Real coordinate space

2014-01-03T13:54:20Z

95.91.230.129: /* Euclidean space */

In [[quantum mechanics]], the '''[[Henry Norris Russell|Russell]]-Saunders''' <ref name="RS">{{cite journal|last=RS|first=
Russell, H. N.|coauthors=Saunders, F. A.|title=New Regularities in the Spectra of the Alkaline Earths|journal=
Astrophysical Journal|year=1925|volume=61|pages=38|doi=10.1086/142872|url=http://adsabs.harvard.edu/doi/10.1086/142872|bibcode = 1925ApJ....61...38R }}</ref> '''term symbol''' is an abbreviated description of the [[angular momentum quantum number]]s in a multi-[[electron]] [[atom]]. Each energy level of a given [[electron configuration]] is described by its own term symbol, assuming [[angular momentum coupling#LS coupling|LS coupling]]. The ground state term symbol is predicted by [[List of Hund's rules|Hund's rules]]. Tables of atomic energy levels identified by their term symbols have been compiled by [[NIST]].<ref name=NIST/>

==Symbol==

The term symbol has the form
::2''S''+1''L''''J''
where
:''S'' is the total [[spin quantum number]]. 2''S'' + 1 is the '''[[spin multiplicity]]''': the maximum number of different possible states of ''J'' for a given (''L'', ''S'') combination.
:''J'' is the [[total angular momentum quantum number]].
:''L'' is the total [[azimuthal quantum number|orbital quantum number]] in [[spectroscopic notation]]. The first 17 symbols of L are:
{| align=center
| align=center width=30px | ''L'' =
| align=center width=30px | 0

| align=center width=30px | 1
| align=center width=30px | 2
| align=center width=30px | 3
| align=center width=30px | 4
| align=center width=30px | 5
| align=center width=30px | 6
| align=center width=30px | 7
| align=center width=30px | 8
| align=center width=30px | 9
| align=center width=30px | 10
| align=center width=30px | 11
| align=center width=30px | 12
| align=center width=30px | 13
| align=center width=30px | 14
| align=center width=30px | 15
| align=center width=30px | 16
| align=left |...
|-
|
| align=center width=30px | ''S''
| align=center width=30px | ''P''
| align=center width=30px | ''D''
| align=center width=30px | ''F''
| align=center width=30px | ''G''
| align=center width=30px | ''H''
| align=center width=30px | ''I''
| align=center width=30px | ''K''
| align=center width=30px | ''L''
| align=center width=30px | ''M''
| align=center width=30px | ''N''
| align=center width=30px | ''O''
| align=center width=30px | ''Q''
| align=center width=30px | ''R''
| align=center width=30px | ''T''
| align=center width=30px | ''U''
| align=center width=30px | ''V''
| align=left | (continued alphabetically)<ref group="note">There is no official convention for naming angular momentum values greater than 20 (symbol Z). Many authors begin using Greek letters at this point (<math>\alpha,\beta, \gamma,</math> ...). The occasions for which such notation is necessary are few and far between, however.</ref>
|}

The nomenclature (''S'', ''P'', ''D'', ''F'') is derived from the characteristics of the spectroscopic lines corresponding to (''s'', ''p'', ''d'', ''f'') orbitals: sharp, principal, diffuse, and fundamental; the rest being named in alphabetical order. When used to describe electron states in an atom, the term symbol usually follows the [[electron configuration]]. For example, one low-lying energy level of the [[carbon]] atom state is written as 1''s''22''s''22''p''2 3''P''2. The superscript 3 indicates that the spin state is a triplet, and therefore ''S'' = 1 (2''S'' + 1 = 3), the ''P'' is spectroscopic notation for ''L'' = 1, and the subscript 2 is the value of ''J''. Using the same notation, the [[ground state]] of carbon is 1''s''22''s''22''p''2 3''P''0.<ref name=NIST>[http://physics.nist.gov/PhysRefData/ASD/levels_form.html NIST Atomic Spectrum Database] To read the carbon atom levels, type "C I" in the Spectrum box and click on Retrieve data. In this database, neutral atoms are identified as I, singly ionized atoms as II, etc.</ref>

==Others==

The term symbol is also used to describe compound systems such as [[meson]]s or atomic nuclei, or even molecules (see [[molecular term symbol]]). In that last case, Greek letters are used to designate the (molecular) orbital angular momenta.

For a given electron configuration
* The combination of an ''S'' value and an ''L'' value is called a '''term''', and has a statistical weight (i.e., number of possible microstates) of (2''S''+1)(2''L''+1);
* A combination of ''S'', ''L'' and ''J'' is called a '''level'''. A given level has a statistical weight of (2''J''+1), which is the number of possible microstates associated with this level in the corresponding term;
* A combination of ''L'', ''S'', ''J'' and ''MJ'' determines a single '''state'''.
As an example, for ''S'' = 1, ''L'' = 2, there are (2×1+1)(2×2+1) = 15 different microstates corresponding to the 3''D'' term, of which (2×3+1) = 7 belong to the 3''D''3 (J = 3) level. The sum of (2''J''+1) for all levels in the same term equals (2''S''+1)(2''L''+1). In this case, ''J'' can be 1, 2, or 3, so 3 + 5 + 7 = 15.

==Term symbol parity==
The parity of a term symbol is calculated as
:<math>P=(-1)^{\sum_i l_i}\ ,\!</math>
where ''li'' is the orbital quantum number for each electron. In fact, only electrons in odd orbitals contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd ''l'' such as in p, f,...) will make an odd term symbol, while an even number of electrons in odd orbitals will make an even term symbol, irrespective of the number of electrons in even orbitals.

When it is odd, the parity of the term symbol is indicated by a superscript letter "o", otherwise it is omitted:
:2P{{su|p=o|b=½}} has odd parity, but 3P0 has even parity.

Alternatively, parity may be indicated with a subscript letter "g" or "u", standing for ''gerade'' (German for "even") or ''ungerade'' ("odd"):
:2P½,u for odd parity, and 3P0,g for even.

==Ground state term symbol==
It is relatively easy to calculate the term symbol for the ground state of an atom using [[List of Hund's rules|Hund's rules]]. It corresponds with a state with maximal ''S'' and ''L''.
#Start with the most stable [[electron configuration]]. Full shells and subshells do not contribute to the overall [[angular momentum]], so they are discarded.
#*If all shells and subshells are full then the term symbol is 1''S''0.
#Distribute the electrons in the available [[atomic orbital|orbital]]s, following the [[Pauli exclusion principle]]. First, fill the orbitals with highest [[magnetic quantum number|''ml'']] value with one electron each, and assign a maximal [[spin quantum number|''ms'']] to them (i.e. +½). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning {{nowrap|''ms'' {{=}} −½}} to them.
#The overall ''S'' is calculated by adding the ''ms'' values for each electron. That is the same as multiplying ½ times the number of '''unpaired''' electrons. The overall ''L'' is calculated by adding the ''ml'' values for each electron (so if there are two electrons in the same orbital, add twice that orbital's ''ml'').
#Calculate [[total angular momentum quantum number|''J'']] as
#*if less than half of the subshell is occupied, take the minimum value {{nowrap|''J'' {{=}} {{!}}''L'' - ''S''{{!}}}};
#*if more than half-filled, take the maximum value {{nowrap|''J'' {{=}} ''L'' + ''S''}};
#*if the subshell is half-filled, then ''L'' will be 0, so {{nowrap|''J'' {{=}} ''S''}}.

As an example, in the case of [[fluorine]], the electronic configuration is 1s22s22p5.

1. Discard the full subshells and keep the 2p5 part. So there are five electrons to place in subshell p ({{nowrap|''l'' {{=}} 1}}).

2. There are three orbitals ({{nowrap|''ml'' {{=}} 1, 0, −1}}) that can hold up to {{nowrap|2(2''l'' + 1) {{=}} 6 electrons}}. The first three electrons can take {{nowrap|''ms'' {{=}} ½ (↑)}} but the Pauli exclusion principle forces the next two to have {{nowrap|''ms'' {{=}} −½ (↓)}} because they go to already occupied orbitals.
{|class=wikitable
|-
|width=30px rowspan=2| ||colspan=3 align=center|''ml''
|-
|align=center width=30px|+1||align=center width=30px|0||align=center width=30px|−1
|-
|''ms'':||align=center|↑↓||align=center|↑↓||align=center|↑
|}

3. {{nowrap|''S'' {{=}} ½ + ½ + ½ − ½ − ½ {{=}} ½}}; and {{nowrap|''L'' {{=}} 1 + 0 − 1 + 1 + 0 {{=}} 1}}, which is "P" in spectroscopic notation.

4. As fluorine 2p subshell is more than half filled, {{nowrap|''J'' {{=}} ''L'' + ''S'' {{=}} {{frac|3|2}}}}. Its ground state term symbol is then {{nowrap|2''S''+1''L''''J'' {{=}} 2P{{frac|3|2}}}}.

==Term symbols for an electron configuration==
To calculate all possible term symbols for a given [[electron configuration]] the process is a bit longer.

* First, calculate the total number of possible microstates ''N'' for a given electron configuration. As before, we discard the filled (sub)shells, and keep only the partially filled ones. For a given orbital quantum number ''l'', t is the maximum allowed number of electrons, {{nowrap|''t'' {{=}} 2(2''l''+1)}}. If there are ''e'' electrons in a given subshell, the number of possible microstates is
::<math>N= {t \choose e} = {t! \over {e!\,(t-e)!}}.</math>

:As an example, lets take the [[carbon]] electron structure: 1s22s22p2. After removing full subshells, there are 2 electrons in a p-level ({{nowrap|''l'' {{=}} 1}}), so we have
::<math>N = {6! \over {2!\,4!}}=15</math>
:different microstates.

* Second, draw all possible microstates. Calculate ''ML'' and ''MS'' for each microstate, with <math>M=\sum_{i=1}^e m_i</math> where ''mi'' is either ''ml'' or ''ms'' for the ''i''-th electron, and ''M'' represents the resulting ''ML'' or ''MS'' respectively:
:{| cellspacing="0"
|-
|  
! colspan="3" align="center" style="border-right: 2px solid windowtext" | ''ml''
| colspan="2" |  
|-
! style="border-bottom: 2px solid windowtext" |  
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | +1
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | 0
! align="center" width="40px" style="border-bottom: 2px solid windowtext; border-right: 2px solid windowtext" | −1
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | ''ML''
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | ''MS''
|-
| rowspan="3" style="border-bottom: 1px solid windowtext"| all up
| align="center" | ↑
| align="center" | ↑
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 1
| align="center" | 1
|-
| align="center" | ↑
| align="center" |
| align="center" style="border-right: 2px solid windowtext" | ↑
| align="center" | 0
| align="center" | 1
|-
| align="center" style="border-bottom: 1px solid windowtext" |
| align="center" style="border-bottom: 1px solid windowtext" | ↑
| align="center" style="border-bottom: 1px solid windowtext; border-right: 2px solid windowtext" | ↑
| align="center" style="border-bottom: 1px solid windowtext" | −1
| align="center" style="border-bottom: 1px solid windowtext" | 1
|-
| rowspan="3" style="border-bottom: 1px solid windowtext" | all down
| align="center" | ↓
| align="center" | ↓
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 1
| align="center" | −1
|-
| align="center" | ↓
| align="center" |
| align="center" style="border-right: 2px solid windowtext" | ↓
| align="center" | 0
| align="center" | −1
|-
| align="center" style="border-bottom: 1px solid windowtext" |
| align="center" style="border-bottom: 1px solid windowtext" | ↓
| align="center" style="border-bottom: 1px solid windowtext; border-right: 2px solid windowtext" | ↓
| align="center" style="border-bottom: 1px solid windowtext" | −1
| align="center" style="border-bottom: 1px solid windowtext" | −1
|-
| rowspan="9" | one up one down
| align="center" | ↑↓
| align="center" |
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 2
| align="center" | 0
|-
| align="center" | ↑
| align="center" | ↓
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 1
| align="center" | 0
|-
| align="center" | ↑
| align="center" |
| align="center" style="border-right: 2px solid windowtext" | ↓
| align="center" | 0
| align="center" | 0
|-
| align="center" | ↓
| align="center" | ↑
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 1
| align="center" | 0
|-
| align="center" |
| align="center" | ↑↓
| align="center" style="border-right: 2px solid windowtext" |
| align="center" | 0
| align="center" | 0
|-
| align="center" |
| align="center" | ↑
| align="center" style="border-right: 2px solid windowtext" | ↓
| align="center" | −1
| align="center" | 0
|-
| align="center" | ↓
| align="center" |
| align="center" style="border-right: 2px solid windowtext" | ↑
| align="center" | 0
| align="center" | 0
|-
| align="center" |
| align="center" | ↓
| align="center" style="border-right: 2px solid windowtext" | ↑
| align="center" | −1
| align="center" | 0
|-
| align="center" |
| align="center" |
| align="center" style="border-right: 2px solid windowtext" | ↑↓
| align="center" | −2
| align="center" | 0
|}

* Third, count the number of microstates for each ''ML''—''MS'' possible combination
:{| cellspacing="0"
|-
| colspan="2" |  
! colspan="3" align="center" | ''MS''
|-
! colspan="2" |  
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | +1
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | 0
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | −1
|-
! rowspan="5" valign="center" | ''ML''
! align="center" width="40px" style="border-right: 2px solid windowtext" | +2
|
| align="center" | 1
|
|-
! align="center" style="border-right: 2px solid windowtext" | +1
| align="center" | 1
| align="center" | 2
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | 0
| align="center" | 1
| align="center" | 3
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | −1
| align="center" | 1
| align="center" | 2
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | −2
| align="center" |
| align="center" | 1
| align="center" |
|}

* Fourth, extract smaller tables representing each possible term. Each table will have the size (2''L''+1) by (2''S''+1), and will contain only "1"s as entries. The first table extracted corresponds to ''ML'' ranging from −2 to +2 (so {{nowrap|''L'' {{=}} 2}}), with a single value for ''MS'' (implying {{nowrap|''S'' {{=}} 0}}). This corresponds to a 1D term. The remaining table is 3×3. Then we extract a second table, removing the entries for ''ML'' and ''MS'' both ranging from −1 to +1 (and so {{nowrap|''S'' {{=}} ''L'' {{=}} 1}}, a 3P term). The remaining table is a 1×1 table, with {{nowrap|''L'' {{=}} ''S'' {{=}} 0}}, i.e., a 1S term.
:{|
|-
| width="150px" |
{| cellspacing="0"
|+ ''S''=0, ''L''=2, ''J''=2 1D2
|-
| colspan="2" |  
! align="center" | ''Ms''
|-
! colspan="2" |  
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | 0
|-
! rowspan="5" valign="center" | ''Ml''
! align="center" width="40px" style="border-right: 2px solid windowtext" | +2
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | +1
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | 0
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | −1
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | −2
| align="center" | 1
|}
| width="250px" |
{| cellspacing="0"
|+ ''S''=1, ''L''=1, ''J''=2,1,03P2, 3P1, 3P0
|-
| colspan="2" |  
! colspan="3" align="center" | ''Ms''
|-
! colspan="2" |  
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | +1
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | 0
! align="center" width="40px" style="border-bottom: 2px solid windowtext" | −1
|-
! rowspan="3" valign="center" | ''Ml''
! align="center" width="40px" style="border-right: 2px solid windowtext" | +1
| align="center" | 1
| align="center" | 1
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | 0
| align="center" | 1
| align="center" | 1
| align="center" | 1
|-
! align="center" style="border-right: 2px solid windowtext" | −1
| align="center" | 1
| align="center" | 1
| align="center" | 1
|}
| width="150px" |
{| cellspacing="0"
|+ ''S''=0, ''L''=0, ''J''=01S0
|-
| colspan="2" |  
! align="center" | ''Ms''
|-
! colspan="2" |  
! align="center" style="border-bottom: 2px solid windowtext" | 0
|-
! valign="center" | ''Ml''
! align="center" width="40px" style="border-right: 2px solid windowtext" | 0
| align="center" | 1
|}
|}
</dl>
* Fifth, applying [[List of Hund's rules|Hund's rules]], deduce which is the ground state (or the lowest state for the configuration of interest.) Hund's rules should not be used to predict the order of states other than the lowest for a given configuration. (See examples at [[Hund's rules#Excited states]].)

* If only two equivalent electrons are involved, there is an "Even Rule" which states
: For two equivalent electrons the only states that are allowed are those for which the sum (L + S) is even.

===Case of three equivalent electrons===
* For three equivalent electrons (with the same orbital quantum number ''l''), there is also a general formula (denoted by ''X(L,S,l)'' below) to count the number of any allowed terms with total orbital quantum number "L" and total total spin quantum number "S".
<math>X(L,S,l)=
\begin{cases}
L-\lfloor\frac{L}{3}\rfloor, & \text{if }S=1/2\text{ and } 0\leq L <l\\
l-\lfloor\frac{L}{3}\rfloor, & \text{if }S=1/2\text{ and } l\leq L \leq 3l-1 \\
\lfloor \frac{L}{3}\rfloor -\lfloor \frac{L-l}{2} \rfloor +\lfloor \frac{L-l+1}{2} \rfloor, & \text{if }S=3/2\text{ and } 0\leq L <l \\
\lfloor \frac{L}{3} \rfloor -\lfloor \frac{L-l}{2} \rfloor, & \text{if }S=3/2\text{ and } l\leq L \leq 3l-3 \\
0, & \text{ other cases}
\end{cases}
</math>

where the [[floor function]] <math>\lfloor x \rfloor </math> denotes the greatest integer not exceeding ''x''.

The detailed proof could be found in Renjun Xu's original paper.<ref name="Xu">{{cite journal|last=Xu|first=Renjun|coauthors=Z. Dai|title=Alternative mathematical technique to determine LS spectral terms|journal=Journal of Physics B: At. Mol. Opt. Phys.|year=2006|volume=39|pages=3221–3239|doi=10.1088/0953-4075/39/16/007|url=http://iopscience.iop.org/0953-4075/39/16/007/|arxiv = physics/0510267 |bibcode = 2006JPhB...39.3221X }}</ref>

* For a general electronic configuration of ''lk'', namely k equivalent electrons occupying one subshell, the general treatment and computer code could also be found in this paper.<ref name="Xu" />

===Alternative method using group theory===
For configurations with at most two electrons (or holes) per subshell, an alternative and much quicker method of arriving at the same result can be obtained from [[group theory]]. The configuration 2p2 has the symmetry of the following direct product in the full rotation group:

:Γ(1) × Γ(1) = Γ(0) + [Γ(1)] + Γ(2),

which, using the familiar labels {{nowrap|Γ(0) {{=}} S}}, {{nowrap|Γ(1) {{=}} P}} and {{nowrap|Γ(2) {{=}} D}}, can be written as
:P × P = S + [P] + D.

The square brackets enclose the anti-symmetric square. Hence the 2p2 configuration has components with the following symmetries:
:S + D (from the symmetric square and hence having symmetric spatial wavefunctions);
:P (from the anti-symmetric square and hence having an anti-symmetric spatial wavefunction).

The Pauli principle and the requirement for electrons to be described by anti-symmetric wavefunctions imply that only the following combinations of spatial and spin symmetry are allowed:
:1S + 1D (spatially symmetric, spin anti-symmetric)
:3P (spatially anti-symmetric, spin symmetric).

Then one can move to step five in the procedure above, applying Hund's rules.

The group theory method can be carried out for other such configurations, like 3d2, using the general formula
:Γ(j) × Γ(j) = Γ(2j) + Γ(2j-2) + ... + Γ(0) + [Γ(2j-1) + ... + Γ(1)].

The symmetric square will give rise to singlets (such as 1S, 1D, & 1G), while the anti-symmetric square gives rise to triplets (such as 3P & 3F).

More generally, one can use
:Γ(j) × Γ(k) = Γ(j+k) + Γ(j+k-1) + ... + Γ(|j-k|)

where, since the product is not a square, it is not split into symmetric and anti-symmetric parts. Where two electrons come from inequivalent orbitals, both a singlet and a triplet are allowed in each case.
<ref>{{cite journal |title = Spin factoring as an aid in the determination of spectroscopic terms | journal = Journal of Chemical Education| volume = 54 | issue = 3 | pages = 147 | year = 1977 | doi = 10.1021/ed054p147|bibcode = 1977JChEd..54..147M |last1 = McDaniel |first1 = Darl H. }}</ref>

== Notes ==
<references group="note" />

== References ==
<references/>

==See also==
* [[Angular quantum number]]s
* [[Angular momentum coupling]]
* [[Molecular term symbol]]

{{DEFAULTSORT:Term Symbol}}
[[Category:Atomic physics]]
[[Category:Theoretical chemistry]]
[[Category:Quantum chemistry]]

Direction cosine

2013-12-25T11:37:59Z

95.91.230.129: /* References */

The '''Automated Readability Index (ARI)''' is a [[readability test]] designed to gauge the understandability of a text. Like the [[Flesch-Kincaid Readability Test|Flesch-Kincaid]] Grade Level, [[Gunning Fog Index]], [[SMOG Index]], [[Fry Readability Formula]], and [[Coleman-Liau Index]], it produces an approximate representation of the [[Grade levels#USA and Canada|US grade level]] needed to comprehend the text.

The formula for calculating the Automated Readability Index is given below:

:<math>
4.71 \left (\frac{\mbox{characters}}{\mbox{words}} \right) + 0.5 \left (\frac{\mbox{words}}{\mbox{sentences}} \right) - 21.43
</math>

where ''characters'' is the number of letters, numbers, and punctuation marks, ''words'' is the number of spaces, and ''sentences'' is the number of sentences. Sentences were counted by hand as each text was typed.

As a rough guide, US grade level 1 corresponds to ages 6 to 8. Reading level grade 8 corresponds to the typical reading level of a 14 year-old US child. Grade 12, the highest US secondary school grade before college, corresponds to the reading level of a 17 year-old.

Unlike the other indices, the ARI, along with the Coleman-Liau, relies on a factor of characters per word, instead of the usual syllables per word. Although opinion varies on its accuracy as compared to the syllables/word and complex words indices, characters/word is often faster to calculate, as the number of characters is more readily and accurately counted by computer programs than syllables. In fact, this index was designed for real-time monitoring of readability on electric typewriters.<ref>{{cite journal
| author = Senter, R.J.
| coauthors = Smith, E.A.
| date = November, 1967
| title = Automated Readability Index.
| url = http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0667273
| publisher = [[Wright-Patterson Air Force Base]]
| id = AMRL-TR-6620
| page = iii
| accessdate = 2012-03-18
}}</ref>

==Notes==

<references/>

==External links==
*[http://www.online-utility.org/english/readability_test_and_improve.jsp Online readability tests] - finds ARI and other indices, suggestions how to improve readability
*[http://www.editcentral.com Readability calculators] - six readability statistics

[[Category:Readability tests]]

[[da:LIX]]
[[no:Lesbarhetsindeks]]
[[sv:LIX]]

formulasearchengine - User contributions [en]

Real coordinate space

Direction cosine