Computational Methods in Organometallic Catalysis. Yu Lan

potential basis sets are widely used in computational organometallic chemistry, which shows a rather high efficiency for transition metals. Pseudo potential basis sets are not to calculate the inner electrons, but to describe the contribution of the inner electrons by a potential, which is placed in Hamiltonian. Pseudo potential basis sets actually include two parts, pseudo potential and the basis sets. The inner electrons adopt effective core potential (ECP), while the outer valence electrons adopt general basis sets. When pseudo potential basis sets are used, inner electrons do not need to be considered directly, so that the consumption of calculation is significantly reduced, which is more obvious for more heavy elements. In transition metal chemistry, relativistic effects mainly come from inner electrons. If the relativistic effect is taken into account in fitting the pseudo potential, that effect can be reflected equivalently in the calculation based on the pseudo potential.

      Generally, the pseudo potential basis sets used in computational organometallic chemistry include three categories: Stuttgart pseudo potentials, Los Alamos National Laboratory (LANL) pseudo potentials, and def series pseudo potentials [66, 77–82]. SDD is the most popular Stuttgart pseudo potential basis set, which often are used to describe the atomic orbitals of transition metals in the fifth and sixth periods. The accuracy of SDD is roughly equivalent to a triple‐zeta basis set.

      LANL series pseudo potentials were proposed by Hay and Wadt in 1980s, which involve relativistic effect for the elements in the fifth and sixth periods. The frequently used double‐zeta LANL series basis set is LANL2DZ [81]. LANL2TZ is restricted from LANL2DZ, which is a quasi‐triple‐zeta basis set [82]. In LANL2TZ+ or LANL2TZ(f) basis set, diffuse function or polarization function is involved, respectively. LANL08 is a completely derestricted basis set based on LANL2DZ. Diffuse function or polarization function also can be involved in LANL08+ or LANL08(f), respectively. Even now, LANL2DZ basis set has been widely used; however, it is not recommended because of its obviously low accuracy. For transition metals, LANL2TZ(f) and LANL08(f) are recommended because of great improvement in accuracy though it is slightly expensive [83].

      The accuracy of def2 series basis sets increases from def2‐SV(P), def2‐SVP, def2‐TZVP, def2‐TZVPP, def2‐QZVP, and def2‐QZVPP [83–87]. For the first four periods, def2 series basis sets are full electronic basis set. From the fifth period, they become pseudo potential basis sets, which are combined with Stuttgart small core pseudo potential. The main advantage of the def2 series basis set is that it covers all elements in the first six periods; therefore, def2 series basis set can be used solidly in most systems instead of a mixed basis set. In computational organometallic chemistry, def2‐TZVP basis set is suggested in energy calculation.

2.4 Solvent Effect

      Solvent effect is very important in organometallic chemistry; therefore, in theoretical calculations, it should be considered in energy calculations. Usually, the solvent effect in homogeneous catalysis calculations is considered by implicit solvent model.

      In implicit solvent model, the solvent environment is simply considered as a polarizable continuous medium; meanwhile, the structure and distribution of solvent molecules close to the solute are not specifically described. The advantage of implicit solvent model is that it can represent the average effect of solvents without the consideration of various possible molecular arrangements of solvent layer as explicit solvent model does, and it does not increase the computational time. Therefore, it is widely used in the field of computational organic and organometallic chemistry. The weakness of the implicit solvent model is that the strong interaction between solvent and solute cannot be represented, such as hydrogen bond. Moreover, the accuracy of solvation energy for ionic solute case is significantly lower than that of neutral solute case.

      In implicit solvent model, the solvent effect can be divided into polar and non‐polar parts. The polar part, which is the main body of the implicit solvent model, reflects the electrostatic interaction between solvent and solute molecule, and also includes the polarization of solute electron distribution contributing by solvent. The nonpolar part reflects various nonelectrostatic interactions between solute molecule and solvent, which includes the solvent exclusion energy, the influence of solute molecule on the entropy effect of solvent, and the dispersion between solute molecule and solvent.

      The frequently used implicit solvent model is a polarizable continuum model (PCM) [88], in which the solvation free energy can be decomposed into three terms

      in which G_es represents the electrostatic energy, G_dr represents the dispersion–repulsion energy, and G_cav represents the hole energy. All those three terms are calculated by a hole defined by a chain of van der Waals spheres centered on the solute atoms. Dielectric formulation (D‐PCM) is the early member of PCM family, which only includes the charge density of the solute wavefunction within the solute surface into the solute–solvent interaction. The integral equation formalism polarizable continuum model (IEF‐PCM), developed by Cances and Mennucci, also includes the charge density of the wavefunction beyond the solute surface into the solute–solvent interaction [89]. The conductor‐like polarizable continuum model (C‐PCM), developed by Barone and Cossi, is the implementation of conductor‐like screening model in the PCM framework, which works well for solvents with a high dielectric constant such as water solvent [90, 91]. In isodensity polarizable continuum model (I‐PCM) and self‐consistent IPCM (SCI‐PCM), the solute cavity can be defined as a surface with a constant electron density (isodensity surface) [92].

In computational organometallic chemistry, SMD (solvation model based on density), proposed by Cramer and Truhlar in 2009, is almost the best implicit solvent model for DFT calculations at present [93]. In the SMD model, the bulk electrostatic contribution is calculated on the basis of the IEF‐PCM protocol.

2.5 How to Choose a Method in Computational Organometallic Chemistry

      2.5.1 Why DFT Method Is Chosen

      The current computational studies of organic and organometallic systems are frequently subjected to a compromise of accuracy and time consumption: while it is always desired to reach the highest possible accuracy in the computational assessment, the necessity to complete the calculations within a reasonable time scale does not always allow for that.

The HF method ignores instantaneous electron correlation; therefore, it can be excluded firstly. Post‐HF methods attempt to treat electron correlation through several methods, which usually can provide a good accuracy in theoretical calculations. However, rather high scaling behaviors of those methods (Table 2.2) restrict the application in computational organometallic chemistry. In fact, some of the post‐HF methods are often used to calculate some small systems to obtain accurate results as a benchmark reference for other computational methods.

Table 2.2 Scaling behaviors of computational methods.

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Methods	Behaviors
HF	N ^4
MP2	N ^5
MP3, CISD, CCSD	N ^5
MP4, CCSD(T)	N ^7
MP5, CISDT	N ^8