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# Module rdc

source code

Module for the calculation of RDCs.

 Functions
float
 ave_rdc_5D(dj, vect, N, A, weights=None) Calculate the ensemble average RDC, using the 5D tensor. source code
float
 ave_rdc_tensor(dj, vect, N, A, weights=None) Calculate the ensemble average RDC, using the 3D tensor. source code
float
 ave_rdc_tensor_dDij_dAmn(dj, vect, N, dAi_dAmn, weights=None) Calculate the ensemble average RDC gradient element for Amn, using the 3D tensor. source code
float
 ave_rdc_tensor_pseudoatom(dj, vect, N, A, weights=None) Calculate the ensemble and pseudo-atom averaged RDC, using the 3D tensor. source code
float
 ave_rdc_tensor_pseudoatom_dDij_dAmn(dj, vect, N, dAi_dAmn, weights=None) Calculate the ensemble and pseudo-atom average RDC gradient element for Amn, using the 3D tensor. source code
float
 rdc_tensor(dj, mu, A) Calculate the RDC, using the 3D alignment tensor. source code
 Variables
__package__ = `'lib.alignment'`

Imports: dot, sum

 Function Details

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

source code

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 or None) - 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

### ave_rdc_tensor_pseudoatom(dj, vect, N, A, weights=None)

source code

Calculate the ensemble and pseudo-atom averaged 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 RDC for each pseudo-atom is calculated and then averaged. The formula for this ensemble and pseudo-atom average RDC value is:

```                    _N_        _M_
\        1 \         T
Dij(theta)  = dj  >  pc . -  >  mu_jcd . Ai . mu_jcd,
/__      M /__
c=1        d=1
```

where:

• i is the alignment tensor index,
• j is the index over spins,
• c is the index over the states or multiple structures,
• d is the index over the pseudo-atoms,
• theta is the parameter vector,
• dj is the dipolar constant for spin j,
• N is the total number of states or structures,
• M is the total number of pseudo-atoms,
• pc is the population probability or weight associated with state c (equally weighted to 1/N if weights are not provided),
• mu_jcd is the unit vector corresponding to spin j, state c, and pseudo-atom d,
• 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` (list of float) - The dipolar constant for spin j.
• `vect` (numpy matrix) - The unit vector matrix. The first dimension corresponds to the structural index, the second dimension to the pseudo-atom index, and the third 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 or None) - The weights for each member of the ensemble (the last member need not be supplied).
Returns: float
The average RDC value.

### ave_rdc_tensor_pseudoatom_dDij_dAmn(dj, vect, N, dAi_dAmn, weights=None)

source code

Calculate the ensemble and pseudo-atom 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_        _M_
dDij(theta)       \        1 \         T   dAi
-----------  = dj  >  pc . -  >  mu_jcd . ---- . mu_jcd,
dAmn           /__      M /__          dAmn
c=1        d=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,
• d is the index over the pseudo-atoms,
• 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,
• M is the total number of pseudo-atoms,
• 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

### 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.

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