Particle-hole eigenvalue

State Particle-hole eigenvalue Inversion eigenvalue Spin... 

It is instructive to compare the approximate weak-coupling theory to essential exact, numerical (density matrix renormalization group) calculations on the same model (namely the Pariser-Parr-Pople model). The numerical calculations are performed on polymer chains with the polyacetylene geometry. Since these chains posses inversion symmetry the many-body eigenstates are either even (Ag) or odd By). As discussed previously, the singlet exciton wave function has either even or odd parity when the particle-hole eigenvalue is odd or even. Conversely, the triplet exciton wavefunction has either even or odd parity when the particle-hole eigenvalue is even or odd. As a consequence, we can express a B state as... 

The projection space can be decomposed for convenience into a primary space, a, and a complementary space, f. The latter space contains operators associated with ionizations coupled to excitations triple products (two-hole particle, 2hp, and two-particle hole, 2ph, subspaces), quintuple products, heptuple products and so on. With this partition of the projection space, the eigenvalue problem can be rewritten as... 

So far we have discussed the pole structure of the extended particle-hole Green s function, which is given by the eigenvalue spectrum of the generalised... 

The complete set of density matrices (eq 1.2) may be subsequently calculated using the eigenvectors. " Only particle-hole and hole-particle components of Iv are computed in the restricted TDHF scheme (Appendix A). Therefore, this non-Hermitian eigenvalue problem of dimension 2M x 2M, M = A cc x Nvir = Nx K — N) in the MO basis set representation may be recast in the form ... 

Figure 3.5. Single-particle energy eigenvalues c/, for the simple example that illustrates the dynamics of electrons and holes in a one-band, one-dimensional crystal.

While all three matrices are interconvertible, the nonnegativity of the eigenvalues of one matrix does not imply the nonnegativity of the eigenvalues of the other matrices, and hence the restrictions Q>0 and > 0 provide two important 7/-representability conditions in addition to > 0. These conditions physically restrict the probability distributions for two particles, two holes, and one particle and one hole to be nonnegative with respect to all unitary transformations of the two-particle basis set. Collectively, the three restrictions are known as the 2-positivity conditions [17]. 

Operator (15.36), thus, has eigenvalue +1/2 depending on whether or not the pairing state is occupied. If only one particle is in that state, then the eigenvalue of operator Qz is zero. It is worthwhile mentioning the fact that vacuum states, both for holes and for particles, are the components of a tensor in the quasispin space... 

Here the Hamiltonian Ho describes the evolution of the polarization in an isolated IQW in the absence of electron-hole populations and corresponds to the Wannier equation (39). The resonant Wannier exciton wave function in momentum space Tk(q) is its eigenfunction with the eigenvalue Ew(k). The Hamiltonian H describes the nonlinear many-particle corrections. It is proportional... 

Here, the new set of the occupied and (active -I- virtual) orbitals is obtained by using the unitary transformations, and a pair of the orbitals whose matrix elements with (U+TV) = A, are called the /-th hole-particle pair orbitals. The importance of a particular hole-particle transition to the overall rectangular matrix T is reflected in the magnitude of the associated eigenvalue A. ... 

We now generalize our previous development to obtain a diagrammatic representation of RS perturbation theory as applied to an N-state system. Consider the problem of finding the perturbation expansion for the lowest eigenvalue of such a system. Here we still have only one hole state, 1>, but there are now N - 1 particle states n>, n = 2, 3,..., AT. We draw the same set of diagrams as before. However, now we can label the particle lines with any index n. For example, the diagram... 

We label the occupied (hole) spin orbitals by ayb c,... and the unoccupied (particle) spin orbitals by r, s, t,. ... The wave function (6.33) is an eigen-fUnction of jFq with an eigenvalue equal to the sum of occupied orbital energies,... 

The first, most primitive, model is the infinite barrier model (IBM). Here the electronic motion is confined by a spherical potential hole with infinitely high barriers. Once the electronic wave functions (spherical Bessel functions) and eigenvalues are known, one can proceed and calculate the dynamic polarizability a co). From this quantity the collective excitations are determined in a straightforward manner (see below). The theoretical prediction [50], shown in Figure 1.2, matches the experimental data (indicated by dots) rather well from very small to mesoscopic particle sizes. The result obtained shows that the IBM, which models the kinetic repulsion of the occupied 4d-shell of atomic Xe, works surprisingly well. This repulsion causes an enhanced electronic density, leading to the blue-shift of the surface-plasmon line. 

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