Gradient matrix

Just like the electric quadrupole moment, the electric field gradient matrix can be written in diagonal form for a suitable choice of coordinate axes. 

Here, the integrand is the off-diagonal gradient matrix element between adiabatic electronic states,... 

Among various initial concentration distributions in a matrix, the sigmoidal drug distribution or a staircase distribution resembling a sigmoidal pattern furnishes zero-order release kinetics for the planar geometry. There are various methods to build concentration gradient matrix systems. The most convenient way to achieve the... 

A point Q0 = QJ0 where all gradient matrix elements are equal to zero is referred to as a stationary point ... 

Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid this parameter is the displacement gradient, V . The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu),y = dujdxj, also denoted Uij in which i denotes the i" displacement element and j denotes a derivative with respect to the y spatial coordinate, i.e. [1],... 

Duffar BV, Weesp, The Netherlands Gradient matrix system ... 

Van Bommel, E.M.G., Ed. Gradient Matrix Systems for Controlled Delivery of Drugs, Company Thesis Duphar B.V., Weesp The Netherlands, 1990. 

The default method for all first-order methods used both the gradient and Hessian matrix in the estimation of the standard errors. Standard errors were also calculated using only the gradient (Matrix = S) or only the Hessian (Matrix = R). The results are presented in Table 7.5 and Table 7.6. 

The constants A, C and D may be determined from comparison between calculations and experiment. The above treatment requires that the Pj s be expressed in the principal axis systems of the electric field gradients. Usually the z-direction may be taken as the direction of the covalent bond to nucleus A. When the population submatrix involving the A valence shell np, np and np orbitals is diagonal then the field gradient matrix is also diagonal - a fact which may be used when... 

S. Bubin, L. Adamowicz. Energy and energy gradient matrix elements with N-partide expHdtly correlated Gaussian basis functions with L = 1. J. Chem. Phys., 128 (2008) 114107. 

In the quantum theory of absorption that we discuss in Chap. 5, the transition gradient matrix element V arises directly, rather than just as an... 

Figure 4.17 illustrates the functions that enter into the transition gradient matrix element for the HOMO—>LUMO excitation of ethylene. Contour plots of the HOMO and LUMO (V a and tpi,) are reproduced from Fig. 4.6B in panels A and B... 

Fig. 4.18 Canonical orientations of orbitals of two carbon atoms. The shaded regions represent boundary surfaces of the wavefunctions as in Fig. 2.5. The Vy z Cartesian coordinate system is centered on atom 1 with the y axis aligned along the interatomic vector. The transition gradient matrix elements VVn and V y are for the orientations shown in A, B and C, respectively. Matrix elements for an arbitrary orientation can be expressed as linear combinations of these canonical matrix elements...

Fig. 9.4 Vector diagrams of contributions to the transition-gradient matrix elements for the first two excitations of trans-butadiene. The initial and final molecular wavefunctions in the four-orbital model used for Fig. 4.19 are indicated. The atoms are labeled 1 by the empty circles...

If p and q are both occupied spinor indices, the first-order density matrix element is a delta function, Dpq = Spq, and the gradient matrix element is zero. This is in accord... 

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