List of Formulas

From FEFF

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 $j$ .

$κ = l Θ(l - j) - (l + 1)Θ(j - l)$

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

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

FMS calculations

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 $cchi(E) = S0 2 * TrG(E)$

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

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]

$\langle nkm|\alpha . A|n'k'm'\rangle$ , $A = u p exp(i ω. r)$ , $| ω | = Δ E / c$ where $u p$ 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 $ξ 0,ξ 1,ξ - 1$ are the usual spherical basis vectors.

The decomposition of $e i k . r$ 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 $e z$ 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]

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

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

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