Cramer solution

Now it is very simple to obtain coefficients Pq and Pj as the Cramer solution of system (5.39). The following expressions for Pq and Pj are thus obtained ... 

Cramer C j and D G Truhlar 1992. AM1-SM2 and PM3-SM3 Parametrized SCF Solvation Models for Free Energies in Aqueous Solution. Journal of Computer-Aided Molecular Design 6 629-666. 

By Cramer s rule, each solution of Eqs. (2-44) is given as the ratio of determinants... 

Solving the normal equations by Cramer s rule leads to the solution set in determinantal fomi... 

One very popular technique is an adaptation of the Born model for orbital-based calculations by Cramer and Truhlar, et. al. Their solvation methods (denoted SMI, SM2, and so on) are designed for use with the semiempirical and ah initio methods. Some of the most recent of these methods have a few parameters that can be adjusted by the user in order to customize the method for a specific solvent. Such methods are designed to predict ACsoiv and the geometry in solution. They have been included in a number of popular software packages including the AMSOL program, which is a derivative of AMPAC created by Cramer and Truhlar. 

This set of equations for the elements of L can be resolved by application of Cramer s rale. Then, a nontrivial solution exists only if the determinant of die coefficients vanishes, or... 

Equation (125) applies for all values of the index k — 1,2,..., m. It is a set of m simultaneous, homogeneous, linear equations for the unknown values of the coefficients c . Following Cramer s rule (Section 7.8), a nontrivial solution exists only if the determinant of the coefficients vanishes. Thus, the secular determinant takes the form... 

If the unit matrix E is of order n, Eq. (67) represents a system of n homogeneous, linear equations in n unknowns. They are usually referred to as the secular equations. According to Cramer s rule [see (iii) of Section 7.8], nontrivial solutions exist only if the determinant of the coefficients vanishes. Thus, for the solutions of physical interest,... 

This result is a system of simultaneous linear, homogeneous equations for the coefficients, cu. Cramer s rule states that a nontrivial solution exists only if... 

Within a similar series of reagents, complexing tendency toward the different cycloamyloses can be qualitatively correlated with the size of the reagent. All three cycloamyloses, for example, are effectively precipitated from aqueous solution by benzene, but only cyclooctaamylose is precipitated by anthracene. Similarly, for cycloheptaamylose, bromobenzene is a more effective precipitant than benzene, whereas the reverse is true for cyclohexaamylose. Discriminating precipitants such as these have been incorporated by French and associates (1949) and by Cramer and Henglein (1958) into schemes for the separation of cyclohexa-, cyclohepta-, and cyclooctaamylose. 

With the realization that the cycloamyloses form stable monomolecular inclusion complexes in solution came the idea that the inclusion process might affect the reactivity of an organic substrate. This idea was initially pursued by Cramer and Dietsche (1959b) who discovered that the rates of hydrolysis of several mandelic acid esters are enhanced by the cycloamyloses. More recently, the inclusion process has been shown to exert both accelerating and decelerating effects on the rates of a variety of organic reactions. The remainder of this article will be devoted to a discussion of these reactions in an attempt to review, compare, and unify the many intriguing facets of cycloamylose catalysis. 

C. J. Cramer and D. G. Truhlar, AM1-SM2 and PM3-SM3 parameterized SCF solvation models for free energies in aqueous solution, J. Comput.-Aided Mol. Design 6 629 (1992). 

G. D. Hawkins, C. J. Cramer, and D. G. Truhlar, Parameterized models of aqueous free energies of solvation based on pairwise descreening of solute atomic charges from a dielectric medium, J. Phys. Chem. 100 19824 (1996) erratum to be published. 

Gaussian elimination is a very efficient method for solving n equations in n unknowns, and this algorithm is readily available in many software packages. For solution of linear equations, this method is preferred computationally over the use of the matrix inverse. For hand calculations, Cramer s rule is also popular. 

Tomasi, J. (1994) Application of continuum solvation models based on a Quantum Mechanical Hamiltonian.,in Cramer, C. J. and Truhlar, D. ( .(cds.). Structure and Reactivity in Aqueous Solution, American Chemical Society, Washington,pp. 10-23. 

In the development of solvation models, Cramer and Tmhalar have made several noteworthy contributions [8-11]. Most of the implicit solvation models do not include the effect of first solvation shell on the solute properties. This can be satisfactorily treated by finding the best effective radii within implicit models. In addition to the first-solvent-shell effects, dispersion interactions and hydrogen bonding are also important in obtaining realistic information on the solvent effect of chemical systems. 

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