30
2 Theory: Bond-Electron-Energy Correlation
or thermal excitation. A Taylor series approximates the pairing potential u(r):
u(r ) =
∂
n u(r )
n!∂r n
r =d
(r − d)
n
= E b +
∂
2 u(r )
2∂r 2
r =d
(r − d)
2
+
∂
3 u(r )
6∂r 3
r =d
(r − d)
3
+ 0
(r − d)
n≥4
(2.3)
The zeroth differential (d, E b ) at equilibrium dominates uniquely the BE shift.
The second-order differential, or the curvature of the potential u(r) corresponding
to the force constant of a dimer undergoing harmonic vibration. The higher-order
nonlinear terms contribute to the transport dynamics like lattice expansion and heat
conductivity. Bond relaxation refers to the response of the coordinates (d, E b ) at
equilibrium to external stimulus such as pressure, temperature, coordination and
chemical environment [2]. In place of what said in the text book, atoms vibrate in the
potential well and meanwhile the potential well glides along the modulation curve
f(x), shown in Fig. 2.2a.
The V cryst (r) experienced by the electron is the sum of u(r) over all nearest neighbors. For the irregularly-coordinated systems, one needs to consider the local perturbation to the crystal potential from V cryst (r) to V cryst (r)(1 + H ) ∼ = E b (1 + H ) without needing considering perturbation to the wavefunction or the high-order potential
terms.
The perturbation shifts the CL positively if H > 0 or negatively if H < 0.
The former is the quantum entrapment (T) and the latter is the polarization (P). A
mixed shift is possible if both the T and P contribute competitively. Perturbation
turns Eq. (2.1) into the following form by replacing the z b with x, representing for
the effect of atomic CN(z) variation or polarization P,
H
= V cr yst (r )(1 + H )
E ν (x) = E ν (0) + α νx (1 + xβ νx /α νx ) + 2xβ νx ν (k, R)
with
⎧
⎨
⎩
E ν (0) = −ν, i|H 0 |ν, i
(Atomic core level)
α νx = −
ν, i
H
ν, i
∝ E b (1 + H ) ∝ E x (Exchange integral)
β νx = −
ν, i
H
ν, j
∝ E b (1 + H ) ∝ E x (Overlap integral)
(2.4)
Perturbation mediates only the integrals without any influencing on the referential
E ν (0).
2 Theory: Bond-Electron-Energy Correlation
or thermal excitation. A Taylor series approximates the pairing potential u(r):
u(r ) =
∂
n u(r )
n!∂r n
r =d
(r − d)
n
= E b +
∂
2 u(r )
2∂r 2
r =d
(r − d)
2
+
∂
3 u(r )
6∂r 3
r =d
(r − d)
3
+ 0
(r − d)
n≥4
(2.3)
The zeroth differential (d, E b ) at equilibrium dominates uniquely the BE shift.
The second-order differential, or the curvature of the potential u(r) corresponding
to the force constant of a dimer undergoing harmonic vibration. The higher-order
nonlinear terms contribute to the transport dynamics like lattice expansion and heat
conductivity. Bond relaxation refers to the response of the coordinates (d, E b ) at
equilibrium to external stimulus such as pressure, temperature, coordination and
chemical environment [2]. In place of what said in the text book, atoms vibrate in the
potential well and meanwhile the potential well glides along the modulation curve
f(x), shown in Fig. 2.2a.
The V cryst (r) experienced by the electron is the sum of u(r) over all nearest neighbors. For the irregularly-coordinated systems, one needs to consider the local perturbation to the crystal potential from V cryst (r) to V cryst (r)(1 + H ) ∼ = E b (1 + H ) without needing considering perturbation to the wavefunction or the high-order potential
terms.
The perturbation shifts the CL positively if H > 0 or negatively if H < 0.
The former is the quantum entrapment (T) and the latter is the polarization (P). A
mixed shift is possible if both the T and P contribute competitively. Perturbation
turns Eq. (2.1) into the following form by replacing the z b with x, representing for
the effect of atomic CN(z) variation or polarization P,
H
= V cr yst (r )(1 + H )
E ν (x) = E ν (0) + α νx (1 + xβ νx /α νx ) + 2xβ νx ν (k, R)
with
⎧
⎨
⎩
E ν (0) = −ν, i|H 0 |ν, i
(Atomic core level)
α νx = −
ν, i
H
ν, i
∝ E b (1 + H ) ∝ E x (Exchange integral)
β νx = −
ν, i
H
ν, j
∝ E b (1 + H ) ∝ E x (Overlap integral)
(2.4)
Perturbation mediates only the integrals without any influencing on the referential
E ν (0).