derivylm Subroutine

public subroutine derivylm(v1, v2, v3, lmax, rabs, ylm, dydth, dydfi, d2ydth2, d2ydfi2, d2ydthdfi)

Calculate the 1st and 2nd derivatives of real spherical harmonics with respect to $\theta$, $\phi$. Use recursion relations for the assoc. Legendre functions $P[l,m]$ to generate the derivatives. These are (taken from Abramowitz and Stegun, Handbook of Mathematical Functions, chapt. 8.): $$\begin{equation} P[l,m+1] = (x^2-1)^{-\frac{1}{2}} ( (l-m)xP[l,m] - (l+m)P[l-1,m] ) \end{equation}$$ $$\begin{equation} (x^2-1)\frac{d}{dx}P[l,m] = (l+m)(l-m+1)(x^2-1)^\frac{1}{2} P[l,m-1] - mxP[l,m] \end{equation}$$ $$\begin{equation} (x^2-1)\frac{d}{dx}P[l,m] = lxP[l,m] - (l+m)P[l-1,m] \end{equation}$$ where $x=\cos{\theta}$, $(x^2-1)^\frac{1}{2} = -\sin{\theta}$, $\frac{d}{d\theta} = -\sin{\theta} \frac{d}{dx}$ Adding these equations: $$\begin{equation} \frac{d}{d\theta}P[l,m](\cos{\theta}) = \frac{1}{2} ( -(l+m)(l-m+1)P[l,m-1] + P[l,m+1] ) \end{equation}$$ It is implied that $P[l,m]=0$ if $m>l$ or $m<-l$. Also, the term $(x^2-1)^\frac{1}{2}$ is ambiguous for real $x$, $0<x<1$; here it is interpreted as $$\begin{equation} (x^2-1)^\frac{1}{2}=-\sin{\theta} \end{equation}$$ but $$\begin{equation} (x^2-1)^{-\frac{1}{2}}=\frac{1}{\sin{\theta}} \end{equation}$$ otherwise the result from Eq.4 (which is cross-checked and correct) does not follow. For the 2nd derivative apply Eq.4 twice. Result: $$\begin{equation} \begin{split} \frac{d^2}{d\theta^2}P[l,m](\cos{\theta}) = \frac{1}{4}((l+m)(l-m+1)(l+m-1)(l-m+2) P[l,m-2]&\\-((l-m)(l+m+1)+(l+m)(l-m+1) )P[l,m]+ P[l,m+2]) \end{split} \end{equation}$$ The $\phi$-derivatives act on \cos{\phi},\sin{\phi} and are trivial. For the associated Legendre functions use the recursion formulas: $$\begin{equation} (l-m+1)P[l+1,m] = (2l+1)\cos{\theta}P[l,m] - (l+m)P[l-1,m] \end{equation}$$ $$\begin{equation} P[l+1,m] = P[l-1,m] - (2*l+1)\sin{\theta}P[l,m-1] \end{equation}$$ ( with $x=\cos{\theta}$. Recursion algorithm for the calculation of $P[l,m]$ and calculation of $Y_l^m$ taken over from subr. ymy of KKR program (implemented there by M. Weinert, B. Drittler). For $m<0$, use $$\begin{equation} P[l,-m] = P[l,m] \frac{(l-m)!}{(l+m)!} \end{equation}$$ Taking into account the lm-prefactors of the spherical harmonics, we construct and use the functions $$\begin{equation} Q[l,m] = \sqrt{\frac{2l+1}{4\pi}} \sqrt{\frac{(l-m)!}{(l+m)!}} P[l,m] \end{equation}$$ whence Eq.4 and Eq.7 become $$\begin{equation} \frac{d}{d\theta}Q[l,m]= \frac{1}{2}(-\sqrt{(l+m)(l-m+1)}Q[l,m-1]+ \sqrt{(l+m+1)(l-m)}Q[l,m+1] ) \end{equation}$$ $$\begin{equation} \begin{split} \frac{d^2}{d\theta^2}Q[l,m] = \frac{1}{4}(\sqrt{(l+m)(l+m-1)(l-m+1)(l-m+2)} Q[l,m-2]&\\+((l-m)(l+m+1)+(l+m)(l-m+1)) Q[l,m]&\\+ \sqrt{(l-m)(l-m-1)(l+m+1)(l+m+2)} Q[l,m+2]) \end{split} \end{equation}$$ Note on sign convension: For the needs of GGA PW91 as implemented here, ylm and derivatives come with a different sign convention compared to the usual in the program: $\sin{\phi}^m \rightarrow (-1)^m \sin{\phi}^m$. Thus some signs change.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=dp)				::	v1
real(kind=dp)				::	v2
real(kind=dp)				::	v3
integer				::	lmax
real(kind=dp)				::	rabs
real(kind=dp)				::	ylm(*)
real(kind=dp)				::	dydth(*)
real(kind=dp)				::	dydfi(*)
real(kind=dp)				::	d2ydth2(*)
real(kind=dp)				::	d2ydfi2(*)
real(kind=dp)				::	d2ydthdfi(*)