maths_fns.rdc

ave_rdc_5D(dj, vect, N, A, weights=None)

Calculate the ensemble average RDC, using the 5D tensor.

This function calculates the average RDC for a set of XH bond vectors from a structural ensemble, using the 5D vector form of the alignment tensor. The formula for this ensemble average RDC value is:

                _N_
                \ 
   Dij(theta) =  >  pc . RDC_ijc (theta),
                /__
                c=1

where:

i is the alignment tensor index,
j is the index over spins,
theta is the parameter vector consisting of the alignment tensor parameters {Axx, Ayy, Axy, Axz, Ayz} for each alignment,
c is the index over the states or multiple structures,
N is the total number of states or structures,
pc is the population probability or weight associated with state c (equally weighted to
RDC_ijc is the back-calculated RDC value for alignment tensor i, spin system j and structure c.

The back-calculated RDC is given by the formula:

   RDC_ijc(theta) = (x_jc**2 - z_jc**2)Axx_i + (y_jc**2 - z_jc**2)Ayy_i + 2x_jc.y_jc.Axy_i + 2x_jc.z_jc.Axz_i + 2y_jc.z_jc.Ayz_i.

Parameters:

dj (float) - The dipolar constant for spin j.
vect (numpy matrix) - The unit XH bond vector matrix. The first dimension corresponds to the structural index, the second dimension is the coordinates of the unit vector.
N (int) - The total number of structures.
A (numpy 5D vector) - The 5D vector object. The vector format is {Axx, Ayy, Axy, Axz, Ayz}.
weights (numpy rank-1 array) - The weights for each member of the ensemble (the last member need not be supplied).

Returns: float

The average RDC value.

ave_rdc_tensor(dj, vect, N, A, weights=None)

source code

Calculate the ensemble average RDC, using the 3D tensor.

This function calculates the average RDC for a set of XH bond vectors from a structural ensemble, using the 3D tensorial form of the alignment tensor. The formula for this ensemble average RDC value is:

                    _N_
                    \             T
   Dij(theta)  = dj  >  pc . mu_jc . Ai . mu_jc,
                    /__
                    c=1

where:

i is the alignment tensor index,
j is the index over spins,
c is the index over the states or multiple structures,
theta is the parameter vector,
dj is the dipolar constant for spin j,
N is the total number of states or structures,
pc is the population probability or weight associated with state c (equally weighted to 1/N if weights are not provided),
mu_jc is the unit vector corresponding to spin j and state c,
Ai is the alignment tensor.

The dipolar constant is defined as:

   dj = 3 / (2pi) d',

where the factor of 2pi is to convert from units of rad.s^-1 to Hertz, the factor of 3 is associated with the alignment tensor and the pure dipolar constant in SI units is:

          mu0 gI.gS.h_bar
   d' = - --- ----------- ,
          4pi    r**3

where:

mu0 is the permeability of free space,
gI and gS are the gyromagnetic ratios of the I and S spins,
h_bar is Dirac's constant which is equal to Planck's constant divided by 2pi,
r is the distance between the two spins.

Parameters:

dj (float) - The dipolar constant for spin j.
vect (numpy matrix) - The unit XH bond vector matrix. The first dimension corresponds to the structural index, the second dimension is the coordinates of the unit vector.
N (int) - The total number of structures.
A (numpy rank-2 3D tensor) - The alignment tensor.
weights (numpy rank-1 array) - The weights for each member of the ensemble (the last member need not be supplied).

Returns: float

The average RDC value.

ave_rdc_tensor_dDij_dAmn(dj, vect, N, dAi_dAmn, weights=None)

source code

Calculate the ensemble average RDC gradient element for Amn, using the 3D tensor.

This function calculates the average RDC gradient for a set of XH bond vectors from a structural ensemble, using the 3D tensorial form of the alignment tensor. The formula for this ensemble average RDC gradient element is:

                     _N_
   dDij(theta)       \             T   dAi
   -----------  = dj  >  pc . mu_jc . ---- . mu_jc,
      dAmn           /__              dAmn
                     c=1

where:

i is the alignment tensor index,
j is the index over spins,
m, the index over the first dimension of the alignment tensor m = {x, y, z}.
n, the index over the second dimension of the alignment tensor n = {x, y, z},
c is the index over the states or multiple structures,
theta is the parameter vector,
Amn is the matrix element of the alignment tensor,
dj is the dipolar constant for spin j,
N is the total number of states or structures,
pc is the population probability or weight associated with state c (equally weighted to 1/N if weights are not provided),
mu_jc is the unit vector corresponding to spin j and state c,
dAi/dAmn is the partial derivative of the alignment tensor with respect to element Amn.

Parameters:

dj (float) - The dipolar constant for spin j.
vect (numpy matrix) - The unit XH bond vector matrix. The first dimension corresponds to the structural index, the second dimension is the coordinates of the unit vector.
N (int) - The total number of structures.
dAi_dAmn (numpy rank-2 3D tensor) - The alignment tensor derivative with respect to parameter Amn.
weights (numpy rank-1 array) - The weights for each member of the ensemble (the last member need not be supplied).

Returns: float

The average RDC gradient element.

rdc_tensor(dj, mu, A)

source code

Calculate the RDC, using the 3D alignment tensor.

The RDC value is:

                          T
   Dij(theta)  = dj . mu_j . Ai . mu_j,

where:

i is the alignment tensor index,
j is the index over spins,
theta is the parameter vector,
dj is the dipolar constant for spin j,
mu_j i the unit vector corresponding to spin j,
Ai is the alignment tensor.

The dipolar constant is defined as:

   dj = 3 / (2pi) d',

where the factor of 2pi is to convert from units of rad.s^-1 to Hertz, the factor of 3 is associated with the alignment tensor and the pure dipolar constant in SI units is:

          mu0 gI.gS.h_bar
   d' = - --- ----------- ,
          4pi    r**3

where:

mu0 is the permeability of free space,
gI and gS are the gyromagnetic ratios of the I and S spins,
h_bar is Dirac's constant which is equal to Planck's constant divided by 2pi,
r is the distance between the two spins.

Parameters:

dj (float) - The dipolar constant for spin j.
mu (numpy rank-1 3D array) - The unit XH bond vector.
A (numpy rank-2 3D tensor) - The alignment tensor.

Returns: float

The RDC value.

Module rdc

ave_rdc_5D(dj, vect, N, A, weights=None)

ave_rdc_tensor(dj, vect, N, A, weights=None)

ave_rdc_tensor_dDij_dAmn(dj, vect, N, dAi_dAmn, weights=None)

rdc_tensor(dj, mu, A)