At first glance, the THO wave functions (9) and (24) look much
more complicated than their HO counterparts (7) and (21). In
particular, in contrast to the HO wave functions, the THO wave functions are not
separable either in the x, y, and z Cartesian coordinates or in the and z axial coordinates. Due to the presence of the Jacobian factor and the
r-dependence of the LST functions, the local-scaling transformation mixes the
x, y and z coordinates and the
and z coordinates. Nevertheless, as
we now proceed to show, the THO wave functions are readily tractable in any
configurational self-consistent calculation. Indeed, the modifications required
to transform a code from the HO to the THO basis are minor.
One of the properties of the HO basis that makes it so useful is the high
accuracy that can be achieved when calculating matrix elements using
Gauss-Hermite and/or Gauss-Laguerre integration formulae [17]. This
feature has been exploited frequently in various mean-field nuclear structure
calculations (see, e.g., Refs. [16,18,15]). To illustrate
how the same methods can be applied in the THO basis, we focus on the specific
example of a diagonal matrix element of a spin and isospin independent potential
function V. This matrix element can be expressed in the axial HO
representation as
The only complication in numerically carrying out the integral (29)
involves determining the inverse LST transformations
z=
and
=
to be inserted into the known function
.
But this only has to be done
once, and, moreover, if Gauss quadratures are used to evaluate the integrals, the
inverse transformation only has to be known at a finite number of
Gauss-quadrature nodes.
Generalization of the above approach to include differential operators, as will
often arise in THO basis configurational calculations, is fairly straightforward.
Such integrals can be done by first transforming derivatives
and
into derivatives
and
,
and then performing the integrations in
the variables
and
over ordinary HO wave functions
(see the next section).