5. Calculating (as per simple LCAO-MO theory involving only an 1s-type AO-pair with a single orbital exponent variable k) the internuclear potential energy U of hydrogen molecular ion (H₂⁺ ion) as a function of its internuclear distance R:
As is nicely explained in Ch. 3 (Section: MO Theory of H₂⁺ molecular ion) of Quantum Chemistry by Ira N. Levine (3rd Ed., Prentice Hall of India, New Delhi, 2003), the internuclear potential energy U (which is also the electronic energy including internuclear repulsion V_NN) of the ground electronic state of H₂⁺ ion (as per the aforesaid simple MO theory) is expressible by the following straightforward relations:
Internuclear Potential Energy  U = (Haa + Hab)/(1 + Sab) + 1/R   where
Overlap integral    Sab = (1 + k*R + k*k*R*R/3) * exp((-1)*k*R)
Coulomb integral    Haa = 0.5*k*k - k - 1/R + (k+1/R) * exp((-2)*k*R)
Resonance Integral  Hab = (-0.5)*k*k*Sab - k*(2-k)*(1+k*R)*exp((-1)*k*R)
[Here k is the optimum orbital exponent for a given distance R, obtainable via variation theory as the optimum value that gives the lowest value of U using a computer-program freely available from the site www.geocities.com/riturajkalita/. The program output displaying the (R, k) values ranging from (0.2, 1.9374) to (10, 0.9991) in steps of 0.2 atomic unit (a. u.) in R is as listed in the file INPE_Res.txt, ] It may be noted here that from the result of the above straightforward calculation, pretty realistic values of the (equilibrium) bond-length R_e and equilibrium bond-dissociation energy D_e for H₂⁺ ion could be easily obtained.
    To do this above calculation very quickly using the programmable-calculator software Assam-Calcu, first enter the commands R = 0.2 and k = 1.9374 therein. Then enter the above three expressions for S_ab, H_aa and H_ab one by one. Then enter the expression for U, getting hereby the value of U for R = 0.2. Next, put R = 0.4 and the corresponding appropriate value for k (it is 1.8327) into Assam-Calcu, and then just re-run the last four commands for values of S_ab, H_aa, H_ab and U. The new value of U for R = 0.4 is thus obtained. Repeat the above process for all other values of R (up to 10.0) for which k is available.

 

 Internuclear   Orbital     Overlap     Coulomb    Resonance  Electronic

  Distance R   Exponent k   Integral    Integral    Integral   Energy U

 (AtomicUnit) (PureNumber)  Sab(a.u.)   Haa(a.u.)   Hab(a.u.)   (a. u.)

 

    0.20000     1.93740

    0.40000     1.83270

    0.60000     1.72620

    0.80000     1.62690

    1.00000     1.53790

    1.20000     1.45980

    1.40000     1.39190

    1.60000     1.33310

    1.80000     1.28250

    2.00000     1.23870

    2.20000     1.20090

    2.40000     1.16830

    2.60000     1.14010

    2.80000     1.11580

    3.00000     1.09490

    3.20000     1.07690

    3.40000     1.06160

    3.60000     1.04860

    3.80000     1.03750

    4.00000     1.02830

    4.20000     1.02070

    4.40000     1.01440

    4.60000     1.00930

    4.80000     1.00520

    5.00000     1.00200

    6.00000     0.99510

    7.00000     0.99530

    8.00000     0.99690

    9.00000     0.99830

   10.00000     0.99910