Term symbol

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In quantum mechanics, the term symbol is an abbreviated description of the (total) angular momentum quantum numbers in a multi-electron atom (however, even a single electron can also be described by a term symbol). Each energy level of an atom with a given electron configuration is described by not only the electron configuration but also its own term symbol, as the energy level also depends on the total angular momentum including spin. The usual atomic term symbols assume LS coupling (also known as Russell-Saunders coupling). The ground state term symbol is predicted by Hund's rules. 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]

LS coupling and symbol[edit]

For 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 or Russell-Saunders 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).[2] This is easily proved 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:
L = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
S P D F G H I K L M N O Q R T U V (continued alphabetically)[note 1]

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, 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.[1]

Terms, levels, and states[edit]

The 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 combination of an S value and an L value is called a term, and has a statistical weight (i.e., number of possible microstates) equal to (2S+1)(2L+1);
  • A combination of S, L and J is called a level. A given level has a statistical weight of (2J+1), which is the number of possible microstates associated with this level in the corresponding term;
  • A combination of S, L, J and MJ determines a single state.

The product (2S+1)(2L+1) for the number of possible microstates with given S and L is also the 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), with good quantum numbers whose corresponding operators are mutually commutative. With given S and L, the eigenstates in this representation span function space with dimension (2S+1)(2L+1), as and . In the coupled representation where total angular momentum (spin + orbital) can be treated, the associated microstates or eigenstates are and these states span the function space with dimension of as . Obviously the dimensions of 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 parity[edit]

The 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 electron is 2(2l+1) which is even, so the summation of li in closed subshell 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:

2Po
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[edit]

It 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.

  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 shells and subshells are full then the term symbol is 1S0.
  2. Distribute the electrons in the available orbitals, following the Pauli exclusion principle. First, fill the orbitals with highest ml value with one electron each, and assign a maximal ms to them (i.e. +½). Once all orbitals in a subshell have one electron, add a second one (following the same order), assigning ms = −½ to them.
  3. 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).
  4. Calculate J as
    • if less than half of the subshell is occupied, take the minimum value J = |LS|;
    • 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.

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 (l = 1).

2. There are three orbitals (ml = 1, 0, −1) that can hold up to 2(2l + 1) = 6 electrons. The first three electrons can take ms = ½ (↑) but the Pauli exclusion principle forces the next two to have ms = −½ (↓) because they go to already occupied orbitals.

ml
+1 0 −1
ms: ↑↓ ↑↓

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

4. As fluorine 2p subshell is more than half filled, J = L + S = 32. Its ground state term symbol is then 2S+1LJ = 2P32.

Term symbols for an electron configuration[edit]

The process to calculate all possible term symbols for a given electron configuration 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, t = 2(2l+1). If there are e electrons in a given subshell, the number of possible microstates is
As an example, lets take the carbon electron structure: 1s22s22p2. After removing full subshells, there are 2 electrons in a p-level (l = 1), so we have
different microstates.
  • Second, draw all possible microstates. Calculate ML and MS for each microstate, with where mi is either ml or ms for the i-th electron, and M represents the resulting ML or MS respectively:
  ml  
  +1 0 −1 ML MS
all up 1 1
0 1
−1 1
all down 1 −1
0 −1
−1 −1
one up
one down
↑↓ 2 0
1 0
0 0
1 0
↑↓ 0 0
−1 0
0 0
−1 0
↑↓ −2 0
  • Third, count the number of microstates for each MLMS possible combination
  MS
  +1 0 −1
ML +2 1
+1 1 2 1
0 1 3 1
−1 1 2 1
−2 1
  • Fourth, extract smaller tables representing each possible term. Each table will have the size (2L+1) by (2S+1), and will contain only "1"s as entries. The first table extracted corresponds to ML ranging from −2 to +2 (so L = 2), with a single value for MS (implying 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 S = L = 1, a 3P term). The remaining table is a 1×1 table, with L = S = 0, i.e., a 1S term.
S = 0, L = 2, J = 2
1D2
  Ms
  0
Ml +2 1
+1 1
0 1
−1 1
−2 1
S=1, L=1, J=2,1,0
3P2, 3P1, 3P0
  Ms
  +1 0 −1
Ml +1 1 1 1
0 1 1 1
−1 1 1 1
S=0, L=0, J=0
1S0
  Ms
  0
Ml 0 1
  • Fifth, applying 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[edit]

  • 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 spin quantum number "S".

where the floor function denotes the greatest integer not exceeding x.

The detailed proof could be found in Renjun Xu's original paper.[3]

  • 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.[3]

Alternative method using group theory[edit]

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 Γ(0) = S, Γ(1) = P and 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(|jk|)

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. [4]

Summary of various coupling schemes and corresponding term symbols[edit]

Basic concepts for all coupling schemes:

  • : individual orbital angular momentum vector for an electron, : individual spin vector for an electron, : individual total angular momentum vector for an electron, .
  • : Total orbital angular momentum vector for all electrons in an atom ().
  • : total spin vector for all electrons ().
  • : total angular momentum vector for all electrons. The way the angular momenta are combined to form depends on the coupling scheme: for LS coupling, for jj coupling, etc.
  • A quantum number corresponding to the magnitude of a vector is a letter without an arrow (ex: l is the orbital angular momentum quantum number for and )
  • The parameter called multiplicity represents the number of possible values of the total angular momentum quantum number J for certain conditions.
  • For a single electron, the term symbol is not written as S is always 1/2 and L is obvious from the orbital type.
  • For two electron groups A and B with their own terms, each term may represent S, L and J which are quantum numbers corresponding to the , and vectors. "Coupling" of terms A and B to form a new term C means finding quantum numbers for new vectors , and . This example is in LS coupling which defines the vectors to be combined. Of course, the angular momentum addition rule is that where X can be s, l, j, S, L, J or any other angular momentum-related quantum number.

LS coupling (Russell-Saunders coupling)[edit]

  • Coupling scheme: and are calculated first then is obtained. In practical point of view, it means L, S and J are obtained by using addition rule of angular momentums with given electronics groups that are to be coupled.
  • Electronic configuration + Term symbol: . is a Term which is from coupling of electrons in group. n,l are principle quantum number, orbital quantum number and means there are N (equivalent) electrons in nl subshell. (2S+1) is multiplicity representing number of possible final total angular momentum quantum number J with given S and L, as long as . For , multiplicity is (2L+1) but (2S+1) is still used in writing Term symbol. Strickly speaking, is called Level and is called Term. Sometimes superscript o is attached to the Term, means the parity of group is odd (P = -1).
  • Example:
    1. : is Level of 3d7 group in which are equivalent 7 electrons are in 3d subshell.
    2. : Terms are assigned for each group (with different principal quantum number n) and rightmost Level is from coupling of Terms of these groups so represents final total spin quantum number S, total orbital angular momentum quantum number L and total angular momentum quantum number J in this atomic energy level.
    3. : There is a space between 5d and . In means and 5d are coupled to get . Final level is from coupling of and 6p.
    4. : There is only one Term which is isolated in left of space. It means is coupled lastly; and 6s are coupled to get then and are coupled to get final Term .

jj Coupling[edit]

  • Coupling scheme: .
  • Electronic configuration + Term symbol:
  • Example:
    1. : There are two groups. One is and the other is . In , there are 2 electrons having same in 6p subshell while there is an electron having in the same subshell for .Coupling of these two group results in J = 3/2.
    2. : 9/2 in () is for 1st group and 2 in () is J2 for 2nd group . Subscript 11/2 of Term symbol is final J for .

J1L2 coupling[edit]

  • Coupling scheme: and .
  • Electronic configuration + Term symbol: where (2S2+1) is multiplicity representing number of possible final total angular momentum quantum number J with given S2 and K, as long as K > S2. For S2 > K, multiplicity is (2K + 1) but (2S2 + 1) is still used in writing Term symbol.
  • Example:
    1. : . 9/2 is K, which comes from coupling of J1 and l2. Subscript 5 in Term symbol is J which is from coupling of K and s2.
    2. : . 7/2 is K, which comes from coupling of J1 and L2. Subscript 7/2 in Term symbol is J which is from coupling of K and S2.

LS1 coupling[edit]

  • Coupling scheme:, .
  • Electronic configuration + Term symbol: where (2S2 + 1) is multiplicity representing number of possible final total angular momentum quantum number J with given S2 and K, as long as K > S2. For S2 > K, multiplicity is (2K+1) but (2S2 + 1) is still used in writing Term symbol.
  • Example:
    1. : . .

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 [1].

See also[edit]

Notes[edit]

  1. ^ 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.

References[edit]

  1. ^ a b NIST Atomic Spectrum Database To read neutral carbon atom levels for example, type "C I" in the Spectrum box and click on Retrieve data.
  2. ^ Levine, Ira N., Quantum Chemistry (4th ed., Prentice-Hall 1991), ISBN 0-205-12770-3
  3. ^ a b Xu, Renjun; Zhenwen, Dai (2006). "Alternative mathematical technique to determine LS spectral terms". Journal of Physics B: Atomic, Molecular and Optical Physics 39: 3221–3239. arXiv:physics/0510267. Bibcode:2006JPhB...39.3221X. doi:10.1088/0953-4075/39/16/007. 
  4. ^ McDaniel, Darl H. (1977). "Spin factoring as an aid in the determination of spectroscopic terms". Journal of Chemical Education 54 (3): 147. Bibcode:1977JChEd..54..147M. doi:10.1021/ed054p147.