The energy of chemical reaction is visualized in real space using the electronic energy density n E (r ជ) associated with the electron density n(r ជ). The electronic energy density n E (r ជ) is decomposed into the kinetic energy density n T (r ជ), the external potential energy density n V (r ជ), and the interelectron potential energy density n W (r ជ). Using the electronic energy density n E (r ជ) we can pick up any point in a chemical reaction system and find how the electronic energy E is assigned to the selected point. We can then integrate the electronic energy density n E (r ជ) in any region R surrounding the point and find out the regional electronic energy E R to the global E. The kinetic energy density n T (r ជ) is used to identify the intrinsic shape of the reactants, the electronic transition state, and the reaction products along the course of the chemical reaction coordinate. The intrinsic shape is identified with the electronic interface S that discriminates the region R D of the electronic drop from the region R A of the electronic atmosphere in the density distribution of the electron gas. If the R spans the whole space, then the integral gives the total E. The regional electronic energy E R in thermodynamic ensemble is realized in electrochemistry as the intrinsic Volta electric potential R and the intrinsic Herring-Nichols work function ⌽ R. We have picked up first a hydrogen-like atom for which we have analytical exact expressions of the relativistic kinetic energy density n T M (r ជ) and its nonrelativistic version n T (r ជ). These expressions are valid for any excited bound states as well as the ground state. Second, we have selected the following five reaction systems and show the figures of the n T (r ជ) as well as the other energy densities along the intrinsic reaction coordinates: a protonation reaction to He, addition reactions of HF to C 2 H 4 and C 2 H 2 , hydrogen abstraction reactions of NH 3 ϩ from HF and NH 3. Valence electrons possess their unique delocalized drop region remote from those heavily localized drop regions adhered to core electrons. The kinetic energy density n T (r ជ) and the tension density ជ S (r ជ) can vividly demonstrate the formation of the chemical bond. Various basic chemical concepts in these chemical reaction systems have been clearly visualized in real three-dimensional space.

Covalent bond describes electron pairing in between a pair of atoms and molecules. The space is partitioned in mutually disjoint regions by using a new concept of the electronic drop region R(D), atmosphere region R(A), and the interface S (Tachibana in J Chem Phys 115:3497-3518, 2001). The covalent bond formation is then characterized by a new concept of the spindle structure. The spindle structure is a geometrical object of a region where principal electronic stress is positive along a line of principal axis of the electronic stress that connects a pair of the R(D)s of atoms and molecules. A new energy density partitioning scheme is obtained using the Rigged quantum electrodynamics (QED). The spindle structure of the stress tensor of chemical bond has been disclosed in the course of the covalent bond formation. The chemical energy density visualization scheme is applied to demonstrate the spindle structures of chemical bonds in H2, C2H6, C2H4 and C2H2 systems. [Figure: see text]. Field theory of the energy density.

How the mode of bonding affects stability and reactivity of molecule on the frame of nonrelativistic limit of the rigged quantum electrodynamics using new indices for description of bond properties related to bond orders have been characterized here. These indices are in close relation with tensorial interpretation of bond that among others allows discriminating covalent bonds using spindle structure concept. The real three-dimensional space representation of new interaction energy density utilized in this study contribute to better understanding of interaction phenomena between atoms and molecules. The differences in reactivity and stabilities of molecules have their root in the redistribution of interaction energy density.

Covalent bond describes electron pairing in between a pair of atoms and molecules. In the current paper, the space is partitioned in mutually disjoint regions by using new concept of the electronic drop region R D , atmosphere region R A , and the interface S [Tachibana, A. J Chem Phys 2001, 115, 3497]. The covalent bond formation is then characterized by a new concept of the spindle structure. The spindle structure is a geometrical object of a region where principal electronic stress is positive along a line of principal axis of the electronic stress that connects a pair of the R D s of atoms and molecules. Using virial theorem of the rigged quantum electrodynamics (QED), a new energy density partitioning scheme is obtained using the kinetic energy densities. In the current paper, a concise judgment condition to identify the given unknown locality as a spindle structure of a covalent bond is formulated in the frame of the nonrelativistic limit of QED. A physical interpretation of the local dynamics of the chosen spindle structure is also guaranteed at the same time as the judgment of the generalized chemical reactivity in terms of the physical quantum mechanical law that is hierarchic and localized. A general frame to elucidate a feature structure and the character of the classification concerning the expression of the internal symmetry of chemical reaction to the chosen spindle structure class is given in a unified way.

Conceptual insights from the density functional theory have been enormously powerful in the fields of physics, chemistry and biology. A natural outcome is the concept of "energy density" as has been developed recently: drop region, spindle structure, interaction energy density. Under external source of electromagnetic fields, charged particles can be accelerated by Lorentz force. Dissipative force can make the state of the charged particles stationary. In quantum mechanics, the energy eigenstate is another rule of the stationary state. Tension density of quantum field theory has been formulated in such a way that it can compensate the Lorentz force density at any point of space-time. This formulation can give mechanical description of local equilibrium leading to the quantum mechanical stationary state. The tension density is given by the divergence of stress tensor density. Electronic spin can be accelerated by torque density derived from the stress tensor density. The spin torque density can be compensated by a force density, called zeta force density, which is the intrinsic mechanism describing the stationary state of the spinning motion of electron.