A posteriori correction

Comparison of these expressions indicates that the activation entropy is related to the steric factor for the reaction. One may interpret the steric factor in terms of the degree of order of molecular configurations required to bring about the reaction, and this viewpoint is generally regarded as more satisfactory from an intellectual viewpoint than is that which regards Ps as an a posteriori correction factor necessary to obtain agreement between theory and experiment. 

As shown by Equations (2.189)—(2.191) the procedure briefly sketched above allows us to take into account all the effects of the solvent, both those intrinsic, i.e. due to the reaction potential, and the others related to the presence of the external field, in a compact and self-consistent form. In this way no a posteriori corrections, such as those usually introduced by cavity factors, are required, but the computed properties can be used as they are and introduced in the expressions linking the microscopic to the macroscopic. 

C can be written as C -I- o(e) provided that s is small compared to the eigenvalues of Hg, which will be the case if there are no intruder states. If the intruder states have been removed by the level shift, it is possible to make an approximate a posteriori correction of the second-order energy to an assumed unshifted calculation without the intruder state. The correction is obtained from Eq. (26) by replacing C with C,... 

Two routes are available to obtain such quantum mechanical description a simulation can be realised with the path integral molecular dynamics method (PIMD), recently applied to liquid water (Billeter, King and van Gunsteren, 1994), or a classical dynamics simulation can be realised, with an a posteriori corrections of the results (Berens, Mackay, White and Wilson, 1983 Berens and Wilson, 1981). The latter choice has been made for the calculation of gaseous CO spectra in liquid argon (Berens and Wlson, 1981). 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

These a posteriori corrections are based on a very simple idea which is suggested by the work of Brandow [10]. Brandow used the Brillouin-Wigner perturbation theory as a starting point for a derivation of the Goldstone linked diagram expansion by elementary time-independent methods . At a NATO Advanced Study Institute held in 1991, Wilson wrote [112] ... 

In the work of Brandow [10], Brillouin-Wigner perturbation theory is used as a step in the theoretical development of first Rayleigh-Schrodinger perturbation theory and then the many-body perturbation theory. In the a posteriori correction developed by the present authors in a paper entitled On the use of Brillouin-Wigner... 

We shall provide an overview of the applications that have been made over the period being review which demonstrate the many-body Brillouin-Wigner approach for each of these methods. By using Brillouin-Wigner methods, any problems associated with intruder states can be avoided. A posteriori corrections can be introduced to remove terms which scale in a non linear fashion with particle number. We shall not, for example, consider in any detail hybrid methods such as the widely used ccsd(t) which employs ccsd theory together with a perturbative estimate of the triple excitation component of the correlation energy. 

A posteriori corrections to multireference limited configuration interaction based on a Brillouin-Wigner perturtxilive analysis... 

A posteriori corrections to second order multireference Brillouin-Wigner perturbation theory based on the identity ... 

Finally, we mention the possibility of a posteriori correction in which we accept the deleterious effect of Cg on the recorded micrograph but attempt to reduce or eliminate it by subsequent digital or analog processing of the image. A knowledge of the wave theory of electron image formation is needed to understand this idea and we therefore defer discussion of it to Section III.B. 

Unlike the BWCCSD method, the MR-BWCCSD theory, the BWCISD theory and the MR-BWCISD theory do not support energies which scale linearly with the number of electrons in the system. In the multireference cases, that is MR-BWCCSD and MR-BWCISD theories, this lack of ex-tensivity arise both from nonlinear terms in the matrix elements of the effective hamiltonian and from diagonalization of the effective hamiltonian when an incomplete model space is employed. By exploiting the known relation between the Brillouin-Wigner and the Rayleigh-Schrddinger denominators it is possible to devise a posteriori corrections to methods formulated within the framework of Brillouin-Wigner perturbation theory. 

A posteriori corrected single reference Brillouin-Wigner configuration interaction, BWCI, can be obtained by introducing a denominator correction... 

The term in is again added in the final iteration. These a posteriori corrected matrix elements are then used to set up and solved the eigenvalue problem for the effective hamiltonian. For the MR-BWCISD method this... 

TABLE 1. Size-consistency error, e, in MR-BWCCSD calculations for the ground state of the F2 molecule, e is given by the difference Emr-bwccsd (F2, r=30 a.u.) -2 ccsd(F). The subscript corrected indicates that the a posteriori correction has been applied whilst the subscript uncorrected designates the uncorrected energy difference. ... 

It can also be seen from Table 4 that the iterative correction gives slightly lower energies than the a posteriori correction. For the larger carbenes with less multireference character the former correction is closer to the CCSD(T) result. 

We have demonstrated that the MR-BWCCSD theory can be subjected to an a posteriori correction (or iteratively corrected) so as to ehmi-nate terms which destroy the extensivity of the method. The origin of these terms has been described in section 8. The practical effects of the correction are described in section 9 where a few examples is given to demonstrate a near complete restoration of extensivity. 

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