We consider adiabatic approximations in a fairly general form. They all boil
down to the fact that in total Hamiltonian
b
Hðr; QÞ ¼ b
T e ðrÞ þ b
T N ðQÞ þ b
V ðr; QÞ
of the stationary Schrödinger equation
b
Hðr; QÞWðr; QÞ ¼ E Á Wðr; QÞ
ð 3:2:1aÞ
or
b
T e ðrÞ þ b
T N ðQÞ þ b
V ðr; QÞ
h
i
Wðr; QÞ ¼ E Á Wðr; QÞ
ð 3:2:1bÞ
produce a separation of the variables (here r(x e , y e , z e ); Q(x N , y N , z N ) are the
coordinates of the fast subsystem (for example, electrons) and the slow subsystem
(for example, nuclei), respectively, W(r,Q) and E are eigenfunctions and the
eigenvalues of the Hamiltonian, b
T and b
V are the operators of the kinetic and
potential energy, respectively).
There are various ways of this separation and, accordingly, various approximations. Often, they all together are not quite accurately are called the adiabatic
approximation or the Born–Oppenheimer approximation. Their essence boils down
to the fact that the wave function of the k-th state is written in the form:
W k ðr; QÞ ¼ U k ðr; QÞvðQÞ;
ð3:2:2Þ
where U k (r,Q) is the wave function of the fast subsystem, and v(Q) is that of the
slow one. In the adiabatic approximation of the 1-st order, the wave functions of the
fast subsystem are sought under the assumption that the slow subsystem does not
move at all, b
T N ðQÞ ¼ 0, and in the coarse adiabatic approximation, its coordinates
are fixed in an equilibrium position, i.e., at b
T N ðQ 0 Þ ¼ 0. So, in the adiabatic
approximation, the wave functions W k (r,Q), called adiabatic, are found as
eigenfunctions of the Schrödinger equation (3.2.1b) with T N (Q) = 0, i.e., as a
solution of the wave equation for an electron moving in the field of immobile nuclei
and having potential energy V(r,Q):
b
T e ðrÞ þ b
V ðr; QÞ
h
i
U k ðr; QÞ ¼ U k ðQÞ Á U k ðr; QÞ
ð 3:2:3Þ
b
V ðr; QÞ is the function of r, and it depends on Q as on a parameter. Accordingly,
U k (r,Q), and the eigenvalue of (3.2.3) U k (Q) is the energy of the k-th state or the
adiabatic terms of the fast subsystem) depend on Q as a parameter too (more deeply
the physical meaning of U k (Q) we will understand below). So each fixed
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3 Theory of Elementary Processes
down to the fact that in total Hamiltonian
b
Hðr; QÞ ¼ b
T e ðrÞ þ b
T N ðQÞ þ b
V ðr; QÞ
of the stationary Schrödinger equation
b
Hðr; QÞWðr; QÞ ¼ E Á Wðr; QÞ
ð 3:2:1aÞ
or
b
T e ðrÞ þ b
T N ðQÞ þ b
V ðr; QÞ
h
i
Wðr; QÞ ¼ E Á Wðr; QÞ
ð 3:2:1bÞ
produce a separation of the variables (here r(x e , y e , z e ); Q(x N , y N , z N ) are the
coordinates of the fast subsystem (for example, electrons) and the slow subsystem
(for example, nuclei), respectively, W(r,Q) and E are eigenfunctions and the
eigenvalues of the Hamiltonian, b
T and b
V are the operators of the kinetic and
potential energy, respectively).
There are various ways of this separation and, accordingly, various approximations. Often, they all together are not quite accurately are called the adiabatic
approximation or the Born–Oppenheimer approximation. Their essence boils down
to the fact that the wave function of the k-th state is written in the form:
W k ðr; QÞ ¼ U k ðr; QÞvðQÞ;
ð3:2:2Þ
where U k (r,Q) is the wave function of the fast subsystem, and v(Q) is that of the
slow one. In the adiabatic approximation of the 1-st order, the wave functions of the
fast subsystem are sought under the assumption that the slow subsystem does not
move at all, b
T N ðQÞ ¼ 0, and in the coarse adiabatic approximation, its coordinates
are fixed in an equilibrium position, i.e., at b
T N ðQ 0 Þ ¼ 0. So, in the adiabatic
approximation, the wave functions W k (r,Q), called adiabatic, are found as
eigenfunctions of the Schrödinger equation (3.2.1b) with T N (Q) = 0, i.e., as a
solution of the wave equation for an electron moving in the field of immobile nuclei
and having potential energy V(r,Q):
b
T e ðrÞ þ b
V ðr; QÞ
h
i
U k ðr; QÞ ¼ U k ðQÞ Á U k ðr; QÞ
ð 3:2:3Þ
b
V ðr; QÞ is the function of r, and it depends on Q as on a parameter. Accordingly,
U k (r,Q), and the eigenvalue of (3.2.3) U k (Q) is the energy of the k-th state or the
adiabatic terms of the fast subsystem) depend on Q as a parameter too (more deeply
the physical meaning of U k (Q) we will understand below). So each fixed
44
3 Theory of Elementary Processes