Matrix first-order approximation

For some direct-gap materials, the quantum electronic selection rules lead to = 0. However, this is only strictly true at / = 0. For 0, it can be assumed, in a first order approximation, that the matrix element involving the top valence and the bottom conduction states is proportional to k that is, Pif k. Within the simplified model of parabolic bands (see Appendix Al), it is obtained that Tuo = Tuog + flp., and therefore Pif k co — cog). Thns, according to Equations (4.31) and (4.32), the absorption coefficient for these transitions (called forbidden direct transitions) has the following spectral dependence ... 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

The first operand between the parentheses in Equation (2) contains the diagonal, the second one the off-diagonal components of the matrix. The diagonal components are also composed of two members, from which the second one gives the first-order approximation of the coupling. 

Suppose Y = f(x, 0, t ) + g(z, e) where nr] — (0, il), (0, ), x is the set of subject-specific covariates x, z, O is the variance-covariance matrix for the random effects in the model (t ), and X is the residual variance matrix. NONMEM (version 5 and higher) offers two general approaches towards parameter estimation with nonlinear mixed effects models first-order approximation (FO) and first-order conditional estimation (FOCE), with FOCE being more accurate and computationally difficult than FO. First-order (FO) approximation, which was the first algorithm derived to estimate parameters in a nonlinear mixed effects models, was originally developed by Sheiner and Beal (1980 1981 1983). FO-approximation expands the nonlinear mixed effects model as a first-order Taylor series approximation about t) = 0 and then estimates the model parameters based on the linear approximation to the nonlinear model. Consider the model... 

In a recent publication we have investigated this first order approximation to the particle-hole self energy for the choice y>) = o ) for the reference state tp) and starting from a Hartree-Fock zeroth order [21]. This particular approximation to the particle-hole self energy is referred to as First Order Static Excitation Potential (FOSEP). In terms of the matrix elements of the Hamiltonian the FOSEP approximation of the primary block H reads... 

The FOSEP approximation has to be compared with two other well-known first order approximation schemes, the Tamm-Dancoff approximation (TDA) [11,30] and the random phase approximation (RPA) [31,32,11,30]. The TDA leads to a hermitian eigenvalue problem of half the dimension of FOSEP. In fact the upper left (ph-ph) block of the FOSEP matrix coincides with... 

The first-order approximation to eq. (6), the so-called adiabatic approximation, consists of neglecting the off-diagonal matrix elements of C (i.e. Cpp t=0 when F r ). It can be shown 7) that, for the groimd state of H2+, the adiabatic correction accoimts for 99.8% of the total correction to the B.O. approximation. A similar statement is correct (8,9) for the ground state of H2. We expect that the adiabatic correction is sufficient for all tiie systems considered here and subsequent discussion will be restricted to the adiabatic correction. 

The matrix represented in (31) is not diagonal, but the terms contributed by the perturbation, namely, LJF AG(L ), are small compared with the (dements of A . Therefore, it is a good first-order approximation to put... 

Although simple models are essential for a fundamental understanding, in the future more realistic models are required, in particular in view of possible applications in cell and tissue engineering. Here we have used cable networks as a first step towards more realistic models for both cell and the matrix. Anisotropic force contraction dipoles are only the first order approximation for the complex mechanical activity of cells and might be extended to more general tensors for mechanical activity and susceptibility. A more sophisticated model would be to replace the force dipoles by whole cell models incorporating the focal adhesion dynamics and stress fibers evolution. 

A first order approximation for Td which is linear in the variables p and q is Td = TrTu + Tr Ta Tu + Tr Tf. The term Tr,i x is the usual Denavit-Hartenberg transformation matrix representing the rigid body motion. If i o is a frame attached to the ground, the transformation from /Eq to i2i+i,2c is given by the recursive relationship i iTd = Td. ... 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

Hisatsune and Linnehan [299] have used infrared measurements to study the decomposition of C104 in a KC1 matrix. Despite the differences in the environment of the perchlorate ion, the kinetics of reaction were similar to those reported by Cordes and Smith [845] for pure KC104. The reaction was second order and E was 185 kJ mole-1. Comparable behaviour was observed for CIOJ in KC1, except that E was lower ( 125 kJ mole-1) and when both ions (CIOJ and C104) were present the reaction was approximately first order. 

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 correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

Fiberglass reinforced plastic (FRP) is used in composite systems. It is particularly important in the process industry because of its corrosion resistance and light weight. The epoxy resin in the FRP matrix begins to melt at approximately 150°C (302°F). This may be used as a first-order failure criterion. Failure of empty 5-mm thick FRP pipes in 2-6 minutes has been reported (SINTEF,... 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

C. Valdemoro, Approximating the second-order reduced density matrix in terms of the first-order one. Phys. Rev. A 45, 4462 (1992). 

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