Quantum-Onsager method

The quantum Onsager model, which has also been termed the Self-Consistent Reaction Field (SCRF) method, is the simplest of the continuum models used in solvation studies. In this model, which dates from the work of Kirkwood[44] and Onsager[45] in the 1930s, the solvent is represented by a continuous rmifonn dielectric with a static dielectric constant, e, surrounding a solute in a spherical cavity[46] - [48]. 

The Quantum-Onsager SCRF Method. In the dipole-in-a-sphere (or quantum-Onsager or Born-Kirkwood-Onsager) SCRF method, the molecular cavity is a sphere of radius a and the interaction between the molecular charge distribution and the reaction field is calculated by approximating the molecular charge distribution as an electric dipole located at the cavity center with electric dipole moment fi. In 1936, Onsager showed that the electric field in the cavity (the reaction field) produced by the polarization of the solvent by fi is (in atomic units)... 

In an SCRF quantum-Onsager calculation, one starts by using a method such as HF, DFT, MP2, or whatever, to calculate an electron probability density p r) for the isolated molecule, preferably at an optimized geometry. is calculated from (13.128) in a wave-function-based method or fi-om (15.119) in DFT] One then calculates the electric dipole moment in vacuum from (13.144) as = —f dt +... 

In the quantum-Onsager SCRF method, an uncharged solute molecule with no permanent dipole moment is unaffected by the solvent. In reality, the quadrupole and higher moments of the solute will interact with the solvent to give a reaction field. 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

The salient features of quantum formulation of Onsager reaction field model (dipole model) is described here. In this method, the reaction field is treated as perturbation to the Hamiltonian of the isolated molecule. If H0 is the Hamiltonian of the isolated molecule and HR[ is the reaction field [21], the Hamiltonian of the whole system (Hlol) is represented as... 

The expansion of the electrostatic potential into spherical harmonics is at the basis of the first quantum-continuum solvation methods (Rinaldi and Rivail, 1973 Tapia and Goschinski, 1975 Hylton McCreery et al., 1976). The starting points are the seminal Kirkwood s and Onsager s papers (Kirkwood 1934 Onsager 1936) the first one introducing the concept of cavity in the dielectric, and of the multipole expansion of the electrostatic potential in that spherical cavity, the second one the definition of the solvent reaction field and of its effect on a point dipole in a spherical cavity. The choice of this specific geometrical shape is not accidental, since multipole expansions work at their best for spherical cavities (and, with a little additional effort, for other regular shapes, such as ellipsoids or cylinders). 

To address the problem of finite system size, the EFP method has also been combined with continuum models in order to model the effects of the neglected bulk solvent [125], The Onsager equation was used to obtain the dipole polarization of the solute molecule (modeled quantum mechanically) and explicit water molecules (modeled by effective fragment potentials) due to the dielectric continuum. Thus the energy becomes... 

In accord with our interest we restrict our exposition in this section to statistical treatments which contain as an element the quantum mechanical cross-section or transition probability discussed in Section IV. Such statistical approaches which have been applied to chemical reactions may be conveniently divided into three categories those based on the Pauli equation or similar considerations (Section V-A), a modified Boltzmann equation (Section V-B), or a quantum statistical formulation of the Onsager theory (Section V-C). These treatments have not had notable success in comparison with experiment, probably because of the implicit Born approximation or its equivalent. It is therefore of considerable importance to extend this type of treatment to cross-sections other than that derived with the Born approximation. The method presented in Section V-C would seem to offer the best hope in this direction. 

New concepts. This last point is the most important. The impact of a model is determined by the quality of the new concepts it introduces points 1-3 are just additional conditions. Onsager s model has introduced new concepts. More than 60 years have shown their validity and their capability of surviving the revolutionary changes in our methods to describe matter. We have paid attention in the preceding discussion to concepts of the cavity and of the reaction field also the concept of the internal field, essential to connect microscopic to macroscopic behavior of matter has recently been introduced into the accurate quantum mechanical realizations of the model. 

The concept of reaction field, originally formulated by Onsager [194], has been proved to be fruitful in the quantum chemical treatment of polar subsystems (solutes) embedded in polarizable environment (solvent) [195]. Simple cavity models, where the solvent is represented by a continuous dielectric medium and the solute is sitting in a cavity inside this dielectric, has numerous application in the framework of semiempirical [196-200] and ab initio [201-205] methods. The utility of this concept in the modelisation of biochemical processes was pointed out by Tapia and his coworkers [206]. 

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