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Application of the Asymptotic Iteration Method to
the Exponential Cosine Screened Coulomb Potential

O. Bayrak, et al. Int. J. Quant. Chem., 107 (2007), p. 1040
http://onlinelibrary.wiley.com/doi/10.1002/qua.21240/epdf

Atomic orbitals

1s
2s                                                  2p  2p  2p
3s                                                  3p  3p  3p
4s                              3d  3d  3d  3d  3d  4p  4p  4p
5s                              4d  4d  4d  4d  4d  5p  5p  5p
6s  4f  4f  4f  4f  4f  4f  4f  5d  5d  5d  5d  5d  6p  6p  6p
7s  5f  5f  5f  5f  5f  5f  5f  6d  6d  6d  6d  6d  7p  7p  7p

https://en.wikipedia.org/wiki/Atomic_orbital#Electron_placement_and_the_periodic_table

Import AIM library





In [1]:



    


# Python program to use AIM tools
from asymptotic import *

Definitions

Variables





In [2]:



    


En, m, hbar, L, r, r0 = se.symbols("En, m, hbar, L, r, r0")
beta, delta, A, A1, A2, A3, A4, A5, A6 = se.symbols("beta, delta, A, A1, A2, A3, A4, A5, A6")

$\lambda_0$ and $s_0$





In [3]:



    


l0 = 2*(beta - (L+1)/r)
s0 = -2*m*En/hbar**2 + A2 - beta**2 + (2*L*beta + 2*beta - A1)/r - A3*r**2 + A4*r**3 - A5*r**4 + A6*r**6

Case: $\delta=0.01$

s states (1s, 2s, 3s, 4s)

Numerical values for variables





In [4]:



    


nL = o* 0
ndelta = o* 1/100
nbeta = o* 6/10

nA, nhbar, nm = o* 1, o* 1, o* 1
nr0 = o* (nL+1)/nbeta

nA1 = 2*nm*nA/nhbar**2
nA2 = nA1*ndelta
nA3 = nA1*ndelta**3/3
nA4 = nA1*ndelta**4/6
nA5 = nA1*ndelta**5/30
nA6 = nA1*ndelta**7/630

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

Initialize AIM solver





In [5]:



    


%%time
# pass lambda_0, s_0 and variable values to aim class
ecsc_d01L0 = aim(l0, s0, pl0, ps0)
ecsc_d01L0.display_parameters()
ecsc_d01L0.display_l0s0(0)
ecsc_d01L0.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 0,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/50,~~A_{3} = 1/1500000,~~A_{4} = 1/300000000,~~A_{5} = 1/150000000000,~~A_{6} = 1/31500000000000000,~~L = 0,~~\beta = 3/5,~~\delta = 1/100,~~\hbar = 1,~~m = 1,~~r_{0} = 5/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-17} r^{6} - 6.66666666666667 \cdot 10^{-12} r^{4} + 3.33333333333333 \cdot 10^{-9} r^{3} - 6.66666666666667 \cdot 10^{-7} r^{2} - 0.34 - \frac{0.8}{r} \right.
                \end{align}
                $









    






CPU times: user 292 ms, sys: 4.11 ms, total: 296 ms
Wall time: 299 ms

Calculation of Taylor series coefficients of $\lambda_0$ and $s_0$





In [6]:



    


%%time
# create coefficients for improved AIM
ecsc_d01L0.c0()
ecsc_d01L0.d0()
ecsc_d01L0.cndn()



    


















    






CPU times: user 12.7 s, sys: 61.9 ms, total: 12.7 s
Wall time: 12.7 s

The solution





In [7]:



    


%%time
ecsc_d01L0.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    






0001   -0.41000091823559637   -0.05000091823559637
0011   -0.49000037958698144   -0.11501346016998136   -0.04477587270206144
0021   -0.49000098757668462   -0.11501346103707918   -0.04561890498780328   -0.02097517821422200
0031   -0.49000098757842334   -0.11501346103707918   -0.04561907951647280   -0.02143487905736983
0041   -0.49000098757842334   -0.11501346103707918   -0.04561907952477892   -0.02143745890259142
0051   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746430420317
0061   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746431125275
0071   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746431125992
0081   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746431125993
0091   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746431125993
0101   -0.49000098757842334   -0.11501346103707918   -0.04561907952477914   -0.02143746431125993
CPU times: user 1.64 s, sys: 4.74 ms, total: 1.64 s
Wall time: 1.64 s

p states





In [8]:



    


%%time

nL = o* 1
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d01L1 = aim(l0, s0, pl0, ps0)
ecsc_d01L1.display_parameters()
ecsc_d01L1.display_l0s0(0)
ecsc_d01L1.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d01L1.c0()
ecsc_d01L1.d0()
ecsc_d01L1.cndn()

# the solution
ecsc_d01L1.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 1,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/50,~~A_{3} = 1/1500000,~~A_{4} = 1/300000000,~~A_{5} = 1/150000000000,~~A_{6} = 1/31500000000000000,~~L = 1,~~\beta = 3/5,~~\delta = 1/100,~~\hbar = 1,~~m = 1,~~r_{0} = 10/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{4}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-17} r^{6} - 6.66666666666667 \cdot 10^{-12} r^{4} + 3.33333333333333 \cdot 10^{-9} r^{3} - 6.66666666666667 \cdot 10^{-7} r^{2} - 0.34 + \frac{0.4}{r} \right.
                \end{align}
                $









    






0001   -0.11000364238680950
0011   -0.11500965662760865   -0.04546075647450115   -0.01101192290578550
0021   -0.11500965664051346   -0.04561102327445309   -0.02127544794802216   -0.00561719116227536
0031   -0.11500965664051346   -0.04561104130934575   -0.02142377036776990   -0.01010666321039609
0041   -0.11500965664051346   -0.04561104131004139   -0.02142437284232580   -0.01039681200617683
0051   -0.11500965664051346   -0.04561104131004141   -0.02142437391168194   -0.01040571761673410
0061   -0.11500965664051346   -0.04561104131004141   -0.02142437391295913   -0.01040586084957102
0071   -0.11500965664051346   -0.04561104131004141   -0.02142437391296036   -0.01040586244329663
0081   -0.11500965664051346   -0.04561104131004141   -0.02142437391296036   -0.01040586245769149
0091   -0.11500965664051346   -0.04561104131004141   -0.02142437391296036   -0.01040586245780550
0101   -0.11500965664051346   -0.04561104131004141   -0.02142437391296036   -0.01040586245780632
CPU times: user 14.2 s, sys: 75.1 ms, total: 14.3 s
Wall time: 14.3 s

d states





In [9]:



    


%%time

nL = o* 2
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d01L2 = aim(l0, s0, pl0, ps0)
ecsc_d01L2.display_parameters()
ecsc_d01L2.display_l0s0(0)
ecsc_d01L2.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d01L2.c0()
ecsc_d01L2.d0()
ecsc_d01L2.cndn()

# the solution
ecsc_d01L2.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 2,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/50,~~A_{3} = 1/1500000,~~A_{4} = 1/300000000,~~A_{5} = 1/150000000000,~~A_{6} = 1/31500000000000000,~~L = 2,~~\beta = 3/5,~~\delta = 1/100,~~\hbar = 1,~~m = 1,~~r_{0} = 5$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{6}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-17} r^{6} - 6.66666666666667 \cdot 10^{-12} r^{4} + 3.33333333333333 \cdot 10^{-9} r^{3} - 6.66666666666667 \cdot 10^{-7} r^{2} - 0.34 + \frac{1.6}{r} \right.
                \end{align}
                $









    






0001   -0.01000812708308532
0011   -0.04558241644527422   -0.01816500193930418
0021   -0.04559483852680523   -0.02136665148838376   -0.00819678662537585
0031   -0.04559483937852977   -0.02139788811597951   -0.01025969975575201   -0.00216772367318591
0041   -0.04559483937855599   -0.02139797707204586   -0.01036636193501593   -0.00427317620770694
0051   -0.04559483937855599   -0.02139797720424042   -0.01036891441427879   -0.00459473205070927
0061   -0.04559483937855599   -0.02139797720438322   -0.01036894955761832   -0.00462738386218829
0071   -0.04559483937855599   -0.02139797720438335   -0.01036894991481571   -0.00462956792124952
0081   -0.04559483937855599   -0.02139797720438335   -0.01036894991786297   -0.00462967663816009
0091   -0.04559483937855599   -0.02139797720438335   -0.01036894991788618   -0.00462968115031828
0101   -0.04559483937855599   -0.02139797720438335   -0.01036894991788634   -0.00462968131623520
CPU times: user 14.4 s, sys: 60.1 ms, total: 14.4 s
Wall time: 14.4 s

f states





In [10]:



    


%%time

nL = o* 3
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d01L3 = aim(l0, s0, pl0, ps0)
ecsc_d01L3.display_parameters()
ecsc_d01L3.display_l0s0(0)
ecsc_d01L3.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d01L3.c0()
ecsc_d01L3.d0()
ecsc_d01L3.cndn()

# the solution
ecsc_d01L3.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 3,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/50,~~A_{3} = 1/1500000,~~A_{4} = 1/300000000,~~A_{5} = 1/150000000000,~~A_{6} = 1/31500000000000000,~~L = 3,~~\beta = 3/5,~~\delta = 1/100,~~\hbar = 1,~~m = 1,~~r_{0} = 20/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{8}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-17} r^{6} - 6.66666666666667 \cdot 10^{-12} r^{4} + 3.33333333333333 \cdot 10^{-9} r^{3} - 6.66666666666667 \cdot 10^{-7} r^{2} - 0.34 + \frac{2.8}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011   -0.02077200253556297
0021   -0.02135467236523013   -0.00957500213995898
0031   -0.02135783380652421   -0.01028569059089555   -0.00334739408749906
0041   -0.02135784009580731   -0.01031225453216720   -0.00440837905404961
0051   -0.02135784010358477   -0.01031273691205531   -0.00454647389449411   -0.00086451571864272
0061   -0.02135784010359230   -0.01031274256592940   -0.00455811983232284   -0.00129390738655260
0071   -0.02135784010359231   -0.01031274261811484   -0.00455879373763631   -0.00139839649909669
0081   -0.02135784010359231   -0.01031274261853315   -0.00455882434905366   -0.00141922509253084
0091   -0.02135784010359231   -0.01031274261853620   -0.00455882554873184   -0.00142264645964549
0101   -0.02135784010359231   -0.01031274261853622   -0.00455882559117361   -0.00142312608631052
CPU times: user 14.5 s, sys: 84.1 ms, total: 14.6 s
Wall time: 14.6 s

Case: $\delta=0.02$

s states (1s, 2s, 3s, 4s)





In [11]:



    


%%time

nL = o* 0
ndelta = o* 2/100
nbeta = o* 6/10

nA, nhbar, nm = o* 1, o* 1, o* 1
nr0 = o* (nL+1)/nbeta

nA1 = 2*nm*nA/nhbar**2
nA2 = nA1*ndelta
nA3 = nA1*ndelta**3/3
nA4 = nA1*ndelta**4/6
nA5 = nA1*ndelta**5/30
nA6 = nA1*ndelta**7/630

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d02L0 = aim(l0, s0, pl0, ps0)
ecsc_d02L0.display_parameters()
ecsc_d02L0.display_l0s0(0)
ecsc_d02L0.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d02L0.c0()
ecsc_d02L0.d0()
ecsc_d02L0.cndn()

# the solution
ecsc_d02L0.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 0,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/25,~~A_{3} = 1/187500,~~A_{4} = 1/18750000,~~A_{5} = 1/4687500000,~~A_{6} = 1/246093750000000,~~L = 0,~~\beta = 3/5,~~\delta = 1/50,~~\hbar = 1,~~m = 1,~~r_{0} = 5/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 4.06349206349206 \cdot 10^{-15} r^{6} - 2.13333333333333 \cdot 10^{-10} r^{4} + 5.33333333333333 \cdot 10^{-8} r^{3} - 5.33333333333333 \cdot 10^{-6} r^{2} - 0.32 - \frac{0.8}{r} \right.
                \end{align}
                $









    






0001   -0.40000728477361900   -0.04000728477361900
0011   -0.48000720213597595   -0.10510358590976973   -0.03509891443195363
0021   -0.48000780260776634   -0.10510358765875565   -0.03602478896382142   -0.01194280694189140
0031   -0.48000780260928268   -0.10510358765875565   -0.03602509984759363   -0.01256447634099594
0041   -0.48000780260928268   -0.10510358765875565   -0.03602509988067940   -0.01257147988622072
0051   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151911691028
0061   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151926161659
0071   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151926183478
0081   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151926183413
0091   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151926183413
0101   -0.48000780260928268   -0.10510358765875565   -0.03602509988068154   -0.01257151926183413
CPU times: user 14.2 s, sys: 87.9 ms, total: 14.3 s
Wall time: 14.3 s

p states





In [12]:



    


%%time

nL = o* 1
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d02L1 = aim(l0, s0, pl0, ps0)
ecsc_d02L1.display_parameters()
ecsc_d02L1.display_l0s0(0)
ecsc_d02L1.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d02L1.c0()
ecsc_d02L1.d0()
ecsc_d02L1.cndn()

# the solution
ecsc_d02L1.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 1,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/25,~~A_{3} = 1/187500,~~A_{4} = 1/18750000,~~A_{5} = 1/4687500000,~~A_{6} = 1/246093750000000,~~L = 1,~~\beta = 3/5,~~\delta = 1/50,~~\hbar = 1,~~m = 1,~~r_{0} = 10/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{4}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 4.06349206349206 \cdot 10^{-15} r^{6} - 2.13333333333333 \cdot 10^{-10} r^{4} + 5.33333333333333 \cdot 10^{-8} r^{3} - 5.33333333333333 \cdot 10^{-6} r^{2} - 0.32 + \frac{0.4}{r} \right.
                \end{align}
                $









    






0001   -0.10002865514124589
0011   -0.10507463825879602   -0.03579323072069503   -0.00135873745798540
0021   -0.10507463828958558   -0.03596756334639395   -0.01225902996251863
0031   -0.10507463828958558   -0.03596759972366887   -0.01248354482989450   -0.00210668416486153
0041   -0.10507463828958558   -0.03596759972693064   -0.01248553165606787   -0.00270082644766119
0051   -0.10507463828958558   -0.03596759972693082   -0.01248554149185499   -0.00275221212022203
0061   -0.10507463828958558   -0.03596759972693082   -0.01248554152458209   -0.00275531078500647
0071   -0.10507463828958558   -0.03596759972693082   -0.01248554152461376   -0.00275541503324959
0081   -0.10507463828958558   -0.03596759972693082   -0.01248554152461363   -0.00275541466208793
0091   -0.10507463828958558   -0.03596759972693082   -0.01248554152461363   -0.00275541467712627
0101   -0.10507463828958558   -0.03596759972693082   -0.01248554152461363   -0.00275541467455309
CPU times: user 14.2 s, sys: 104 ms, total: 14.3 s
Wall time: 14.3 s

d states





In [13]:



    


%%time

nL = o* 2
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d02L2 = aim(l0, s0, pl0, ps0)
ecsc_d02L2.display_parameters()
ecsc_d02L2.display_l0s0(0)
ecsc_d02L2.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d02L2.c0()
ecsc_d02L2.d0()
ecsc_d02L2.cndn()

# the solution
ecsc_d02L2.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 2,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/25,~~A_{3} = 1/187500,~~A_{4} = 1/18750000,~~A_{5} = 1/4687500000,~~A_{6} = 1/246093750000000,~~L = 2,~~\beta = 3/5,~~\delta = 1/50,~~\hbar = 1,~~m = 1,~~r_{0} = 5$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{6}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 4.06349206349206 \cdot 10^{-15} r^{6} - 2.13333333333333 \cdot 10^{-10} r^{4} + 5.33333333333333 \cdot 10^{-8} r^{3} - 5.33333333333333 \cdot 10^{-6} r^{2} - 0.32 + \frac{1.6}{r} \right.
                \end{align}
                $









    






0001   -0.00006339996825397
0011   -0.03583507674588912   -0.00868897746921428
0021   -0.03585065879061275   -0.01225429060076955
0031   -0.03585066078101929   -0.01230975836654534   -0.00223472639055766
0041   -0.03585066078116743   -0.01231013226301407   -0.00251494090101816
0051   -0.03585066078116744   -0.01231013388871822   -0.00253671995721804
0061   -0.03585066078116744   -0.01231013389352322   -0.00253791125947495
0071   -0.03585066078116744   -0.01231013389352513   -0.00253794497576190
0081   -0.03585066078116744   -0.01231013389352512   -0.00253794462146054
0091   -0.03585066078116744   -0.01231013389352512   -0.00253794463712601
0101   -0.03585066078116744   -0.01231013389352512   -0.00253794463582518
CPU times: user 14.2 s, sys: 51.9 ms, total: 14.3 s
Wall time: 14.3 s

f states





In [14]:



    


%%time

nL = o* 3
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d02L3 = aim(l0, s0, pl0, ps0)
ecsc_d02L3.display_parameters()
ecsc_d02L3.display_l0s0(0)
ecsc_d02L3.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d02L3.c0()
ecsc_d02L3.d0()
ecsc_d02L3.cndn()

# the solution
ecsc_d02L3.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 3,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/25,~~A_{3} = 1/187500,~~A_{4} = 1/18750000,~~A_{5} = 1/4687500000,~~A_{6} = 1/246093750000000,~~L = 3,~~\beta = 3/5,~~\delta = 1/50,~~\hbar = 1,~~m = 1,~~r_{0} = 20/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{8}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 4.06349206349206 \cdot 10^{-15} r^{6} - 2.13333333333333 \cdot 10^{-10} r^{4} + 5.33333333333333 \cdot 10^{-8} r^{3} - 5.33333333333333 \cdot 10^{-6} r^{2} - 0.32 + \frac{2.8}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011   -0.01132519188794263
0021   -0.01203112762565044   -0.00097325854057537
0031   -0.01203810015165261   -0.00208826388479452
0041   -0.01203813587478997   -0.00218898426014016
0051   -0.01203813601144682   -0.00219590266554967
0061   -0.01203813601180090   -0.00219624824558798
0071   -0.01203813601180083   -0.00219625629775609
0081   -0.01203813601180083   -0.00219625615937815
0091   -0.01203813601180083   -0.00219625616550524
0101   -0.01203813601180083   -0.00219625616520729
CPU times: user 15 s, sys: 67.9 ms, total: 15.1 s
Wall time: 15 s

Case: $\delta=0.06$

s states (1s, 2s, 3s)





In [15]:



    


%%time

nL = o* 0
ndelta = o* 6/100
nbeta = o* 6/10

nA, nhbar, nm = o* 1, o* 1, o* 1
nr0 = o* (nL+1)/nbeta

nA1 = 2*nm*nA/nhbar**2
nA2 = nA1*ndelta
nA3 = nA1*ndelta**3/3
nA4 = nA1*ndelta**4/6
nA5 = nA1*ndelta**5/30
nA6 = nA1*ndelta**7/630

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d06L0 = aim(l0, s0, pl0, ps0)
ecsc_d06L0.display_parameters()
ecsc_d06L0.display_l0s0(0)
ecsc_d06L0.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d06L0.c0()
ecsc_d06L0.d0()
ecsc_d06L0.cndn()

# the solution
ecsc_d06L0.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 0,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 3/25,~~A_{3} = 9/62500,~~A_{4} = 27/6250000,~~A_{5} = 81/1562500000,~~A_{6} = 243/27343750000000,~~L = 0,~~\beta = 3/5,~~\delta = 3/50,~~\hbar = 1,~~m = 1,~~r_{0} = 5/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 8.88685714285714 \cdot 10^{-12} r^{6} - 5.184 \cdot 10^{-8} r^{4} + 4.32 \cdot 10^{-6} r^{3} - 0.000144 r^{2} - 0.24 - \frac{0.8}{r} \right.
                \end{align}
                $









    






0001   -0.36019019990476190   -0.00019019990476190
0011   -0.44020020067474932   -0.06742077904740583   -0.00235883382553319
0021   -0.44020051019603707   -0.06742085544919782   -0.00534645930707232
0031   -0.44020051019805567   -0.06742085544937821   -0.00534742751186176
0041   -0.44020051019805581   -0.06742085544937025   -0.00534914623749730
0051   -0.44020051019805581   -0.06742085544937028   -0.00534927749258486
0061   -0.44020051019805581   -0.06742085544937030   -0.00534926008962952
0071   -0.44020051019805581   -0.06742085544937030   -0.00534924157448826
0081   -0.44020051019805581   -0.06742085544937030   -0.00534923034085046
0091   -0.44020051019805581   -0.06742085544937030   -0.00534922365520876
0101   -0.44020051019805581   -0.06742085544937030   -0.00534921957213672
CPU times: user 14 s, sys: 55.8 ms, total: 14 s
Wall time: 14 s

p states





In [16]:



    


%%time

nL = o* 1
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d06L1 = aim(l0, s0, pl0, ps0)
ecsc_d06L1.display_parameters()
ecsc_d06L1.display_l0s0(0)
ecsc_d06L1.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d06L1.c0()
ecsc_d06L1.d0()
ecsc_d06L1.cndn()

# the solution
ecsc_d06L1.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 1,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 3/25,~~A_{3} = 9/62500,~~A_{4} = 27/6250000,~~A_{5} = 81/1562500000,~~A_{6} = 243/27343750000000,~~L = 1,~~\beta = 3/5,~~\delta = 3/50,~~\hbar = 1,~~m = 1,~~r_{0} = 10/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{4}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 8.88685714285714 \cdot 10^{-12} r^{6} - 5.184 \cdot 10^{-8} r^{4} + 4.32 \cdot 10^{-6} r^{3} - 0.000144 r^{2} - 0.24 + \frac{0.4}{r} \right.
                \end{align}
                $









    






0001   -0.06072319390476190
0011   -0.06677739399642305   -0.00337878003854047
0021   -0.06677739500471407   -0.00438839669293946
0031   -0.06677739500504026   -0.00438200224405734
0041   -0.06677739500503967   -0.00438257211597066
0051   -0.06677739500503965   -0.00438272846521701
0061   -0.06677739500503965   -0.00438276794658649
0071   -0.06677739500503965   -0.00438278026748762
0081   -0.06677739500503965   -0.00438278558504989
0091   -0.06677739500503965   -0.00438278881793095
0101   -0.06677739500503965   -0.00438279124909896
CPU times: user 14 s, sys: 71.9 ms, total: 14.1 s
Wall time: 14 s

d states





In [17]:



    


%%time

nL = o* 2
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d06L2 = aim(l0, s0, pl0, ps0)
ecsc_d06L2.display_parameters()
ecsc_d06L2.display_l0s0(0)
ecsc_d06L2.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d06L2.c0()
ecsc_d06L2.d0()
ecsc_d06L2.cndn()

# the solution
ecsc_d06L2.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 2,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 3/25,~~A_{3} = 9/62500,~~A_{4} = 27/6250000,~~A_{5} = 81/1562500000,~~A_{6} = 243/27343750000000,~~L = 2,~~\beta = 3/5,~~\delta = 3/50,~~\hbar = 1,~~m = 1,~~r_{0} = 5$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{6}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 8.88685714285714 \cdot 10^{-12} r^{6} - 5.184 \cdot 10^{-8} r^{4} + 4.32 \cdot 10^{-6} r^{3} - 0.000144 r^{2} - 0.24 + \frac{1.6}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011   -0.00203308283873400
0021   -0.00226463207592012
0031   -0.00226185506260991
0041   -0.00226181491889181
0051   -0.00226184451860893
0061   -0.00226185853338824
0071   -0.00226186473712927
0081   -0.00226186760243209
0091   -0.00226186882886936
0101   -0.00226186913428076
CPU times: user 14.5 s, sys: 52 ms, total: 14.6 s
Wall time: 14.5 s

f states





In [18]:



    


%%time

nL = o* 3
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d06L3 = aim(l0, s0, pl0, ps0)
ecsc_d06L3.display_parameters()
ecsc_d06L3.display_l0s0(0)
ecsc_d06L3.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d06L3.c0()
ecsc_d06L3.d0()
ecsc_d06L3.cndn()

# the solution
ecsc_d06L3.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 3,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 3/25,~~A_{3} = 9/62500,~~A_{4} = 27/6250000,~~A_{5} = 81/1562500000,~~A_{6} = 243/27343750000000,~~L = 3,~~\beta = 3/5,~~\delta = 3/50,~~\hbar = 1,~~m = 1,~~r_{0} = 20/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{8}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 8.88685714285714 \cdot 10^{-12} r^{6} - 5.184 \cdot 10^{-8} r^{4} + 4.32 \cdot 10^{-6} r^{3} - 0.000144 r^{2} - 0.24 + \frac{2.8}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011  Nothing to show!
0021  Nothing to show!
0031  Nothing to show!
0041  Nothing to show!
0051  Nothing to show!
0061  Nothing to show!
0071  Nothing to show!
0081  Nothing to show!
0091  Nothing to show!
0101  Nothing to show!
CPU times: user 14.6 s, sys: 68.1 ms, total: 14.6 s
Wall time: 14.6 s

Case: $\delta=0.1$

s states (1s, 2s)





In [19]:



    


%%time

nL = o* 0
ndelta = o* 10/100
nbeta = o* 6/10

nA, nhbar, nm = o* 1, o* 1, o* 1
nr0 = o* (nL+1)/nbeta

nA1 = 2*nm*nA/nhbar**2
nA2 = nA1*ndelta
nA3 = nA1*ndelta**3/3
nA4 = nA1*ndelta**4/6
nA5 = nA1*ndelta**5/30
nA6 = nA1*ndelta**7/630

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d10L0 = aim(l0, s0, pl0, ps0)
ecsc_d10L0.display_parameters()
ecsc_d10L0.display_l0s0(0)
ecsc_d10L0.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d10L0.c0()
ecsc_d10L0.d0()
ecsc_d10L0.cndn()

# the solution
ecsc_d10L0.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 0,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/5,~~A_{3} = 1/1500,~~A_{4} = 1/30000,~~A_{5} = 1/1500000,~~A_{6} = 1/3150000000,~~L = 0,~~\beta = 3/5,~~\delta = 1/10,~~\hbar = 1,~~m = 1,~~r_{0} = 5/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{2}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-10} r^{6} - 6.66666666666667 \cdot 10^{-7} r^{4} + 3.33333333333333 \cdot 10^{-5} r^{3} - 0.000666666666666667 r^{2} - 0.16 - \frac{0.8}{r} \right.
                \end{align}
                $









    






0001   -0.32085133404642149
0011   -0.40088577765048470   -0.03491913851838471
0021   -0.40088476962677383   -0.03491568414251706
0031   -0.40088476912279443   -0.03491582022156478
0041   -0.40088476911837347   -0.03491585275850849
0051   -0.40088476911813149   -0.03491586008204345
0061   -0.40088476911809068   -0.03491585698246674
0071   -0.40088476911807737   -0.03491584821085167
0081   -0.40088476911807091   -0.03491583784620531
0091   -0.40088476911806703   -0.03491583082401232
0101   -0.40088476911806466   -0.03491583191898983
CPU times: user 14.3 s, sys: 40 ms, total: 14.3 s
Wall time: 14.3 s

p states





In [20]:



    


%%time

nL = o* 1
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d10L1 = aim(l0, s0, pl0, ps0)
ecsc_d10L1.display_parameters()
ecsc_d10L1.display_l0s0(0)
ecsc_d10L1.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d10L1.c0()
ecsc_d10L1.d0()
ecsc_d10L1.cndn()

# the solution
ecsc_d10L1.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 1,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/5,~~A_{3} = 1/1500,~~A_{4} = 1/30000,~~A_{5} = 1/1500000,~~A_{6} = 1/3150000000,~~L = 1,~~\beta = 3/5,~~\delta = 1/10,~~\hbar = 1,~~m = 1,~~r_{0} = 10/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{4}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-10} r^{6} - 6.66666666666667 \cdot 10^{-7} r^{4} + 3.33333333333333 \cdot 10^{-5} r^{3} - 0.000666666666666667 r^{2} - 0.16 + \frac{0.4}{r} \right.
                \end{align}
                $









    






0001   -0.02312735427961765
0011   -0.03245650968532679
0021   -0.03245503087632942
0031   -0.03245501230562262
0041   -0.03245501152851536
0051   -0.03245501121012633
0061   -0.03245501051909821
0071   -0.03245500965032106
0081   -0.03245500892248537
0091   -0.03245500856874680
0101   -0.03245500861256962
CPU times: user 14.2 s, sys: 39.9 ms, total: 14.3 s
Wall time: 14.3 s

d states





In [21]:



    


%%time

nL = o* 2
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d10L2 = aim(l0, s0, pl0, ps0)
ecsc_d10L2.display_parameters()
ecsc_d10L2.display_l0s0(0)
ecsc_d10L2.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d10L2.c0()
ecsc_d10L2.d0()
ecsc_d10L2.cndn()

# the solution
ecsc_d10L2.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 2,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/5,~~A_{3} = 1/1500,~~A_{4} = 1/30000,~~A_{5} = 1/1500000,~~A_{6} = 1/3150000000,~~L = 2,~~\beta = 3/5,~~\delta = 1/10,~~\hbar = 1,~~m = 1,~~r_{0} = 5$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{6}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-10} r^{6} - 6.66666666666667 \cdot 10^{-7} r^{4} + 3.33333333333333 \cdot 10^{-5} r^{3} - 0.000666666666666667 r^{2} - 0.16 + \frac{1.6}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011  Nothing to show!
0021  Nothing to show!
0031  Nothing to show!
0041  Nothing to show!
0051  Nothing to show!
0061  Nothing to show!
0071  Nothing to show!
0081  Nothing to show!
0091  Nothing to show!
0101  Nothing to show!
CPU times: user 14.5 s, sys: 76.1 ms, total: 14.6 s
Wall time: 14.6 s

f states





In [22]:



    


%%time

nL = o* 3
nr0 = o* (nL+1)/nbeta

pl0 = {beta:nbeta, L:nL}
ps0 = {hbar:nhbar, m:nm, delta:ndelta,
       beta:nbeta, L:nL, r0:nr0, 
       A1:nA1, A2:nA2, A3:nA3, 
       A4:nA4, A5:nA5, A6:nA6}

# pass lambda_0, s_0 and variable values to aim class
ecsc_d10L3 = aim(l0, s0, pl0, ps0)
ecsc_d10L3.display_parameters()
ecsc_d10L3.display_l0s0(0)
ecsc_d10L3.parameters(En, r, nr0, nmax=101, nstep=10, dprec=500, tol=1e-101)

# create coefficients for improved AIM
ecsc_d10L3.c0()
ecsc_d10L3.d0()
ecsc_d10L3.cndn()

# the solution
ecsc_d10L3.get_arb_roots(showRoots='-r', printFormat="{:22.17f}")



    


















    







$\displaystyle \text{Values of the parameters of } \lambda_0:$









    







$\displaystyle L = 3,~~\beta = 3/5$









    







$\displaystyle \text{Values of the parameters of } s_0:$









    







$\displaystyle A_{1} = 2,~~A_{2} = 1/5,~~A_{3} = 1/1500,~~A_{4} = 1/30000,~~A_{5} = 1/1500000,~~A_{6} = 1/3150000000,~~L = 3,~~\beta = 3/5,~~\delta = 1/10,~~\hbar = 1,~~m = 1,~~r_{0} = 20/3$









    







$\displaystyle            
                \begin{align}
                \lambda_0 &= \left[2 \beta - \frac{2 \left(L + 1\right)}{r} \right.\\ 
                      s_0 &= \left[A_{2} - A_{3} r^{2} + A_{4} r^{3} - A_{5} r^{4} + A_{6} r^{6} - \frac{2 En m}{\hbar^{2}} - \beta^{2} + \frac{- A_{1} + 2 L \beta + 2 \beta}{r} \right.\\ 
                \lambda_0 &= \left[\frac{6}{5} - \frac{8}{r} \right.\\ 
                      s_0 &= \left[- 2.0 En + 3.17460317460317 \cdot 10^{-10} r^{6} - 6.66666666666667 \cdot 10^{-7} r^{4} + 3.33333333333333 \cdot 10^{-5} r^{3} - 0.000666666666666667 r^{2} - 0.16 + \frac{2.8}{r} \right.
                \end{align}
                $









    






0001  Nothing to show!
0011  Nothing to show!
0021  Nothing to show!
0031  Nothing to show!
0041  Nothing to show!
0051  Nothing to show!
0061  Nothing to show!
0071  Nothing to show!
0081  Nothing to show!
0091  Nothing to show!
0101  Nothing to show!
CPU times: user 14.5 s, sys: 80 ms, total: 14.6 s
Wall time: 14.5 s

















In [ ]:

Content source: mkarakoc/aim

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