We first give a short review of the ``local-current approximation'' (LCA), derived from a general variation principle, which serves as a semiclassical description of strongly collective excitations in finite fermion systems starting from their quantum-mechanical mean-field ground state. We illustrate it for the example of coupled translational and compressional dipole excitations in metal clusters. We then discuss collective electronic dipole excitations in C$_{60}$ molecules (Buckminster fullerenes). We show that the coupling of the pure translational mode (``surface plasmon'') with compressional volume modes in the semiclasscial LCA yields semi-quantitative agreement with microscopic time-dependent density functional (TDLDA) calculations, while both theories yield qualitative agreement with the recent experimental observation of a ``volume plasmon''.
Early in the history of nuclear physics, the (isovector) giant dipole resonance (GDR) provided one of the first manifestations of strongly collective excitations in finite fermion systems. Two classical models were suggested to describe the physics of the GDR: a) the model of Goldhaber and Teller, 1 in which protons and neutrons are both incompressible fluids undergoing a relative translational oscillation, and b) the model of Steinwedel and Jensen 2 (previously also proposed by Migdal 3 ), in which protons and neutrons are both locally decompressed or compressed with opposite phases, such that the total nuclear density remains constant and a dipole oscillation results (cf. Fig. 1 below). Detailed analysis of experimental data revealed later that a suitable combination of both models was necessary to interpret these data, 4,5 so that the GDR could be classically best understood in terms of coupled translational and compressional dipole modes. These early classical models were later refined by the so-called "fluid dynamics" 6 and the "sum rule approach" 7 based on the selfconsistent mean-field description of collective excitations in the random phase approximation (RPA). 8 These semiclassical models were successfully used to describe collective excitations not only in nuclei, but also in metal clusters, 9,11,10 where a similar coupling between translational and compressional dipole modes has been shown to well describe the collective optical response. 12,13,14 In this paper, we review the "local current approximation" (LCA), which encompasses both the fluid-dynamical and sum rule approaches and can be derived from a variational principle on the same footing as the RPA, and quote some of its results for metal clusters. We then apply the LCA to collective electronic excitations in C 60 molecules, for which recent experiments 15 have revealed a "volume plasmon", a broad high-energy shoulder in the photo-ionization cross section (otherwise dominated by the "surface plasmon" 16,17 ) which again can be semiclassically understood as a compressional component of the collective dipole excitation.
The stationary Schrödinger equation Ĥ|ν = ( T + V )|ν = E ν |ν for a many-body system can be cast into the following well-known “equations of motion”: 8
where the operators O † ν and O ν are defined by
|0 is the ground state and ω ν = E ν -E 0 (ν > 0) are the excitation energies. As shown in Ref. 18 , Eq. ( 1) can be rederived by the following variational principle:
where the “moments” m 1 and m 3 -cf. (15) for their names -are defined by
As long as Q is taken to be the most general (nonlocal) hermitean operator, the system (3,4) is equivalent to the exact stationary Schrödinger equation. Successive orthogonalization of Q1 , Q2 ,… yields the exact excitation spectrum E 3 ( Qν ) = ω ν (ν = 1, 2, . . .). With Qν ∝ O † ν +O ν , which may be interpreted as a set of generalized coordinates (cf. Refs. 8,13 ), we are brought back to (1).
In the selfconsistent microscopic mean-field approaches, one replaces |0 either by a Slater determinant (Hartree-Fock theory, HF) or by the Kohn-Sham (KS) ground state in terms of the local density ρ(r) (density functional theory, DFT). If the operator Q in (4) is replaced by a one-particle-one-hole (1p1h) operator, its variation (3) (4).] In the framework of DFT (using the local density approximation, LDA, for the exchange-correlation energy), the RPA is often also referred to as the time-dependent LDA (TDLDA).
In the sum rule and fluid dynamical approaches, one approximates the collective excitation energies ω ν by the energy E 3 [Q ν ], defined as in (3,4) but in terms of suitable local model operators Q ν (r), such as Q d = z to describe pure translations (Goldhaber-Teller model), the monopole operator Q 0 = r 2 for radial compressions (“breathing mode”); Q 2 = r 2 Y 20 for quadrupole oscillations, etc.
The local current approximation (LCA) consists in the following assumptions. One uses for |0 in (4) the uncorrelated HF or KS ground state, like in RPA or TDLDA, and takes the operator Q in (3,4) to be a local function Q(r). For local and spin-less (e.g. Coulomb) two-body interactions V , one then obtains
where u(r) is a local displacement field which is proportional to the collective current, see (12) below. (This justifies the name of LCA.) Note that if (and possibly a current density according to the “current-DFT” 20 ) in terms of the ground-state HF (or KS) wave functions φ i (r). The variation δE 3 [Q]/δQ(r) = 0 leads to fluid dynamical eigenvalue equations: 18
yielding the spectrum ω ν and eigenmodes u ν (r). Eq. ( 6) represents three coupled nonlinear fourth-order partial differential equations for u j (r), which in general are extremely hard to solve. Because of their dependence on the wave functions φ i , one also speaks of “quantum fluid dynamics” which includes the effects of zero sound. 6 A practical way (“finite-basis LCA”) to solve (6) approximately consists in expand
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