External potentials

Form of the functional

W-SLDA codes minimize functional of the generic form:

$$E=\int \mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)\,d^3r-\sum_{\sigma}\int\left(\mu_{\sigma}-V_{\sigma}^{\textrm{(ext)}}(r)\right)n_{\sigma}(r)\,d^3r\\-\int\left(\Delta^{\textrm{(ext)}}(r)\nu^*(r)+\textrm{h.c.}\right)d^3r-\sum_{\sigma}\int \vec{v}_{\sigma}^{\textrm{(ext)}}(r)\cdot\vec{j}_{\sigma}(r)\,d^3r$$

where:

• $\mathcal{E}_{\textrm{edf}}(n,\nu,\ldots)$ is energy density functional which defines the physical system,
• $V_{\sigma}^{\textrm{(ext)}}(r)$ is spin dependent external potential, and $\mu_{\sigma}$ are chemical potentials (Lagrange multipliers) for constraining particle number.
• $\Delta^{\textrm{(ext)}}(r)$ is external pairing potential,
• $\vec{v}_{\sigma}^{\textrm{(ext)}}(r)$ is external velocity field.

W-SLDA toolkit provides flexible framework that allows for custom definition of all these terms. User must provide body of following functions contained in file: problem-definition.h. Each function can be parametrized by [User defined parameters](User defined parameters).

Definition of the external potential $V_{\sigma}^{\textrm{(ext)}}(r)$

/** 
 * EXTERNAL POTENTIAL V_ext
 * @param ix x-coordinate from range [0,NX), to convert to Cartesian use: x = DX*(ix-NX/2)
 * @param iy y-coordinate from range [0,NY), to convert to Cartesian use: y = DY*(iy-NY/2),
 *           NOTE: in case of 1d code iy=0
 * @param iz z-coordinate from range [0,NZ), to convert to Cartesian use: z = DZ*(iz-NZ/2)
 *           NOTE: in case of 1d and 2d codes iz=0
 * @param it iteration number
 * @param spin spin indicator, value from set {SPINA,SPINB}
 * @param params array of input parameters, before call of this routine the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * @return value of the external potential V_spin(x,y,z)
 * */
double v_ext(int ix, int iy, int iz, int it, int spin, double *params, size_t extra_data_size, void *extra_data)
{
    // ADD HERE FORMULA FOR V_ext(r)
    double V_ext = 0.0;

    return V_ext; 
}

Definition of the external pairing potential $\Delta^{\textrm{(ext)}}(r)$

/** 
 * EXTERNAL PAIRING POTENTIAL Delta_ext
 * @param ix x-coordinate from range [0,NX), to convert to Cartesian use: x = DX*(ix-NX/2)
 * @param iy y-coordinate from range [0,NY), to convert to Cartesian use: y = DY*(iy-NY/2)
 *           NOTE: in case of 1d code iy=0
 * @param iz z-coordinate from range [0,NZ), to convert to Cartesian use: z = DZ*(iz-NZ/2)
 *           NOTE: in case of 1d and 2d codes iz=0
 * @param it iteration number
 * @param delta - value of delta computed self-consistently for given iteration it. 
 * @param params array of input parameters, before call of this routine the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * @return value of external pairing potential Delta_{ext}(x,y,z)
 * */
double complex delta_ext(int ix, int iy, int iz, int it, double complex delta, double *params, size_t extra_data_size, void *extra_data)
{
    // ADD HERE FORMULA FOR Delta_ext(r)
    double complex D_ext = 0.0 + I*0.0;

    return D_ext; 
}

Note: Due to technical reasons this function differs with respect to return type between st-wslda and td-wslda codes. Namely:

• st-wslda: return type must be C99 double complex
• td-wslda: return type must be compatible with CUDA Complex

Definition of the external velocity field $\vec{v}_{\sigma}^{\textrm{(ext)}}(r)$

/** 
 * EXTERNAL VELOCITY FIELD vec[v]_ext
 * @param ix x-coordinate from range [0,NX), to convert to Cartesian use: x = DX*(ix-NX/2)
 * @param iy y-coordinate from range [0,NY), to convert to Cartesian use: y = DY*(iy-NY/2)
 *           NOTE: in case of 1d code iy=0
 * @param iz z-coordinate from range [0,NZ), to convert to Cartesian use: z = DZ*(iz-NZ/2)
 *           NOTE: in case of 1d and 2d codes iz=0
 * @param it iteration number
 * @param spin spin indicator, value from set {SPINA,SPINB}
 * @param coordinate - Cartesian coordinate of the external velocity vector that should be computed, value from set {XAXIS, YAXIS, ZAXIS}
 *                     NOTE: for 1d code only XAXIS is requested, for 2d code XAXIS and YAXIS are requested.
 * @param params array of input parameters, before call of this routine the params array is processed by process_params() routine
 * @param extra_data_size size of extra_data in bytes, if extra_data size=0 the optional data is not uploaded
 * @param extra_data optional set of data uploaded by load_extra_data()
 * @return value of the external velocity vector v_ext(x,y,z)
 * */
double velocity_ext(int ix, int iy, int iz, int it, int spin, int coordinate, double *params, size_t extra_data_size, void *extra_data)
{
    // ADD HERE FORMULAS FOR vec{v}_ext=(vx, vy, vz)
    double v_ext;
    if(coordinate==XAXIS) v_ext=0.0;
    if(coordinate==YAXIS) v_ext=0.0;
    if(coordinate==ZAXIS) v_ext=0.0;

    return v_ext; 
}