Stress tensor quantum

Any stationary state is seen to be brought about by the force balance W = —Wq. The quantum force is a function of the pressure potential, or stress tensor, f dp/p that produces inner forces in the continuum. At equilibrium, in stationary states, the potential energy remains constant, i.e. 

In our first simple example the electrostatic potential set up by CsCl is almost but not quite a minimal surface [10]. The reason is that the Coulomb electrostatic energy is only a part of the whole electromagnetic field. Two body, three and higher order, non-additive van der Waals interactions contribute to the complete field, distributed within the crystal. This leads one to expect that the condition that the stress tensor of the field is zero, as for soap films, yields the condition for equilibrium of the crystal. Precisely that condition is that for the existence of a minimal surface. Strictly speaking the minimal surface might be defined by the condition that the electromagnetic stress tensor is zero. But in any event, we see in this manner that the occurrence of minimal surfaces, should be a consequence of equilibrium (cf. Chapter 3,3.2.4). Indeed a statement of equilibrium may well be equivalent to quantum statistical mechanics. 

The derivation of eqn (5.27) also entails the vanishing of a surface term, one involving the quantum stress tensor. The stress tensor was first introduced in its relativistic form in 1927 by Schrodinger and its properties were later discussed by Pauli (1933). The derivation of eqn (5.27) follows that given by Pauli (1958). 

The stress tensor is a symmetric dyadic and its mathematical properties were reviewed at the end of Chapter 5. It has the dimensions of pressure, force/unit area, or, equivalently, of an energy density. The quantum stress tensor plays a dominant role in the description of the mechanical properties of an atom in a molecule and in the local mechanics of the charge density. 

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... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

Schweitz s representation of the quantum stress tensor a (r) in terms of the flux density operator acting on the momentum in Equation (19) makes clear its interpretation as a momentum flux density. Schweitz does not, however, consider how the surface flux virial in the quantum case, Zs or i ,s, may be related to the pv product. This, as demonstrated in the following section, has been accomplished using the atomic statement of the virial theorem [12]. 

All atomic properties are determined by an integration of a corresponding density over the basin of the atom. This is possible even for a property like the energy which involves two-electron interactions because the potential as well as the kinetic energy are expressible in terms of quantum mechanical stress tensor, a quantity which in turn in completely determined by the one-electron density matrix. 

General Extensions. - Bader applied ideas of AIM to the atomic force microscope (AFM). In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is an IAS. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach is contrasted with literature results that equate it to the Hellmann-Feynman forces exerted on the nuclei of the atoms in the tip. 

It is a great pleasure to dedicate this chapter to Professor Deb on his 70th birthday. Professor Deb has made very important contributions to quantum fluid dynamics. He introduced stress tensors to density functional theory. This chapter can be considered as an extension of his ideas to excited states. 

The study of the density functional theory by Hohenberg-Kohn has invoked the new idea of energy in terms of electron density. A natural outcome is the concept of energy density, which has been published in a recent paper of stress tensor and the references cited therein. Electronic stress tensor plays a very significant role as has been originally formulated and reviewed by PauU in his textbook of quantum mechanics. 

This is called the quantum electron spin vorticity principle the time evolution of the electron spin s is driven by the antisymmetric component of the electronic stress tensor x through the vorticity rot . The quantum electron spin vorticity principle is schematically shown in Figure 12.1. 

For future technology of spintronics and photonics, the interaction of chirality of electron spin with another particle, such as nucleus and photon, should play an important role. Furthermore, the general relativity has recently been of vital importance to our daily lives, particularly for ultra-high-precision communication with an artificial satellite (e.g., GPS). The intrinsic formulations of tlie quantum electron spin vorticity principle and the energy density concept presented in this paper should help us understand the importance of stress tensor in modeling of materials of technological importance and chemical reactions. 

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. 

All the nuclear CPs except the 3, -3 CPs of the hydrogen atoms corresponding to Vp(r) and VV dr) coincide with the nuclear positions. The latter are displaced 0.033 and 0.018 bohr towards the C nucleus, respectively. Feir) and F (r) are the Ehrenfest force derived fixrm the quantum stress tensor and Eq. 6.10, respectively. All distances in a.u... 

Keywords Chemical reaction prediction. Conceptual density functional theory, Ehrenfest force, Electronic stress tensor, Reaction force partitioning. Quantum theory of atoms in molecules... 

The use of the correspondence principle to define a quantum mechanical analog of the stress tensor in classical mechanics goes back to Schrodinger and Pauli [33, 34]. The interest of electronic structure theorists in the stress tensor has been episodic, starting with the rise of computational density functional theory in the... 

That is, when one defines the stress tensor, one must make a choice of gauge [55, 61-67]. This ambiguity is closely related to the well-known ambiguity in the definition of the local kinetic energy [64, 68-71] and arises because there are infinitely many ways to define quantum mechanical operators that correspond to a given classical observable in the 0 limit [72, 73]. In practice, all of the... 

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