226
15 Random-Phase Approximation (RPA)
M
x m
y m
= ω m
x m
− y m
(15.10)
where ω m denotes the eigenvalue, and the eigenvector consists of 1 p-1h components,
x m , and 1h-1 p components, y m :
X m =
x m
y m
The difference to the usual eigenvalue problem is, of course, the minus sign in
front of the 1h-1 p components on the right-hand side of the eigenvalue equation.
Using a compact matrix notation, the individual equations, m = 1, 2, . . . , can be
aggregated as
MX = SX
(15.11)
where and X denote the diagonal matrix of eigenvalues and the eigenvector matrix,
respectively.
The structure of the RPA matrices M and S according to Eq. (15.7) has some
specific implications. There are two related sets of solutions referred to as excitation
solutions, m ∈ {+}, and de-excitation solutions, m ∈ {−}. The excitation solutions
relate to the upper left block, A, which is identical with the TDA (or ADC(1))
secular matrix, presented in Eq. (14.30) of Sect. 14.2. Supposing for a moment that
the coupling block can be disregarded, B = 0, the excitation solutions simply become
the TDA solutions. Likewise, the de-excitation solutions are related to the lower right
block, A
∗ , of the RPA secular matrix. However, they merely repeat the excitation
solutions in a somewhat different shape. Let
ω + , X + =
x
y
(15.12)
be the eigenvalue and eigenvector of an excitation solution. It is easily shown that
there is a corresponding de-excitation solution with the eigenvector
X − =
y
∗
x
∗
(15.13)
and an eigenvalue ω − , being (if real) the negative of ω + :
ω − = −ω
∗
+
(15.14)
What about normalization of the pseudo-eigenvectors? As the reader should check,
two eigenvectors belonging to different eigenvalues obey the following pseudoorthogonality relation
X
†
m SX m = (x
†
m , y
†
m )
x m
− y m
= x
†
m x m − y
†
m y m = 0 for m = m
(15.15)
15 Random-Phase Approximation (RPA)
M
x m
y m
= ω m
x m
− y m
(15.10)
where ω m denotes the eigenvalue, and the eigenvector consists of 1 p-1h components,
x m , and 1h-1 p components, y m :
X m =
x m
y m
The difference to the usual eigenvalue problem is, of course, the minus sign in
front of the 1h-1 p components on the right-hand side of the eigenvalue equation.
Using a compact matrix notation, the individual equations, m = 1, 2, . . . , can be
aggregated as
MX = SX
(15.11)
where and X denote the diagonal matrix of eigenvalues and the eigenvector matrix,
respectively.
The structure of the RPA matrices M and S according to Eq. (15.7) has some
specific implications. There are two related sets of solutions referred to as excitation
solutions, m ∈ {+}, and de-excitation solutions, m ∈ {−}. The excitation solutions
relate to the upper left block, A, which is identical with the TDA (or ADC(1))
secular matrix, presented in Eq. (14.30) of Sect. 14.2. Supposing for a moment that
the coupling block can be disregarded, B = 0, the excitation solutions simply become
the TDA solutions. Likewise, the de-excitation solutions are related to the lower right
block, A
∗ , of the RPA secular matrix. However, they merely repeat the excitation
solutions in a somewhat different shape. Let
ω + , X + =
x
y
(15.12)
be the eigenvalue and eigenvector of an excitation solution. It is easily shown that
there is a corresponding de-excitation solution with the eigenvector
X − =
y
∗
x
∗
(15.13)
and an eigenvalue ω − , being (if real) the negative of ω + :
ω − = −ω
∗
+
(15.14)
What about normalization of the pseudo-eigenvectors? As the reader should check,
two eigenvectors belonging to different eigenvalues obey the following pseudoorthogonality relation
X
†
m SX m = (x
†
m , y
†
m )
x m
− y m
= x
†
m x m − y
†
m y m = 0 for m = m
(15.15)