Electron-hole density matrix

The above mentioned positivity conditions state that the 2-RDM D, the electron-hole density matrix G, and the two-hole density matrix Q must be positive semidefinite. A matrix is positive semidefinite if and only if all of its eigenvalues are nonnegative. The solution of the corresponding eigenproblems is readily carried out [73]. For D, it yields the following set of eigenvalues ... 

Use of the time-independent Keldysh form [76] in the NEGF theory enables us to calculate one-electron (hole) density matrix of any nonequilibrium steady state. In the NEGF theory, the lesser and greater Green s functions, GV(E) are defined as follows ... 

As a matter of fact, the hole and particle occupancies are identical for any bipartite networks treated within Jt-approximation, up to FCl/PPP. This is a simple corollary of the generalized pairing theorem of McLachlan [94] stating that the jr-electron charge density matrix of the alternant hydrocarbons is of the form... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The HRO s are holes density operators and operate by first filling orbitals with electrons (i.e. they annihilate holes) and then removing electrons from orbitals (i.e. they create holes). These operators generate the Holes Reduced Density Matrix (HRDM) which in our notation takes the form ... 

Yaron, D., Moore, E.E., Shuai, Z., Bredas, J.L. Comparison of density matrix renormalization group calculations with electron-hole models of exciton binding in conjugated polymers. J. Chem. Phys. 1998, 108(17), 7451. 

The pair density or second-order density matrix, obtained from a single determinantal function composed of orthogonal spin functions i is given in eqn (E1.4). Comparison of that expression for the pair density with that given in eqn (E7.10) yields for the Fermi hole for a reference electron of a spin at Tj... 

Hence, the integral of the lesser/greater Green s function is equal to the density matrix of the electron/hole. The density matrix of an electron is represented by... 

For an intraband transition between the states s) and s ) of the electron-hole pair one may simple average t(r) c(r) and ipl(r)ipv(r) over the unit cell using the Bloch functions at the band extremes, since the principal term does not already vanish. As a result, the corresponding matrix element of the charge density is given by the sum of the electron and hole contributions ... 

Also, for a spin-unpolarized two-electron system like the Hooke s atom, the exact parallel hole is made up entirely of exchange. This can be seen most easily in the spin-decomposed second-order density matrix. Since the ground state wavefunction is a spin singlet, it contains no contribution in which both electrons have the same spin. Therefore... 

The exchange-correlation energy can thus be obtained by integrating the electron-electron interaction over the A variable and subtracting the Coulomb part. The right-hand side of eq. (B.18) can be written in terms of the second-order reduced density matrix eq. (6.14), and the definition of the exchange-correlation hole in eq. (6.21) allows the Coulomb energy to be separated out. 

Faster computers and development of better numerical algorithms have created the possibility to apply RPA in combination with semiempirical Hamiltonian models to large molecular sterns. Sekino and Bartlett - derived the TDHF expressions for frequency-dependent off-resonant optical polarizabilities using a perturbative expansion of the HF equation (eq 2.8) in powers of external field. This approacii was further applied to conjugated polymer (iialns. The equations of motion for the time-dependent density matrix of a polyenic chain were first derived and solved in refs 149 and 150. The TDHF approach based on the PPP Hamiltonian - was subsequently applied to linear and nonlinear optical response of neutral polyenes (up to 40 repeat units) - and PPV (up to 10 repeat units). " The electronic oscillators (We shall refer to eigenmodes of the linearized TDHF eq with eigenfrequencies Qv as electronic oscillators since they represent collective motions of electrons and holes (see Section II))... 

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