The NS MMQ 2-site model

This is the numerical model for 2-site exchange for proton-heteronuclear SQ, ZQ, DQ and MQ CPMG data, as derived in (Korzhnev et al., 2004a,2005a,2004b).
It is selected by setting the model to `NS MMQ 2-site'.
The simple constraint
*p*_{A} > *p*_{B} is used to halve the optimisation space, as both sides of the limit are mirror image spaces.
Different sets of equations are used for the different data types.

The basic evolution matrices for single, zero and double quantum CPMG-type data for this model are

R_{2eff} = - log, |
(11.56) |

where
**M**_{A}(0) is proportional to the vector
[*p*_{A}, *p*_{B}]^{T} and

M_{A}(T_{relax}) = A_{±}A_{}A_{}A_{±}M_{A}(0) |
(11.57) |

The evolution matrix
**A** is defined as

A_{±} = e^{a±⋅τCPMG}, |
(11.58) |

where

a_{±} = . |
(11.59) |

For different data types
*Δω* is defined as:
*Δω* (^{15}N SQ-type data);
*Δω*^{H} (^{1}H SQ-type data);
*Δω*^{H} - *Δω* (ZQ-type data); and
*Δω*^{H} + *Δω* (DQ-type data).

The equation for the exchange process for multiple quantum CPMG-type data is

R_{2eff} = - logRe⋅AB + CD⋅, |
(11.60) |

where *T* is the constant time interval, and the matrices
**A**,
**B**,
**C**, and
**D** are dually defined.
When *n* is even, they are defined as

When *n* is odd, they are defined as

When *n* is zero, to avoid matrix powers of zero they are defined as

The
**M** matrices are defined as:

M_{j} = exp(m_{j}δ), |
(11.64) |

where 2*δ* is the spacing between successive 180^{o} pulses and where
The references for this model are:

For the model it is assumed that
R_{2, DQ}^{A} = R_{2, ZQ}^{A} = R_{2A}^{0} and
R_{2, DQ}^{B} = R_{2, ZQ}^{B} = R_{2B}^{0}.
The references for this model are:

More information about the NS MMQ 2-site model is available from:

- the relax wiki at http://wiki.nmr-relax.com/NS_MMQ_2-site,
- the API documentation at http://www.nmr-relax.com/api/3.2/lib.dispersion.ns_mmq_2site-module.html,
- the relaxation dispersion page of the relax website at http://www.nmr-relax.com/analyses/relaxation_dispersion.html#NS_MMQ_2-site.

The relax user manual (PDF), created 2020-08-26.