therefore be chosen sufficiently large to ensure that the total charge on the optically
active moiety is converged. This can be tested with little computational effort by
extending the QM region stepwise and calculating NPA charges [82] from the HF
ground-state density [54]. Note, that DFT (in particular LDA, GGA) strongly
overestimates inter-residual charge exchange [54]. Directly considering the excitation energy as criterion for the appropriate extend of the QM region is problematic.
Apart from size-consistency issues, the successive substitution of MM point charges
by a QM density causes fluctuations in the electrostatic potential and excitation
energies that converge very slowly with the size of the QM region and cannot be
achieved efficiently. It is therefore more practical to focus on a converged charge
exchange and improve the electrostatic description of the surrounding protein or
solvent environment by other means, e.g., by using a polarisable force field.
4.3.3 Environment polarisation
The development of polarisable force fields is a very active field of research and has
grown considerably in the last 10 years. Apart from geometry-dependent terms, the
explicit treatment of polarisation is regarded as the key ingredient to improve the
accuracy of conventional force fields. Excited-state QM/MM studies can take profit
from this development as polarisation also improves the electrostatic representation
of the solvent/protein environment and allows a self-consistent mutual polarisation
between QM and polarisable MM regions. The calculation of vertical excitation
energies based on the ground-state adopted polarisable MM region already reduces
the error in the electrostatic potential when compared to different fixed point charge
models from various force fields [83, 84]. If the polarisable environment is allowed
to respond to the excitation-induced charge re-distribution in the QM region, the
problem arises that the wave functions of the fully self-consistent solution for the
different electronic states are not orthogonal anymore [85, 86]. A simple solution
consists in taking the vertical excitation energy of the ground-state adapted system
as a lower bound and that of the excited-state adapted one as an upper bound. In the
next step, the polarisable region is allowed to respond to the new QM charge
distribution, but the QM calculation is not iterated, hence preserving orthogonality.
This reduces the difference between the upper and lower bound. The average of the
two results can then be considered as the best approximation [84, 86, 87].
A straight-forward approach to explicit polarisation is to introduce another
(QM2) layer around the optically active region (QM1). As the computational
demand for calculating a ground-state charge distribution is lower than that for
the accurate calculation of excitation energies, a lower-level method and further
approximations to obtain a linear-scaling can be used [88, 89]. The X-POL method
[90], e.g., uses iterative AM1 fragment calculations to obtain a self-consistent
charge distribution. The effect of protein polarisation is rather long-range. It is
therefore not sufficient to apply the polarisation method only within a layer of a few
A ˚ in thickness around the QM1 region [83]. Therefore, further approximations are
4 Theoretical Methods
55
active moiety is converged. This can be tested with little computational effort by
extending the QM region stepwise and calculating NPA charges [82] from the HF
ground-state density [54]. Note, that DFT (in particular LDA, GGA) strongly
overestimates inter-residual charge exchange [54]. Directly considering the excitation energy as criterion for the appropriate extend of the QM region is problematic.
Apart from size-consistency issues, the successive substitution of MM point charges
by a QM density causes fluctuations in the electrostatic potential and excitation
energies that converge very slowly with the size of the QM region and cannot be
achieved efficiently. It is therefore more practical to focus on a converged charge
exchange and improve the electrostatic description of the surrounding protein or
solvent environment by other means, e.g., by using a polarisable force field.
4.3.3 Environment polarisation
The development of polarisable force fields is a very active field of research and has
grown considerably in the last 10 years. Apart from geometry-dependent terms, the
explicit treatment of polarisation is regarded as the key ingredient to improve the
accuracy of conventional force fields. Excited-state QM/MM studies can take profit
from this development as polarisation also improves the electrostatic representation
of the solvent/protein environment and allows a self-consistent mutual polarisation
between QM and polarisable MM regions. The calculation of vertical excitation
energies based on the ground-state adopted polarisable MM region already reduces
the error in the electrostatic potential when compared to different fixed point charge
models from various force fields [83, 84]. If the polarisable environment is allowed
to respond to the excitation-induced charge re-distribution in the QM region, the
problem arises that the wave functions of the fully self-consistent solution for the
different electronic states are not orthogonal anymore [85, 86]. A simple solution
consists in taking the vertical excitation energy of the ground-state adapted system
as a lower bound and that of the excited-state adapted one as an upper bound. In the
next step, the polarisable region is allowed to respond to the new QM charge
distribution, but the QM calculation is not iterated, hence preserving orthogonality.
This reduces the difference between the upper and lower bound. The average of the
two results can then be considered as the best approximation [84, 86, 87].
A straight-forward approach to explicit polarisation is to introduce another
(QM2) layer around the optically active region (QM1). As the computational
demand for calculating a ground-state charge distribution is lower than that for
the accurate calculation of excitation energies, a lower-level method and further
approximations to obtain a linear-scaling can be used [88, 89]. The X-POL method
[90], e.g., uses iterative AM1 fragment calculations to obtain a self-consistent
charge distribution. The effect of protein polarisation is rather long-range. It is
therefore not sufficient to apply the polarisation method only within a layer of a few
A ˚ in thickness around the QM1 region [83]. Therefore, further approximations are
4 Theoretical Methods
55