Determination of Segment Properties
After the segmented surface is obtained by the union of all the vdW spheres outlined around each atom, where each atom possesses a 'COSMO radii', the molecule is then immersed in a homogeneous conductor having infinite dielectric constant (Klamt, 2005). In such a scenario, the screening charges from the conductor migrate to the segmented surface to negate the underlying molecular charge distribution. This is physically modeled by placing a screening charge from the conductor on each segment. The total energy of solvation (Esolv) for the entire molecule is then determined from the interaction of the screening charges with the solute nuclei, the solute electronic density and within themselves. The segment properties are now calculated by defining the energy of solvation (Esolv), which solves the interaction of the segments with the solute nuclei (Z) (first term of Equation 3.1), solute electron density (second term of Equation 3.1) and interaction within the segments (third term of Equation 3.1), as given below. qa represents the screening charge of ath segment.
where:
Here B, C, D and r represent vector quantities and represent the contributions due to Coulomb's law. Za and ra denote the molecular charge and position of nucleus A, respectively. Va, possessing a unit of inverse of length, signifies the distance between the segment and the inherent charge density p(r) of the molecule. Daa represents the energy required to provide a charge on each segment.
The expression for the self-energy of a segment was made by assuming the molecule as a sphere of radius R. In such a situation, the total charge Q is distributed homogenously across its surface. Hence, the electrostatic energy of the sphere is given by
As per the COSMO protocol, the surface has to divide into n homogeneously similar segments. This implies that the energy then becomes the total of the sum of segment interaction energies and the segment self-energies; this is given by
Here, Eap is the interaction energy between segments, while E0,elf is the selfenergy of segment. The latter reflects the energy required to maintain a charge of a given density on a particular segment. qa denotes the charge of segment a, while rap provides the distance between segments a and p. It is assumed that the area and charge of each segment are similar and the summations as above are exact. However, we still do not have an exact expression for E0,elf, since the energy of the sphere must be the same and will not depend on the distribution of charge on the sphere. An explicit way to determine the same is to an expression for can be obtained by equating Equations 3.11 and 3.12. Therefore, by rearranging the above equations, we obtain (Burnett, 2012):
By reorganizing the terms, we obtain:
Here, R =
The energy of solvation (Esolv) is then minimized with respect to the segments' charges (qj, as given below:
As the molecule is placed on the continuum with infinite dielectric constant, the charges in the nearby region tend to cancel the electron density p(r) of the solute molecule. This causes the molecule to polarize. Hence, this calculation needs to be performed in an iterative manner, so as to achieve consistency. At each step, the screening charge q is found for the respective p(r), after which p(r) is updated with new values of q. This procedure, which utilizes
FIGURE 3.2
Gaussian03-generated COSMO surfaces for three molecules: Acetone (739 segments), water (292 segments) and toluene (1053 segments). Surfaces colored by screening charge density, where blue regions have negative screening charge values, green regions are neutral and red regions have positive values.
an iterative method, is called 'self-consistent field' approach. This process of generating and identifying the set of p(r) and q that are self-consistent is called a 'conductor-like screening model' (COSMO).
In addition to q and area of the segments a and r, it also provides the potential for each segment ф. The calculation also provides us with the total energy of the molecule in a conductor, which we will denote as (ECOSMO). Sample screening charges for different molecules are depicted in Figure 3.2. It should be noted that the positive screening charges depict the negative regions of the molecule. Likewise, the negative screening charges represent the positive regions of the molecule. The screening charge distribution of the molecules is drawn by using the COSMO builder within TURBOMOLE. The package can also evaluate the segment potential for the segments, which we shall discuss in the ensuing section.
For the acetone COSMO file, 'nps' represents the number of segments, which is 739. The values of area (A2) = 371.79 and volume (A3) = 561.92 represent the total cavity area and volume, respectively, within the conductor, where acetone molecule is placed. The next section denotes the position of all the 10 atoms, along with the total screening charge on each atom 'COSMOCharge' and the last column represents the division of this 'COSMOCharge' with the area corresponding to each atom, called 'area'. The last column is the most desirable quantity, as it reflects the screening charge density. It is interesting to note that except for oxygen atom 'O3' (COSMOCharge = 0.24046), all other atoms posses a negative charge. This follows the obvious trend, as negatively charged atoms possess positive screening charges, and vice versa. Further, the positive charge on the oxygen atom is then divided into a number of segments for which information is available in the concluding part of the COSMO file. For example, information for the segments of 'C1' and 'H9' atoms of acetone is depicted in Figure 3.3. The screening charges q of the solvent medium are usually calculated by a scaling q*, so that q = /(e) q*, where /(e) is the scaling factor given by
COSMO-SAC: A Predictive Model for Calculating Thermodynamics
61
FIGURE 3.3
An excerpt of a COSMO file, as generated from Gaussian03 version C.02 for acetone molecule.
(Continued)
Phase Equilibria in Ionic Liquid Facilitated Liquid-Liquid Extractions
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FIGURE 3.3 (Continued)
An excerpt of a COSMO file, as generated from Gaussian03 version C.02 for acetone molecule.
Thus, a simpler boundary condition of a conductor appears in the above equation and also in Figure 3.3 (fepsi = 1.00). It should be noted that a conductor has an infinite supply of charge, so as to screen the entire solute molecule. Hence, there is a point of thought that the selection of solvent as a conductor can alter the charges, as evident in Figure 3.2. Therefore, a decision has to be arrived for the selection of a solvent in such a manner so that it is able to screen the entire solute component. This brings out to the point that it should possess sufficient charge, and thus, for obvious reason, it has to be a perfect conductor. Here, the COSMO approximation is exact in the limit of e = <» and is within 0.5% accuracy for strong dielectrics such as water with a permeability of (e = 80). Even for a lower dielectric limit of solvents such as e = 2, COSMO coincides with the exact dielectric model within 10%. Hence, the COSMO approach is becoming a standard Continuum Solvation Model (CSM) in quantum chemical codes.
It should be noted that the self-consistent field (SCF) calculation or the COSMO scheme has to be done once for each component. Once these charges are obtained, then a framework has to be created so as to restore these charges when placed in an actual solvent of a known dielectric constant. This is where the derivation of chemical potentials and activity coefficients is performed. Our aim will then be to create a repository of COSMO files, which will include ions, solutes, solvents and inorganic compounds, subject to their dissociation constants. A sample description of an input for generating the COSMO file is given in Figure 3.4.
The different terms and their meaning are as follows:
'% mem = 540 MW' refers to the total amount of internal memory required to perform this calculation. The route section with '# P BVP86/SVP/DGA1'
FIGURE 3.4
Input file for COSMO file generation in Gaussian03.
indicates the following form 'DFT Theory/Basis Set/optional'. So, the form of functional used for DFT is the P BVP86, while the basis set used is SVP or Split Valence Polarized. The orbital coefficients for SVP can be obtained by the GFprint and GFInput commands in Gaussian03. The density gradient approximation or 'optional' used is DGA1. It expands the electron density of the atoms in the form of atom-centered functions for saving computational time. It only integrates the coulomb interaction in place of all the electron integrals. The 'SCF = tight' indicates a full convergence of energy.
