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

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


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

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


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

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


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


1 Answer 1


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.


You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .