Perfect transferability---an unattainable limit

While known quantum mechanical relationships require more information than is contained in just p(r) for the determination of the energy, only a portion of the information contained in the one-density matrix is required for this purpose (Bader 1980). The one-density matrix is a function of six variables. The physical information of F (r, r ) is however, contained in the neighbourhood of its diagonal elements r = r. To establish this, one expresses r (r, r ) in a new system of coordinates defined by 

Each term in eqn (6.95) has a direct physical interpretation when the derivatives are evaluated after expressing F in terms of its natural orbital expansion. Retaining the same ordering of terms as appears in eqn (6.95), this yields 

The terms in the expansion derived from derivatives with respect to X are identical to those obtained by taking the corresponding derivatives of the charge density p(X) itself. The dyadic Wp is the Hessian matrix of p, whose eigenvectors and eigenvalues determine the properties of the critical points in the charge distribution. The trace of this term is the Laplacian of the charge density, V p. 

The derivatives with respect to x sample the off-diagonal behaviour of F and generate terms related to the current density j and the quantum stress tensor er. The first-order term is proportional to the current density, and this vector field is the x complement of the gradient vector field Vp. The second-order term is proportional to the stress tensor. Considered as a real symmetric matrix, its eigenvalues and eigenvectors will characterize the critical points in the vector field J and its trace determines the kinetic energy densities jK(r) and G(r). The cross-term in the expansion is a dyadic whose trace is the divergence of the current density. 

The second-order expansion given in eqn (6.96) recovers all of the physical quantities needed to describe a quantum system and determine its properties the charge density p and its gradient vector field Vp define atoms and determine many of their properties in a stationary state the current density determines the system s magnetic properties and the change in p in a time-dependent system and, finally, the stress tensor determines the local and average mechanical properties of the system. Thus, one does not need all the 

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