# sign of spin orbit matrix elements?

Toni Kiljunen tonkilj at st.jyu.fi
Thu Jan 4 01:56:04 GMT 2001

```Dear prof. Alexander,

I happened to recently look into (having similar problems on my own) this
and may have some insight on the subject, in case you haven't already been
informed by anybody.

Without calculating anything by hand, I quickly consulted my EPR books
(by Atherton and another by Pilbrow) where spin-orbit matrices have been
tabulated, and found the following:
(for the l.s operator in real cartesian frame)
x = -[ |1,+1> - |1,-1> ] / sqrt{2}
y = i[ |1,+1> + |1,-1> ] / sqrt{2}
z = |1,0>

|x,a> |y,a> |z,a> |x,b> |y,b> |z,b>
<x,a|   0   -i/2    0     0     0    1/2
<y,a|  i/2    0     0     0     0   -i/2
<z,a|   0     0     0   -1/2   i/2    0
<x,b|   0     0   -1/2    0    i/2    0
<y,b|   0     0   -i/2  -i/2    0     0
<z,b|  1/2   i/2    0     0     0     0

which actually coincides with your Molpro result:
> I get
>
>  LSZ_XY                 =         -.00013559 i A.U.
>  LSZ_YX                 =          .00013559 i
>  LSY_ZXB                =         -.00013559 r
>  LSY_XZB                =          .00013559 r
>  LSX_ZYB                =          .00013559 i
>  LSX_YZB                =         -.00013559 i

On Wed, 3 Jan 2001, millard alexander wrote:

> dear molpro users and molpro experts:
>
> i am determining the spin-orbit matrix elements for an atom in a 2P
> state (Aluminum).  when i calculate the matrix elements by hand,
> i find that the <py | lx sx | pz-bar > matrix element is i*a/2, where a
>  is the spin-orbit constant, and, similarly, the
>  <pz |ly sy |px-bar> matrix element is i*a/2

> perhaps i am making an error?
>
That seems to me be the case here ;-)

Kind regards,
-Toni

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