Spectral term

There are four quantum numbers for describing the state of an electron, they are principal quantum number n, which takes the value of 1, 2, 3, 4, azimuthal quantum number, 

In a multi-electronic atom, the following quanmm numbers can also be used to describe the energy levels, and the relationships between the quantum number of electrons are as follows. 

The spectral term is a symbol which combines the azimuthal quantum number I and magnetic quantum number m to describe the energy level relationship between electronic configurations. 

Seven orbitals are present in the 4f shell (/ = 3). Their magnetic quanmm numbers are —3, -2, —1,0, 1,2, and 3, respectively. When lanthanide elements are in their ground states, the distribution of the 4f electrons in the orbitals are as shown in Table 1.2. Here, A represents the energy difference between the ground state and the J multiple state that lies right above the ground state 4/ is the spin-orbital coupling coefficient. 

In this table, Ml is the total magnetic quantum number of the ion. Its maximum is the total orbital angular quantum number L. Ms is the total spin quantum number along the magnetic field direction. Its maximum is the total spin quantum number 5. / = L 5, is the total angular momentum quantum number of the ion and is the sum of the orbital and spin momentum. For the first seven ions (from La + to Eu +), J =L —S, for the last eight ions (from Gd + to Lu +), J = L + S. The spectral term consists of three quantum numbers, L, S, and J and may be expressed as. The value of L is indicated by S, P, D, F, G, H, and I for L = 0, 

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The bond diagrams provide an obvious simple method of determining the allowed spectral terms for equivalent electrons with Russell-Saunders coupling, which may be convenient for the reason that it separates states of different multiplicity at the start. 

FIGURE 18.4 A completeTanabe-Sugano diagram for a t/2 metal ion in an octahedral field. When A = 0, the spectral terms (shown on the left vertical axis) are those of the free gaseous ion. Spectroscopic transitions occur between states having the same multiplicity. 

In his last paper Dunham obtained a formula for values of spectral terms for a particular isotopic species i in a particular electronic state, which we suppose generally to be of symmetry class 2 or 0 implying neither net electronic orbital nor net intrinsic electronic angular momentum ... 

Two further comments about formula 8 and its parameters are noteworthy. The number of terms with coefficients l i in the double sum is formally doubly infinite so that formula 8 can represent spectral terms involving arbitrarily large values of V and 7 with data of finite extent the sums become truncated in a systematic and consistent manner conforming to parameters of a minimum number required according to that intermediate model. Instead of exact equalities in formulae 10 and 11, the approximations arise because each term coefficient constitutes a sum of contributions [5],... 

In an analogous manner, this approach is extensible to treat simultaneously data of multiple isotopic variants of a particular diatomic species such a treatment might be based on apphcation of differences of spectral terms according to formula 19, with empirical parameters and A therein as explained at that point. In such a treatment there is convenience in distributing a factor reduced mass p between vibrational and rotational quanmm numbers, or rather their respective functionals, in the following form, known as mass-reduced quantum numbers [68],... 

Although authors [74,75] have claimed that a generator-coordinate theory yields an expression for spectral terms exactly of Dunham s form, as in formula 8, in which however term coefficients encompass intrinsically elfects of at... 

Data are often normalized so that the area under the curve is preserved. This area is given by the dc spectral term, that is, for /t = 0. To preserve the area in the discrete inverse-filtered result, every term should be multiplied by the dc spectral components of the impulse response function (if the impulse response function has not been normalized earlier). We would then have ford... 

The two postulates are compatible with the observation that the frequencies of the spectral lines emitted by an atom can be represented as the differences between pairs of a set of frequency values, called the term values or spectral terms of the atom. These term values are now seen to be the values of the energy of the stationary states divided by h (to give frequency, in sec- 1), or by he (to give wave number, in cm- 1, as is customarily given in tables of term values). 

Hybrid states of this sort are to be formed from structures with the same value of J and also with the same parity. The parity of a configuration is even in case that it involves an even number of electrons in orbitals with odd value of l (p,/, etc.) and odd in case that it involves an odd number of electrons in orbitals with odd l. In tables of spectral terms the parity is often indicated by use of a superscript ° on the symbols of states with odd parity. In the above example of neutral osmium the two configurations considered have even parity. 

In helium, the para-state is one group or system of terms in the spectrum of helium that is due to atoms in which the spin of the two electrons are opposing each other. Another group of spectral terms, the orthohelium terms, is given by those helium atoms whose two electrons have parallel spins. Because of the Pauli Exclusion Principle, a helium atom in its ground state must be in a para-state. 

Usually the dependence of the radial integrals of electronic transitions on spectral terms is neglected, and their value averaged with respect to these terms is employed. However, there are cases when the role of this dependence may increase substantially and may even become decisive. This is the case, when the radial integral itself becomes small and then it is rather sensitive even to insignificant improvements of wave function. 

The coupled energy states in RS coupling are called multiplets and are described by spectral terms of the form 2,s + X, where 25+1 is the spin multiplicity and S is the total spin quantum number. 

The electron configuration d3 produces a number of states, one of which has the spectral term 2G. Describe the splitting and/or re-labeling of this 2G state under the following perturbations and summarize your discussion in the form of a correlation... 

In Oh symmetry the ground-state configuration of this low-spin complex would be t2g. Determine the ground-state and excited-state spectral terms in Oh symmetry. 

Determine the ground-state electron configuration and spectral term of the following octahedral complexes low-spin Fe(CN)g- low-spin Cr(CO)6 high-spin Cr(H20)g+. 

In table I, the systems of regularities now known for arc and spark spectra of ten elements in the fourth period are represented, the numbers, 1,2,3, 4, etc., indicating the maximum multiplicities in the spectral terms or atomic energy levels. 

The virtual presence of all the spectral terms in the effective time makes this approach more adequate than the superparamagnetic blocking model. In the latter, the effective relaxation time of magnetization is identified just with the inverse of the decrement X], which is the smallest at , 0. 

Electronic configurations and spectral terms of ground state trivalent lanthanide ions are listed in Table 1.2. Figure 1.6 shows the energy level diagram for trivalent lanthanide ions. 

Table 1.2 Electronic configurations and spectral terms of trivalent lanthanide ions in the gronnd state [5].

Table 1.7 Ground state spectral terms and electronic repulsive energy related constants of trivalent lanthanide ions [15].

Trivalent ions 4f electrons Total angnlar momentnm L Ground state spectral term (eV) (eV) e 63... 

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