词条 | Term symbol | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
释义 |
The use of the word term for an energy level is based on the Rydberg-Ritz combination principle, an empirical observation that the wavenumbers of spectral lines can be expressed as the difference of two terms. This was later explained by the Bohr quantum theory, which identified the terms (multiplied by hc, where h is the Planck constant and c the speed of light) with quantized energy levels and the spectral wavenumbers (again multiplied by hc) with photon energies. Tables of atomic energy levels identified by their term symbols have been compiled by the National Institute of Standards and Technology. In this database, neutral atoms are identified as I, singly ionized atoms as II, etc.[1] Neutral atoms of the chemical elements have the same term symbol for each column in the s-block and p-block elements, but may differ in d-block and f-block elements, if the ground state electron configuration changes within a column. Ground state term symbols for chemical elements are given below. LS coupling and symbolFor light atoms, the spin-orbit interaction (or coupling) is small so that the total orbital angular momentum L and total spin S are good quantum numbers. The interaction between L and S is known as LS coupling, Russell-Saunders coupling or Spin-Orbit coupling. Atomic states are then well described by term symbols of the form 2S+1LJ where S is the total spin quantum number. 2S + 1 is the spin multiplicity, which represents the number of possible states of J for a given L and S, provided that L ≥ S. (If L < S, the maximum number of possible J is 2L + 1).[1] This is easily proven by using Jmax = L + S and Jmin = |L - S|, so that the number of possible J with given L and S is simply Jmax - Jmin + 1 as J varies in unit steps. J is the total angular momentum quantum number. L is the total orbital quantum number in spectroscopic notation. The first 17 symbols of L are:
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 from G onwards, except that J is omitted. 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 1s22s22p2 3P2. The superscript 3 indicates that the spin state is a triplet, and therefore S = 1 (2S + 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 1s22s22p2 3P0.[3] Small letters refer to individual orbitals or one-electron quantum numbers, whereas capital letters refer to many-electron states or to their quantum numbers. Terms, levels, and statesThe term symbol is also used to describe compound systems such as mesons or atomic nuclei, or molecules (see molecular term symbol). For molecules, Greek letters are used to designate the component of orbital angular momenta along the molecular axis. For a given electron configuration
The product (2S+1)(2L+1) as a number of possible microstates with given S and L is also a number of basis states in the uncoupled representation, where S, mS, L, mL (mS and mL are z-axis components of total spin and total orbital angular momentum respectively) are good quantum numbers whose corresponding operators mutually commute. With given S and L, the eigenstates in this representation span function space of dimension (2S+1)(2L+1), as and . In the coupled representation where total angular momentum (spin + orbital) is treated, the associated microstates (or eigenstates) are and these states span the function space with dimension of as . Obviously the dimension of function space in both representation must be the same. As an example, for S = 1, L = 2, there are (2×1+1)(2×2+1) = 15 different microstates (= eigenstates in the uncoupled representation) corresponding to the 3D term, of which (2×3+1) = 7 belong to the 3D3 (J = 3) level. The sum of (2J+1) for all levels in the same term equals (2S+1)(2L+1) as the dimensions of both representations must be equal as described above. In this case, J can be 1, 2, or 3, so 3 + 5 + 7 = 15. Term symbol parityThe parity of a term symbol is calculated as where li is the orbital quantum number for each electron. means even parity while is for odd parity. In fact, only electrons in odd orbitals (with l odd) contribute to the total parity: an odd number of electrons in odd orbitals (those with an odd l such as in p, f,...) correspond to an odd term symbol, while an even number of electrons in odd orbitals correspond to an even term symbol. The number of electrons in even orbitals is irrelevant as any sum of even numbers is even. For any closed subshell, the number of electrons is 2(2l+1) which is even, so the summation of li in closed subshells is always an even number. The summation of quantum numbers over open (unfilled) subshells of odd orbitals (l odd) determines the parity of the term symbol. If the number of electrons in this reduced summation is odd (even) then the parity is also odd (even). 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 symbolIt is relatively easy to calculate the term symbol for the ground state of an atom using Hund's rules. It corresponds with a state with maximum S and L.
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(2l + 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.
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|2S+1LJ {{=}} 2P{{frac|3|2}}}}. Atomic term symbols of the chemical elementsIn the periodic table, because atoms of elements in a column 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 may share the same ground state term symbol for the column. Thus, hydrogen and the alkali metals are all 2S{{frac|1|2}}, the alkali earth metals are 1S0, the boron column elements are 2P{{frac|1|2}}, the carbon column elements are 3P0, the pnictogens are 4S{{frac|3|2}}, the chalcogens are 3P2, the halogens are 2P{{frac|3|2}}, and the inert gases are 1S0, per the rule for full shells and subshells stated above. Term symbols for the ground states of most chemical elements[4] are given in the collapsed table below (with citations for the heaviest elements here). In the d-block and f-block, the term symbols are not always the same for elements in the same column of the periodic table, because open shells of several d or f electrons have several closely spaced terms whose energy ordering is often perturbed by addition of an extra complete shell to form the next element in the column. For example, the table shows that the first pair of vertically adjacent atoms with different ground-state term symbols are V and Nb. The 6D1/2 ground state of Nb corresponds to an excited state of V 2112 cm-1 above the 4F3/2 ground state of V, which in turn corresponds to an excited state of Nb 1143 cm-1 above the Nb ground state.[3] These energy differences are small compared to the 15158 cm-1 difference between the ground and first excited state of Ca,[3] which is the last element before V with no d electrons. {{Periodic table (term symbol)}}Term symbols for an electron configurationThe process to calculate all possible term symbols for a given electron configuration is somewhat longer.
As an example, consider the carbon electron structure: 1s22s22p2. After removing full subshells, there are 2 electrons in a p-level ({{nowrap|l {{=}} 1}}), so there are different microstates.
Case of three equivalent electrons
where the floor function denotes the greatest integer not exceeding x. The detailed proof can be found in Renjun Xu's original paper.[5]
Alternative method using group theoryFor 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.[6] Summary of various coupling schemes and corresponding term symbolsBasic concepts for all coupling schemes:
LS coupling (Russell-Saunders coupling)
jj Coupling
J1L2 coupling
LS1 coupling
Most famous coupling schemes are introduced here but these schemes can be mixed together to express energy state of atom. This summary is based on [https://www.nist.gov/pml/pubs/atspec/index.cfm]. Racah notation and Paschen notationThese are notations for describing states of singly excited atoms, especially noble gas atoms. Racah notation is basically a combination of LS or Russell-Saunder coupling and J1L2 coupling. LS coupling is for a parent ion and J1L2 coupling is for an coupling of the parent ion and the excited electron. The parent ion is an unexcited part of the atom. For example, in Ar atom excited from a ground state …3p6 to an excited state …3p54p in electronic configuration, 3p5 is for the parent ion while 4p is for the excited electron.[8] In Racah notation, states of excited atoms are denoted as . Quantities with a subscript 1 are for the parent ion, n and l are principal and orbital quantum numbers for the excited electron, K and J are quantum numbers for and where and are orbital angular momentum and spin for the excited electron respectively. “o” represents a parity of excited atom. For an inert (noble) gas atom, usual excited states are Np5nl where N = 2, 3, 4, 5, 6 for Ne, Ar, Kr, Xe, Rn, respectively in order. Since the parent ion can only be 2P1/2 or 2P3/2, the notation can be shortened to or , where nl means the parent ion is in 2P3/2 while nl′ is for the parent ion in 2P1/2 state. Paschen notation is a somewhat odd notation; it is an old notation made to attempt to fit an emission spectrum of neon to a hydrogen-like theory. It has a rather simple structure to indicate energy levels of an excited atom. The energy levels are denoted as n’l#. l is just an orbital quantum number of the excited electron. n'l is written in a way that 1s for (n = N + 1, l = 0), 2p for (n = N + 1, l = 1), 2s for (n = N + 2, l = 0), 3p for (n = N + 2, l = 1), 3s for (n = N + 3, l = 0), etc. Rules of writing n'l from the lowest electronic configuration of the excited electron are: (1) l is written first, (2) n' i
See also
Notes1. ^Levine, Ira N., Quantum Chemistry (4th ed., Prentice-Hall 1991), {{ISBN|0-205-12770-3}} 2. ^There is no official convention for naming angular momentum values greater than 20 (symbol Z). Many authors begin using Greek letters at this point ( ...). The occasions for which such notation is necessary are few and far between, however. 3. ^1 2 3 NIST Atomic Spectrum Database To read neutral carbon atom levels for example, type "C I" in the Spectrum box and click on Retrieve data. 4. ^{{cite web |url=https://physics.nist.gov/PhysRefData/ASD/ionEnergy.html |title=NIST Atomic Spectra Database Ionization Energies Form |publisher=National Institute of Standards and Technology (NIST) |website=NIST Physical Measurement Laboratory |quote=This form provides access to NIST critically evaluated data on ground states and ionization energies of atoms and atomic ions. |date=October 2018 |access-date=28 January 2019}} 5. ^1 {{cite journal |last1=Xu |first1=Renjun |last2=Zhenwen |first2=Dai |title=Alternative mathematical technique to determine LS spectral terms |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |year=2006 |volume=39 |issue=16 |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 }} 6. ^{{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. }} 7. ^{{cite web |url=https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas/atomic-spectroscopy-2#node095 |title= Atomic Spectroscopy - Different Coupling Scheme 9. Notations for Different Coupling Schemes |date=1 November 2017 |website=NIST Physical Measurement Laboratory |publisher=National Institute of Standards and Technology (NIST) |access-date=31 January 2019 }} 8. ^{{Cite web|url=https://www.physics.utoronto.ca/~phy326/hene/HeNeAppendices.pdf|title=APPENDIX 1 - Coupling Schemes and Notation|last=|first=|date=|website=|publisher=University of Toronto: Advanced Physics Laboratory - Course Homepage|access-date=5 Nov 2017}} References 3 : Atomic physics|Theoretical chemistry|Quantum chemistry |
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