Orthogonal second order

Orthogonal Second-order Design (Box-Benken Design)... 

This example refers to response dependence on two factors (k=2). Orthogonal second-order design in this case, according to Table 2.164, has nine design points (N=9). The design matrix with outcomes of design points is shown in Table 2.165. The same case has been elaborated in the previous section, in Example 2.43, by application of rotatable second-order design. However, the connection between coded and real values of factors for the same null point is now different ... 

To process data obtained by application of orthogonal second-order designs, regression coefficient significances are checked by expression (2.144), along with previous calculations of variance or error in determining regression coefficients ... 

When nc=0 or n0=l, as is the case in orthogonal second-order designs, B4 and Hartley s designs, Eq. (2.157) becomes simpler ... 

Orthogonal second-order design Equal replication (2.170) and (2.128)... 

We have found that the structure of the empty (8,8) CNT at the ground state is the Kekule structure with tripling of the translational period and two essentially different bonds (Fig. 1). The structure of C20 is shown in Fig. 2. It has Di symmetry with three orthogonal second order symmetry axes, one of which is shown in Fig. 2 by the dashed line. 

Fundamental support for the d correction was expressed by Morokuma et who observed that in energy-partitioned SCF calculations the term became excessive for basis sets subject to large BSSE. They argued that a virtuals-only scheme is the proper one to correct for this. It was noted that the virtuals of the ghost to be included should really be those of the counterpoise-corrected instead of the isolated ghost. This recipe is difficult to implement. However, in their example the difference from the simple recipe was extremely small. Similarly. Hayes and Stone state that, while only the virtuals of the partner are available in their non-orthogonal second-order perturbation theory, the occupied orbitals may become available in higher orders. 

For the second-order correction, we can recognize that the only possible normalized trial function l) orthogonal to Ei]) in this 2x2 case is... 

Note that the choice of non-orthogonal versus orthogonal basis functions has no consequence for the numerical variational solutions (cf. Coulson s treatment of He2, note 76), but it undermines the possibility of physical interpretation in perturbative terms. While a proper Rayleigh-Schrodinger perturbative treatment of the He- He interaction can be envisioned, it would not simply truncate at second order as assumed in the PMO analysis of Fig. 3.58. Note also that alternative perturbation-theory formulations that make no reference to an... 

Figure 5.4 Isomers of the NH3- HF complex (left) and leading donor-acceptor interactions (right), with estimated second-order stabilization energies in parentheses. (Note that a second nF —> uhN stabilization [with orthogonal nF directed along the H—F axis] competes with that shown in (b) to control the HF- -HNH2 geometry.)...

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

Recently, a unitarily invariant decomposition of Hermitian second-order matrices of arbitrary symmetry under permutation of the indices within the row or column subsets of indices has been reported by Alcoba [77]. This decomposition, which generalizes that of Coleman, also presents three components that are mutually orthogonal with respect to the trace scalar product [77] ... 

The second term in equation (3) ensures the orthogonality of the transformed radial orbital obtained. The criterion for determining the optimal values of integer parameters k and m and the freely variable parameter B (or A) is the maximum of the averaged energy correction, expressed in the second order of perturbation theory. 

Differential scanning calorimetry (DSC) Since lc s form phases in a thermodynamic sense, a transition from one phase to another is accompanied by a phase-transition enthalpy. Nevertheless, there are phase transitions of second-order character which can hardly be detected by DSC since there is no phase-transition enthalpy but just a change in heat capacity. A typical example is the transition from orthogonal phases to tilted phases. 

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