Research Journ al of Chemical Sciences ___ ______________________________ ______ ____ ___ ISSN 22 31 - 606X Vol. 3 ( 4 ), 19 - 24 , April (201 3 ) Res. J. Chem. Sci. International Science Congress Association 19 Studies on Atomic Spectroscopic Terms and Term Symbols for Non - equivalent Electrons of d 2 s 1 p 1 Configuration Using Russell - Saunders Coupling Scheme Meena Parmeshwar Lal 1 *, Kumar Nitin 1 , Meena Anoop Singh 1 and Meena Kamod Singh 2 1 Department of Chemistry, M .L.S. University, Udaipur, Rajasthan - 313001, INDIA 2 Department of Chemistry, M.L.V. Govt. College, Bhilwara, Rajasthan - 311001, INDIA Available online at: www.isca.in Received 11 th January 201 3 , revised 11 th February 201 3 , accepted 13 rd March 201 3 Abstract Russell - Saunders (R - S) coupling scheme is a n important coupling scheme to determine the terms and to assign the term symbols to the terms to the valence electrons of free atoms. The atomic term symbols provide the inf ormation about the energy of atomic valence electrons , total spin, total orbital angular momentum, total angular momentum , spectral and magnetic properties of atom . In this proposed work computation is done to determine all the possible terms for the non - equivalent valence electrons of d 2 s 1 p 1 configuration manually and the term symbols are assigned to the terms using Russell - Saunders coupling scheme. The possible microstates calculated for the non - equivalent electrons of d 2 s 1 p 1 configuration are 540 and the total numbers of atomic spectroscopic terms determined from these microstates are 38 (17 types) , 6 are quintets (5 types), 17 are triplets (6 types) and 13 are singlet’s (6 types). The ground state term for the non - equivalent electrons of d 2 s 1 p 1 confi guration is quintet G ( 5 G) and the ground state is quintet G two ( 5 G 2 ) . Key w ords: Term symbol, m icrostate, quintet , triplet and singlet . Introduction Non - equivalent electrons are those which have different values of either n or l such as 3p 1 3d 1 , 4f 1 5d 1 or 4f 2 5d 1 configuration. The coupling of angular momentum of non equivalent electrons is straight forward than for equivalent electrons 1 . There are three types of interactions between the electrons viz. spin - spin coupling, orbit - orbit coupling and spin - orbit coupling. It is assumed that: spin - spin coupling � orbit - orbit coupling � spin - orbit coupling and two principle coupling schemes Russell - Saunders coupling and j - j coupling adopted for arising or splitting of state in free atom and it is found tha t the Russell Saunders scheme gives a good approximation for the first row transition series elements where spin - orbit (j - j) coupling scheme can generally be ignored, however for elements with atomic number greater than thirty, spin - orbit coupling becomes more significant due to higher nuclear charge and the j - j coupling scheme is used 2 - 4 . However, for heavier atoms it is still convenient to use Russell - Saunders scheme 5 . These interactions (coupling) produced by the electron’s orbital and spin angular momen tum give rise to a series of energy levels or states which are characterized by energy, orbital angular momentum, spin angular momentum and total angular momentum and expressed by proper term symbols and defined by new quantum numbers L, M L , M S and S and for multi electron systems these are obtained summing vectorially the quantum numbers for the individual electrons . The terms have orbital degeneracy (2L+1) and spin degeneracy (2S+1) so that the total degeneracy is equal to multiplication of above two an d it is characterized the different possible combination of orbital and spin which individual electron can acquired may be called microstate 6 . Total number of microstates increases with the increase the number of electrons in orbital and for the non equiv alent electronic system the microstates are much greater than the similar equivalent electronic system. The formulation of hole can be used for the sub shell that is more than half full. When a sub shell is more than half full, it is simpler and more conve nient to work out the terms by considering the holes that are vacancies in the various orbital’s rather than larger number of electrons actually present. By considering holes the terms which arise for pair of atoms with p n and p 6 - n , d n and d 10 - n and f n and f 14 - n arrangements give rise the identical terms 7 . The term is applied for energy associated with the state of an atom involved in a transition and t erm symbols are abbreviated description of the energy, angular momentum and spin multiplicity of an atom i n particular state and a complete term symbol is represented by (2S+1) L J 1,5,7,8,9 . For a particular electronic configuration of an atom the energy states which are degenerate when the ion is free of perturbing influences must be break up into two or more non equivalent states when the ion or atom is introduced into a lattice. These splitting are by purely electrostatic forces 10 . The term splits up into singlet, doublet, triplet, quartet, quintet, sextet etc. due to spin - spin coupling and orbit - orbit coupl ing which further split up into different states due to orbit - spin coupling that give different values of J 11 .The ground state term and order of stability of other terms (excited states or terms) can be determine by applying Hund’s rule 1, 5,7,9,10,12 . Research Journal of Chemical Scie nces ___ _ _ _______________________________ ______________ _ ________ ISSN 22 31 - 606X Vol. 3 ( 4 ), 19 - 24 , April (201 3 ) Res. J. Chem. Sci. International Science Congress Association 20 Meth odology Calculation of Total number of Microstates for Non - equivalent Electrons of d 2 s 1 p 1 Configuration: When placing electrons in orbital’s there is usually more than one way to accomplish this, particularly when the electrons are going into a degenerate set of orbital’s. The orbital’s can be considered to be boxes, two boxes per orbital corresponding to the two different values of the electron spin. The question of arranging x electrons in a degenerate set of ‘r’ orbital’s is equivalent to asking how man y ways are there to distribute ‘n’ indistinguishable objects among n boxes (where n would equal 2r). The answer is given by the expression 13 . n ! Number of ways of filling electr ons N= x! (n! - x!) n= 2(2l+1) or two wise of the Total No. of orbital’s, x= Total No. of electron in sub Shell For d 2 p 1 s 1 system n= 10, 6 & 2 and x= 2 and 1, So, 10! 6! 2! N= X X 2! (10! - 2! ) 1! (6! - 1!) 1! (2! - 1!) 10x9x8x7x6x5x4x3x2x1 6x5x4x3x2x1 2x1 N = X X 2x1x8x7x6x5x4x3x2x1 1x5x4x3x2x1 1x1 N = 45x6x2= 540 Microstates Possible Spin States for the Non - equivalent Electrons of d 2 s 1 p 1 Configuration that Allowed by the Pauli Principle : Possible spin states with total number of microstates can be determined by arranging the electrons in orbital’s according to Pauli principle and all possible spin states for the equivalent or non - equivalent electrons of a particular c onfiguration can be a rranged systematically with their total spin and total number of microstates. For non equivalent electrons of d 2 s 1 p 1 c onfiguration possible spin states are arranged in table - 1. Statistics of Microstates : Statistically the microstates for the equivalent o r non - equivalent electrons of a particular c onfiguration can be represented numerically and symbolically. For the non equivalent electrons of d 2 s 1 p 1 c onfiguration the microstates are represented statistically numerically table - 2 and symbolically t able - 3. Determination of Orbital Angular Momentum Quantum Number (L), l - l Coupling : It is a vector sum of all the l values of all electrons. The coupling of orbital momentum of equivalent or non - equivalent electrons is referred as l - l coupling, it gives a resulta nt L of magnitude [L (L+1)] 1/2 h/2π =L*h/2π 1 . It defines the state of free atom or ion as a whole while ‘l ’ defines the state of the electron only, L is always an integer including zero, it is quantized so the only permissible arrangements are those where the resultant is whole number of quanta. The orientations of l 1, l 2 l 3 …….. l n which they can take up are governed by values that the quantum number L can take. L is associated with the total orbital angular momentum for the four electrons of d 2 s 1 p 1 confi guration and restricted to the values L=(l 1 +l 2 ), (l 1 +l 2 - 1), (l 1 +l 2 - 2),…..,|(l 1 - l 2 )| 1,5,7,10,12 . The space quantization of L produces 2L+1 components with M L =L, L - 1, ….. - L analogous of l . In the present case L=5,4,3,2,1,0 and magnitudes of L are 30 1/2 h/ 2π , 20 1/2 h/2π , 12 1/2 h/2π , 6 1/2 h/2π , 2 1/2 h/2π and 0 respectively. The spectroscopic terms of the atom are labeled S, P, D, F, G, H, corresponding to L=5, 4,3,2,1 and 0 respectively . It follows that the d 2 s 1 p 1 configuration gives rise to S, P, D , F, G, and H terms . Table - 1 Statistics of microstates (numerically) of non - equivalent electrons of d 2 s 1 p 1 configuration S.N. Total spin Spin orientation of electrons in orbital’s Microstates Total microstates 1 +2 ↑ ↑ ↑ ↑ 30 30 2 +1 ↑ ↑ ↓ ↑ 30 165 3 ↑ ↑ ↑ ↓ 30 4 ↑ ↑ ↑ ↓ 30 5 ↑ ↓ ↑ ↑ 60 6 ↑↓ ↑ ↑ 15 7 - 1 ↑↓ ↓ ↓ 15 75 8 ↓ ↓ ↓ ↑ 30 9 ↓ ↓ ↑ ↓ 30 10 - 2 ↓ ↓ ↓ ↓ 30 30 11 0 ↓ ↓ ↑ ↑ 30 240 12 ↑ ↓ ↓ ↑ 60 13 ↑ ↓ ↑ ↓ 60 14 ↑ ↓ ↑ ↑ 60 15 ↑↓ ↑ ↓ 15 16 ↑↓ ↓ ↑ 15 Total numb er of microstates - 540 Research Journal of Chemical Scie nces ___ _ _ _______________________________ ______________ _ ________ ISSN 22 31 - 606X Vol. 3 ( 4 ), 19 - 24 , April (201 3 ) Res. J. Chem. Sci. International Science Congress Association 21 Determination of Total Spin Angular Momentum Quantum Number (S) s - s Coupling : It represents the total spin of atom, while the ‘s’ represents spin state of an electron and the coupling between the spin momentum is referred as s - s cou pling. The vector for each electron is always of magnitude of 3/4 1/2 h/2π according to [s(s+1)] 1/2 h/2π = s* h/2π 1 . The vector can only take up orientations relative to each other such that the magnitude of s - s coupling can as follow [S (S+1)] 1/2 h/2π = S*h/2π. S= (s 1 +s 2 ), (s 1 +s 2 - 1), (s 1 +s 2 - 2)….…, | (s 1 +s 2 )| 1,5,7,10,12 . In the present case of d 2 s 1 p 1 configuration for the four electrons the values of S= 2, 1 or 0 and the vector sums giving resultant S vectors of magnitude 6 1/2 h/2π, 2 1/2 h/2π and zero. Determination of M L and M S : M L = ∑m l = the component of the total ang ular momentum along a given axis and total values of are M L =2L+1, M L = +L …0… - L. Total possible values of M L for d 2 s 1 p 1 configuration are 2x5+1= 11 which are ranged from +5 to - 5. M S = ∑m s , It defines spin state for given ‘S’ value and it is equal to (2S+1) . M S =+S…0… - S. Total M S values for the non equivalent electrons of d 2 s 1 p 1 configuration are 2x2+1=5 which are ranged from +2 to - 2. Determination of (J) l - s Coupling : It is a resultant of the orbital angular momentum vector and the electron spin angular m omentum vector. The coupling between L and S is due to spin – orbit interaction caused by the positive charge Ze on the nucleus and is proportional to (Z 4 ). 1 This coupling gives total angular momentum vector J [J (J+1)] 1/2 h/2π = J*h/2π, where J are restricted to the values J = L+S , L+S - 1,…..L - S, from which it follows that if L ≥ S the J can take 2S+1 values , but if Lit can take 2L+1 values. When L=0 J can take only one value 1, 7,9,14 . The vector sum can be made only in certain ways and the values of ‘J’ may be either 1+1/2 or 1 - 1/2. 1 - 1/2 is of lower energy state since in 1 - 1/2 state the orbital and spin are opposed 8,14 . Resolving the Chart of Microstate into appropriate atomic states : An atomic state forms an array o f microstate consisting 2S+1 columns and 2L+1 rows. Thus, for a 3 H State requires two columns or (11x3) array and 5 G State requires (9x5) array 5, 7 . By removing each state from the microstate table we can draw a microstate sub table for each state 15 - 18 . D rawing a Complete Matrix Table : A complete matrix table for d 2 s 1 p 1 configuration can be drawn t able - 4 that includes all terms, term symbols, total microstates, multiplicity, total J values and values taken up by J . 15 - 18 . Result s and Discussion The atomi c spectroscopic terms determined for d 2 s 1 p 1 configuration are 17 types, among these 5 types are quintet, 6 types are triplet and 6 types are singlet. These terms are 3 H, 1 H, 5 G, 3 G(3 - Terms), 1 G(2 - Terms), 5 F, 3 F(4 - Terms), 1 F(3 - Terms), 5 D(2 - Terms), 3 D(5 - T erms), 1 D(3 - Terms), 5 P, 3 P(4 - Terms), 1 P(3 - Terms), 5 S, 3 S (2 - Terms), 1 S and the stability order of these atomic spectroscopic terms is 5 �G 5 �F 5 D(2)� 5 �P 5 �S 3 �H 3 G(3)� 3 F(4)� 3 D(5)� 3 P(4)� 3 S(2)� 1 �H 1 G(2)� 1 F(3)� 1 D(3)� 1 P(3)� 1 S and the ground state term is quintet G ( 5 G) and ground state is quintet G two ( 5 G 2 ). Conclusion Total atomic spectroscopic terms determined for the non equivalent electrons of d 2 s 1 p 1 configuration of free atom are 38 (17 types) which grouped into 17 energy states due to the s - s and l - l coupling, among these 5 are quintets, 6 are triplets and 6 are singlet, which further split up into different energy states due to spin – orbit (s - l) coupling. The splitting of terms of non equivalent electrons of d 2 s 1 p 1 configuration and their stabi lity order according to Hund’s rule can be drawn graphically f igure - 1 . Table - 2 Possible spin states for the non - equivalent electrons of d 2 s 1 p 1 configuration M S M L ↓ +2 +1 0 - 1 - 2 Total 5 1 2 1 4 4 1 5 8 5 1 20 3 2 10 16 10 2 40 2 4 17 26 17 4 68 1 5 22 34 22 5 88 0 6 25 38 25 6 100 - 1 5 22 34 22 5 88 - 2 4 17 26 17 4 68 - 3 2 10 16 10 2 40 - 4 1 5 8 5 1 20 - 5 1 2 1 4 Total 30 135 210 135 30 540 Research Journal of Chemical Scie nces ___ _ _ _______________________________ ______________ _ ________ ISSN 22 31 - 606X Vol. 3 ( 4 ), 19 - 24 , April (201 3 ) Res. J. Chem. Sci. International Science Congress Association 22 Table - 3 Statistics of microstates (symbolically) of non - equivalent electrons of d 2 s 1 p 1 configuration M S M L ↓ +2 +1 0 - 1 - 2 Total microstates 5 | || | 4 4 | ||||| |||||||| ||||| | 20 3 || |||||||||| |||||||||||||||| |||||||||| || 40 2 |||| |||||||||||||| ||| |||||||||||||||||||||||||| ||||||||||||||||| |||| 68 1 ||||| |||||||||||||||||||||| |||||||||||||||||||||||||||||||||| |||||||||||||||||||||| ||||| 88 0 |||||| ||||||||||||||||||||||||| |||||||||||||||||||||||||||||||||||||| ||||||||||||||||||||||||| |||||| 100 - 1 ||||| |||||||||||||||||||||| |||||||||||||||||||||||||||||||||| |||||||||||||||||||||| ||||| 88 - 2 |||| ||||||||||||||||| |||||||||||||||||||||||||| ||||||||||||||||| |||| 68 - 3 || |||||||||| |||||||||||||||| |||||||||| || 40 - 4 | ||||| |||||||| ||||| | 20 - 5 | || | 4 Total number of microstates - 540 Table - 4 Complete matrix table for the non - equivalent electrons of d 2 s 1 p 1 configuration S.N L L Label S 2S+1 Term symbol Total values of J Several possible states Array Micro states 1 5 H 1 0 3 1 3 H 1 H J=3 J=1 3 H 6 , 3 H 5 , 3 H 4 1 H 5 11x3 11x1 33 11 2 4 G 2 1 0 5 3 1 5 G 3 G(3) 1 G(2) J=5 J=3 J=1 5 G 6 , 5 G 5 , 5 G 4 , 5 G 3 , 5 G 2 3 G 5 , 3 G 4 , 3 G 3 1 G 4 9x5 (9x3)3 (9x1)2 45 81 18 3 3 F 2 1 0 5 3 1 5 F 3 F(4) 1 F(3) J=5 J=3 J=1 5 F 5 , 5 F 4 , 5 F 3 , 5 F 2 , 5 F 1 3 F 4 , 3 F 3 , 3 F 2 3 F 1 7x5 (7x3)4 (7x1)3 35 84 21 4 2 D 2 1 0 5 3 1 5 D(2) 3 D(5) 1 D(3) J=5 J=3 J=1 5 D 4 , 5 D 3 , 5 D 2 , 5 D 1 , 5 D 0 3 D 3 , 3 D 2 , 3 D 1 1 D 2 (5x5)2 (5x3)5 (5x1)3 50 75 15 5 1 P 2 1 0 5 3 1 5 P 3 P(4) 1 P(3) J=5 J=3 J=1 5 P 3 , 5 P 2 , 5 P 1 3 P 2 , 3 P 1 , 3 P 0 1 P 0 3x5 (3x3)4 (3x1)3 15 36 9 6 0 S 2 1 0 5 3 1 5 S 3 S(2) 1 S J=5 J=3 J= 1 5 S 2 3 S 1 1 S 0 1x5 91x3)2 1x1 5 6 1 Total number of microstates - 540 Research Journal of Chemical Scie nces ___ _ _ _______________________________ ______________ _ ________ ISSN 22 31 - 606X Vol. 3 ( 4 ), 19 - 24 , April (201 3 ) Res. J. Chem. Sci. International Science Congress Association 23 Figure - 1 Splitting of the terms of the non - equivalent electrons of d 2 s 1 p 1 configuration Acknowledge ment The Authors are thankful to the members of department of Chemistry, M.L.V. Govt. College, Bhilwara (Rajasthan) and Department of Chemistry, M.L.S. University, Udaipur (Rajasthan) for helping and encouraging to us to do this work. References 1. Hollas J .M., Modern Spectroscopy, Wiley India Pvt. Ltd., ( 2010 ) 2. Mehlman G., Balloffet and Estewa J.M., Far - Ultraviolet Absorption Spectra with Auto - Ionized Levels of Beryllium and Magnesium, Astrophisic J ., 157 ,957 - 956 , (1969) 3. Wopg E.Y., Configuration Interaction of the Pr 3+ Ion , J. Chem. Phys ., 38(4) , 976 (1963) 4. Lever A.B.P., The Electronic Spectra of Transition Metal Complexes with F (Free Ion) Ground Terms Derivation of Dq and B., J. Chem. Ed ., 45 , 711 - 712 (1968) 5. 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