Reduced linear equations method

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

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. 

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. 

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. 

The calculated total concentration of component j T ) is then compared to the total analytical (input) concentration of component to calculate the residual in the mass balance. From this point an iterative algorithm based on the Newton-Rapshon method and Gausian elimination (to convert non-linear equations to linear) is used to refine the initial estimates of each component concentration. At each refinement the residual in the mass balance is reduced until some acceptable limit is reached. 

The resulting C + E equations are nonlinear in unknowns nj, nj, and tt but In nj are iteration variables since nj occur in logarithmic terms. These equations are linearized using first-order Taylor Series (Newton-Raphson method), in the variables An , A (In nj), and ir, and with n nj are reduced to S + 1 + E linear equations in unknowns AN, A (In N), and tt. When extended to include P mixed phases, we nave shown that they are nearly identical to the equations of the RAND Method and have the same coefficient matrix. 

The round-off error propagation associated with the use of Shacham and Kehat s direct method for the solution of large sparse systems of linear equations is investigated. A reordering scheme for reducing error propagation is proposed as well as a method for iterative refinement of the solution. Accurate solutions for linear systems, which contain up to 500 equations, have been obtained using the proposed method, in very short computer times. 

The linear operator A acts from the space M to the space D. It is easier to solve the inverse problem if the corresponding operator equation describes the transformation within the same vector space, for example, within a space of the model parameters M. By introducing the Euler equation, we have reduced the general linear inverse problem (4.1) to the linear operator equation (4.4) determined for the elements of the model parameter space. We will consider below several methods of solving this linear equation. 

This equation is to be formulated for all grid points (i, j). A system of linear equations for the unknown temperatures at time tk+l, that has to be solved for every time step, is obtained. Each equation contains five unknowns, only the temperature at the previous time tk is known. A good solution method has been presented by P.W. Peaceman and H.H. Rachford [2.69]. It is known as the alternating-direction implicit procedure (ADIP). Here, instead of the equation system (2.305) two tridiagonal systems are solved, through which the computation time is reduced, see also [2.53]. 

Now we solve first the non-linear Eq. (81a) using, for example, the iteration method to obtain the value of for the assumed values of the parameters K "/ and K as well as of the heterogeneity parameters. Next, we solve the second non-linear equation (81b) for the same parameters K , Kg , kT/ci, (i=A,C), and for the new calculated value of In the same way as before, we reduce the number of the intrinsic equilibrium constants from four to two. 

These thirty linear equations are reduced to seven normal equations as indicated above. By solving these, the atomic weights of the seven elements are obtained with the errors of observation evenly distributed among them according to the method of least squares. 

Equation 38 then is solved for the diameter. This linear fit method gives excellent results for the optimal diameter at reduced densities of about 2.4 and higher. 

Finally, linear equations (75b) and (79) can be solved by different methods. For example, using inverse matrices reduce Eqs. (75) and (80) to a traditional form of constrained inversion. Alternatively (Section 4.8), other numerical or computer techniques, such as, SVD, conjugated gradients, iterations, generic inversion, etc. can be used for solving Eqs. (75b) and (80). 

Three methods which do not require solution of the nonlinear partial differential equation are presented for estimating extractor performance. The choice of method depends on the value of the dimensionless outlet solute concentration, oj. If (oj + 1)/oJ is close to 1, the reaction is effectively irreversible and the pseudosteady-state solution of the advancing front model satisfactorily predicts performance after normalization to include solute solubility in the globule. If (oj + 1)/oJ is not close to 1, the advancing front results will still apply, provided that the amount of solute extracted by reaction is small and membrane solubility controls. When oj is small enough so that (oj + 1) is close to 1, then the reversible reaction model can be reduced to a linear equation with an analytical solution. Otherwise, for oj values when neither (oj + 1) nor (oj + 1)/o is nearly 1, a reasonable first approximation is made by adjusting the actual concentration of internal reagent to an effective concentration which equals the amount consumed to reach equilibrium. 

Abstract The modified equation-of-motion coupled cluster approach of Sekino and Bartlett is extended to computations of the mixed electric-dipole/magnetic-dipole polarizability tensor associated with optical rotation in chiral systems. The approach - referred to here as a linearized equation-of-motion coupled cluster (EOM-CCl) method - is a compromise between the standard EOM method and its linear response counterpart, which avoids the evaluation of computationally expensive terms that are quadratic in the field-perturbed wave functions, but still yields properties that are size-extensive/intensive. Benchmark computations on five representative chiral molecules, including (P)-hydrogen peroxide, (5)-methyloxirane, (5 )-2-chloropropioniuile, (/ )-epichlorohydrin, and (75,45)-norbornenone, demonstrate typically small deviations between the EOM-CCl results and those from coupled cluster linear response theory, and no variation in the signs of the predicted rotations. In addition, the EOM-CCl approach is found to reduce the overall computing time for multi-wavelength-specific rotation computations by up to 34%. 

In this chapter, we develop analytical solution methods, which have very close analogs with methods used for linear ODEs. A few nonlinear difference equations can be reduced to linear form (the Riccati analog) and the analogous Euler-Equidimensional finite-difference equation also exists. For linear equations, we again exploit the property of superposition. Thus, our general solutions will be composed of a linear combination of complementary and particular solutions. 

The method discussed in Section 3 for the derivation of the generalized Boltzmann equation can be carried over, with some simple modifications, to derive an analogous kinetic equation for d, t). However, due to the special role of the tagged particle and the fact that the corresponding initial N-particle distribution is proportional to Ui, one obtains a linear equation for 4>d (vi, t) that for low densities reduces to the Lorentz-Boltzmann equation " ... 

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