabilities program in Molpro [104]). Properties for unfragmented calixarenes were obtained with the finite-field (FF) approach of a field strength 0.003 a.u. and by comparing the 3- and 5-point formula to control the accuracy of the FF procedure (1*s* orbitals of nonhydrogen atoms were frozen in the MP2 method to speed up the calculations after verification that this approximation has a very small effect on these properties). Because of the testing character of this investigation, we used a small cc-pVDZ basis, for which the second-order properties for the whole molecules can be obtained in a manageable computational time.
