Density matrix element

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

The oharge- and exohange-density matrix elements in the AO basis are ... 

The two-eleetron density matrix elements are given in similar fashion ... 

Density matrix element in AO basis Matrix eigenvalue van der Waals parameter Dielectric constant... 

An alternative relax-and-drive procedure can be based on a strictly unitary treatment where the advance from Iq to t is done with a norm-conserving propagation such as provided by the split-operator propagation technique.(49, 50) This however is more laborious, and although it conserves the norm of the density matrix, it is not necessarily more accurate because of possible inaccuracies in the individual (complex) density matrix elements. It can however be used to advantage when the dimension of the density matrix is small and exponentiation of matrices can be easily done.(51, 52)... 

Within the MC-AFDF ADMA method, the management of multiple index assignments ofbasis orbitals and individual density matrix elements requires a series of index conversion relations. These relations are briefly reviewed below, using the notations of the original reference [143]. 

Let us investigate the change of the various components of the density matrix under propagation from time t to t + 5t. We first consider the action of jSfo- Due to this operator, the density matrix element p (t) attains a phase factor... 

Thus the density matrix elements take on the familiar form of a Gaussian with a shifted center. We can now find the one-particle density matrix element by... 

It is reasonable (but not necessarily correct ) to assign any electrons associated with a particular diagonal element, to that atom on which the basis function ( ), is located. It is also reasonable to assign electrons associated with off-diagonal elements P v, where both ( ), and ( )v reside on the same atom, to that atom. However, it is not apparent how to partition electrons from density matrix elements P, v where ( ) i and ( )vreside on different atoms. Mulliken provided a recipe. Give each atom half of the total. Very simple but completely arbitrary ... 

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