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# 11.2.11 Poisson Boundary Conditions

(February 4, 2022)

Each of the implicit solvation models described above is designed for isotropic, bulk solvation—conditions that can be qualitatively described using a scalar dielectric constant, $\varepsilon$. For an anisotropic environment, such as a liquid/vapor interface, a more general approach is to solve Poisson’s equation with a spatially-varying dielectric function, $\varepsilon(\mathbf{r})$. Such an approach can be used, for example, to model the air/water interface by describing certain regions of space using $\varepsilon=1$ (air) and other regions using $\varepsilon=78$ (water).222, 221 The atomistic region is described at the SCF level. Construction of the dielectric function $\varepsilon(\mathbf{r})$ is based on a solute cavity, somewhat analogous to the solute cavity in a PCM calculation but where the dielectric changes smoothly (rather than abruptly) from $\varepsilon=1$ inside the cavity (in the atomistic QM region) to $\varepsilon=\varepsilon_{\rm solvent}$ outside, in the continuum solvent. Options for modeling a liquid/vapor interface are also available; see Section 11.2.11.3. In contrast to PCMs, which must approximate the “volume polarization” due to penetration of the tails of the QM charge density beyond the solute cavity, this effect is described exactly under Poisson boundary conditions. The price to be paid for lifting this approximation is that charge densities must be discretized and integrated over three-dimensional space rather than simply a two-dimensional solute cavity surface.

Q-Chem’s implementation of the Poisson Equation Solver (PEqS) method was introduced in Ref. 222 and refined in Refs. 221 and 861. Support for dielectric media with finite ionic strength was added in Ref. 1077. Users of this methodology are asked to cite at least Refs. 221 and and 861.

## 11.2.11.1 Theory

The most general formulation of Poisson’s equation, for a dielectric function rather than a dielectric constant, is

 $\hat{\bm{\nabla}}\bm{\cdot}\left[\varepsilon(\mathbf{r})\hat{\bm{\nabla}}% \varphi_{\rm tot}(\mathbf{r})\right]=-4\pi\rho_{\rm sol}(\mathbf{r})\;.$ (11.26)

The solute’s charge density will be separated into nuclear and electronic components,

 $\rho_{\rm sol}(\mathbf{r})=\rho_{\rm nuc}(\mathbf{r})+\rho_{\rm elec}(\mathbf{% r})\;.$ (11.27)

The total electrostatic potential $\varphi_{\rm tot}$ is comprised of the solute’s electrostatic potential $\varphi_{\rm elec}+\varphi_{\rm nuc}$, which is obtained from the density in Eq. (11.27), plus the polarization potential induced in the medium:

 $\varphi_{\rm tot}(\mathbf{r})=\varphi_{\rm elec}(\mathbf{r})+\varphi_{\rm nuc}% (\mathbf{r})+\varphi_{\rm pol}(\mathbf{r})\;.$ (11.28)

To compute $\rho_{\rm elec}$, the electronic contribution to the electrostatic potential in vacuum is first evaluated on a Cartesian grid,

 $\varphi_{\rm elec}(\mathbf{r}_{i})=\sum_{\mu\nu}^{N_{\rm basis}}P_{\mu\nu}% \Bigl{\langle}g_{\mu}(\mathbf{r})\Bigl{|}\frac{1}{|\mathbf{r}-\mathbf{r}_{i}|}% \Bigr{|}g_{\nu}(\mathbf{r})\Bigr{\rangle}\;.$ (11.29)

Here P is the one-electron density matrix, $g_{\mu}$ and $g_{\nu}$ are atom-centered Gaussian basis functions, and the integration is over $\mathbf{r}$. The electronic part of the solute’s charge density is then obtained from

 $\rho_{\rm elec}(\mathbf{r})=-\frac{1}{4\pi}\hat{\nabla}^{2}\varphi_{\rm elec}(% \mathbf{r})\;,$ (11.30)

where in practice the Laplacian of $\varphi_{\rm elec}$ is evaluated using an eighth-order finite difference scheme.221 The nuclear charge density $\rho_{\rm nuc}$ is described classically,

 $\rho_{\rm nuc}(\mathbf{r})=-\sum_{A}^{N_{\rm atoms}}Z_{\!A}\ \delta\left(% \mathbf{r}-\mathbf{R}_{A}\right)\;,$ (11.31)

but to avoid numerical problems the nuclear charges are smeared out over one Cartesian grid voxel in practice. Operationally, this is achieved by adding $Z_{A}/dV$ to the grid point nearest $\mathbf{R}_{A}$, where $dV$ is the volume of a Cartesian voxel. The exact nuclear electrostatic potential in vacuum is computed according to

 $\varphi_{\rm nuc}(\mathbf{r})=-\frac{1}{4\pi}\sum_{A}^{N_{\rm atoms}}\frac{Z_{% \!A}}{|\mathbf{r}-\mathbf{R}_{A}|}\;.$ (11.32)

The Dirac delta function in Eq. (11.31) has dimensions of (volume)${}^{-1}$ and therefore incorporates the aforementioned factor of $dV^{-1}$

Having computed the electronic and nuclear charge densities and the electrostatic potentials in vacuum, one has in hand an exact solution of Eq. (11.26) for the case where $\varepsilon(\mathbf{r})=1$. The induced polarization charge density and corresponding electrostatic potential are then obtained iteratively, following closely the procedure outlined in Refs. 321 and 41. Equation (11.26) is rewritten to appear as Poisson’s equation in vacuum with an additional source charge density:

 \displaystyle\begin{aligned} \displaystyle\hat{\nabla}^{2}\varphi_{\rm tot}(% \mathbf{r})&\displaystyle=-4\pi\left[\frac{\rho_{\rm sol}(\mathbf{r})}{% \varepsilon(\mathbf{r})}+\frac{\hat{\bm{\nabla}}\ln\varepsilon(\mathbf{r})\bm{% \cdot}\hat{\bm{\nabla}}\varphi_{\rm tot}(\mathbf{r})}{4\pi}\right]\\ &\displaystyle=-4\pi\bigl{[}\rho_{\rm sol}(\mathbf{r})+\rho_{\rm pol}(\mathbf{% r})\bigr{]}\;,\end{aligned} (11.33)

where the polarization charge density is

 $\rho_{\rm pol}(\mathbf{r})=\rho_{\rm iter}(\mathbf{r})+\left(\frac{1-% \varepsilon(\mathbf{r})}{\varepsilon(\mathbf{r})}\right)\rho_{\rm sol}(\mathbf% {r})\;.$ (11.34)

The iterative charge density $\rho_{\rm iter}(\mathbf{r})$ takes the form

 $\rho_{\rm iter}^{(k+1)}(\mathbf{r})=\frac{\hat{\bm{\nabla}}\ln\varepsilon(% \mathbf{r})\bm{\cdot}\hat{\bm{\nabla}}\varphi^{(k)}_{\rm tot}(\mathbf{r})}{4\pi}$ (11.35)

at iteration $k+1$. This function is nonzero only in transition regions where the dielectric is being interpolated from vacuum to solvent. Note also that the second term in Eq. (11.34) vanishes wherever $\varepsilon(\mathbf{r})=1$. The quantity $\rho_{\mathrm{iter}}(\mathbf{r})$ is iterated to convergence by performing updates $\rho_{\mathrm{iter}}^{(k)}(\mathbf{r})\rightarrow\rho_{\mathrm{iter}}^{(k+1)}(% \mathbf{r})$ using Eq. (11.35), taking $\varphi^{(0)}_{\mathrm{tot}}(\mathbf{r})$ to be the solute’s electrostatic potential in vacuum. Each time $\rho_{\rm iter}(\mathbf{r})$ is updated, a new polarization charge density is generated and Eq. (11.33) is solved to obtain a new total electrostatic potential. To stabilize the iterative updates of $\rho_{\rm iter}(\mathbf{r})$, we use a damping procedure

 \displaystyle\begin{aligned} \displaystyle\rho_{\rm iter}^{(k+1)}(\mathbf{r})&% \displaystyle=\frac{\eta}{4\pi}\bigl{[}\hat{\bm{\nabla}}\ln\varepsilon(\mathbf% {r})\bm{\cdot}\hat{\bm{\nabla}}\varphi^{(k)}_{\rm tot}(\mathbf{r})\bigr{]}+(1-% \eta)\rho_{\rm iter}^{(k)}(\mathbf{r})\\ &\displaystyle=\eta\rho_{\rm iter}^{(k+1)}+(1-\eta)\rho_{\rm iter}^{(k)}(% \mathbf{r})\end{aligned} (11.36)

rather than using Eq. (11.35) as written. Following Refs. 321 and 41, we use $\eta=0.6$.

Once the total charge density and electrostatic potential are iterated to self-consistency, the free energy of solvation,

 $G_{\rm pol}=\frac{1}{2}\int d\textbf{r}\;\varphi_{\rm pol}(\textbf{r})\;\rho_{% \rm sol}(\mathbf{r})\;,$ (11.37)

is computed, where

 $\varphi_{\rm pol}(\textbf{r})=\varphi_{\rm tot}(\textbf{r})-\varphi_{\rm sol}(% \mathbf{r})\;.$ (11.38)

Since $\varphi_{\mathrm{tot}}$ and $\rho_{\mathrm{tot}}$ are computed self-consistently at each SCF iteration, the polarization effects must be incorporated into the Fock matrix. This is accomplished by adding the correction term $\Delta\hat{F}=\delta G_{\mathrm{pol}}/\delta\rho_{\mathrm{elec}}$. The matrix form of this correction in the atomic orbital basis is

 $\Delta F_{\mu\nu}=\int d\textbf{r}\;\varphi_{\rm pol}(\textbf{r})\;g_{\mu}(% \textbf{r})\;g_{\nu}(\textbf{r})\;.$ (11.39)

For additional details and a full description of the algorithm, see Ref. 221.

## 11.2.11.2 Nonequilibrium Solvation for Vertical Ionization

Nonequilibrium solvation within the framework of PCMs is discussed in Section 11.2.3.3, and that methodology776, 1261 has been adapted for vertical ionization processes in conjunction with Poisson boundary conditions.222, 221 A state-specific approach has been implemented that requires solution of a nonlinear Schrödinger equation, Eq. (11.7). For the reference state, corresponding to $i=0$ in Eq. (11.7), this is equivalent to solving Eq. (11.26) for the induced polarization potential and computing the solvation free energy and Fock matrix correction according to Eqs. (11.37) and (11.39), respectively. For the ionized state, denoted by a subscript $i$, the solvent polarization is partitioned into fast and slow components. Within the Marcus partitioning scheme,1261 Eq. (11.33) becomes221

 $-\frac{1}{4\pi}\hat{\nabla}^{2}\varphi_{{\rm tot},i}(\mathbf{r})=\rho_{{\rm sol% },i}(\mathbf{r})+\rho^{\rm slow}_{{\rm pol},0}(\mathbf{r})+\rho^{\rm fast}_{{% \rm pol},i}(\mathbf{r})\;,$ (11.40)

where the fast polarization charge density of the ionized state, $\rho^{\rm fast}_{{\rm pol},i}$, is given by

 $\rho^{\rm fast}_{{\rm pol},i}(\mathbf{r})=\rho_{{\rm iter},i}(\mathbf{r})+% \left(\frac{1-\varepsilon_{\rm opt}(\mathbf{r})}{\varepsilon_{\rm opt}(\mathbf% {r})}\right)\bigl{[}\rho_{{\rm sol},i}(\mathbf{r})+\rho^{\rm slow}_{{\rm pol},% 0}(\mathbf{r})\bigr{]}$ (11.41)

and the fast polarization potential, $\varphi^{\rm fast}_{{\rm pol},i}$, is

 $\varphi^{\rm fast}_{{\rm pol},i}(\textbf{r})=\varphi_{{\rm tot},i}(\textbf{r})% -\varphi_{{\rm sol},i}(\mathbf{r})-\varphi^{\rm slow}_{{\rm pol},0}(\mathbf{r}% )\;.$ (11.42)

Subscripts $i$ and 0 refer to the ionized state and the reference state, respectively. The quantity $\varepsilon_{\rm opt}$ in Eq. (11.41) is the solvent’s optical dielectric constant, and $\rho_{{\rm iter},i}$ is computed using Eq. (11.36) but with a dielectric function constructed using $\varepsilon_{\rm opt}$ and with $\varphi^{\rm fast}_{{\rm tot},i}$ substituted in place of $\varphi_{\rm tot}$. For the Marcus partitioning scheme, the slow component of the induced solvent polarization charge density is computed as the three-dimensional (volume charge rather than surface charge) analogue of Eq. (11.9).

## 11.2.11.3 Job Control for the Poisson Equation Solver

A calculation using Poisson boundary conditions is requested by setting SOLVENT_METHOD = PEQS in the $rem section. For computational efficiency, the Poisson Equation Solver (PEqS) is typically not engaged until the error in the vacuum SCF calculation falls below a specified threshold, taken to be $10^{3}~{}E_{h}$ by default and user-controllable by means of the$rem variable PEQS_SWITCH.

Note:  When PEqS solvation is requested, symmetry and the standard nuclear orientation are disabled automatically, equivalent to SYM_IGNORE = TRUE and NO_REORIENT = TRUE. Poisson’s equation is solved on a three-dimensional Cartesian grid and the energy need not be rotationally invariant unless the grid spacing is very small.

PEQS_SWITCH

PEQS_SWITCH
Inclusion of solvent effects begins when the SCF error falls below 10${}^{-\rm PEQS\_SWITCH}$.
TYPE:
INTEGER
DEFAULT:
3
OPTIONS:
$n$ Corresponding to 10${}^{-n}$
RECOMMENDATION:
Use the default unless solvent effects need to be incorporated earlier in the SCF procedure.

Further job control for the Poisson equation solver (PEqS) routines is set up in two sections: $peqs_grid and$peqs. (An optional third section, $epsilon, is discussed later.) The$peqs_grid section is used to define the Cartesian discretization grid, as follows:

$peqs_grid DimX <Nx> <Xmin> <Xmax> DimY <Ny> <Ymin> <Ymax> DimZ <Nz> <Zmin> <Zmax>$end


The Cartesian grid must be orthorhombic and Nx, Ny, and Nz refer to the number of grid points for the $x$, $y$, and $z$ axes.

Note:  A fourth-order multigrid algorithm is used to accelerate convergence of the conjugate gradient PEqS routine that solves Eq. (11.33).221 This requires that the number of grid points be odd, with the additional constraint that the quantities $N_{x}-1$, $N_{y}-1$, and $N_{z}-1$ must all be divisible by 8.

Values in the last two columns of the $peqs_grid section specify the minimum and maximum values of $x$, $y$, and $z$, in units of Ångstroms. The length of the grid in the $x$ direction is $L_{x}=x_{\rm max}-x_{\rm min}$ and the grid spacing is $\Delta x=L_{x}/(N_{x}-1)$. The volume element discussed in the context of Eq. (11.31) is $dV=\Delta x\Delta y\Delta z$. The$peqs section controls other aspects of a PEqS calculation, including construction of the solute cavity and other required parameters for interfacial or nonequilibrium solvation. The format is:

$peqs <Keyword> <parameter/option>$end


Available keywords are described below.

MaxIter
Sets the maximum number of iterations used in the conjugate gradient solver.
INPUT SECTION: $peqs TYPE: INTEGER DEFAULT: 500 OPTIONS: User-defined. RECOMMENDATION: Use the default unless the calculation fails to converge. SolverThresh The electrostatic potential is considered converged when the error falls below 10${}^{-\rm PEQSolverThresh}$. INPUT SECTION:$peqs
TYPE:
INTEGER
DEFAULT:
5
OPTIONS:
$n$ Corresponding to 10${}^{-n}$
RECOMMENDATION:
Use the default unless a tighter convergence criterion is desired, at greater computational cost.

PolarIterScale
Specifies the mixing parameter $\eta$ that is used in Eq. (11.36) to stabilize iterative solution for the polarization charge density.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: 0.6 OPTIONS: $\eta$ Desired value of the mixing parameter (unit-less). RECOMMENDATION: Use the default, which was tested in Refs. 321 and 41, unless the calculation proves difficult to converge. BatchSize Evaluate electrostatic potential integrals in batches over small parts of the Cartesian grid, rather than computing all integrals in a single batch. INPUT SECTION:$peqs
TYPE:
INTEGER
DEFAULT:
5000
OPTIONS:
$n$ Corresponding to a batch size of $n$ grid points.
RECOMMENDATION:
For large grids ($\geq 10^{6}$ grid points), a batch size $n\approx N_{x}N_{y}N_{z}/5$ is recommended.

SolventDielectric
Sets the dielectric value for the solvent.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: 78.39 OPTIONS: $\varepsilon$ Desired value of $\varepsilon$ (dimensionless). RECOMMENDATION: The default value corresponds to water at 25${}^{\circ}$C. OpticalDielectric Sets the optical dielectric value of the solvent. INPUT SECTION:$peqs
TYPE:
FLOAT
DEFAULT:
1.7778
OPTIONS:
$\varepsilon_{\rm opt}$ Desired value of $\varepsilon_{\rm opt}$ (dimensionless).
RECOMMENDATION:
The optical dielectric is equal to the square of the index of refraction. The default value corresponds to water at 25${}^{\circ}$C.

SoluteCavity
Specifies the type of solute cavity, which determines the form of the dielectric function $\varepsilon(\mathbf{r})$.
INPUT SECTION: $peqs TYPE: STRING DEFAULT: Vacuum OPTIONS: Vacuum Perform calculation in vacuum, $\varepsilon(\mathbf{r})\equiv 1$. RigidVDW Create a cavity using spherically symmetric error functions. Spherical Create a single spherical cavity around the solute. Arbitrary Create a user-defined dielectric cavity as obtained from the$epsilon input section.
RECOMMENDATION:
None.

In conventional PCMs the solute cavity is a rigid two-dimensional surface constructed from a union of atomic spheres, the radii of which are generally take to be equal to the van der Waals radius ($r_{\rm vdW}$) scaled by a factor of 1.2; see Section 11.2.4.457 This cavity is rigid in the sense that once the atomic coordinates are specified, it remains unchanged during the SCF cycles. (The isodensity cavity discussed in Section 11.2.6 is an exception, but is not available for the PEqS method.) Furthermore, there is an abrupt and discontinuous change in the dielectric constant at the solute/continuum boundary. A three-dimensional analogue of this cavity, using continuous and differentiable spherically-symmetric error functions, has been implemented for PEqS calculations, following the procedure in Ref. 321. The permittivity function, which depends parametrically on the nuclear positions $\mathbf{R}_{A}$, is

 $\varepsilon(\mathbf{r};\{\mathbf{R}_{A}\})=\bigl{(}\varepsilon_{\rm solvent}-1% \bigr{)}\ \left\{\prod_{A}^{N_{\rm atoms}}h(d_{A},\Delta;\lvert\mathbf{r}-% \mathbf{R}_{A}\rvert)\right\}\;,$ (11.43)

where

 $h(d_{A},\Delta;\lvert\mathbf{r}-\mathbf{R}_{A}\rvert)=\frac{1}{2}\left[1+{\rm erf% }\left(\frac{\lvert\mathbf{r}-\mathbf{R}_{A}\rvert-d_{A}}{\Delta}\right)\right% ]\;.$ (11.44)

Equation (11.43) smoothly interpolates between $\varepsilon=1$ and $\varepsilon=\varepsilon_{\rm solvent}$, over a length scale of $\approx 4\Delta$. The value of $\Delta$ is specified using the RigidScale keyword. The quantity $d_{A}$ in Eqs. (11.43) and (11.44) sets radius of the atomic sphere for atom $A$. By default, $d_{A}=1.2\,r_{\rm vdW}$ but this can be controlled as described below.

RigidScale
Sets the length scale on which the error function employed in the rigid vdW cavity construction interpolates the dielectric value from vacuum to solvent.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: 0.265 Å OPTIONS: $\Delta$ Specifies the desired value (in Å); see Eq. (11.44). RECOMMENDATION: Use the default value, which was tuned in Ref. 321 so that errors in small-molecule solvation energies were $\approx 1$ kcal/mol. VDWType Specifies details for the rigid vdW cavity construction. INPUT SECTION:$peqs
TYPE:
STRING
DEFAULT:
Scaled
OPTIONS:
Unscaled $d_{A}=r_{\mathrm{vdW}}$ Scaled $d_{A}=r_{\mathrm{vdW}}\times scale$ Shifted $d_{A}=(r_{\mathrm{vdW}}+shift)\times scale$
RECOMMENDATION:
None. The values of $scale$ and $shift$ are set with the VDWScale and VDWShift keywords.

VDWScale
Sets the empirical scale factor applied to the atomic van der Waals radius.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: 1.2 OPTIONS: $scale$ Specifies the desired dimensionless scaling factor for the atomic vdW radii. RECOMMENDATION: Use the default value. VDWShift Adjusts the center of the spherically-symmetric error functions when constructing the rigid vdW cavity. INPUT SECTION:$peqs
TYPE:
FLOAT
DEFAULT:
0.0 Ångstroms
OPTIONS:
$shift$ Specifies the desired shift (in Å).
RECOMMENDATION:
None. If VDWType is set to Shifted, the vdW scale factor is set to 1.0 by default. This can be adjusted using the VDWScale keyword.

Setting the SoluteCavity keyword to Spherical requests the construction of a spherical cavity around the solute, and the dielectric is smoothly interpolated from vacuum to solvent using a hyperbolic tangent function:222

 $\varepsilon(r)=\frac{1}{2}\Bigl{\{}(\varepsilon_{\mathrm{solvent}}+1)+(% \varepsilon_{\mathrm{solvent}}-1)\ \mbox{tanh}\bigl{[}\alpha(r-r_{\mathrm{mid}% })\bigr{]}\Bigr{\}}\;.$ (11.45)

Here, $r_{\mathrm{mid}}$ is the distance where the dielectric assumes the value $\varepsilon(r_{\mathrm{mid}})=(\varepsilon_{\mathrm{solvent}}+1)/2$. The value of $r_{\mathrm{mid}}$ is taken to be the sum of the sphere radius, $R$, and half of the interpolation length, $L$: $r_{\mathrm{mid}}=R+L/2$. The sphere radius and interpolation length are controlled with the SphereRadius and InterpolLength keywords described below. The parameter $\alpha$ controls the sharpness of the switching process, with $\alpha=4/L$ by default.

SphereRadius
Sets the radius of the spherical solute cavity.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: No default. OPTIONS: $R$ Desired spherical cavity radius (in Å). RECOMMENDATION: None. See the Supporting Information of Ref. 222 for more information. InterpolLength Sets the length scale on which the dielectric is smoothly interpolated from vacuum to solvent. INPUT SECTION:$peqs
TYPE:
FLOAT
DEFAULT:
No default.
OPTIONS:
$L$ Desired interpolation length (in Å).
RECOMMENDATION:
None.

InterpolScale
For a given interpolation length ($L$), InterpolScale ($\alpha$) sets the sharpness of the dielectric transition from vacuum to solvent.
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: $\alpha$ = $4/L$ OPTIONS: $\alpha$ Desired interpolation scale factor (in Å${}^{-1}$). RECOMMENDATION: Use the default unless a broader (smaller $\alpha$) or narrower (larger $\alpha$) transition region is desired. Interface Perform a solvation calculation at a solvent/vacuum interface. INPUT SECTION:$peqs
TYPE:
STRING
DEFAULT:
False
OPTIONS:
True Modify the dielectric function to simulate an interface between solvent and vacuum. False Perform a solvation calculation in bulk solvent.
RECOMMENDATION:
The user will also need to specify the length scale on which the dielectric is smoothly interpolated from the bulk solvent value to vacuum, and the location of the Gibbs dividing surface (see below).

By setting Interface = True, the dielectric function on the Cartesian grid is further modified to mimic a solvent/vacuum interface. First, the solute cavity is constructed as specified by the SoluteCavity keyword as discussed above. Then, the dielectric function is smoothly interpolated in the $z$ direction across the Gibbs dividing surface ($z\equiv z_{\mathrm{GDS}}$) using the following hyperbolic tangent switching function:222, 221

 $\varepsilon(z)=\frac{1}{2}\Bigl{\{}(\varepsilon_{\rm solvent}+1)+(1-% \varepsilon_{\rm solvent})\ \mbox{tanh}\bigl{[}\beta(z-z_{\mathrm{GDS}})\bigr{% ]}\Bigr{\}}\;.$ (11.46)

This interpolates the dielectric function over the interface length $L_{\rm interface}$, centered at the Gibbs dividing surface, $z\equiv z_{\mathrm{GDS}}$. Both of these values are controlled by the keywords InterfaceLength and GibbsDS, respectively. Similar to the parameter $\alpha$ used in the spherical cavity construction, the parameter $\beta=4/L_{\rm interface}$ controls the sharpness of the interpolation across the Gibbs dividing surface.

InterfaceLength
Sets the length scale over which the dielectric function is smoothly transitioned from bulk solvent to vacuum in the $z$ direction
INPUT SECTION: $peqs TYPE: FLOAT DEFAULT: None OPTIONS: $L_{\rm interface}$ Desired interface length (in Å). RECOMMENDATION: This sets the value $\beta=4/L_{\rm interface}$ in Eq. (11.46). See the Supporting Information of Ref. 222 for a full description of the solvent/vacuum interface construction. GibbsDS Sets the location of the Gibbs dividing surface, in the $z$ direction. INPUT SECTION:$peqs
TYPE:
FLOAT
DEFAULT:
None
OPTIONS:
$z_{\mathrm{GDS}}$ Desired location of the Gibbs dividing surface (in Å).
RECOMMENDATION:
Consult the literature. One such way to determine this value is to compute a density profile of the solvent as a function of $z$ and set this location to be where the solvent density has decreased to 50% of the bulk value. Usually $z_{\mathrm{GDS}}\approx L_{\rm interface}/2$.

NonequilJob
Obtain the nonequilibrium free energy of solvation for a vertical ionization process.
INPUT SECTION: $peqs TYPE: STRING DEFAULT: False OPTIONS: True Compute the nonequilibrium free energy for a vertical ionization process. False Compute the equilibrium solvation free energy. RECOMMENDATION: None. NonequilPartition Specifies the manner in which the solvent response is partitioned into fast and slow components. INPUT SECTION:$peqs
TYPE:
STRING
DEFAULT:
Marcus
OPTIONS:
Marcus Employ the Marcus partitioning scheme. Pekar Employ the Pekar partitioning scheme.
RECOMMENDATION:
Use the default. Although the fast and slow solvation responses are different between the two approaches, the total solvation free energy is the same,1261 but the Pekar scheme is computationally more expensive than the Marcus scheme.

NonequilState
Specifies the state of interest for a nonequilibrium vertical ionization.
INPUT SECTION: $peqs TYPE: STRING DEFAULT: Reference OPTIONS: Reference The reference (initial) state, from which an electron will be removed. Ionized The final (ionized) state. RECOMMENDATION: None. Both values will be needed in a compound input job, in order to compute the non-equilibrium response to vertical ionization; see Example 11.2.11.4. Lastly, by setting SoluteCavity to Arbitrary in the$peqs input section, the user may choose to specify a completely user-defined permittivity function $\varepsilon(\mathbf{r})$,31 which must be generated externally and input into Q-Chem pointwise on the PEqS grid, using a $epsilon section in the input file. The format for that section consists of Cartesian grid points $(x_{i},y_{i},z_{i})$ following by $\varepsilon(x_{i},y_{i},z_{i})$, as shown below. $epsilon
x1  y1  z1    eps(x1,y1,z1)
x1  y1  z2    eps(x1,y1,z2)
. . .
x1  y1  zNz   eps(z1,y1,zNz)
x1  y2  z1    eps(x1,y2,z1)
. . .
x1  y2  zNz   eps(z1,y2,zNz)
. . .
. . .
xNx  yNy  zNz    eps(xNx,yNy,zNz)
$end  ## 11.2.11.4 Examples The following example computes the solvation free energy of a water molecule immersed in water. The Cartesian grid is cubic with a side length of 15.0 Å and 73 grid points in each direction. A rigid vdW cavity is used based on scaled vdW radii for the atomic spheres. The dielectric is not set, so takes the default value of 78.39. Example 11.19 Free energy of solvation of water in water. $molecule
0 1
O     0.053004    -0.020947    -0.034784
H     0.003424     0.185855     0.910594
H    -0.844842     0.146674    -0.358413
$end$rem
EXCHANGE          wB97X-V
BASIS             6-31+G*
SCF_CONVERGENCE   5
THRESH            14
SOLVENT_METHOD    PEQS
PEQS_SWITCH       0
$end$peqs
SOLUTECAVITY RIGIDVDW
$end$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end  View output The next example illustrates calculation of the solvation free energy of a water molecule at a water/vacuum interface, with the Gibbs dividing surface placed at $z=0.50$ Å. The length of the interface is set to $L_{\rm interface}=2.75$ Å and the dielectric is interpolated from bulk solvent to vacuum in the positive $z$ direction across the Gibbs dividing surface. The Cartesian grid, solute cavity, and solvent dielectric are the same as in the previous example. Example 11.20 Free energy of solvation of water at a water/vacuum interface. $molecule
0 1
O     0.053004    -0.020947    -0.034784
H     0.003424     0.185855     0.910594
H    -0.844842     0.146674    -0.358413
$end$rem
SCF_CONVERGENCE   5
THRESH            14
EXCHANGE          wB97X-V
BASIS             6-31+G*
SOLVENT_METHOD    PEQS
PEQS_SWITCH       0
$end$peqs
SOLUTECAVITY RIGIDVDW
INTERFACE TRUE
INTERFACELENGTH 2.75
GIBBSDS 0.50
INTERFACEDIRECTION POSITIVE
$end$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end  View output The final example illustrates a nonequilibrium (NonEquilJob = True) solvation calculation for the vertical ionization of $\rm H_{2}O^{-}$ in bulk water. This is a compound job that first calculates the equilibrium solvation free energy of the anionic state (NonEquilState = Reference), then computes the nonequilibrium energy correction for the ionized state (NonEquilState = Ionized). The two jobs are separated by “@@@”. The Cartesian grid and solvent dielectric are the same as the previous examples, but the solute cavity is chosen to be spherical with a radius of 2.0 Å. The Marcus scheme (NonEquilPartition = Marcus) is used to partition the solvent response into fast and slow components. Since this is the default method, the NonEquilPartition keyword is omitted. Example 11.21 Nonequilibrium free energy of solvation for the vertical ionization of $\rm H_{2}O^{-}$ $molecule
-1 2
O     0.053004    -0.020947    -0.034784
H     0.003424     0.185855     0.910594
H    -0.844842     0.146674    -0.358413
$end$rem
SCF_CONVERGENCE   5
THRESH            14
EXCHANGE          HF
BASIS             6-31++G*
SOLVENT_METHOD    PEQS
PEQS_SWITCH       0
$end$peqs
SOLUTECAVITY   SPHERICAL
SPHERERADIUS   2.00
NONEQUILJOB    TRUE
NONEQUILSTATE  REFERENCE
$end$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
$end @@@$molecule
0 1
O     0.053004    -0.020947    -0.034784
H     0.003424     0.185855     0.910594
H    -0.844842     0.146674    -0.358413
$end$rem
SCF_CONVERGENCE   5
THRESH            14
EXCHANGE          HF
BASIS             6-31++G*
SOLVENT_METHOD    PEQS
PEQS_SWITCH       0
$end$peqs
SOLUTECAVITY   SPHERICAL
SPHERERADIUS   2.00
NONEQUILJOB    TRUE
NONEQUILSTATE  IONIZED
$end$peqs_grid
DimX 73 -7.50 7.50
DimY 73 -7.50 7.50
DimZ 73 -7.50 7.50
\$end


View output