Difference of Chirality of the Electron Between Enantiomers of H 2 X 2
99
The stress tensor is known to classify the chemical bond [13, 14]. In the following,
we call only ̂ t
k
e (x) the spin torque, and the sum of the terms in the right-hand side is
called the torque for the spin. Our equation of motion for the spin is recently shown
to be derived from the spin vorticity principle in a sophisticated way [15].
One may wonder whether this new contribution disturbs the consistency between
experimental observations and a prediction by quantum mechanics. The expectation value of the zeta force is zero after the integration over the whole space, since
the zeta force density operator is given as the gradient form of the zeta potential.
Hence, contributions from the zeta force are considered to be negligible in past experiments. However, the contribution from the zeta force give a nonzero effect in a local
region, even after the integration over a restricted local region. Hence, the zeta force
is observable quantity if an experimental setup is carefully designed for this purpose.
In addition, the equation of motion from quantum field theory has another advantage over that from quantum mechanics. In a time-independent stationary state of
the electron spin. the spin torque and zeta force are canceled out with each other and
the torque for the spin is zero at any point. Hence in quantum field theory a local
picture of the spin dynamics can be correctly described. In quantum mechanics, any
local spin dynamics cannot be predicted. The Heisenberg equation of the spin gives
generically nonzero torque for a local region even for a spin stationary state. This
is because the expectation value of quantum mechanics is defined as the integration
over the whole space and hence local description is theoretically out of scope.
3 Computational Details
Our equation is defined in quantum field theory, and hence a state should also be
prepared in the theory. However, a generic state based on quantum field theory is
not available for our purpose, since most computation code is based on quantum
mechanics. Hence in this work we use wave functions derived from ordinary electronic structure computations as a substitution. With the usage of these wave functions, computations of physical quantities, parity-violating energy, zeta potential,
and so on, are performed by QEDynamics program package [16–18].
For ordinary electronic structure computations, we use DIRAC14 program package [19]. Four-component wave function by relativistic quantum mechanics can be
computed by this code, which is indispensable for the study of spin. The dyall.ae2z
basis set [20] is used for large components of all atoms. The small component
basis set is generated by restricted kinetic balance in the code. The effect of threecomponent vector potential is ignored in our calculations, since it is quantitatively
small for states by quantum mechanics computations [21]. The structure of H 2 X 2
molecules are determined as follows. First, the geometrical optimization computations are performed by Hartree-Fock computations with Dirac-Coulomb Hamiltonian. The internuclear lengths between heavy atoms X, 1.390 Å for oxygen atoms,
2.058 Å for sulfur atoms, 2.333 Å for selenium atoms, and 2.729 Å for tellurium
atoms, while the internuclear lengths between X and H atoms, 0.9439 Å for oxygen
99
The stress tensor is known to classify the chemical bond [13, 14]. In the following,
we call only ̂ t
k
e (x) the spin torque, and the sum of the terms in the right-hand side is
called the torque for the spin. Our equation of motion for the spin is recently shown
to be derived from the spin vorticity principle in a sophisticated way [15].
One may wonder whether this new contribution disturbs the consistency between
experimental observations and a prediction by quantum mechanics. The expectation value of the zeta force is zero after the integration over the whole space, since
the zeta force density operator is given as the gradient form of the zeta potential.
Hence, contributions from the zeta force are considered to be negligible in past experiments. However, the contribution from the zeta force give a nonzero effect in a local
region, even after the integration over a restricted local region. Hence, the zeta force
is observable quantity if an experimental setup is carefully designed for this purpose.
In addition, the equation of motion from quantum field theory has another advantage over that from quantum mechanics. In a time-independent stationary state of
the electron spin. the spin torque and zeta force are canceled out with each other and
the torque for the spin is zero at any point. Hence in quantum field theory a local
picture of the spin dynamics can be correctly described. In quantum mechanics, any
local spin dynamics cannot be predicted. The Heisenberg equation of the spin gives
generically nonzero torque for a local region even for a spin stationary state. This
is because the expectation value of quantum mechanics is defined as the integration
over the whole space and hence local description is theoretically out of scope.
3 Computational Details
Our equation is defined in quantum field theory, and hence a state should also be
prepared in the theory. However, a generic state based on quantum field theory is
not available for our purpose, since most computation code is based on quantum
mechanics. Hence in this work we use wave functions derived from ordinary electronic structure computations as a substitution. With the usage of these wave functions, computations of physical quantities, parity-violating energy, zeta potential,
and so on, are performed by QEDynamics program package [16–18].
For ordinary electronic structure computations, we use DIRAC14 program package [19]. Four-component wave function by relativistic quantum mechanics can be
computed by this code, which is indispensable for the study of spin. The dyall.ae2z
basis set [20] is used for large components of all atoms. The small component
basis set is generated by restricted kinetic balance in the code. The effect of threecomponent vector potential is ignored in our calculations, since it is quantitatively
small for states by quantum mechanics computations [21]. The structure of H 2 X 2
molecules are determined as follows. First, the geometrical optimization computations are performed by Hartree-Fock computations with Dirac-Coulomb Hamiltonian. The internuclear lengths between heavy atoms X, 1.390 Å for oxygen atoms,
2.058 Å for sulfur atoms, 2.333 Å for selenium atoms, and 2.729 Å for tellurium
atoms, while the internuclear lengths between X and H atoms, 0.9439 Å for oxygen