5 Introduction to Quantum Vibrational Spectroscopy
95
Therefore, MM is the only method of computational chemistry presently capable
of treating multiscale chemical systems. Examples include large biological systems,
solvated systems involving a large solvent volume, as well as composite materials.
The unmatched affordability of MM makes it useful for molecular dynamics simulations. It is possible, e.g. to obtain vibrational spectra of the molecular models
treated by molecular dynamics by calculating the dipole moment autocorrelation
function. From the point of view of NIR spectroscopy, however, the MM potentials are too approximate to yield useful results. Briefly mentioned here should be
hybrid quantum mechanics/molecular mechanics (QM/MM) approaches, in which
only the chemically most relevant part of the molecular system is treated quantum
mechanically while MM potentials are considered as sufficiently accurate to model
all remaining interactions. These QM/MM schemes enable a more accurate treatment of the potential in key molecular fragments important from the point of view
of a particular study.
5.3.2.6 The Fundamental Dilemma in Computational Chemistry; Cost
Versus Accuracy Factor
With few exceptions, in computational chemistry, a higher accuracy can only be
achieved with a significant increase in the demand for resources, understood mostly
as calculation time or/and memory requirements. The nominal complexity of a
method is limited to the number of electrons/atoms in the systems and scales
distinctly different among the methods presented here. From the point of view of
practical applications in spectroscopy, this should be a fundamental consideration
as the application of higher levels of theory to the molecular system of interest may
become prohibitively expensive. In the most straightforward case, the computational
complexity of MM simulations is proportional to the square of the number of treated
atomic centers N, O(N
2 ), whereas advanced implementations are capable of reducing
the scaling to O(N log N). The simplest ab initio HF method formally scales as
O(N
4 ). However, those schemes are widely regarded as not being sufficiently accurate
for spectroscopic applications. The significant improvement in accuracy of post-HF
approaches comes at a steep increase in their complexity, e.g., starting from O(N
5 )
for MP2, O(N
7 ) for CCSD(T), and O(N
8 ) scaling for CCSDT. The CI formalism
elevates this trend further, with CISD O(N
6 ), CISDTQ O(N
10 ), while FCI is known
to scale factorial with respect to the system size. In addition, post-HF methods require
a larger number of functions describing the distribution of each electron (i.e., basis
sets of one-electron functions) to provide accurate results. This gives an answer to the
question that may arise at some point, about the root cause for numerous approximations that have been introduced to quantum theory in practical implementations. Such
consideration explains the impact that DFT has in the field of practical applications,
as it scales as O(N
3 ), proportionally to the spatial dimensionality of the electron
density function. Calculations performed with popular hybrid functionals such as
B3LYP nominally scale as O(N
4 ) but their practical effectiveness is enhanced by
a decisively more rapid basis set convergence typical for DFT in comparison with
95
Therefore, MM is the only method of computational chemistry presently capable
of treating multiscale chemical systems. Examples include large biological systems,
solvated systems involving a large solvent volume, as well as composite materials.
The unmatched affordability of MM makes it useful for molecular dynamics simulations. It is possible, e.g. to obtain vibrational spectra of the molecular models
treated by molecular dynamics by calculating the dipole moment autocorrelation
function. From the point of view of NIR spectroscopy, however, the MM potentials are too approximate to yield useful results. Briefly mentioned here should be
hybrid quantum mechanics/molecular mechanics (QM/MM) approaches, in which
only the chemically most relevant part of the molecular system is treated quantum
mechanically while MM potentials are considered as sufficiently accurate to model
all remaining interactions. These QM/MM schemes enable a more accurate treatment of the potential in key molecular fragments important from the point of view
of a particular study.
5.3.2.6 The Fundamental Dilemma in Computational Chemistry; Cost
Versus Accuracy Factor
With few exceptions, in computational chemistry, a higher accuracy can only be
achieved with a significant increase in the demand for resources, understood mostly
as calculation time or/and memory requirements. The nominal complexity of a
method is limited to the number of electrons/atoms in the systems and scales
distinctly different among the methods presented here. From the point of view of
practical applications in spectroscopy, this should be a fundamental consideration
as the application of higher levels of theory to the molecular system of interest may
become prohibitively expensive. In the most straightforward case, the computational
complexity of MM simulations is proportional to the square of the number of treated
atomic centers N, O(N
2 ), whereas advanced implementations are capable of reducing
the scaling to O(N log N). The simplest ab initio HF method formally scales as
O(N
4 ). However, those schemes are widely regarded as not being sufficiently accurate
for spectroscopic applications. The significant improvement in accuracy of post-HF
approaches comes at a steep increase in their complexity, e.g., starting from O(N
5 )
for MP2, O(N
7 ) for CCSD(T), and O(N
8 ) scaling for CCSDT. The CI formalism
elevates this trend further, with CISD O(N
6 ), CISDTQ O(N
10 ), while FCI is known
to scale factorial with respect to the system size. In addition, post-HF methods require
a larger number of functions describing the distribution of each electron (i.e., basis
sets of one-electron functions) to provide accurate results. This gives an answer to the
question that may arise at some point, about the root cause for numerous approximations that have been introduced to quantum theory in practical implementations. Such
consideration explains the impact that DFT has in the field of practical applications,
as it scales as O(N
3 ), proportionally to the spatial dimensionality of the electron
density function. Calculations performed with popular hybrid functionals such as
B3LYP nominally scale as O(N
4 ) but their practical effectiveness is enhanced by
a decisively more rapid basis set convergence typical for DFT in comparison with