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Test of Computational Approaches for Gold-Thiolate Clusters Calculation using Lomonosov Supercomputer

High-level procedures (MP2, CCSD, CCSD(T)) and reliable experimental data have been used to assess the performance of a variety of exchange-correlation functionals for the calculation of structures and energies of small models of thiolate-protected gold clusters. Clusters represent rather complicated objects for examination, therefore the simple models including Au2, AuS were considered to find an appropriate method to calculate Au-Au and Au-S interactions in protected clusters. The mean unsigned errors of the quantum chemical methods were evaluated via reliable experimental bond distances and dissociation energies of Au2 and AuS. Based on the calculation, the SVWN5, TPSS+D3, PBE96+D3, and PBE0+D3 were found to give the most reliable results and can be recommended for calculation of the structure and properties of thiolate-protected gold clusters. The influence of the relativistic corrections calculated in Dirac-Coulomb-Breit framework and inclusion of dispersion corrections on the structure and energy of thiolate-protected gold clusters have been analyzed.


Introduction
Supercomputer simulation based on quantum chemical methods is effectively applied to predict and examine the structure and properties of different compounds including nanoclusters, materials, metal-organic systems, and drugs [1,2]. Thiolate-protected gold clusters, Au n (SR) m , are popular objects in theoretical and nano science. The quantum chemical study of clusters has a number of challenges: (i) To describe Au-Au and Au-S interactions correctly, the approach should take into account relativistic effects, dispersion interactions, and others features; (ii) High-level ab initio procedures are not feasible for Au n (SR) m , since they become very resourceintensive; (iii) Application of the Au n (SR) m data in solid or liquid phases as a benchmark of theoretical methods is difficult due to that a majority of quantum chemical methods are related to a gas phase. The problem of an extensive global search of potential energy surface to locate global minima can be solved by the genetic algorithm with DFT calculation [3].
We propose new insights to choose the quantum chemical protocol of calculation of gold protected clusters and similar metal-organic systems performing benchmark study of the simple fragments of cluster. Clusters represent rather complicated objects for examination, therefore the simple models including Au 2 , AuS were considered.
The Dirac-Coulomb-Breit Hamiltonian (DCB) with separate spin-free and spin-dependent components [10] was employed. The energy-optimized extended Gaussian basis set of triplepolarized quality of the large component, and the corresponding kinetically balanced basis for the small component [11] was used (BS-4). The dispersion corrections with the Becke-Johnson damping (DFT+D3) [12] was also calculated.
For each protocol mean unsigned errors (MUEs) were established. The DFT MUEs were compared with the MP2, CCSD, and CCSD(T) MUEs and divided into three groups: (i) accurate procedure if MUE is less than CCSD(T)/BS-1 MUE; (ii) less accurate procedure if MUE is in the range between CCSD(T)/BS-1 and CCSD(T)/BS-2; and (iii) erroneous procedure if MUE is more than MUE of CCSD(T)/BS-2.

Results and Discussion
The calculated R e values of Au 2 molecule are compared to the experimental data 2.4719 [4]; the MUEs are shown on Fig. 1. Evidently, CCSD(T)/BS-1 and MP2/BS-1 protocols give the most accurate Au-Au distance amongab initio methods. For all ab initio methods, the MUEs increase in the order BS-1 <BS-2 <BS-3. Thus, the expanding of a basis set slightly improve the distance calculated at CCSD(T) level only. In contrast, the Re calculated by different functionals slightly varies from the basis sets. The first group of the high accurate functionals contains SVWN5/BS-1, SVWN5/BS-2, SVWN5/BS-3, and M11/BS-1. The homogeneous electron gas approximation to XC energy could be provided the advance of SVWN5. The second group is represented by PW91, PBE96, BP86, TPSS, PBE0, BHandH, MPW1K, B3P86, B3PW91, HSE06, TPSSh. Other functionals including BLYP, B3LYP, M06, M06L, M11L have high MUEs, more than 0.08 , and failed to describe the experimental Au-Au distance. It should be noted that the majority of protocols tend to overestimate R e in Au 2 , except MP2/BS-1 and SVWN5/BS-1. Dispersion corrections slightly effect the R e . The Au-Au distance calculated by DCB framework is equal to 2.530 ; MUE=0.058 .
Among ab initio methods, CCSD(T)/BS-1 and MP2/BS-1 successfully predict the dissociation energy of Au 2 (Fig. 1) The dispersion corrections slightly influence the results. Relativistic DCB dissociation energy (2.30 eV) is completely in line with the experimental ones.
The equilibrium distance and dissociation energy of AuS are calculated using CCSD, CCSD(T), MP2, DFT, DCB. The experimental value of R e , 2.156 [5] was used as a benchmark. MP2/BS-1 accurately predicts the Au-S distance among the considered ab initio methods. For MP2, CCSD, CCSD(T) the error increases in the order BS-1 <BS-2 <BS-3. The Au-S distance calculated using DFT also depends on the basis set and a functional type. The first group of accurate procedure (with MUEs less than 0.06 ) contains all considered functionals with BS-1 excluding BLYP, HCTH, M11L, BHandH, B97, B98, B3P86, B3LYP, M06. The dispersion and relativistic corrections slightly influence R e value. For instance, the Au-S distance calculated in DCB approximation has the same error (0.07 ) as the first group of the DFT methods.
All  The calculation of Au-Au and Au-S distances in Au 2 and AuS reveals the functionals which are close in accuracy to high-level ab initio procedures and experimental data. Due to SVWN5, B3PW91, PBE96, PBE0, B3LYP, TPSS, TPSSh, M06, M06L, M11, M11L, PW91, BP86 procedure have the least errors in the predicted properties of Au 2 and AuS, they are further applied to benchmark calculation of the cyclic gold-thiolate complexes.
Finally, the Au 20 (SR) 16 structure is calculated by SVWN5, PBE96, TPSS, PBE96+D3, TPSS+D3, PBE0, PBE0+D3 methods using BS-2, BS-3 basis set and by DCB framework with BS-4 basis set to understand the accuracy and time of the DFT procedure to predict the interatomic distances in real thiol protected gold cluster. The main interatomic distances and average calculation time of one optimization step of calculation are collected in Tab. 1. The structure obtained at TPSS/BS-2 method proved to be more close to the experimental one than the structure predicted in PBE96/BS-2 and SVWN5/BS-2 approaches. It should be noted that dispersion corrections improve convergence of the calculated interatomic distances and the experimental ones using the PBE96 functional. The account of relativistic effects in DCB framework also improves the geometrical parameters, but the average calculation time of one optimization step is sufficiently longer than time of PBE96/BS-2 and PBE96/BS-3 protocols.

Conclusions
The structures and energies of small models of thiolate-protected gold clusters were calculated by ab initio methods (MP2, CCSD, CCSD(T)) and different exchange-correlation functionals including dispersion and relativistic corrections. To find the accurate method for the study of Au-Au and Au-S interactions, the diatomic Au 2 and AuS molecules were considered.
All calculated Au-Au bonds in the Au 20 (SR) 16 are slightly larger than the corresponding values in crystal structure. The account of dispersion corrections and relativistic effects improves the geometrical parameterizes. So, SVWN5, TPSS+D3, PBE96+D3, PBE0+D3 using BS-2 and BS-3 could be recommended for Au n (SR) m calculation, because they describe Au-Au and Au-S interactions more accurately.
The obtained results illustrated the complications of Au n (SR) m for theoretical investigation and provide new information to theorists and chemists studying structure and properties of protected gold clusters or other complicated chemical systems including self-assembled monolayers, ligand-protected metal clusters, and organometallic complexes.