2.2 Molecular Dynamics and the Separation of Variables
33
Ψ (x, y, t) =
i
A i ψ A,i (x)e
−iE A,i t/
⎡
⎣
j
B j ψ B, j (y)e
−iE B, j t/
⎤
⎦ .
(2.39)
Note that, if x and y are not separable, it is always possible to write ˆ
H (x, y) =
ˆ
H A (x) + ˆ
H B (y) + ˆ
H AB (x, y), and since the eigenvectors of ˆ
H A and ˆ
H B have anyway
to represent complete sets we may write
Ψ (x, y, 0) =
i j
C i j ψ A,i (x)ψ B, j (y)
(2.40)
where C i j =
ψ A,i ψ B, j |Ψ
. To put the above equation in the form of (2.38) one
would have to evaluate the coefficients A i and B j in such a way that A i B j = C i j .
But that system has, in general, no solution (more equations than unknowns).
As a simple example of separation of variables we may consider two molecules,
A and B, very far from each other so that they do not interact. Then, the total Hamiltonian is just the sum of the molecular Hamiltonians of A and B. The wavefunctions
are products as in Eq. (2.36) or (2.39): molecule A behaves in a way that is completely independent to what happens to B and vice versa. Moreover, their energies
are separately conserved. However, if A and B get closer, their mutual interaction
is no longer negligible, so the time-dependent wavefunction is not in the form of
Eq. (2.39). Nevertheless, if the coupling between A and B is small, it may be treated
as a perturbation, taking the products ψ A,i (x)ψ B, j (y) as zero-order stationary states.
If the system, at time t = 0, is found in one of these zero-order states, it will evolve in
time, populating the other states. This means that energy can be exchanged between
A and B: an energy transfer between systems that are approximately independent,
but coupled by a small perturbation, is typically governed by the Fermi golden rule
(see Sect. 3.11 and Chap. 6).
2.2.2 Separation of Translation and Rotation
The state of an isolated molecule does not change if the system as a whole (or,
equivalently, the fixed reference frame) is translated or rotated. In other words, the
molecular Hamiltonian is invariant with respect to rotations and translations so that it
commutes with the respective operators, which are represented by the total linear and
angular momentum ˆ
P and ˆ
J (generators of infinitesimal translations and rotations,
respectively). As a consequence, P and J are conserved quantities, as much as in
classical mechanics (for the relationship between infinitesimal transformations and
constants of motion, see for instance, Merzbacher [1]).
The translational invariance, which implies the conservation of the total momentum, leads to the separation of the center of mass coordinates. To this aim, a transformation of the Cartesian coordinates of electrons and nuclei has to be done. As such
33
Ψ (x, y, t) =
i
A i ψ A,i (x)e
−iE A,i t/
⎡
⎣
j
B j ψ B, j (y)e
−iE B, j t/
⎤
⎦ .
(2.39)
Note that, if x and y are not separable, it is always possible to write ˆ
H (x, y) =
ˆ
H A (x) + ˆ
H B (y) + ˆ
H AB (x, y), and since the eigenvectors of ˆ
H A and ˆ
H B have anyway
to represent complete sets we may write
Ψ (x, y, 0) =
i j
C i j ψ A,i (x)ψ B, j (y)
(2.40)
where C i j =
ψ A,i ψ B, j |Ψ
. To put the above equation in the form of (2.38) one
would have to evaluate the coefficients A i and B j in such a way that A i B j = C i j .
But that system has, in general, no solution (more equations than unknowns).
As a simple example of separation of variables we may consider two molecules,
A and B, very far from each other so that they do not interact. Then, the total Hamiltonian is just the sum of the molecular Hamiltonians of A and B. The wavefunctions
are products as in Eq. (2.36) or (2.39): molecule A behaves in a way that is completely independent to what happens to B and vice versa. Moreover, their energies
are separately conserved. However, if A and B get closer, their mutual interaction
is no longer negligible, so the time-dependent wavefunction is not in the form of
Eq. (2.39). Nevertheless, if the coupling between A and B is small, it may be treated
as a perturbation, taking the products ψ A,i (x)ψ B, j (y) as zero-order stationary states.
If the system, at time t = 0, is found in one of these zero-order states, it will evolve in
time, populating the other states. This means that energy can be exchanged between
A and B: an energy transfer between systems that are approximately independent,
but coupled by a small perturbation, is typically governed by the Fermi golden rule
(see Sect. 3.11 and Chap. 6).
2.2.2 Separation of Translation and Rotation
The state of an isolated molecule does not change if the system as a whole (or,
equivalently, the fixed reference frame) is translated or rotated. In other words, the
molecular Hamiltonian is invariant with respect to rotations and translations so that it
commutes with the respective operators, which are represented by the total linear and
angular momentum ˆ
P and ˆ
J (generators of infinitesimal translations and rotations,
respectively). As a consequence, P and J are conserved quantities, as much as in
classical mechanics (for the relationship between infinitesimal transformations and
constants of motion, see for instance, Merzbacher [1]).
The translational invariance, which implies the conservation of the total momentum, leads to the separation of the center of mass coordinates. To this aim, a transformation of the Cartesian coordinates of electrons and nuclei has to be done. As such