Term symbol

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In quantum mechanics, the symbol of the term is a brief description of the quantum number of angular momentum (total) in a multi-electron atom (however, even one electron can be represented by the term symbol). Each atomic energy level with a given electron configuration is explained not only by the electron configuration but also the term symbol itself, since the energy level also depends on the total angular momentum including the spin. Common atomic term symbols consider LS coupling (also known as Russell-Saunders coupling or Spin-Orbit coupling). The symbol of a basic country term is predicted by the Hund rule. The atomic energy level table identified by its long-term symbol has been prepared by the National Institute of Standards and Technology. In this database, neutral atoms are identified as I, single ionized atoms as II, etc.

The use of the term term for energy levels is based on the Rydberg-Ritz combination principle, the empirical observation that the spectral line wave numbers can be expressed as the difference of the two terms hc , where h is Planck's constant and c the speed of light) quantized energy and spectral numbers (again multiplied by hc ) with photon energy.

Video Term symbol

LS kopling dan simbol

For light atoms, the spin-orbit (or coupling) interaction is small so that the total momentum angular momentum L and the total rotation S is a good quantum number. The interactions between L and S are known as LS couplings, Russell-Saunders Coupling Spin-Orbit . The status of the atom is then well illustrated by the form symbols

^{2 S 1} L

Where

S is the total quantum spin count. 2S 1 is a rotating multiplicity, representing the number of possible states J to be given L and S , provided that < i> L> = S . (If L & lt; S , the maximum number possible J is 2L 1 ). This is easily proven by using J _max = L S and J _min = | L - S | , so the number of possibilities J provided L and S is just J _max - J _min 1 because J varies in step units.

J is the total quantum total momentum quantum.

L is the total quantum number of orbital in spectroscopic notation. The first 17 symbols of L are:

The nomenclature ( S , P , D , F ) is derived from the characteristics of the spectroscopic line corresponding to ( s , p , d , f ) orbital: sharp, principal, diffuse, and fundamental; the rest are named in alphabetical order, except that J is omitted. When used to describe the state of electrons in an atom, the term symbol usually follows the electron configuration. For example, one low energy level of the state of a carbon atom is written as 1 s ² 2 s ² 2 p ^{2Ã, 3} P ₂ . Superscript 3 shows that spin states are triplets, and therefore S = 1 (2 S 1 = 3), P is a spectroscopic notation for < i> L = 1, and subscript 2 is the value J . Using the same notation, the ground state of carbon is 1 s ² 2 s ² 2 p ^{2Ã, 3} P ₀ .

Maps Term symbol

Requirements, levels and status

The term symbol is also used to describe a compound system such as a meson or an atomic nucleus, or molecule (see the symbol of a molecular term). For molecules, Greek letters are used to designate orbital corner momentum components along the molecular axis.

For certain electron configurations

• The combined value of S and the value L is called term , and has statistical weight (that is, the number of possible microstates) equal to (2 S 1) (2 L 1);
• The combination of S , L and J is called level . The given level has statistical weight (2 J 1), which is the number of microstate possibilities associated with this level in the appropriate terms;
• The combination S , L , J and M _J specify one < b> circumstances .

For example, for S = 1, L = 2 , there is (2ÃÆ'â € 1 1) (2ÃÆ' â € "2 1) = 15 various microstate (= eigenstates in uncoupled representation) corresponding to ³ D term , which (3) <3> 3 <3> <3> . The number of (2J 1) for all levels in the same time period (2 S 1) (2 L 1) as the second dimension of the representation must be the same as described above. In this case, J can be 1, 2, or 3, so 3 5 7 = 15.

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The term parity symbol

Paritas simbol istilah dihitung sebagai

${\ displaystyle P = (- 1) ^ {\ sum _ {i} l_ {i}} \, \!} Â Â$

di mana l _i adalah bilangan kuantum orbital untuk setiap elektron. ${\ displaystyle P = 1} Â Â$ berarti bahkan paritas saat ${\ displaystyle P = -1} Â$ adalah untuk paritas ganjil. Faktanya, hanya elektron dalam orbital ganjil (dengan l ganjil) berkontribusi por total total paras: jumlah ganjil elektron dalam orbital ganjil (yang memiliki ganjil l seperti dalam p, f,...) sesuai dengan simbol istilah aneh, sementara jumlah genap dalam orbital ganjil sesuai dengan simbol istilah genap. Jumlah electricity dalam orbital genap tidak relevan karena jumlah bilangan genap pun genap. Untuk setiap subdued tertutup, jumlah electricity adalah 2 (2l 1) yang genap, sehingga penjumlahan l _i dalam subkontak tertutup selalu nomor genap. Penjumlahan bilangan kuantum ${\ displaystyle \ sum _ {i} l_ {i}} Â$ pada subshells terbuka (tidak terisi) dari orbital ganjil ( l ganjil) menentukan paritas simbol jangka. Jika jumlah electricity dalam ini dikurangi penjumlahan adalah ganjil (genap) maka paritasnya juga ganjil (genap).

When odd, the parity of the term symbol is indicated by a superscript letter "o", if not omitted:

² P ^o
_{Ã,½} has an odd parity, but ³ P < sub> 0 even has parity.

Alternatively, parity may be indicated by the subscript letter "g" or "u", standing for gerade (German for "even") or ungerade ("weird"):

² P _{Ã,½, u} for odd parity, and ³ P _{0, g} for even.

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The basic terms of the term symbol

It is relatively easy to calculate the term symbol for the ground state of an atom using the Hund rule. This corresponds to a state with a maximum of S and L .

1. 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 the shells and subshells are full then the term symbol is ¹ S ₀ .
2. Distribute electrons in available orbital, following the Pauli exclusion principle. First, fill in the orbital with the highest _l value with each one electron, and set the maximum m _s to they (ie ½). After all the orbital in the subshell has one electron, add one second (follow the same sequence), assign m _s = -Ã,½ to them.
3. The overall S is calculated by adding m _s value for each electron. According to the first rule of Hund, the ground state has all unpaired electrons rotating parallel to the same m _s value, which is conventionally selected as ½. Overall S then Ã,½ times the number of unpaired electrons. The overall L is calculated by adding the m _l value for each electron (so if there are two electrons in the same orbitals, add twice the orbitals m _l ).
4. Calculate J as
   • if less than half of the subshell is occupied, take the minimum value J = | L - S | ;
   • if more than half filled, take the maximum value J = L S ;
   • if the subshell is half-filled, then L will be 0, so J = S .

For example, in the case of fluorine, the electronic configuration is 1s ² 2s ² 2p ⁵ .

1. Remove the complete sub-tables and save the 2p ⁵ section. So there are five electrons to place in subshell p ( l = 1 ).

2. There are three orbitals ( m _l = 1, 0, -1 ) that can store up to 2 (2 l 1) = 6 electrons . The first three electrons can take m _s = Ã,½ (?) but the Pauli exclusion principle forces the next two people to have m _s = -Ã,½ (?) because they go to a pre-populated orbital.

3. S = Ã,½ Ã,½ Ã,½ - Ã,½ - Ã,½ = Ã,½ ; and L = 1 0 - 1 1 0 = 1 , which is "P" in spectroscopic notation.

4. As 2p fluorine subpol over half filled, S = ³ / ₂ . The symbol of the state term is then ^{2 S 1} L _J = < soup> 2 P _{³ / 2} .

The periodic table of common column term symbols

In the periodic table, since elements in columns usually have the same outer electron structure, and always have the same electron structure in the "s-block" and "p-block" elements (see block (periodic table)), all elements can share symbols the same state term for the column. Thus, hydrogen and alkali metals are all ² S _{¹ / 2} , alkaline earth metal ¹ S ₀ , elemental boron column ² P _{¹ /sub > 2} , carbon column element ³ P ₀ , pnictogen is ⁴ S _{³ / 2} , chromogen is ³ P ₂ , halogen is ² P _{³ / 2} , and the inert gas is ¹ S ₀ , as per the rules for the full shell and the subshell mentioned above.

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The term symbol for electron configuration

di mana fungsi lantai ${\ displaystyle \ lfloor x \ rfloor}  ÂÂ$ menunjukkan bilangan bulat terbesar yang tidak melebihi x .

Detailed evidence can be found on original Renjun Xu paper.

• For general electronic configuration l ^k , the equivalent electron k that occupies a subshell, general treatment and computer code can also be found in this paper..

Alternate method using group theory

For configurations with at most two electrons (or holes) per subshell, alternative and faster methods to arrive at the same result can be obtained from group theory. The 2p ² configuration has the symmetry of the following direct products in the full rotation group:

? ⁽¹⁾ ÃÆ'â € "? ⁽¹⁾ =? ⁽⁰⁾ [? ⁽¹⁾ ]? ⁽²⁾ ,

which, using the familiar label ? ⁽⁰⁾ = S , ? ⁽¹⁾ = P and ? ⁽²⁾ = D , can be written as

P ÃÆ'â € "P = S [P] D.

Square brackets attach an anti-symmetrical square. Then the 2p ² configuration has the following symmetry component:

S D (from a symmetrical square and hence has a symmetrical spatial wave function);

P (from the anti-symmetric square and hence has an anti-symmetrical spatial function).

The Pauli principle and the requirements for electrons to be explained by the anti-symmetrical wave function imply that only the following combinations of spatial and rotary symmetry are allowed:

¹ S ¹ D (spatial symmetry, anti-symmetrical spin)

³ P (symmetrical spacial antacymetric spin).

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

Group theory methods can be performed for other such configurations, such as 3d ² , using the general formula

? ^(j) ÃÆ'â € "? ^(j) =? ^(2j) ? ^{(2j- 2)} ...? ⁽⁰⁾ [? ^(2j-1) ...? ⁽¹⁾ ].

A symmetrical square will produce a singlet (like ¹ S, ¹ D, & ¹ G), while an anti-symmetric square raises a triplet (like ³ P & amp; ³ F).

More general, which can be used

? ^{( j )} ÃÆ' - ? ^{( k )} =? ^{( j k )} ? ^{( j k -1)} ...?

where, since the product is not square, it is not divided into symmetrical and anti-symmetrical parts. Where two electrons come from unbalanced orbitals, both singlets and triplets are allowed in each case.

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Summary of various coupling schemes and corresponding term symbols

Source of the article : Wikipedia