Let μ(t) be a time dependent vector defined within an arbitrary fixed frame F as
μ(t) = δ_{x}, δ_{y}, δ_{z}  (12.1) 
where δ_{i} is the time dependent direction cosine between the unit vector and the axis i of frame F. Key for understanding the statistical mechanics of a second rank rotational process is the time dependence of the outer product
P(t)  = μ(t)⊗μ(t),  (12.2) 
= .  (12.3) 
Assuming statistical mechanical ensemble averaging, the observable expected value of the matrix P(t) is a matrix which defines the ordering of the vector μ(t) within the frame F. This order matrix is
where
S_{ij}(t) = .  (12.5) 
Because of the symmetry S_{ij}(t) = S_{ji}(t), the order matrix has 6 unique elements.
Assuming that the time dependent process modulating μ(t) is much faster than the evolution period t_{max} of the observed physical interaction, for example the weak molecular alignment process which induces residual dipolar couplings (RDCs) and pseudocontact shifts (PCSs) in NMR, the order matrix which gives rise to the nonisotropic effect is equation 12.4c at t_{max} = ∞. Hence the nonzero order matrix is
S(∞) = .  (12.6) 
Let the frame C(t) be time dependent within an arbitrary fixed frame F. After a time period t the shift from C(0) to C(t) is given by the rotation
R(t) = ≡,  (12.7) 
where rotation matrix element c_{ij} is equivalent to the direction cosine δ_{ij} between axis i of C(t) and axis j of C(0). For second rank physical processes modulated by rotational motions, analogously to the outer product expected value of 12.4b, the time dependence of the process is governed by the outer product
This is a rank4, three dimensional rotational tensor defining the ordering of the frame C(t) after a period t within the original frame C(0). This is the definition of the second degree frame order tensor.
The matrix form of the second degree frame order tensor in rank2, 9D Kronecker product notation is
where
≡.  (12.10) 
This is a rank2, 3D order matrix of rank2, 3D order matrices. To see this, the T_{14} rank4 matrix transpose of Daeg^{(2)} in Kronecker product notation is
The 3D matrix in the top left corner is the ordering of the xaxis with itself, the central matrix is the ordering of the yaxis with itself, and the bottom right is the ordering of the zaxis with itself. The offdiagonal 3D matrices are the crosscorrelations between the three axes. Using the notation e_{x}, e_{y} and e_{z} for the orthogonal axis system of the time dependent frame C(t), the second degree frame order matrix can be written as
Daeg^{T14}(t) = .  (12.12) 
If the rank2, 3D order matrix between the axes A and B is denoted as
S_{AB}(t) = ,  (12.13) 
then the frame order matrix is
Daeg^{T14}(t) = .  (12.14) 
The frame order matrix is diagonally symmetric, as can be seen in the T_{14} transpose of the matrix in rank2, 9D Kronecker product form (equation 12.11, hence for the second degree frame order matrix there are 45 unique elements. For the 9D Kronecker product notation of equation 12.9, this transformed diagonal symmetry can be schematically represented as
When rotational symmetries are present in the time modulation of the frame C(t) then, according to Perrin (1936), the averages of the double products where an index appears only once is zero. In this case, the active frame order matrix elements are
This matrix consists of 15 unique elements. It is the weighted sum of the three rank4 identity matrices I_{1}, I_{2} and I_{3}.
According to Spencer (1980), the rank4 identity matrices are defined as
where δ_{ij} is the Kronecker delta and e_{i} are the axes. In general, the identity matrix is
I = λδ_{ij}δ_{kl} + μδ_{ik}δ_{jl} + νδ_{il}δ_{kj}e_{i}⊗e_{j}⊗e_{k}⊗e_{l}.

(12.17) 
Expanding 12.16a to 12.16c to 9D Kronecker product matrix form,
The identity matrices are related to each other via the rank4 matrix transposes
In the case of unrestricted motions, the time limits of the frame order matrix are
Daeg^{(2)}(t = 0) = I_{1},  (12.20) 
and
Daeg^{(2)}(t = ∞) = I_{2}.  (12.21) 
The rank4, 3D frame order tensor of equation 12.8 on page was derived for second order rotational physical processes. However this can be generalised for physical processes of all orders. The tensor power of the time dependent rotation matrix R(t) is defined as
R^{⊗n}(t)R(t)⊗^{ ... }⊗R(t),  (12.22) 
where the outer product is repeated n times. Therefore let the frame order tensor be defined as
where n is the order of the physical process. The rank of the 3D tensors is 2n. The first few frame order tensors of rank2, rank4, rank6, and rank8 are
In index and direction cosine notation,
The rotation matrices of the general frame order tensor of equation 12.23 can be decomposed into a time dependent and time independent component. The original frame F can be defined as the motional eigenframe of the system and a new arbitrary frame F' introduced. The forward rotation from the reference frame F' to the motional eigenframe F will be denoted as R_{eigen}. The rotation matrix decomposition is
R'(t) = R_{eigen}⋅R(t)⋅R_{eigen}^{T}.  (12.26) 
Hence the second degree frame order tensor is
Daeg^{(2)} = ,  (12.27) 
Using the mixed product property
AC⊗BD = (A⊗B)(C⊗D),  (12.28) 
the arbitrary frame, second degree frame order matrix is
Generalising from the 2 to the n ^{th}order, the generalised frame order tensor rotation is
For the modelling aspect of the frame order theory, one more rotation is required. In equation 12.30, it is assumed that the starting position for the moving rigid body is that of its motional average. However in the initial 3D structure, this is not the case and an additional rotation to the average position R_{ave} is required. Taking this into account, the generalised frame order tensor is defined as
Daeg^{(n)}(t) = R_{eigen}^{⊗n}⋅⋅R_{eigen}^{T⊗n}⋅R_{ave}^{T⊗n},  (12.31) 
where R_{eigen} is the eigenframe rotation matrix, R(t) is the time dependent rotation matrix, R_{ave} is the rotation from the average domain position to the motional eigenframe, and ⊗n is the n ^{th} tensor power. In applications to physical processes which require numerical integration, prerotating the rigid body by R_{ave} to the average position is equivalent but more numerically efficient. Therefore the R_{ave} can be dropped and equation 12.30 used instead.
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