Quadratic variants

Thus, we can expect further improvements in the description of multiple bond breaking by the QMMCC method. This statement parallels similar findings by Head-Gordon et al. (27, 28, 128, 129), who considered the quadratic variant of the ECC theory of Arponen and Bishop (114 -123), in which the energy is calculated by imposing the stationary conditions for the asymmetric, doubly connected, energy functional... 

New noniterative coupled-cluster methods for bond breaking the method of moments of coupled-cluster equations and its quadratic variant and the renormalized and completely renormalized coupled-cluster approaches... 

The results of the LECCSD, BECCSD, QECCSD, and full ECCSD calculations for the STO-3G N2 molecule are shown in Tables 4 and 5 and Figures 5 and 6. The results in Table 4 and Figure 5 clearly demonstrate that the complete ECCSD formalism of Piecuch and Bartlett [62], in which all nonlinear terms in E and T are included, and its bilinear and quadratic variants, BECCSD and QECCSD, respectively, defined by the truncated... 

A disadvantage of all these limited Cl variants is that they are not size-consistent.The Quadratic Configuration Interaction (QCI) method was developed to correct this deficiency. The QCISD method adds terms to CISD to restore size consistency. QCISD also accounts for some correlation effects to infinite order. QCISD(T) adds triple substitutions to QCISD, providing even greater accuracy. Similarly, QCISD(TQ) adds both triples and quadruples from the full Cl expansion to QCISD. 

Note that there are n + m equations in the n + m unknowns x and A. In Section 8.6 we describe an important class of NLP algorithms called successive quadratic programming (SQP), which solve (8.17)—(8.18) by a variant of Newton s method. 

The method of moments of coupled-cluster equations (MMCC) is extended to potential energy surfaces involving multiple bond breaking by developing the quasi-variational (QV) and quadratic (Q) variants of the MMCC theory. The QVMMCC and QMMCC methods are related to the extended CC (ECC) theory, in which products involving cluster operators and their deexcitation counterparts mimic the effects of higher-order clusters. The test calculations for N2 show that the QMMCC and ECC methods can provide spectacular improvements in the description of multiple bond breaking by the standard CC approaches. 

Therefore, the controller is a linear time-invariant controller, and no online optimization is needed. Linear control theory, for which there is a vast literature, can equivalently be used in the analysis or design of unconstrained MPC (Garcia and Morari, 1982). A similar result can be obtained for several MPC variants, as long as the objective function in Eq. (4). remains a quadratic function of Uoptfe+ -iife and the process model in Eq. (22) remains linear in Uoptfe+f-ife. Incidentally, notice that the appearance of the measured process output y[ ] in Eq. (22) introduces the measurement information needed for MPC to be a feedback controller. This is in the spirit of classical hnear optimal control theory, in which the controlled... 

The first variant to look at is therefore to start the process slightly further up the chain of derivatives. If we start at the linear B-spline we get a basis function of support 3, if we start at the quadratic we get one of support 4, and so on. If we call the original UP function UP0, we can call the others UPi, UP2 etc. 

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