Applications of Computational Chemistry to Selected Problems …
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and many validation studies have been conducted [39–42]. These studies have shown
that no functional is fully reliable, which is due to the challenges associated with
accurately describing dispersion, self-interaction correction, dynamical correlation,
and nondynamical correlation [28]. Nevertheless, B3LYP has traditionally been one
of the most popular functionals in the chemistry community [43, 44]. Whereas criticism against B3LYP is sometimes harsh, relatively good performance of B3LYP
has also been pointed out for many applications. For example, an important question
in relation to the studies of transition-metal complexes is how well DFT works in
predicting the relative stability of different spin states. Verma et al. have recently performed benchmark calculations on 14 iron complexes using 20 exchange–correlation
functionals [45]. In their benchmark study, PW6B95 predicted the correct ground
spin states for all 14 complexes examined. B3LYP, MPW1B95, and MN15 also performed well, producing correct answers for 13 complexes. This number decreased
to 12 when the OPBE, OLYP, and M06-L functionals were used. The use of a larger
basis set allows the calculated energy to be closer to the convergent result for a given
functional and reduces the occurrence of poor descriptions of electronic distributions, which is therefore recommended. However, in practice, large basis sets slow
down DFT computation significantly, and geometry is not as sensitive to the basis
set used as energy is. Therefore, geometry optimization is typically performed with
a small basis set (e.g., of double-ζ quality), which is followed by single-point energy
evaluation with a larger basis set. It should also be noted that larger basis sets do not
necessarily guarantee higher accuracy of energy values because of the approximate
nature of exchange–correlation functionals. In many cases, effective-core potential
basis sets are employed for transition metals. Semiempirical (or semiclassical) dispersion corrections such as DFT-D3 are often tacked onto the DFT energy to improve
the description of noncovalent interactions between fragments in the system [46].
When the target system is not very large and contains no more than ~150 atoms
(with the number depending on the available computers), DFT can easily be applied to
the system in its entirety (Fig. 1a). Today, the majority of DFT calculations that probe
the mechanisms of homogeneous catalytic reactions employ this kind approach.
However, the system would contain more atoms than can be handled by this DFT
approach. One possible solution to this problem is to use the QM/QM’ method in
which two different levels of quantum mechanics are used in combination (Fig. 1b).
QM and QM’ denote higher- and lower-level QM methods, respectively. For example,
a smaller basis set may be used for the QM’ calculation than for the QM calculation,
which can reduce the computational cost compared with the case of full QM calculation. Alternatively, semiempirical MO or density-functional tight-binding (DFTB)
methods may be used for the lower-level QM’ calculation. Any other combination
of two (or even more) different levels of QM methods is also possible.
The use of QM’ in QM/QM’ permits computational treatments of systems that
are larger than what can be treated by pure QM. However, QM/QM’ is still too
expensive when studying enzymes that contain thousands of atoms. In such cases, it
is advantageous to use the hybrid QM/MM method (Fig. 1c) [47–49], because MM
is computationally much less costly than QM’. When studying enzymatic systems,
DFT is usually used for the QM part of QM/MM, whereas force-field parameters that
465
and many validation studies have been conducted [39–42]. These studies have shown
that no functional is fully reliable, which is due to the challenges associated with
accurately describing dispersion, self-interaction correction, dynamical correlation,
and nondynamical correlation [28]. Nevertheless, B3LYP has traditionally been one
of the most popular functionals in the chemistry community [43, 44]. Whereas criticism against B3LYP is sometimes harsh, relatively good performance of B3LYP
has also been pointed out for many applications. For example, an important question
in relation to the studies of transition-metal complexes is how well DFT works in
predicting the relative stability of different spin states. Verma et al. have recently performed benchmark calculations on 14 iron complexes using 20 exchange–correlation
functionals [45]. In their benchmark study, PW6B95 predicted the correct ground
spin states for all 14 complexes examined. B3LYP, MPW1B95, and MN15 also performed well, producing correct answers for 13 complexes. This number decreased
to 12 when the OPBE, OLYP, and M06-L functionals were used. The use of a larger
basis set allows the calculated energy to be closer to the convergent result for a given
functional and reduces the occurrence of poor descriptions of electronic distributions, which is therefore recommended. However, in practice, large basis sets slow
down DFT computation significantly, and geometry is not as sensitive to the basis
set used as energy is. Therefore, geometry optimization is typically performed with
a small basis set (e.g., of double-ζ quality), which is followed by single-point energy
evaluation with a larger basis set. It should also be noted that larger basis sets do not
necessarily guarantee higher accuracy of energy values because of the approximate
nature of exchange–correlation functionals. In many cases, effective-core potential
basis sets are employed for transition metals. Semiempirical (or semiclassical) dispersion corrections such as DFT-D3 are often tacked onto the DFT energy to improve
the description of noncovalent interactions between fragments in the system [46].
When the target system is not very large and contains no more than ~150 atoms
(with the number depending on the available computers), DFT can easily be applied to
the system in its entirety (Fig. 1a). Today, the majority of DFT calculations that probe
the mechanisms of homogeneous catalytic reactions employ this kind approach.
However, the system would contain more atoms than can be handled by this DFT
approach. One possible solution to this problem is to use the QM/QM’ method in
which two different levels of quantum mechanics are used in combination (Fig. 1b).
QM and QM’ denote higher- and lower-level QM methods, respectively. For example,
a smaller basis set may be used for the QM’ calculation than for the QM calculation,
which can reduce the computational cost compared with the case of full QM calculation. Alternatively, semiempirical MO or density-functional tight-binding (DFTB)
methods may be used for the lower-level QM’ calculation. Any other combination
of two (or even more) different levels of QM methods is also possible.
The use of QM’ in QM/QM’ permits computational treatments of systems that
are larger than what can be treated by pure QM. However, QM/QM’ is still too
expensive when studying enzymes that contain thousands of atoms. In such cases, it
is advantageous to use the hybrid QM/MM method (Fig. 1c) [47–49], because MM
is computationally much less costly than QM’. When studying enzymatic systems,
DFT is usually used for the QM part of QM/MM, whereas force-field parameters that