8.5 Empirical Correlations (Legon and Demaison 2011)
221
8.5.2 Dihedral Angles
As observed in many molecules (Juanes et al. 2017), the popular MP2 and B3LYP
methods fail to deliver accurate dihedral angles, the error being sometimes as large
as several degrees. It may be easily explained by the fact that it requires much less
energy to modify a dihedral angle than a bond angle (it requires about 4.2 kJmol
–1
to distorts a ∠(CCC) bond angle by 10° and only 0.8 kJ/mol to distort a τ (CCCC)
dihedral angle by 10°) (Hargittai and Levy 1999). For the molecules investigated
up to now, the accuracy was found to be much less sensitive to the basis set than to
the method, and the CCSD/cc-pVTZ level of theory was found to be a significant
improvement over the MP2 method. However, although the CCSD method can be
easily used in the case of small molecules, it is significantly more expensive than the
MP2 method.
There is sometimes one way to solve this problem when the rotational constants,
A, B, C, are sensitive to the torsional angles. For instance, in diphenyldisulfide,
C 6 H 5 SSC 6 H 5 , for the angle τ (CSSC), we have (in MHz.degree
−1 ) ∂A/∂τ = −16.5;
∂B/∂τ = 5.7; and ∂C/∂τ = 2.3, whereas approximate values of the rovibrational
corrections are (in MHz) A e − A 0 = 4.8 (0.4% of A 0 ); B e − B 0 = 2.2 (0.7% of
B 0 ); and C e − C 0 = 1.8 (0.6% of C 0 ) (Demaison et al. 2019). Thus, even using the
ground-state rotational constants, it is possible to get a rather accurate estimation of
the dihedral angle.
A powerful method to estimate the dihedral angles in large molecules such as
proteins is the use of the
3 J-coupling constants derived from the analysis of a nuclear
magnetic resonance (NMR) spectrum. When placed in a magnetic field, nuclei with
a non-zero nuclear spin (such as
1 H,
13 C, or
19 F) absorb electromagnetic radiation at
a frequency characteristic of the isotope. It is the chemical shift of the nucleus. Some
of the most useful information for structure determination in NMR spectrum comes
from J-coupling or scalar coupling (a special case of spin-spin coupling) between
active nuclei. This coupling arises from the interaction of different spin states through
the chemical bonds of a molecule and results in the splitting of NMR signals. This
J-coupling contains information about bond distances and angles.
In saturated X–C–C–Y units, the coupling constant
3 J XY depends primarily on the
dihedral angle τ (X–C–C–Y) and can be described by the Karplus equation (1959)
3 J XY = A cos 2τ + B cos τ + C
(8.15)
τ is the dihedral angle, and A, B, and C are empirically derived parameters whose
values depend on the atoms involved. This relationship is used with
3 J(H, H) coupling
constants where the superscript “3” indicates that the H atom is coupled to another H
atom three bonds away, via H–C–C–H bonds (these H are called vicinal). Almost all
3 J are positive, but their magnitude varies widely (from ~0 to 25 Hz). There are many
variants of this equation (Haasnoot et al. 1980). One weak point of this method is
that the measurements are rarely performed in gas phase, which limits the accuracy.
Furthermore, electronegative substituents decrease the coupling constants.
221
8.5.2 Dihedral Angles
As observed in many molecules (Juanes et al. 2017), the popular MP2 and B3LYP
methods fail to deliver accurate dihedral angles, the error being sometimes as large
as several degrees. It may be easily explained by the fact that it requires much less
energy to modify a dihedral angle than a bond angle (it requires about 4.2 kJmol
–1
to distorts a ∠(CCC) bond angle by 10° and only 0.8 kJ/mol to distort a τ (CCCC)
dihedral angle by 10°) (Hargittai and Levy 1999). For the molecules investigated
up to now, the accuracy was found to be much less sensitive to the basis set than to
the method, and the CCSD/cc-pVTZ level of theory was found to be a significant
improvement over the MP2 method. However, although the CCSD method can be
easily used in the case of small molecules, it is significantly more expensive than the
MP2 method.
There is sometimes one way to solve this problem when the rotational constants,
A, B, C, are sensitive to the torsional angles. For instance, in diphenyldisulfide,
C 6 H 5 SSC 6 H 5 , for the angle τ (CSSC), we have (in MHz.degree
−1 ) ∂A/∂τ = −16.5;
∂B/∂τ = 5.7; and ∂C/∂τ = 2.3, whereas approximate values of the rovibrational
corrections are (in MHz) A e − A 0 = 4.8 (0.4% of A 0 ); B e − B 0 = 2.2 (0.7% of
B 0 ); and C e − C 0 = 1.8 (0.6% of C 0 ) (Demaison et al. 2019). Thus, even using the
ground-state rotational constants, it is possible to get a rather accurate estimation of
the dihedral angle.
A powerful method to estimate the dihedral angles in large molecules such as
proteins is the use of the
3 J-coupling constants derived from the analysis of a nuclear
magnetic resonance (NMR) spectrum. When placed in a magnetic field, nuclei with
a non-zero nuclear spin (such as
1 H,
13 C, or
19 F) absorb electromagnetic radiation at
a frequency characteristic of the isotope. It is the chemical shift of the nucleus. Some
of the most useful information for structure determination in NMR spectrum comes
from J-coupling or scalar coupling (a special case of spin-spin coupling) between
active nuclei. This coupling arises from the interaction of different spin states through
the chemical bonds of a molecule and results in the splitting of NMR signals. This
J-coupling contains information about bond distances and angles.
In saturated X–C–C–Y units, the coupling constant
3 J XY depends primarily on the
dihedral angle τ (X–C–C–Y) and can be described by the Karplus equation (1959)
3 J XY = A cos 2τ + B cos τ + C
(8.15)
τ is the dihedral angle, and A, B, and C are empirically derived parameters whose
values depend on the atoms involved. This relationship is used with
3 J(H, H) coupling
constants where the superscript “3” indicates that the H atom is coupled to another H
atom three bonds away, via H–C–C–H bonds (these H are called vicinal). Almost all
3 J are positive, but their magnitude varies widely (from ~0 to 25 Hz). There are many
variants of this equation (Haasnoot et al. 1980). One weak point of this method is
that the measurements are rarely performed in gas phase, which limits the accuracy.
Furthermore, electronegative substituents decrease the coupling constants.