Interatomic Potentials Including Chemistry

Abstract

Beginning from two theories, classical and quantum mechanical, as realized in terms of Newton’s second law and the time-independent Schrödinger equation, we put forth a framework for understanding the development of atomistic potentials that include chemistry. Our analysis introduces, explains, and exploits the Fragment Hamiltonian approach to the electronic structure of molecular and condensed matter systems. Illustrations of the Fragment Hamiltionian display the roles of various physical concepts in the formation of these atomistic potentials. Electron density fluctuations are clearly seen as essential to the realistic description of interatomic interactions over a large range of nuclear (ionic) configurations. Finally, we present a novel approach to the parameterization of interatomic potentials that explicitly include the effect of charge fluctuations, the environment-dependent dynamic charge potential.

Appendix

The functional forms for the environment dependent charges as well as the EDD-Q potential form for water and silica are given below:

Silica

The charge on a silica atom is obtained as follows:

$$q^{\rm{Si}} = q_{res}^{\rm{Si}} + \sum\limits_{{B = 1}}^{N_{O}} {\varepsilon_{B} \,dq_{{B}}^{\rm{SiO}} } ,\,{\text{where}}\,\varepsilon_{B} = \left\{ {\begin{array}{*{18}l} {1,R_{\rm{SiO}} \le r_{c} } \\ {0,\text{ otherwise}} \\ \end{array} } \right\}$$

(3.165)

where the bond charge is defined by

$$dq_{{B}}^{\rm{SiO}} =\Delta q\left[ {1 - \tan h_3\left( {\frac{{R_{AB} - r_{c1} }} {{r_{0} }}} \right)} \right] + \sum\limits_{C \ne B}^{L_{\rm O}} {\varepsilon_{C} dq_{{BC}}^{\rm{OSiO}} }$$

(3.166)

and

$$\begin{aligned} dq_{BC}^{{\rm OSiO}} & = \left[ {b_{1} \sin^{2} \theta_{BAC} \exp \left( { - \theta_{BAC} } \right) + b_{2} } \right]\tan h_4\left( {\frac{{R_{AB} - r_{c2} }}{{r_{0} }}} \right) \\ & \quad + \left[ {p_{1} \exp \left( { - \theta_{BAC}^{2} \sin^{2} \theta_{BAC} } \right)} \right]\tan h_2\left( {\frac{{R_{AB} - r_{c3} }}{{r_{0} }}} \right) \\ \end{aligned}$$

(3.167)

In the above equations, r ₀ equals unity and has dimensions of Å, R _AB and R _AC are the distances of the Si atom A from oxygens B and C respectively, and θ _BAC is the angle formed by those three atoms; the various parameters are given in Table 3.9.

Table 3.9 Charge parameters for the EDD-Q potential

Similarly, the net charge on an oxygen atom having L _Si silicon neighbors is given by

$$q^{\text{O}} = q_{res}^{\text{O}} - \sum\limits_{{A = 1 }}^{L_{\text{Si}}} {\varepsilon_{A} dq_{A}^{\text{SiO}} } ,$$

(3.168)

where we have used the definition of bond charge from (3.166). As is obvious from (3.165) and (3.168), a net residual charge q _res is associated with isolated silicon or oxygen atoms.

Equations (3.169) and (3.170) was used as the basis for our potential function with the functional form and the parameters of ϕ _AB and $\mathcal{A}_A$ chosen to yield the ground state (as well as the deformed) geometry and energetics of the model clusters as predicted by DFT. The effective charges on atoms were scaled by a constant multiplicative factor while evaluating the Coulombic term in the energy expression. In other words, we use a screened value for the charge rather than the Mulliken populations, such that the ratio of the effective atomic charge and the Mulliken population is a constant.

$$\bar E_{A} = F(q_{A}) + \frac{1}{2}\sum\limits_{B \ne A}^{N} {\phi_{AB} } + \frac{1}{2}\sum\limits_{B \ne A}^{N} {\frac{{q_{A} q_{B} }}{{R_{AB} }}}$$

(3.169)

where R _AB was the distance of separation between atoms A and B and F(q _A ) was expressed as

$$F\left( {q_{A} } \right) = \mathcal{A}_{A} q_{A} \ln q_{A}^{2}$$

(3.170)

The functional forms of ϕ _AB , $\mathcal{A}_A$, F(q _A ) and the total energy expression are given below in (3.171)–(3.176), and Table 3.10 contains the values of the various parameters. ϕ is given by

$$\phi_{\text {SiSi}} = \frac{{a_{\text {SiSi}} }}{{r_{\text {SiSi}}^{12} }}$$

(3.171)

$$\phi_{\text {OO}} = \frac{{a_{\text {OO}} }}{{r_{\text {OO}}^{12} }}$$

(3.172)

$$\phi_{\text {SiO}} = \frac{{a_{\text {SiO}} }}{{r_{\text {SiO}}^{20} }} - \frac{{c_{\text {SiO}} }}{{r_{\text {SiO}}^{6} }}$$

(3.173)

and

$$F\left( {q_{A} } \right) = \left( {q^{A}_{res} - q^{A} } \right)\left[ {\mathcal{A}_{A}^{1} \ln \left( {\frac{{q^{A}_{res} - q_{A} }}{{d_{A} }}} \right)^{2} } \right]$$

(3.174)

where for silicon atom A,

$$\mathcal{A}_{A}^{1} = \mathcal{A}_{\text{Si}}^{C} \left\{ {1 + \sum\limits_{{B = 1}}^{L_{\text O}} {\varepsilon_{B} f^{\text{Si}} \left[ {1 - \tan h_\eta \left( {\frac{{r_{AB} - r_{ae} }}{{r_{0} }}} \right)} \right]} } \right\}$$

(3.175)

and for oxygen atom B,

$$\mathcal{A}_{B}^{1} = \mathcal{A}_{\text{O}}^{C} .$$

(3.176)

$\mathcal{A}_{\text{Si}}^{0}$, $\mathcal{A}_{\text{O}}^{0}$, $\mathcal{A}_{\text{Si}}^{C}$, and $\mathcal{A}_{\text{Si}}^{C}$ are parameters given in Table 3.10. The potential energy of atom A is

$$\bar E_{A} = F(q_{A} ) + \frac{1}{2}\sum\limits_{B \ne A}^{N} {\phi_{AB} } + \frac{1}{2}\sum\limits_{B \ne A}^{N} {\frac{{q_{A}^{eff} q_{B}^{eff} }}{{R_{AB} }}}$$

(3.177)

where q _A is the charge as obtained from (A3.1) to (A3.4) and $q_{A}^{eff}$ is given by

$$q_{A}^{eff} = {\raise0.7ex\hbox{${q_{A} }$} \!\mathord{\left/ {\vphantom {{q_{A} } {s_{q} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${s_{q} }$}}$$

(3.178)

with s _q given in Table 3.10.

Table 3.10 Potential parameters for the EDD-Q potential

Water

Water polymorphs are characterized by the formation of hydrogen bonds between neighboring water molecules. In this regard, in the EDD-Q formulation, the net atomic charge on oxygen and hydrogen atoms constituting a water monomer depends both on the two underlying OH bond distance(s) and HOH bond angle as well as the hydrogen bond distance between neighboring water monomers.

First, we consider the functional form for calculating atomic charges within a monomer (H₁OH₂) that consists of two OH bonds (r_OH1 and r_OH2) and a bond angle (θ) expressed in radians.

The charge (expressed in e) on the hydrogen atom (q_H1) is given by

$$ \begin{aligned} q_{\rm{H1}} &= \left[ {\alpha (\theta )\exp ( - 2r_{\rm{OH1}} ) + \beta (\theta )r_{\rm{OH1}} \exp ( - r_{\rm{OH1}} ) + c(\theta )r_{\rm{OH1}} } \right]\sin^{2} (\theta ) \\ &\qquad + \,(r_{\rm{OH1}} - r_{\rm{OH2}} )d(\theta )\sin^{2} (\theta ), \\ \end{aligned} $$

(3.179)

where α, β, c, d are functions of θ (given below in (3.180)). The charge on the other hydrogen atom within a water monomer can also be obtained in a similar fashion.

$$ \begin{aligned} \alpha (\theta ) = \alpha_{1} \exp (\theta ) + \alpha_{2} \exp (\theta^{2} /4) + \alpha_{3} \theta \hfill \\ \beta (\theta ) = \beta_{1} \theta \exp ( - \theta ) + \beta_{2} \theta \exp ( - 2\theta ) + \beta_{3} \theta^{2} \hfill \\ c(\theta ) = c_{1} \theta \exp ( - \theta ) + c_{2} \theta \exp ( - 2\theta ) + c_{3} \hfill \\ d(\theta ) = d_{1} (\theta - \theta^{2} ) + d_{2} \theta^{3} \hfill \\ \end{aligned} $$

(3.180)

To prevent energy discontinuities at the neighbor cutoff distance (=1.5 Å), a switching function S(t) is used for modulating the calculated hydrogen charge as given below:

$$ \begin{aligned} q_{\rm{H1}} = q_{\rm{H1}} S(r_{\rm{OH1}} - t_{cut} ) \hfill \\ S(t) = 0.5(1 - \tan h (t/t_{1} )) \hfill \\ \end{aligned} $$

(3.181)

The charge on the oxygen atom in the water monomer is given by

$$ q_{\rm{O}} = - (q_{\rm{H1}} + q_{\rm{H2}} ). $$

(3.182)

The corresponding parameters for the water monomer charges are given in Table 3.11.

Table 3.11 Charge parameters for the water monomer

To account for charge transfer between neighboring monomers, a net intermolecular charge transfer (dq) between two monomers is obtained as follows:

$$ dq = a\exp ( - br_{\rm{HB}} ), $$

(3.183)

where r _HB is the hydrogen bond distance between donor hydrogen atom and acceptor oxygen atoms belong to neighboring monomers respectively. The neighbor cutoff distance s_cut between two monomers is set to 2.5 Å and thus dq is modulated by the switching function S as defined before.

$$ dq = dqS(r_{\rm{HB}} - r_{im} ). $$

(3.184)

Further, dq is partitioned among the respective monomer atoms as follows:

$$\begin{aligned} dq^{{\text{O}}_{donor}} = - 0.5dq \hfill \\ dq^{{\text{O}}_{acceptor}} = 0.75dq \hfill \\ dq^{{\text{H}}_{donor}} = - 0.4dq \hfill \\ \end{aligned}$$

(3.185)

In addition, for the acceptor molecule, $dq^{{\text{H}}_{acceptor}}$ is obtained by partitioning [$dq - dq^{{\text{O}}_{acceptor}}$] equally among the two hydrogen atoms, while within the donor molecule the other hydrogen atom acquires an additional charge equaling [$dq^{{\text{O}}_{donor}} + dq^{{\text{H}}_{donor}} - dq]$.

Thus the total charge for any atom is given by a sum of its ‘monomer’ charge plus additional charge transfer that is acquired due to hydrogen bonding with neighboring monomers. Note that an equivalent dq arises for every hydrogen bonded interaction a monomer participates in Table 3.12.

Table 3.12 Water dimer charge parameters

Similar to the Hamiltonian defined for silica, the energy of an atom i is given by

$$\bar E_{A} = F(q_{A} ) + \frac{1}{2}\sum\limits_{B \ne A} {\Phi _{AB} } + \frac{1}{2}\sum\limits_{B \ne A} {\frac{{q_{A} q_{B} }}{{R_{AB} }}}$$

(3.186)

Here,

$$F(q_{A} ) = {\mathcal{A}}_{A}^{m} q_{A} \ln (q_{A}^{2} ) + {\mathcal{A}}_{A}^{d} dq_{A} \ln (dq_{A}^{2} )$$

(3.187)

For an oxygen atom, ${\mathcal{A}}_{A}^{m}$ is defined in terms of the monomer bond angle and bond distances as follows:

$$ {\mathcal{A}}_{{\rm O}}^{m} = - 2A_{E{\rm O}} \sin^{2} (\theta )\exp \left( - \frac{{r_{0}^{2}}}{2}(r_{{\rm OH}1} - r_{{{\rm OH}}2} )^{2} \sin^{2} (\theta )\right), $$

(3.188)

while for a hydrogen atom, ${\mathcal{A}}_{A}^{m}$ depends on the number of hydrogen bonds (L _OH) that the hydrogen atom (H) is involved with. Further, the index ‘s’ in (3.189) refers to the oxygen atom in the monomer to which H is associated with. $\theta_{u{{\rm H}}s}$ refers to the angle between the O_s–H bond (within the monomer) and the respective hydrogen bonds that H forms with other O_u atoms that belong to neighboring monomers.

$${\mathcal{A}}_{\text{H}}^{m} = {\mathcal{A}}_{E\text{H}} \left(1 + \eta \sum\limits_{u = 1}^{{L_{\text{OH}} }}\right) {\left[ {\left( \exp \left( - 2\left(1 + \cos \left(\theta_{u\text{H}s} \right)\right)^{2} \right)\right)\left( {1 - \tan h \left(\frac{{r_{u\text{H}} - r_{D} }}{{t_{2} }}\right)} \right)} \right]} .$$

(3.189)

Finally, $\Phi _{ij}$ is given below:

$$\begin{aligned}\Phi _{\text{OO}} & = a_{\text{OO}} \exp ( - 4r_{\text{OO}} ) \\ \Phi _{\text{OH}} & = 2\left[ {a_{\text{OH}} r_{\text{OH}} + b_{\text{OH}} \exp ( - r_{\text{OH}} ) + \frac{{C_{\text{OH}} }}{{r_{\text{OH}}^{24} }}} \right]S(r_{\text{OH}} - r_{cut} ) \\ \Phi _{\text{HH}} & = 2a_{\text{HH}} \exp ( - 2r_{\text{HH}} )S(r_{\text{HH}} - H_{cut} ). \\ \end{aligned}$$

(3.190)

Tables 3.13 and 3.14 provides the corresponding potential parameters.

Table 3.13 Potential parameters (part I)

Table 3.14 Potential parameters (part II)

For the two systems, the charge and the other potential parameters, while parameterized with respect to ab initio data, were selected in an ad hoc fashion. In order to provide a more streamlined approach, a more systematic approach leading to a more intuitive functional form is proposed as noted below. Future development of EDD-Q potentials will be based on these functional forms (3.191–3.199).

As discussed before, for a bond formed between atoms A and B, the bond-charge ($q_{AB}^{bond}$) is a function of the number of nearest neighbors of A($L_{A}$) and B($L_{B}$), the interatomic distance ($R_{AB}$), and the bond-angles that arise due to the remaining nearest neighbors of A and B ($\theta_{BAC}$ and $\theta_{ABC'}$), where C and C′ represent the neighbors of A and B respectively. Note the qualitative similarity of the functional forms for charges (3.191–3.198) to the multipole expansion used commonly to express the electrostatic potentials that arise from a distribution of charges.

$$q_{A} = \sum\limits_{B = 1}^{{L_{A} }} {q_{AB}^{bond} }$$

(3.191)

$$q_{AB}^{bond} = \left[ {q_{AB} + q_{BA} } \right]$$

(3.192)

$$q_{AB} = \frac{1}{2}\left[ {\frac{1}{{L_{A} }}q_{AB}^{S} (R_{AB} ) + \sum\limits_{C \ne B}^{{L_{A} }} {\Delta q(R_{AB} ,\theta_{BAC} )} } \right]$$

(3.193)

$${\text{if}}\, L_{A} = 1,\ q_{AB} = q_{BA}$$

$$q_{AB}^{S} (R_{AB} ) = \sum\limits_{p} {\frac{{\mathcal{A}_{p} }}{{R_{AB}^{p} }}} ;p > 2$$

(3.194)

$$\Delta q(R_{AB} ,\theta_{BAC} ) = f^{A} (\theta_{BAC} ,R_{AB} ) + g^{A} (R_{AB})$$

(3.195)

$$f^{A} (\theta ,r) = w^{A} (\theta )y(r)$$

(3.196)

$$w^{A} (\theta) = \left[ {u_{A}^{f} + \sum\limits_{p} {C_{p}^{A} } \cos^{p} (p\theta );p > 0} \right] \,$$

(3.197)

$$y(R) = \left[ {u_{A}^{e} + \sum\limits_{p} {\frac{{B_{p} }}{{R_{AB}^{p} }};p > 2} } \right]$$

$$g^{A} (r) = u_{A}^{g} + \sum\limits_{p} {\frac{{D_{p}^{A} }}{{R_{AB}^{p} }};p > 2}$$

(3.198)

The EDD-Q Hamiltonian which consists of a Coulombic term, a 2-body term (ɸ) and an embedding term (F(q)) are given below. F(q) consists of three contributions; the most important of the three terms is the second term, which can be correlated to the ‘energy-cost’ for embedding a bond that is formed between pairs of atoms.

$$\begin{aligned} \bar E & = \sum\limits_{A} {\bar E_{A} } \\ \bar E_{A} & = \frac{1}{2}\sum\limits_{B \ne A} {\frac{{q_{A} q_B}}{{R_{AB} }} + } F(q_{A} ) + \frac{1}{2}\sum\limits_{B \ne A} {\phi_{AB} (R_{AB} )} \\ F(q_{A} ) & = M_{1} q_{A}^{2} \ln (q_{A}^{2} ) + M_{2} \sum\limits_{B \ne A} {q_{AB} \ln (q_{AB}^{2} )} + M_{3} \sum\limits_{B \ne A} {h_{AB} q_{AB}^{2} \ln (q_{AB}^{2} )} \\ h_{AB} & = \sum\limits_{C \ne B}^{L_{A}} {\left[ {1 - \left( {h^{(1)} \cos (\theta_{BAC} ) + h^{(2)} \cos (2\theta_{BAC} ) + h^{(3)} \cos (3\theta_{BAC} )} \right)} \right]} \\ \phi_{AB} & = \frac{{T_{AB} }}{{R_{AB}^{12} }} - \frac{{S_{AB} }}{{R_{AB}^{6} }} \\ \end{aligned}$$

(3.199)