Notation term symbols

Dimensionless groups for a proeess model ean be easily obtained by inspeetion from Table 13-2. Eaeh of the three transport balanees is shown (in veetor/tensor notation) term-by-term under the deseription of the physieal meanings of the respeetive terms. The table shows how various well-known dimensionless groups are derived and gives the physieal interpretation of the various groups. Table 13-3 gives the symbols of the dimensions of the terms in Table 13-2. 

In an atomic term symbol the value of L = 1,2,..., etc. is coded according to spectroscopic notation as 5, P, D,F,G,... etc. (alphabetically for L > 3). Likewise, each atomic state can be characterized by a total electronic spin quantum number S,... 

A.12-1 for a brief review of atomic-spectroscopic notation). In a given term-symbol, T will be S, P, D, F, or G etc. depending on whether the total electronic orbital angular-momentum quantum number L is 0, 1,... 

A term symbol is written in the short-hand notation... 

We add a brief remark on the notation for the terms, A few lines above we have used capital letters S, P, D,. . , instead of small letters (p. 126). This is customary in the case of several electrons for the purpose of indicating the total orbital moment. It is also customary to define the multiplet character, that is to say, the number of terms belonging to a particular multiplet (all with the same principal and azimuthal quantum numbers), by attaching this number to the term symbol as a left-hand upper index also, the multiplicity written down is always the one which occurs when I is large, viz. 25+1. In fact,... 

We are now in a position to write full term symbols which include information about 5, L and J. The notation for a full term symbol is ... 

When / = 0, L is denoted with an S when / = 1, L is denoted with a P, and so on. Similar term symbols are used to notate the angular momenta in many-electron atoms. For the hydrogenlike 2p sublevels with = and, the term symbols are 2 P3/2 and 2 Pi/2 respectively, where the first digit indicating n = 2 is useful for specifying the principal quantum number of the valence electron in hydrogenlike and alkali atom states. 

The Racah notation for rare gas excited states neatly summarizes all of these pertinent angular momenta in the term symbol nllK]j. As two examples of Ne multiplets, we consider the 3s[X]j and the np K]j states. The former are the lowest-lying (2p) (3s) excited states with L = 0 and S = Since one always has Lg = I, Sc = i, and or f in the rare-gas atoms, the possible K values are from Eq. 9.14... 

For convenience and avoidance of confusion, notations and symbols such as used in the report to the International Commission on Radiation Units by Brownell, G. L., Berman, M. and Robertson, J. S. (1968), Nomenclature for tracer kinetics. International Journal of Applied Radiation and Isotopes, 19, 249, should be considered for use as standard terms for turnover models. 

Note the ground-state atomic term symbol for Cr is F, which splits into T2, Ti and A2 for an octahedral transition metal complex. For more information regarding term symbol notation and absorption spectra for transition metal complexes, see Shiiver et al. Inorganic Chemistry, 4th ed., W. H. Freeman New York, 2006. http //www.scribd.com/doc/6672586/Electronic-Spectroscopy-l Note excited electrons give off their energy via infrared emission and thermal interactions with the corundum crystal lattice, referred to as electron-phonon (lattice vibrations) interactions. 

A formal point of objection is the improper use of percentage notation, which is open to cumbersome handling as well as to error of interpretation. In good mathematical practices, the percentage symbol is the abbreviation of a dimensionless factor (% = 1/100 = 0.01 = 10-2). The abbreviation should never be used in the definitions of formulas and calculations these must be carried through in terms of fractions. Only in the final presentation may a percentage (99.5%) be used in place of the actual fraction (0.995). 

Here, we have borrowed Hiitter and Ottinger s use of a o symbol to indicate a kinetic interpretation of the stochastic term, but adopted a more explicit notation for its use. The o is used here to indicate that the function to its left should be evaluated at a midstep position, as in a Stratonovich SDE, but that the random quantity to the right of the diamond should be evaluated by evaluating the function Cp (X) at the beginning of each timestep, as in an Ito SDE. This notation is similar to that used by Peters [13] to denote a mixed interpretation that is identical to the kinetic interpretation defined above, which Peters indicates by using a Stratonovich circle in the position where we use a diamond. 

Any reaction in Eqs. (1) may be written as a conventional chemical equation by setting it equal to zero and transposing the negative terms to the other side of the equation. This notation has been discussed by Aris (14). Chemical equality, denoted by the symbol has been shown by Sellers (15) to be a group equivalence, thus satisfying ordinary rules of mathematical equality. Except when specific reservations are stated, every reaction is assumed to be reversible, that is, to be capable of any real rate of advancement, positive or negative. 

To further illustrate Dirac notation for some simple formulas in Euclidean 3-space, we can rewrite analogs of (9.20a-e) in Dirac notation, all in terms of underlying Dirac objects y) (using boldface symbols to stress the association with ordinary vectors) ... 

We can represent a sum of terms using a shorthand notation involving the summation symbol Z. For example, the sum of terms... 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

To make matters even more complicated, it is much more convenient in crystallography to use an entirely different system of symmetry symbols, termed the Herman-Maugin (as opposed to Schoenflies) notation (Table 8.2). 

This procedure of obtaining the Green function 7Z(u) is called hereafter the method of the Lanczos continued fractions (LCF) in the symbolic notation from the rhs of Eq. (224). Then, we say that the quantity 7 LC (u) is the infinite-order LCF for the exact Green function lZ u) from Eq. (49). A truncation at the nth term in Eq. (230) leads to the approximation ... 

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