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 SQ, ZD and DQ equations

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

$${\frac{{1}}{{T_{\textrm{relax}}}}} {\frac{{\mathbf{M}_A(T_{\textrm{relax}})}}{{\mathbf{M}_A(0)}}}$$ (11.56)

where M_A(0) is proportional to the vector [p_A, p_B]^T and

$$\left(\vphantom{ \mathbf{A_{\pm}}\mathbf{A_\mp}\mathbf{A_\mp}\mathbf{A_{\pm}} }\right. \mp \mp \left.\vphantom{ \mathbf{A_{\pm}}\mathbf{A_\mp}\mathbf{A_\mp}\mathbf{A_{\pm}} }\right)^{n}_{}$$ (11.57)

The evolution matrix A is defined as

A_± = e^{a_±⋅τ_CPMG}, (11.58)

where

$$\begin{pmatrix} -\textrm{k}_{\textrm{AB}}-\mathrm{R}_{\mathrm{2A}... ...}_{\textrm{BA}}\pm\imath\Delta\omega - \mathrm{R}_{\mathrm{2B}}^0 \end{pmatrix}$$ (11.59)

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

The MQ equations

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

$${\frac{{1}}{{T}}} \left\{\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pma... ...ot \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\end{pmatrix}\right] }\right. \left[\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0\en... ...ght) \cdot \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\end{pmatrix}}\right. {\frac{{0.5}}{{p_{\textrm{A}}}}} \begin{pmatrix}1&0\end{pmatrix} \left(\vphantom{ \mathbf{AB} + \mathbf{CD} }\right. \left.\vphantom{ \mathbf{AB} + \mathbf{CD} }\right) \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\end{pmatrix} \left.\vphantom{ \frac{0.5}{p_{\textrm{A}}} \begin{pmatrix}1&0\en... ...ght) \cdot \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\end{pmatrix}}\right] \left.\vphantom{ Re \left[ \frac{0.5}{p_{\textrm{A}}} \begin{pmat... ...t \begin{pmatrix}p_{\textrm{A}}\\ p_{\textrm{B}}\end{pmatrix}\right] }\right\}$$ (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_jδ), (11.64)

where 2δ 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_2A⁰ and R_{2, DQ}^B = R_{2, ZQ}^B = R_2B⁰. The references for this model are:

• Korzhnev et al. (2004a)
• Korzhnev et al. (2004b)
• Korzhnev et al. (2005a)

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.