Computational Methods in Organometallic Catalysis. Yu Lan

      where T_e′ is the kinetic energy of noninteracting electrons whose density is the same as the density of the real electrons, the true interacting electrons. V_ne is the nuclear–electron attraction term. V_ee is the classical electron–electron repulsion [37]. The last term is called the exchange–correlation functional, and is a catch‐all term to account for all other aspects of the true system. However, it offers no guidance as to the form of that functional.

      The exchange–correlation functional is generally written as a sum of two components, an exchange part and a correlation part. This is an assumption, an assumption that we have no way of knowing is true or not. These component functionals are usually written in terms of an energy density ε

      The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist, which permit the calculation of certain physical quantities quite accurately. One of the initial simple approximations of exchange–correlation functional is the local‐density approximation (LDA), in which the exchange–correlation functional of uniform electron gas with same density is used as the approximation of the corresponding nonuniform system [38]. Unexpectedly, such a simple approximation often yields good results, which directly led to the widespread application of DFT currently. If the electron densities of different spin components are further considered, the local spin density approximation (LSDA) can be obtained. Despite the great success of L(S)DA, there are many shortcomings, such as systematic overestimation of binding energies.

To make improvements over the L(S)DA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not just the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA). It is at this point that the form of the functionals begins to cause the eyes to glaze over and the acronyms to appear to be random samplings from an alphabet soup. The method of constructing GGA exchange–correlation functional can be divided into two ways. One is the group headed by Becke, which believes that “everything is allowed.” Any formation of exchange–correlation functionals for any reason can be chosen, while the quality of this formation only depends on the comparison with real system. Another group, led by Perdew, believes that the development of exchange–correlation functionals should be based on certain physical laws, such as scaling relations and progressive behavior.

      2.2.2 Jacob's Ladder of Density Functionals

Indeed, there is no unified standard for the classification of density functionals in the physical chemistry field. In 2001, J. P. Perdew et al. proposed using “Jacob's ladder” to classify the level of density functionals [39, 40]. As shown in Scheme 2.1, the ground in the “Jacob's ladder” is HF theory, which is an imprecise method with neither exchange energy nor correlation energy. In fact, HF calculation is rarely used in theoretical and computational chemistry nowadays.

Scheme 2.1 Jacob's ladder of density functionals.

      The first rung in “Jacob's ladder” is the density functional based on L(S)DA, the variable in which kind of functionals is the local spin density. The exchange functional of L(S)DA can be written as analytic expressions, which is often called Slater or Dirac exchange functional. However, the correlation functional of L(S)DA has no analytic expression, and can only be fitted by a functional with parameters from the results of high‐level calculations on some uniform electron gases. L(S)DA has achieved surprising success in the early works on the computational study of solid‐state physics. However, it is failure in computational chemistry because L(S)DA usually overestimates the bonding energy.

      2.2.3 The Second Rung in “Jacob's Ladder” of Density Functionals

The second rung in “Jacob's ladder” of density functionals is the GGA. The variables in this kind of functionals are local spin density and its gradient. Therefore, there are no analytic expressions for both exchange functionals and correlation functionals of GGA density functionals. The first successful GGA density functional for chemical calculation was Becke–Lee–Yang–Parr hybrid DFT functional (BLYP) [41, 42]. Thereinto, B was Becke88 exchange functional, while LYP was Lee–Yang–Parr correlation functional. Around 2000, the commonly used GGA functionals in computational organometallic chemistry are the Perdew–Burke–Ernzerh (PBE) and BP86 functionals [43, 44]. Although these functionals are seldom used at present, many of popular functionals are developed on the basis of these functionals.

      2.2.4 The Third Rung in “Jacob's Ladder” of Density Functionals

      The third rung in “Jacob's ladder” of density functionals is meta‐GGA functionals. The variables with more functionals than GGA are the kinetic energy density or the second derivative of the local spin density. The most common meta‐GGA involved are M06‐L, TPSS, and VSXC [45–47], which are often used in computational organometallic chemistry currently.

      2.2.5 The Fourth Rung in “Jacob's Ladder” of Density Functionals

      The fourth rung in “Jacob's ladder” of density functionals is hybrid‐GGA and hybrid‐meta‐GGA. This kind of functionals are the most popular functional in computational chemistry currently, into which HF exchange is introduced. In the field of computational organometallic chemistry, the commonly used hybrid‐GGA functionals involves B3LYP [42, 48], B97 [49], O3LYP [50], PBE0 [51], mPW1PW [52], X3LYP [53], etc.; the commonly used hybrid‐meta‐GGA functionals involves M05, M05‐2X, M06, M06‐HF, M06‐2X, TPSSh, MPW1K, etc. [46, 54–58].

      Undoubtedly, B3LYP is the most widely used functional in computational chemistry, which combines exact HF exchange with Becke's gradient‐corrected exchange, the LYP correlation functional, and VWN for the local correlation terms, as the following function:

      According to Becke's parametrization against the G1 database, a₀, a_x, and a_c are taken as 0.20, 0.72, and 0.81, respectively.

      2.2.6 The Fifth Rung in “Jacob's Ladder” of Density Functionals

      In the fifth rung of “Jacob's ladder,” the information of virtual orbital is