New Numerical Approach to Calculate Microstates of Equivalent and Non-Equivalent Electrons

: A term symbol is used to describe atomic microstate states, which give the multiplicity and total angular momentum of the atomic state. Russel Sauder coupling scheme is used to generate terms of equivalent and non-equivalent electronic configurations. For equivalent electrons, the terms are calculated using Pauli’s principle, and the number of terms is limited and is calculated by the combination rule. The total possible electrons and total available electrons are used in the combination formula. In case of non-equivalent electrons, the number of terms are found by the permutation rule. The number of terms for equivalent electrons is less than the terms for non-equivalent electrons. The number of possible microstates for p 2 and d 5 configurations are 15 and 252 respectively. While the number of final microstates for 1p2p and 3d4d configurations are 36 and 100. In the proposed study, a Python programme was developed that generates the microstate according to filled and half-filled subshell electronic configurations for equivalent, non-equivalent, and combinations of both. Examples of microstates for non-equivalent electrons of configuration 1s2s, sp, sd, ss, 2p3p, pd, pf, 3d4d, df, 4f5f and for equivalent electrons of configuration s u , p v , d x , and f y are presented.


INTRODUCTION
Atomic microstates are identified by a spectral term that specifies their multiplicity and overall angular momentum.Term symbols provide information on the spectral and magnetic properties of various elements [1].These term values are helpful to identify coefficients of fractional parentage of wavefunctions of elements having complex structures like Praseodymium, Tantalum etc. Henry Russell and Frederick Saunders introduced the Russell-Saunders (which is abbreviated as R-S) scheme for the first time in 1923.Initially, it was used to electrons in half-filled orbits of atoms with lower atomic numbers since the spin-orbit coupling is less effective than the electrostatic effect [2].However, spin-orbit coupling due to higher nuclear charge seems to be more important for those

INTRODUCTION
Atomic microstates are identified by a spectral term that specifies their multiplicity and overall angular momentum.Term symbols provide information on the spectral and magnetic properties of various elements [1].These term values are helpful to identify coefficients of fractional parentage of wavefunctions of elements having complex structures like Praseodymium, Tantalum etc. Henry Russell and Frederick Saunders introduced the Russell-Saunders (which is abbreviated as R-S) scheme for the first time in 1923.Initially, it was used to electrons in half-filled orbits of atoms with lower atomic numbers since the spin-orbit coupling is less effective than the electrostatic effect [2].However, spin-orbit coupling due to higher nuclear charge seems to be more important for those elements that have a higher atomic number.The R-S technique is still useful for rare Earth elements and heavier transition elements.The complex spectra of structures containing valence electrons in distinct sub shells were successfully interpreted by using a vector model for terms that were established before the quantum mechanical approach [3] elements that have a higher atomic number.The R-S technique is still useful for rare Earth elements and heavier transition elements.The complex spectra of structures containing valence electrons in distinct sub shells were successfully interpreted by using a vector model for terms that were established before the quantum mechanical approach [3].The three vectors, and are produced as a result of the R-S coupling, which is the basis for the present nomenclature for the specific energy level (2S+1) LJ, where 2S+1 is the multiplicity or spin multiplicity of a term, the orbital angular momentum vectors of the valence electrons are vectorially added together to form L and spin angular momentum vectors of valence electrons are vectorially summed to generate S, and vectorial sum of L and S is J.A given term produces the number of microstates which are simply (2S + 1) × (2L + 1) and by applying Hund's rule, ground state term and the order of stability can be determined.A state of definite J can be achieved when L and S coupled together and J's allowed values range from |L + S| to |L -S| [4,5].
(1) (2) Each term spit into (2J + 1) terms with energy difference proportional to the applied field intensity (Zeeman Effect) and a quantum number M J that can have the values J, J-1, J-2, …., -J is used to define states.As a result, J can take 2S + 1 values when L ≥ S but 2L + 1 values when L < S and J can only take one value when L = 0.The valence electrons' energy levels may be defined if the possible L, S and J values are known [6].Energy associated with the state of an atom taking part in a transition is described by an R-S spectral term, and energy levels in an atom with many electrons are briefly described by term symbols.
When an ion or atom is placed into a lattice, electronic repulsion splits the degenerate state into two or more states.Equivalent electrons are ones whose l and n values are the same such as np 2 , nd 6 , nf 4 , or nf 6 configuration [7].As a result, identical terms are produced for nf 4 and nf 10 configurations.The number of microstates for the sub shell increases when the number of electrons in orbital of the incomplete sub shell increases, but the nonequivalent electronic system produces a number of microstates that is much greater than that of the similar equivalent electronic system [8][9][10][11][12][13].In 2019, Javaid et al. [14] evaluated 187 spectral terms with 457 J values and 106 wavefunctions for 4f2 5d2 configuration of Praseodymium II by using Russell -Saunders technique.Zafar et al. [15] determined 46 orthonormal wave functions for 4f 3 6s 2 ground state configuration of Praseodymium I using spectral terms in 2020.
The main purpose of our study is to develop a machine algorithm using the Python language to evaluate term symbols of equivalent and nonequivalent electronic configurations.This program provides a user input interface for saving datasets into the program directory.It asks either to enter '0' for equivalent electron configuration or to hit '1' for non-equivalent electron configuration.

METHODOLOGY
In this article, the term symbols and the microstates of filled and half-filled subshell electronic configurations are calculated.Microstates are the numerous ways in which electrons can be arranged in a set of orbitals, each of which has a unique energy.The total number of microstates (W) of a system is the total number of definite arrangements for "e" number of electrons to be placed in "n" number of possible orbital positions.The number of microstates for equivalent electrons can be calculated by using a simple expression: where e denotes the number of electrons, and n is the total available orbitals [16].
Spectral Term corresponds to energy states and provides knowledge of angular momenta.For the non-equivalent electrons, there are more available microstates than for the equivalent electrons.Some of the available microstates for non-equivalent electrons are forbidden for equivalent electrons because of Pauli's principle [17,18].In Russell Saunders Coupling Scheme, term symbols are provided by (2S+1) LJ, where S shows the total spin angular momentum, L denotes the orbital angular momentum and J symbolizes the total angular momentum [19][20][21].L can have the following values in a term symbol: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…, and the English capital letters "S, P, D, F, G, H, I, K, K" are used to represent each value of L. 2S+1 denotes the spin multiplicity of the spectral terms like singlet, doublet, triplet and so on.The Russell-Saunders technique makes the assumption that spin-orbit coupling < orbit-orbit coupling < spin-spin coupling [22,23].
A computer algorithm was designed that generates terms of equivalent electrons and non equivalent electrons of any configuration.A user inputs either '0 and 1' for generation of microstates of equivalent and non-equivalent electrons.An algorithm flowchart has been given in Figure 1.It further requires orbital numbers (e.g., 0, 1, 2, 3 for s, p, d, f respectively) as input.These inputs must be given for the task's completion, or it may lead to the failure of the program.Permitted terms for the configuration of s u , p v , d x , and f y where u=1 and 2, v = 1 to 5, x = 1 to 9 and y =1 to 13 are found by considering these equivalent configuration.The configurations 1s2s, sp, sd, ss, 2p3p, pd, pf, 3d4d, df, 4f5f have non-equivalent electrons.Term symbols for these non equivalent electronic system are generated by calculating total orbital angular momentum and total spin angular momentum.For non equivalent electrons, Pauli's Exclusion Principle is not taken into account [24,25].
vectorially added together to form L and spin angular momentum vectors of valence electrons are vectorially summed to generate S, and vectorial sum of L and S is J.A given term produces the number of microstates which are simply (2S + 1) × (2L + 1) and by applying Hund's rule, ground state term and the order of stability can be determined.A state of definite J can be achieved when L and S coupled together and J's allowed values range from |L + S| to |L -S| [4,5].
using spectral terms in 2020.
The main purpose of our study is to develop a machine algorithm using the Python language to evaluate term symbols of equivalent and nonequivalent electronic configurations.This program provides a user input interface for saving datasets into the program directory.It asks either to enter '0' for equivalent electron configuration or to hit '1' for non-equivalent electron configuration.

METHODOLOGY
In this article, the term symbols and the microstates of filled and half-filled subshell electronic configurations are calculated.Microstates are the Advancing Atomic Modeling with P numerous ways in which electrons can be arranged in a set of orbitals, each of which has a unique energy.The total number of microstates (W) of a system is the total number of definite arrangements for "e" number of electrons to be placed in "n" number of possible orbital positions.The number of microstates for equivalent electrons can be calculated by using a simple expression: where e denotes the number of electrons, and n is the total available orbitals [16].
Spectral Term corresponds to energy states and provides knowledge of angular momenta.For the non-equivalent electrons, there are more available microstates than for the equivalent electrons.Some of the available microstates for non-equivalent electrons are forbidden for equivalent electrons because of Pauli's principle [17,18].In Russell Saunders Coupling Scheme, term symbols are provided by (2S+1) LJ, where S shows the total spin angular momentum, L denotes the orbital angular momentum and J symbolizes the total angular momentum [19][20][21].L can have the following values in a term symbol: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,…, and the English capital letters "S, P, D, F, G, H

RESULTS AND DISCUSSION
The filling of electrons in a shell depends on the nature of electrons, the electrons having same principal quantum number and same orbital quantum number face the limitations in filling the energy level slots, As they have to follow the Pauli's principle, no two electrons having same four set of quantum number can occupy the same energy level.Hence the number of microstates in case of equivalent electrons are less than that are available for non-equivalent electrons.
We used the theory of the coupling of orbital and spin angular momenta.Using a Python programme, we evaluated the term symbols for both equivalent and non-equivalent electrons.These configurations s u , p v , d x , and f y have equivalent electrons (where u=1 and 2, v = 1 to 5, x = 1 to 9 and y =1 to 13) and the microstates for these configurations are calculated by our program and are given in Table 1.Moreover, the configurations 1s2s, sp, sd, ss, 2p3p, pd, pf, 3d4d, df, 4f5fhave non-equivalent electrons, the microstates for these configurations are found by using combination rule for two non-equivalent orbitals.Table 2. shows the some of the microstates for two non-equivalent orbitals.

SULTS AND DISCUSSION
ng of electrons in a shell depends on the f electrons, the electrons having same l quantum number and same orbital number face the limitations in filling gy level slots, As they have to follow the principle, no two electrons having same of quantum number can occupy the nergy level.Hence the number of tes in case of equivalent electrons are n that are available for non-equivalent s.
used the theory of the coupling of orbital n angular momenta.Using a Python me, we evaluated the term symbols for equivalent and non-equivalent s.These configurations s u , p v , d x , and f quivalent electrons (where u=1 and 2, v = 1 to 5, x = 1 to 9 and y =1 to 13) and the microstates for these configurations are calculated by our program and are given in Table 1.Moreover, the configurations 1s2s, sp, sd, ss, 2p3p, pd, pf, 3d4d, df, 4f5f have nonequivalent electrons, the microstates for these configurations are found by using combination rule for two non-equivalent orbitals.Table 2. shows the some of the microstates for two nonequivalent orbitals.
For example l1 = 1, l2 = 1 then L= 2, 1, 0 using L = |l1 + l2|,…, |l1 -l2| so the states D, P, and S will be generated.To find the multiplicity, we find the total spin generated by two electrons each having spin half.The total spin will be 1 and 0, that would lead to a multiplicity of 3 and 1.Therefore, the terms generated by two nonequivalent p-electrons are 3 D, 3 P, 3 S, 1   total spin generated by two electrons each having spin half.The total spin will be 1 and 0, that would lead to a multiplicity of 3 and 1.Therefore, the terms generated by two non-equivalent p-electrons are 3 D, 3 P, 3 S, 1 D, 1 P, and 1 S.The GitHub program repository can be accessed at https://github.com/AhmedAliRajput/Term-Symbol-Calculation.git.

CONCLUSIONS
A computer program is developed in Python to generate term values of various electronic configurations of atoms and ions.Russell Saunder coupling scheme is implemented for equivalent, non-equivalent, and combinations of both electrons in open and close shells of atoms and ions.It is very complicated to calculate terms using pen and paper, even for three equivalent electrons in a d shell.With the help of this program the task can be completed in a few seconds.This term calculator is useful for students working in the field of spectroscopy or quantum chemistry.For example, if students require to determine the microstates of np 2 configuration that is the case of equivalent electrons, generates 3 microstates that accommodates 18 electrons, wherease for np-(n+1)p configuration of nonequivalent electrons, program will generate 6 microstates that accommodate 36 electrons.If microstates for higher orbitals is to determine, it would cost a lot of time, therefore, this programs would be handy tool to deal with such lengthy calculations.The code can be downloaded from https://github.com/AhmedAliRajput/Term-Symbol-Calculation.git.

CONFLICT OF INTEREST
The authors declare no conflict of interest.

Table 1 . Term symbols for equivalent electrons Orbitals No. of Electrons Total No. of Microstates Final No. of Microstates Term Symbols
D,1P, and 1 S.The GitHub program repository can be