Computational Methods in Organometallic Catalysis. Yu Lan
which each atomic orbital is represented by a single mathematical function. The atomic orbitals used in this procedure are represented by what is known as the basis set. Following this idea, the mathematical form of atomic orbitals should be considered, when it is used to be linear combinations for the construction of molecular orbitals. One choice would be to simply use the hydrogenic wavefunctions adapted for other atoms, which is called a Slater-type orbital (STO). These wavefunctions have radial forms possessing terms such as rn−1e−ζr (ζ = Z/n), whose function form has clear physical meaning (Scheme 2.3a). However, it is very difficult to evaluate the complex two‐electron integrals.
Scheme 2.3 The combination of Gaussian‐type orbitals (GTOs) for the construction of Slater‐type orbital (STO). (a) STO. (b) GTO. (c) The combination of GTOs.
Following a suggestion of Boys, Pople decided to use a combination of Gaussian‐type functions to mimic the STO, which was named Gaussian‐type orbital (GTO). These orbitals have a different spatial function, X′Y‴Z′e−ζr2; therefore, the integrals required to build the Fock matrix can be evaluated exactly (Scheme 2.3b). The tradeoff is that GTOs do differ in shape from the STOs, particularly at the nucleus where the STO has a cusp, but the GTO is continually differentiable. The computational advantage is so substantial that it is more efficient to represent a single atomic orbital as a combination of several GTOs rather than a single STO (Scheme 2.3c). When a few GTOs with differing shapes are added, the result is a function that resembles an STO. Now, the basis set is not just the atomic orbitals, but is instead all the GTOs that are used to make up the atomic orbitals. The minimum Gaussian‐type basis set is STO‐3G, in which “STO” is the abbreviation of STO, and “3G” means that each STO is obtained by a linear combination of three GTOs [66, 67].
The minimum basis set has one basis function for every formally occupied or partially occupied orbital in the atom, which is referred to as a single‐zeta (SZ) basis set. The use of the term zeta here reflects that each basis function mimics a single STO, which is defined by its exponent, zeta (ζ). The minimum basis set is usually inadequate, failing to allow the core electrons to get close enough to the nucleus and the valence electrons to delocalize. An obvious solution is to double the size of the basis set, creating a double‐zeta (DZ) basis. Further improvement can be had by choosing a triple‐zeta (TZ) or even larger basis set [68].
2.3.2 Pople's Basis Sets
As most of chemistry focuses on the action of the valence electrons, Pople developed the split‐valence basis sets, single zeta in the core and double zeta in the valence region, which won him the 1998 Nobel prize in chemistry. A double‐zeta split‐valence basis set for carbon has three s basis functions and two p basis functions for a total of nine functions, a triple‐zeta split valence basis set has four s basis functions and three p functions for a total of 13 functions, and so on.
For the vast majority of basis sets, including the split‐valence sets, the basis functions are not made up of a single Gaussian‐type function. Rather, a group of Gaussian‐type functions are contracted together to form a single basis function. An example is split‐valence basis set 6‐31G, which is popular in computational organic chemistry. In this basis set, the left value means that each core basis function comprises six Gaussian functions. Meanwhile, the valence space is split into two basis functions, which referred to the inner and outer parts of valence space. The inner basis function is composed of three contracted Gaussian‐type functions, and each outer basis function is a single Gaussian‐type function. Thus, for carbon, the core region is a single s basis function made up of six s‐GTOs. The carbon valence space has two s and two p basis functions. The inner basis functions are made up of three Gaussians, and the outer basis functions are each composed of a single Gaussian‐type function. Therefore, the carbon 6‐31G basis set has nine basis functions made up of 22 Gaussian‐type functions. This type of split‐valence basis sets involves 3‐21G, 4‐31G, 6‐31G, 6‐311G, etc. [69–72]. The accuracy of those basis sets depends mainly on the number of basis functions, and secondly on the number of Gaussians. However, the time consumed in calculation increases accordingly with the improvement of accuracy.
2.3.3 Polarization Functions
A critical problem with a simple split‐valence basis set, such as 6‐31G, is that the flexibility of wavefunctions is insufficient to distort to the actual shape. Extending the basis set by including a set of functions that mimic the atomic orbitals with angular momentum one greater than in the valence space greatly improves the basis flexibility. These added basis functions are called polarization functions. For second and third periodic elements, adding polarization functions means adding a set of d GTOs; therefore, a basis set involving polarization functions for heavy atoms can be written as 6‐31G(d) or 6‐31G*. For hydrogen, polarization functions are a set of p functions. Therefore, a basis set involving polarization functions for all atoms can be written as 6‐31G(d,p) or 6‐31G**. As adding multiple sets of polarization functions has become broadly implemented, the use of asterisks has been abandoned in favor of explicit indication of the number of polarization functions within parentheses, that is, 6‐311G(2df,2p) means that two sets of d functions and a set of f functions are added to heavy atoms and two sets of p functions are added to the hydrogen atoms. The polarization functions are simply mathematical tools that allow to give the basis set more flexibility, and thus produce a better calculation.
2.3.4 Diffuse Functions
For anions or molecules with many adjacent lone pairs, the basis set must be augmented with diffuse functions to allow the electron density to expand into a larger volume. For split‐valence basis sets, this is designated by “+” as in 6‐31+G(d). The diffuse functions added are a full set of additional functions of the same type as are present in the valence space. So, for carbon, the diffuse functions would be an added s basis function and a set of p basis functions. If a molecule involving hydride, diffuse functions for hydrogen atom is necessary, which would be an added s basis function, the corresponding split‐valence basis set can be written as 6‐31++G(d).
2.3.5 Correlation‐Consistent Basis Sets
The split‐valence basis sets developed by Pople are widely used. The correlation‐consistent basis sets developed by Dunning are popular alternatives. The split‐valence basis sets were constructed by minimizing the energy of the atom at the HF level with respect to the contraction coefficients and exponents. The correlation‐consistent basis sets were constructed to extract the maximum electron correlation energy for each atom. The correlation‐consistent basis sets are designated as “cc‐pVNZ,” to be read as correlation‐consistent polarized split valence N‐zeta, where N designates the degree to which the valence space is split [73–76]. The “cc‐p,” stands for “correlation consistent polarized” and the “V” indicates they are valence only basis sets. As N increases, the number of polarization functions also increases. So, for example, the cc‐pVDZ basis set for carbon is double‐zeta in the valence space and includes a single set of d functions, and the cc‐pVTZ basis set is triple‐zeta in the valence space and has two sets of d functions and a set of f functions. The addition of diffuse functions to the correlation‐consistent basis sets is designated with the prefix aug‐, as in aug‐cc‐pVTZ. Usually, the correlation‐consistent basis sets are used in post‐HF level calculations, which revealed a significantly better accuracy.
2.3.6 Pseudo