Quadrupole Model

The quadrupole model¹ works similarly to the dipole model. The traceless quadrupole moments of the format [ \(Q_{xx}\), \(Q_{yy}\), \(Q_{zz}\), \(Q_{xy}\), \(Q_{xz}\), \(Q_{yz}\) ] are predicted using atomic charges as latent variables.

The model_params can be set, and are listed below along with their default values.

Parameter	Default	Description
`max_quadrupole`	`False`	When set to a number, exclude quadrupoles above this value in loss function
`quad_scale`	`1`	Quadrupole unit during training
`use_quad_per_atom`	`False`	Use the per-atom quadrupole in place of total quadrupole in loss function
`log_quad_per_atom`	`True`	Log the per-atom quadrupole error, automatically enabled with `use_quad_per_atoms=True`
`use_quad_weight`	`False`	Scale the energy loss according to the `'d_weight'` Tensor in the dataset
`use_l2`	`False`	Include L2 regularization in loss
`quad_loss_multiplier`	`1`	Weight of quadrupole loss
`l2_loss_multiplier`	`1`	Weight of l2

Variants

There are different variants of quadrupole and quadrupole-quadrupole models available.

atomic charge (AC) model (AC_quadrupole_model)
atomic charge dipole-quadrupole (AC-DQ) model (AC_dipole-quadrupole_model)
AC+AD dipole-quadrupole model (AC_AD_dipole_quadrupole_model)

Dipole-quadrupole models

The dipole-quadrupole model fits the dipole and quadrupole of the inputs to the atomic charges and predicts dipoles and quadrupoles. The AC+AD dipole-quadrupole model uses a contribution from atomic dipoles for dipole prediction.

The dipole moment is expressed as

\[ \boldsymbol{\mu}(\boldsymbol{r}) = \sum_i q_i \boldsymbol{r}_i \]

The traceless quadrupole tensor is defined as

\[ Q'_{\alpha\beta}(\boldsymbol{r}) = \sum_i q_i \left(3 r_{\alpha i} r_{\beta i} - \delta_{\alpha\beta} |\boldsymbol{r}_i|^2 \right) \]

where \(q_i\) is the atomic charge and \(\boldsymbol{r}_i\) is the position vector of atom \(i\) and \(\alpha,\beta\) run over \(\{x,y,z\}\). This is one of the three conventions of traceless quadrupoles found in the literature.

Usage

The dipole-quadrupole models are compatible with PiNet2.

These models can be selected by changing name of the model in model_params (See overview for further instructions.).

The models trained on the QM9 dataset can be found in the PiNet2-DQ repository

Parameters

Parameter	Default	Description
`max_quadrupole`	`False`	When set to a number, exclude quadrupoles above this value in loss function
`quad_scale`	`1`	Quadrupole unit during training
`quad_unit`	`1`	Output unit of quadrupole during prediction
`vector_quadrupole`	`True`	Toggle whether to use scalar or vector quadrupole predictions, should be the same as quadrupole moment of the dataset
`charge_neutrality`	`True`	Enable charge neutrality
`neutral_unit`	`'system'`	If charge neutrality is applied, set the charge neutral unit. Choose `'system'` for system neutrality, or `'water_molecule'` for neutrality per water molecule
`regularization`	`True`	Enable regularization of the interaction prediction, only available for models containing the 'R' term
`use_quad_per_atom`	`False`	Use the per-atom quadrupole in place of total quadrupole in loss function
`log_quad_per_atom`	`True`	Log the per-atom quadrupole error, automatically enabled with `use_quad_per_atoms=True`
`use_quad_weight`	`False`	Scale the energy loss according to the `'d_weight'` Tensor in the dataset
`use_l2`	`False`	Include L2 regularization in loss
`quad_loss_multiplier`	`1`	Weight of quadrupole loss
`l2_loss_multiplier`	`1`	Weight of l2

Model specifications

pinn.models.AC_quadrupole_model

pinn.models.AC_dipole_quadrupole.py

pinn.models.AC_AD_dipole_quadrupole.py

¹ K. Muuga, L. Knijff, and C. Zhang, “Molecular electrostatic potentials from machine learning models for dipole and quadrupole predictions,” AI Sci. 2(2), 025002 (2026). ↩