# List of Formulas

• The list of formulas used in FEFF

The corresponding soubroutines are quoted in the square bracket.

## Parameter of angular momentum and quantum number kappa

The quantum number κ is a function of the orbital angular momentum quantum number l and the total angular momentum quantum number j.

κ = lΘ(lj) − (l + 1)Θ(jl)

where Θ(x) is equal to one if x is greater than zero and is equal to zero if x is less than zero. N.B. In the above equation the argument of Θ(x) can never equal zero since l is integral and j is half-integral.

The above equation is equivalent to:

$\kappa = -j - 1/2 \qquad \textrm{ if }\;\; j > l$

$\kappa = +j + 1/2 \qquad \textrm{ if }\;\; j < l$

## Density of States calculations

### subroutine ff2rho

if (msapp != 1)

$\rho^c_{l}(E) = \rho^c_{l}(E)+\rm{Im}[\chi(E)]\rho^{sc}_l(E)\,$

else $\rho^c_{l}(E)\,$

### subroutine fmsdos

$\rm{gtr}_l(E) =Tr G_{L,L'}(E)=\sum_{m=-l}^{l} \rm{gg}_{L,L}(E)$

## XAS calculations

### subroutine setkap(ihole, kinit, linit)

The contribution to the XAS from a given site $i\,$ and orbital angular momentum $l\,$.

$\mu_{li}(E)=\mu{li}^{0\prime}(E)[1+\chi_{li}^\prime(E)]$. where $\mu_{li}^{0\prime}(E)$ is the smoothly varying atomic background contributions and $\chi_{li}^\prime(E)$ is the fine-structure or XAFS spectrum. A prime to denote final-state quantities calculated in the presence of a screened core hole.

### subroutine ff2xmu

Adds the contributions from each path and absorber, including Debye-Waller factors. Writes down main output: chi.dat and xmu.dat cchi(E) = S02 * TrG(E)

$\mu=\frac{\rm{rchtot}(E)}{\rm{xsedge}}=\rm{Im}[\rm{xsec}(E)+\rm{xsnorm}(E)*\chi_a(E)+\chi_i(E)]/\rm{xsedge}$

## Solving Dirac equations [afovrg]

The spin-orbit eigenfunction (eigenstates of $j^2,\,j_z,\,and\, P$)

$\chi_{km}(\theta,\phi)=\sum_{\sigma}Y_l^{m-\sigma}(\theta,\phi)\phi^\sigma .\langle lm-\sigma\frac{1}{2}\sigma|l\frac{1}{2}jm\rangle$

## Matrix Elements for Electromagnetic Multipole Transitions (Grant) [xmult]

$\langle nkm|\alpha . A|n'k'm'\rangle$, A = upexp(iω.r), | ω | = ΔE / c where up is a polarization vector, and ω is the propagation vector.

Simplify the problem by taking ω to define the z-axis $\vec{\omega} = \omega e_z$, so that we can write the polarization vector $u_p = \frac{1}{\sqrt{2}}\left[e_x+i p e_y\right]=-p\xi_p$, $p=\pm1$, where ξ01 − 1 are the usual spherical basis vectors.

The decomposition of eik.r in terms of the irreducible tensors $C_0^{(l)}$ $e^{i k.r}=\sum_{l=0}^{\infty}i^l(2l+1)j_l(\omega r) C_0^{(l)}(\Omega)$, where the angles involved are spherical polar coordinates with ez as polar axis.

The vector operator α is a tensor of order 1. The product of two tensors can be decomposed in terms of the irreducible tensors. $\alpha . A=\alpha.u_p \exp(i\omega . r)=\sum_{L=1}^\infty\sum_{l=L-1}^{L+1}a_{lL}(r)\begin{pmatrix}0&X_p^{(l1)L}\\X_p^{(l1)L}&0\end{pmatrix}$, where $a_{lL}(r)=(-1)^L[L]^{1/2}\,i^l\begin{pmatrix}l&1&L\\0&-p&p\end{pmatrix}j_l(\omega r)$

In the form of Wigner-Eckart theorem: $\langle nkm|a_{lL}(r)X_p^{(l1)L}\otimes\begin{pmatrix}0&1\\1&0\end{pmatrix}|n'k'm'\rangle=(-1)^{j-m}\begin{pmatrix}j&l&j'\\-m&p&m'\end{pmatrix}R_{k,k'}^{l,L}$

In the dipole approximation it is enough to keep only one term (l = 0, L = 1).

Note: the usual notation of atomic physics $[k,l,\,...]=(2k+1)(2l+1)\,...$.

### subroutine xmult(k, kp, ls, lb, xm1, xm2)

• input: $k=k, kp=k', ls=l, lb=L\,$
• output: $xm1=C_{k,k'}(1), xm2=C_{k,k'}(-1)\,$

See Grant eq. 6.30. calculate the factors

$\langle k|\alpha\cdot A( l, L)|k' \rangle=(-1)^{j-m}\begin{pmatrix}j & L &j'\\ -m & p & m'\end{pmatrix}R_{k,k'}$

$R_{k,k'}^{l,L} = \int dr (C_{k,k'}(1)\,P_k Q_{k'}+ C_{k,k'}(-1)\,Q_kP_{k'})j_l(wr)$

xm1, xm2 both either real or pure imaginary

$\beta=\pm 1$ corresponds to the upper(lower) component of Dirac spinor.

• calculate xm1 (β = 1 in eq.6.30)
• calculate xm2 (β = − 1 in eq.6.30)

$C_{k,k'}^{l,L}(\beta)=\sqrt{6}\left[j,L,j',\lambda,\lambda'\right]^{1/2}(-1)^{\lambda}\begin{Bmatrix}\lambda & l & \lambda' \\ \frac{1}{2} &\frac{1}{2} &1 \\ j & j' & L\end{Bmatrix}\begin{pmatrix}\lambda &l &\lambda' \\ 0 &0 & 0\end{pmatrix}\times\delta(\lambda,j-\frac{1}{2}\alpha\beta)\delta(\lambda',j'+\frac{1}{2}\alpha'\beta)$

## Performs radial integration for multipole matrix element or central atom absorption [radint]

if ifl = 2

$\rm{xirf}=2\times\int_0^\infty dr\,dR_{k,k^\prime}^R(r) \int_0^{r} dr^\prime\,dR_{k,k^\prime}^N(r^\prime)$

### subroutine xrci ( mult, xm, dgc0, dpc0, p, q, bf, value)

• output: value

r-dependent multipole matrix element (before r-integration)

$dR_{k,k^\prime}^R(r)=P_k Q_{k^\prime}^R \left[C_{k,k^\prime}^{0,1}(-1)j_0(kr)+C_{k,k^\prime}^{2,1}(-1)j_2(kr)\right] +Q_k P_{k^\prime}^R\left[C_{k,k^\prime}^{0,1}(1)j_0(kr)+C_{k,k^\prime}^{2,1}(1)j_2(kr)\right]$

dgc0*q* (xm(2)*bf(0) + xm(4)*bf(2)) + dpc0*p * (xm(1)*bf(0) + xm(3)*bf(2))

for double radial integral (use the irregular components of the final state)

$dR_{k,k^\prime}^N(r)=P_k Q_{k^\prime}^N \left[C_{k,k^\prime}^{0,1}(-1)j_0(kr)+C_{k,k^\prime}^{2,1}(-1)j_2(kr)\right] +Q_k P_{k^\prime}^N\left[C_{k,k^\prime}^{0,1}(1)j_0(kr)+C_{k,k^\prime}^{2,1}(1)j_2(kr)\right]$

## Radial Integration Routine

### csomm (dr,dp,dq,dpas,da,m,np)

• output: da

$\rm{da}=\int_{r=0}^{dr(np)} (\rm{dp}+\rm{dq})dr^m$ Modified to use complex p and q. SIZ 4/91

integration by the method of simpson of (dp+dq)*dr**m from 0 to r=dr(np)

dpas=exponential step; for r in the neighborhood of zero (dp+dq)=cte*r**da

$r=\exp(-8.8+.05n), dr=.05\times r dn$

## Wigner 3j symbol [cwig3j]

$J=j_1+j_2:\quad \begin{pmatrix} j_1 & j_2 & J\\ m_1 & m_2 & M\end{pmatrix}\equiv\frac{(-1)^{j_1-j_2+M}}{\sqrt{2J+1}}\langle j_1 j_2 m_1 m_2|JM\rangle$

### function cwig3j (j1,j2,j3,m1,m2,ient)

• input: $j1=j_1, j2=j_2, j3=J, m1=m1, m2=m2, ient\,$
• output: wigner 3j coefficient for integers (ient=1) or semiintegers (ient=2) other arguments should be multiplied by ient

## Orthogonality relations

$\sum_{m_1=-j_1}^{+j_1}\sum_{m_2=-j_2}^{+j_2}\begin{pmatrix}j_1 & j_2 & j_3\\ m_1 & m_2 & m_3\end{pmatrix}\begin{pmatrix}j_1 & j_2 & j_3'\\ m_1 & m_2 & m_3'\end{pmatrix}=\frac{1}{2j_3+1}\,\delta_{j_3j_3'}\delta_{m_3m_3'}$

## Composition relation for the spherical harmonics

$\int Y_{l_1}^{m_1}Y_{l_2}^{m_2}Y_{l_3}^{m_3} d\Omega = \left[\frac{(2l_1+1)(2l_2+1)(2l_3+1)}{4\pi}\right]^{\frac{1}{2}}\begin{pmatrix}l_1 & l_2 & l_3\\ 0 & 0 & 0\end{pmatrix}\begin{pmatrix}l_1 & l_2 & l_3\\ m_1 & m_2 & m_3\end{pmatrix}$

## 6j symbols [sixj]

$J=j+j'+j'':\quad a)\,j'+j=g',\,g'j''=J\quad b)\,j+j''=g'',\,j'+g''=J\quad$

$\langle j',(jj'')g'';JM|(j'j)g',j'';J'M'\rangle=\delta_{JJ'}\delta_{MM'}\sqrt{(2g'+1)(2g''+1)}(-)^{j+j'+j''+J}\begin{Bmatrix}j' & j & g' \\ j'' & J & g''\end{Bmatrix}$

## 9j symbols [ninej]

$J=j_1+j_2+j_3+j_4:\quad a)\,j_1+j_2=J_{12},\,j_3+j_4=J_{34},\,J_{12}+J_{34}=J\quad b)\,j_1+j_3=J_{13},\,j_2+j_4=J_{24},\,J_{13}+J_{24}=J\quad$

$\langle (j_1 j_2)J_{12},(j_3 j_4)J_{34};JM|(j_1 j_3)J_{34},(j_2 j_4)J_{24};J'M'\rangle=\delta_{JJ'}\delta_{MM'}\sqrt{(2J_{12}+1)(2J_{34}+1)(2J_{13}+1)(2J_{24}+1)}\begin{Bmatrix}j_1 & j_2 & J_{12} \\ j_3 & j_4 & J_{34}\\ J_{13} & J_{24} & J\end{Bmatrix}$