Atomic state function

T. Zalandauskas, G. Gaigalas, Routines for transformation of atomic state functions from LS- to jj-coupled basis, Lithmnian J. Phys., 43, 15-26 (2003). 

The atomic state function is given by a linear combination of CSFs as shown... 

In our case of nuclear quadrupole coupling k takes on the value 2. When open shell atoms are treated relativistically a specific atomic state function rPJM) (ASF) has definite angular momentum J, Mj and parity P therefore being eigenfunction to the operators and... 

D. R. Hartree, Reports on Progress in Physics 11, 113 (1948) this survey is brought up to date in D. R. Hartree, The Calculation of Atomic Structures Wiley and Sons, New York, and Chapman and Hall, London, 1957. See also R. S. Knox, Bibliography of Atomic Wave Functions/ in Solid State Physics (Seitz and Turnbull, eds.), Academic Press, New York, 1957, Yol. 4, p. 413. 

Fieschi, R., and Lowdin, P.-O., "Atomic state wave functions, generated by projection operators."... 

The goodness of the PP representation can be checked by comparing the all-electron and PP orbital energies and relative stability of atomic states. The comparison is shown in Table 4, and is seen to be very satisfying. For a balanced treatment, also the carbon and oxygen atoms were treated by a PP, as described in previous work5.3d functions were not introduced in the sulphur basis set, mainly because they were not deemed necessary for the illustrative purposes of this chapter. Also, the derivation of a PP representation for polarization functions is not a straightforward matter. The next section is devoted to the discussion of this point. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

An independent estimate of the amount of p character of these bonds can be made with use of the assumption that a linear extrapolation of the low-lying vibrational energy levels (as indicated by the Morse potential function) will lead to the energy level of the atomic state involved in the bond. The equation... 

A second simplihcation results from introducing the Born-Oppenheimer separation of electronic and nuclear motions for convenience, the latter is most often considered to be classical. Each excited electronic state of the molecule can then be considered as a distinct molecular species, and the laser-excited system can be viewed as a mixture of them. The local structure of such a system is generally described in terms of atom-atom distribution functions t) [22, 24, 25]. These functions are proportional to the probability of Ending the nuclei p and v at the distance r at time t. Building this information into Eq. (4) and considering the isotropy of a liquid system simplifies the theory considerably. 

What is obviously needed is a generally accepted recipe for how atomic states should be dealt with in approximate density functional theory and, indeed, a few empirical rules have been established in the past. Most importantly, due to the many ways atomic energies can be obtained, one should always explicitly specify how the calculations were performed to ensure reproducibility. From a technical point of view (after considerable discussions in the past among physicists) there is now a general consensus that open-shell atomic calculations should employ spin polarized densities, i. e. densities where not necessarily... 

The proton transfer may occur rapidly after the excitation and form a tautomer, when either acidic or basic moieties of the same molecule become stronger acids or bases in the excited state. The majority of reactions of this type involve the proton transfer from an oxygen donor to an oxygen or nitrogen acceptor, although a few other cases are known, where a nitrogen atom can function as a donor and a carbon atom as the acceptor. Usually an intramolecular hydrogen bond between the two moieties of a molecule facilitates the proton transfer. 

Based on the same underlying principles as the molecular-based quantum methods, solid-state DFT represents the bulk material using periodic boundary conditions. The imposition of these boundary conditions means that it becomes more efficient to expand the electron density in periodic functions such as plane waves, rather than atom-based functions as in the molecular case. The efficiency of the calculations is further enhanced by the use of pseudo-potentials to represent the core electrons and to make the changes in the electron density... 

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