18
K. P. Kepp
states that often play an important role in the chemical process [13, 120, 121]. It
makes a substantial difference if one does not include a systematic effect that favors
one of these spin states consistently. Examples include hydrogenases and heme proteins, where the spin states are close in energy and important for the mechanism [14,
95, 121]. The HS state is generally more entropic than the LS state, and one can
expect errors of 10–20 kJ/mol [36] per metal site systematically underestimating the
importance of the HS state if entropy is neglected. Neglect of vibrational entropy
can lead to the erroneous assignment of a LS state as the ground state. Similar errors
will occur in estimates of the best theoretical method based on comparing energies
without entropy directly to experimental spin states, which always represent free
energies that include the entropy effect.
The question then arises whether one can model this vibrational entropy with
decent accuracy. Standard approaches involve the computation of the harmonic vibration frequencies of the molecule, which is already required to obtain the ZPE, which,
incidentally, also favors the HS state’s longer and weaker bonds. Once this calculation
has been carried out, it is straightforward to estimate the vibrational entropy by using
thermodynamic state functions and the calculation of the vibration partition function
Q vib . Most quantum-chemistry programs can routinely perform this computation.
For single molecules, the vibrational entropies correlate decently with experimental
S SCO with errors translating into typically 5 kJ/mol for T S SCO , partly because
the electronic configurational entropy is relatively similar for the systems [36]. The
estimates neglect differential entropy contributions from solvent–solute and crystal packing, i.e., they represent only the contribution from single SCO molecules. In
terms of intermolecular contributions, both high-frequency and low-frequency modes
contribute to the entropy [122–124]. One can expect the soft vibrational modes to
be associated with large relative errors for the computed estimates. However, importantly, the high-frequency (M–L stretch) frequencies of the first coordination sphere
of the single molecule dominates the entropy effect as shown by Raman spectroscopy
[122], and these are well modeled by DFT [36, 92, 125]. Thus including entropy estimates for the first coordination sphere is much better than omitting them, as they
improve the G SCO (T ) and thus T ½ substantially. However, given the current limitations in the accuracy of these calculations, simply adding a constant contribution
of T S [52] may be a reasonable approach for many transition metal systems.
The entropy contribution to the balance between the spin states depends greatly on
the nature of the ligand and the metal ion, with stronger ligands showing much larger
entropy effects than weak ligands [52]. Typical T S contributions of 5–30 kJ/mol are
estimated for mononuclear complexes at room temperature [52]. This range is similar
to the experimental range seen for Fe(II) and Fe(III) SCO systems [36]. The small
effects are typical of weak field or weakly bound ligands. For nitrogen-donor ligands
as are commonly found in SCO systems, the entropy contribution ranges typically
from 10 to 25 kJ/mol [36, 52]. The entropy contribution is relatively insensitive to
the theoretical method used as long as the geometry and vibrational frequencies are
reasonable. This makes the entropy contribution more straightforward to estimate
than the electronic energy contribution, which is discussed in more detail below.
K. P. Kepp
states that often play an important role in the chemical process [13, 120, 121]. It
makes a substantial difference if one does not include a systematic effect that favors
one of these spin states consistently. Examples include hydrogenases and heme proteins, where the spin states are close in energy and important for the mechanism [14,
95, 121]. The HS state is generally more entropic than the LS state, and one can
expect errors of 10–20 kJ/mol [36] per metal site systematically underestimating the
importance of the HS state if entropy is neglected. Neglect of vibrational entropy
can lead to the erroneous assignment of a LS state as the ground state. Similar errors
will occur in estimates of the best theoretical method based on comparing energies
without entropy directly to experimental spin states, which always represent free
energies that include the entropy effect.
The question then arises whether one can model this vibrational entropy with
decent accuracy. Standard approaches involve the computation of the harmonic vibration frequencies of the molecule, which is already required to obtain the ZPE, which,
incidentally, also favors the HS state’s longer and weaker bonds. Once this calculation
has been carried out, it is straightforward to estimate the vibrational entropy by using
thermodynamic state functions and the calculation of the vibration partition function
Q vib . Most quantum-chemistry programs can routinely perform this computation.
For single molecules, the vibrational entropies correlate decently with experimental
S SCO with errors translating into typically 5 kJ/mol for T S SCO , partly because
the electronic configurational entropy is relatively similar for the systems [36]. The
estimates neglect differential entropy contributions from solvent–solute and crystal packing, i.e., they represent only the contribution from single SCO molecules. In
terms of intermolecular contributions, both high-frequency and low-frequency modes
contribute to the entropy [122–124]. One can expect the soft vibrational modes to
be associated with large relative errors for the computed estimates. However, importantly, the high-frequency (M–L stretch) frequencies of the first coordination sphere
of the single molecule dominates the entropy effect as shown by Raman spectroscopy
[122], and these are well modeled by DFT [36, 92, 125]. Thus including entropy estimates for the first coordination sphere is much better than omitting them, as they
improve the G SCO (T ) and thus T ½ substantially. However, given the current limitations in the accuracy of these calculations, simply adding a constant contribution
of T S [52] may be a reasonable approach for many transition metal systems.
The entropy contribution to the balance between the spin states depends greatly on
the nature of the ligand and the metal ion, with stronger ligands showing much larger
entropy effects than weak ligands [52]. Typical T S contributions of 5–30 kJ/mol are
estimated for mononuclear complexes at room temperature [52]. This range is similar
to the experimental range seen for Fe(II) and Fe(III) SCO systems [36]. The small
effects are typical of weak field or weakly bound ligands. For nitrogen-donor ligands
as are commonly found in SCO systems, the entropy contribution ranges typically
from 10 to 25 kJ/mol [36, 52]. The entropy contribution is relatively insensitive to
the theoretical method used as long as the geometry and vibrational frequencies are
reasonable. This makes the entropy contribution more straightforward to estimate
than the electronic energy contribution, which is discussed in more detail below.