Permanent Multipole#
Definitions and Units#
The electrostatic potential at r due to the charge distribution nearby is
Tinker uses a variable electric (in chgpot module) to represent the the factor \(1/(4\pi\epsilon_0)\). Its default magnitude is 332.063713, which is a constant defined by variable coulomb (in units module), and its units are kcal/mol Å/e2. The default value can be modified by the ELECTIRIC keyword.
Note
Should the value of coulomb documented here be out-dated and become inconsistent with our code, please send us a pull request.
Expanding \(1/|r-s|\) in Taylor series, \(4\pi\epsilon_0\phi(r)\) can be rewritten as
where three pairs of square brackets give a set of definitions of monopole (charge, C), dipole (D), and quadrupole moments (Q*), respectively. The units of the multipole moments used in Tinker parameter files and internal calculation are different.
Multipole |
Parameter Units |
Internal Units |
---|---|---|
Charge |
e |
e |
Dipole |
e Bohr |
e Å |
Quadrupole |
e Bohr2 |
e Å2 |
In addition to different units, the quadrupole moments in Tinker parameter files use what is traditionally called traceless quadrupole \(\Theta\) that has a different definition than Q*. The third term in (2) can be rewritten as
hence the traceless quadrupole can be defined as
It is easy to confirm that \(\sum_k^{x,y,z}(3 s_k s_k - s^2)=0\), thus,
Internally, Tinker scales \(\Theta\) by 1/3
so that the energy expression is the same as if we were using Q*.
Energy Torque Gradient#
Potential energy
Potential energy with discretized charge distribution in (3)
Distance
Pairwise (atoms 1 and 2) quadrupole energy
Multipoles
T matrix
The upper left 4×4 elements of \(T_{12}\)
In the EWALD summation, \(1/r^k\) terms will have different forms (Bn). Neverthelss, they are still connected through derivatives.
Non-EWALD |
EWALD |
---|---|
\(1/r\) |
\(B_0=\mathrm{erfc}(\alpha r)/r\) |
\(1/r^3\) |
\(B_1\) |
\(3/r^5\) |
\(B_2\) |
\(15/r^7\) |
\(B_3\) |
\(105/r^9\) |
\(B_4\) |
\(945/r^{11}\) |
\(B_5\) |
The Bn terms are related to the (complementary) Boys functions and (complementary) error functions. For \(x>0\) and \(n\ge 0\),
The Boys functions can be generated through upward and downward recursions
Energy, torque, and force
Terms |
Energy |
Torque |
Force |
---|---|---|---|
C |
\(\phi C\) |
N/A |
\(\phi' C\) |
D |
\(\phi' D\) |
\(\phi'\times D\) |
\(\phi'' D\) |
Q |
\(\phi'' Q\) |
\(\phi''\times Q\) |
\(\phi''' Q\) |
where \(\epsilon_{ijk}\) is the Levi-Civita symbol.
Reference: [6].