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# 11.2.2 Kirkwood-Onsager Multipole Expansion Method

(July 14, 2022)

The simplest implicit solvation model available in Q-Chem is the multipolar expansion model, 605 Kirkwood J. G.
J. Chem. Phys.
(1934), 2, pp. 767.
, 825 Mikkelsen K. V. et al.
J. Chem. Phys.
(1988), 89, pp. 3086.
, 490 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
which has also been called the Kirkwood-Onsager model or sometimes the generalized Kirkwood model. 490 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
In this approach, the solute is placed inside of a spherical cavity of radius $a$ that is surrounded by a homogeneous dielectric medium whose dielectric constant is $\varepsilon$, and these constitute the only parameters of the model. The term Onsager model is sometimes synonymous with a point-dipole approximation for the solute’s charge density, but Q-Chem’s version uses a single-center multipole expansion of the density (which can be extended to arbitrarily high order), in order to obtain an essentially exact description of the solute’s electrostatic potential. The model then consists of using Kirkwood’s analytic expressions for the solvation energy of each spherical harmonic function, in a spherical cavity inside of a dielectric continuum. 490 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.

Regarding the cavity radius $a$, Onsager’s original suggestion is based on the molar volume $V_{m}$ of the pure solute,

 $a=(3V_{m}/4\pi N_{A})^{1/3}$ (11.1)

where $N_{A}$ is Avogadro’s constant. This was later shown to be a poor choice in the context of modern quantum chemistry calculations. 490 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
It is also common to add 0.5 Å to the value of $a$ in Eq. (11.1) in order to account for the first solvation shell, 1300 Wong M. W., Frisch M. J., Wiberg K. B.
J. Am. Chem. Soc.
(1991), 113, pp. 4776.
or to set $a$ equal to the maximum distance between the solute center of mass and the solute atoms, plus the relevant van der Waals radii. A third option is to set $2a$ (the cavity diameter) equal to the largest solute–solvent internuclear distance, plus the van der Waals radii of the relevant atoms. Clearly, there is quite a bit of arbitrariness in this choice and solvation energies are quite sensitive to the value of $a$, 490 Herbert J. M.
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
(2021), 11, pp. e1519.
and the PCMs that are described in Section 11.2.3 have largely made the multipolar expansion method obsolete, since the PCMs employ the exact electron density and can be used (if desired) with a spherical cavity, although the more typical choice is a molecule-shaped van der Waals cavity.

The Kirkwood-Onsager SCRF is requested by setting SOLVENT_METHOD = KIRKWOOD in the $rem section. Some additional options can be specified in the$solvent section, as described below, of which only CavityRadius is required. Energies and analytic gradients for the Kirkwood-Onsager solvent model are available for Hartree-Fock, DFT, and CCSD calculations. It is often advisable to perform a gas-phase calculation of the solute molecule first, which can serve as the initial guess for a subsequent Kirkwood-Onsager implicit solvent calculation. For coupled-cluster calculations using this model, one may set CC_SAVEAMPL = TRUE to retain the CC amplitudes from the gas-phase calculation, which will save some time in the subsequent solution-phase calculation.

Note:  For CCSD calculations the Kirkwood-Onsager model works only with CCMAN2 = FALSE.

The following job-control options belong in the $solvent section, not the$rem section. As with other parts of the Q-Chem input file, this input section is not case-sensitive.

Sets the radius of the spherical solute cavity.
INPUT SECTION: $solvent TYPE: FLOAT DEFAULT: No default. OPTIONS: $a$ Cavity radius in Å. RECOMMENDATION: None Dielectric Sets the dielectric constant of the solvent continuum. INPUT SECTION:$solvent
TYPE:
FLOAT
DEFAULT:
78.39
OPTIONS:
$\varepsilon$ Use a (dimensionless) value of $\varepsilon$.
RECOMMENDATION:
The default value corresponds to water at 25${}^{\circ}$C.

MultipoleOrder
Determines the order to which the multipole expansion of the solute charge density is carried out.
INPUT SECTION: $solvent TYPE: INTEGER DEFAULT: 15 OPTIONS: $\ell$ Include up to $\ell$th order multipoles. RECOMMENDATION: Use the default. The multipole expansion is usually converged by order $\ell$ = 15. Example 11.1 Multipole expansion model at the Hartree-Fock level; H${}_{2}$O in acetonitrile $molecule
0 1
O       0.00000000     0.00000000     0.11722303
H      -0.75908339     0.00000000    -0.46889211
H       0.75908339     0.00000000    -0.46889211
$end$rem
METHOD            HF
BASIS             6-31g**
SOLVENT_METHOD    Onsager
$end$solvent
Dielectric        35.9   !   Acetonitrile
MultipoleOrder    15     !   this is the default value
$end  Example 11.2 Kirkwood-Onsager SCRF applied to hydrogen fluoride in water, performing a gas-phase calculation first. $molecule
0 1
H       0.000000     0.000000    -0.862674
F       0.000000     0.000000     0.043813
$end$rem
METHOD          HF
BASIS           6-31G*
$end @@@$molecule
$end$rem
$end$solvent