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# 11.2.8 COSMO

(December 20, 2021)

According to Table 11.3, COSMO and C-PCM appear to differ only in the dielectric screening factor, $f_{\varepsilon}$ in Eq. (11.3). Indeed, surface charges in either model are computed according to

 $\mathbf{q}=-f_{\varepsilon}\mathbf{S}^{-1}\mathbf{v}\;.$ (11.25)

As discussed in Section 11.2.4, the user can choose between various values of $f_{\varepsilon}$, including the original value $f_{\varepsilon}=(\varepsilon-1)/(\varepsilon+1/2)$ that was suggested by Klamt and co-workers, 571 Klamt A., Jonas V.
J. Chem. Phys.
(1996), 105, pp. 9972.
or else $f_{\varepsilon}=(\varepsilon-1)/\varepsilon$ as is typically used in C-PCM calculations. 1137 Truong T. N., Stefanovich E. V.
Chem. Phys. Lett.
(1995), 240, pp. 253.
, 230 Cossi M. et al.
J. Comput. Chem.
(2003), 24, pp. 669.
, 627 Lange A. W., Herbert J. M.
Chem. Phys. Lett.
(2011), 509, pp. 77.
. More importantly, however, COSMO differs from C-PCM in that the former includes an ad hoc correction for outlying charge that goes beyond Eq. (11.25), whereas C-PCM consists of nothing more than induced surface charges computed (self-consistently) according to Eq. (11.25). This correction, which is common to many implementations of COSMO, 461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
involves the use of two separate solute cavities. It is worth noting that Eq. (11.25) was later shown to implicitly include an outlying charge correction, 202 Chipman D. M.
J. Chem. Phys.
(1999), 110, pp. 8012.
by virtue of the fact that it is derivable from the SS(V)PE model, 626 Lange A. W., Herbert J. M.
J. Chem. Phys.
(2011), 134, pp. 204110.
, 461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
and the latter was developed specifically with an eye towards the treatment of outlying charge. As such, there is little theoretical justification for the additional explicit correction for outlying charge, despite its success in practice. 572 Klamt A., Moya C., Palomar J.
J. Chem. Theory Comput.
(2015), 11, pp. 4220.
See Ref.  461 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
for a discussion of these issues.

In any case, the nature of the a posteriori correction for the outlying charge proceeds as follows. Upon solution of Eq. (11.25), the outlying charge correction in COSMO 571 Klamt A., Jonas V.
J. Chem. Phys.
(1996), 105, pp. 9972.
, 56 Baldridge K., Klamt A.
J. Chem. Phys.
(1997), 106, pp. 6622.
is obtained by first defining a larger cavity that is likely to contain essentially all of the solute’s electron density; in practice, this typically means using atomic radii of $1.95R$, where $R$ denotes the original atomic van der Waals radius that was used to compute $\mathbf{q}$. (Note that unlike the PCMs described in Sections 11.2.3 and 11.2.4, where the atomic radii have default values but a high degree of user-controllability is allowed, the COSMO atomic radii are parameterized for this model and are fixed.) A new set of charges, $\mathbf{q}^{\prime}=-f_{\varepsilon}(\mathbf{S}^{\prime})^{-1}\mathbf{v}^{\prime}$, is then computed on this larger cavity surface, and the charges on the original cavity surface are adjusted to new values, $\mathbf{q}^{\prime\prime}=\mathbf{q}+\mathbf{q}^{\prime}$. Finally, a corrected electrostatic potential on the original surface is computed according to $\mathbf{v}^{\prime\prime}=-f_{\varepsilon}\mathbf{S}\mathbf{q}^{\prime\prime}$. It is this potential that is used to compute the solute–continuum electrostatic interaction (polarization energy), $G_{\rm pol}=\tfrac{1}{2}\sum_{i}q_{i}^{\prime\prime}v_{i}^{\prime\prime}$. (For comparison, when the C-PCM approach described in Section 11.2.3 is used, the electrostatic polarization energy is $G_{\rm pol}=\tfrac{1}{2}\sum_{i}q_{i}v_{i}$, computed using the original surface charges $\mathbf{q}$ and surface electrostatic potential $\mathbf{v}$.) With this outlying charge correction, Q-Chem’s implementation of COSMO resembles the one in Turbomole. 994 Schäfer A. et al.
Phys. Chem. Chem. Phys.
(2000), 2, pp. 2187.

A COSMO calculation is requested by setting SOLVENT_METHOD = COSMO in the $rem section, in addition to normal job control variables. The keyword Dielectric in the$solvent section is used to set the solvent’s static dielectric constant, as described above for other solvation models.