Quadratic truncation

An alternative quadratic truncation test (QT) has also been suggested with associated asymptotic superlinear convergence.116 This criterion monitors, instead of the relative residual, the sufficient decrease of the quadratic model, 4 (p). Specifically, it checks whether qk p) has decreased sufficiently from one inner iteration to the next, in relation to the progress realized per inner iteration ... 

The left side is expanded in a binomial series, which is truncated after the quadratic term. Combination leads to... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

The exchange repulsion energy in EFP2 is derived as an expansion in the intermolecular overlap. When this overlap expansion is expressed in terms of frozen LMOs on each fragment, the expansion can reliably be truncated at the quadratic term [44], This term does require that each EFP carries a basis set, and the smallest recommended basis set is 6-31-1— -G(d,p) [45] for acceptable results. Since the basis set is used only to calculate overlap integrals, the computation is very fast and quite large basis sets are realistic. 

The improvement in the fit from the quadratic polynomial applied to the nonlinear data indicated that the square term was indeed an important factor in fitting that data. In fact, including the quadratic term gives well-nigh a perfect fit to that data set, limited only by the computer truncation precision. The coefficient obtained for the quadratic term is comparable in magnitude to the one for linear term, as we might expect from the amount of curvature of the line we see in Anscombe s plot [7], The coefficient of the quadratic term for the normal data, on the other hand, is much smaller than for the linear term. 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

Considering now the calculated energies of the truncated expansions as functions of the estimated normalization deficiencies, one finds that, in all cases, the energies approach the flill-SDTQ value for the limit Ac (Ntr)-- 0 from above along a linear or weakly quadratic curve. By means of an extrapolation of this curve one can then determine the degree of truncation necessary for the error not to exceed the desired threshold, say 1 mh, without having to calculate a Cl wavefimctions larger than the truncated one. 

Truncating the Taylor series at the quadratic terms (assuming these terms dominate because only small displacements from the equilibrium geometry are of interest), one has the so-called harmonic potential ... 

Carbon dioxide. Collision-induced absorption in carbon dioxide shows a discernible density dependence beyond density squared, even at densities as low as 20 amagats [34]. Over a range of densities up to 85 amagats the variation of the absorption with density may be closely represented by a (truncated) virial series (as in Eq. 1.2, with I(v) replaced by a(v)) of just two terms, one quadratic and the other cubic in density. The coefficient of g3 is negative. Relative to the leading quadratic coefficient, it is,... 

When the relation (A.26) is truncated, the method of unitary transformation reduces to that of the perturbations. It is sometimes possible to resum (A.26) exactly This is the case when we apply the transformation (2.35) to the hamiltonian (2.15) containing the linear and quadratic excitation-vibration couplings. The use of (A.26) allows us to obtain the relations (2.38)—(2.39). 

The accuracy of the averaged model truncated at order p9(q 0) thus depends on the truncation of the Taylor series as well as on the truncation of the perturbation expansion used in the local equation. The first error may be determined from the order pq 1 term in Eq. (23) and may be zero in many practical cases [e.g. linear or second-order kinetics, wall reaction case, or thermal and solutal dispersion problems in which / and rw(c) are linear in c] and the averaged equation may be closed exactly, i.e. higher order Frechet derivatives are zero and the Taylor expansion given by Eq. (23) terminates at some finite order (usually after the linear and quadratic terms in most applications). In such cases, the only error is the second error due to the perturbation expansion of the local equation. This error e for the local Eq. (20) truncated at 0(pq) may be expressed as... 

Truncated Newton methods were introduced in the early 1980s111-114 and have been gaining popularity ever since.82-109 110 115-123 Their basis is the following simple observation. An exact solution of the Newton equation at every step is unnecessary and computationally wasteful in the framework of a basic descent method. That is, an exact Newton search direction is unwarranted when the objective function is not well approximated by a convex quadratic and/or the initial point is distant from a solution. Any descent direction will suffice in that case. As a solution to the minimization problem is approached, the quadratic approximation may become more accurate, and more effort in solution of the Newton equation may be warranted. 

The performance and large-scale feasibility of a TN algorithm depend on the precise formulation of a truncation criterion and implementation of the inner PCG loop. The PCG process can be terminated when either one of the following conditions is satisfied (1) the residual rk is sufficiently small, (2) the quadratic model qk p ) as defined in Eq. [35] is sufficiently reduced, or (3) a direction of negative curvature d is encountered (i.e., < 0). A negative... 

Laidig and Bartlett/li8(a)/ have implemented the theory in its linearized form. They applied it to the symmetric dissociating potential curve for the ground state of water and the symmetric abstraction from BeH2. Each T is truncated at the two-body level. The results were compared to the corresponding MR-CISD values. To our knowledge, a quadratic extension of the formalism has not been applied computationally. A spin-adapted version of the linearized model has also been recently formu1 ated/118[Pg.332]

When the electron is partially delocalized, one should switch to the adiabatic representation in which the upper and lower CT surface are split by an energy gap depending on P. If this energy gap is expanded in P with truncation after the second-order term, we come to the model of a donor-acceptor complex whose dipolar polarizabilities are different in the ground and excited states. The solute-solvent interaction energy then attains the energy of solute polarization that is quadratic in P... 

In principle, a Cl approach provides an exact solution of the many-electron problem. In practice, however, only a finite set of Slater determinants can be handled in the linear expansion. A common procedure is to retain all Slater determinants that differ from the HF determinant by one or two excitations (although one-electron excitations do not couple directly to the ground state they couple with two-electron excitations, which in turn affect the ground state indirectly). Unfortunately, such a procedure is not size consistent. For example, the energy of two highly separated monomers will not be twice that of a single monomer in such a truncated Cl calculation. Fortunately, a slightly modified approach called quadratic Cl has recently been developed (Pople et al., 1987) that is size consistent. 

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