Nonphysical transformation

So far we have mainly described perturbations (and integrations) with respect to real physical properties, such as a reaction coordinate or the temperature. However, in the section on umbrella sampling we provided several applications of theories derived by Zwanzig (Eq. [22]) and Kirkwood (Eq. [33]) for fluids that are based on nonphysical transformations of the Hamilton-... 

Further developments of these ideas took place in computational structural biology, where nonphysical local transformations were implemented within the framework of thermodynamic cycles. These nonphysical transformations were introduced in 1981 by Warshel, who studied ionization in acidic residues in proteins pK calculations). Although the cycle included nonphysical transformations, they were not carried out by the perturbation technique. A year later Warshel used the perturbation method together with umbrella sampling to study the solvation free energy contribution to an electron transfer reaction coordinate, using two spheres for donor and acceptor in water the perturbation, however, was performed along a physical path. Warshel also modeled some enzymatic reactions that involve nonphysical processes. ... 

In other words, the relative free energy of binding is obtained from the free energy differences of the nonphysical transformations. This process can involve the annihilation and creation of atoms. In practice, considering for example the solvent case, one defines a linear hybrid potential energy function E (x, X),... 

Figure 3 A thermodynamic cycle for the binding of two ligands, LI and LI, to a protein P. In the experiment the ligands are transferred from the solvent to the active site, and the difference AAF = AFj - is measured. In simulations the nonphysical transformation of LI - L2 is carried out in the protein and in solution, and the corresponding free energies AFp and AF are calculated. The thermodynamic cycle leads to the desired AAF in terms of the latter free energy differences, AAF = AFp - AF. ...

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

Schell (1965) recognized that the major deficiency of the Wiener inverse filter is the nonphysical nature of the partially negative solutions that it is prone to generate. He sought to extrapolate the band-limited transform O(co) by seeking a nonnegative physical solution 6 + (x) through minimization of... 

For a basic deconvolution problem involving band-limited data, the trial solution d(0) may be the inverse- or Wiener-filtered estimate y(x) (x) i(x). Application of a typical constraint may involve chopping off the nonphysical parts. Transforming then reveals frequency components beyond the cutoff, which are retained. The new values within the bandpass are discarded and replaced by the previously obtained filtered estimate. The resulting function, comprising the filtered estimate and the new superresolving frequencies, is then inverse transformed, and so forth. 

Hence, after a decade of false starts, chemists finally learned that the correct basis set should consist of functions that could represent the atomic Hartree-Fock orbitals plus allow for contraction and polarization corrections in the region where they are largest. Similarly it was realized that the Hartree-Fock virtual molecular orbitals were too diffuse for representing the correction to the SCF wavefunction due to electron correlation. Rather, correlation effects are best represented using excitations to nonphysical molecular orbitals that are of the same size as the occupied MOs. Initially this was learned by transforming existing wavefunctions to natural orbital form. Later, MCSCF orbital optimizations were used to obtain these localized correlating orbitals. 

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