Slater decomposition

Analogous to the Schmidt decomposition, it can be proved that every 1 ) can be represented in an appropriately chosen basis in in a form of Slater decomposition [45],... 

Fig. 7. C1 isotope effect in decarboxylation reactions. Theoretical curves for transition state which has lost a C—C stretching frequency. Curve S— Slater decomposition coordinate, curve N—molecular fragment coordinate...

We now discuss (ii), the evaluation of operator expectation values with the reference ho- We are interested in multireference problems, where may be extremely complicated (i.e., a very long Slater determinant expansion) or a compact but complex wavefunction, such as the DMRG wavefunction. By using the cumulant decomposition, we limit the terms that appear in the effective Hamiltonian to only low-order (e.g., one- and two-particle operators), and thus we only need the one- and two-particle density matrices of the reference wavefunction to evaluate the expectation value of the energy in the energy expression (7). To solve the amplitude equations, we further require the commutator of which, for a two-particle effective Hamiltonian and two-particle operator y, again involves the expectation value of three-particle operators. We therefore invoke the cumulant decomposition once more, and solve instead the modihed amplitude equation... 

Moritz, G., Reiher, M. Decomposition of density matrix renormalization group states into a Slater determinant basis. J. Chem. Phys. 2007, 126(24), 244109. 

The local spin values obtained in a decomposition scheme may either be local expectation values as evident from the last section. Since we are interested merely in Slater determinants describing local high-spin centers, it is most suitable to employ local (Sza) values as properties to be constrained in such a multiplier scheme. If we recall that the (Sza) expectation values should be equal to the ideal Msa and denote ideal values as M al values, the Lagrangian can be chosen to be (135,136)... 

When P(E) is not perturbed by the reaction, so that the distribution of critically energized molecules is that characteristic of equilibrium, the RRK model leads to a specific first-order rate constant of the form k = A exp —E /RT) where A is the frequency of internal energy transfer between oscillators. The Slater formulation in these circumstances gives k = V exp —E /RT ), both results being similar in form to the Arrhenius equation. The A factor in the RRK model represents the frequency of energy transfer between oscillators, which Jor weakly coupled oscillators would be of the order of their beat frequencies, or about 10 to 10 sec In the Slater model, V represents a weighted rms frequency of the normal frequencies which describe the decomposition [Eq. (X.6.1)]... 

While the Slater model does not lend itself to a simple solution in terms of the quantum theory, the fact that it agrees in form with the simple quantum model of Rice-Ramsperger and Kassel suggests that we can write the following expression for the mean rate of decomposition of a critically energized molecule of energy E E ... 

The transmission coefficient corrects for the number of complexes that pass over the barrier and are reflected back again before being deactivated to final products. Generally it is assumed that ic = 1. The RRK and Slater models for spontaneous decompositions need similar correction. 

Other minimal basis sets have been used in S.C.F. calculations for chemical reactions. I must mention a very accurate study of the transition state for the geometrical isomerization of cyclopropane, and a search for nondynamical pathways on the potential surface for this reaction when a minimal basis set of Slater orbitals was used 16 0-162) Calculations with a basis of this kind were also performed for the coplanar decomposition of cyclobutane 163>. Minimal basis sets based on gaussian... 

It seems worthwhile to examine critically this transcription of the Slater method into the standard absolute reaction rate theory. In the simple unimolecular bond break, it does appear reasonable that the coordinate q between the tvfo atoms A and B must reach and go beyond a critical extension q0 in order that decomposition takes place. In Slater s calculations account is taken of the different energies involved in stretching q to q0. In regarding q as the mode of decomposition in the transition state method, one must, however, first look at the potential energy surface. The decomposition path involves passage over the lowest possible barrier between reactants and products. It does not seem reasonable to assume that this path necessarily only involves motion of the atoms A and B at the activated complex. Possibly, a more reasonable a priori formulation in a simple decomposition process would be to choose q as the coordinate which tears the two decomposition fragments apart. Such a coordinate would lead roughly to the relation... 

Bone and Coward 21 found float the rate of decomposition of methane in the presence of porcelain was very low at temperatures below 700° C. Large surface exposure promoted the reaction, which apparently was reversible. Using as a basis the amount of dissociation in an empty tube Slater 82 found that silica, magnesia, alumina and baryta did not accelerate the reaction, and that copper, carborundum, graphite, charcoal, and iron did. The temperatures were between 900° and 1000° C., rather high for good comparison of catalytic surfaces. 

The theory of this type of decomposition has been developed by O. K. Rice, H. C. Rams-berger, and L. S. Kassel more recently, N. B. Slater has treated the problem in more exact and elegant terms. The treatment is based on the supposition that if too much energy gets into a particular mode of vibration, then vibration of the molecule in this mode leads to dissociation of the molecule. 

The classical mechanical model of unimolecular decomposition developed by Slater (1959) is based on the normal mode harmonic oscillator Hamiltonian, Eq. (2.47). 

The first and the second reasons are related to each other. To understand their relation and their difference we have to go back to Wigner and Seitz s paper D, where the Fermi correlation is defined. The wave function used there is the simplest one compatible with the Pauli principle, namely a single Slater determinant. Alternatively to the decomposition of n into and Tit introduced in Section II. A, one can break down the functions q and 51 in the following way ... 

Tritium specific activity determinations are usually carried out with larger uncertainties than mass spectrometric deuterium determinations. Calculations of k kj ratios with the use of equation 203 are therefore not very reliable. Nevertheless, the data obtained in the high-temperature region indicate clearly that the critical coordinate for reaction 197 is the C—H rather than the C—C distance and support the Slater assumption based on theoretical considerations concerning unimolecular decomposition of cyclopropane. 

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