Potentials scalar

To solve the electrical problem at the electrodes, two variables need to be found (i.e., the electrical potential (scalar) and the current density (vector)). By analogy, two variables are also needed to find the ionic flux and the potential distribution. The vector relationship is given by Ohm s law (3.10), while the scalar relation is provided by expression (3.5), which can be re-written as ... 

Equation (4.10a) is the electrostatic potential, scalar relativistic (SR), and spin-orbit (SO) terms of the ZORA Hamiltonian in the absence of electromagnetic fields. The remaining part, (4.10b) and (4.10c), represents the hyper-fine terms due to the presence of the nuclear magnetic moments, JIa and ps. 

Calculation based on modem theoretical HF cluster and full potential scalar relativistic linearized augmented plane w ave [13]. 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

Note that for potentials that depend only on the scalar distance r between the colliding particles, the amplitude y (9) does not depend on the azimuthal angle associated with the direction of observation. 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

In an Abelian theory [for which I (r, R) in Eq. (90) is a scalar rather than a vector function, Al=l], the introduction of a gauge field g(R) means premultiplication of the wave function x(R) by exp(igR), where g(R) is a scalar. This allows the definition of a gauge -vector potential, in natural units... 

Knowledge of the spatial dimensions of a molecule is insufficient to imderstand the details of complex molecular interactions. In fact, molecular properties such as electrostatic potential, hydrophilic/lipophilic properties, and hydrogen bonding ability should be taken into account. These properties can be classified as scalar isosurfaces), vector field, and volumetric properties. 

Besides molecular orbitals, other molecular properties, such as electrostatic potentials or spin density, can be represented by isovalue surfaces. Normally, these scalar properties are mapped onto different surfaces see above). This type of high-dimensional visualization permits fast and easy identification of the relevant molecular regions. 

The representation of molecular properties on molecular surfaces is only possible with values based on scalar fields. If vector fields, such as the electric fields of molecules, or potential directions of hydrogen bridge bonding, need to be visualized, other methods of representation must be applied. Generally, directed properties are displayed by spatially oriented cones or by field lines. 

For an irrotational, incompressible, and frictionless fluid flow there exists a scalar velocity potential 4> such that the velocity vector V is... 

X is the scalar distance between the solute molecule and the center of the imaginary membrane, with the LJ parameters of the solute used as reducing parameters. The residual chemical potential for a pure fluid (which would correspond to component 2 in its pure state at the state conditions of cell A) can then, for example, be found using the expression... 

Forces are vector quantities and the potential energy t/ is a scalar quantity. For a three-dimensional problem, the link between the force F and the potential U can be found exactly as above. We have... 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

The vector potential is not uniquely defined since the gradient of any scalar function may be added (the curl of a derivative is always zero). It is convention to select it as... 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

In the Hamiltonian conventionally used for derivations of molecular magnetic properties, the applied fields are represented by electromagnetic vector and scalar potentials [1,20] and if desired, canonical transformations are invoked to change the magnetic gauge origin and/or to introduce electric and magnetic fields explicitly into the Hamiltonian, see e.g. refs. [1,20,21]. Here we take as our point of departure the multipolar Hamiltonian derived in ref. [22] without recourse to vector and scalar potentials. 

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