# 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:

## Density of States calculations

### subroutine ff2rho

if (msapp != 1)

else

### subroutine fmsdos

## FMS calculations

## XAS calculations

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

The contribution to the XAS from a given site and orbital angular momentum .

. where is the smoothly varying atomic background contributions and 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)

## Solving Dirac equations [afovrg]

The spin-orbit eigenfunction (eigenstates of )

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

, *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
, so that we can write the polarization vector
, , where ξ_{0},ξ_{1},ξ_{ − 1} are the usual spherical basis vectors.

The decomposition of *e*^{ik.r} in terms of the irreducible tensors
,
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. , where

In the form of Wigner-Eckart theorem:

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

Note: the usual notation of atomic physics .

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

- input:
- output:

See Grant eq. 6.30. calculate the factors

xm1, xm2 both either real or pure imaginary

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)

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

if ifl = 2

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

- output: value

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

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)

## Radial Integration Routine

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

- output: da

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

## Wigner 3j symbol [cwig3j]

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

- input:
- output: wigner 3j coefficient for integers (ient=1) or semiintegers (ient=2) other arguments should be multiplied by ient

## Orthogonality relations

## Composition relation for the spherical harmonics

## 6j symbols [sixj]

## 9j symbols [ninej]