Second-density-derivative tensor

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

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. 

In addition, we used Bader s atoms-in-molecules (AIM) theory [56,57] to help analyze some of the results. For convenience, we give here a very brief overview of this approach. According to the AIM theory, every chemical bond has a bond critical point at which the first derivative of the charge density, p(r), is zero. The (> r) topology is described by a real, symmetric, second-rank Hessian-of-/3(r) tensor, and the tensor trace is related to the bond interaction energy by a local expression of the virial theorem ... 

When the second order approximations to the pressure tensor and the heat flux vector are inserted into the general conservation equation, one obtains the set of PDEs for the density, velocity and temperature which are called the Burnett equations. In principle, these equations are regarded as valid for non-equilibrium flows. However, the use of these equations never led to any noticeable success (e.g., [28], pp. 150-151) [39], p. 464), merely due to the severe problem of providing additional boundary conditions for the higher order derivatives of the gas properties. Thus the second order approximation will not be considered in further details in this book. 

The elastic constant tensor is a 6 x 6 matrix that contains the second derivatives of the energy density with respect to external strain ... 

Abstract Several potentially useful scalar and vector fields that have been scarcely or even never used to date in Quantum Chemical Topology are defined, computed, and analyzed for a few small molecules. The fields include the Ehrenfest force derived from the second order density matrix, which does not show many of the spurious features encountered when it is computed from the electronic stress tensor, the exchange-correlation (xc) potential, the potential acting on one electron in a molecule, and the additive and effective energy densities. The basic features of the topology of some of these fields are also explored and discussed, paying attention to their possible future interest. 

The current density second-rank tensor [3, 15, 71, 75] associated with the magnetic-field induced current density / is a function of position defined all over the molecular domain by the derivative... 

One can perform a QTAIM-Uke analysis based on the Ehrenfest force, since F(r) also defines a vector field with a (divergent) maximum at the nucleus. For any given point in space, Tq, one can follow the force-ascent lines to a nucleus, and decide that this atom exerts more force on the point Fq than the other atoms in the system. One must be cautious, however, because of the occasional presence of nonnuclear minima in the Ehrenfest potential [9, 10]. That is, in the same way that one may draw analogies between the stress tensor and the second derivative of the density, ff (r) VV p(r), one may draw analogies between the Ehrenfest force and the... 

Here, represents the Cauchy stress tensor, p is the mass density, and ft and m, are the body forces and displacements in the i direction within a bounded domain Q. The two dots over the displacements indicate second derivative in time. The indices i and j in the subscripts represent the Cartesian coordinates x, y, and z. When a subscript follows a comma, this indicates a partial derivative in space with respect to the corresponding index. For the special case of elastic isotropic solids, the stress tensor can be expressed in terms of strains following Hooke s law of elasticity, and the strains, in turn, can be expressed in terms of displacements. The resulting expression for the stress tensor is... 

The electric field gradient (EFG) is a ground state property of solids that sensitively depends on the asymmetry of the electronic charge density near the probe nucleus. The EFG is defined as the second derivative of the electrostatic potential at the nucleus position written as a traceless tensor. A nucleus with a nuclear spin number / > 1 has a nuclear quadrupole moment (Q) that interacts with the EFG which originates from the nonspherical charge distribution surrounding this nucleus. This interaction... 

Atomic basis functions in quantum chemistry transform like covariant tensors. Matrices of molecular integrals are therefore fully covariant tensors e.g., the matrix elements of the Fock matrix are F v = (Xn F Xv)- In contrast, the density matrix is a fully contravariant tensor, P = (x IpIx )- This representation is called the covariant integral representation. The derivation of working equations in AO-based quantum chemistry can therefore be divided into two steps (1) formulation of the basic equations in natural tensor representation, and (2) conversion to covariant integral representation by applying the metric tensors. The first step yields equations that are similar to the underlying operator or orthonormal-basis equations and are therefore simple to derive. The second step automatically yields tensorially correct equations for nonorthogonal basis functions, whose derivation may become unwieldy without tensor notation because of the frequent occurrence of the overlap matrix and its inverse. 

---