# Induced Dipole¶

## Energy¶

\(\mu\) is the induced dipole in the external field *E*.
The induced field due to the induced dipole is \(E^u=-T\mu\), and the
induced dipole is proportional to the total field \(E^t\):

where \(\alpha\) is the polarizability. Defining \(\tilde{T}=\alpha^{-1}+T\), the induced dipole is the solution to the linear equation

The polarization energy is given by

On the right-hand side of (6):

the 1st term is the contribution from the external field;

the 2nd term is the mutual and self polarization energy.

Finally, the polarization energy is

## Gradient¶

With limited numerical precision, the solution to linear equation (5) cannot be fully precise:

The gradient of the induced dipole can be written in

and the polarization gradient is

only if the convergence of (8) is tight that \(\epsilon\) and \(\epsilon'\) terms will drop.

## Conjugate Gradient¶

Tinker uses the following Conjugate Gradient algorithm (C.G.) with a sparse
matrix preconditioner (denoted as *M*) [2]
to obtain the induced dipoles.
Related Tinker variables and routines are tabulated.

C.G. Terms |
Tinker variables and routines |
---|---|

\(\gamma\) |
a |

\(\beta\) |
b |

\(r\) |
rsd |

\(M r\) |
zrsd |

\(p\) |
conj |

\(\tilde{T} p\) |
vec |

\(-T\) |
ufield() |

\(M\) |
uscale() |

## Polarization Model: AMOEBA (Thole Damping 2)¶

AMOEBA force field adopts two polarization schemes, *d* and *p*, for the
external field due to the permanent multipoles, and a third scheme *u* for
mutual induced dipole interactions. Both *d* and *u* schemes are group-based.
The *p* scheme is atomic connectivity-based. Tinker uses C.G. iterations to
solve the following linear equations

and defines the polarization energy as

From an optimizational perspective, (9) is the minimum of the target function

whereas the way C.G. coded in Tinker is to solve the minimum of another target function

The difference in two target functions is usually negligible unless other loose convergence methods are used to compute the induced dipoles.

In the Thole damping model, a charge distribution \(\rho\) is used as a replacement for the point dipole model. AMOEBA adopts the second functional form

from paper [3], where *u* is the polarizability-scaled
distance. The electrostatic field and potential at distance *r* can be obtained
from Gauss’s law,

where \(\lambda_1\) serves as the \(B_0\) term in EWALD quadrupole interactions. \(\lambda_n\) terms are also related via derivatives

Thus,