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Mathematical and Theoretical Physics

Research Article Volume 1 Issue 6

A new study of energy levels of hydrogenic atoms and some molecules for new more general exponential screened coulomb potential

Abdelmadjid Maireche

Department of Physics, University of M'sila, Algeri

Correspondence:

Received: August 14, 2018 | Published: November 29, 2018

citation: Maireche A. A new study of energy levels of hydrogenic atoms and some molecules for new more general exponential screened coulomb potential. Open Acc J Math Theor Phy. 2018;1(6):232-237 DOI: 10.15406/oajmtp.2018.01.00040

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Abstract

In present research paper, the solutions of the modified Schrodinger (MSE) with new more general exponential screened coulomb (NMGESC) potential, have been presented by means generalized Bopp’s shift method and standard perturbation theory, in the noncommutative three dimensional space phase (NC: 3D-RSP). The bound state energy eigenvalues, in terms of the generalized the hypergeometric function, the discreet atomic quantum numbers (j=| l-s |,......( l+s )( n,l )andm), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaacIcaca WGQbGaeyypa0JcdaabdaqaaKqzGeGaamiBaiaac2cacaWGZbaakiaa wEa7caGLiWoajugibiaacYcacaGGUaGaaiOlaiaac6cacaGGUaGaai Olaiaac6cakmaabmaabaqcLbsacaWGSbGaey4kaSIaam4CaaGccaGL OaGaayzkaaWaaeWaaeaajugibiaad6gacaGGSaGaamiBaaGccaGLOa GaayzkaaqcLbsaqaaaaaaaaaWdbiaadggacaWGUbGaamizaiaaykW7 caWGTbWdaiaacMcacaGGSaaaaa@57D5@ two infinitesimal parameters ( Θ,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaKqzGe GaeuiMdeLaaiilaiabeE8aJbGccaGLOaGaayzkaaaaaa@3DD4@ which are induced by position-position, in addition to, the dimensional parameters of NMGESC potential and the corresponding noncommutative Hamiltonian operator were obtained for hydrogenic atoms and the molecules ( CO,NO ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaKqzGe Gaam4qaiaad+eacaGGSaGaamOtaiaad+eaaOGaayjkaiaawMcaaKqz GeGaaiOlaaaa@3F2A@ We have also shown that, the total complete degeneracy of energy levels of NMGESC potential equals the new values 2 n 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaikdaca WGUbGcdaahaaWcbeqaaKqzadGaaGOmaaaajugibiaac6caaaa@3D74@ Furthermore, the global group symmetry (NC: 3D-RSP) corresponding NMGESC potential reduce to the new subgroup (NC: 3D-RS) symmetries.

Keywords: schrödinger equation, hydrogenic atoms, more general exponential screened coulomb potential, noncommutative space and phase, star product and generalized Bopp’s shift method

Abbreviations

NMGESC, new more general exponential screened coulomb potential; NC: 3D–RSP, noncommutativity three dimensional real space phase; CCRs, canonical commutations relations; NNCCRs, new noncommutative canonical commutations relations; SP, Schrödinger picture; HP, Heisenberg picture; MSE, modified Schrödinger equation

Introduction

The more general exponential screened coulomb (MGESC) potential is known to describe adequately the effective potential of a many–body system of a variety of fields such as the atomic, solid state, plasma and quantum field theory.1–4 In particularity, this potential used to calculate the bounded state eigen values of molecules ( CO,NO ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaKqzGe Gaam4qaiaad+eacaGGSaGaamOtaiaad+eaaOGaayjkaiaawMcaaKqz GeGaaiOlaaaa@3F2A@ The noncommutativity of space–time, which known firstly by Heisenberg and was formalized by Snyder at 1947, suggest by the physical recent results in string theory. Very recently, several authors have attempted to obtain either the exact or approximate solutions of the non–relativistic Schrodinger equation or two relativistic (Klein–Gordon and Dirac) equations for different potentials in NC space. We want to extended, the study of Ita et al.,3 to the case of extended quantum mechanics to the possibility of finding other applications and more profound interpretations in the sub-atomics scales on based to the works5–19 and our previously works20–40 in this context. The no relativistic energy levels for hydrogenic atoms and molecules ( CO,NO ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaKqzGe Gaam4qaiaad+eacaGGSaGaamOtaiaad+eaaOGaayjkaiaawMcaaKqz GeGaaiilaaaa@3F28@ which interacted with NMGESC potential in the context of NC space have not been obtained yet. The purpose of the present paper is to attempt study the MSE with NMGESC potential (see below):

V mg ( r )= V 0 r ( 1+( 1+αr )exp( 2αr ) ) V mg ( r ^ )= V 0 α V 0 r ^ 2 V 0 α 3 r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadAfakm aaBaaaleaajugWaiaad2gacaWGNbaaleqaaOWaaeWaaeaajugibiaa dkhaaOGaayjkaiaawMcaaKqzGeGaeyypa0JaeyOeI0IcdaWcaaqaaK qzGeGaamOvaOWaaSbaaSqaaKqzadGaaGimaaWcbeaaaOqaaKqzGeGa amOCaaaakmaabmaabaqcLbsacaaIXaGaey4kaSIcdaqadaqaaKqzGe GaaGymaiabgUcaRiabeg7aHjaadkhaaOGaayjkaiaawMcaaKqzGeGa ciyzaiaacIhacaGGWbGcdaqadaqaaKqzGeGaeyOeI0IaaGOmaiabeg 7aHjaadkhaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaKqzGeGaeyOK H4QaamOvaOWaaSbaaSqaaKqzadGaamyBaiaadEgaaSqabaGcdaqada qaaKqzGeGabmOCayaajaaakiaawIcacaGLPaaajugibiabg2da9iaa dAfakmaaBaaaleaajugWaiaaicdaaSqabaqcLbsacqaHXoqycqGHsi slkmaalaaabaqcLbsacaWGwbGcdaWgaaWcbaqcLbmacaaIWaaaleqa aaGcbaqcLbsaceWGYbGbaKaaaaGaeyOeI0IaaGOmaiaadAfakmaaBa aaleaajugWaiaaicdaaSqabaqcLbsacqaHXoqykmaaCaaaleqabaqc LbmacaaIZaaaaKqzGeGabmOCayaajaGcdaahaaWcbeqaaKqzadGaaG Omaaaaaaa@7EEC@  (1)

in (NC: 3D-RSP) symmetries using the generalized Bopp’s shift method which depend on the concepts that we present below in the third section. The new structure of extended quantum mechanics based to new NC canonical commutations relations (NNCCRs) in both Schrödinger and Heisenberg pictures ((SP) and (HP)), respectively, as follows (Throughout this paper, the natural units c==1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadogacq GH9aqpcqWIpecAcqGH9aqpcaaIXaaaaa@3D3B@  will be used):5–23

{ [ x i , p j ]=[ x i ( t ), p j ( t ) ]=i δ ij   [ x i , x j ]=[ x i ( t ), x j ( t ) ]=0 [ p i , p j ]=[ p i ( t ), p j ( t ) ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGabaabaeqaba WaamWaaeaacaWG4bWaaSbaaSqaaKqzadGaamyAaaWcbeaakiaacYca caWGWbWaaSbaaSqaaiaadQgaaeqaaaGccaGLBbGaayzxaaGaeyypa0 ZaamWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG 0baacaGLOaGaayzkaaGaaiilaiaadchadaWgaaWcbaGaamOAaaqaba GcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH 9aqpcaWGPbGaeqiTdq2aaSbaaSqaaKqzadGaamyAaiaayIW7caWGQb aaleqaaOGaaeiiaaqaamaadmaabaGaamiEamaaBaaaleaacaWGPbaa beaakiaacYcacaWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGLBbGaay zxaaGaeyypa0ZaamWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOWa aeWaaeaacaWG0baacaGLOaGaayzkaaGaaiilaiaadIhadaWgaaWcba GaamOAaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfa caGLDbaacqGH9aqpcaaIWaaabaWaamWaaeaacaWGWbWaaSbaaSqaai aadMgaaeqaaOGaaiilaiaadchadaWgaaWcbaGaamOAaaqabaaakiaa wUfacaGLDbaacqGH9aqpdaWadaqaaiaadchadaWgaaWcbaGaamyAaa qabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacaGGSaGaamiCamaa BaaaleaacaWGQbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaa Gaay5waiaaw2faaiabg2da9iaaicdaaaGaay5Eaaaaaa@8214@ { [ x ^ i , p ^ j ]=[ x ^ i ( t ) , p ^ j ( t ) ]=i δ ij    [ x ^ i , x ^ j ]=[ x ^ i ( t ) , x ^ j ( t ) ]=i θ ij   [ p ^ i , p ^ j ]= [ p ^ i ( t ) , p ^ j ( t ) ]=i θ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHshI3daGaba abaeqabaWaamWaaeaaceWG4bGbaKaadaWgaaWcbaGaamyAaaqabaGc daWfGaqaaiaacYcaaSqabeaacqGHxiIkaaGcceWGWbGbaKaadaWgaa WcbaGaamOAaaqabaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaaiqa dIhagaqcamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamiDaaGaay jkaiaawMcaamaaxacabaGaaiilaaWcbeqaaiabgEHiQaaakiqadcha gaqcamaaBaaaleaacaWGQbaabeaakmaabmaabaGaamiDaaGaayjkai aawMcaaaGaay5waiaaw2faaiabg2da9iaadMgacqaH0oazdaWgaaWc baGaamyAaiaayIW7caWGQbaabeaakiaabccacaqGGaaabaWaamWaae aaceWG4bGbaKaadaWgaaWcbaGaamyAaaqabaGcdaWfGaqaaiaacYca aSqabeaacqGHxiIkaaGcceWG4bGbaKaadaWgaaWcbaGaamOAaaqaba aakiaawUfacaGLDbaacqGH9aqpdaWadaqaaiqadIhagaqcamaaBaaa leaacaWGPbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaamaaxa cabaGaaiilaaWcbeqaaiabgEHiQaaakiqadIhagaqcamaaBaaaleaa caWGQbaabeaakmaabmaabaGaamiDaaGaayjkaiaawMcaaaGaay5wai aaw2faaiabg2da9iaadMgacqaH4oqCdaWgaaWcbaGaamyAaiaayIW7 caWGQbaabeaakiaabccaaeaadaWadaqaaiqadchagaqcamaaBaaale aacaWGPbaabeaakmaaxacabaGaaiilaaWcbeqaaiabgEHiQaaakiqa dchagaqcamaaBaaaleaacaWGQbaabeaaaOGaay5waiaaw2faaiabg2 da9iaabccadaWadaqaaiqadchagaqcamaaBaaaleaacaWGPbaabeaa kmaabmaabaGaamiDaaGaayjkaiaawMcaamaaxacabaGaaiilaaWcbe qaaiabgEHiQaaakiqadchagaqcamaaBaaaleaacaWGQbaabeaakmaa bmaabaGaamiDaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabg2da9i aadMgadaqdaaqaaiabeI7aXbaadaWgaaWcbaGaamyAaiaayIW7caWG QbaabeaaaaGccaGL7baaaaa@9761@ (2)

However, the new operators ξ ^ ( t )=[ x ^ i ( t ) p ^ i ( t ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH+oaEgaqcam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maadmaabaGabmiE ayaajaWaaSbaaSqaaiaadMgaaeqaaOWaaeWaaeaacaWG0baacaGLOa GaayzkaaGaeyikIOTabmiCayaajaWaaSbaaSqaaiaadMgaaeqaaOWa aeWaaeaacaWG0baacaGLOaGaayzkaaaacaGLBbGaayzxaaaaaa@4A2F@ in (HP) are depending to the corresponding new operators ξ ^ =[ x ^ i p ^ i ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH+oaEgaqcai abg2da9maadmaabaGabmiEayaajaWaaSbaaSqaaiaadMgaaeqaaOGa eyikIODcLbsaceWGWbGbaKaakmaaBaaaleaajugWaiaadMgaaSqaba aakiaawUfacaGLDbaaaaa@447B@ in (SP) from the following projections relations:20

ξ( t )=exp(i H ^ mg ( t t 0 ))ξexp(i H ^ mgi ( t t 0 )) ξ ^ ( t )=exp(i H ^ ncni ( t t 0 ))* ξ ^ *exp(i H ^ ncni ( t t 0 )) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH+oaEdaqada qaaiaadshaaiaawIcacaGLPaaacqGH9aqpciGGLbGaaiiEaiaaccha caGGOaGaamyAaiqadIeagaqcamaaBaaaleaajugWaiaad2gacaaMb8 Uaam4zaaWcbeaakmaabmaabaGaamiDaiabgkHiTiaadshadaWgaaWc baqcLbmacaaIWaaaleqaaaGccaGLOaGaayzkaaGaaiykaiabe67a4j GacwgacaGG4bGaaiiCaiaacIcacqGHsislcaWGPbGabmisayaajaWc daWgaaqaaKqzadGaamyBaiaaygW7caWGNbGaaGzaVlaadMgaaSqaba GcdaqadaqaaiaadshacqGHsislcaWG0bWcdaWgaaqaaKqzadGaaGim aaWcbeaaaOGaayjkaiaawMcaaiaacMcacqGHshI3cuaH+oaEgaqcam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iGacwgacaGG4bGa aiiCaiaacIcacaWGPbGabmisayaajaWcdaWgaaqaaKqzadGaamOBai aaykW7caWGJbGaeyOeI0IaamOBaiaaykW7caWGPbaaleqaaOWaaeWa aeaacaWG0bGaeyOeI0IaamiDamaaBaaaleaajugWaiaaicdaaSqaba aakiaawIcacaGLPaaacaGGPaGaaiOkaiqbe67a4zaajaGaaiOkaiGa cwgacaGG4bGaaiiCaiaacIcacqGHsislcaWGPbGabmisayaajaWcda WgaaqaaKqzadGaamOBaiaaygW7caaMi8Uaam4yaiabgkHiTiaad6ga caaMb8UaaGjcVlaadMgaaSqabaGcdaqadaqaaiaadshacqGHsislca WG0bWaaSbaaSqaaKqzadGaaGimaaWcbeaaaOGaayjkaiaawMcaaiaa cMcaaaa@9F85@ (3)

Here ξ=( x i p i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH+oaEcqGH9a qpdaqadaqaaiaadIhadaWgaaWcbaqcLbmacaWGPbaaleqaaOGaeyik IOTaamiCamaaBaaaleaajugWaiaadMgaaSqabaaakiaawIcacaGLPa aaaaa@4482@ and ξ( t )=( x i ( t ) p i ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH+oaEdaqada qaaiaadshaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadIhadaWg aaWcbaGaamyAaaqabaGcdaqadaqaaiaadshaaiaawIcacaGLPaaacq GHOiI2jugibiaadchalmaaBaaabaqcLbmacaWGPbaaleqaaOWaaeWa aeaajugibiaadshaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4BF7@ , while the dynamics of new systems dξ( t ) dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaKqzGe Gaaeizaiabe67a4PWaaeWaaeaajugibiaadshaaOGaayjkaiaawMca aaqaaKqzGeGaaeizaiaabshaaaaaaa@40AF@  are described from the following motion equations in extended quantum mechanics: 20

dξ( t ) dt =[ ξ( t ), H ^ mg ] d ξ ^ ( t ) dt =[ ξ ^ ( t ) , H ^ ncmg ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaabs gacqaH+oaEdaqadaqaaiaadshaaiaawIcacaGLPaaaaeaacaqGKbGa aeiDaaaacqGH9aqpdaWadaqaaiabe67a4naabmaabaGaamiDaaGaay jkaiaawMcaaiaacYcaceWGibGbaKaadaWgaaWcbaqcLbmacaWGTbGa am4zaaWcbeaaaOGaay5waiaaw2faaiabgkDiEpaalaaabaqcLbsaca qGKbGafqOVdGNbaKaakmaabmaabaqcLbsacaWG0baakiaawIcacaGL PaaaaeaajugibiaabsgacaqG0baaaOGaeyypa0ZaamWaaeaacuaH+o aEgaqcamaabmaabaGaamiDaaGaayjkaiaawMcaamaaxacabaGaaiil aaWcbeqaaKqzadGaey4fIOcaaKqzGeGabmisayaajaGcdaWgaaWcba qcLbmacaWGUbGaam4yaiabgkHiTiaad2gacaWGNbaaleqaaaGccaGL BbGaayzxaaaaaa@6845@ (4)

the two operators H ^ mg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada WgaaWcbaGaamyBaiaadEgaaeqaaaaa@3ABB@ and H ^ ncmg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada WgaaWcbaGaamOBaiaadogacqGHsislcaWGTbGaam4zaaqabaaaaa@3D83@ are presents the ordinary and new quantum Hamiltonian operators for NMGESC potential in the quantum mechanics and it’s extension, respectively, while d ξ ^ ( t ) dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaabs gacuaH+oaEgaqcamaabmaabaGaamiDaaGaayjkaiaawMcaaaqaaiaa bsgacaqG0baaaaaa@3EFE@  are describe the dynamics of systems in (NC: 3D–RSP). The very small two parameters θ μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeI7aXP WaaWbaaSqabeaajugWaiabeY7aTjabe27aUbaaaaa@3EEC@ and θ ¯ μν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaKqzGe GaeqiUdehaaOWaaWbaaSqabeaajugWaiabeY7aTjabe27aUbaaaaa@3EFD@ (compared to the energy) are elements of two anti symmetric real matrixes ( θ μv , θ ¯ μv )=( θ vμ , θ ¯ vμ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGOaGaeqiUde 3aaWbaaSqabeaacqaH8oqBcaWG2baaaOGaaiilaiqbeI7aXzaaraWa aWbaaSqabeaacqaH8oqBcaWG2baaaOGaaiykaiabg2da9iabgkHiTi aacIcacqaH4oqCdaahaaWcbeqaaiaadAhacqaH8oqBaaGccaGGSaGa fqiUdeNbaebadaahaaWcbeqaaiaadAhacqaH8oqBaaGccaGGPaaaaa@5081@  and ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabgE HiQaGaayjkaiaawMcaaaaa@3A4C@  denote to the new star product, which is generalized between two arbitrary functions ( f,g ) ( x,p )( f ^ , g ^ )( x ^ , p ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadA gacaqGSaGaam4zaaGaayjkaiaawMcaaiaabccadaqadaqaaiaadIha caGGSaGaamiCaaGaayjkaiaawMcaaiabgkziUoaabmaabaGabmOzay aajaGaaiilaiqadEgagaqcaaGaayjkaiaawMcaamaabmaabaGabmiE ayaajaGaaiilaiqadchagaqcaaGaayjkaiaawMcaaaaa@4B19@ to the new form f ^ ( x ^ , p ^ ) g ^ ( x ^ , p ^ )( fg )( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGMbGbaKaada qadaqaaiqadIhagaqcaiaacYcaceWGWbGbaKaaaiaawIcacaGLPaaa ceWGNbGbaKaadaqadaqaaiqadIhagaqcaiaacYcaceWGWbGbaKaaai aawIcacaGLPaaacqGHHjIUdaqadaqaaiaadAgacqGHxiIkcaWGNbaa caGLOaGaayzkaaWaaeWaaeaacaWG4bGaaiilaiaadchaaiaawIcaca GLPaaaaaa@4CA4@  in ordinary 3-dimensional space-phase:6–21

( f,g ) ( x,p )( f ^ , g ^ )( x ^ , p ^ )( fg )( x,p )=( fg i 2 θ μν μ x f ν x g i 2 θ ¯ μν μ p f ν p gg )( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadA gacaqGSaGaam4zaaGaayjkaiaawMcaaiaabccadaqadaqaaiaadIha caGGSaGaamiCaaGaayjkaiaawMcaaiabgkziUoaabmaabaGabmOzay aajaGaaiilaiqadEgagaqcaaGaayjkaiaawMcaamaabmaabaGabmiE ayaajaGaaiilaiqadchagaqcaaGaayjkaiaawMcaaiabggMi6oaabm aabaGaamOzaiabgEHiQiaadEgaaiaawIcacaGLPaaadaqadaqaaiaa dIhacaGGSaGaamiCaaGaayjkaiaawMcaaiabg2da9maabmaabaGaam OzaiaadEgacqGHsisldaWcaaqaaiaadMgaaeaacaaIYaaaaiabeI7a XnaaCaaaleqabaGaeqiVd0MaeqyVd4gaaOGaeyOaIy7aa0baaSqaai abeY7aTbqaaiaadIhaaaGccaWGMbGaaGPaVlabgkGi2oaaDaaaleaa cqaH9oGBaeaacaWG4baaaOGaam4zaiabgkHiTmaalaaabaGaamyAaa qaaiaaikdaaaWaa0aaaeaacqaH4oqCaaWaaWbaaSqabeaacqaH8oqB cqaH9oGBaaGccqGHciITdaqhaaWcbaGaeqiVd0gabaGaamiCaaaaki aadAgacaaMc8UaeyOaIy7aa0baaSqaaiabe27aUbqaaiaadchaaaGc caWGNbGaam4zaaGaayjkaiaawMcaamaabmaabaGaamiEaiaacYcaca WGWbaacaGLOaGaayzkaaaaaa@86E6@ (5)

where the notion ( μ x , μ p )f( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabgk Gi2oaaDaaaleaacqaH8oqBaeaacaWG4baaaOGaaiilaiabgkGi2oaa DaaaleaacqaH8oqBaeaacaWGWbaaaaGccaGLOaGaayzkaaGaamOzam aabmaabaGaamiEaiaacYcacaWGWbaacaGLOaGaayzkaaaaaa@47BB@  denote to the ( p μ , x μ )f( x,p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaamaala aabaGaeyOaIylabaGaeyOaIyRaamiCamaaCaaaleqabaGaeqiVd0ga aaaakiaacYcadaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaahaa WcbeqaaiabeY7aTbaaaaaakiaawIcacaGLPaaacaWGMbWaaeWaaeaa caWG4bGaaiilaiaadchaaiaawIcacaGLPaaaaaa@4AA7@ . The effects of (space-space) and (phase-phase) noncommutativity properties, respectively induce the second and the third terms in the above equation. The organization scheme of the recently work is given as follows: In next section, we briefly review the ordinary SE with MGESC potential on based to ref.3 The Section 3 is devoted to studying the MSE by applying the generalized Bopp's shift method for NMGESC potential. In the next subsection, by applying standard perturbation theory to find the quantum spectrum of n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaWbaaS qabeaacaWG0bGaamiAaaaaaaa@3ADA@  excited levels in for spin-orbital interaction in the framework of the global group (NC-3D: RSP) and then, we derive the magnetic spectrum for NMGESC potential. In the fourth section, we resume the global spectrum and corresponding NC Hamiltonian operator for NMGESC potential and corresponding energy levels of hydrogenic atoms and the molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaado eacaWGpbGaaiilaiaad6eacaWGpbaacaGLOaGaayzkaaaaaa@3D50@ . Finally, the concluding remarks have been presented in the last section.

Overview of the eigenfunctions and the energy eigenvalues for MGESC potential for hydrogenic atoms and molecules (CO, NO):

In this section, we shall recall here the time independent SE for a MGESC potential V mg ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaea qbaaGcbaGaamOvamaaBaaaleaacaWGTbGaam4zaaqabaGcdaqadaqa aiaadkhaaiaawIcacaGLPaaaaaa@3FD9@ , which studied by Ita et al.,2 and generalized to new form by Ita et al.,3 also in ref.: 3,4

V mg ( r )=( a r )( 1+( 1+b )exp( 2b ) ) V mg ( r )= V 0 r ( 1+( 1+αr )exp( 2αr ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamOCaaGaayjkaiaawMca aiabg2da9maabmaabaGaeyOeI0YaaSaaaeaacaWGHbaabaGaamOCaa aaaiaawIcacaGLPaaadaqadaqaaiaaigdacqGHRaWkdaqadaqaaiaa igdacqGHRaWkcaWGIbaacaGLOaGaayzkaaGaciyzaiaacIhacaGGWb WaaeWaaeaacqGHsislcaaIYaGaamOyaaGaayjkaiaawMcaaaGaayjk aiaawMcaaiabgkziUkaadAfadaWgaaWcbaqcLbmacaWGTbGaam4zaa WcbeaakmaabmaabaGaamOCaaGaayjkaiaawMcaaiabg2da9iabgkHi TmaalaaabaGaamOvamaaBaaaleaajugWaiaaicdaaSqabaaakeaaca WGYbaaamaabmaabaGaaGymaiabgUcaRmaabmaabaGaaGymaiabgUca Riabeg7aHjaadkhaaiaawIcacaGLPaaaciGGLbGaaiiEaiaacchada qadaqaaiabgkHiTiaaikdacqaHXoqycaWGYbaacaGLOaGaayzkaaaa caGLOaGaayzkaaaaaa@70EB@ (6)

where a V 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbGaeyOKH4 QaamOvamaaBaaaleaacaaIWaaabeaaaaa@3C68@  and bαr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGIbGaeyOKH4 QaeqySdeMaamOCaaaa@3D3E@  are the strength coupling constant (the potential depth of the MGESC potential) and the screened parameter (adjustable positive parameter), respectively. The part with exp. term of eq. (6) can be expanded in the power series of r up to the second term:

1 r e 2αr 1 r ( 12αr+2 α 2 r 2 )=2α+ 1 r +2 α 2 r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiaaig daaeaacaWGYbaaaiaadwgadaahaaWcbeqaaiabgkHiTiaaikdacqaH XoqycaWGYbaaaOGaeyyrIa0aaSaaaeaacaaIXaaabaGaamOCaaaada qadaqaaiaaigdacqGHsislcaaIYaGaeqySdeMaamOCaiabgUcaRiaa ikdacqaHXoqydaahaaWcbeqaaiaaikdaaaGccaWGYbWaaWbaaSqabe aacaaIYaaaaaGccaGLOaGaayzkaaGaeyypa0JaeyOeI0IaaGOmaiab eg7aHjabgUcaRmaalaaabaGaaGymaaqaaiaadkhaaaGaey4kaSIaaG Omaiabeg7aHnaaCaaaleqabaGaaGOmaaaakiaadkhaaaa@5AA1@ (7)

Inserting eq. (7) into eq. (6), explicit form of MGESC potential is obtained as:

V( r ) V 0 α 2 V 0 r 2 V 0 α 3 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaeWaae aacaWGYbaacaGLOaGaayzkaaGaeyyrIaKaamOvamaaBaaaleaacaaI Waaabeaakiabeg7aHjabgkHiTmaalaaabaGaaGOmaiaadAfadaWgaa WcbaGaaGimaaqabaaakeaacaWGYbaaaiabgkHiTKqzGeGaaGOmaiaa dAfakmaaBaaaleaacaaIWaaabeaakiabeg7aHTWaaWbaaeqabaqcLb macaaIZaaaaOGaamOCamaaCaaaleqabaqcLbmacaaIYaaaaaaa@4F23@ (8)

If we insert this potential into the Schrödinger equation (3):

( d 2 d r 2 + 2 r d dr l(l+1) r 2 ) R nl ( r )+2μ[ E nl + V 0 r ( 1+( 1+αr )exp( 2αr ) ) ] R nl ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaamaala aabaGaamizamaaCaaaleqabaGaaGOmaaaaaOqaaiaadsgacaWGYbWa aWbaaSqabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGOmaaqaai aadkhaaaWaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaeyOeI0Ya aSaaaeaacaWGSbGaaiikaiaadYgacqGHRaWkcaaIXaGaaiykaaqaai aadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaamOu amaaBaaaleaacaWGUbGaamiBaaqabaGcdaqadaqaaiaadkhaaiaawI cacaGLPaaacqGHRaWkcaaIYaGaeqiVd02aamWaaeaacaWGfbWaaSba aSqaaiaad6gacaWGSbaabeaakiabgUcaRmaalaaabaGaamOvamaaBa aaleaacaaIWaaabeaaaOqaaiaadkhaaaWaaeWaaeaacaaIXaGaey4k aSYaaeWaaeaacaaIXaGaey4kaSIaeqySdeMaamOCaaGaayjkaiaawM caaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0IaaGOmaiabeg7a HjaadkhaaiaawIcacaGLPaaaaiaawIcacaGLPaaaaiaawUfacaGLDb aacaWGsbWaaSbaaSqaaiaad6gacaWGSbaabeaakmaabmaabaGaamOC aaGaayjkaiaawMcaaiabg2da9iaaicdaaaa@755E@ (9)

Here μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeY7aTb aa@3A19@  is the reduced mass of molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaKqzGe Gaam4qaiaad+eacaGGSaGaamOtaiaad+eaaOGaayjkaiaawMcaaaaa @3DE9@  or the reduced mass of electron ant it’s nucleus for hydrogenic atoms. The electronic radial wave functions are shown as a function of the Laguerre polynomial in terms of some parameters:3

R nl ( ν )= N n,l ( 2β ) 1 2 α exp( ν 2 ) ν α 1 2 L n 2α+1 ( ν ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaSbaaS qaaiaad6gacaWGSbaabeaakmaabmaabaGaeqyVd4gacaGLOaGaayzk aaGaeyypa0JaamOtamaaBaaaleaacaWGUbGaaiilaiaadYgaaeqaaO WaaeWaaeaacaqGYaGaeqOSdigacaGLOaGaayzkaaWaaWbaaSqabeaa daWcaaqaaiaaigdaaeaacaaIYaaaaiabgkHiTiabeg7aHbaakiGacw gacaGG4bGaaiiCamaabmaabaGaeyOeI0YaaSaaaeaacqaH9oGBaeaa caaIYaaaaaGaayjkaiaawMcaaiabe27aUnaaCaaaleqabaGaeqySde MaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaaaaGccaWGmbWaa0ba aSqaaiaad6gaaeaacaaIYaGaeqySdeMaey4kaSIaaGymaaaakmaabm aabaGaeqyVd4gacaGLOaGaayzkaaaaaa@61DD@ (10)

where r= ( 2β ) 1 ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGYbGaeyypa0 ZaaeWaaeaacaaIYaGaeqOSdigacaGLOaGaayzkaaWaaWbaaSqabKqa GfaajugWaiabgkHiTiaaigdaaaGccqaH9oGBaaa@42E4@ , therefore, the complete wave function Ψ( r,θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHOoqwdaqada qaaiaadkhacaGGSaGaeqiUdeNaaiilaiabew9aMbGaayjkaiaawMca aaaa@40C1@  and the energy E nl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad6gacaWGSbaabeaaaaa@3AAE@  of the potential in eq. (6) are given by:3

Ψ( r,θ,ϕ )= N nl ( 2β ) α1/2 exp( ν 2 ) ν α1/2 L n 2α+1 ( ν ) Y l m ( θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHOoqwdaqada qaaiaadkhacaGGSaGaeqiUdeNaaiilaiabew9aMbGaayjkaiaawMca aiabg2da9iaad6eadaWgaaWcbaGaamOBaiaadYgaaeqaaOWaaeWaae aacaqGYaGaeqOSdigacaGLOaGaayzkaaWaaWbaaeqabaqcLbmacqaH XoqycqGHsislcaaIXaGaai4laiaaikdaaaGcciGGLbGaaiiEaiaacc hadaqadaqaaiabgkHiTmaalaaabaGaeqyVd4gabaGaaGOmaaaaaiaa wIcacaGLPaaacqaH9oGBdaahaaWcbeqaaiabeg7aHjabgkHiTiaaig dacaGGVaGaaGOmaaaakiaadYealmaaDaaabaqcLbmacaWGUbaaleaa jugWaiaaikdacqaHXoqycqGHRaWkcaaIXaaaaOWaaeWaaeaacqaH9o GBaiaawIcacaGLPaaacaWGzbWcdaqhaaqaaKqzadGaamiBaaWcbaqc LbmacaWGTbaaaOWaaeWaaeaacqaH4oqCcaGGSaGaeqy1dygacaGLOa Gaayzkaaaaaa@73CE@ (11)and

E nl = V 0 e α r 0 +2μ ( V 0 + V 0 e α r 0 n+l+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad6gacaWGSbaabeaakiabg2da9iabgkHiTiaadAfadaWgaaWc baGaaGimaaqabaGccaWGLbWaaWbaaSqabeaajugWaiabgkHiTiabeg 7aHjaadkhalmaaBaaameaajugWaiaaicdaaWqabaaaaOGaey4kaSIa aGOmaiabeY7aTnaabmaabaWaaSaaaeaacaWGwbWaaSbaaSqaaKqzad GaaGimaaWcbeaakiabgUcaRiaadAfadaWgaaWcbaqcLbmacaaIWaaa leqaaOGaamyzamaaCaaaleqabaqcLbmacqGHsislcqaHXoqycaWGYb WcdaWgaaadbaqcLbmacaaIWaaameqaaaaaaOqaaiaad6gacqGHRaWk caWGSbGaey4kaSIaaGymaaaaaiaawIcacaGLPaaadaahaaWcbeqaaK qzadGaaGOmaaaaaaa@60C3@ (12)

With r 0 =1.21282 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaaicdaaeqaaOGaeyypa0Jaaeymaiaab6cacaqGYaGaaeymaiaa bkdacaqG4aGaaeOmaaaa@3FB4@  and r 0 =1.1508 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaaicdaaeqaaOGaeyypa0Jaaeymaiaab6cacaqGXaGaaeynaiaa bcdacaqG4aaaaa@3F00@ for (CO and NO) molecules, for hydrogenic atoms, r 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaSbaaS qaaiaaicdaaeqaaaaa@39B1@  can be present the average dimension between the electron and the nucleus, N nl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6ealm aaBaaabaqcLbmacaWGUbGaamiBaaWcbeaaaaa@3C7F@ is the normalization constant, α= 1 2 4l(l+1)+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqycqGH9a qpdaWcaaqaaiaaigdaaeaacaaIYaaaamaakaaabaGaaGinaiaadYga caGGOaGaamiBaiabgUcaRiaaigdacaGGPaGaey4kaSIaaGymaaWcbe aaaaa@434E@ , β 2 = 2μ E nl α 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGydaahaa WcbeqaaiaaikdaaaGccqGH9aqpcqGHsisldaWcaaqaaiaaikdacqaH 8oqBcaWGfbWaaSbaaSqaaiaad6gacaWGSbaabeaaaOqaaiabeg7aHn aaCaaaleqabaGaaGOmaaaaaaaaaa@4449@ and Y l m ( θ,ϕ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGzbWaa0baaS qaaiaadYgaaeaacaWGTbaaaOWaaeWaaeaacqaH4oqCcaGGSaGaeqy1 dygacaGLOaGaayzkaaaaaa@4083@  are the well-known spherical harmonic functions.

Method and theoretical approach

In this section, we shall give an overview or a brief preliminary for a NMGESC potential V nc-mg (r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaSbaaS qaaiaab6gacaqGJbGaaeylaiaad2gacaqGNbaabeaakiaacIcacaWG YbGaaiykaaaa@3F98@ , in (NC: 3D-RSP) symmetries. To perform this task the physical form of modified Schrödinger equation (MSE), it is necessary to replace ordinary three-dimensional Hamiltonian operators H ^ ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada qadaqaaiaadchadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamiEamaa BaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaaa@3F24@ , ordinary complex wave function Ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHOoqwdaqada qaamaaxacabaGaamOCaaWcbeqaaKqzadGaeyOKH4kaaaGccaGLOaGa ayzkaaaaaa@3F51@  and ordinary energy E nl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad6gacaWGSbaabeaaaaa@3AAE@  by new three Hamiltonian operators H ^ ncmg ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada WgaaWcbaGaamOBaiaadogacqGHsislcaWGTbGaam4zaaqabaGcdaqa daqaaiqadchagaqcamaaBaaaleaacaWGPbaabeaakiaacYcaceWG4b GbaKaadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaa@4420@ , new complex wave function Ψ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacuqHOoqwgaWeam aabmaabaWaa8XaaeaaceWGYbGbambaaiaawgoiaaGaayjkaiaawMca aaaa@3DD6@  and new values E ncmg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad6gacaWGJbGaeyOeI0IaamyBaiaadEgaaeqaaaaa@3D70@ , respectively. In addition to replace the ordinary old product by new star product ( ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabgE HiQaGaayjkaiaawMcaaaaa@3A4C@ , which allow us to constructing the MSE in (NC-3D: RSP) symmetries as:21–28

H ^ mg ( p i , x i )Ψ( r )= E nl Ψ( r ) H ^ ( p ^ i , x ^ i )Ψ( r ^ )= E ncmg Ψ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada WgaaWcbaGaamyBaiaadEgaaeqaaOWaaeWaaeaacaWGWbWaaSbaaeaa caWGPbaabeaacaGGSaGaamiEamaaBaaabaGaamyAaaqabaaacaGLOa GaayzkaaGaeuiQdK1aaeWaaeaadaWhcaqaaiaadkhaaiaawEniaaGa ayjkaiaawMcaaiabg2da9iaadweadaWgaaWcbaGaamOBaiaadYgaae qaaOGaeuiQdK1aaeWaaeaadaWhcaqaaiaadkhaaiaawEniaaGaayjk aiaawMcaaiabgkDiElqadIeagaqcamaabmaabaGabmiCayaajaWaaS baaSqaaiaadMgaaeqaaOGaaiilaiqadIhagaqcamaaBaaaleaacaWG PbaabeaaaOGaayjkaiaawMcaaiabgEHiQiabfI6aznaabmaabaWaaC biaeaaceWGYbGbaKaaaSqabeaacqGHsgIRaaaakiaawIcacaGLPaaa cqGH9aqpcaWGfbWaaSbaaSqaaiaad6gacaWGJbGaeyOeI0IaamyBai aadEgaaeqaaOGaeuiQdK1aaeWaaeaadaWfGaqaaiqadkhagaqcaaWc beqaaiabgkziUcaaaOGaayjkaiaawMcaaaaa@6EA4@ (13)

The Bopp’s shift method employed in the solutions enables us to explore an effective way of obtaining the modified potential in extended quantum mechanics, it based on the following new commutators:28–34

[ x ^ i , x ^ j ]=[ x ^ i ( t ), x ^ j ( t ) ]=i θ ij   and  [ p ^ i , p ^ j ]=[ p ^ i ( t ), p ^ j ( t ) ]=i θ ¯ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWadaqaaiqadI hagaqcamaaBaaaleaacaWGPbaabeaakiaacYcaceWG4bGbaKaadaWg aaWcbaGaamOAaaqabaaakiaawUfacaGLDbaacqGH9aqpdaWadaqaai qadIhagaqcamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamiDaaGa ayjkaiaawMcaaiaacYcaceWG4bGbaKaadaWgaaWcbaGaamOAaaqaba GcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH 9aqpcaWGPbGaeqiUde3aaSbaaSqaaiaadMgacaWGQbaabeaakiaabc cacaqGGaGaaeyyaiaab6gacaqGKbGaaeiiaiaabccadaWadaqaaiqa dchagaqcamaaBaaaleaacaWGPbaabeaakiaacYcaceWGWbGbaKaada WgaaWcbaGaamOAaaqabaaakiaawUfacaGLDbaacqGH9aqpdaWadaqa aiqadchagaqcamaaBaaaleaacaWGPbaabeaakmaabmaabaGaamiDaa GaayjkaiaawMcaaiaacYcaceWGWbGbaKaadaWgaaWcbaGaamOAaaqa baGcdaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUfacaGLDbaacq GH9aqpcaWGPbWaa0aaaeaacqaH4oqCaaWaaSbaaSqaaiaadMgacaWG Qbaabeaaaaa@70A5@ (14)

The new generalized positions and momentum coordinates ( x ^ i , p ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiqadI hagaqcamaaBaaaleaacaWGPbaabeaakiaacYcaceWGWbGbaKaadaWg aaqaaiaadMgaaeqaaaGaayjkaiaawMcaaaaa@3E52@  in (NC: 3D-RSP) are depended with corresponding usual generalized positions and momentum coordinates ( x i i , p i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI hadaWgaaWcbaGaamyAaaqabaGcdaWgaaWcbaGaamyAaaqabaGccaGG SaGaamiCamaaBaaabaGaamyAaaqabaaacaGLOaGaayzkaaaaaa@3F56@  in ordinary quantum mechanics by the following, respectively:30–36

( x i , p i )( x ^ i , p ^ i )=( x i θ ij 2 p j , p i + θ ¯ ij 2 x j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI hadaWgaaWcbaGaamyAaaqabaGccaGGSaGaamiCamaaBaaabaGaamyA aaqabaaacaGLOaGaayzkaaGaeyO0H49aaeWaaeaaceWG4bGbaKaada WgaaWcbaGaamyAaaqabaGccaGGSaGabmiCayaajaWaaSbaaeaacaWG PbaabeaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadIhadaWgaa WcbaGaamyAaaqabaGccqGHsisldaWcaaqaaiabeI7aXnaaBaaaleaa caWGPbGaamOAaaqabaaakeaacaaIYaaaaiaadchadaWgaaWcbaGaam OAaaqabaGccaqGSaGaamiCamaaBaaabaGaamyAaaqabaGaey4kaSYa aSaaaeaadaqdaaqaaiabeI7aXbaadaWgaaWcbaGaamyAaiaadQgaae qaaaGcbaGaaGOmaaaacaWG4bWaaSbaaSqaaiaadQgaaeqaaaGccaGL OaGaayzkaaaaaa@5DB6@ (15)

The above equation allows us to obtain the two operators r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGYbGbaKaada ahaaqabeaajugWaiaaikdaaaaaaa@3AE7@  and p ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGWbGbaKaada ahaaqabeaajugWaiaaikdaaaaaaa@3AE5@  in (NC-3D: RSP):35–38

( r 2 , p 2 )( r ^ 2 , p ^ 2 )=( r 2 L Θ  ,  p 2 + L θ ¯    ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadk hadaahaaWcbeqaaiaaikdaaaGccaGGSaGaamiCamaaCaaabeqaaiaa ikdaaaaacaGLOaGaayzkaaGaeyO0H49aaeWaaeaaceWGYbGbaKaada ahaaWcbeqaaiaaikdaaaGccaGGSaGabmiCayaajaWaaWbaaeqabaGa aGOmaaaaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaadkhadaahaa WcbeqaaiaaikdaaaGccqGHsislieqaceWFmbGbaSaaiiqacuGFyoqu gaWcaiaabccacaqGSaGaaeiiaiaadchadaahaaqabeaacaaIYaaaai abgUcaRiqadYeagaWcamaaFiaabaWaa0aaaeaacqaH4oqCaaaacaGL xdcacaqGGaGaaeiiaaGaayjkaiaawMcaaaaa@5867@ (16)

The two couplings LΘ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqacaWFmbGaeu iMdefaaa@3A22@  and L θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieqaceWFmbGbaS aadaWhcaqaamaanaaabaacceGae4hUdehaaaGaay51Gaaaaa@3C3D@  are ( L x Θ 12 + L y Θ 23 + L z Θ 13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadY eadaWgaaWcbaGaamiEaaqabaGccqqHyoqudaWgaaWcbaGaaGymaiaa ikdaaeqaaOGaey4kaSIaamitamaaBaaaleaacaWG5baabeaakiabfI 5arnaaBaaaleaacaaIYaGaaG4maaqabaGccqGHRaWkcaWGmbWaaSba aSqaaiaadQhaaeqaaOGaeuiMde1aaSbaaSqaaiaaigdacaaIZaaabe aaaOGaayjkaiaawMcaaaaa@4A9F@  and ( L x θ ¯ 12 + L y θ ¯ 23 + L z θ ¯ 13 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadY eadaWgaaWcbaGaamiEaaqabaGcdaqdaaqaaiabeI7aXbaadaWgaaWc baGaaGymaiaaikdaaeqaaOGaey4kaSIaamitamaaBaaaleaacaWG5b aabeaakmaanaaabaGaeqiUdehaamaaBaaaleaacaaIYaGaaG4maaqa baGccqGHRaWkcaWGmbWaaSbaaSqaaiaadQhaaeqaaOWaa0aaaeaacq aH4oqCaaWaaSbaaSqaaiaaigdacaaIZaaabeaaaOGaayjkaiaawMca aaaa@4B8F@ , respectively and ( L x , L y and L z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadY eadaWgaaWcbaGaamiEaaqabaGccaGGSaGaamitamaaBaaaleaacaWG 5baabeaakiaaykW7caWGHbGaamOBaiaadsgacaaMc8UaamitamaaBa aaleaacaWG6baabeaaaOGaayjkaiaawMcaaaaa@45F4@  are the three components of angular momentum operator L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGmbGbaSaaaa a@38B7@  while Θ ij =θij/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHyoqudaWgaa WcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaeqiUdeNaamyAaiaadQga caGGVaGaaGOmaaaa@4166@ . Thus, the reduced Schrödinger equation (without star product) can be written as:

H ^ ( p ^ i , x ^ i )Ψ( r ^ )= E ncmg Ψ( r ^ )H( p ^ i , x ^ i )ψ( r )= E ncmg ψ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGibGbaKaada qadaqaaiqadchagaqcamaaBaaaleaacaWGPbaabeaakiaacYcaceWG 4bGbaKaadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGHxi IkcqqHOoqwdaqadaqaamaaxacabaGabmOCayaajaaaleqabaGaeyOK H4kaaaGccaGLOaGaayzkaaGaeyypa0JaamyramaaBaaaleaacaWGUb Gaam4yaiabgkHiTiaad2gacaWGNbaabeaakiabfI6aznaabmaabaWa aCbiaeaaceWGYbGbaKaaaSqabeaacqGHsgIRaaaakiaawIcacaGLPa aacqGHshI3caWGibWaaeWaaeaaceWGWbGbaKaadaWgaaWcbaGaamyA aaqabaGccaGGSaGabmiEayaajaWaaSbaaSqaaiaadMgaaeqaaaGcca GLOaGaayzkaaGaeqiYdK3aaeWaaeaaceWGYbGbaSaaaiaawIcacaGL PaaacqGH9aqpcaWGfbWaaSbaaSqaaiaad6gacaWGJbGaeyOeI0Iaam yBaiaadEgaaeqaaOGaeqiYdK3aaeWaaeaaceWGYbGbaSaaaiaawIca caGLPaaaaaa@6CC6@ (17)

the new operator of Hamiltonian H ncmgi ( p ^ i , x ^ i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad6gacaWGJbGaeyOeI0IaamyBaiaadEgacaWGPbaabeaakmaa bmaabaGabmiCayaajaWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiqadI hagaqcamaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaaa@44FE@ can be expressed as:

H mg ( p i , x i ) H ncmgi ( p ^ i , x ^ i )H( x ^ i = x i θ ij 2 p j , p ^ i = p i + θ ¯ ij 2 x j ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamiCamaaBaaaleaacaWG PbaabeaakiaacYcacaWG4bWaaSbaaSqaaiaadMgaaeqaaaGccaGLOa GaayzkaaGaeyO0H4TaamisamaaBaaaleaacaWGUbGaam4yaiabgkHi Tiaad2gacaWGNbGaamyAaaqabaGcdaqadaqaaiqadchagaqcamaaBa aaleaacaWGPbaabeaakiaacYcaceWG4bGbaKaadaWgaaWcbaGaamyA aaqabaaakiaawIcacaGLPaaacqGHHjIUcaWGibWaaeWaaeaaceWG4b GbaKaadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWG4bWaaSbaaSqa aiaadMgaaeqaaOGaeyOeI0YaaSaaaeaacqaH4oqCdaWgaaWcbaGaam yAaiaadQgaaeqaaaGcbaGaaGOmaaaacaWGWbWaaSbaaSqaaiaadQga aeqaaOGaaiilaiqadchagaqcamaaBaaabaGaamyAaaqabaGaeyypa0 JaamiCamaaBaaabaGaamyAaaqabaGaey4kaSYaaSaaaeaacuaH4oqC gaqeamaaBaaabaGaamyAaiaadQgaaeqaaaqaaiaaikdaaaGaamiEam aaBaaaleaacaWGQbaabeaaaOGaayjkaiaawMcaaaaa@6F2C@ (18)

Now, we want to find to the NMGESC potential V mg ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGabmOCayaajaaacaGLOaGa ayzkaaaaaa@3D53@ :

V mg ( r ) V mg ( r ^ )= V 0 α 2 V 0 r ^ 2 V 0 α 3 r ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamOCaaGaayjkaiaawMca aiabgkDiElaadAfadaWgaaWcbaGaamyBaiaadEgaaeqaaOWaaeWaae aaceWGYbGbaKaaaiaawIcacaGLPaaacqGH9aqpcaWGwbWaaSbaaSqa aiaaicdaaeqaaOGaeqySdeMaeyOeI0YaaSaaaeaacaaIYaGaamOvam aaBaaaleaacaaIWaaabeaaaOqaaiqadkhagaqcaaaacqGHsislcaaI YaGaamOvamaaBaaaleaacaaIWaaabeaakiabeg7aHnaaCaaaleqaba GaaG4maaaakiqadkhagaqcamaaCaaaleqabaGaaGOmaaaaaaa@5611@ (19)

After straightforward calculations, we can obtain the important term ( V 0 r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabgk HiTmaalaaabaGaamOvamaaBaaaleaacaaIWaaabeaaaOqaaiqadkha gaqcaaaaaiaawIcacaGLPaaaaaa@3D2C@ , which will be use to determine the NMGESC potential in (NC: 3D- RSP) symmetries as:

V 0 r V 0 r ^ = V 0 r V 0 L Θ 2 r 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsisldaWcaa qaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGYbaaaiabgkDi ElabgkHiTmaalaaabaGaamOvamaaBaaaleaacaaIWaaabeaaaOqaai qadkhagaqcaaaacqGH9aqpcqGHsisldaWcaaqaaiaadAfadaWgaaWc baGaaGimaaqabaaakeaacaWGYbaaaiabgkHiTmaalaaabaGaamOvam aaBaaaleaacaaIWaaabeaaieqakiqa=XeagaWcaiqbfI5arzaalaaa baGaaGOmaiaadkhadaahaaWcbeqaaiaaiodaaaaaaaaa@4E5B@ (20)

By making the substitution above equation into eq. (19), we find the global our working new Hamiltonian operator H nc-mg ( r ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaab6gacaqGJbGaaeylaiaad2gacaWGNbaabeaakmaabmaabaGa bmOCayaajaaacaGLOaGaayzkaaaaaa@3FCC@  satisfies the equation in (NC: 3D-RSP) symmetries:

H mg ( p i , x i ) H ncmg ( r ^ )= H mg ( p i , x i )+( 2 V 0 α 3 V 0 r 3 ) L Θ + L θ ¯ 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamiCamaaBaaabaqcLbma caWGPbaakeqaaiaacYcacaWG4bWaaSbaaeaajugWaiaadMgaaOqaba aacaGLOaGaayzkaaGaeyO0H4TaamisamaaBaaaleaacaWGUbGaam4y aiabgkHiTiaad2gacaWGNbaabeaakmac0dyadaqaiqpGcKa9aoOCay ac0dycaaGaiqpGwIcacGa9aAzkaaGamqpGg2da9iaadIeadaWgaaWc baGaamyBaiaadEgaaeqaaOWaaeWaaeaacaWGWbWaaSbaaeaajugWai aadMgaaOqabaGaaiilaiaadIhadaWgaaqaaKqzadGaamyAaaGcbeaa aiaawIcacaGLPaaacqGHRaWkdaqadaqaaiaaikdacaWGwbWcdaWgaa qaaKqzadGaaGimaaWcbeaakiabeg7aHnaaCaaaleqabaGaaG4maaaa kiabgkHiTmaalaaabaGaamOvamaaBaaaleaacaaIWaaabeaaaOqaai aadkhadaahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaacbeGa b8htayaalaGafuiMdeLbaSaacqGHRaWkdaWcaaqaaiqa=XeagaWcam aaFiaabaWaa0aaaeaaiiqacqGF4oqCaaaacaGLxdcaaeaacaaIYaGa eqiVd0gaaaaa@7CAE@ (21)

where the operator H mg ( p i , x i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamiCamaaBaaabaqcLbma caWGPbaakeqaaiaacYcacaWG4bWcdaWgaaGcbaqcLbmacaWGPbaake qaaaGaayjkaiaawMcaaaaa@4383@ is just the ordinary Hamiltonian operator with MGESC potential in commutative space:

H mg ( p i , x i )= p 2 2μ + V 0 α 2 V 0 r 2 V 0 α 3 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacaWGNbaabeaakmaabmaabaGaamiCaSWaaSbaaOqaaKqz adGaamyAaaGcbeaacaGGSaGaamiEaSWaaSbaaOqaaKqzadGaamyAaa GcbeaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadchadaahaaqa beaacaaIYaaaaaqaaiaaikdacqaH8oqBaaGaey4kaSIaamOvaSWaaS baaeaajugWaiaaicdaaSqabaGccqaHXoqycqGHsisldaWcaaqaaiaa ikdacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOCaaaacqGHsi slcaaIYaGaamOvamaaBaaaleaacaaIWaaabeaakiabeg7aHnaaCaaa leqabaGaaG4maaaakiaadkhadaahaaWcbeqaaiaaikdaaaaaaa@5ADA@ (22)

while the rest two terms are proportional’s with two infinitesimals parameters ( Θand θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabfI 5arjaaysW7caWGHbGaamOBaiaadsgacaaMe8UafqiUdeNbaebaaiaa wIcacaGLPaaaaaa@427E@ and then we can considered as a perturbations terms H per-mg ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaabchacaqGLbGaaeOCaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaadkhaaiaawIcacaGLPaaaaaa@40B5@  in (NC: 3D-RSP) symmetries as:

H per-mg ( r )=( 2 V 0 α 3 V 0 r 3 ) L Θ + L θ ¯ 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaabchacaqGLbGaaeOCaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaadkhaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiaaikdaca WGwbWaaSbaaSqaaiaaicdaaeqaaOGaeqySde2aaWbaaSqabeaacaaI ZaaaaOGaeyOeI0YaaSaaaeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaa GcbaGaamOCaSWaaWbaaeqabaqcLbmacaaIZaaaaaaaaOGaayjkaiaa wMcaaGqabiqa=XeagaWcaiqbfI5arzaalaGaey4kaSYaaSaaaeaace WFmbGbaSaadaWhcaqaamaanaaabaacceGae4hUdehaaaGaay51Gaaa baGaaGOmaiabeY7aTbaaaaa@5874@ (23)

The exact modified spin-orbital spectrum for NMGESC potential in global (NC: 3D- RSP) symmetries

In this subsection, we apply the same strategy, which we have seen in our previously works,36–40 under such particular choice, one can easily reproduce both couplings ( L Θ and  L θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaamaaxa cabaacbeqcLbsacaWFmbaaleqabaqcLbsacqGHsgIRaaGcdaWfGaqa aKqzGeGaeuiMdefaleqabaqcLbsacqGHsgIRaaGaaGPaVlaadggaca WGUbGaamizaabaaaaaaaaapeGaaiiOaiaaykW7k8aadaWfGaqaaKqz GeGaa8htaaWcbeqaaKqzGeGaeyOKH4kaaOWaaCbiaeaadaqdaaqaaK qzGeGaeqiUdehaaaWcbeqaaKqzGeGaeyOKH4kaaaGccaGLOaGaayzk aaaaaa@52E2@  to the new physical forms ( γΘ L S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo7aNj aaykW7cqqHyoqukmaaxacabaqcLbsacaWGmbaaleqabaqcLbsacqGH sgIRaaGcdaWfGaqaaKqzGeGaam4uaaWcbeqaaKqzGeGaeyOKH4kaaa aa@4571@ and γ θ ¯ L S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo7aNj aaykW7cuaH4oqCgaqeaOWaaCbiaeaajugibiaadYeaaSqabeaajugi biabgkziUcaakmaaxacabaqcLbsacaWGtbaaleqabaqcLbsacqGHsg IRaaaaaa@45C8@ ), respectively, to obtain the new forms of H so-mg ( r,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWcdaWgaa qaaKqzadGaae4Caiaab+gacaqGTaGaamyBaiaabEgaaSqabaGcdaqa daqaaKqzGeGaamOCaiaacYcacqqHyoqucaGGSaGafqiUdeNbaebaaO GaayjkaiaawMcaaaaa@4642@  for 3D- NMGESC potential as follows:

H so-mg ( r,Θ, θ ¯ )γ{ ( 2 V 0 α 3 V 0 r 3 )Θ+ θ ¯ 2μ } L S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaKqzadGaae4Caiaab+gacaqGTaGaamyBaiaabEgaaSqabaGcdaqa daqaaKqzGeGaamOCaiaacYcacqqHyoqucaGGSaGafqiUdeNbaebaaO GaayjkaiaawMcaaKqzGeGaeyyyIORaeq4SdCMcdaGadaqaamaabmaa baGaaGOmaiaadAfadaWgaaWcbaGaaGimaaqabaGccqaHXoqydaahaa WcbeqaaiaaiodaaaGccqGHsisldaWcaaqaaiaadAfadaWgaaWcbaGa aGimaaqabaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaOGaay jkaiaawMcaaiabfI5arjabgUcaRmaalaaabaWaa0aaaeaacqaH4oqC aaaabaGaaGOmaiabeY7aTbaaaiaawUhacaGL9baacaaMc8+aaCbiae aacaWGmbaaleqabaGaeyOKH4kaaOWaaCbiaeaacaWGtbaaleqabaGa eyOKH4kaaaaa@661E@ (24)

Here γ 1 137 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeo7aNj abgIKi7QWaaSaaaeaajugibiaaigdaaOqaaKqzGeGaaGymaiaaioda caaI3aaaaaaa@3FF1@  is a new constant, which play the role of fine structure constant, we have chosen the two vectors Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbfI5arz aalaaaaa@39EC@ and θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWhcaqaamaana aabaacceqcLbsacqWF4oqCaaaakiaawEniaaaa@3BEE@  parallel to the spin S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaKqzGe Gaam4uaaWcbeqaaKqzadGaeyOKH4kaaaaa@3C9F@  of hydrogenic atoms. Furthermore, the above perturbative terms H per-mg ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaKqzadGaaeiCaiaabwgacaqGYbGaaeylaiaad2gacaWGNbaaleqa aOWaaeWaaeaajugibiaadkhaaOGaayjkaiaawMcaaaaa@4287@  can be rewritten to the following new form:

H somg ( r,Θ, θ ¯ )= γ 2 { ( 2 V 0 α 3 V 0 r 3 )Θ+ θ ¯ 2μ }( J L S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaadohacaWGVbGaeyOeI0IaamyBaiaadEgaaeqaaOWaaeWaaeaa caWGYbGaaiilaiabfI5arjaacYcacuaH4oqCgaqeaaGaayjkaiaawM caaiabg2da9maalaaabaGaeq4SdCgabaGaaGOmaaaadaGadaqaamaa bmaabaGaaGOmaiaadAfadaWgaaWcbaGaaGimaaqabaGccqaHXoqyda ahaaWcbeqaaiaaiodaaaGccqGHsisldaWcaaqaaiaadAfadaWgaaWc baGaaGimaaqabaaakeaacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaO GaayjkaiaawMcaaiabfI5arjabgUcaRmaalaaabaWaa0aaaeaacqaH 4oqCaaaabaGaaGOmaiabeY7aTbaaaiaawUhacaGL9baacaaMc8+aae WaaeaadaWfGaqaaiaadQeaaSqabeaacqGHsgIRaaGccqGHsisldaWf GaqaaiaadYeaaSqabeaacqGHsgIRaaGccqGHsisldaWfGaqaaiaado faaSqabeaacqGHsgIRaaGcdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@6B6E@ (25)

This operator traduces the coupling between spin S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaKqzGe Gaam4uaaWcbeqaaKqzadGaeyOKH4kaaaaa@3C9F@  and orbital momentum L S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiaadY eaaSqabeaajugWaiabgkziUcaakmaaxacabaGaam4uaaWcbeqaaiab gkziUcaaaaa@3F21@ . The set ( H somg ( r,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGOaGaamisam aaBaaaleaacaWGZbGaam4BaiabgkHiTiaad2gacaWGNbaabeaakmaa bmaabaGaamOCaiaacYcacqqHyoqucaGGSaGafqiUdeNbaebaaiaawI cacaGLPaaaaaa@455F@ , J 2 , L 2 , S 2 and J z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaqGkbWaaWbaaS qabeaajugWaiaabkdaaaGccaGGSaGaamitamaaCaaaleqabaGaaGOm aaaakiaacYcacaWGtbWaaWbaaSqabeaacaaIYaaaaOGaaGPaVlaadg gacaWGUbGaamizaiaaysW7caWGkbWaaSbaaSqaaiaadQhaaeqaaOGa aiykaaaa@4835@ forms a complete of conserved physics quantities and for S = 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWhdaqaaKqzGe Gaam4uaaGccaGLHdcacqGH9aqpdaWfGaqaaKqzGeGaaGymaiaac+ca caaIYaaaleqabaqcLbmacqGHsgIRaaaaaa@4227@ , the eigen values of the spin orbital coupling operator are k ± 1 2 { ( l± 1 2 )(l± 1 2 +1)+l(l+1) 3 4 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiabgglaXcqabaGccqGHHjIUdaWcbaWcbaGaaGymaaqaaiaaikda aaGcdaGadaqaamaabmaabaGaamiBaiabgglaXoaalaaabaGaaGymaa qaaiaaikdaaaaacaGLOaGaayzkaaGaaiikaiaadYgacqGHXcqSdaWc aaqaaiaaigdaaeaacaaIYaaaaiabgUcaRiaaigdacaGGPaGaey4kaS IaamiBaiaacIcacaWGSbGaey4kaSIaaGymaiaacMcacqGHsisldaWc aaqaaiaaiodaaeaacaaI0aaaaaGaay5Eaiaaw2haaaaa@55FC@ corresponding: j=l+1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGQbGaeyypa0 JaamiBaiabgUcaRiaaigdacaGGVaGaaGOmaaaa@3DC6@ (spin up) and j=l1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGQbGaeyypa0 JaamiBaiabgkHiTiaaigdacaGGVaGaaGOmaaaa@3DD1@  (spin down), respectively then one can form a diagonal ( 3×3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaaio dacqGHxdaTcaaIZaaacaGLOaGaayzkaaaaaa@3CEE@ matrix, with diagonal elements are ( H somg ) 11 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI eadaWgaaWcbaGaam4Caiaad+gacqGHsislcaWGTbGaam4zaaqabaaa kiaawIcacaGLPaaadaWgaaWcbaGaaGymaiaaigdaaeqaaaaa@40B9@ , ( H somg ) 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI eadaWgaaWcbaGaam4Caiaad+gacqGHsislcaWGTbGaam4zaaqabaaa kiaawIcacaGLPaaadaWgaaWcbaGaaGOmaiaaikdaaeqaaaaa@40BB@  and ( H somg ) 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI eadaWgaaWcbaGaam4Caiaad+gacqGHsislcaWGTbGaam4zaaqabaaa kiaawIcacaGLPaaadaWgaaWcbaGaaG4maiaaiodaaeqaaaaa@40BD@  for NMGESC potential in (NC: 3D-RSP) symmetries, as:

( H somg ) 11 =γ k + ( ( 2 V 0 α 3 V 0 2 r 3 )Θ+ θ ¯ 2μ )if j=l+1/2  ( H somg ) 22 =γ k ( ( 2 V 0 α 3 V 0 2 r 3 )Θ+ θ ¯ 2μ )if j=l1/2 ( H somg ) 33 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaamaabmaaba GaamisamaaBaaaleaacaWGZbGaam4BaiabgkHiTiaad2gacaWGNbaa beaaaOGaayjkaiaawMcaamaaBaaaleaacaaIXaGaaGymaaqabaGccq GH9aqpcqaHZoWzcaWGRbWaaSbaaSqaaiabgUcaRaqabaGcdaqadaqa amaabmaabaGaaGOmaiaadAfadaWgaaWcbaGaaGimaaqabaGccqaHXo qydaahaaWcbeqaaiaaiodaaaGccqGHsisldaWcaaqaaiaadAfadaWg aaWcbaGaaGimaaqabaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaG 4maaaaaaaakiaawIcacaGLPaaacqqHyoqucqGHRaWkdaWcaaqaaiqb eI7aXzaaraaabaGaaGOmaiabeY7aTbaaaiaawIcacaGLPaaacaaMc8 UaaeyAaiaabAgacaqGGaGaamOAaiabg2da9iaadYgacqGHRaWkcaaI XaGaai4laiaaikdacaqGGaaabaWaaeWaaeaacaWGibWaaSbaaSqaai aadohacaWGVbGaeyOeI0IaamyBaiaadEgaaeqaaaGccaGLOaGaayzk aaWaaSbaaSqaaiaaikdacaaIYaaabeaakiabg2da9iabeo7aNjaadU gadaWgaaWcbaGaeyOeI0cabeaakmaabmaabaWaaeWaaeaacaaIYaGa amOvamaaBaaaleaacaaIWaaabeaakiabeg7aHnaaCaaaleqabaGaaG 4maaaakiabgkHiTmaalaaabaGaamOvamaaBaaaleaacaaIWaaabeaa aOqaaiaaikdacaWGYbWaaWbaaSqabeaacaaIZaaaaaaaaOGaayjkai aawMcaaiabfI5arjabgUcaRmaalaaabaGafqiUdeNbaebaaeaacaaI YaGaeqiVd0gaaaGaayjkaiaawMcaaiaaykW7caqGPbGaaeOzaiaabc cacaWGQbGaeyypa0JaamiBaiabgkHiTiaaigdacaGGVaGaaGOmaiaa ysW7aeaadaqadaqaaiaadIeadaWgaaWcbaGaam4Caiaad+gacqGHsi slcaWGTbGaam4zaaqabaaakiaawIcacaGLPaaadaWgaaWcbaGaaG4m aiaaiodaaeqaaOGaeyypa0JaaGimaaaaaa@9C91@ (26)

After profound calculation, one can show that, the new radial function R nl ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbWaaSbaaS qaaiaad6gacaWGSbaabeaakmaabmaabaGaamOCaaGaayjkaiaawMca aaaa@3D45@  satisfying the following differential equation for NMGESC potential:

( d 2 d r 2 + 2 r d dr l(l+1) r 2 ) R nl ( r )+2μ[ E nl + V 0 r ( 1+( 1+αr )exp( 2αr ) )( 2 V 0 α 3 V 0 r 3 ) L Θ L θ ¯ 2μ ] R nl ( r )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaamaala aabaGaamizamaaCaaaleqabaGaaGOmaaaaaOqaaiaadsgacaWGYbWa aWbaaSqabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaGOmaaqaai aadkhaaaWaaSaaaeaacaWGKbaabaGaamizaiaadkhaaaGaeyOeI0Ya aSaaaeaacaWGSbGaaiikaiaadYgacqGHRaWkcaaIXaGaaiykaaqaai aadkhadaahaaWcbeqaaiaaikdaaaaaaaGccaGLOaGaayzkaaGaamOu amaaBaaaleaacaWGUbGaamiBaaqabaGcdaqadaqaaiaadkhaaiaawI cacaGLPaaacqGHRaWkcaaIYaGaeqiVd02aamWaaeaacaWGfbWaaSba aSqaaiaad6gacaWGSbaabeaakiabgUcaRmaalaaabaGaamOvamaaBa aaleaacaaIWaaabeaaaOqaaiaadkhaaaWaaeWaaeaacaaIXaGaey4k aSYaaeWaaeaacaaIXaGaey4kaSIaeqySdeMaamOCaaGaayjkaiaawM caaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0IaaGOmaiabeg7a HjaadkhaaiaawIcacaGLPaaaaiaawIcacaGLPaaacqGHsisldaqada qaaiaaikdacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaeqySde2aaWba aSqabeaacaaIZaaaaOGaeyOeI0YaaSaaaeaacaWGwbWaaSbaaSqaai aaicdaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaG4maaaaaaaakiaa wIcacaGLPaaaieqaceWFmbGbaSaadaWfGaqaaiabfI5arbWcbeqaai abgkziUcaakiabgkHiTmaalaaabaGab8htayaalaWaaCbiaeaadaqd aaqaaiabeI7aXbaaaSqabeaacqGHsgIRaaaakeaacaaIYaGaeqiVd0 gaaiaaykW7aiaawUfacaGLDbaacaWGsbWaaSbaaSqaaiaad6gacaWG SbaabeaakmaabmaabaGaamOCaaGaayjkaiaawMcaaiabg2da9iaaic daaaa@9021@ (27)

The two terms which composed the expression of H per-mg ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaabchacaqGLbGaaeOCaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaadkhaaiaawIcacaGLPaaaaaa@40B5@  are proportional with two infinitesimals parameters ( Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHyoquaaa@394B@  and θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiqbeI7aXz aaraaaaa@3A31@ ), thus, in what follows, we proceed to solve the modified radial part of the MSE that is, equation (27) by applying standard perturbation theory for their exact solutions at first order of two parameters Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI5arb aa@39DA@  and θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaKqzGe GaeqiUdehaaaaa@3A2A@ .

The exact modified spin-orbital spectrum for NMGESC potential in extended global (NC: 3D- RSP) symmetries
The purpose here is to give a complete prescription for determine the energy level of n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad6gakm aaCaaaleqabaqcLbmacaWG0bGaamiAaaaaaaa@3CA1@ excited states, of hydrogenic atoms with NMGESC potential, we first find the corrections E u-mg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaabwhacaqGTaGaamyBaiaadEgaaeqaaaaa@3C50@  and E d-mg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaabsgacaqGTaGaamyBaiaadEgaaeqaaaaa@3C3F@  for hydrogenic atoms which have j=l+1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadQgacq GH9aqpcaWGSbGaey4kaSIaaGymaiaac+cacaaIYaaaaa@3E55@  (spin up) and j=l1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGQbGaeyypa0 JaamiBaiabgkHiTiaaigdacaGGVaGaaGOmaaaa@3DD1@  (spin down), respectively, at first order of two parameters Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabfI5arb aa@39DA@  and θ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaiabeI 7aXbaaaaa@399B@  obtained by applying the standard perturbation theory to find the following:

E umg =γ N nl 2 ( 2β ) 22α k + 0 + exp( ν ) ν 2α+1 [ L n 2α+1 ( ν ) ] 2 ( ( 2 V 0 α 3 V 0 r 3 )Θ+ θ ¯ 2μ )dν E dmg =γ N nl 2 ( 2β ) 2α2 k 0 + exp( ν ) ν 2α+1 [ L n 2α+1 ( ν ) ] 2 ( ( 2 V 0 α 3 V 0 r 3 )Θ+ θ ¯ 2μ )dν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadweada WgaaWcbaGaamyDaiabgkHiTiaad2gacaWGNbaabeaakiabg2da9iab eo7aNjaaykW7caWGobWaaSbaaSqaaiaad6gacaWGSbaabeaakmaaCa aaleqabaGaaGOmaaaakmaabmaabaGaaeOmaiabek7aIbGaayjkaiaa wMcaamaaCaaaleqabaGaeyOeI0IaaGOmaiabgkHiTiaaikdacqaHXo qyaaGccaWGRbWaaSbaaSqaaiabgUcaRaqabaGcdaWfGaqaamaaxaba baWaaubmaeqabeqabiabgUIiYdaabaqcLbmacaaIWaaakeqaaaqabe aajugWaiabgUcaRiabg6HiLcaakiGacwgacaGG4bGaaiiCamaabmaa baGaeyOeI0IaeqyVd4gacaGLOaGaayzkaaGaeqyVd42aaWbaaSqabe aacaaIYaGaeqySdeMaey4kaSIaaGymaaaakmaadmaabaGaamitamaa DaaaleaacaWGUbaabaGaaGOmaiabeg7aHjabgUcaRiaaigdaaaGcda qadaqaaiabe27aUbGaayjkaiaawMcaaaGaay5waiaaw2faamaaCaaa leqabaGaaGOmaaaakmaabmaabaWaaeWaaeaacaaIYaGaamOvamaaBa aaleaacaaIWaaabeaakiabeg7aHnaaCaaaleqabaGaaG4maaaakiab gkHiTmaalaaabaGaamOvamaaBaaaleaacaaIWaaabeaaaOqaaiaadk hadaahaaWcbeqaaiaaiodaaaaaaaGccaGLOaGaayzkaaGaeuiMdeLa ey4kaSYaaSaaaeaacuaH4oqCgaqeaaqaaiaaikdacqaH8oqBaaGaaG PaVdGaayjkaiaawMcaaiaaykW7caWGKbGaeqyVd4gabaGaamyramaa BaaaleaacaWGKbGaeyOeI0IaamyBaiaadEgaaeqaaOGaeyypa0Jaeq 4SdCMaaGPaVlaad6eadaWgaaWcbaGaamOBaiaadYgaaeqaaOWaaWba aSqabeaacaaIYaaaaOWaaeWaaeaacaqGYaGaeqOSdigacaGLOaGaay zkaaWaaWbaaSqabeaacqGHsislcaaIYaGaeqySdeMaeyOeI0IaaGOm aaaakiaadUgadaWgaaWcbaGaeyOeI0cabeaakmaaxacabaWaaCbeae aadaqfWaqabeqabeGaey4kIipaaeaajugWaiaaicdaaOqabaaabeqa aKqzadGaey4kaSIaeyOhIukaaOGaciyzaiaacIhacaGGWbWaaeWaae aacqGHsislcqaH9oGBaiaawIcacaGLPaaacqaH9oGBdaahaaWcbeqa aiaaikdacqaHXoqycqGHRaWkcaaIXaaaaOWaamWaaeaacaWGmbWaa0 baaSqaaiaad6gaaeaacaaIYaGaeqySdeMaey4kaSIaaGymaaaakmaa bmaabaGaeqyVd4gacaGLOaGaayzkaaaacaGLBbGaayzxaaWaaWbaaS qabeaacaaIYaaaaOWaaeWaaeaadaqadaqaaiaaikdacaWGwbWaaSba aSqaaiaaicdaaeqaaOGaeqySde2aaWbaaSqabeaacaaIZaaaaOGaey OeI0YaaSaaaeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOC amaaCaaaleqabaGaaG4maaaaaaaakiaawIcacaGLPaaacqqHyoqucq GHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmaiabeY7aTbaacaaM c8oacaGLOaGaayzkaaGaaGPaVlaadsgacqaH9oGBaaaa@DC41@ (28)

Now, we can write the above two equations to the new form:

E umg =γ N nl 2 ( 2β ) 2α2 k + { Θ T 1 ( n,α )+Θ T 2 ( n,α )+ θ ¯ 2μ T 3 ( n,α ) } E dmg =γ N nl 2 ( 2β ) 2α2 k { Θ T 1 ( n,α )+Θ T 2 ( n,α )+ θ ¯ 2μ T 3 ( n,α ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadweada WgaaWcbaGaamyDaiabgkHiTiaad2gacaWGNbaabeaakiabg2da9iab eo7aNjaaykW7caWGobWaaSbaaSqaaiaad6gacaWGSbaabeaakmaaCa aabeqaaiaaikdaaaWaaeWaaeaacaqGYaGaeqOSdigacaGLOaGaayzk aaWaaWbaaeqabaGaeyOeI0IaaGOmaiabeg7aHjabgkHiTiaaikdaaa Gaam4AamaaBaaaleaacqGHRaWkaeqaaOWaaiWaaeaacqqHyoqucaWG ubWaaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacaWGUbGaaiilaiabeg 7aHbGaayjkaiaawMcaaiabgUcaRiabfI5arjaadsfadaWgaaWcbaGa aGOmaaqabaGcdaqadaqaaiaad6gacaGGSaGaeqySdegacaGLOaGaay zkaaGaey4kaSYaaSaaaeaacuaH4oqCgaqeaaqaaiaaikdacqaH8oqB aaGaamivamaaBaaaleaacaaIZaaabeaakmaabmaabaGaamOBaiaacY cacqaHXoqyaiaawIcacaGLPaaaaiaawUhacaGL9baaaeaacaWGfbWa aSbaaSqaaiaadsgacqGHsislcaWGTbGaam4zaaqabaGccqGH9aqpcq aHZoWzcaaMc8UaamOtamaaBaaaleaacaWGUbGaamiBaaqabaGcdaah aaqabeaacaaIYaaaamaabmaabaGaaeOmaiabek7aIbGaayjkaiaawM caamaaCaaabeqaaiabgkHiTiaaikdacqaHXoqycqGHsislcaaIYaaa aiaadUgadaWgaaWcbaGaeyOeI0cabeaakmaacmaabaGaeuiMdeLaam ivamaaBaaaleaajugWaiaaigdaaSqabaGcdaqadaqaaiaad6gacaGG SaGaeqySdegacaGLOaGaayzkaaGaey4kaSIaeuiMdeLaamivamaaBa aaleaacaaIYaaabeaakmaabmaabaGaamOBaiaacYcacqaHXoqyaiaa wIcacaGLPaaacqGHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmai abeY7aTbaacaWGubWaaSbaaSqaaiaaiodaaeqaaOWaaeWaaeaacaWG UbGaaiilaiabeg7aHbGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaaaa@A67C@ (29)

Moreover, the expressions of the three factors T 1 ( n,α ), T 2 ( n,α )and T 3 ( n,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaSbaaS qaaiaaigdaaeqaaOWaaeWaaeaacaWGUbGaaiilaiabeg7aHbGaayjk aiaawMcaaiaacYcacaWGubWaaSbaaSqaaiaaikdaaeqaaOWaaeWaae aacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaiaadggacaWGUbGa amizaiaaykW7caWGubWaaSbaaSqaaiaaiodaaeqaaOWaaeWaaeaaca WGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaaaa@5093@ are given by:

T 1 ( n,α )=2 V 0 α 3 T 3 ( n,α ) =2 V 0 α 3 0 + ν ( 2α+2 )1 exp( ν ) [ L n 2α+1 ( ν ) ] 2 dν  T 2 ( n,α )= V 0 ( 2β ) 3 0 + ν ( 2α1 )1 exp( ν ) [ L n 2α+1 ( ν ) ] 2 dν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadsfada WgaaWcbaGaaGymaaqabaGcdaqadaqaaiaad6gacaGGSaGaeqySdega caGLOaGaayzkaaGaeyypa0JaaGOmaiaadAfadaWgaaWcbaGaaGimaa qabaGccqaHXoqydaahaaWcbeqaaiaaiodaaaGccaWGubWaaSbaaSqa aiaaiodaaeqaaOWaaeWaaeaacaWGUbGaaiilaiabeg7aHbGaayjkai aawMcaaiaabccacqGH9aqpcaaIYaGaamOvamaaBaaaleaacaaIWaaa beaakiabeg7aHnaaCaaaleqabaGaaG4maaaakmaaxacabaWaaCbeae aadaqfWaqabSqabeqaniabgUIiYdaaleaacaaIWaaabeaaaeqabaGa ey4kaSIaeyOhIukaaOGaeqyVd42aaWbaaSqabeaadaqadaqaaiaaik dacqaHXoqycqGHRaWkcaaIYaaacaGLOaGaayzkaaGaeyOeI0IaaGym aaaakiGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0IaeqyVd4gaca GLOaGaayzkaaWaamWaaeaacaWGmbWaa0baaSqaaiaad6gaaeaacaaI YaGaeqySdeMaey4kaSIaaGymaaaakmaabmaabaGaeqyVd4gacaGLOa GaayzkaaaacaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaaGPa VlaadsgacqaH9oGBcaqGGaaabaGaamivamaaBaaaleaajugWaiaaik daaSqabaGcdaqadaqaaiaad6gacaGGSaGaeqySdegacaGLOaGaayzk aaGaeyypa0JaeyOeI0IaamOvamaaBaaaleaajugWaiaaicdaaSqaba GcdaqadaqaaiaaikdacqaHYoGyaiaawIcacaGLPaaadaahaaWcbeqa aiaaiodaaaGcdaWfGaqaamaaxababaWaaubmaeqaleqabeqdcqGHRi I8aaWcbaGaaGimaaqabaaabeqaaiabgUcaRiabg6HiLcaakiabe27a UnaaCaaaleqabaWaaeWaaeaacaaIYaGaeqySdeMaeyOeI0IaaGymaa GaayjkaiaawMcaaiabgkHiTiaaigdaaaGcciGGLbGaaiiEaiaaccha daqadaqaaiabgkHiTiabe27aUbGaayjkaiaawMcaamaadmaabaGaam itamaaDaaaleaacaWGUbaabaGaaGOmaiabeg7aHjabgUcaRiaaigda aaGcdaqadaqaaiabe27aUbGaayjkaiaawMcaaaGaay5waiaaw2faam aaCaaaleqabaGaaGOmaaaakiaaykW7caWGKbGaeqyVd4gaaaa@B083@ (30)

To evaluate the above factors T 1 ( n,α ), T 2 ( n,α )and T 3 ( n,α ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGubWaaSbaaS qaaiaaigdaaeqaaOWaaeWaaeaacaWGUbGaaiilaiabeg7aHbGaayjk aiaawMcaaiaacYcacaWGubWaaSbaaSqaaiaaikdaaeqaaOWaaeWaae aacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaiaadggacaWGUbGa amizaiaaykW7caaMc8UaamivamaaBaaaleaacaaIZaaabeaakmaabm aabaGaamOBaiaacYcacqaHXoqyaiaawIcacaGLPaaacaGGSaaaaa@52CE@  we apply the following special integration:41

0 + t . ε1. exp( ωt ) L m λ ( ωt ) L n β ( ωt )dt= ω ε Γ( nε+β+1 )Γ( m+λ+1 ) m!n!Γ( 1ε+β )Γ( 1+λ ) F 3 2 ( m,ε,εβ;n+ε,λ+1;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaamaaxa babaWaaubmaeqaleqabeqdcqGHRiI8aaWcbaGaaGimaaqabaaabeqa aiabgUcaRiabg6HiLcaakiaadshadaqhaaWcbaGaaiOlaaqaaiabew 7aLjabgkHiTiaaigdacaGGUaaaaOGaciyzaiaacIhacaGGWbWaaeWa aeaacqGHsislcqaHjpWDcaWG0baacaGLOaGaayzkaaGaamitamaaDa aaleaacaWGTbaabaGaeq4UdWgaaOWaaeWaaeaacqaHjpWDcaWG0baa caGLOaGaayzkaaGaaGPaVlaadYeadaqhaaWcbaGaamOBaaqaaiabek 7aIbaakmaabmaabaGaeqyYdCNaamiDaaGaayjkaiaawMcaaiaadsga caWG0bGaeyypa0ZaaSaaaeaacqaHjpWDdaahaaWcbeqaaiabgkHiTi abew7aLbaakiabfo5ahnaabmaabaGaamOBaiabgkHiTiabew7aLjab gUcaRiabek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaacqqHtoWrda qadaqaaiaad2gacqGHRaWkcqaH7oaBcqGHRaWkcaaIXaaacaGLOaGa ayzkaaaabaGaamyBaiaabgcacaWGUbGaaeyiaiabfo5ahnaabmaaba GaaGymaiabgkHiTiabew7aLjabgUcaRiabek7aIbGaayjkaiaawMca aiabfo5ahnaabmaabaGaaGymaiabgUcaRiabeU7aSbGaayjkaiaawM caaaaadaWgbaWcbaGaaG4maaqabaGccaWGgbWaaSbaaSqaaiaaikda aeqaaOWaaeWaaeaacqGHsislcaWGTbGaaiilaiabew7aLjaacYcacq aH1oqzcqGHsislcqaHYoGycaGG7aGaeyOeI0IaamOBaiabgUcaRiab ew7aLjaacYcacqaH7oaBcqGHRaWkcaaIXaGaai4oaiaaigdaaiaawI cacaGLPaaaaaa@A168@ (31)

where F 3 2 ( m,ε,εβ;n+ε,λ+1;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWgbaWcbaGaaG 4maaqabaGccaWGgbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacqGH sislcaWGTbGaaiilaiabew7aLjaacYcacqaH1oqzcqGHsislcqaHYo GycaGG7aGaeyOeI0IaamOBaiabgUcaRiabew7aLjaacYcacqaH7oaB cqGHRaWkcaaIXaGaai4oaiaaigdaaiaawIcacaGLPaaaaaa@4FCC@ is obtained from the generalized hyper geometric function. F p q ( α 1 ,..., α p , β 1 ,...., β q ,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWgbaWcbaGaam iCaaqabaGccaWGgbWaaSbaaSqaaiaadghaaeqaaOWaaeWaaeaacqaH XoqydaWgaaWcbaGaaGymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUa Gaaiilaiabeg7aHnaaBaaaleaacaWGWbaabeaakiaacYcacqaHYoGy daWgaaWcbaGaaGymaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaai OlaiaacYcacqaHYoGydaWgaaWcbaGaamyCaaqabaGccaGGSaGaamOE aaGaayjkaiaawMcaaaaa@5136@ for p=3andq=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaadchacq GH9aqpcaaIZaGaaGPaVlaadggacaWGUbGaamizaiaaykW7caWGXbGa eyypa0JaaGOmaaaa@43AB@ while Γ( x )= 0 + z x1 e z dz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHtoWrdaqada qaaiaadIhaaiaawIcacaGLPaaacqGH9aqpdaWdXbqaaiaadQhadaah aaWcbeqaaiaadIhacqGHsislcaaIXaaaaOGaamyzamaaCaaaleqaba GaeyOeI0IaamOEaaaaaeaacaaIWaaabaGaey4kaSIaeyOhIukaniab gUIiYdGccaWGKbGaamOEaaaa@4B03@ denote to the usual Gamma function. After straightforward calculations, we can obtain the explicitly results:

T 1 ( n,α )=2 V 0 α 3 T 3 ( n,α )=2 V 0 α 3 Γ( n )Γ( n+2α+2 ) n ! 2 Γ( 0 )Γ( 2α+2 ) F 3 2 ( n,2α+2,1;n+2α+2,2α+2;1 ) T 2 ( n,α )= V 0 ( 2β ) 3 Γ( n+1 )Γ( n+2α+2 ) n ! 2 Γ( 3 )Γ( 2α+2 ) F 3 2 ( n,2α1,2;n+2α1,2α+2;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadsfada WgaaWcbaGaaGymaaqabaGcdaqadaqaaiaad6gacaGGSaGaeqySdega caGLOaGaayzkaaGaeyypa0JaaGOmaiaadAfadaWgaaWcbaGaaGimaa qabaGccqaHXoqydaahaaWcbeqaaiaaiodaaaGccaWGubWaaSbaaSqa aiaaiodaaeqaaOWaaeWaaeaacaWGUbGaaiilaiabeg7aHbGaayjkai aawMcaaiabg2da9iaaikdacaWGwbWaaSbaaSqaaiaaicdaaeqaaOGa eqySde2aaWbaaSqabeaacaaIZaaaaOWaaSaaaeaacqqHtoWrdaqada qaaiaad6gaaiaawIcacaGLPaaacqqHtoWrdaqadaqaaiaad6gacqGH RaWkcaaIYaGaeqySdeMaey4kaSIaaGOmaaGaayjkaiaawMcaaaqaai aad6gacaqGHaWaaWbaaSqabeaacaqGYaaaaOGaeu4KdC0aaeWaaeaa caaIWaaacaGLOaGaayzkaaGaeu4KdC0aaeWaaeaacaaIYaGaeqySde Maey4kaSIaaGOmaaGaayjkaiaawMcaaaaadaWgbaWcbaGaaG4maaqa baGccaWGgbWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaacqGHsislca WGUbGaaiilaiaaikdacqaHXoqycqGHRaWkcaaIYaGaaiilaiaaigda caGG7aGaeyOeI0IaamOBaiabgUcaRiaaikdacqaHXoqycqGHRaWkca aIYaGaaiilaiaaikdacqaHXoqycqGHRaWkcaaIYaGaai4oaiaaigda aiaawIcacaGLPaaaaeaacaWGubWaaSbaaSqaaiaaikdaaeqaaOWaae WaaeaacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaiabg2da9iab gkHiTiaadAfadaWgaaWcbaGaaGimaaqabaGcdaqadaqaaiaaikdacq aHYoGyaiaawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaGcdaWcaaqa aiabfo5ahnaabmaabaGaamOBaiabgUcaRiaaigdaaiaawIcacaGLPa aacqqHtoWrdaqadaqaaiaad6gacqGHRaWkcaaIYaGaeqySdeMaey4k aSIaaGOmaaGaayjkaiaawMcaaaqaaiaad6gacaqGHaWaaWbaaSqabe aacaqGYaaaaOGaeu4KdC0aaeWaaeaacaaIZaaacaGLOaGaayzkaaGa eu4KdC0aaeWaaeaacaaIYaGaeqySdeMaey4kaSIaaGOmaaGaayjkai aawMcaaaaadaWgbaWcbaGaaG4maaqabaGccaWGgbWaaSbaaSqaaiaa ikdaaeqaaOWaaeWaaeaacqGHsislcaWGUbGaaiilaiaaikdacqaHXo qycqGHsislcaaIXaGaaiilaiabgkHiTiaaikdacaGG7aGaeyOeI0Ia amOBaiabgUcaRiaaikdacqaHXoqycqGHsislcaaIXaGaaiilaiaaik dacqaHXoqycqGHRaWkcaaIYaGaai4oaiaaigdaaiaawIcacaGLPaaa aaaa@CA43@ (32)

 We have Γ( 0 )=( 1 )!= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHtoWrdaqada qaaiaaicdaaiaawIcacaGLPaaacqGH9aqpdaqadaqaaiabgkHiTiaa igdaaiaawIcacaGLPaaacaqGHaGaeyypa0JaeyOhIukaaa@42D1@ , Γ( n+1 )=n! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHtoWrdaqada qaaiaad6gacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeyypa0JaamOB aiaabgcaaaa@3FF2@ and Γ( 3 )=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHtoWrdaqada qaaiaaiodaaiaawIcacaGLPaaacqGH9aqpcaaIYaaaaa@3D44@ , allow us the two to obtain the exact modifications E u-mg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaabwhacaqGTaGaamyBaiaadEgaaeqaaaaa@3C50@  and E d-mg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaabsgacaqGTaGaamyBaiaadEgaaeqaaaaa@3C3F@ of n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaWbaaS qabeaacaWG0bGaamiAaaaaaaa@3ADA@  excited states of hydrogenic atoms with NMGESC potential, which produced by modified spin-orbital effect H somg ( r,Θ, θ ¯ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaadohacaWGVbGaeyOeI0IaamyBaiaadEgaaeqaaOWaaeWaaeaa caWGYbGaaiilaiabfI5arjaacYcacuaH4oqCgaqeaaGaayjkaiaawM caaaaa@44B3@  as:

E u-mg = 1 2 γΘ N nl 2 ( 2β ) 12α V 0 k f( n,α ) E d-mg = 1 2 γΘ N nl 2 ( 2β ) 12α V 0 k + f( n,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadweada WgaaWcbaGaaeyDaiaab2cacaWGTbGaam4zaaqabaGccqGH9aqpcqGH sisldaWcaaqaaiaaigdaaeaacaaIYaaaaiabeo7aNjaaykW7cqqHyo qucaWGobWaaSbaaSqaaiaad6gacaWGSbaabeaakmaaCaaaleqabaGa aGOmaaaakmaabmaabaGaaeOmaiabek7aIbGaayjkaiaawMcaamaaCa aaleqabaGaaGymaiabgkHiTiaaikdacqaHXoqyaaGccaWGwbWaaSba aSqaaiaaicdaaeqaaOGaam4AamaaBaaaleaacqGHsislaeqaaOGaam OzamaabmaabaGaamOBaiaacYcacqaHXoqyaiaawIcacaGLPaaaaeaa caWGfbWaaSbaaSqaaiaabsgacaqGTaGaamyBaiaadEgaaeqaaOGaey ypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaacqaHZoWzcaaM c8UaeuiMdeLaamOtamaaBaaaleaacaWGUbGaamiBaaqabaGcdaahaa WcbeqaaiaaikdaaaGcdaqadaqaaiaabkdacqaHYoGyaiaawIcacaGL PaaadaahaaWcbeqaaiaaigdacqGHsislcaaIYaGaeqySdegaaOGaam OvamaaBaaaleaacaaIWaaabeaakiaadUgadaWgaaWcbaGaey4kaSca beaakiaadAgadaqadaqaaiaad6gacaGGSaGaeqySdegacaGLOaGaay zkaaaaaaa@7C2B@ (33)

Where f( n,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaeWaae aacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaaaa@3D8A@  is given by:

f( n,α )= Γ( n+2α+2 ) n!Γ( 2α+2 ) F 3 2 ( n,2α1,2;n+2α1,2α+2;1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaeWaae aacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaaiabg2da9maalaaa baGaeu4KdC0aaeWaaeaacaWGUbGaey4kaSIaaGOmaiabeg7aHjabgU caRiaaikdaaiaawIcacaGLPaaaaeaacaWGUbGaaeyiaiabfo5ahnaa bmaabaGaaGOmaiabeg7aHjabgUcaRiaaikdaaiaawIcacaGLPaaaaa WaaSraaSqaaiaaiodaaeqaaOGaamOramaaBaaaleaacaaIYaaabeaa kmaabmaabaGaeyOeI0IaamOBaiaacYcacaaIYaGaeqySdeMaeyOeI0 IaaGymaiaacYcacqGHsislcaaIYaGaai4oaiabgkHiTiaad6gacqGH RaWkcaaIYaGaeqySdeMaeyOeI0IaaGymaiaacYcacaaIYaGaeqySde Maey4kaSIaaGOmaiaacUdacaaIXaaacaGLOaGaayzkaaaaaa@6AAD@ (34)

Thus, the extended global quantum group symmetry (NC: 3D-RSP) reduce to new quantum subgroup symmetry (NC: 3D-RS).

The exact modified magnetic spectrum for NMGESC potential in extended global (NC: 3D- RSP) symmetries:
Further to the important previously obtained results, now, we consider another physically meaningful phenomena produced by the effect of NMGESC potential related to the influence of an external uniform magnetic field B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaea qbaaGcbaGabmOqayaalaaaaa@3B43@ , to avoid the repetition in the theoretical calculations, it’s sufficient to apply the following replacements:

{ Θ χ B θ ¯ σ ¯ B ( ( 2 V 0 α 3 V 0 2 r 3 )Θ+ θ ¯ 2μ )( ( 2 V 0 α 3 V 0 2 r 3 )χ+ σ ¯ 2μ ) B L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGabaabaeqaba GafuiMdeLbaSaacqGHsgIRcqaHhpWyceWGcbGbaSaaaeaacuaH4oqC gaqegaWcaiabgkziUoaanaaabaGaeq4WdmhaaiqadkeagaWcaaaaca GL7baacqGHshI3daqadaqaamaabmaabaGaaGOmaiaadAfadaWgaaWc baGaaGimaaqabaGccqaHXoqydaahaaWcbeqaaiaaiodaaaGccqGHsi sldaWcaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaaIYaGa amOCamaaCaaaleqabaGaaG4maaaaaaaakiaawIcacaGLPaaacqqHyo qucqGHRaWkdaWcaaqaaiqbeI7aXzaaraaabaGaaGOmaiabeY7aTbaa aiaawIcacaGLPaaacqGHshI3daqadaqaamaabmaabaGaaGOmaiaadA fadaWgaaWcbaGaaGimaaqabaGccqaHXoqydaahaaWcbeqaaiaaioda aaGccqGHsisldaWcaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaake aacaaIYaGaamOCamaaCaaaleqabaGaaG4maaaaaaaakiaawIcacaGL PaaacqaHhpWycqGHRaWkdaWcaaqaamaanaaabaGaeq4Wdmhaaaqaai aaikdacqaH8oqBaaaacaGLOaGaayzkaaGabmOqayaalaWaaCbiaeaa caWGmbaaleqabaGaeyOKH4kaaaaa@76DC@ (35)

Here χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiabeE8aJb aa@3A1A@ and σ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaiabeo 8aZbaaaaa@39A8@  are two infinitesimal real proportional’s constants, and we choose the arbitrary external magnetic field B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaceWGcbGbaSaaaa a@38AD@  parallel to the (Oz) axis, which allow us to introduce the new modified magnetic Hamiltonian H mmg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacqGHsislcaWGTbGaam4zaaqabaaaaa@3C8A@  in (NC: 3D-RSP) symmetries as:

H mmg =( ( 2 V 0 α 3 V 0 2 r 3 )χ+ σ ¯ 2μ ) modz MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad2gacqGHsislcaWGTbGaam4zaaqabaGccqGH9aqpdaqadaqa amaabmaabaGaaGOmaiaadAfadaWgaaWcbaGaaGimaaqabaGccqaHXo qydaahaaWcbeqaaiaaiodaaaGccqGHsisldaWcaaqaaiaadAfadaWg aaWcbaGaaGimaaqabaaakeaacaaIYaGaamOCamaaCaaaleqabaGaaG 4maaaaaaaakiaawIcacaGLPaaacqaHhpWycqGHRaWkdaWcaaqaamaa naaabaGaeq4WdmhaaaqaaiaaikdacqaH8oqBaaaacaGLOaGaayzkaa GaeyynHa8aaSbaaSqaaiGac2gacaGGVbGaaiizaiabgkHiTiaadQha aeqaaaaa@58AC@ (36)

Here modz B J z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGH1ecWdaWgaa WcbaGaciyBaiaac+gacaGGKbGaeyOeI0IaamOEaaqabaGccqGHHjIU ceWGcbGbaSaadaWfGaqaaiaadQeaaSqabeaacqGHsgIRaaGccqGHsi slcqGH1ecWdaWgaaWcbaGaamOEaaqabaaaaa@47D1@ denote to the modified Zeeman effect while z S B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGH1ecWdaWgaa WcbaGaamOEaaqabaGccqGHHjIUcqGHsisldaWfGaqaaiaadofaaSqa beaacqGHsgIRaaGcceWGcbGbaSaaaaa@4152@  is the ordinary Hamiltonian operator of Zeeman Effect. To obtain the exact noncommutative magnetic modifications of energy E mag-mg ( n,m,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab2gacaqGHbGaae4zaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaad6gacaGGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGLPa aaaaa@448D@ , we just replace k + MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaSbaaS qaaiabgUcaRaqabaaaaa@39D2@  and Θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqqHyoquaaa@394B@  in the eq. (33) by the following parameters: m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbaaaa@38C6@  and χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHhpWyaaa@398B@ , respectively:

E magmg ( n,m,α )= 1 2 γχ N nl 2 ( 2β ) 12α V 0 k f( n,α )Bm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad2gacaWGHbGaam4zaiabgkHiTiaad2gacaWGNbaabeaakmaa bmaabaGaamOBaiaacYcacaWGTbGaaiilaiabeg7aHbGaayjkaiaawM caaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaeq4S dCMaaGPaVlabeE8aJjaad6eadaWgaaWcbaGaamOBaiaadYgaaeqaaO WaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaqGYaGaeqOSdigacaGL OaGaayzkaaWaaWbaaSqabeaacaaIXaGaeyOeI0IaaGOmaiabeg7aHb aakiaadAfadaWgaaWcbaGaaGimaaqabaGccaWGRbWaaSbaaSqaaiab gkHiTaqabaGccaWGMbWaaeWaaeaacaWGUbGaaiilaiabeg7aHbGaay jkaiaawMcaaiaadkeacaWGTbaaaa@6479@ (37)

We have lm+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaliabgkHiTiaadY gacqGHKjYOcaWGTbGaeyizImQaey4kaSIaamiBaaaa@3FEC@ , which allow us to fixing ( 2l+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIYaGaamiBai abgUcaRiaaigdaaaa@3B1E@ ) values for discreet number m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaad2gaaa a@3955@ .

Results

In the light of the results of the preceding sections, let us resume the modified eigenenergies E nc -umg ( n,j,l,s,m,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab6gacaqGJbGaaeiiaiaab2cacaqG1bGaamyBaiaabEgaaeqa aOWaaeWaaeaacaWGUbGaaiilaiaadQgacaGGSaGaamiBaiaacYcaca WGZbGaaiilaiaad2gacaGGSaGaeqySdegacaGLOaGaayzkaaaaaa@4A27@ and E nc -dmg ( n,j,l,s,m,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab6gacaqGJbGaaeiiaiaab2cacaqGKbGaamyBaiaadEgaaeqa aOWaaeWaaeaacaWGUbGaaiilaiaadQgacaGGSaGaamiBaiaacYcaca WGZbGaaiilaiaad2gacaGGSaGaeqySdegacaGLOaGaayzkaaaaaa@4A18@ of a hydrogenic atoms with spin S = 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiaado faaSqabeaacqGHsgIRaaGccqGH9aqpdaWfGaqaaiaaigdacaGGVaGa aGOmaaWcbeqaaiabgkziUcaaaaa@4052@  for MSE with NMGESC potential obtained in this paper, the total energies corresponding n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaWbaaS qabeaacaWG0bGaamiAaaaaaaa@3ADA@  excited states in (NC: 3D-RSP) symmetries are determined on based to our original results presented on the Eqs. (33) and (37), in addition to the ordinary energy E nl MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaad6gacaWGSbaabeaaaaa@3AAE@ MGESC potential, which presented in the eq. (13):

E nc -umg ( n,j,l,s,m,α )= E nl 1 2 χγ N nl 2 ( 2β ) 12α V 0 k + f( n,α )( Θ k + +Bmχ ) E nc -dmg ( n,j,l,s,m,α )= E nl 1 2 χγ N nl 2 ( 2β ) 12α V 0 k f( n,α )( Θ k +Bmχ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadweada WgaaWcbaGaaeOBaiaabogacaqGGaGaaeylaiaabwhacaWGTbGaae4z aaqabaGcdaqadaqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSbGaai ilaiaadohacaGGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGLPaaa cqGH9aqpcaWGfbWaaSbaaSqaaiaad6gacaWGSbaabeaakiabgkHiTm aalaaabaGaaGymaaqaaiaaikdaaaGaeq4XdmMaaGPaVlabeo7aNjaa d6eadaWgaaWcbaGaamOBaiaadYgaaeqaaOWaaWbaaSqabeaacaaIYa aaaOWaaeWaaeaacaqGYaGaeqOSdigacaGLOaGaayzkaaWaaWbaaSqa beaacaaIXaGaeyOeI0IaaGOmaiabeg7aHbaakiaadAfadaWgaaWcba GaaGimaaqabaGccaWGRbWaaSbaaSqaaiabgUcaRaqabaGccaWGMbWa aeWaaeaacaWGUbGaaiilaiabeg7aHbGaayjkaiaawMcaamaabmaaba GaeuiMdeLaam4AamaaBaaaleaacqGHRaWkaeqaaOGaey4kaSIaamOq aiaad2gacqaHhpWyaiaawIcacaGLPaaaaeaacaWGfbWaaSbaaSqaai aab6gacaqGJbGaaeiiaiaab2cacaqGKbGaamyBaiaabEgaaeqaaOWa aeWaaeaacaWGUbGaaiilaiaadQgacaGGSaGaamiBaiaacYcacaWGZb Gaaiilaiaad2gacaGGSaGaeqySdegacaGLOaGaayzkaaGaeyypa0Ja amyramaaBaaaleaacaWGUbGaamiBaaqabaGccqGHsisldaWcaaqaai aaigdaaeaacaaIYaaaaiabeE8aJjaaykW7cqaHZoWzcaWGobWaaSba aSqaaiaad6gacaWGSbaabeaakmaaCaaaleqabaGaaGOmaaaakmaabm aabaGaaeOmaiabek7aIbGaayjkaiaawMcaamaaCaaaleqabaGaaGym aiabgkHiTiaaikdacqaHXoqyaaGccaWGwbWaaSbaaSqaaiaaicdaae qaaOGaam4AamaaBaaaleaacqGHsislaeqaaOGaamOzamaabmaabaGa amOBaiaacYcacqaHXoqyaiaawIcacaGLPaaadaqadaqaaiabfI5arj aadUgadaWgaaWcbaGaeyOeI0cabeaakiabgUcaRiaadkeacaWGTbGa eq4XdmgacaGLOaGaayzkaaaaaaa@B0CC@ (38)

This is the main goal of this work, It’s clearly, that the obtained eigenvalues of energies are real’s and then the noncommutative diagonal Hamiltonian H ncmg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiaad6gacaWGJbGaeyOeI0IaamyBaiaadEgaaeqaaaaa@3D73@  is Hermitian, furthermore it’s possible to writing the three elements: α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaaa@3973@ as follows:

( H ncmg ) 11 = Δ nc 2μ + H intumg ( H ncmg ) 22 = Δ nc 2μ + H intdmg ( H ncmg ) 33 = Δ 2μ V 0 r ( 1+( 1+αr )exp( 2αr ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaamaabmaaba GaamisamaaBaaaleaacaWGUbGaam4yaiabgkHiTiaad2gacaWGNbaa beaaaOGaayjkaiaawMcaamaaBaaaleaacaaIXaGaaGymaaqabaGccq GH9aqpcqGHsisldaWcaaqaaiabfs5aenaaBaaaleaacaWGUbGaam4y aaqabaaakeaacaaIYaGaeqiVd0gaaiabgUcaRiaadIeadaWgaaWcba GaciyAaiaac6gacaGG0bGaeyOeI0IaamyDaiaad2gacaWGNbaabeaa aOqaamaabmaabaGaamisamaaBaaaleaacaWGUbGaam4yaiabgkHiTi aad2gacaWGNbaabeaaaOGaayjkaiaawMcaamaaBaaaleaacaaIYaGa aGOmaaqabaGccqGH9aqpcqGHsisldaWcaaqaaiabfs5aenaaBaaale aacaWGUbGaam4yaaqabaaakeaacaaIYaGaeqiVd0gaaiabgUcaRiaa dIeadaWgaaWcbaGaciyAaiaac6gacaGG0bGaeyOeI0Iaamizaiaad2 gacaWGNbaabeaaaOqaamaabmaabaGaamisamaaBaaaleaacaWGUbGa am4yaiabgkHiTiaad2gacaWGNbaabeaaaOGaayjkaiaawMcaamaaBa aaleaacaaIZaGaaG4maaqabaGccqGH9aqpcqGHsisldaWcaaqaaiab fs5aebqaaiaaikdacqaH8oqBaaGaeyOeI0YaaSaaaeaacaWGwbWaaS baaSqaaiaaicdaaeqaaaGcbaGaamOCaaaadaqadaqaaiaaigdacqGH RaWkdaqadaqaaiaaigdacqGHRaWkcqaHXoqycaWGYbaacaGLOaGaay zkaaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGHsislcaaIYaGaeqyS deMaamOCaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaaaa@8E68@ (39)

Where

Δ nc 2μ = Δ θ ¯ L σ ¯ L 2μ H intumg = V 0 r ( 1+( 1+αr )exp( 2αr ) ) +γ( k + Θ+χ modz )( 2 V 0 α 3 V 0 r 3 ) H intdmg V 0 r ( 1+( 1+αr )exp( 2αr ) ) +γ( k Θ+χ modz )( 2 V 0 α 3 V 0 r 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaamaalaaaba GaeuiLdq0aaSbaaSqaaiaad6gacaWGJbaabeaaaOqaaiaaikdacqaH 8oqBaaGaeyypa0ZaaSaaaeaacqqHuoarcqGHsislcuaH4oqCgaqeam aaxacabaGaamitaaWcbeqaaiabgkziUcaakiabgkHiTmaanaaabaGa eq4WdmhaamaaxacabaGaamitaaWcbeqaaiabgkziUcaaaOqaaiaaik dacqaH8oqBaaaabaGaamisamaaBaaaleaaciGGPbGaaiOBaiaacsha cqGHsislcaWG1bGaamyBaiaadEgaaeqaaOGaeyypa0JaeyOeI0YaaS aaaeaacaWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOCaaaadaqa daqaaiaaigdacqGHRaWkdaqadaqaaiaaigdacqGHRaWkcqaHXoqyca WGYbaacaGLOaGaayzkaaGaciyzaiaacIhacaGGWbWaaeWaaeaacqGH sislcaaIYaGaeqySdeMaamOCaaGaayjkaiaawMcaaaGaayjkaiaawM caaiaabccacqGHRaWkcqaHZoWzdaqadaqaaiaadUgadaWgaaWcbaGa ey4kaScabeaakiabfI5arjabgUcaRiabeE8aJjaaykW7cqGH1ecWda WgaaWcbaGaciyBaiaac+gacaGGKbGaeyOeI0IaamOEaaqabaaakiaa wIcacaGLPaaadaqadaqaaiaaikdacaWGwbWaaSbaaSqaaiaaicdaae qaaOGaeqySde2aaWbaaSqabeaacaaIZaaaaOGaeyOeI0YaaSaaaeaa caWGwbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaamOCamaaCaaaleqaba GaaG4maaaaaaaakiaawIcacaGLPaaaaeaacaWGibWaaSbaaSqaaiGa cMgacaGGUbGaaiiDaiabgkHiTiaadsgacaWGTbGaam4zaaqabaGccq GHsisldaWcaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWG YbaaamaabmaabaGaaGymaiabgUcaRmaabmaabaGaaGymaiabgUcaRi abeg7aHjaadkhaaiaawIcacaGLPaaaciGGLbGaaiiEaiaacchadaqa daqaaiabgkHiTiaaikdacqaHXoqycaWGYbaacaGLOaGaayzkaaaaca GLOaGaayzkaaGaaeiiaiabgUcaRiabeo7aNnaabmaabaGaam4Aamaa BaaaleaacqGHsislaeqaaOGaeuiMdeLaey4kaSIaeq4XdmMaaGPaVl abgwtiapaaBaaaleaaciGGTbGaai4BaiaacsgacqGHsislcaWG6baa beaaaOGaayjkaiaawMcaamaabmaabaGaaGOmaiaadAfadaWgaaWcba GaaGimaaqabaGccqaHXoqydaahaaWcbeqaaiaaiodaaaGccqGHsisl daWcaaqaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGYbWaaW baaSqabeaacaaIZaaaaaaaaOGaayjkaiaawMcaaaaaaa@C5AB@ (40)

Thus, the ordinary kinetic term for MGESC potential Δ 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsisldaWcaa qaaiabfs5aebqaaiaaikdacqaH8oqBaaaaaa@3CA9@  and ordinary interaction V 0 r ( 1+( 1+αr )exp( 2αr ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsisldaWcaa qaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGYbaaamaabmaa baGaaGymaiabgUcaRmaabmaabaGaaGymaiabgUcaRiabeg7aHjaadk haaiaawIcacaGLPaaaciGGLbGaaiiEaiaacchadaqadaqaaiabgkHi TiaaikdacqaHXoqycaWGYbaacaGLOaGaayzkaaaacaGLOaGaayzkaa aaaa@4D18@ >are replaced by new modified form of kinetic term Δ nc 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiabfs 5aenaaBaaaleaacaWGUbGaam4yaaqabaaakeaacaaIYaGaeqiVd0ga aaaa@3DCD@  and new modified interactions modified to the new form ( H intumg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiGacMgacaGGUbGaaiiDaiabgkHiTiaadwhacaWGTbGaam4zaaqa baaaaa@3F6B@ and H intdmg MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibWaaSbaaS qaaiGacMgacaGGUbGaaiiDaiabgkHiTiaadsgacaWGTbGaam4zaaqa baaaaa@3F5A@ ). On the other hand, it is evident to consider the quantum number m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGTbaaaa@38C6@  takes ( 2l+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaaik dacaWGSbGaey4kaSIaaGymaaGaayjkaiaawMcaaaaa@3CA7@  values and we have also two values for j=l± 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGQbGaeyypa0 JaamiBaiabgglaXoaalaaabaGaaGymaaqaaiaaikdaaaaaaa@3E2F@ , thus every state in usually three dimensional space of energy for NMGESC potential will be 2( 2l+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIYaWaaeWaae aacaaIYaGaamiBaiabgUcaRiaaigdaaiaawIcacaGLPaaaaaa@3D63@ sub-states. To obtain the total complete degeneracy of energy level of the NMGESC potential in noncommutative three-dimension spaces-phases, we need to sum for all allowed values of l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbstHrhAaeXatLxBI9gBaerbd9wDYLwzYbItLDharuavP1wzZbIt LDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4 rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9 pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaaea qbaaGcbaGaamiBaaaa@3B5B@ . Total degeneracy is thus,

2 i=0 n1 ( 2l+1 ) 2 n 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaaIYaWaaabCae aadaqadaqaaiaaikdacaWGSbGaey4kaSIaaGymaaGaayjkaiaawMca aaWcbaGaamyAaiabg2da9iaaicdaaeaacaWGUbGaeyOeI0IaaGymaa qdcqGHris5aOGaeyyyIORaaGOmaiaad6gadaahaaWcbeqaaiaaikda aaaaaa@4959@ (41)

Note that the obtained new energy eigen values ( E nc -umg ( n,j,l,s,m,α )and E nc-dmg ( n,j,l,s,m,α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadw eadaWgaaWcbaGaaeOBaiaabogacaqGGaGaaeylaiaabwhacaWGTbGa am4zaaqabaGcdaqadaqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSb GaaiilaiaadohacaGGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGL PaaacaWGHbGaamOBaiaadsgacaaMc8UaamyramaaBaaaleaacaqGUb Gaae4yaiaab2cacaqGKbGaaeyBaiaadEgaaeqaaOWaaeWaaeaacaWG UbGaaiilaiaadQgacaGGSaGaamiBaiaacYcacaWGZbGaaiilaiaad2 gacaGGSaGaeqySdegacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@619E@ and E nc -mg ( n,j,l,s,α,m ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab6gacaqGJbGaaeiiaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSbGaaiilaiaadohaca GGSaGaeqySdeMaaiilaiaad2gaaiaawIcacaGLPaaaaaa@4931@  now depend to new discrete atomic quantum numbers ( n,j,l,s )andm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaad6 gacaGGSaGaamOAaiaacYcacaWGSbGaaiilaiaadohaaiaawIcacaGL PaaacaWGHbGaamOBaiaadsgacaaMc8UaamyBaaaa@4477@ in addition to the parameter α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHXoqyaaa@3973@ of the potential. It is pertinent to note that when the molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaado eacaWGpbGaaiilaiaad6eacaWGpbaacaGLOaGaayzkaaaaaa@3D50@  have S 1/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiaado faaSqabeaacqGHsgIRaaGccqGHGjsUdaWfGaqaaiaaigdacaGGVaGa aGOmaaWcbeqaaiabgkziUcaaaaa@4113@ , the total operator can be obtains from the interval | ls |j| l+s | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaabdaqaaiaadY gacqGHsislcaWGZbaacaGLhWUaayjcSdGaeyizImQaamOAaiabgsMi JoaaemaabaGaamiBaiabgUcaRiaadohaaiaawEa7caGLiWoaaaa@4812@ , which allow us to obtaining the eigenvalues of the operator ( J 2 L 2 S 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaamaaFm aabaGaamOsaaGaayz4GaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Ya a8XaaeaacaWGmbaacaGLHdcadaahaaWcbeqaaiaaikdaaaGccqGHsi sldaWhdaqaaiaadofaaiaawgoiamaaCaaaleqabaGaaGOmaaaaaOGa ayjkaiaawMcaaaaa@45C5@  as k( j,l,s )j(j+1)+l(l+1)s(s+1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaeWaae aacaWGQbGaaiilaiaadYgacaGGSaGaam4CaaGaayjkaiaawMcaaiab ggMi6kaadQgacaGGOaGaamOAaiabgUcaRiaaigdacaGGPaGaey4kaS IaamiBaiaacIcacaWGSbGaey4kaSIaaGymaiaacMcacqGHsislcaWG ZbGaaiikaiaadohacqGHRaWkcaaIXaGaaiykaaaa@50AF@  and then the non-relativistic energy spectrum E nc -mg ( n,j,l,s,m,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab6gacaqGJbGaaeiiaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSbGaaiilaiaadohaca GGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGLPaaaaaa@4931@ reads:

E nc -mg ( n,j,l,s,m,α )= V 0 e α r 0 +2μ ( V 0 + V 0 e α r 0 n+l+1 ) 2 1 2 γχ N nl 2 ( 2β ) 12α V 0 f( n,α )( Θk( j,l,s )+Bmχ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaSbaaS qaaiaab6gacaqGJbGaaeiiaiaab2cacaWGTbGaam4zaaqabaGcdaqa daqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSbGaaiilaiaadohaca GGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGLPaaacqGH9aqpcqGH sislcaWGwbWaaSbaaSqaaiaaicdaaeqaaOGaamyzamaaCaaaleqaba GaeyOeI0IaeqySdeMaamOCamaaBaaameaacaaIWaaabeaaaaGccqGH RaWkcaaIYaGaeqiVd02aaeWaaeaadaWcaaqaaiaadAfadaWgaaWcba GaaGimaaqabaGccqGHRaWkcaWGwbWaaSbaaSqaaiaaicdaaeqaaOGa amyzamaaCaaaleqabaGaeyOeI0IaeqySdeMaamOCamaaBaaameaaca aIWaaabeaaaaaakeaacaWGUbGaey4kaSIaamiBaiabgUcaRiaaigda aaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0YaaS aaaeaacaaIXaaabaGaaGOmaaaacqaHZoWzcaaMc8Uaeq4XdmMaamOt amaaBaaaleaacaWGUbGaaGPaVlaadYgaaeqaaOWaaWbaaSqabeaaca aIYaaaaOWaaeWaaeaacaqGYaGaeqOSdigacaGLOaGaayzkaaWaaWba aSqabeaacaaIXaGaeyOeI0IaaGOmaiabeg7aHbaakiaadAfadaWgaa WcbaGaaGimaaqabaGccaWGMbWaaeWaaeaacaWGUbGaaiilaiabeg7a HbGaayjkaiaawMcaamaabmaabaGaeuiMdeLaaGPaVlaadUgadaqada qaaiaadQgacaGGSaGaamiBaiaacYcacaWGZbaacaGLOaGaayzkaaGa ey4kaSIaamOqaiaad2gacqaHhpWyaiaawIcacaGLPaaaaaa@92B0@ (42)

Paying attention to the behavior of the spectrums (38) and (42) ( E nc -umg ( n,j,l,s,m,α )and E nc -dmg ( n,j,l,s,m,α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadw eadaWgaaWcbaGaaeOBaiaabogacaqGGaGaaeylaiaabwhacaWGTbGa ae4zaaqabaGcdaqadaqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSb GaaiilaiaadohacaGGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGL PaaacaWGHbGaamOBaiaadsgacaaMc8UaamyramaaBaaaleaacaqGUb Gaae4yaiaabccacaqGTaGaaeizaiaad2gacaWGNbaabeaakmaabmaa baGaamOBaiaacYcacaWGQbGaaiilaiaadYgacaGGSaGaam4CaiaacY cacaWGTbGaaiilaiabeg7aHbGaayjkaiaawMcaaaGaayjkaiaawMca aaaa@6241@ , it is possible to recover the results of commutative space (12) when we consider ( Θ,χ )( 0,0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabfI 5arjaacYcacqaHhpWyaiaawIcacaGLPaaacqGHsgIRdaqadaqaaiaa icdacaGGSaGaaGimaaGaayjkaiaawMcaaaaa@42D5@ . Finlay, we can say that the results we have obtained in our recently research are more profound than the results listed in our reference.20

Conclusion

 In this paper three-dimensional MSES for NMGESC potential has been solved via Bopp’s shift method and standard perturbation theory in (NC: 3D-RSP) symmetries, we resume the main obtained results:

    1. The exact energy spectrum ( E nc -umg ( n,j,l,s,m,α ), E nc-dmg ( n,j,l,s,m,α ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadw eadaWgaaWcbaGaaeOBaiaabogacaqGGaGaaeylaiaabwhacaWGTbGa am4zaaqabaGcdaqadaqaaiaad6gacaGGSaGaamOAaiaacYcacaWGSb GaaiilaiaadohacaGGSaGaamyBaiaacYcacqaHXoqyaiaawIcacaGL PaaacaGGSaGaamyramaaBaaaleaacaqGUbGaae4yaiaab2cacaqGKb GaaeyBaiaadEgaaeqaaOWaaeWaaeaacaWGUbGaaiilaiaadQgacaGG SaGaamiBaiaacYcacaWGZbGaaiilaiaad2gacaGGSaGaeqySdegaca GLOaGaayzkaaaacaGLOaGaayzkaaaaaa@5E01@ for n th MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaWbaaS qabeaacaWG0bGaamiAaaaaaaa@3ADA@ excited levels, for hydrogenic atoms,
    2. Ordinary interaction ( V 0 r ( 1+( 1+αr )exp( 2αr ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsisldaWcaa qaaiaadAfadaWgaaWcbaGaaGimaaqabaaakeaacaWGYbaaamaabmaa baGaaGymaiabgUcaRmaabmaabaGaaGymaiabgUcaRiabeg7aHjaadk haaiaawIcacaGLPaaaciGGLbGaaiiEaiaacchadaqadaqaaiabgkHi TiaaikdacqaHXoqycaWGYbaacaGLOaGaayzkaaaacaGLOaGaayzkaa aaaa@4D18@ were replaced by new modified interactions 3) ( H intumg and H intdmg ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaadI eadaWgaaWcbaGaciyAaiaac6gacaGG0bGaeyOeI0IaamyDaiaad2ga caWGNbaabeaakiaaykW7caWGHbGaamOBaiaadsgacaaMc8Uaamisam aaBaaaleaaciGGPbGaaiOBaiaacshacqGHsislcaWGKbGaamyBaiaa dEgaaeqaaaGccaGLOaGaayzkaaaaaa@4E66@ for hydrogenic atoms.
    3. The ordinary kinetic term Δ 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsisldaWcaa qaaiabfs5aebqaaiaaikdacqaH8oqBaaaaaa@3CA9@ modified to the new form Δ nc 2μ = Δ θ ¯ L σ ¯ L 2μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiabfs 5aenaaBaaaleaacaWGUbGaam4yaaqabaaakeaacaaIYaGaeqiVd0ga aiabg2da9maalaaabaGaeuiLdqKaeyOeI0IafqiUdeNbaebadaWfGa qaaiaadYeaaSqabeaacqGHsgIRaaGccqGHsisldaqdaaqaaiabeo8a ZbaadaWfGaqaaiaadYeaaSqabeaacqGHsgIRaaaakeaacaaIYaGaeq iVd0gaaaaa@4E59@  for NMGESC potential,
    4. Spectra of bounded diatomic molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaado eacaWGpbGaaiilaiaad6eacaWGpbaacaGLOaGaayzkaaaaaa@3D50@  were studied analytically. NMGESC potential has been used to model the molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaado eacaWGpbGaaiilaiaad6eacaWGpbaacaGLOaGaayzkaaaaaa@3D50@
    5. We have shown that, the group symmetry (NC: 3D-RSP) corresponding NMGESC potential reduce to the symmetry sub-group (NC: 3D-RS).
    6. It has been shown that, the MSE presents useful rich spectrums for improved understanding of hydrogenic atoms and molecules ( CO,NO ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiaado eacaWGpbGaaiilaiaad6eacaWGpbaacaGLOaGaayzkaaaaaa@3D50@  influenced by the NMGESC potential and we have seen also that the modified of spin-orbital and modified Zeeman effect were appears du the presence of the two infinitesimal parameters ( Θ,χ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjY=Pj0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9vqaq pepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaiabfI 5arjaacYcacqaHhpWyaiaawIcacaGLPaaaaaa@3D3B@  which are induced by position-position noncommutativity property of space.

Acknowledgements

None.

Conflict of interest

Author declares that there is no conflicts of interest.

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