Auto-correlation vector

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Linear representations are by far the most frequently used descriptor type. Apart from the already mentioned structural keys and hashed fingerprints, other types of information are stored. For example, the topological distance between pharmacophoric points can be stored [179, 180], auto- and cross-correlation vectors over 2-D or 3-D information can be created [185, 186], or so-called BCUT [187] values can be extracted from an eigenvalue analysis of the molecular adjacency matrix. 

General Random Rotation in one Plane. Shimizu has pointed out that an analysis by Kubo of the shape of spectral lines can be applied very siny>ly to this situation. If the angular velocity of the rotor at time / is cdf), its angular displacement since / = 0 is cofrO dr and the normalized auto-correlation function of a vector rotating with it is... 

Here, it is assumed that the distance r is a constant independent of time. After the time-consuming calculations for Equations (3.3)-(3.5) and (3.11), Ti, T2, and NOE can finally be expressed by the auto-correlation functions G (t) of the orientation functions F, which describe the random time fluctuation of the C—H vector, or by the spectral densities Jq(o)) that are the Fourier transforms of G,(t) with frequency <0 as follows ... 

The probability density function T(X X) can be any function which satisfies the condition that if T(X X) is not zero T(X X ) is also not zero. Computationally feasible functions are chosen as the probability density function T X X) for the computational convenience. The vector set X, generated with this algorithm has the serial correlation, which means that the sequential position vectors such as X,- and X,+i are correlated. Thus, the error estimation may be modified with the auto-correlation time [20, 21] in the practical QMC calculations. In addition, the modification to the central limit theorem is also proposed in recent QMC study [22]. 

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