44
T. Onishi
MO41 = 0.17φ H1
1s + 0.31φ H1
1s − 0.13φ H2
1s − 0.24φ H2
1s
−0.17φ H3
1s − 0.31φ H3
1s + 0.13φ H4
1s + 0.24φ H4
1s
+0.17φ Mg1
3s − 0.17φ Mg3
3s
(2.9)
MO42 = −0.17φ H1
1s − 0.36φ H1
1s + 0.17φ H2
1s + 0.36φ H2
1s
− 0.17φ H3
1s − 0.36φ H3
1s + 0.17φ H4
1s + 0.36φ H4
1s
(2.10)
In MO39, H1, H2, H3 and H4 1s orbitals are overlapped, and H1, H2, H3 and H4
have one lobe. From chemical bonding rule, it is found that σ-type covalent bonding
is formed between hydride ions. MO42 is inversion σ-type covalent bonding to
MO39. In MO39, MO40 and MO41, hydrogen 1s orbitals slightly overlap with
magnesium 3s orbitals. From chemical bonding rule, it is also found that slight
covalent bonding is formed between hydride ion and magnesium.
2.4.3 Hydride Ion Conduction
Figure 2.16a, b show the potential energy curves of K 2 Mg 4 H 3 and KMg 4 H 3 models,
when displacing hydride ion along the diagonal line. In K 2 Mg 4 H 3 model, local
minima are found between midpoint and lattice position (d = 0.7 and 2.1 Å), and
local maximum is found at the midpoint (d = 1.4 Å). The activation energy for
hydride ion conduction is 0.61 eV, whereas the partial activation energy crossing
a midpoint (local maximum) is 0.23 eV. On the other hand, in KMg 4 H 3 model,
local minima are found between midpoint and lattice position (d = 0.6 and 2.2 Å),
and local maximum is found at the midpoint (d = 1.4 Å). The activation energies
crossing a midpoint and lattice position are 0.39 and 0.40 eV, respectively. It is
because the total energy at lattice position is almost the same as local maximum.
Figure 2.17 depicts the shapes of selected MOs related to hydrogen 1s orbitals
in K 2 Mg 4 H 3 model. In MO40s and MO41s, hydrogen 1s orbitals are overlapped.
From chemical bonding rule, it is found that covalent bonding is formed between
hydride ions. The wave functions of MO39s at lattice position, local minimum and
local maximum are expressed as
MO39 (Lattice) = − 0.12φ K1 (3pz) +0.12φ K2 (3pz)
− 0.18φ H1
1s
−0.29φ H1
1s
−0.13φ H2
1s
−0.19φ H2
1s
− 0.18φ H3
1s
−0.29φ H3
1s
− 0.13φ Mg1
3s
−0.12φ Mg2
3s
−0.12φ Mg3
3s
− 0.13φ Mg4 (3s
)
(2.11)
T. Onishi
MO41 = 0.17φ H1
1s + 0.31φ H1
1s − 0.13φ H2
1s − 0.24φ H2
1s
−0.17φ H3
1s − 0.31φ H3
1s + 0.13φ H4
1s + 0.24φ H4
1s
+0.17φ Mg1
3s − 0.17φ Mg3
3s
(2.9)
MO42 = −0.17φ H1
1s − 0.36φ H1
1s + 0.17φ H2
1s + 0.36φ H2
1s
− 0.17φ H3
1s − 0.36φ H3
1s + 0.17φ H4
1s + 0.36φ H4
1s
(2.10)
In MO39, H1, H2, H3 and H4 1s orbitals are overlapped, and H1, H2, H3 and H4
have one lobe. From chemical bonding rule, it is found that σ-type covalent bonding
is formed between hydride ions. MO42 is inversion σ-type covalent bonding to
MO39. In MO39, MO40 and MO41, hydrogen 1s orbitals slightly overlap with
magnesium 3s orbitals. From chemical bonding rule, it is also found that slight
covalent bonding is formed between hydride ion and magnesium.
2.4.3 Hydride Ion Conduction
Figure 2.16a, b show the potential energy curves of K 2 Mg 4 H 3 and KMg 4 H 3 models,
when displacing hydride ion along the diagonal line. In K 2 Mg 4 H 3 model, local
minima are found between midpoint and lattice position (d = 0.7 and 2.1 Å), and
local maximum is found at the midpoint (d = 1.4 Å). The activation energy for
hydride ion conduction is 0.61 eV, whereas the partial activation energy crossing
a midpoint (local maximum) is 0.23 eV. On the other hand, in KMg 4 H 3 model,
local minima are found between midpoint and lattice position (d = 0.6 and 2.2 Å),
and local maximum is found at the midpoint (d = 1.4 Å). The activation energies
crossing a midpoint and lattice position are 0.39 and 0.40 eV, respectively. It is
because the total energy at lattice position is almost the same as local maximum.
Figure 2.17 depicts the shapes of selected MOs related to hydrogen 1s orbitals
in K 2 Mg 4 H 3 model. In MO40s and MO41s, hydrogen 1s orbitals are overlapped.
From chemical bonding rule, it is found that covalent bonding is formed between
hydride ions. The wave functions of MO39s at lattice position, local minimum and
local maximum are expressed as
MO39 (Lattice) = − 0.12φ K1 (3pz) +0.12φ K2 (3pz)
− 0.18φ H1
1s
−0.29φ H1
1s
−0.13φ H2
1s
−0.19φ H2
1s
− 0.18φ H3
1s
−0.29φ H3
1s
− 0.13φ Mg1
3s
−0.12φ Mg2
3s
−0.12φ Mg3
3s
− 0.13φ Mg4 (3s
)
(2.11)