THO and Gauss integration formulae

 

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 $\rho$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 $\rho$ 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

$$\langle\varphi_{\alpha }\vert V\vert\varphi_{\alpha }\rangle ... ...ho) \varphi^2_{n_{z}}(z){\varphi_{n_{\rho }}^{m_{l}}}^2(\rho),$$ (27)

and in the THO representation as

 

$$\langle\psi_{\alpha }\vert V\vert\psi_{\alpha }\rangle \int\limits_{-\infty }^{\infty}dz\int\limits_{0}^{\infty } \rho \;d\rho V(z,\rho)$$ =

$$\times D(z,\rho) \textstyle{\ \varphi^2_{n_{z}}\left(\frac{z}{{r}}f_z({r... ...ht) {\varphi_{n_{\rho }}^{m_{l}}}^2\left(\frac{\rho }{{r}}f_\rho({r} )\right)}.$$ (28)

The way to calculate the second matrix element (28) is by first transforming to the $\rho^{\prime}$and $z^{\prime}$ variables (22). This absorbs the Jacobian $D(z,\rho)$and leads to an integral over HO wave functions that is almost identical to (27), namely

 

$$\langle\psi_{\alpha }\vert V\vert\psi_{\alpha }\rangle \int\limits_{-\infty }^{\infty}dz^{\prime}\int\limits_{0}^{\infty... ...prime}V\left(z(z^{\prime},\rho^{\prime}),\rho(z^{\prime}, \rho^{\prime})\right)$$ =

$$\times \varphi^2_{n_{z}}(z^{\prime}) {\varphi_{n_{\rho }}^{m_{l}}} ^2(\rho^{\prime}).$$ (29)

The only complication in numerically carrying out the integral (29) involves determining the inverse LST transformations z= $z(z^{\prime},\rho^{\prime})$ and $\rho$= $\rho(z^{\prime},\rho^{\prime})$to be inserted into the known function $V(z,\rho)$. 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 $\partial/\partial z$ and $\partial/\partial\rho$ into derivatives $\partial/\partial z^{\prime}$and $\partial/\partial\rho^{ \prime}$, and then performing the integrations in the variables $z^{\prime}$ and $\rho^{\prime}$ over ordinary HO wave functions (see the next section).