Резюме. In this article j-j coupled atomic terms for (n-1)f\(^{X}\) nd\(^{1}\) (x=1 and 3) configurations using microstate building method were computed and the j-j terms for (n-1)f\(^{1}\) nd\(^{1}\) configuration correlated pictorially with the L-S terms. The j-j coupled atomic terms derived for (n-1)f\(^{1}\) nd\(^{1}\) configuration are [(7/2, 5/2) or (f\(_{7/2}\))\(^{1}\) (d\(_{5/2}\))\(^{1}\)], [(7/2, 3/2) or (f\(_{7/2}\))\(^{1}\) (d\(_{3/2}\))\(^{1}\)], [(5/2, 5/2) or (f\(_{5/2}\))\(^{1}\)(d\(_{5/2}\))\(^{1}\)] and[(5/2, 3/2) or (f\(_{5/2}\))\(^{1}\) (d\(_{3/2}\))\(^{1}\)], and for (n-1)f\(^{3}\) nd\(^{1}\) configuration are [(7/2, 7/2, 7/2, 5/2) or (f\(_{7/2}\))\(^{3}\) (d\(_{5/2}\))\(^{1}\)], [(7/2, 7/2, 7/2, 3/2) or (f\(_{7/2}\))\(^{3}\)(d\(_{3/2}\))\(^{1}\)], [(7/2, 7/2, 5/2, 5/2) or (f\(_{7/2}\))\(^{2}\) (f\(_{5/2}\))\(^{1}\)(d\(_{5/2}\))\(^{1}\)], [(7/2, 7/2, 5/2, 3/2) or (f\(_{7/2}\))\(^{2}\) (f\(_{5/2}\))\(^{1}\) (d\(_{3/2}\))\(^{1}\)], [(7/2, 5/2, 5/2, 5/2) or (f\(_{7/2}\))\(^{1}\) (f\(_{5/2}\))\(^{2}\) (d\(_{5/2}\))\(^{1}\)], [(7/2, 5/2, 5/2, 3/2) or (f\(_{7/2}\))\(^{1}\) (f\(_{ 5/2}\))\(^{2}\) (d\(_{3/2}\))\(^{1}\)], [(5/2, 5/2, 5/2, 5/2) or (f\(_{5/2}\))\(^{3}\) (d\(_{5/2}\))\(^{1}\)] and [(5/2, 5/2, 5/2, 3/2) or (f\(_{5/2}\))\(^{3}\) (d\(_{3/2}\))\(^{1}\)].

Ключови думи: angular momentum; jj coupling; j-j coupled term; L-S coupling;microstate

Introduction

Obtaining of the atomic terms and states from a given atomic configuration is by now a conventional topic in inorganic chemistry (Li et al., 2008; Atkins et al., 2006; Hoescroff & Sharpe, 2008). Atomic terms derived by coupling angular momentum that described by two schemes one is L-S and another is \(\mathrm{j}-\mathrm{j}\) scheme. In the L-S coupling scheme, it is assumed that electronic repulsion is much greater than spin-orbit interaction and it is appropriate where, the inter-electronic repulsion is much larger than the spin orbit interaction. Number of methods have been developed to derive the L-S terms for electronic configuration via Partitioning technique (Olson, 2011), Partial term method (Kiremire, 1990), Partitioning total spin method (Chen. 1989), Group representation method (Guofan & Ellzey, 1987), Numerical algorithm method (Kiremire, 1987), Spin factoring method (McDaniel, 1977), Hyde method (Hyde, 1975), Ford method (Ford, 1972), Group theoretical method (Judd, 1967; Wybourne, 1966), Generating functions derived via group theory method (Curl & Kilpatrick, 1960), Slater graphics (Slater, 1960), Quantum mechanical method (Russell & Saunders, 1925), Vector model (Landé, 1921) and Microstate building method by electronic arrangement (Meena et al., 2012; Meena et al., 2013).

j-j coupling becomes more significant than the L-S coupling in heavy atoms and L-S coupling gradually change to \(\mathrm{j}-\mathrm{j}\) coupling in going from light to heavy atom. \(\mathrm{j}-\mathrm{j}\) coupled atomic terms can be obtained by different ways such as electronic arrangement (Meena et al., 2015); Orofino & Faria, 2010; Novak, 1999; Campbell, 1998; Gauerke & Campbell, 1994; Haigh, 1995; Rubio & Perez, 1986; Richtmyer et al., 1969; Tuttle, 1967). The j-j coupling scheme becomes more treatable in chemistry when the microstates are built by arranging electrons and this makes easy to get the terms also. The notations for L-S terms are universally standardized but not for the \(\mathrm{j}-\mathrm{j}\) coupling terms. The common notation systems for the \(\mathrm{j}-\mathrm{j}\) coupling terms that are designated by the j’s as \(\left[\left(\mathrm{j}_{1}, \mathrm{j}_{2} \ldots .\right)_{\mathrm{J}}\right]\) (Haigh, 1995) and \(\left[\left(\mathrm{j}_{1}\right)^{\mathrm{a}}\left(\mathrm{j}_{2}\right)^{\mathrm{b}}\left(\mathrm{j}_{3}\right)^{\mathrm{c}} \ldots\right]\) (Rubio & Perez, 1986; Richtmyer et al., 1969; Tuttle, 1967) but in some systems the terms are designated by \(l^{\prime}\) s as \(\left[\left(l_{\ell-1 / 2}\right)^{\mathrm{a}}\left(l_{\ell+1 / 2}\right)^{\mathrm{b}}\left(l^{\prime}{ }_{\ell^{\prime}-1 / 2}\right)^{\mathrm{c}} \ldots\right]\) (Ga (lℓ+1/2) b (l’ℓ’-1/2) c…] (Gauerke & Campbell, 1994) where \(l\) and \(l^{\prime}\) are letter designations for orbital’s , and \(\ell\) and \(\ell\) ' are the numerical values of the angular momentum quantum numbers for the electrons. In this article I used both \(\left(\mathrm{j}_{1}, \mathrm{j}_{2}, \mathrm{j}_{3}\right)_{\mathrm{J}}\) a nd \(\left[\left(l_{\ell-1 / 2}\right)^{\mathrm{a}}\left(l_{\ell+1 / 2}\right)^{\mathrm{b}}\left(l^{\prime}{ }_{\ell^{\prime}-1 / 2}\right)^{\mathrm{c}} \ldots\right]\) notations to represent the \(\mathrm{j}-\mathrm{j}\) c…] notations to represent the j-j coupled atomic terms.

Methodology

Number of microstates for electrons of \((n-1) f^{*} n d^{l}(n=1 \& 3)\) configuration

Total number of microstates for any electronic configuration is given by an expression, \(\mathrm{N}=\tfrac{\mathrm{n}!}{\mathrm{x}(\mathrm{n}-\mathrm{x})!}\) Where \(\mathrm{n}=2(2 l+1)\) and \(\mathrm{x}=\) total number of electrons in sub shell.

For \((\mathrm{n}-1) \mathrm{f}^{\mathrm{l}} \mathrm{nd}^{1}\) configuration, \(\mathrm{n}=14 \& 10\) and \(\mathrm{x}=1 \& 1, \mathrm{~N}=\tfrac{14!}{1!(14-1)!} \mathrm{x}\) \(\tfrac{10!}{1!(10-1)!}=140\), and for \((n-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration, \(\mathrm{n}=14\) & 10 and \(\mathrm{x}=3 \& 1, \mathrm{~N}=\) \(\tfrac{14!}{3!(14-3)!} \times \tfrac{10!}{1!(10-1)!}=3640\)

Number of terms for \((n-1) f^{x} n d^{l}(x=1 \& 3)\) configuration

Number of terms for a particular electronic state with equivalent electrons are \((\mathrm{x}+1)\), for subshell less than half filled, ' x ' for half filled subshell, and \((4 \ell+3-\mathrm{x})\) for subshell more than half filled, where ' \(x\) ' represent total number of electrons in partially filled subshell and ' \(\ell\) ' is numerical value of the angular momentum quantum numbers for electrons (Gauerke & Campbell, 1994). But for the electronic state with nonequivalent electrons \(\left[(\mathrm{n}-1) \mathrm{f}^{\mathrm{x}} \mathrm{nd}^{\mathrm{y}}\right]\) the number of terms are \([(\mathrm{x}+1)(\mathrm{y}+1)]\). In

\((\mathrm{n}-1) \mathrm{f}^{\mathrm{x}} \mathrm{nd}^{1}(\mathrm{x}=1 \& 3)\) configurations total \(\mathrm{j}-\mathrm{j}\) terms are four and eight. The possible \(j\) values for f electrons are \(\mathrm{j}=7 / 2\) or 5/2 and for d electron 5/2 or 3/2, and the \(\mathrm{j}-\mathrm{j}\) terms \(\left(\mathrm{j}_{1}, \mathrm{j}_{2} \ldots\right)\) derived for \((\mathrm{n}-1) \mathrm{f}^{1} \mathrm{nd}^{1}\) configuration are \((7 / 2,5 / 2)\), (7/2, 5/2), (7/2, 3/2), (5/2, 5/2) and \((5 / 2,3 / 2)\) and for \((\mathrm{n}-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration the terms are (7/2, 7/2, 7/2, 5/2), (7/2, \(7 / 2,7 / 2,3 / 2),(7 / 2,7 / 2,5 / 2,5 / 2),(7 / 2,7 / 2,5 / 2,3 / 2),(7 / 2,5 / 2,5 / 2,5 / 2),(7 / 2,5 / 2\), \(5 / 2,3 / 2)\), (5/2, 5/2, 5/2, 5/2) and (5/2, 5/2, 5/2,3/2). j-j terms in the form of \(\left[\left(l_{\ell-1 / 2}\right)^{i}\right.\) \(\left.\left(l_{\ell+1 / 2}\right)^{n-i}\right]\) notation will be \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) and \(\left[\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) for \((\mathrm{n}-1) \mathrm{f}^{1} \mathrm{nd}^{1}\) configuration and \(\left[\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\), \(\left[\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right],\left[\left(\mathrm{f}_{5 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) and \(\left[\left(\mathrm{f}_{5 / 2}\right)^{3}\right.\) \(\left(\mathrm{d}_{3 / 2}\right)^{1}\) ] for \((\mathrm{n}-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration.

Number of microstates for particular \(j-j\) atomic term

Number of microstates for a particular \(\mathrm{j}-\mathrm{j}\) term of the form \(\left[\left(1_{\ell-1 / 2}\right)^{\mathrm{i}}\left(1_{\ell+1 / 2}\right)^{\mathrm{n-}}\right]\) or \(\left(\mathrm{j}_{1}, \mathrm{j}_{2}, \mathrm{j}_{3}, \mathrm{j}_{4}\right)\) for each sub set of equivalent electrons is given by \(\tfrac{(2 \ell)!(2 \ell+2)!}{\mathrm{i}!(2 \ell-\mathrm{i})!(\mathrm{n}-\mathrm{i})!(2 \ell+2+\mathrm{i}-\mathrm{n})!}\) and for nonequivalent electrons \(\tfrac{(2 \ell)!(2 \ell+2)!}{\mathrm{i}!(2 \ell-\mathrm{i})!(\mathrm{n}-\mathrm{i})!(2 \ell+2+\mathrm{i}-\mathrm{n})!} x \tfrac{(2 \ell)!(2 \ell+2)!}{\mathrm{i}!(2 \ell-\mathrm{i})!(\mathrm{n}-\mathrm{i})!(2 \ell+2+\mathrm{i}-\mathrm{n})!}\).

For the term \((7 / 2,5 / 2)\) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) of \((\mathrm{n}-1) \mathrm{f}^{1} \mathrm{nd}^{1}\) configuration total microstates will be

\[ \tfrac{(2 \times 3)!(2 \times 3+2)!}{0!(2 \times 3-0)!(1-0)!(2 \times 3+2+0-1)!} \times \tfrac{(2 \times 2)!(2 \times 2+2)!}{0!(2 \times 2-0)!(1-0)!(2 \times 2+2+0-1)!}=48 \]

And for the term (7/2, 7/2, 7/2, 5/2) or [(f7/2) ,7/2,7/2,5/2 or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) of \((\mathrm{n}-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration total microstates will be \[ \tfrac{(2 \times 3)!(2 \times 3+2)!}{0!(2 \times 3-0)!(3-0)!(2 \times 3+2+0-3)!} \times \tfrac{(2 \times 2)!(2 \times 2+2)!}{0!(2 \times 2-0)!(1-0)!(2 \times 2+2+0-1)!}=336 \]

Computed microstates for \(\mathrm{j}-\mathrm{j}\) coupled atomic terms of \((\mathrm{n}-1) \mathrm{f}^{\mathrm{x}} \mathrm{nd}^{1}(\mathrm{n}=1 \& 3)\) configurations are given in Tables 1 and 2.

J levels for \(j-j\) coupled electronic terms for \((n-1) f^{1}\) nd \({ }^{1}\) configuration

J levels for \(\mathrm{j}-\mathrm{j}\) coupled terms are obtained by eliminating the microstates associated with the particular J level from microstate tables starting from maximum \(\mathrm{M}_{\mathrm{J}}\) value and further moving to the lowest J level. From the Table 3 for the \(\mathrm{j}-\mathrm{j}\) coupled term \((7 / 2,5 / 2)\) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) for \((\mathrm{n}-1) \mathrm{f}^{1} \mathrm{nd}^{1}\) configuration the maximum \(\mathrm{M}_{\mathrm{J}}\) values is 6 which results into \(\mathrm{J}=6\) level for this term , when the 13 microstates associated with \(\mathrm{J}=6\) level are eliminated, and the next maximum \(\mathrm{M}_{\mathrm{J}}\) obtained is 5 that yield another J level by elimination of 11 microstates associated with this \(\mathrm{J}=5\) level, and further elimination of 9, 7, 5 and 3 microstates produce 4, 3, 2 and 1 J levels for this term. The same procedure is applied for other \(\mathrm{j}-\mathrm{j}\) coupled terms for (\(\mathrm{n}-1\) ) \(\mathrm{f}^{\mathrm{l}} \mathrm{nd}^{1}\) configuration as illustrated in Table 4 for (7/2, 3/2) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\right.\) \(\left.\left(\mathrm{d}_{3 / 2}\right)^{1}\right]\) term, in Table 5 for \((5 / 2,5 / 2)\) or \(\left[\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) term, in Table 6 for \((5 / 2,3 / 2)\) or \(\left[\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) term.

J levels for \(j-j\) coupled atomic terms for \((n-1) f^{\beta}\) nd \({ }^{l}\) configuration

For \(\mathrm{j}-\mathrm{j}\) coupled term \((7 / 2,7 / 2,7 / 2,5 / 2)\) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) of \((\mathrm{n}-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration the maximum \(\mathrm{M}_{\mathrm{J}}\) values is 10 which results into \(\mathrm{J}=10\) level for this term, when the 21 microstates associated with \(\mathrm{J}=10\) level are eliminated, and the next maximum \(\mathrm{M}_{\mathrm{J}}\) obtained is 9 that yield another J level by elimination of 19 microstates associated with this \(\mathrm{J}=9\) level, and further elimination of \(34,45,52,55,45,35,20\), 9 and 1 microstates produce 8(2), 7(3), 6(4) , 5(5) , 4(5) , 3(5) , 2(4) , 1(3) and 0 J levels for this term as shown in Table 7.

The same procedure is applied for other \(\mathrm{j}-\mathrm{j}\) coupled terms for \((\mathrm{n}-1) \mathrm{f}^{3} \mathrm{nd}^{1}\) configuration as illustrated in Table 8 for (7/2, 7/2, 7/2, 7/2, 3/2) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) term, in Table 9 for (\(7 / 2,7 / 2,5 / 2,5 / 2\) ) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) term, in Table 10 for (\(7 / 2,7 / 2,5 / 2\), 3/2) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) term, in Table 11 for (7/2, 5/2, 5/2, 5/2) ) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\right.\) \(\left.\left(\mathrm{d}_{5 / 2}\right)^{1}\right]\) term, in Table 12 for (\(7 / 2,5 / 2,5 / 2,3 / 2\) ) or \(\left[\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) term, in Table 13 for (5/2, 5/2, 5/2, 5/2) or \(\left[\left(\mathrm{f}_{5 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right]\) term and in Table 14 for (5/2, 5/2, 5/2, 3/2) or \(\left[\left(\mathrm{f}_{5 / 2}\right)^{3}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right]\) term.

Results and discussion

For \((\mathrm{n}-1) \mathrm{f}^{\mathrm{l}} \mathrm{nd}^{1}\) configuration four \(\mathrm{j}-\mathrm{j}\) coupled atomic terms derived which are \(\left[\left\{(7 / 2,5 / 2) \text { or }\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right\}_{6,5,4,3,2,1}\right],\left[\left\{(7 / 2,3 / 2) \text { or }\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}_{5,4,3,2}\right],[\{(5 / 2,5 / 2)\) or \(\left.\left.\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right\}_{5,4,3,2,1,0}\right]\) and \(\left[\left\{(5 / 2,3 / 2) \text { or }\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}_{4,3,2,1}\right]\), and for the \((\mathrm{n}-1)\) \(\mathrm{f}^{3} \mathrm{nd}^{1}\) configuration eight j-j terms derived which are \(\left[\left\{(7 / 2,7 / 2,7 / 2,5 / 2)\right.\right.\) or \(\left(\mathrm{f}_{7 / 2}\right)^{3}\) \(\left.\left.\left(\mathrm{d}_{5 / 2}\right)^{1}\right\}_{10,9,8(2), 7(3), 6(4), 5(5), 4(5), 3(5), 2(4), 1(3), 0}\right],\left[\left\{(7 / 2,7 / 2,7 / 2,3 / 2) \text { or }\left(\mathrm{f}_{7 / 2}\right)^{3}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}_{9,8,7(2), 6(3) \text {, }}\right.\) 5(3), 4(4), 3(4), 2(3), 1(2), 0] \(\left[\left\{(7 / 2,7 / 2,5 / 2,5 / 2)\right.\right.\) or \(\left.\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right\}{ }_{11,10(2), 9(4), 8(6), 7(9), 6(12), 5(14),}\) 4(15), 3(14), 2(12), 1(8), 0(3) ], [{(7/2, 7/2, 5/2, 3/2) or (f7/2) ,3(14),2(f5/2) 1 (d3/2) ,0(3), \(\left[\left\{(7 / 2,7 / 2,5 / 2,3 / 2)\right.\right.\) or \(\left.\left(\mathrm{f}_{7 / 2}\right)^{2}\left(\mathrm{f}_{5 / 2}\right)^{1}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}{ }_{10,9(2), 8(3), 7(7), 6(8), 5(10), 4(11),}\) \(3(11), 2(10), 1(5), 0(2)]\), [\{(7/2,5/2, 5/2) 2 (d5/2) or \(\left.\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right\}_{10,9(2), 8(4), 7(6), 6(9), 5(11), 4(12), 3(12),}\) 2(10), 1(7), 0(2) , \(\left[\left\{(7 / 2,5 / 2,5 / 2,3 / 2) \text { or }\left(\mathrm{f}_{7 / 2}\right)^{1}\left(\mathrm{f}_{5 / 2}\right)^{2}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}_{9,8(2), 7(4), 6(6), 5(8), 4(9), 3(9), 2(8), 1(5), 0(2)}\right]\), \(\left[\left\{(5 / 2,5 / 2,5 / 2,5 / 2) \text { or }\left(\mathrm{f}_{5 / 2}\right)^{3}\left(\mathrm{~d}_{5 / 2}\right)^{1}\right\}_{7,6,5(2), 4(3), 3(3), 2(3), 1(2), 0}\right]\) and \([\{(5 / 2,5 / 2,5 / 2,3 / 2)\) or \(\left.\left.\left(\mathrm{f}_{5 / 2}\right)^{3}\left(\mathrm{~d}_{3 / 2}\right)^{1}\right\}_{6,5,4(2), 3(3), 2(2), 1(2), 0}\right]\).

Conclusion

In this article I have derived \(\mathrm{j}-\mathrm{j}\) coupled atomic terms and electronic states (\(\mathrm{J}-\) levels) for nonequivalent electrons of \((n-1) \mathrm{f}^{\mathrm{x}} \mathrm{nd}^{1}(\mathrm{x}=1 \& 3)\) configurations using simple method and the \(\mathrm{j}-\mathrm{j}\) terms and electronic states for \((\mathrm{n}-1) \mathrm{f}^{1} \mathrm{nd}^{1}\) configuration correlated pictorially with L-S terms as shown in Fig. 1.

Acknowledgement. The author is thankful to Dr. K. S. Meena, Assistant professor in Chemistry, M. L. V. Govt. College, Bhilwara (Rajasthan) for providing necessary assistance for this work.

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