78
4 Molecular Crystals
frequency, ω, equals to
1
2
ω, the estimation of vibrational zero-point energy results
in obtaining the average angular frequency
ω =
ωg(ω)dω,
(4.13)
where g(ω) is the vibrational density of states. For simple ionic crystals, this can be
done through analyzing moments of g(ω):
M n =
ω
n g(ω)dω,
(4.14)
some (but n = 1) of which are available from experimental heat capacity at low temperatures [44]. Similar attempts were also reported for crystals of relatively simple
molecules [45, 46]. In general, the information about the vibrational density of states
is necessary. Although scattering techniques can, in principle, be utilized, more difficulties are encountered for systems consisting of plural atomic species than the case
of the structural study discussed in Sect. 3.1.1. Indeed, this issue involves not only the
difference in scattering power of atoms but also ways of involvement in vibrational
modes, i.e., eigenvectors of lattice vibration. Computational means would now be
accessible rather easily.
4.4.2 Lattice Sum
Unless the cohesive energy is calculated based on the quantum mechanics, often
called first principle or ab initio calculation, estimating cohesive energy is through
a summation of interatomic potentials, v(|r|), which is a function of the interatomic
distance |r|. The calculations involve terms of the form
i = j
v(|r i − r j |) =
1
2
m
n =m
h,k,l
v(|ha + kb + l c + r n − r m |),
(4.15)
where indexes i and j are for atoms in the system, indexes m and n for atoms in a
unit cell spanned by lattice vectors a, b and c. Sums of the form
h,k,l
v(|ha + kb + l c + r n − r m |)
(4.16)
are generally called the lattice sum.
Although we encounter no difficulty if the convergence is reasonably quick, functions of distance r with negative powers usually need special care because of slow
convergence. Representative is the electrostatic (Coulomb) potential for ions. If the
crystal is one-dimensional with a as the interionic distance, the sum, i.e., electrostatic
energy is
4 Molecular Crystals
frequency, ω, equals to
1
2
ω, the estimation of vibrational zero-point energy results
in obtaining the average angular frequency
ω =
ωg(ω)dω,
(4.13)
where g(ω) is the vibrational density of states. For simple ionic crystals, this can be
done through analyzing moments of g(ω):
M n =
ω
n g(ω)dω,
(4.14)
some (but n = 1) of which are available from experimental heat capacity at low temperatures [44]. Similar attempts were also reported for crystals of relatively simple
molecules [45, 46]. In general, the information about the vibrational density of states
is necessary. Although scattering techniques can, in principle, be utilized, more difficulties are encountered for systems consisting of plural atomic species than the case
of the structural study discussed in Sect. 3.1.1. Indeed, this issue involves not only the
difference in scattering power of atoms but also ways of involvement in vibrational
modes, i.e., eigenvectors of lattice vibration. Computational means would now be
accessible rather easily.
4.4.2 Lattice Sum
Unless the cohesive energy is calculated based on the quantum mechanics, often
called first principle or ab initio calculation, estimating cohesive energy is through
a summation of interatomic potentials, v(|r|), which is a function of the interatomic
distance |r|. The calculations involve terms of the form
i = j
v(|r i − r j |) =
1
2
m
n =m
h,k,l
v(|ha + kb + l c + r n − r m |),
(4.15)
where indexes i and j are for atoms in the system, indexes m and n for atoms in a
unit cell spanned by lattice vectors a, b and c. Sums of the form
h,k,l
v(|ha + kb + l c + r n − r m |)
(4.16)
are generally called the lattice sum.
Although we encounter no difficulty if the convergence is reasonably quick, functions of distance r with negative powers usually need special care because of slow
convergence. Representative is the electrostatic (Coulomb) potential for ions. If the
crystal is one-dimensional with a as the interionic distance, the sum, i.e., electrostatic
energy is