276
14 Principles: Bond-Band-Barrier Correlation
Setting the radii of O
−2 and Cu
+ equal to the standard Goldschmidt radii of 1.32
Å and 0.53 Å, respectively, then the ionic bond length equals the standard bulk value
of 1.85 Å for the Cu 2 O. The effective CNs of Cu
+2 and Cu
+ are taken as 4 and 6,
respectively. So, the corresponding contracting coefficients are Q 1 = 0.12 and Q 2 =
0.04, respectively. Thus, the lengths of the contracting ionic bonds are:
BL1 = 1.85 × 0.88 = 1.628
Å
BL2 = 1.85 × 0.96 = 1.776
Å
The bond angle BA12 is constrained to be 104.5° or less, because of the smaller
repulsion between the bonding orbitals. BA33 can be any value greater than 109.5°
due to the strong repulsion between the lone pair induced dipoles. In calculations
using this model for the dynamic processes, the contracting coefficient Q 2 is taken as
an adjustable variable, while the Q 1 (= 0.12) is always assumed as a constant because
it forms immediately upon oxygen molecule dissociates. Thus, the variables of Q 2 ,
DCu x , and BA12 are independent variables in determining the collective motion of
the atoms in the complex unit cell during the reaction.
Change of the variables (setting DCu x = 0.25 ± 0.25 Å, BA12 ≤ 104.5°, Q 2
= 0.04 ± 0.04) is independent and restricted to finite intervals. This differs from
the atomic-dislocation wise in which one must consider the atomic dislocation of a
certain atom once in one direction.
The advantage of such a set of variables is that the number of the adjustable variables is reduced from the conventional five to two (for stable system) or three (for
dynamic system), and the bond geometry is constrained by the principle of tetrahedron formation. As will be demonstrated, any single variation of these variables
dislocates almost all the atoms in the unit cell. These variables also reconcile all
the simultaneously geometrical variations due to bond formation without dislocating
individual atoms independently. In a real reacting system, any individual atomicshift, in principle, will affect other atoms of the entire system. In other words, the
bond geometry is more realistic and convenient than the atomic-disposition wise. It
never happens that atoms dislocate independently during reaction.
14.2.5.3 Bond Relaxation Versus Atomic Dislocation
Variation of the bond relaxation in geometry and length (Q 1 = 0.12, Q 2 , DCu x ,
and BA12) determines all the structural parameters (D 12 , DO x , DO z , DCu x , and
DCu z ) used in calculation. DCu x , together with DCu z , determines the orientation
and the distance between Cu
p and O
−2 . The parameter spaces of atomic-position and
bond-geometry are interchangeable as illustrated in Fig. 14.6.
• Coordination System and Constraints
14 Principles: Bond-Band-Barrier Correlation
Setting the radii of O
−2 and Cu
+ equal to the standard Goldschmidt radii of 1.32
Å and 0.53 Å, respectively, then the ionic bond length equals the standard bulk value
of 1.85 Å for the Cu 2 O. The effective CNs of Cu
+2 and Cu
+ are taken as 4 and 6,
respectively. So, the corresponding contracting coefficients are Q 1 = 0.12 and Q 2 =
0.04, respectively. Thus, the lengths of the contracting ionic bonds are:
BL1 = 1.85 × 0.88 = 1.628
Å
BL2 = 1.85 × 0.96 = 1.776
Å
The bond angle BA12 is constrained to be 104.5° or less, because of the smaller
repulsion between the bonding orbitals. BA33 can be any value greater than 109.5°
due to the strong repulsion between the lone pair induced dipoles. In calculations
using this model for the dynamic processes, the contracting coefficient Q 2 is taken as
an adjustable variable, while the Q 1 (= 0.12) is always assumed as a constant because
it forms immediately upon oxygen molecule dissociates. Thus, the variables of Q 2 ,
DCu x , and BA12 are independent variables in determining the collective motion of
the atoms in the complex unit cell during the reaction.
Change of the variables (setting DCu x = 0.25 ± 0.25 Å, BA12 ≤ 104.5°, Q 2
= 0.04 ± 0.04) is independent and restricted to finite intervals. This differs from
the atomic-dislocation wise in which one must consider the atomic dislocation of a
certain atom once in one direction.
The advantage of such a set of variables is that the number of the adjustable variables is reduced from the conventional five to two (for stable system) or three (for
dynamic system), and the bond geometry is constrained by the principle of tetrahedron formation. As will be demonstrated, any single variation of these variables
dislocates almost all the atoms in the unit cell. These variables also reconcile all
the simultaneously geometrical variations due to bond formation without dislocating
individual atoms independently. In a real reacting system, any individual atomicshift, in principle, will affect other atoms of the entire system. In other words, the
bond geometry is more realistic and convenient than the atomic-disposition wise. It
never happens that atoms dislocate independently during reaction.
14.2.5.3 Bond Relaxation Versus Atomic Dislocation
Variation of the bond relaxation in geometry and length (Q 1 = 0.12, Q 2 , DCu x ,
and BA12) determines all the structural parameters (D 12 , DO x , DO z , DCu x , and
DCu z ) used in calculation. DCu x , together with DCu z , determines the orientation
and the distance between Cu
p and O
−2 . The parameter spaces of atomic-position and
bond-geometry are interchangeable as illustrated in Fig. 14.6.
• Coordination System and Constraints