and second on time:
i h
1
T t
ð Þ
dT t
ð Þ
dt
¼ E
ð1:8Þ
The solution of the latter has the form T t
ð Þ ¼ Ae
ÀiEt= h , where E, is energy of
stationary state, one of the eigenvalues of the Hamiltonian operator defined for the
system in time-independent (stationary) state. Finally, the solution to the
Schrödinger equation has the form:
Wðx; tÞ ¼ wðxÞTðtÞ ¼ wðxÞe
ÀiEt= h
ð1:9Þ
One can easily see that although the potential of V(r) is independent of time, the
wave function oscillates in time with the frequency depending on the energy corresponding to the current stationary state of the system (however, at the same time,
the probability distribution Pðx; tÞ ¼ Wðx; tÞ
j
j
2 ¼ wðxÞ
j
j
2 , and therefore also the
distribution of total electron density, is constant in time).
Since the solution of the time-dependent part of Schrödinger equation is identical
in each case, in order to obtain full information about the system’s state and its
evolution over time, it is sufficient to find a solution for time-independent
Schrödinger equation, of the general form:
^
Hwðr 1 ; r 2 ; . . .; r n ; R a ; R b ; . . .; R m Þ ¼ Ewðr 1 ; r 2 ; . . .; r i ; R a ; R b ; . . .; R m Þ
ð1:10Þ
where r i and R a —the coordinates of electrons and atomic nuclei, respectively, n—
number of electrons, m—number of atomic nuclei, and respective total energy
operator assumes the form:
b
H ¼ b
T þ b
V ¼ À
h
2
2
X
a
1
m a
r
2
a À
h
2
2m e
X
i
r
2
i þ
X
a
X
a [ b
Z a Z b e
2
r ab
À
X
a
X
i
Z a e
2
r ia
þ
X
j
X
i [ j
e
2
r ij
ð1:11Þ
Except for a few simplest cases, finding an exact, analytical solution of this
equation is impossible, and thus, some simplifications are usually necessary to be
used and solutions are obtained in numerical form.
1.2.1.2 Born–Oppenheimer Approximation, Potential Energy
Hypersurface
The first simplification, bringing us closer to the solution of the time-independent
Schrödinger equation for a multi-electron system, is called Born–Oppenheimer
approximation (BOA). This approximation is rooted directly in the simple observation
8
A. Koleżyński
i h
1
T t
ð Þ
dT t
ð Þ
dt
¼ E
ð1:8Þ
The solution of the latter has the form T t
ð Þ ¼ Ae
ÀiEt= h , where E, is energy of
stationary state, one of the eigenvalues of the Hamiltonian operator defined for the
system in time-independent (stationary) state. Finally, the solution to the
Schrödinger equation has the form:
Wðx; tÞ ¼ wðxÞTðtÞ ¼ wðxÞe
ÀiEt= h
ð1:9Þ
One can easily see that although the potential of V(r) is independent of time, the
wave function oscillates in time with the frequency depending on the energy corresponding to the current stationary state of the system (however, at the same time,
the probability distribution Pðx; tÞ ¼ Wðx; tÞ
j
j
2 ¼ wðxÞ
j
j
2 , and therefore also the
distribution of total electron density, is constant in time).
Since the solution of the time-dependent part of Schrödinger equation is identical
in each case, in order to obtain full information about the system’s state and its
evolution over time, it is sufficient to find a solution for time-independent
Schrödinger equation, of the general form:
^
Hwðr 1 ; r 2 ; . . .; r n ; R a ; R b ; . . .; R m Þ ¼ Ewðr 1 ; r 2 ; . . .; r i ; R a ; R b ; . . .; R m Þ
ð1:10Þ
where r i and R a —the coordinates of electrons and atomic nuclei, respectively, n—
number of electrons, m—number of atomic nuclei, and respective total energy
operator assumes the form:
b
H ¼ b
T þ b
V ¼ À
h
2
2
X
a
1
m a
r
2
a À
h
2
2m e
X
i
r
2
i þ
X
a
X
a [ b
Z a Z b e
2
r ab
À
X
a
X
i
Z a e
2
r ia
þ
X
j
X
i [ j
e
2
r ij
ð1:11Þ
Except for a few simplest cases, finding an exact, analytical solution of this
equation is impossible, and thus, some simplifications are usually necessary to be
used and solutions are obtained in numerical form.
1.2.1.2 Born–Oppenheimer Approximation, Potential Energy
Hypersurface
The first simplification, bringing us closer to the solution of the time-independent
Schrödinger equation for a multi-electron system, is called Born–Oppenheimer
approximation (BOA). This approximation is rooted directly in the simple observation
8
A. Koleżyński