UniformGradient.hpp

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 * [7] Lamichhane, Newman & Gezelter, J. Chem. Phys. 141, 134110 (2014).
 * [8] Bhattarai, Newman & Gezelter, Phys. Rev. B 99, 094106 (2019).
 */
 
/*! \file perturbations/UniformGradient.hpp
    \brief Uniform Electric Field Gradient perturbation
*/
 
#ifndef PERTURBATIONS_UNIFORMGRADIENT_HPP
#define PERTURBATIONS_UNIFORMGRADIENT_HPP
 
#include "brains/ForceModifier.hpp"
#include "brains/SimInfo.hpp"
 
namespace OpenMD {
 
  //! Applies a uniform electric field gradient to the system
  /*! The gradient is applied as an external perturbation.  The user specifies
 
  \code{.unparsed}
    uniformGradientStrength = c;
    uniformGradientDirection1 = (a1, a2, a3)
    uniformGradientDirection2 = (b1, b2, b3);
  \endcode
 
    in the .omd file where the two direction vectors, \f$ \mathbf{a} \f$
    and \f$ \mathbf{b} \f$ are unit vectors, and the value of \f$ g \f$
    is in units of \f$ V / \AA^2 \f$
 
    The electrostatic potential corresponding to this uniform gradient is
 
    \f$ \phi(\mathbf{r})  = - \frac{g}{2} \left[
    \left(a_1 b_1 - \frac{\cos\psi}{3}\right) x^2
    + (a_1 b_2 + a_2 b_1) x y + (a_1 b_3 + a_3 b_1) x z +
    + (a_2 b_1 + a_1 b_2) y x
    + \left(a_2 b_2 - \frac{\cos\psi}{3}\right) y^2
    + (a_2 b_3 + a_3 b_2) y z + (a_3 b_1 + a_1 b_3) z x
    + (a_3 b_2 + a_2 b_3) z y
    + \left(a_3 b_3 - \frac{\cos\psi}{3}\right) z^2 \right] \f$
 
    where \f$ \cos \psi = \mathbf{a} \cdot \mathbf{b} \f$.  Note that
    this potential grows unbounded and is not periodic.  For these reasons,
    care should be taken in using a Uniform Gradient with point charges.
 
    The corresponding field is:
 
    \f$ \mathbf{E} = \frac{g}{2} \left( \begin{array}{c}
    2\left(a_1 b_1 - \frac{\cos\psi}{3}\right) x +  (a_1 b_2 + a_2 b_1) y
    + (a_1 b_3 + a_3 b_1) z  \\
    (a_2 b_1 + a_1 b_2)  x + 2 \left(a_2 b_2 - \frac{\cos\psi}{3}\right) y
    + (a_2 b_3 + a_3 b_2) z \\
    (a_3 b_1 + a_1 b_3) x +  (a_3 b_2 + a_2 b_3) y
    + 2 \left(a_3 b_3 - \frac{\cos\psi}{3}\right) z \end{array} \right) \f$
 
    The field also grows unbounded and is not periodic.  For these reasons,
    care should be taken in using a Uniform Gradient with point dipoles.
 
    The corresponding field gradient is:
 
    \f$ \nabla \mathbf{E} = \frac{g}{2} \left( \begin{array}{ccc}
    2\left(a_1 b_1 - \frac{\cos\psi}{3}\right)  &
    (a_1 b_2 + a_2 b_1) & (a_1 b_3 + a_3 b_1) \\
    (a_2 b_1 + a_1 b_2)  & 2 \left(a_2 b_2 - \frac{\cos\psi}{3}\right) &
    (a_2 b_3 + a_3 b_2) \\
    (a_3 b_1 + a_1 b_3) & (a_3 b_2 + a_2 b_3) &
    2 \left(a_3 b_3 - \frac{\cos\psi}{3}\right) \end{array} \right) \f$
 
    which is uniform everywhere.
 
    The uniform field gradient applies a force on charged atoms,
    \f$ \mathbf{F} = C \mathbf{E}(\mathbf{r}) \f$.
    For dipolar atoms, the gradient applies both a potential,
    \f$ U = -\mathbf{D} \cdot \mathbf{E}(\mathbf{r}) \f$, a force,
    \f$ \mathbf{F} = \mathbf{D} \cdot \nabla \mathbf{E} \f$, and a torque,
    \f$ \mathbf{\tau} = \mathbf{D} \times \mathbf{E}(\mathbf{r}) \f$.
 
    For quadrupolar atoms, the uniform field gradient exerts a potential,
    \f$ U = - \mathsf{Q}:\nabla \mathbf{E} \f$, and a torque
    \f$ \mathbf{F} = 2 \mathsf{Q} \times \nabla \mathbf{E} \f$
 
  */
  class UniformGradient : public ForceModifier {
  public:
    UniformGradient(SimInfo* info);
 
  private:
    void modifyForces() override;
    void initialize();
 
    bool initialized;
    bool doUniformGradient;
    bool doParticlePot;
 
    Globals* simParams;
    Mat3x3d Grad_;
    Vector3d a_, b_;
    RealType g_, cpsi_;
  };
}  // namespace OpenMD
 
#endif