11.2.9.1 Further technical information

The weight at a grid point is defined

$$w({\bf r}) = \frac{P_C({\bf r})}{\sum_A P_A({\bf r})}$$ (7)

where

$$P_A({\bf r}) = \prod_{B\ne A} s(\nu_{AB})$$ (8)

in which

$$\nu_{AB} = \mu_{AB} + a_{AB}(1-\mu_{AB}^2)$$ (9)

$$\mu_{AB} = \frac{\vert{\bf {r -A}}\vert - \vert{\bf {r -B}}\vert} {\vert{\bf {A -B}}\vert}$$ (10)

$$= \frac{(r_A - r_B)}{R_{AB}}$$ (11)

Note that ${\bf r}$ is tied to atom $C$, i.e., ${\vec\nabla}_X{\bf r}^\dagger=\delta_{CX}{\bf 1}$. We define

$$y(\nu)=1-\nu^2$$ (12)

and $s$ chosen such that

$$\frac{\partial s(\nu)}{\partial \nu} = A y(\nu)^{m_\mu}, s(-1)=1, s(1)=0,$$ (13)

i.e.,

$$s(\nu) = 1/2 - 1/2 \nu \sum_{l=0}^m (2l-1)!! y(\nu)^l / (2^l l!)$$ (14)

We require the derivatives of the grid weights with respect to nuclear coordinates.

$$\frac{\partial w}{\partial {\bf X}} = \frac{\frac{\partial P_C({\bf r})} {\partial {\bf X}}}{\sum_A P_A... ...{\partial P_A({\bf r})} {\partial {\bf X}}}{\left(\sum_A P_A({\bf r})\right)^2}$$ (15)

$$= w\left(\frac{1}{P_C({\bf r})}\frac{ \partial P_C({\bf r})} {\part... ... \frac{\partial P_A({\bf r})}{\partial {\bf X}}} {\sum_A P_A({\bf r}) } \right)$$ (16)

So, we apply the chain rule recursively; firstly, the derivative of $P$ in terms of the derivatives of $\mu$:

$$\frac{\partial P_A}{\partial {\bf X}} = P_A \sum_{B \ne A}\frac{1}{s(\nu_{AB})} \frac{\partial s(\nu_{AB})}{\partial {\bf X}}$$ (17)

$$= P_A \sum_{B \ne A}\frac{1}{s(\nu_{AB})} \frac{\partial s(\nu_{AB})}{\partial \mu_{AB}} \frac{\partial \mu_{AB}}{\partial {\bf X}}$$ (18)

Next, differential properties of $s$:

$$\frac{\partial s}{\partial \nu} = A*y^{m_\mu}$$ (19)

$$\frac{\partial \nu}{\partial \mu} = 1 -2 \mu a$$ (20)

Hence

$$\frac{\partial s}{\partial \mu} = (1 -2 \mu a) (A*y^{m_\mu})$$ (21)

Finally, derivatives of $\mu$:

$$\frac{\partial \mu_{AB}}{\partial {\bf X}} = (\delta_{CX}-\... ...\delta_{AX}-\delta_{BX}) \frac{(r_A-r_B)({\bf A-B})}{R_{AB}^3}$$ (22)