  Index

# _atom_rho_multipole_kappa_

Names:
'_atom_rho_multipole_kappa' '_atom_rho_multipole_kappa_prime0' '_atom_rho_multipole_kappa_prime1' '_atom_rho_multipole_kappa_prime2' '_atom_rho_multipole_kappa_prime3' '_atom_rho_multipole_kappa_prime4'

Definition:

```   Gives the radial function expansion-contraction coefficients
(kappa = _atom_rho_multipole_kappa and
kappa'(l) = _atom_rho_multipole_kappa_prime[l])
for the atom specified in _atom_rho_multipole_atom_label.

The coefficients refer to the  multipole formalism described by
Hansen & Coppens [1978, equation (2)] which gives the electron
density at position vector r with respect to an atomic
nucleus as:

rho(r) = Pc*rho_core(r) + Pv*kappa^3^*rho_valence(kappa*r)
+ sum{kappa'(l)^3^*R(kappa'(l),l,r)}
*sum{P(l,m)*d(l,m,theta,phi)}

where:
Pc     = _atom_rho_multipole_coeff_Pc
Pv     = _atom_rho_multipole_coeff_Pv
P(0,0) = _atom_rho_multipole_coeff_P00
Pc + Pv + P(0,0) = Z (the atomic number) for a neutral atom
P(l,m) = _atom_rho_multipole_coeff_P[lm],

d(l,m,theta,phi) is the spherical harmonic of order l,m at the
position (theta, phi) with respect to spherical coordinates
centred on the atom. The spherical coordinates are related
to the local Cartesian axes defined in category
ATOM_LOCAL_AXES, z is the polar axis from which the angle
theta is measured, and the angle phi is measured from the
x axis in the xy plane with the y axis having a value of
phi = +90 degrees.

R(kappa'(l),l,r) is defined in the _atom_rho_multipole_radial_*
items.

rho_core(r) and rho_valence(kappa*r) are the spherical core and
valence densities, respectively. They are obtained from
atomic orbital analytic wavefunctions such as those tabulated
by Clementi & Roetti (1974). They are also the Fourier
transforms of the X-ray scattering factors given in
_atom_rho_multipole_scat_core and
_atom_rho_multipole_scat_valence.

The order, l, of kappa' refers to the order of the multipole
function, 0 <= l <= 4.  The values of kappa' are normally
constrained to be equal.

Ref:  Clementi, E. & Roetti, C. (1974). At. Data Nucl. Data
Tables, 14, 177-478.
Hansen, N. K. & Coppens, P.  (1978).
Acta Cryst. A34, 909-921.

```

Appears in list containing _atom_rho_multipole_atom_label

Type: numb

Category: atom_rho_multipole