7.50. Implicit Solvation Models

Implicit solvation models play an important role in quantum chemistry. Without resorting to placing multiple solvation shells of solvent molecules implicit solvent models are able to mimic the effect of a specific solvent on the solute.

The implicit solvent models available in ORCA are

1. C-PCM[76] : The Conductor-like Continuum Polarization Model

2. SMD[558] : The Solvation Model based on Density

3. OpenCOSMO-RS[293] : Interface to the open source implementation of the COSMO-RS model

4. ALPB/ddCOSMO/CPCM-X : The solvation models available in XTB

Regarding C-PCM and SMD, they are natively implemented in ORCA. Points 1, 2 and 3 are covered in this section. For more information on point 4, check Section ONIOM Methods.

7.50.1. The Conductor-like Polarizable Continuum Model (C-PCM)

The conductor-like polarizable continuum model (C-PCM) is an implementation of the conductor-like apparent surface charge methods. In these models the solute is placed in a cavity of roughly molecular shape. The solvent reaction field is described by apparent polarization charges on the cavity surface, which are in turn determined by the solute. These charges can be treated as punctual (point charges) or be modelled as spherical Gaussians [904]. The cavity in ORCA is constructed differently depending on how the charges are treated. In the case of using point charges, the cavity is generated through the GEPOL[649, 650, 651] algorithm, either as solvent-excluding surface (SES), or solvent-accessible surface (SAS). When Gaussian charges are considered, the user can choose between a scaled vdW surface or the GEPOL SES, and the charge positions are determined following a Lebedev quadrature approach. This scheme is known as Gaussian Charge Scheme[287] and more details on how to use it are given in Section Use of the Gaussian Charge Scheme.

The ORCA C-PCM implementation closely follows the C-PCM[76] paper. The molecular Hamiltonian of the isolated system is perturbed by the solvent:

\[\hat{H} = \hat{H}^0 + \hat{V}\]

where \(\hat{H}^0\) is the Hamiltonian of the isolated molecule, whereas \(\hat{V}\) describes the solute – solvent interactions. The SCF procedure leads to the variational minimization of the free energy of the solute, \(G\):

\[G = \left\langle\Psi \left| \hat{H}^0 \right| \Psi \right\rangle+ \frac{1}{2} \left\langle\Psi \left| \hat{V} \right| \Psi \right\rangle\]

Using the conductor-like boundary condition the electrostatic potential can be determined by

\[V(\vec{r}) + \sum_i^{N_q} V_{q_i} (\vec{r}) = 0\]

where \(V\) and \(V_{q_i}\) are the electrostatic potential due to the solute and to the polarization charges, \(\vec{r}\) is a point on the cavity surface, and \(N_q\) is the total number of solvation charges. The vector of polarization charge can then be determined by

(7.316)\[\mathbf{AQ = -V} \]

where the vector \(\mathbf{V}\) contains the electrostatic potential due to the solute at the position of the charges. The elements of the matrix \(\mathbf{A}\) have a different functional form depending on how the charges are treated. If we use point charges:

\[A_{ii} = 1.07 \sqrt{ \frac{4\pi}{S_i} }\]

(7.317)\[ A_{ij} = \frac{1}{r_{ij} } \]

in which \(S_i\) is the area of the surface element \(i\), and \(r_{ij}=\left| \vec{r}_i - \vec{r}_j \right|\). When Gaussian charges are considered:

\[A_{ii} = \frac{\zeta_i\sqrt{2/\pi} }{F_i}\]

\[A_{ij} = \frac{\mathrm{erf}\left(\zeta_{ij}r_{ij}\right)}{r_{ij} }\]

Here, \(\zeta_i\) is the exponent of the Gaussian charge \(i\) (\(i\) belongs to sphere \(I\)). This quantity is calculated as \(\zeta_i=\zeta/(R_I\sqrt{w_i})\), where \(R_I\) is the radius of sphere \(I\), \(w_i\) is the weight of the Lebedev point \(i\), and \(\zeta\) is a width parameter optimized for each particular Lebedev grid [904]. On the other hand, \(\zeta_{ij}=\zeta_i\zeta_j/\sqrt{\zeta_i^2+\zeta_j^2}\). The function \(F_i\), known as switching function, measures the contribution of the Gaussian charge \(i\) to the solvation energy. This function is calculated as

\[F_i=\prod^{\mathrm{atoms} }_{J,i\notin J}g(\vec{r}_i,\vec{R}_J)\]

where \(g(\vec{r}_i,\vec{R}_J)\) is the elementary switching function. In ORCA we use the improved Switching/Gaussian (ISWIG) function for \(g(\vec{r}_i,\vec{R}_J)\) proposed in ref. [494]:

\[g(\vec{r}_i,\vec{R}_J)=1-\frac{1}{2}\left\{\mathrm{erf}\left[\zeta_i\left(R_J-r_{iJ}\right)\right]+\mathrm{erf}\left[\zeta_i\left(R_J+r_{iJ}\right)\right]\right\}\]

If \(g(\vec{r}_i,\vec{R}_J)<10^{-7}\) the value of \(g\) is set equal to 0.

If we consider a solvent with a dielectric constant \(\varepsilon\), eq. (7.316) reads as

(7.318)\[ \mathbf{AQ} = -f(\varepsilon)\mathbf{V} \]

where \(f(\varepsilon)=(\varepsilon - 1)/(\varepsilon + x)\) is a scaling function, and \(x\) is in the range 0-2. In C-PCM \(x\) is equal to 0.

The C-PCM model can be used via

! CPCM(solvent)

where solvent is one of the available solvents in Table 7.27

Table 7.27 List of available solvents for the different implicit solvation methods in ORCA. The data for the dielectric constant used within C-PCM is that at 293.15,[368] except for ammonia, which has a boiling point of 239.81 K. For the rest of solvation models, see the corresponding sources.[558][814]

Solvent	C-PCM	SMD	COSMO-RS	ALPB	ddCOSMO	CPCM-X
1,1,1-trichloroethane	X	X
1,1,2-trichloroethane	X	X
1,2,4-trimethylbenzene	X	X				X
1,2-dibromoethane	X	X	X
1,2-dichloroethane	X	X	X			X
1,2-ethanediol	X	X
1,4-dioxane / dioxane	X	X	X	X	X
1-bromo-2-methylpropane	X	X
1-bromooctane / bromooctane	X	X	X
1-bromopentane	X	X
1-bromopropane	X	X
1-butanol / butanol	X	X	X			X
1-chlorohexane / chlorohexane	X	X	X			X
1-chloropentane	X	X
1-chloropropane	X	X
1-decanol / decanol	X	X	X			X
1-fluorooctane	X	X	X			X
1-heptanol / heptanol	X	X	X			X
1-hexanol / hexanol	X	X	X			X
1-hexene	X	X
1-hexyne	X	X
1-iodobutane	X	X
1-iodohexadecane / hexadecyliodide	X	X	X			X
1-iodopentane	X	X
1-iodopropane	X	X
1-nitropropane	X	X
1-nonanol / nonanol	X	X	X			X
1-octanol / octanol	X	X	X	X	X	X
1-pentanol / pentanol	X	X	X			X
1-pentene	X	X
1-propanol / propanol	X	X	X			X
2,2,2-trifluoroethanol	X	X
2,2,4-trimethylpentane / isooctane	X	X	X			X
2,4-dimethylpentane	X	X
2,4-dimethylpyridine	X	X
2,6-dimethylpyridine	X	X	X			X
2-bromopropane	X	X
2-butanol / secbutanol	X	X	X			X
2-chlorobutane	X	X
2-heptanone	X	X
2-hexanone	X	X
2-methoxyethanol / methoxyethanol	X	X	X			X
2-methyl-1-propanol / isobutanol	X	X	X			X
2-methyl-2-propanol	X	X
2-methylpentane	X	X
2-methylpyridine / 2methylpyridine	X	X	X			X
2-nitropropane	X	X
2-octanone	X	X
2-pentanone	X	X
2-propanol / isopropanol	X	X	X			X
2-propen-1-ol	X	X
e-2-pentene	X	X
3-methylpyridine	X	X
3-pentanone	X	X
4-heptanone	X	X
4-methyl-2-pentanone / 4methyl2pentanone	X	X	X			X
4-methylpyridine	X	X
5-nonanone	X	X
acetic acid / aceticacid	X	X	X			X
acetone	X	X	X	X	X
acetonitrile / mecn / ch3cn	X	X	X	X	X	X
acetophenone	X	X	X			X
ammonia	X		X
aniline	X	X	X	X	X	X
anisole	X	X	X			X
benzaldehyde	X	X	X	X	X
benzene	X	X	X	X	X	X
benzonitrile	X	X	X			X
benzyl alcohol / benzylalcohol	X	X	X			X
bromobenzene	X	X	X			X
bromoethane	X	X	X			X
bromoform	X	X	X			X
butanal	X	X
butanoic acid	X	X
butanone	X	X	X			X
butanonitrile	X	X
butyl ethanoate / butyl acetate / butylacetate	X	X	X			X
butylamine	X	X
n-butylbenzene / butylbenzene	X	X	X			X
sec-butylbenzene / secbutylbenzene	X	X	X			X
tert-butylbenzene / tbutylbenzene	X	X	X			X
carbon disulfide / carbondisulfide / cs2	X	X	X	X	X	X
carbon tetrachloride / ccl4	X	X	X			X
chlorobenzene	X	X	X			X
chloroform / chcl3	X	X	X	X	X	X
a-chlorotoluene	X	X
o-chlorotoluene	X	X
conductor	X				X
m-cresol / mcresol	X	X	X			X
o-cresol	X	X
cyclohexane	X	X	X			X
cyclohexanone	X	X	X			X
cyclopentane	X	X
cyclopentanol	X	X
cyclopentanone	X	X
decalin	X	X	X			X
cis-decalin	X	X
n-decane / decane	X	X	X			X
dibromomethane	X	X				X
dibutylether	X	X	X			X
o-dichlorobenzene / odichlorobenzene	X	X	X			X
e-1,2-dichloroethene	X	X
z-1,2-dichloroethene	X	X
dichloromethane / ch2cl2 / dcm	X	X	X	X	X	X
diethyl ether / diethylether	X	X	X	X	X	X
diethyl sulfide	X	X
diethylamine	X	X
diiodomethane	X	X
diisopropyl ether / diisopropylether	X	X	X			X
cis-1,2-dimethylcyclohexane	X	X
dimethyl disulfide	X	X
n,n-dimethylacetamide / dimethylacetamide	X	X	X			X
n,n-dimethylformamide / dimethylformamide / dmf	X	X	X	X	X	X
dimethylsulfoxide / dmso	X	X	X	X	X	X
diphenylether	X	X	X			X
dipropylamine	X	X
n-dodecane / dodecane	X	X	X			X
ethanethiol	X	X
ethanol	X	X	X	X	X	X
ethyl acetate / ethylacetate / ethanoate	X	X	X	X	X	X
ethyl methanoate	X	X
ethyl phenyl ether / ethoxybenzene	X	X	X			X
ethylbenzene	X	X	X			X
fluorobenzene	X	X	X			X
formamide	X	X
formic acid	X	X
furan / furane			X	X	X
n-heptane / heptane	X	X	X			X
n-hexadecane / hexadecane	X	X	X	X	X	X
n-hexane / hexane	X	X	X	X	X	X
hexanoic acid	X	X
iodobenzene	X	X	X			X
iodoethane	X	X
iodomethane	X	X
isopropylbenzene	X	X	X			X
p-isopropyltoluene / isopropyltoluene	X	X				X
mesitylene	X	X	X			X
methanol	X	X	X	X	X	X
methyl benzoate	X	X
methyl butanoate	X	X
methyl ethanoate	X	X
methyl methanoate	X	X
methyl propanoate	X	X
n-methylaniline	X	X
methylcyclohexane	X	X
n-methylformamide / methylformamide	X	X	X			X
nitrobenzene / phno2	X	X	X			X
nitroethane	X	X	X			X
nitromethane / meno2	X	X	X	X	X	X
o-nitrotoluene / onitrotoluene	X	X				X
n-nonane / nonane	X	X	X			X
n-octane / octane	X	X	X			X
n-pentadecane / pentadecane	X	X	X			X
octanol(wet) / wetoctanol / woctanol				X	X
pentanal	X	X
n-pentane / pentane	X	X	X			X
pentanoic acid	X	X
pentyl ethanoate	X	X
pentylamine	X	X
perfluorobenzene / hexafluorobenzene	X	X	X			X
phenol	X		X	X	X
propanal	X	X
propanoic acid	X	X
propanonitrile	X	X
propyl ethanoate	X	X
propylamine	X	X
pyridine	X	X	X			X
tetrachloroethene / c2cl4	X	X	X			X
tetrahydrofuran / thf	X	X	X	X	X	X
tetrahydrothiophene-s,s-dioxide /	X	X				X
/ tetrahydrothiophenedioxide / sulfolane	X	X				X
tetralin	X	X	X			X
thiophene	X	X
thiophenol	X	X
toluene	X	X	X	X	X	X
trans-decalin	X	X
tributylphosphate	X	X	X			X
trichloroethene	X	X
triethylamine	X	X	X			X
n-undecane / undecane	X	X	X			X
water / h2o	X	X	X	X	X	X
xylene	X	X				X
m-xylene	X	X
o-xylene	X	X
p-xylene	X	X

The parameters can be more accurately defined using the %cpcm block input. The available options are as follows

%cpcm  epsilon             80.0    # Dielectric constant
       refrac               1.0    # Refractive index
       rsolv                1.3    # Solvent probe radius
       rmin                 0.5    # Minimal GEPOL sphere radius
       pmin                 0.1    # Minimal distance between two surface points
       fepstype            cpcm    # Epsilon function type: cpcm, cosmo
       xfeps                0.0    # X parameter for the feps scaling function
       surfacetype vdw_gaussian    # Cavity surface: gepol_ses, gepol_sas
                                                  vdw_gaussian, gepol_ses_gaussian
       ndiv                   5    # Maximum depth for recursive sphere generation
       num_leb              302    # Lebedev points for the Gaussian charge scheme
       radius[N]            1.3    # Atomic radius for atomic number N in Angstrom
       AtomRadii(N,1.4)            # Atomic radius for the Nth atom in Angstrom
       scale_gauss          1.2    # Scaling factor for the atomic radii in the
                                     Gaussian charge scheme
       cut_area             0.0    # Cutoff for the area of a surface segment in a.u.
                                     Only valid for the Gaussian charge scheme
       cut_swf             1e-7    # Cutoff for the switching function
                                     Only valid for the Gaussian charge scheme
       thresh_h             5.0    # Threshold for the charge density on a hydrogen
                                     atom in charges/Å^2 (isodensity scheme)
       thresh_noth          5.0    # Threshold for the charge density on non-hydrogen
                                     atoms in charges/Å^2 (isodensity scheme)
       CPCMccm                0    # Coupled-cluster/C-PCM scheme
       cds_cpcm               0    # Use of the GVDW_nel or GSES_nel scheme
       end

Regarding the parameters shown above, some of them can be just used within a given type of cavity surface and charge scheme, as it is mentioned in subsections Use of the Gaussian Charge Scheme, and Use of the Point Charge Scheme. ORCA supports two types of solvation charge schemes: (i) the point charge scheme, (ii) the Gaussian charge scheme. The default solvation scheme from ORCA 5.0 on is the Gaussian charge scheme with a vdW-type cavity (”surfacetype vdw_gaussian”). Older ORCA versions considered “surfacetype gepol_ses” and the point charge scheme as defaults. So, if one wants to reproduce the results obtained with ORCA versions older than ORCA 5.0, please use the tag “surfacetype gepol_ses” in the “%cpcm” block.

The different charge schemes are described in the subsections Use of the Gaussian Charge Scheme, and Use of the Point Charge Scheme. The availability of the analytical gradient and Hessian for the different combinations of charge scheme and surface is shown in Table Table 7.28.

Table 7.28 Available type of gradients and Hessians within the C-PCM in ORCA. The tag “YES” means that the feature is implemented, and “NO” that it isn’t. For clarity, we denote by “*” the default scheme in ORCA (surfacetype vdw_gaussian).

surfacetype	Charge type	Gradient		Hessian
		Analytical	Numerical	Analytical	Numerical
gepol_ses	Point	Yes	Yes	Yes	Yes
gepol_sas	Point	Yes	Yes	Yes	Yes
vdw_gaussian^*	Gaussian	Yes	Yes	Yes	Yes
gepol_ses_gaussian	Gaussian	No	Yes	No	Yes

Note: If the user wants to turn off the C-PCM, one has to write the following tag in the simple input:

! NOCPCM

This is needed, for instance, in the context of concatenated calculations using the $new_job feature, where the previous calculation involved solvation, but one wants to turn it off for the next one.

7.50.1.1. Use of the Gaussian Charge Scheme

The Gaussian charge scheme avoids the Coulomb singularity present in conventional point charge surface element models. This approach, when applied together with a switching function, results in a smooth solvation potential and, more importantly, on smooth derivatives of this quantity with respect to external perturbations. Then, it is highly recommended to adopt this approach within the C-PCM. The Gaussian charge scheme can be used with two types of solute cavity surfaces: (1) a scaled vdW surface, (2) a solvent-excluded surface (SES). To assign the radii for the different atoms we follow the scheme proposed in ref. [494]. That is, we use Bondi radii [114] for all elements, except for hydrogen where we adopt 1.1 Å. For 16 of the main-group elements in the periodic table, where Bondi’s radii are not defined, we adopt the radii proposed in ref. [552] by Mantina et al. This is the case for elements: Be, B, Al, Ca, Ge, Rb, Sr, Sb, Cs, Ba, Bi, Po, At, Rn, Fr, Ra. For the elements that are not covered neither by Bondi nor by Mantina, we consider a radius of 2 Å.

• Scaled vdW cavity

The Gaussian charge scheme with a scaled vdW-type cavity is now the default in ORCA, so one just needs to add the C-PCM tag in the %cpcm block in the input file. That would correspond to (although the user does not need to write that, as it is internally processed by ORCA):

%cpcm
surfacetype vdw_gaussian
end

In this case, the radius \(R_I\) of atom \(I\) for the scaled vdW cavity is calculated as

\[R_I=f_{scal}R_I^{\mathrm{vdW} }\]

where \(R_I^{\mathrm{vdW} }\) is the vdW radius of atom \(I\) and \(f_{scal}\) is a scaling factor. This parameter is by default equal to 1.2, as suggested in ref. [494]. However, the user can modify its value through the scale_gauss tag in the %cpcm block in the input file. The number of C-PCM charges per atom is such that we have an approximate density of 5.0 charges/Å\(^2\) on the surface of the cavity. This number corresponds approximately to 110 points on a hydrogen atom (110/\((4\pi\times 1.32^2)\approx 5.0\)Å\(^{-2}\)). ORCA will choose then different levels of discretization (Lebedev grids) depending on the radii of the atoms. This scheme is called isodensity scheme. The threshold for the number of charges per unit of area on hydrogens and non-hydrogen atoms can be specified by the user via thresh_h and thresh_noth, respectively. Alternatively, the user can request a fixed number of Lebedev points per sphere in the cavity (independent of the radius of the sphere). This can be done via the num_leb tag. This parameter can adopt the following values: 50, 110, 194, 302, 434, 590, 770, 974, and 1202. ORCA versions older than ORCA 6.0 use this last scheme.

The analytical gradient, as well as the analytical Hessian are available for this solvation method.

• Solvent-excluded surface

The GEPOL-generated SES can be used together with the Gaussian charge scheme. In this case, the Gaussian charges are not only placed on the surface of the atomic spheres but also on the surface of the new spheres generated through the GEPOL algorithm. To use this approach we should modify the ORCA input file as follows

%cpcm
surfacetype gepol_ses_gaussian
end

The SES is in general recommended when the solute is explicitly solvated by few solvation layers. In this case, the additional GEPOL spheres prevent the solvent to fill the space between explicit solvent molecules or between those molecules and the solute. The radius of the solvent sphere that rolls over the solute vdW-type surface to generate the SES is controlled by the parameter “rsolv”. This radius has a default value of 1.30 Å, but the user can change it by specifying another value for “rsolv” in the %cpcm block. The minimum radius for an added GEPOL sphere is controlled by “rmin”.

Neither the analytical gradient nor the analytical Hessian are available for this strategy. They should be computed numerically. Due to the interdependency between GEPOL spheres and the atoms present in our system, the analytical gradient computed using the SES does not converge as smooth as compared to that using the vdW-type cavity (see ref. [287]) and can lead to wrong minima. The Hessian is affected in the same way. Then, ORCA 6.0 does not support anymore the analytical gradient/Hessian for such surface to prevent inaccurate results.

7.50.1.2. Use of the Point Charge Scheme

Within this scheme, the solvation charges are treated as punctual and no switching function is used to accept/discard charges in the intersection between the spheres that form the solute cavity. These two facts lead to a discontinuous potential energy surface (numerical instabilities in the SCF energy and its derivatives). Then, the point charge scheme is not recommended within the C-PCM. If the user still wants to use this charge scheme it can be used together with two different type of surfaces for the solute cavity: (1) the SES, and (2) the SAS.

• Solvent-excluded surface

In the same way as for the Gaussian charge scheme with the SES, this surface is generated for the point charge scheme through the GEPOL algorithm. However, while for the case of surfacetype gepol_ses_gaussian, the surface is discretized using Lebedev-type grids, in the point charge scheme, the surface is divided in spherical triangles called “tesserae”. The level of tesselation is controlled by the tag ndiv. The radii used for the solute atoms in order to construct the GEPOL cavity are those optimized for the COSMO-RS model[445] for H, C, N, O, F, S, Cl, Br, and I, while for the rest of elements scaled Bondi radii are used.

In order to use the point charge scheme with the SES, we should add the following tag in the %cpcm block:

%cpcm
surfacetype gepol_ses
end

• Solvent-accessible surface

Here, the surface of the solute cavity is generated by following the center of a sphere with radius “rsolv” (representing the solvent molecule) rolling over the surface of the vdW surface of the solute. The discretization of the resulting surface is done via tesserae, as done for the SES. In order to use the point charge scheme together with the SAS, one should add the following tag in the %cpcm block:

%cpcm
surfacetype gepol_sas
end

• How to circumvent the numerical instabilities in the point charge scheme

Numerical instabilities are implicit in the point charge scheme due to the “punctual” nature of the charges and to the fact that no switching function is considered in the intersection between the spheres that form the solute cavity. At the same time, there is no way to predict, in advance if there will be discontinuities in the SCF energy and/or in its derivatives. However, if the user still wants to use this charge scheme, no matter with which type of surface (SES or SAS), there are two things that one can try to minimize the aforementioned problems:

• Change ndiv : The parameter “ndiv” controls the number of triangles per sphere in the solute cavity. By changing the number of triangles, it also changes the number of triangles in the intersection between spheres and, then, this can solve those situations where two point charges get too close making the elements \(A_{ij}\) in eq (7.317) to diverge (and the solvation charges to suddenly have very large values). However, this strategy can be a solution for a particular case, but will never ensure that for the same system with another geometry the discontinuities in the SCF energy do not show (as it does not prevent point charges to be too close from each other).

• Increase pmin : This parameter removes those charges that are at a distance lower than pmin from each other. The default value for pmin is of 0.1 a.u. (\(\approx 0.0529\) Å). Then, by incrasing pmin one removes all pairs of charges that are too close from each other in the intersection between the spheres. Note: Be careful when increasing pmin. Although this prevents sudden jumps in the SCF energy, it can lead to “biasing” the solute-solvent interaction, as one is removing a significant number of charges that represent the effect of the solvent.

7.50.1.3. Calculation of the free energy of solvation within the C-PCM

The solvation free energy, \(\Delta G_{solv}\), is defined as the free energy of transfer of a solute from the gas phase to the condensed phase. This quantity can be written as (see eq 27 in ref. [287])

(7.319)\[ \Delta G_{solv}=E_{solv}(\vec{R}_l,\vec{R}_v)+\Delta G_{el}+\Delta G_{cav}+\Delta G_{disp} \]

The first term, \(E_{solv}(\vec{R}_v,\vec{R}_l)\), corresponds to the difference between the liquid-phase expectation value of the gas-phase Hamiltonian, \(E(\vec{R}_l)\), and the gas-phase potential energy surface \(E(\vec{R}_v)\)

\[E_{solv}(\vec{R}_v,\vec{R}_l)=E(\vec{R}_l)-E(\vec{R}_v)\]

The second term in eq (7.319) accounts for the electronic-polarization contribution to \(\Delta G_{solv}\) and is calculated from the charges spread over the surface of the solute cavity (vdW-type or the SES). Finally, \(\Delta G_{cav}\) is the cavitation energy, that is, the reversible work required to create a cavity inside the bulk of the solvent in order to accommodate the solute, while \(\Delta G_{disp}\) accounts for the changes in the dispersion energy occuring when solvating the solute. The sum of these last two terms correspond to the non-electrostatic contribution, \(\Delta G_{nel}\), to \(\Delta G_{solv}\)

\[\Delta G_{nel}=\Delta G_{cav}+\Delta G_{disp}\]

In ORCA, if a system is solvated within the C-PCM (defining !CPCM(solvent name) in the input file, within either the point charge scheme or the Gaussian charge scheme) there is no “non-electrostatic solvation contribution” neither to the SCF energy, nor to its derivatives. In order to have a rough idea of the value of \(\Delta G_{nel}\), one can estimate this quantity through empirical equations available in the literature obtained from experimental data. For instance, \(\Delta G_{nel}\) can be estimated from the free energy of hydration for linear-chain alkanes[852]

(7.320)\[ \Delta G_{\text{nel} } = 1.321 + 0.0067639\times\mathrm{SASA} \]

where SASA is the solvent-accessible surface area. However, although eq (7.320) may give a good estimation of \(\Delta G_{nel}\) for organic molecules in water, it is a bad approximation for non-electrostatic solvation effects when considering solvents different than water (as well as for many solutes in water). There are two alternatives to this approximation that one can adopt in order to include non-electrostatic solvation effects in ORCA calculations:

1. The use of the SMD model in combination with the C-PCM (see section The SMD Solvation Model).

2. The use of Gaussian charges to compute the electrostatic contribution to \(\Delta G_{solv}\) together with an equation for \(\Delta G_{\text{nel} }\) calculated from the contribution of the solute atoms to the SASA.[287]

Focusing on the second strategy, the cavitation energy, \(\Delta G_{cav}\) is calculated through the equation proposed by Pierotti and Claverie.[180, 680] In this case, \(\Delta G_{cav}\) is expanded in powers of the ratio \(\overline{\mathbf{R} }\) between the radius of the solute sphere \(R_I\) and that of the solvent spheres \(R_S\). Then, the cavitation energy for the solute sphere centered at atom \(I\) reads as,

\[\Delta G_{cav,I}=-\mathrm{ln}(1-y)+\left(\frac{3y}{1-y}\right)\overline{\mathbf{R} }+\left[\frac{3y}{1-y}+\frac{9}{2}\left(\frac{y}{1-y}\right)^2\right]\overline{\mathbf{R} }^2+\frac{yP}{\rho_Sk_BT}\overline{\mathbf{R} }^3\]

with \(y=8\pi\rho_SR_S^3/6\), being \(\rho_S\) the density of the solvent, \(\overline{\mathbf{R} }=R_I/R_S\), \(T\) the temperature and \(P\) the pressure. If we consider the solvent-accessible surface (SAS), then, the total \(\Delta G_{cav}\) is the sum of the \(\Delta G_{cav,I}\) weighted by a factor that depends on the exposed \(\mathrm{SASA}_I\) of sphere \(I\)

(7.321)\[ \Delta G_{cav}=\sum_{i=1}^{N_{\mathrm{sph} }}\frac{\mathrm{SASA}_I}{4\pi R_I^2}\Delta G_{cav,I}=\sum_{i=1}^{N_{el} }\frac{\mathrm{SASA}_i}{4\pi R_I^2}\Delta G_{cav,i} \]

Here, the summation over the total number of spheres (\(N_{\mathrm{sph} }\)) that conform the SAS is replaced by a summation over the total number of surface elements (\(N_{\mathrm{el} }\)) in which the SAS is divided into.

With respect to \(\Delta G_{disp}\), this quantity is assumed to depend linearly on the contribution of each surface element to the SASA

(7.322)\[ \Delta G_{disp}=\sum_{i=1}^{N_{\mathrm{el} }}\sigma_I\mathrm{SASA}_i \]

Here, the factor \(\sigma_I\) is the atomic surface tension of the sphere \(I\) to which the surface element \(i\) belongs to.

This strategy, that considers eqs (7.321) and (7.322) to compute \(\Delta G_{nel}\) assumes the use of Gaussian charges for the calculation of \(\Delta G_{el}\) in eq (7.319). When a vdW-type cavity is used to account for electrostatic solvation effects, then the solvation approach is called GVDW_nel, while when a SES-type cavity is used we refer to the solvation approach as GSES_nel (“G” for Gaussian charges and “nel” because of the inclusion of non-electrostatic solvation effects). Details on the GVDW_nel and GSES_nel models can be found in ref. [287].

The GVDW_nel and GSES_nel models have been parametrized for solutes containing H, C, N and O atoms in the following solvents: benzene, chloroform, cyclohexane, octanol, toluene and water. The parameters for the \(\sigma_I\) atomic surface tensions in eq (7.322) together with the radius of the solvent molecules \(R_S\) used to generate the SAS are provided in ref. [287]. The use of the GVDW_nel and GSES_nel is controlled by the string cds_cpcm to add in the %cpcm block. This flag should be equal to 2 within these two models (by default, when we use C-PCM together with the Gaussian Charge Scheme without non-electrostatic solvation effects, cds_cpcm is equal to 0). In order to calculate \(\Delta G_{solv}\) for the GVDW_nel scheme one has to do the following steps:

1. Optimize the geometry of the solute (anisole in this case) in vacuum. For a DFT calculation at the B3LYP/def2-TZVP level of theory (with RIJCOSX), our input file would read like:

   ! B3LYP def2-TZVP D3BJ tightopt
   
   %maxcore 4000
   
   * xyz 0 1
   C   0.66588156410505    1.19833878437922    -0.02647672759883
   C   1.95470174000504    0.69483871574862    -0.01927812843331
   C   2.17893173780041   -0.68097630146538     0.00933914777704
   C   1.09270919891774   -1.54252176637089     0.03141990372830
   C  -0.21073409652496   -1.05179390481975     0.02471602253843
   C  -0.42517201252444    0.32533458015547    -0.00499572635290
   H   0.47565631823749    2.26336213776820    -0.05012843684387
   H   2.79210971512409    1.38085300778245    -0.03749594945361
   H   3.18771596563920   -1.07149825338966     0.01433233618265
   H   1.25061585302875   -2.61357041387417     0.05495238973649
   H  -1.03877793934640   -1.74453554370491     0.04346162651229
   O  -1.65366310927299    0.90979347913477    -0.01693978807213
   C  -2.79791686738707    0.07370821312998     0.01117694976431
   H  -2.83142307710305   -0.58944196707467    -0.85867919998175
   H  -3.65619850989594    0.74176224021803    -0.00958531946332
   H  -2.82883648080292   -0.53125300761732     0.92288089996024
   *
   

2. Use the resulting structure (anisole.xyz) as input structure for the subsequent geometry optimization in solution. In this case, we consider water as the solvent. The input file looks like:

   ! B3LYP def2-TZVP D3BJ tightopt CPCM(Water)
   
   %cpcm
   cds_cpcm 2
   end
   
   %maxcore 4000
   
   * xyzfile 0 1 anisole.xyz
   

3. Search for the “FINAL SINGLE POINT ENERGY” in the output file for the solvated system (we call it \(E_s\)):

   -------------------------   --------------------
   FINAL SINGLE POINT ENERGY      -346.723542928530
   -------------------------   --------------------
   

4. Search for the “FINAL SINGLE POINT ENERGY” in the output file for the system in vacuum (we call it \(E_0\)):

   -------------------------   --------------------
   FINAL SINGLE POINT ENERGY      -346.719657762242
   -------------------------   --------------------
   
   

5. Calculate \(\Delta G_{solv}=E_s-E_0=-0.003885\) a.u. If we convert it to kcal/mol and round it to two decimal digits, we have \(\Delta G_{solv}=-2.44\) kcal/mol. This quantity is in very good agreement with its experimental counterpart (-2.45 kcal/mol[558]).

6. In case the user is interested in the value of \(\Delta G_{nel}\), this quantity is printed in the “CPCM Solvation Model Properties” block:

   CPCM Solvation Model Properties:
   Surface-charge          :  -0.02580685887229
   Corrected charge        :   0.00000000000000
   Outlying charge corr.   :   0.00005046198558 Eh    0.00137 eV
   Free-energy (cav+disp)  :   0.00272597237192 Eh    0.07418 eV
   

Here, \(\Delta G_{nel}\) corresponds to “Free-energy (cav+disp) “ and is calculated through eqs (7.321) and (7.322). The term called “Surface-charge” corresponds to the solute net charge that lies outside the C-PCM cavity. It is basically the sum of the C-PCM charges (without the scaling by \(f(\varepsilon)\)). The effect of this excess of charge on the energy and the C-PCM charges themselves can be corrected by the so-called outlying charge correction. In ORCA, this correction is calculated through a Lagrangian-based algorithm,[705] but only printed for information purposes. That is, the outlying charge effect is neither added to the SCF energy nor to its derivatives. In any case, the corrected total charge is printed in “Corrected charge”, while the correction term for the energy is printed in “Outlying charge corr.”. To further help the user, ORCA also prints a file with extension .cpcm_corr, where the corrected C-PCM charges are provided.

If we want to use, instead, the GSES_nel model, one just needs to add surfacetype gepol_ses_gaussian in the %cpcm block.

Note: The analytical Hessian is not available for the GVDW_nel and GSES_nel models as the second derivative of \(\Delta G_{nel}\) with respect to nuclear displacements is not implemented.

7.50.2. The Conductor-like Screening Solvation Model (COSMO)

Note

The COSMO solvation model has been removed from ORCA v4.0.0 !!!

Please use the C-PCM solvation model in combination with the COSMO epsilon function if required!
As a short form to use the C-PCM model with the COSMO epsilon function, you can specify the solvent via !CPCMC(solvent) .

7.50.3. The SMD Solvation Model

The SMD solvation model has been proposed by the Cramer and Truhlar groups,[558] and is based on the quantum mechanical charge density of a solute molecule interacting with a continuum description of the solvent. In the model the full solute electron density is used without defining partial atomic charges and the solvent is not represented explicitly but rather as a dielectric medium with the surface tension at the solute–solvent boundary. SMD is a universal solvation model, in the sense that it is applicable to any charged or uncharged solute in any solvent or liquid medium for which a few key descriptors are known. In particular, these descriptors are the dielectric constant, refractive index, bulk surface tension, and acidity and basicity parameters. Neglecting the concentration contribution, the model separates the observable solvation free energy into two main components,

(7.323)\[ \Delta G_{\mathrm{S} } = \Delta G_{\mathrm{ENP} } + \Delta G_{\mathrm{CDS} }. \]

In ORCA, the first component is the bulk electrostatic contribution arising from a self-consistent reaction field treatment that involves the electrostatic interaction using the Conductor-like Polarizable Continuum Model (C-PCM). However, the radii are set to “intrinsic atomic Coulomb radii”. The second component, called the cavity-dispersion solvent-structure (CDS) term, is the contribution resulting from short-range interactions between the solute and solvent molecules in the first solvation shell. This contribution is a sum of terms that are proportional (with geometry-dependent proportionality constants called atomic surface tensions) to the solvent-accessible surface areas of the individual atoms of the solute. The CDS contribution to the free energy of solvation is given by

\[\Delta G_{\mathrm{CDS} }=\sum_{k}^{\text{atoms} }\sigma_{k}A_{k}(\textbf{R},{R_{Z_{k} }+r_{s} }) + \sigma^{\mathrm{[M]} }\sum_{k}^{\text{atoms} }A_{k}(\textbf{R},{R_{Z_{k} }+r_{s} }),\]

where \(\sigma_{k}\) and \(\sigma^{\mathrm{[M]} }\) are the atomic surface tension of atom \(k\) and the molecular surface tension, respectively, and \(A_{k}\) is the solvent-accessible surface area (SASA). The SASA depends on the geometry R, the set \({R_{Z_{k} }}\) of all atomic van der Waals radii, and the solvent radius \(r_{s}\), which is added to each of the atomic van der Waals radii. In the program Bondi radii are used for CDS contribution.

More details can be found in the original paper of Marenich et al. [558], which should be cited in publications using results of SMD calculations.

SMD can be employed in single point calculations and geometry optimizations, using single-determinant SCF (HF and DFT) and CASSCF methods. In post-SCF methods the result is corrected in the reference wave function. The SMD solvation model is invoked in the input file via

! SMD(solvent)

where solvent is one of the 179 solvents in the SMD library (see Table Table 7.27). Alternatively, one can request the SMD model via the %cpcm block by writing:

%cpcm   smd true             # turn on SMD
        SMDsolvent "solvent" # specify the name of solvent from the list
 end

Independently on the way the user invokes the SMD model, ORCA automatically sets a number of default SMD parameters for the chosen solvent. If required, the user can also manually specify the solvent descriptors used in an SMD calculation in the %cpcm block.

%cpcm   soln               # index of refraction at optical frequencies at 293 K
        soln25             # index of refraction at optical frequencies at 298 K
        sola               # Abraham's hydrogen bond acidity
        solb               # Abraham's hydrogen bond basicity
        solg               # relative macroscopic surface tension
        solc               # aromaticity, fraction of non-hydrogenic solvent atoms
                           # that are aromatic carbon atoms
        solh               # electronegative halogenicity, fraction of non-hydrogenic
                           # solvent atoms that are F, Cl, or Br
 end

Let’s consider the following input for a water molecule solvated by water,

! B3LYP def2-SVP tightscf SMD(water)

* xyz 0 1
O  -0.00000018976103    0.00606010894837    0.00000000004527
H   0.76098169249695   -0.58891312953082   -0.00000000000022
H  -0.76098151333900   -0.58891299029372   -0.00000000000022
*

Before the SCF part starts, the program prints the SMD information. This part reads as:

--------------------
CPCM SOLVATION MODEL
--------------------
CPCM parameters:
  Epsilon                                         ...      78.3550
  Refrac                                          ...       1.3328
  Rsolv                                           ...       1.3000
  Surface type                                    ... GAUSSIAN VDW
  Discretization scheme                           ... Constant charge density
    Threshold for H atoms                         ...       5.0000 (charges/Ang^2)
    Threshold for non-H atoms                     ...       5.0000 (charges/Ang^2)
  Epsilon function type                           ...         CPCM
Solvent:                                          ... WATER
SMD-CDS solvent descriptors:
  Soln                                            ...       1.3328
  Soln25                                          ...       1.3323
  Sola                                            ...       0.0000
  Solb                                            ...       0.0000
  Solg                                            ...       0.0000
  Solc                                            ...       0.0000
  Solh                                            ...       0.0000
Radii:
  Scheme                                          ... Element-dependent radii
  Radius for O  used is    2.8724 Bohr (=   1.5200 Ang.)
  Radius for H  used is    2.2677 Bohr (=   1.2000 Ang.)
Calculating surface                               ...        done! (  0.0s)
Cavity surface points                             ...          244
Cavity Volume                                     ...     130.0616
Cavity Surface-area                               ...     129.3354
Calculating surface distance matrix               ...        done! (  0.0s)
Performing Cholesky decomposition & store         ...        done! (  0.0s)
Overall time for CPCM initialization              ...                 0.0s

After the SCF is converged, the SMD contribution to the total energy is printed (this term is labelled as “SMD CDS” term).

               *****************************************************
               *                     SUCCESS                       *
               *           SCF CONVERGED AFTER  10 CYCLES          *
               *****************************************************

Recomputing exchange energy using gridx3       ... done (    0.056 sec)
Old exchange energy                            :     -1.791736873 Eh
New exchange energy                            :     -1.791694663 Eh
Exchange energy change after final integration :      0.000042211 Eh
Total energy after final integration           :    -76.335230273 Eh

SMD CDS free energy correction energy :                 1.44927     Kcal/mol
Total Energy after SMD CDS correction =               -76.332920707 Eh
             **** ENERGY FILE WAS UPDATED (water.en.tmp) ****

----------------
TOTAL SCF ENERGY
----------------

Total Energy       :        -76.33292070695327 Eh           -2077.12437 eV

Components:
Nuclear Repulsion  :          9.11286213981824 Eh             247.97359 eV
Electronic Energy  :        -85.43200537714571 Eh           -2324.72305 eV
One Electron Energy:       -123.06209110061025 Eh           -3348.68974 eV
Two Electron Energy:         37.63008572346453 Eh            1023.96669 eV
CPCM Dielectric    :         -0.01612924659271 Eh              -0.43890 eV
SMD CDS (Gcds)     :          0.00230956608299 Eh               0.06285 eV

Notes:

• If one is interested in the calculation of free energies of solvation using the SMD model, one just needs to compute \(\Delta G_{\mathrm{S}}\) according to eq (7.323). However, here one should take into account that the SMD model considers the same concentration (1 mol/L) in both the gaseous and solution phases. Then, if a gas-phase standard state of 1 atm is considered, and we want to compare the calculated \(\Delta G_{\mathrm{S} }\) with its experimental counterpart, a concentration term equal to 1.89 kcal/mol has to be added to the calculated \(\Delta G_{\mathrm{S} }\).

7.50.4. Dynamic Radii Adjustment for Continuum Solvation (DRACO)

The DRACO scheme is an approach that improves the performance of implicit solvation models, in particular the accuracy of the calculated solvation free energies.[688] It is based on a dynamic scaling of the original static radii used to describe the atoms/spheres that define the cavity of the solute.

In this approach, the original radii \(r_i\) of an atom \(i\) is scaled according to

(7.324)\[ r_{i\text{scale}} = f_{i\text{scale}}r_i \]

with the scaling factor \(f_{i\text{scale}}\) determined from

(7.325)\[ f_{i\text{scale}} = \text{erf}(a_Z(q_{\text{eff},i}-b_Z)) + 1 \]

Here, \(q_{\text{eff},i}\) corresponds to the effective partial charge and is defined as

(7.326)\[ q_{\text{eff},i} = q_i + k_Zq_i\text{CN}_i \]

where \(q_i\) is the atomic partial charge. The parameters \(a_Z\), \(b_Z\) and \(k_Z\) are element-specific parameters, and \(\text{CN}_i\) is the fractional coordination number.

For oxygen atoms, the \(f_{i\text{scale}}\) is corrected via a term that depends on the Abraham’s hydrogen bond acidity (\(\alpha\)):

(7.327)\[ f_{i\text{scale}}^{\text{O}} = f_{i\text{scale}} +c_{\text{O}}(0.43-\alpha) \]

with \(c_{\text{O}}\) a parameter that is different for pure C-PCM or SMD. The correction in eq (7.327) is only applied for solvents with \(\alpha<0.43\).

DRACO is parametrized for C-PCM and SMD for the following solvents: acetonitrile, DMSO, methanol, and water. The element-specific parameters in eqs (7.325),(7.326), and (7.327) are available in https://github.com/grimme-lab/DRACO.

The use of DRACO with C-PCM or SMD is triggered via the following tags in the simple input:

! CPCM(solvent) DRACO

or

! SMD(solvent) DRACO

Alternatively, it can also be requested in the %cpcm block:

%cpcm
DRACO true
end

In ORCA, the default scheme to calculate the partial charges in eq (7.326) is the electronegativity-equilibration (EEQ) charge model (D4 case),[132]. However, one can also request the Charge Extended Hückel (CEH) model.[541] The charge scheme within DRACO is controlled by the following tag in the %cpcm block:

%cpcm
draco_charges ceh  # default = eeq
end

If CEH charges are requested, ORCA needs to run an XTB calculation to generate them. Here, one should have at least the version 6.7.1 of XTB (older versions will not generate the CEH charges, see section Program Components). Regarding the fractional coordination number in eq (7.326), it is calculated as described in GFN2-XTB calculations.[70]

Note: DRACO can only be used, for the moment, in single-point energy calculations.

7.50.5. OpenCOSMO-RS

ORCA is interfaced to openCOSMO-RS,[293] an open source implementation of the COSMO-RS model.[447, 448] This model is widely used in both academia and industry to predict fluid phase thermodynamics.

The main idea behind COSMO-RS is that the interaction between molecules in the liquid phase can de depicted as an ensemble of interacting surface segments. The properties on the surface segments are calculated via QM calculations of the solute and the solvent in a perfect conductor (\(\epsilon = \infty\)), and the interaction free energies between segments are functions of a set of descriptors. Among them, the most relevant one is the screening charge density (\(\sigma\)).

A complete description of the COSMO-RS model used in ORCA is provided in ref [293], and the code is available in https://github.com/TUHH-TVT. Since openCOSMO-RS has been parametrized for ORCA 6.0, the executable is shipped together with the ORCA 6.0 binaries and called openCOSMORS. ORCA will then call internally this executable whenever it is needed.

COSMO-RS calculations within ORCA are a special type of calculation. By requesting COSMO-RS, ORCA runs a set of single-point-energy calculations for both the solute and the solvent and then calls the openCOSMO-RS executable, which expects an input geometry for the solute and the solvent and calculates the free energy of solvation of the solute in the solvent. A workflow of a calculation requesting COSMO-RS in ORCA is the following:

1. Single-point-energy calculation of the solute in the gas-phase

2. Single-point-energy calculation of the solute in a conductor (\(\epsilon = \infty\))

3. Single-point-energy calculation of the solvent in a conductor (\(\epsilon = \infty\))

4. Do COSMO-RS via the openCOSMO-RS executable and compute solvation properties

Points 1 to 3 correspond to DFT calculations at the BP86/def2-TZVPD level. This is the level of theory used to parametrize COSMO-RS for ORCA 6.0. As it is shown later, the user can change the functional and basis set, but this is not recommended!

Regarding calculations 2 and 3, to produce smooth \(\sigma\) profiles, ORCA discards the surface segments with an area < 0.01 Å\(^2\). For these two calculations, the radii of several elements used to construct the cavity of solute and solvent are different than the defaults employed within non-COSMO-RS calculations (using C-PCM). This is due to the fact that the provided openCOSMO-RS binaries involve a special parametrization of COSMO-RS for ORCA 6.0, and this does not only affect the COSMO-RS parameters, but also the radii of several elements used in the C-PCM part.

After point 4 is done, the free energy of solvation (\(\Delta G_{\mathrm{S}}\)) of the solute in the solvent is printed in the ORCA output file. In ORCA 6.0 the use of openCOSMO-RS is restricted to the calculation of \(\Delta G_{\mathrm{S}}\).

7.50.5.1. How to Run a ORCA/COSMO-RS Calculation

ORCA/COSMO-RS calculations are controlled through the %cosmors block. These type of calculations require two input structures, one for the solute and one for the solvent. The solute coordinates are provided in the input file as done for any other type of calculation. However, regarding the structure for the solvent, there are two options:

1. Retrieve the structure of the solvent from a database

2. Provide the structure in a separate file

To use strategy 1, we need to request the solvent via:

COSMORS(Solvent)

in the simple input or using the %cosmors block:

%cosmors
solvent "Solvent"
end

The list of internal solvents available in ORCA are shown in Table 7.27. For instance, for a water molecule solvated by acetonitrile, the input file looks like:

!COSMORS(Acetonitrile)
* xyz 0 1
  O    0.00000006589375       0.00157184228646       0.00000000004493
  H    0.77316868532439      -0.58666889665624      -0.00000000000005
  H   -0.77316876182122      -0.58666895650640      -0.00000000000005
*

If the user wants to provide a structure for the solvent (strategy 2), then a separate file with extension .cosmorsxyz should be available. The name of this file (without extension) is controlled by the tag solventfilename in the %cosmors block. For instance, if we want to calculate the free energy of solvation of acetone in water, the ORCA input file would look like this:

%cosmors
solventfilename "water"
end

* xyz 0 1
  H    1.99757808828569      0.25022586507917      0.72957579847856
  C    1.44666788654273      0.06088176074125     -0.18975506731892
  H    1.67115809398389      0.82690156694531     -0.93346420198265
  H    1.76410010378270     -0.89580894800398     -0.61702931492419
  C   -0.03218812888253     -0.00668250402989      0.08117960952503
  O   -0.47126229461002     -0.06383203315654      1.21590571064657
  C   -0.93671505604705     -0.00759565576779     -1.12174123739234
  H   -0.88761294036774      0.97611724740670     -1.59852429171257
  H   -0.58940208311204     -0.73299176093053     -1.85998395179737
  H   -1.96332366957567     -0.22151553828368     -0.83026305352213
*

The structure of the solvent (water) is the one in the water.cosmorsxyz file:

3
0 1
  O    0.00000006589375       0.00157184228646       0.00000000004493
  H    0.77316868532439      -0.58666889665624      -0.00000000000005
  H   -0.77316876182122      -0.58666895650640      -0.00000000000005

where the first line corresponds to the number of atoms, and in the second line the charge and multiplicity are provided.

The output for COSMO-RS is printed in the ORCA output file after the line that reads OPENCOSMO-RS CALCULATION. First of all, the information regarding the level of theory, the solute and the solvent is printed. For the example above (acetone in water), it reads as,

------------------------------------------------------------------------------
                          OPENCOSMO-RS CALCULATION
------------------------------------------------------------------------------

----------------------
GENERAL INFORMATION
----------------------
Calculation method                          ... DFT
Functional                                  ... BP86
Basis set                                   ... DEF2-TZVPD

----------------------
SOLUTE INFORMATION
----------------------
Number of atoms                             ...     10
Total charge                                ...      0
Multiplicity                                ...      1

CARTESIAN COORDINATES (ANGSTROEM)
  H      1.997578    0.250226    0.729576
  C      1.446668    0.060882   -0.189755
  H      1.671158    0.826902   -0.933464
  H      1.764100   -0.895809   -0.617029
  C     -0.032188   -0.006683    0.081180
  O     -0.471262   -0.063832    1.215906
  C     -0.936715   -0.007596   -1.121741
  H     -0.887613    0.976117   -1.598524
  H     -0.589402   -0.732992   -1.859984
  H     -1.963324   -0.221516   -0.830263

----------------------
SOLVENT INFORMATION
----------------------
Solvent name                                ... water
Number of atoms                             ...      3
Total charge                                ...      0
Multiplicity                                ...      1

CARTESIAN COORDINATES (ANGSTROEM)
  O      0.000000    0.001572    0.000000
  H      0.773169   -0.586669   -0.000000
  H     -0.773169   -0.586669   -0.000000

After these lines, ORCA prints the final single point energy for each of the QM calculations, together with the output file to which the ORCA output is redirected:

----------------------------------------------
Single Point Calculation (solute / gas-phase)
----------------------------------------------
Output single point calculation redirected to >test.solute_vac.lastout

FINAL SINGLE POINT ENERGY (Solute-gas-phase)     -193.233975714789

----------------------------------------------
Single Point Calculation (solute / CPCM)
----------------------------------------------
Output single point calculation redirected to >test.solute_cpcm.lastout

FINAL SINGLE POINT ENERGY (Solute-CPCM)     -193.243987123912

----------------------------------------------
Single Point Calculation (solvent / CPCM)
----------------------------------------------
Output single point calculation redirected to >test.solvent_cpcm.lastout

FINAL SINGLE POINT ENERGY (Solvent-CPCM)      -76.479155022866

Once this information is printed, ORCA calls the openCOSMO-RS executable and \(\Delta G_{\mathrm{S}}\) is printed in the following block:

----------------------
SOLVATION DATA
----------------------
Reference temperature              :                 298.15 K
Free energy of solvation (dGsolv)  :       -0.006626234385 Eh             -4.158026 kcal/mol

Note: In order to calculate the free energy of the solvated solute (Gsolv), one should add
      the computed dGsolv to the "Final Gibbs free energy" (Gvac = H-T*S) of the solute in gas-phase.
      That is: Gsolv = Gvac + dGsolv. Here, the Gvac has been calculated previously after a frequency
      calculation of the solute at a certain level of theory and printed in the "THERMOCHEMISTRY" block.

As pointed out in the last paragraph of the COSMO-RS output, to calculate the Gibbs free energy of the solute solvated by the given solvent, one should add the calculated \(\Delta G_{\mathrm{S}}\) to the Final Gibbs free energy of the solute in the gas-phase.

The parameters that can be defined in the %cosmors block in the ORCA input file are the following:

%cosmors
  aeff                 5.92500 # Effective contact area between surface segments (Å^2)
  lnalpha              0.20200 # Logarithm of the misfit prefactor
  lnchb                0.16600 # Hydrogen bond (HB) strength parameter
  chbt                 1.50    # Parameter for the temperature dependence of the HB
  sigmahb              9.61e-3 # HB threshold parameter (e/Å^2)
  rav                  0.50    # Radius to average ideal screening charges in Å
  fcorr                2.40    # Parameter adjusted from dielectric screening energies
  ravcorr              1.00    # Additional radius to calculate the misfit energy in Å
  astd                 41.6240 # Standard surface area (normalization factor) in Å^2
  zcoord               10.0    # Coordination number
  dgsolv_eta          -4.44800 # Offset for the solv. energy calculation
  dgsolv_omegaring     0.26300 # Correction for solv. energy of molecules with rings
  temp                 298.15  # Reference temperature in Kelvin
  dftfunc              "BP86"  # String for the DFT functional
  dftbas               "def2-TZVPD"  # String for the basis set
  solvent              "THF"   # Solvent from the internal database
  solventfilename      "water" # Name of the .cosmorsxyz solvent file
end

It is not recommended to change the defaults of the COSMO-RS parameters.

Note: The workflow explained above for ORCA/openCOSMO-RS calculations involves the same structure for the solute in the gas-phase and in solution. However, these structures may differ substantially depending on the type of solute and solvent. In ORCA, it is possible to optimize the structures for each of the three calculations needed in the ORCA/openCOSMO-RS workflow. That is, (1) the solute in gas-phase, (2) the solute in a conductor, and (3) the solvent in a conductor. To do that, one needs to add the level of optimization via the dftfunc tag, which is exclusive for the DFT functional, but as an exception, can be extended with the optimization tag:

%cosmors
  dftfunc  "BP86 tightopt"
end

7.50.6. Implicit Solvation in Coupled-Cluster Methods

The coupled-cluster Lagrangian, \(\mathcal{L}\), for a system implicitly solvated reads as follows,[133, 142, 285]

(7.328)\[ \mathcal{L}(\Lambda,T)=\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}H_0\mathrm{e}^T|\psi_0\rangle+\frac{1}{2}\mathbf{\bar{Q} }(\Lambda,T)\cdot\mathbf{\bar{V} }(\Lambda,T) \]

where \(\psi_0\) is the reference wave function, and \(H_0\) is the Hamiltonian for the isolated molecule. The operator \(T\) for CCSD is defined in terms of single and double excitations (\(T=T_1+T_2\)), and \(\Lambda\) is the de-excitation operator, defined in terms of the Lagrange multipliers:

(7.329)\[ T=T_1+T_2=\sum_{ia}t_a^ia_a^+a_i+\sum_{ijab}t_{ab}^{ij}a_a^+a_b^+a_ja_i \]

(7.330)\[ \Lambda=\sum_{ia}\lambda_i^aa_i^+a_a+\frac{1}{2}\sum_{ijab}\lambda_{ij}^{ab}a_i^+a_aa_j^+a_b \]

Here, \(t_a^i\) and \(t_{ab}^{ij}\) are the singles and doubles wave function amplitudes and \(a_i\) and \(a_a^+\) are standard fermion annihilation and creation operators, respectively. Canonical occupied orbitals are denoted by the symbols \(i,j,k,\dots\), virtual orbitals by the symbols \(a,b,c,\dots\), and we use the symbols \(p,q,r\dots\) for general orbital indices.

The quantities \(\mathbf{\bar{Q} }\) and \(\mathbf{\bar{V} }\) are the CC expectation values of the C-PCM operators \(\mathbf{Q}\) and \(\mathbf{V}\), which are the solvation charges vector and solute potential vector defined at the position the charges, respectively.

(7.331)\[ \mathbf{\bar{Q} }(\Lambda,T)=\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}\mathbf{Q}\mathrm{e}^T|\psi_0\rangle \]

(7.332)\[ \mathbf{\bar{V} }(\Lambda,T)=\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}\mathbf{V}\mathrm{e}^T|\psi_0\rangle \]

Equation (7.328) can be rewritten by introducing the normal product form of an operator:

(7.333)\[ X_N=X-\langle\psi_0|X|\psi_0\rangle=X-X_0 \]

If one uses this result in eq (7.328), together with the fact that \(\mathbf{Q}\) and \(\mathbf{V}\) are related through eq (7.318), then eq (7.328) reads as,

(7.334)\[\begin{split} \begin{split} \mathcal{L}(\Lambda,T)=&\langle\psi_0|H_0|\psi_0\rangle+\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}H_{0N}\mathrm{e}^T|\psi_0\rangle+\frac{1}{2}\mathbf{Q}_0\cdot\mathbf{V}_0+\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)+\frac{1}{2}\mathbf{\bar{Q} }_N(\Lambda,T)\cdot\mathbf{\bar{V} }_N(\Lambda,T)=\\ &=E_0+\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}H_{0N}\mathrm{e}^T|\psi_0\rangle+\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)+\frac{1}{2}\mathbf{\bar{Q} }_N(\Lambda,T)\cdot\mathbf{\bar{V} }_N(\Lambda,T) \end{split} \end{split}\]

Here, \(\mathbf{Q}_0\) and \(\mathbf{V}_0\) are the \(\mathbf{Q}\) and \(\mathbf{V}\) vectors calculated with the \(\psi_0\) wave function, and \(E_0\) is the reference energy (\(E_0=\langle\psi_0|H_0|\psi_0\rangle+\frac{1}{2}\mathbf{Q}_0\cdot\mathbf{V}_0\)). Different approximations can be adopted in eq (7.334) depending on how one calculates its last term \(\frac{1}{2}\mathbf{\bar{Q} }_N(\Lambda,T)\cdot\mathbf{\bar{V} }_N(\Lambda,T)\).

In ORCA there are three different CCSD/CPCM approaches: (i) the PTE scheme, (ii) the PTE(S) scheme, and the (iii) the PTES scheme, being the last one the default. Here, the acronym PTE stands for “perturbation theory and energy” and “S” for singles. The choice of any of these approaches is controlled via the tag “CPCMccm”. Information about which CCSD/C-PCM is used by ORCA in a calculation is printed in the “ORCA-MATRIX DRIVEN CI” block in the output file in the line starting by “CPCM scheme”:

--------------------------------------------------------------------------------
                             ORCA-MATRIX DRIVEN CI
--------------------------------------------------------------------------------

--------------------------------
AUTOMATIC CHOICE OF INCORE LEVEL
--------------------------------

Memory available                           ...   2000.00 MB
Memory needed for S+T                      ...      9.26 MB
Memory needed for J+K                      ...     18.57 MB
Memory needed for DIIS                     ...    129.61 MB
Memory needed for 3-ext                    ...     72.69 MB
Memory needed for 4-ext                    ...    486.14 MB
Memory needed for triples                  ...      0.00 MB
 -> Final InCoreLevel    ... 5

Wavefunction type
-----------------
Correlation treatment                      ...      CCSD
Single excitations                         ... ON
Orbital optimization                       ... OFF
Calculation of Lambda equations            ... ON
Calculation of Brueckner orbitals          ... OFF
Perturbative triple excitations            ... OFF
CPCM scheme                                ... PTE(S)
Calculation of F12 correction              ... OFF

In the following subsections, we describe the different CCSD/C-PCM approaches available in ORCA and how to use them.

7.50.6.1. PTE scheme

In the “perturbation theory energy” (PTE) scheme, the last term in eq (7.334) is equal to zero (this term does not depend on \(\Lambda\) and \(T\)),

(7.335)\[ \mathcal{L}(\Lambda,T)=E_0+\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}H_{0N}\mathrm{e}^T|\psi_0\rangle+\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T) \]

The potential \(\mathbf{\bar{V} }_N\) can be written as follows:

(7.336)\[ \mathbf{\bar{V} }_N(\Lambda,T)=\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}\sum_{pq}\mathbf{v}_{pq}\{p^+q\}\mathrm{e}^T|\psi_0\rangle=\sum_{pq}\mathbf{v}_{pq}\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}\{p^+q\}\mathrm{e}^T|\psi_0\rangle=\sum_{pq}\mathbf{v}_{pq}\Gamma_{pq} \]

where we have used that \(\mathbf{V}_N=\sum_{pq}\mathbf{v}_{pq}\{p^+q\}\) (second-quantized form of a normal ordered operator), with \(\mathbf{v}_{pq}\) the components of the solute potential in the MO basis. The matrix \(\Gamma\) is the CCSD relaxed one-electron density matrix. Then, the contribution to the equations for the \(T\) amplitudes comes from the derivative of \(\mathbf{\bar{V} }_N(\Lambda,T)\) with respect to the \(\Lambda\) amplitudes (\(\mathbf{Q}_0\) does not depend on the Lagrange multipliers). In this context, the Hamiltonian \(H_{0N}\) contains a term that depends on the elements of the Fock matrix (\(\sum_{pq}f_{pq}\{p^+q\}\)) and that has the same functional form as \(\mathbf{V}_N\). As the Fock matrix is updated in the reference calculation with a C-PCM term that reads as (in AO basis) \(F_{\mu\nu}^{\mathrm{CPCM} }=\mathbf{Q}_0\cdot\mathbf{V}_{\mu\nu}\), then the term \(\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)\) is added implicitly to eq (7.335).

Once the \(T\) amplitudes are obtained, the total energy, \(E\), is calculated as

(7.337)\[ E=E_0+\langle\psi_0|\mathrm{e}^{-T}(H_{0N}+\mathbf{Q}_0\cdot\mathbf{V}_N)\mathrm{e}^T|\psi_0\rangle=E_0+\sum_{ia}F_{ia}t_a^i+\frac{1}{4}\sum_{ijab}\langle ij||ab\rangle\left(t_{ab}^{ij}+2t_a^it_b^j\right) \]

Then, the C-PCM contribution to the CC energy within the PTE scheme occurs through the term \(\frac{1}{2}\mathbf{Q}_0\cdot\mathbf{V}_0\) in \(E_0\) and implicitly through the Fock matrix elements \(F_{ia}\) (\(F_{ia}=F_{ia}^0+\mathbf{Q}_0\cdot\mathbf{v}_{ia}\)).

The PTE scheme corresponds to “CPCMccm 0”, and is implemented for canonical-CCSD (RHF and UHF) and DLPNO-CCSD (RHF and UHF). For instance, for a DLPNO-CCSD calculation (closed-shell reference) of a system solvated by water using the PTE scheme, the input file looks like:

! DLPNO-CCSD cc-pVTZ cc-PVTZ/C TightSCF CPCM(Water)

%cpcm
CPCMccm 0
end

* xyzfile 0 1 water.xyz

7.50.6.2. PTE(S) scheme

In this scheme (where the “S” stands for singles), the last term in eq (7.334) depends on the \(T\) amplitudes, but not on the \(\Lambda\) amplitudes,

(7.338)\[ \mathcal{L}(\Lambda,T)=E_0+\langle\psi_0|(1+\Lambda)\mathrm{e}^{-T}H_{0N}\mathrm{e}^T|\psi_0\rangle+\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)+\frac{1}{2}\mathbf{\bar{Q} }_N(T)\cdot\mathbf{\bar{V} }_N(T) \]

Again, in the same way as for the PTE scheme, the C-PCM contribution to the equations for the \(T\) amplitudes comes from the term \(\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)\) in eq (7.338), which is implictly added to the Fock matrix elements in the MO basis. The last term in eq (7.338) does not depend on the \(\Lambda\) amplitudes and then does not contribute to the equations for the \(T\) amplitudes. However, this term depends on the \(T\) amplitudes through the elements \(\gamma_{ai}\) of the density matrix \(\Gamma\),

(7.339)\[ \gamma_{ai}=t_a^i+\sum_{me}\left(t_{ae}^{im}-t_e^it_a^m\right)\lambda_m^e-\frac{1}{2}\sum_{mnef}\lambda_{mn}^{ef}\left(t_{ef}^{in}t_a^m+t_{af}^{mn}t_e^i\right) \]

and then it contributes to the final energy \(E\) in the following way:

(7.340)\[ \frac{1}{2}\mathbf{\bar{Q} }_N(T)\cdot\mathbf{\bar{V} }_N(T)=\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right) \]

That gives the final equation for the total energy of our system,

(7.341)\[\begin{split} \begin{split} E&=E_0+\langle\psi_0|\mathrm{e}^{-T}(H_{0N}+\mathbf{Q}_0\cdot\mathbf{V}_N)\mathrm{e}^T|\psi_0\rangle+\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right)=\\ &=E_0+\sum_{ia}F_{ia}t_a^i+\frac{1}{4}\sum_{ijab}\langle ij||ab\rangle\left(t_{ab}^{ij}+2t_a^it_b^j\right)+\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right)\end{split} \end{split}\]

Therefore, the CC energy for a solvated system within the PTE(S) scheme involves three C-PCM contributions: (1) the term \(\frac{1}{2}\mathbf{Q}_0\cdot\mathbf{V}_0\) included in \(E_0\), (2) the term \(\sum_{ia}\mathbf{Q}_0\cdot\mathbf{v}_{ia}t_a^i\) that occurs implicitly through \(\sum_{ia}F_{ia}t_a^i\) and (3) the term \(\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right)\).

This scheme is available in ORCA for canonical-CCSD (RHF and UHF) and DLPNO-CCSD (RHF and UHF). In order to use it, the user needs to add the tag “CPCMccm 1” in the %cpcm block.

! DLPNO-CCSD cc-pVTZ cc-PVTZ/C TightSCF CPCM(Water)

%cpcm
CPCMccm 1
end

* xyzfile 0 1 water.xyz

7.50.6.3. PTES scheme

In this scheme, both \(\mathbf{\bar{Q} }_N\) and \(\mathbf{\bar{V} }_N\) in the last term in eq (7.334) depend on the \(T\) amplitudes but just one of them depends on the \(\Lambda\) amplitudes,

(7.342)\[ \mathcal{L}(\Lambda,T)=E_0+\langle\psi_0|(1+\Lambda)\mathrm{e}^TH_{0N}\mathrm{e}^T|\psi_0\rangle+\mathbf{Q}_0\cdot\mathbf{\bar{V} }_N(\Lambda,T)+\frac{1}{2}\mathbf{\bar{Q} }_N(T)\cdot\mathbf{\bar{V} }_N(\Lambda,T) \]

The C-PCM terms enter these equations on the one hand, implicitly, through the elements of the Fock matrix (like for the PTE and PTE(S) schemes), and on the other hand, explicitly through the derivatives of \(\frac{1}{2}\mathbf{\bar{Q} }_N(T)\cdot\mathbf{\bar{V} }_N(\Lambda,T)\) with respect to \(\Lambda\). If we call \(\mathcal{L}_{\mathrm{CPCM} }\) the last C-PCM term in eq (7.342), then the contribution from this term to the \(T\) amplitudes equations read as:

(7.343)\[ \frac{\partial\mathcal{L}_{\mathrm{CPCM} }}{\partial \lambda_i^a}=\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left[-\sum_jt_a^j\mathbf{v}_{ji}+\sum_bt_b^i\mathbf{v}_{ab}+\mathbf{v}_{ia}+\sum_{bj}\left(t_{ba}^{ji}-t_a^jt_b^i\right)\mathbf{v}_{bj}\right] \]

(7.344)\[ \frac{\partial\mathcal{L}_{\mathrm{CPCM} }}{\partial \lambda_{ij}^{ab} }=\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left[-\frac{1}{2}\sum_kt_{ab}^{kj}\mathbf{v}_{ki}+\frac{1}{2}\sum_ct_{ac}^{ij}\mathbf{v}_{bc}-\frac{1}{2}\sum_{ck}\left(t_{ab}^{kj}t_c^i+t_{cb}^{ij}t_a^k\right)\mathbf{v}_{ck}\right] \]

The contribution to the energy is the same as that for the PTE(S) scheme, but with different values for the \(T\) amplitudes (as the equations to calculate them differ slightly from those for the PTE(S) scheme).

(7.345)\[\begin{split} \begin{split} E&=E_0+\langle\psi_0|\mathrm{e}^{-T}(H_{0N}+\mathbf{Q}_0\cdot\mathbf{V}_N)\mathrm{e}^T|\psi_0\rangle+\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right)=\\ &=E_0+\sum_{ia}F_{ia}t_a^i+\frac{1}{4}\sum_{ijab}\langle ij||ab\rangle\left(t_{ab}^{ij}+2t_a^it_b^j\right)+\frac{1}{2}\left(\sum_{ai}t_a^i\mathbf{q}_{ai}\right)\cdot\left(\sum_{ai}t_a^i\mathbf{v}_{ai}\right)\end{split} \end{split}\]

This scheme is the default CCSD/C-PCM approach in ORCA and is available in ORCA for canonical-CCSD (RHF and UHF) and DLPNO-CCSD (RHF and UHF). In this case, the tag “CPCMccm” in the %cpcm block is equal to 2. However, as the PTES scheme is the default in ORCA, the user just needs to add the information about the solvent in the input file, in order to use this approach.

! DLPNO-CCSD cc-pVTZ cc-PVTZ/C TightSCF CPCM(Water)

* xyz 0 1
O  -0.00000018976103    0.00606010894837    0.00000000004527
H   0.76098169249695   -0.58891312953082   -0.00000000000022
H  -0.76098151333900   -0.58891299029372   -0.00000000000022
*

Notes regarding the use of the CCSD/C-PCM schemes:

• For calculations within the PTE(S) and the PTES schemes, the explicit C-PCM contribution to the total energy (see eqs. (7.341) and (7.345)) is printed in the “COUPLED CLUSTER ENERGY” block after the equations for the “T” amplitudes converge. In this case, this energy is labelled as “C-PCM corr. term” and is already included in the correlation energy, “E(CORR) “. For the input example from above, this information looks like:

  ----------------------
  COUPLED CLUSTER ENERGY
  ----------------------
  
  E(0)                                       ...    -76.066903687
  E(CORR)(strong-pairs)                      ...     -0.267203798
  E(CORR)(weak-pairs)                        ...     -0.000106106
  E(CORR)(corrected)                         ...     -0.267309904
  C-PCM corr. term (included in E(CORR))     ...     -0.000003158
  E(TOT)                                     ...    -76.334213591
  Singles Norm <S|S>**1/2                    ...      0.017784967
  T1 diagnostic                              ...      0.006287935
  

  This contribution does not represent the whole C-PCM contribution to the correlation energy, as this one also occurs, implicitly, through the “T” amplitudes.

• The C-PCM contribution to the \(\Lambda\) equations is implemented in ORCA for the PTE(S) and PTES schemes. Then, the user can request unrelaxed densities.