SMx models

Using the PCM8/ST model, Orozco et al. [207] arrive at values similar to those found by the SMx models. Interestingly, in this case it is the AMI-based model that is more accurate than the PM3-based one. This illustrates the subtle balancing that goes into the parameterization of models that include electrostatic and non-electrostatic effects simultaneously. 

In order to be more generally applicable, the SMx models of Cramer and Truhlar address the issue of data scarcity by making the atomic surface tensions a function of quantifiable solvent properties, i.e.,... 

Unfortunately, a detailed comparison of the continuum models is available only at the semiempirical level.54,55 Because the SMx models are specially parametrized to describe free energy of hydration, it is not surprising that they are the best for reproducing this value. A detailed discussion of the advantages and limitations of different types of solvation models with regard to the various types of approximations and different types of organic molecules can be found in references 53-55. 

Cramer considers the development of the SMX models as his most important scientific contribution. This work drove the field forward. Tomasi and Barone moved to put in non-electrostatic components in a more accurate way, Cramer claims. He is somewhat disappointed in the limited use of the SM models by other computational chemists. What s driving the train is that people use what s in the code that they bought. To the extent that PCM is a very successful solvation model, it s in part because it is in Gaussian. The SMX models sadly have never been available in Gaussian. ... 

The SMx Approach Generalized Born Electrostatics Augmented by First-Hydration-Shell Effects Each of the foregoing solvation models, when implemented at the semiempirical level, resembles closely its implementations employing ah initio molecular orbital theory—indeed, the ab initio versions often predate the semiempirical. On the other hand, the generalized Born model, discussed with respect to Equation [16] for the case of molecular mechanics,has certain properties that make it particularly appropriate i" to the semiempirical level. 27,202,203 Qur own SMx models, where SM denotes solvation model, take advantage of this, and we now review these models. 

The CDS parameters, on the other hand, are expected neither to be solvent-independent nor to be clearly related to any particular solvent bulk observable, especially insofar as they correct for errors in the NDDO wavefunc-tion and its impact on the ENP terms. The CDS parameters also make up empirically for the errors that inevitably occur when a continuous charge distribution is modeled by a set of atom-centered nuclear charges and for the approximate nature of the generalized Born approach to solving the Poisson equation. Hence, the CDS parameters must be parameterized separately against available experimental data for every solvent. This requirement presents an initial barrier to developing new solvent parameter sets, and at present, published SMx models are available for water only (although a hexadecane parameter seH will be available soon). 

The SMx aqueous solvation models, of which the most successful are called AM1-SM2,27 AMl-SMla, and PM3-SM3, °- adopt this quantum statistical approach, which takes account of the ENP and CDS terms on a consistent footing. The NDDO models employed are specified as the first element (AMI or PM3) of each identifier. It is worth emphasizing that the SMx models specifically calculate the absolute free energy of solvation—a quantity not easily obtained with other approaches. We have reviewed the development and performance of the models elsewhere.203 We anticipate our further observations later in this chapter by noting that the mean unsigned error in predicted free energies of solvation is about 0.6-0.9 kcal/mol for the SMx models for a data set of 150 neutral solutes that spans a wide variety of functionalities. A number of examples are provided later in this chapter. 

The effect of induced dipoles in the medium adds an extra term to the molecular Hamilton operator. = -r R (16.49) where r is the dipole moment operator (i.e. the position vector). R is proportional to the molecular dipole moment, with the proportional constant depending on the radius of the originally implemented for semi-empirical methods, but has recently also been used in connection with ab initio methods." Two other widely available method, the AMl-SMx and PM3-SMX models have atomic parameters for fitting the cavity/dispersion energy (eq. (16.43)), and are specifically parameterized in connection with AMI and PM3 (Section 3.10.2). The generalized Bom model has also been used in connection with force field methods in the Generalized Bom/Surface Area (GB/SA) model. In this case the Coulomb interactions between the partial charges (eq. (2.19)) are combined... 

Generalized Bom (GB) approach. The most common implicit models used for small molecules are the Conductor-Like Screening Model (COSMO) [77,78], the DPCM [79], the Conductor-Like Modification to the Polarized Continuum Model (CPCM) [80,81], the Integral Equation Formalism Implementation of PCM (IEF-PCM) [82] PB models, and the GB SMx models of Cramer and Truhlar [23,83-86]. The newest Minnesota solvation models are the SMD universal Solvation Model based on solute electron density [26] and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [87-89] with semiempirical terms that account for local electrostatics [90]. Further details on these methods can be found in Chapter 11 of Reference [23]. 

The SMx models are available in the program AMSOL (comp.chem.umn.edu/ amsol/). TTie programs SPARTAN, PC SPARTAN, and MaeSPARTAN have the SM2 model for aqueous solutions. SPARTAN and PC SPARTAN Pro have AM1-SM5.4 for aqueous solutions. The SM5.0R model is available in the program OMNISOL (comp.chem.umn.edu/omnisol). 

Simple corrections for non-electrostatic interactions have been suggested, wherein atomic-specific parameters are used to describe cavitation, Pauli repulsion, and dispersion [2, 3, 20, 84], These non-electrostatic interactions are then added to Gpoi to obtain the total solvation energy. The most successful examples of this approach are the so-called SMx models of Cramer and Truhlar [27], most of which are not actually PCMs per se but rather generalized Bom models. However, one such model ("SMD") has recently been parameterized for use with lEF-PCM electrostatics [55] and exhibits mean errors of < 1 kcal/mol as compared to experimental solvation... 

It is here important to reeall that such improvements are not limited to BE solvation methods for example, Rivail and the Nancy group have recently extended their multipole-expansion formalism to permit the analytic computation of first and second derivatives of the solvation free energy for arbitrary cavity shapes, thereby facilitating the assignment of stationary points on a solvated potential energy surface. Analytic gradients for SMx models at ab initio theory have been recently described (even if they have been available longer at the semiempirical level ), and they have been presented also for finite difference solutions of the Poisson equation and for finite element solutions. 

The basic equations of the SMx models follow. The standard-state free energy... 

The NDDO approximation of the Hartree-Fock equation for the SMx models can be written as follows ... 

The last term in equation 17, which is required for accurate calculations of the absolute free energies of solvation, is modified differently for various SMx models. For example, in SM2 and SM3 versions it has the following form... 

Rather, different approaches are based on the so-called generalized Born model in which the charge distribution of the solute is represented by point charges on aU atoms and the solute-solvent interaction is defined by a sum of the Born contributions of each atom in which the reciprocal of the sphere radius in O Eq. 15.2 is replaced by a parameterized empirical formula. The most elaborated applications of this model is found in the series of SMx models of Cramer and Truhlar (x = 1 to 8 nowadays, corresponding to successive improvements) (Cramer and Truhlar 1991, 2008). 

During the last 40 years it has been possible to witness an important evolution on the way the environment around a solute molecule is described. The reaction field approach, the effect a continuous dielectric medium has on the charge distribution of a molecule that polarizes back the dielectric and generates a reaction potential, is a standard scheme to consider the solvent effects on many molecular properties. Most modem continuum models obtain through a self-consistent cycle the wave function of the molecule affected by the reaction potential thus the self-consistent reaction field acronym (SCRF). Solvatochromic effects have been more or less successfully explained using from Onsager s to more refined models like Nancy SCRF [44], Tomasi s polarizable continuum model (PCM) [45], Cramer and Tmhlar s SMx models [46]. 

Development of these models is currently very rapid. In the late 1980s these methods were in their infancy. Now the better models can reproduce observed heats of solution with an average error of about 1 kcal mol. Three models that show considerable promise are the COSMO technique of Klamt and Schiiurmann, the Tomasi model as implemented by Orozco et al., and the SMx models of Cramer and Truhlar. These models have several advantages over discrete solvent molecules, the most important being that averaging over many configurations does not need to be performed. Possibly the main disadvantage is that covalent effects are not included. This, however, seems to be a minor point. 

The various SMx models are all available in the semiempirical package AMSOL. ... 

Here, we note only that the properties computed from these models are extremely sensitive to how the molecular cavity (solute/continuum interface) is constructed, and that nonelectrostatic solvation effects are typically (though not always ) neglected in the PCMs such as COSMO, GCOSMO, lEF-PCM, and SS(V)PE that are derived from Poisson s equation for continuum electrostatics. In contrast, such effects are built into the empirical SMx models and are crucial to accurate prediction of solvation free energies. " It is unclear how the neglect of nonelectrostatic effects might impact the calculation of anion VDEs cavitation effect should cancel, but dispersion effects may not, as the anion is intrinsically more polarizable. One may hope that these effects will disappear if a sufficiently large number of explicit solvent molecules is included as part of the QM solute. 

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