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I have been working on a toy HF code for some time. From this reference I am trying to work out an analytic expression for the first derivatives of overlap integrals. The original expression for overlap is given as: $$ S_x(a_x,b_x) = \sqrt{\frac{\pi}{\alpha + \beta}}*\sum_{i_x=0}^{a_x}\binom{a_x}{i_x}(P_x-A_x)^{(a_x-i_x)}*\sum_{j_x=0}^{b_x}\binom{b_x}{j_x}(P_x-B_x)^{(b_x-j_x)}*\frac{(i_x+j_x-1)!!}{2[\alpha+\beta]^{(\frac{i_x+j_x}{2})}} $$

Working through the expression and ignoring the constants, I got to this step

$$ \frac{dS_x}{dA_x}=(a_x - i_x) (P_x-A_x)^{(a_x-i_x-1)}(P_x-B_x)^{(b_x-j_x)}\frac{d}{dA_x}\left(P_x-A_x\right) + (b_x - j_x) (P_x-B_x)^{(b_x-j_x-1)}(P_x-A_x)^{(a_x-i_x)}\frac{d}{dA_x}\left(P_x-B_x\right) $$

where

$$ P_x = \frac{\alpha*A_x + \beta * B_x}{(\alpha + \beta)} $$

However, I am not sure if this expression is correct. Some pointers to the exact analytic expressions would be much appreciated.

Edit:

Using Mathematica to work out the expressions, I got these two expressions:

$$ \frac{dS_x}{dA_x} = \left(\frac{\alpha}{\alpha+\beta}\right)^{(b_x-j_x)}*\left(\frac{\beta}{\alpha+\beta}\right)^{(a_x-i_x)}*(A_x-B_x)^{(b_x-j_x-1)}*(B_x-A_x)^{(a_x-i_x)}*\left(a_x+b_x-(i_x+j_x)\right) $$

and

$$ \frac{dS_x}{dB_x} = -\left(\frac{\alpha}{\alpha+\beta}\right)^{(b_x-j_x)}*\left(\frac{\beta}{\alpha+\beta}\right)^{(a_x-i_x)}*(A_x-B_x)^{(b_x-j_x-1)}*(B_x-A_x)^{(a_x-i_x)}*\left(a_x+b_x-(i_x+j_x)\right) $$

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1 Answer 1

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You don't have to write special code for derivative integrals. Since you know that the Cartesian Gaussian basis functions are of the form $ \chi_x = (x-x_0)^l \exp[-\alpha (x-x_0)^2]$ (same for the other directions), you know that

$ \displaystyle \frac {\partial \chi_x} {\partial x_0} = l(x-x_0)^{l-1} \exp[-\alpha (x-x_0)^2] - 2\alpha (x-x_0)^{l+1} \exp[-\alpha (x-x_0)^2]$.

This means that the derivative of a function with angular momentum $l$ is a linear combination of functions with angular momentum $l+1$ and $l-1$. You can thereby compute the derivative of any integral by combining the original integrals for different angular momenta.

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