Electrons quantities

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

The determination of electronic quantities is preliminary to the study of molecular dynamics and radiative properties. 

An important property of the density matrices p and P, with and without spin, is that they provide simple and general expressions for the expectation values of all 1-electron quantities, for any type of system and any type of wavefunction thus the total potential and kinetic energies (not depending on spin) are expressible as... 

This section is based mainly on work published by Miller and Schafer17 and Hickman and Morgner.22 As in the preceding section, the intention is to show how the electronic quantities V0(R), F+(/ ), and T(R) can be used to calculate—this time quantum mechanically—observable quantities. 

Matrix elements of the operators of the interaction energy between two shells of equivalent electrons may be expressed, with the aid of the CFP, in terms of the corresponding two-electron quantities. Substituting in such formula the explicit expression for the two-electron matrix element, after a number of mathematical manipulations and using the definition of submatrix elements of operators composed of unit tensors, we get convenient expressions for the matrix elements in the case of two shells of equivalent electrons. The corresponding details may be found in [14], here we present only final results. 

Submatrix elements of operators (22.17) and (22.18) for the case of a subshell of equivalent electrons, due to their one-electron character, are simply proportional to the corresponding one-electron quantities, where the submatrix elements of the operators, composed of unit tensors (7.2), serve as the coefficients of proportionality, i.e. 

Operators of electronic transitions, except the third form of the Ek-radiation operator for k > 1, may be represented as the sums of the appropriate one-electron quantities (see (13.20)). Their matrix elements for complex electronic configurations consist of the sums of products of the CFP, 3n./-coefficients and one-electron submatrix elements. The many-electron part of the matrix element depends only on tensorial properties of the transition operator, whereas all pecularities of the particular operator are contained in its one-electron submatrix element. 

The third form of the Efc-transition operator contains even two-particle terms [77], therefore its matrix elements are expressed in terms of the corresponding two-electron quantities. Unfortunately, these matrix elements are very cumbersome, and therefore are of little use. For the special case of transitions in shell lN they were established in [171]. Submatrix elements of /c-transitions with relativistic corrections (4.18)-(4.20) and (4.22) are considered in detail for one-electron and many-electron configurations in [172],... 

The calculation of the tr energy can be carried out by replacing it by a mono-electronic quantity in the Hamiltonian. 

These laws are now known to be the result of the fact that electricity is composed of individual particles, the electrons. Quantity of... 

With the foregoing multiparticle perspective in mind, it is of interest to combine analysis and calculation to determine the extent to which 7)y, strictly speaking a many-electron quantity, may be cast as an effective one-particle (i.e., orbital) matrix element. where DA in this context denotes the donor and acceptor orbi-... 

As for the effect of concentration of the oxidant, we speculate that the difference of concentration polarization between oxygen and 3 mM hydrogen peroxide can be attributed to the difference between their diffusion coefficients or the concentration decrease of hydrogen peroxide caused by non-electrochemical decomposition. This is because the total electron quantities of the oxygen saturated solution and the 3 mM hydrogen peroxide solution are almost the same. [21, 33]... 

Weakly interacting systems — as is implied by their name — are expected to change only to certain extent in the complex formation. Using some one-electron quantities (including those related to LMOs) comparative studies can thus be done by exploiting transferability. 

The kinetic energy terms are representative one-electron quantities of molecular orbitals. Although the results are given for molecules, their transferable properties could be shown for weakly interacting systems (dimer of water, e.g.) as well. 

Calculated DFT properties listed in Table 1 were obtained from the fit of the ground-state potential energy curves to 12 points calculated around the energy minimum [32]. Dissociation energy has been corrected for basis set superposition error by a standard counterpoise technique. The local approximation to the exchange and correlation gives the best fit to bond distances, theoretical values differ by no more than 0.03 A (4%) from the experimental ones (see Table 1). Vibrational frequencies are also predicted to lie within 1 % off the experiment. One should remember, however, that other advanced quantum chemical methods give equally satisfactory results for these, basicaly one-electron quantities and that inclusion of nonlocal effects does not improve the DFT predictions. The dipole moment, fi, is much more sensitive... 

Indeed, as for any electronic quantity, the value of He actually depends on R. However, this dependence is usually weak. Equation (3.3.5) holds even if the independence of He is a linear function of the fi-centroid (Halevi, 1965) defined by... 

Perhaps the simplest electronic quantity of chemical interest is the total n-electron probability density—the charge density". The R matrix provides an algebraic representation of the total electronic charge density. 

The most important difference between the data supplied to lister and to scount is that the columns of perms must form a group and must not be redundant. This is simply because scount does not maintain a list and therefore cemnot discard repetitions. It is trivial to make the changes to scount (analogous to those in lister) which will make the function work in an identical way for the one-electron quantities. 

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