Reduced Linear Equations

A feature common to all propagator or response function methods is that the response function is given as the product of a property gradient vector T (Pa) with the inverse of the principal propagator matrix hiuS — E) and another property gradient vector 

In actual calculations, however, the inverse of the principal propagator is never evaluated. Instead, the solution vector is obtained as a solution of the corresponding set of coupled linear equations 

The number of trial vectors required for a converged solution is normally orders of magnitude smaller than the dimension of the principal propagator. 

In a given iteration n the expansion of the solution vector, Elq. (13.9), is inserted in the linear equations Elq. (13.8), which are then premultiplied with the trial vectors hi 6 . This transforms the original hnear equations, Elq. (13.8), to the basis of the trial vectors 6i 6 , which is called the reduced space. 

The linear equations in the reduced space are then solved with standard techniques, meaning by calculating the inverse of (hujS — E ). In order to check whether the solution vector is already converged in this iteration one compares the norm of a residual vector, defined as, 

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G. D. Purvis and R. J. Bartlett, J. Chem. Phys., 75, 1284 (1981). The Reduced Linear Equation Method in Coupled Cluster Theory. 

Obviously the inversion of the very large matrix Rff E) is one of the difficult problems that has to be addressed. An inversion could be performed by employing a reduced linear equation (RLE) scheme but rapidly becomes impractical with increasing basis sets. A number of approximate treatments have been proposed with varying success. The order concept can be preserved with the identity... 

In the following sections some examples will be given of the calculation of the electromagnetic molecular properties introduced in Chapters 4 to 8 with some of the ab initio methods described in Chapters 10 to 12. The examples are neither meant to give an exhaustive overview of the performance of the different ab initio methods nor the molecular properties. But before doing so we have to discuss one important practical issue in all quantum chemical calculations, the one-electron basis set, and the more technical question of how the response functions or propagators are evaluated in actual calculations, i.e. the reduced linear equations algorithm. 

This reduces the calculation at each step to solution of a set of linear equations. The program description and listing are given in Appendix H. 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

Occasionally some nonlinear algebraic equations can be reduced to linear equations under suitable substitutions or changes of variables. In other words, certain curves become the graphs of lines if the scales or coordinate axes are appropriately transformed. 

Hie quasi steady state approximation can be conveniently applied to equations 19 to 21, without any significant loss of accuracy, due to tlie high reactivity of tlie reacting species in aqueous solution. Hms, the system of ordinary differential equations is readily reduced to a system of algebraic non linear equations. 

The above theory is usually called the generalized linear response theory because the linear optical absorption initiates from the nonstationary states prepared by the pumping process [85-87]. This method is valid when pumping pulse and probing pulse do not overlap. When they overlap, third-order or X 3 (co) should be used. In other words, Eq. (6.4) should be solved perturbatively to the third-order approximation. From Eqs. (6.19)-(6.22) we can see that in the time-resolved spectra described by x"( ), the dynamics information of the system is contained in p(Af), which can be obtained by solving the reduced Liouville equations. Application of Eq. (6.19) to stimulated emission monitoring vibrational relaxation is given in Appendix III. 

A simple example will show how higher-degree linear equations reduce to a system of first-order equations. 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

A stepwise multivariate LR analysis of the log PS values of the 23 diffusion compounds and 50 descriptors yielded a linear equation that consisted of 10 descriptors. After considering the relevance in physical meaning of each descriptor and statistical significance, the 10-descriptor model was reduced to a 3-descriptor model (Eq. 68) ... 

The fastest HPLC separations are achieved using the maximum available pressure drop. Using reduced variables, Equation 9.6 illustrates a linear relationship between retention time and mobile phase viscosity for packed columns and fixed values of AP (pressure drop), Areq (required efficiency for a given separation) and (a constant that describes the permeability of the packed bed) [4]... 

It should be noted that the system of linear equations expressed by (54) represents the response variable from the three batches. The required condition for the three batches to have the same intercept is that 5i = 52 = 0. The three batches will have the same slope if and only if Ai = A2 = 0. Thus, the problem for testing poolabil-ity reduces to fit the regression model (54) and test the following hypotheses hy. 5i = 52 = 0 and h2 = A2 = 0. Therefore, if the null hypothesis h2 is not rejected at the 0.25 significance level, it implies that the slopes of the three batches are the same, that is, Pi = p2 = = p. Similarly, if the null hypothesis hi is not rejected, the intercepts of the three batches are the same, that is, ai = a2 = a3 = a. If that is the case, the shelf life is determined by a model with a single intercept and a single slope. 

If w is complex one can, of course, first reduce it to the preceding case by writing the 2n equations for its real and imaginary parts. Usually, however, one is only interested in the quantities uv u they obey a set of n2 linear equations, from which one can again find n2 approximate equations for... 

Thanks to this lemma the problem reduces to solving the linear equation (5.2). Of course, this can be done explicitly only in rare cases, but we are interested in an expansion for small fluctuations. Suppose... 

If the exponent a turns out to be less than its classical value found in the linear approximation, ao = 1 (d 2), the relevant reaction rate K(t) is reducing at long times so that in the long-time limit K(oo) = 0 (the reaction rate s zerofication). On the other hand, when searching the asymptotic solution of non-linear equations, the asymptotic relation (6.2.4) permits to replace tK(t)n(t) for a. 

Two empirical increment systems (Table 4.22) derived from experimental data as collected in Table 4.23 permit prediction of alkanol carbon-13 shifts. One is related to the shift value B of the hydrocarbon R —H and involves, as usual, addition of the increments Z, = <5,(r oh) — r)(- R > according to eq. (4.8 a) [268]. The other employs a linear equation (4.8 b), correlating the shifts of an alkanol R — OH and the corresponding methyl-alkane R —CH3 by a constant bk and a slope ak, which is 0.7-0.8 for a and about 1 for ji and y positions [269]. Specific parameter sets characterize primary, secondary, and tertiary alcohols (Table 4.22). The magnitudes of Z) increments in eq. (4.8 a) decrease successively from primary to tertiary alcohols (Table 4.22), obviously as a result of reduced populations of conformers with yliauche interactions in the conformational equilibrium when the degree of alkylation increases. 

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