Function for the calculation of the trasformed relaxation hessians.
The trasformed relaxation hessians
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Data structure: self.data.d2ri_prime
Dimension: 3D, (parameters, parameters, relaxation data)
Type: Numeric 3D matrix, Float64
Dependencies: self.data.jw, self.data.djw, self.data.d2jw
Required by: self.data.d2ri
Formulae (a hessian matrix is symmetric)
~~~~~~~~
Components
~~~~~~~~~~
Dipolar
~~~~~~~
1 / mu0 \ 2 (gH.gN.h_bar)**2
d = - . | ---- | . ----------------
4 \ 4.pi / <r**6>
3 / mu0 \ 2 (gH.gN.h_bar)**2
d' = - - . | ---- | . ----------------
2 \ 4.pi / <r**7>
21 / mu0 \ 2 (gH.gN.h_bar)**2
d" = -- . | ---- | . ----------------
2 \ 4.pi / <r**8>
CSA
~~~
(wN.csa)**2
c = -----------
3
2.wN**2.csa
c' = -----------
3
2.wN**2
c" = -------
3
R1()
~~~~
J_R1_d = J(wH-wN) + 3J(wN) + 6J(wH+wN)
dJ(wH-wN) dJ(wN) dJ(wH+wN)
J_R1_d_prime = --------- + 3 . ------ + 6 . ---------
dJj dJj dJj
d2J(wH-wN) d2J(wN) d2J(wH+wN)
J_R1_d_2prime = ---------- + 3 . ------- + 6 . ----------
dJj.dJk dJj.dJk dJj.dJk
J_R1_c = J(wN)
dJ(wN)
J_R1_c_prime = ------
dJj
d2J(wN)
J_R1_c_2prime = -------
dJj.dJk
R2()
~~~~
J_R2_d = 4J(0) + J(wH-wN) + 3J(wN) + 6J(wH) + 6J(wH+wN)
dJ(0) dJ(wH-wN) dJ(wN) dJ(wH) dJ(wH+wN)
J_R2_d_prime = 4 . ----- + --------- + 3 . ------ + 6 . ------ + 6 . ---------
dJj dJj dJj dJj dJj
d2J(0) d2J(wH-wN) d2J(wN) d2J(wH) d2J(wH+wN)
J_R2_d_2prime = 4 . ------- + ---------- + 3 . ------- + 6 . ------- + 6 . ----------
dJj.dJk dJj.dJk dJj.dJk dJj.dJk dJj.dJk
J_R2_c = 4J(0) + 3J(wN)
dJ(0) dJ(wN)
J_R2_c_prime = 4 . ----- + 3 . ------
dJj dJj
d2J(0) d2J(wN)
J_R2_c_2prime = 4 . ------- + 3 . -------
dJj.dJk dJj.dJk
sigma_noe()
~~~~~~~~~~~
J_sigma_noe = 6J(wH+wN) - J(wH-wN)
dJ(wH+wN) dJ(wH-wN)
J_sigma_noe_prime = 6 . --------- - ---------
dJj dJj
d2J(wH+wN) d2J(wH-wN)
J_sigma_noe_2prime = 6 . ---------- - ----------
dJj.dJk dJj.dJk
Spectral density parameter - Spectral density parameter
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1()
------- = d . J_R1_d_2prime + c . J_R1_c_2prime
dJj.dJk
d2R2() d c
------- = - . J_R2_d_2prime + - . J_R2_c_2prime
dJj.dJk 2 6
d2sigma_noe()
------------- = d . J_sigma_noe_2prime
dJj.dJk
Spectral density parameter - Chemical exchange
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1() d2R2() d2sigma_noe()
-------- = 0 , -------- = 0 , ------------- = 0
dJj.dRex dJj.dRex dJj.dRex
Spectral density parameter - CSA
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1()
-------- = c'. J_R1_c_prime
dJj.dcsa
d2R2() c'
-------- = - . J_R2_c_prime
dJj.dcsa 6
d2sigma_noe()
------------- = 0
dJj.dcsa
Spectral density parameter - Bond length
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1()
------ = d'. J_R1_d_prime
dJj.dr
d2R2() d'
------ = - . J_R2_d_prime
dJj.dr 2
d2sigma_noe()
------------- = d' . J_sigma_noe_prime
dJj.dr
Chemical exchange - Chemical exchange
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1() d2R2() d2sigma_noe()
------- = 0 , ------- = 0 , ------------- = 0
dRex**2 dRex**2 dRex**2
Chemical exchange - CSA
~~~~~~~~~~~~~~~~~~~~~~~
d2R1() d2R2() d2sigma_noe()
--------- = 0 , --------- = 0 , ------------- = 0
dRex.dcsa dRex.dcsa dRex.dcsa
Chemical exchange - Bond length
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1() d2R2() d2sigma_noe()
------- = 0 , ------- = 0 , ------------- = 0
dRex.dr dRex.dr dRex.dr
CSA - CSA
~~~~~~~~~
d2R1()
------- = c" . J_R1_c
dcsa**2
d2R2() c"
------- = - . J_R2_c
dcsa**2 6
d2sigma_noe()
------------- = 0
dcsa**2
CSA - Bond length
~~~~~~~~~~~~~~~~~
d2R1() d2R2() d2sigma_noe()
------- = 0 , ------- = 0 , ------------- = 0
dcsa.dr dcsa.dr dcsa.dr
Bond length - Bond length
~~~~~~~~~~~~~~~~~~~~~~~~~
d2R1()
------ = d" . J_R1_d
dr**2
d2R2() d"
------ = - . J_R2_d
dr**2 2
d2sigma_noe()
------------- = d" . J_sigma_noe
dr**2
|
