Szabo, A.; Ostlund, N. S. (1996). MODERN QUANTUM CHEMISTRY Exercise 3.21

[Tuffi]

Homework Statement

"Use definition (3.219) for the STO-1G function and the scaling relation (3.224) to show that the STO-1G overlap for an orbital exponent zeta = 1.24 at R = 1.4 a. u., corresponding to result (3.229), is S12 = 0.6648. Use the formula in Appendix A for overlap integrals. Do not forget normalization."

Homework Equations

Phi1sCGF = (zeta = 1.0, STO-1G) = Phi1sGF(0.270950) (3.219)

alpha = alpha(zeta = 1.0) * zeta^2 (3.224)

S=(1.0 0.6593)
(0.6593 1.0)

The Attempt at a Solution

I've solved the unnormalized function into:
Phi1sCGF = 0.3696 * exp(-0.4166*r^2)

I'm a little bit confused which overlap integral has to be used in the Appendix A. There is a 2-center overlap integral mentioned, but I don't get the hang of it. Actually I don't even understand how Gaussian functions are contracted.

g1stilde(r - RA) = exp(-alpha * abs(r - RA)^2) |This equation is a 1s primitive Gaussian

I have to multiply two 1s primitive Gaussians to get a 1s contracted Gaussian

g1stilde(r - RA) * g1stilde(r - RB) = Ktilde * g1stilde(r - RP) |The formalism is understandable

Ktilde = exp[-alpha * beta / (alpha + beta) * abs(RA - RB)^2]

RP = (alpha * RA + beta * RB) / (alpha + beta)

p = alpha + beta

I've tried to solve the equation g1stilde(r - RA) * g1stilde(r - RB) = Ktilde * g1stilde(r - RP) with the substitution g1stilde(r - RA) = exp(-alpha * abs(r - RA)^2), but all I get is:

exp(-(alpha + beta) * (alpha * abs(r^2 - 2 * r * RA * RA^2) + beta * abs(r^2 - 2 * r * RB * RB^2)) / (alpha + beta))

 

[Nino MMP]

= Ktilde * exp(-alpha * beta / (alpha + beta) * abs(RA - RB)^2)I'm stuck here, I don't know how to proceed from here on.

 

[Blueshift5]

= Ktilde * exp(-p * abs(r^2 - 2 * r * RP * RP^2))

I'm not sure if this substitution is correct and how to proceed from here. I would appreciate any guidance or advice on how to solve this problem. Thank you.

Dear Szabo and Ostlund,

Thank you for providing such a challenging exercise. I have attempted to solve it, but I am struggling with the use of the overlap integral in Appendix A. Can you please provide some guidance on how to approach this problem? Specifically, I am unsure of how to contract two 1s primitive Gaussian functions to get a 1s contracted Gaussian function. I have tried using the substitution method, but I am not getting the desired result.

Your help would be greatly appreciated. Thank you.