236
15 Random-Phase Approximation (RPA)
where f
(C)
1 derives from the diagrams (C7)–(C10), while the other two contributions
are related to diagrams (C5) and (C11), respectively. The explicit expressions are
listed in Appendix A.9.
The unitary transformation Y may be viewed as transforming the excited “RPA
states” |
R P A
m
to associated intermediate 1 p-1h states |
R P A
ak , so that
Y ak,m = =
R P A
ak |
R P A
m
(15.63)
Could one possibly construct those intermediate RPA states from CE states
c
†
a c k |
R P A
0
based on a RPA ground state |
R P A
0
yet to be determined? What would
the presumed RPA ground state look like, and would it be of any use? In Sect. 16.3,
we come back to this issue.
Exercises
15.1 Formulate in the time representation the RPA recursion relation according to
Fig. 15.2 for the RPA polarization propagator,
R P A
rs,r s (t, t
). Derive Eqs. (15.1)–
(15.3) using appropriate Fourier transforms.
15.2 Determine the two eigenvectors of the 2 × 2 RPA model according to
Eqs. (15.21,15.22); verify that the pseudo-orthonormalization scheme (15.17)
applies.
15.3 Schematic model:
(a) TDA variant: Consider a manifold of m “elementary excitations” | k , k =
1, . . . , m with “unperturbed” energies k , interacting via a uniform “perturbation,” w = = k | ˆ
U | l , k, l = 1, . . . , m. Inspect the eigenvalue equation for
the secular matrix and design a graphical scheme to determine the eigenvalues
(excitation energies) e k . The graphical solution allows for an obvious distinction of m − 1 ordinary solutions and a particular (“plasmon”) solution.
(b) The model can be simplified even further by supposing that the unperturbed energies are degenerate: k = e, k = 1, . . . , m. Determine and analyze
the analytical solutions for the eigenvalues and eigenvectors of the particular
and ordinary solutions.
(c) RPA variant: Augment the TDA excitations with a corresponding set of “deexcitations,” | k , k = 1, . . . , m, and suppose the same type of uniform coupling,
w = = k | ˆ
U | l , k, l = 1, . . . , m,
between
excitations
and
de-excitations (sub-block B of the RPA secular matrix). Transform the RPA
eigenvalue equations in a form allowing for graphical solutions and analyze
the various solutions.
15.4 Prove the identity (15.49) for a specific 1 p-1h vector component, say aαkα
(where a and k denote spatial orbitals). Bring the left side of Eq. (15.49)
into spin-free form and replace the matrix element p ak of ˆ
p z with d ak =
φ a |ˆ z|φ k using the relation −i p ak /m e = ( a − k )d ak + +φ a |[ ˆ
w, ˆ
z]|φ k (see
Eqs. (14.73)–(14.75) and Exercise 14.3).
15.5 In a similar way like in Exercise 15.4, show that the dipole sum rule (15.46)
derives directly from the RPA expression (15.44).
15 Random-Phase Approximation (RPA)
where f
(C)
1 derives from the diagrams (C7)–(C10), while the other two contributions
are related to diagrams (C5) and (C11), respectively. The explicit expressions are
listed in Appendix A.9.
The unitary transformation Y may be viewed as transforming the excited “RPA
states” |
R P A
m
to associated intermediate 1 p-1h states |
R P A
ak , so that
Y ak,m = =
R P A
ak |
R P A
m
(15.63)
Could one possibly construct those intermediate RPA states from CE states
c
†
a c k |
R P A
0
based on a RPA ground state |
R P A
0
yet to be determined? What would
the presumed RPA ground state look like, and would it be of any use? In Sect. 16.3,
we come back to this issue.
Exercises
15.1 Formulate in the time representation the RPA recursion relation according to
Fig. 15.2 for the RPA polarization propagator,
R P A
rs,r s (t, t
). Derive Eqs. (15.1)–
(15.3) using appropriate Fourier transforms.
15.2 Determine the two eigenvectors of the 2 × 2 RPA model according to
Eqs. (15.21,15.22); verify that the pseudo-orthonormalization scheme (15.17)
applies.
15.3 Schematic model:
(a) TDA variant: Consider a manifold of m “elementary excitations” | k , k =
1, . . . , m with “unperturbed” energies k , interacting via a uniform “perturbation,” w = = k | ˆ
U | l , k, l = 1, . . . , m. Inspect the eigenvalue equation for
the secular matrix and design a graphical scheme to determine the eigenvalues
(excitation energies) e k . The graphical solution allows for an obvious distinction of m − 1 ordinary solutions and a particular (“plasmon”) solution.
(b) The model can be simplified even further by supposing that the unperturbed energies are degenerate: k = e, k = 1, . . . , m. Determine and analyze
the analytical solutions for the eigenvalues and eigenvectors of the particular
and ordinary solutions.
(c) RPA variant: Augment the TDA excitations with a corresponding set of “deexcitations,” | k , k = 1, . . . , m, and suppose the same type of uniform coupling,
w = = k | ˆ
U | l , k, l = 1, . . . , m,
between
excitations
and
de-excitations (sub-block B of the RPA secular matrix). Transform the RPA
eigenvalue equations in a form allowing for graphical solutions and analyze
the various solutions.
15.4 Prove the identity (15.49) for a specific 1 p-1h vector component, say aαkα
(where a and k denote spatial orbitals). Bring the left side of Eq. (15.49)
into spin-free form and replace the matrix element p ak of ˆ
p z with d ak =
φ a |ˆ z|φ k using the relation −i p ak /m e = ( a − k )d ak + +φ a |[ ˆ
w, ˆ
z]|φ k (see
Eqs. (14.73)–(14.75) and Exercise 14.3).
15.5 In a similar way like in Exercise 15.4, show that the dipole sum rule (15.46)
derives directly from the RPA expression (15.44).