15.1 Derivation and Properties of the RPA Equations
227
Compatible with the orthogonalization, the pseudo-eigenvectors can be normalized
according to
X
†
m X m = |x m |
2
− | y m |
2
=
1, for m ∈ {+}
−1, for m ∈ {−}
(15.16)
Here, one has to distinguish the excitation and de-excitation solutions as in the first
case (usually) the norm of the 1 p-1h components is larger than that of the 1h-1 p
components, whereas the opposite applies to the de-excitation solutions. The full
pseudo-orthonormalization conditions can be comprised in the following compact
matrix expression
X
† SX = S
(15.17)
As a simple consequence, SX
† SX = 1, and the inverse of X is given by
X
−1
= SX
† S
(15.18)
Now we may come back to the original form (15.4) of the RPA propagator. The
solution of the RPA pseudo-eigenvalue problem established by Eqs. (15.11, 15.17)
allows us to write the RPA secular matrix as
M = SX X
−1
= SX S X
† S
(15.19)
where Eq. (15.18) has been used to arrive at the final expression. Inserting this expression for M in Eq. (15.4) leads to the form
R P A
(ω) = X S (ω1 − )
−1 X
†
(15.20)
in which the RPA propagator
R P A
(ω) is expressed entirely in terms of the RPA
(pseudo-) eigenvalues and eigenvectors. In that sense, the inversion of the ωdependent matrix in Eq. (15.4) is equivalent to solving the RPA pseudo-eigenvalue
problem.
The RPA pseudo-eigenvalue problem (15.11) can be reformulated as an ordinary
eigenvalue problem of the non-hermitian matrix M
= SM. Being eigenvalues of
a non-hermitian matrix, the RPA energies ω m are not necessarily real numbers. A
complex eigenvalue has of course no physical meaning and would indicate a failure
of the RPA in the treatment of the concerned excitation.
The possibility of complex eigenvalues as well as other features of the RPA
mathematics can nicely be demonstrated by means of a simple model. We shall
consider the 2 × 2 RPA-type matrix
M =
a b
b a
(15.21)
227
Compatible with the orthogonalization, the pseudo-eigenvectors can be normalized
according to
X
†
m X m = |x m |
2
− | y m |
2
=
1, for m ∈ {+}
−1, for m ∈ {−}
(15.16)
Here, one has to distinguish the excitation and de-excitation solutions as in the first
case (usually) the norm of the 1 p-1h components is larger than that of the 1h-1 p
components, whereas the opposite applies to the de-excitation solutions. The full
pseudo-orthonormalization conditions can be comprised in the following compact
matrix expression
X
† SX = S
(15.17)
As a simple consequence, SX
† SX = 1, and the inverse of X is given by
X
−1
= SX
† S
(15.18)
Now we may come back to the original form (15.4) of the RPA propagator. The
solution of the RPA pseudo-eigenvalue problem established by Eqs. (15.11, 15.17)
allows us to write the RPA secular matrix as
M = SX X
−1
= SX S X
† S
(15.19)
where Eq. (15.18) has been used to arrive at the final expression. Inserting this expression for M in Eq. (15.4) leads to the form
R P A
(ω) = X S (ω1 − )
−1 X
†
(15.20)
in which the RPA propagator
R P A
(ω) is expressed entirely in terms of the RPA
(pseudo-) eigenvalues and eigenvectors. In that sense, the inversion of the ωdependent matrix in Eq. (15.4) is equivalent to solving the RPA pseudo-eigenvalue
problem.
The RPA pseudo-eigenvalue problem (15.11) can be reformulated as an ordinary
eigenvalue problem of the non-hermitian matrix M
= SM. Being eigenvalues of
a non-hermitian matrix, the RPA energies ω m are not necessarily real numbers. A
complex eigenvalue has of course no physical meaning and would indicate a failure
of the RPA in the treatment of the concerned excitation.
The possibility of complex eigenvalues as well as other features of the RPA
mathematics can nicely be demonstrated by means of a simple model. We shall
consider the 2 × 2 RPA-type matrix
M =
a b
b a
(15.21)