Statistical theory of g-strain

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab  

And a related way to express their interdependence is by means of their correlation coefficient rab  

If two random variables are nncorrelated, then both their covariance Cab and their correlation coefficient rab are equal to zero. If two random variables are fully correlated, then the absolute value of their covariance is C,J = cacb, and the absolute value of their correlation coefficient is unity rab = 1. A key point to note for our EPR linewidth theory to be developed is that two fully correlated variables can be fully positively correlated rab = 1, or fully negatively correlated rab = -1. Of course, if two random variables are correlated to some extent, then 0 Cab oacb, and 0 IrJ 1. 

We know that a model in which the principal values of the g tensor are random variables, leading to Equation 9.1, falls short of describing experimental data in detail. Therefore, we now expand the model as follows the random variables are the principal values of a physical entity that is characterized by a tensor in 3-D space, but 

It is precisely this variance of (g) that we are after, because its square root gives us the angular dependent linewidth. A general expression in matrix notation can be derived for the variance (Hagen et al. 1985c)  

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This chapter considers the distribution of spin Hamiltonian parameters and their relation to conformational distribution of biomolecular structure. Distribution of a g-value or g-strain leads to an inhomogeneous broadening of the resonance line. Just like the g-value, also the linewidth, W, in general, turns out to be anisotropic, and this has important consequences for powder patterns, that is, for the shape of EPR spectra from randomly oriented molecules. A statistical theory of g-strain is developed, and it is subsequently found that a special case of this theory (the case of full correlation between strain parameters) turns out to properly describe broadening in bioEPR. The possible cause and nature of strain in paramagnetic proteins is discussed. 

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