Sturmians and Generalized Sturmians in Quantum Theory

Abstract

The theory of Sturmians and generalized Sturmians is reviewed. It is shown that when generalized Sturmians are used as basis functions, calculations on the spectra and physical properties of few-electron atoms can be performed with great ease and good accuracy. The use of many-center Coulomb Sturmians as basis functions in calculations on N-electron molecules is also discussed. Basis sets of this type are shown to have many advantages over other types of ETO’s, especially the property of automatic scaling.

Appendix: Angular and Hyperangular Integrations

In a 3-dimensional space, the volume element is given by dx ₁dx ₂dx ₃ in Cartesian coordinates or by r ²dr dΩ in spherical polar coordinates. Thus, we can write

$$ {\text{d}}{x_1}{\text{d}}{x_2}{\text{d}}{x_3} = {r^2}{\text{d}}r\,{\text{d}}\Omega, $$

(188)

where dΩ is the element of solid angle. Similarly, in a d-dimensional space we can write

$$ {\text{d}}{x_1}{\text{d}}{x_2} \cdots {\text{d}}{x_d} = {r^{{d - 1}}} {\text{d}}r\,{\text{d}}\Omega, $$

(189)

where r is the hyperradius and where dΩ is the element of generalized solid angle. We will now prove a general theorem for angular and hyperangular integration [24]:

Theorem 1 (Angular integration theorem).

Let

$$ I(n) \equiv \int {{\text{d}}\Omega {{\left( {\frac{{{x_1}}}{r}} \right)}^{{{n_1}}}}{{\left( {\frac{{{x_2}}}{r}} \right)}^{{{n_2}}}} \cdots {{\left( {\frac{{{x_d}}}{r}} \right)}^{{{n_d}}}}}, $$

(190)

where x ₁ , x ₂ ,…, x _d are the Cartesian coordinates of a d-dimensional space, dΩ is the generalized solid angle, r is the hyperradius, defined by

$$ {r^2} \equiv \sum\limits_{{j = 1}}^d {x_j^2}, $$

(191)

and where the n _j ’s are positive integers or zero. Then

$$ {I(n) = \left\{ \!\!\!\!\!\!{\frac{{{\pi^{{d/2}}}}}{{{2^{{(n/2 - 1)}}}\Gamma \left( {\frac{{d + n}}{2}} \right)}}\prod\limits_{{j = 1}}^d {({n_j} - 1)!!} }\quad {{\text{if}}\,\,{\text{all}}\,\,{\text{the}}\,\,{n_j}{\text{s}}\,\,{\text{are}}\,\,{\text{even}}} \\cr (0\qquad {\text{otherwise}}\right)}, $$

(192)

where

$$ {n \equiv \sum\limits_{{j = 1}}^d {{n_j}}}. $$

(193)

Proof. Consider the integral

$$ {\int\limits_0^{\infty } {{\text{d}}r\,\,{r^{{d - 1}}}{{\text{e}}^{{ - {r^2}}}}\int {{\text{d}}\Omega \,x_1^{{n1}}\,x_2^{{n2}} \cdots \,\,x_d^{{nd}} = \prod\limits_{{j = 1}}^d {\int\limits_{{ - \infty }}^{\infty } {{\text{d}}{x_j}\,\,x_j^{{{n_j}}}{{\text{e}}^{{ - x_j^2}}}}}}}}. $$

(194)

If n _j is zero or a positive integer, then

$$ \int\limits_{{ - \infty }}^{\infty } {{\text{d}}{x_j}\,\,x_j^{{nj}}\,{{\text{e}}^{{ - x_j^2}}} = } \left\{ {\frac{{({n_j} - 1)!!\sqrt {\pi } }}{{{2^{{{n_j}/2}}}}}} \quad \qquad{{\text{if}}\ {n_{{j\,}}}\ {\text{is}}\ {\text{even}}} \\cr (0 \qquad \qquad \qquad{{\text{if}}\ {n_j}\ {\text{is}}\ {\text{odd}}}\right), $$

(195)

so that the right-hand side of (194) becomes

$$ {\prod\limits_{{j = 1}}^d {\int\limits_{{ - \infty }}^{\infty } {{\text{d}}{x_j}} \,\,\,} x_j^{{{n_j}}}{{\text{e}}^{{ - x_j^2}}}} = \left\{ {\frac{{{\pi^{{d/2}}}}}{{{2^{{n/2}}}}}\prod\limits_{{j = 1}}^d {({n_j} - 1)!!} }\quad{{\text{if}}\,\,{\text{all}}\,\,{\text{the}}\,\,{n_j}{\text{s}}\,\,{\text{are}}\,\,{\text{even}}} 0\quad{\text{otherwise}} \right. $$

(196)

The left-hand side of (5) can be written in the form

$$ {\int\limits_0^{\infty } {{\text{d}}r\,\,{r^{{d + n - 1}}}\,\,{{\text{e}}^{{ - {r^2}}}}} \int {{\text{d}}\Omega } {\left( {\frac{{{x_1}}}{r}} \right)^{{{n_1}}}}\,{\left( {\frac{{{x_2}}}{r}} \right)^{{{n_2}}}} \cdots {\left( {\frac{{{x_d}}}{r}} \right)^{{{n_d}}}} = \frac{{I(n)}}{2}\Gamma \left( {\frac{{d + n}}{2}} \right)}. $$

(197)

Substituting (196) and (197) into (194), we obtain (192). Q.E.D.

Comments

In the special case where d = 3, (192) becomes

$$ {\int {{\text{d}}\Omega {{\left( {\frac{{{x_1}}}{r}} \right)}^{{{n_1}}}}{{\left( {\frac{{{x_2}}}{r}} \right)}^{{{n_2}}}}\,\,{{\left( {\frac{{{x_d}}}{r}} \right)}^{{{n_3}}}}} = \left\{ {\frac{{4\pi }}{{(n + 1)!!}}\prod\limits_{{j = 1}}^3 {({n_j} - 1)!!} } \quad{{\text{all}}\,\,\,{n_j}{\text{s}}\,\,\,{\text{even}}} \\\hskip-6.8pc 0\quad{\text{otherwise}} \right.} $$

(198)

Let us now consider a general polynomial (not necessarily homogeneous) of the form:

$$ {P({x}) = \sum\limits_n {{c_{{n\,\,\,\,\,}}}x_1^{{{n_1}}}\,\,x_2^{{{n_2}}} \ldots x_d^{{{n_d}}}}}. $$

(199)

Then we have

$$ {\int {{\text{d}}\Omega \,} \,\,P(x) = \sum\limits_n {{c_{{n\,\,\,\,}}}\int {{\text{d}}\Omega \,} \,\,x_1^{{{n_1}}}\,\,x_2^{{{n_2}}} \ldots x_d^{{{n_d}}}} = \sum\limits_n {{c_n}\,\,{r^n}\,\,I({n})}}. $$

(200)

It can be seen that (192) can be used to evaluate the generalized angular integral of any polynomial whatever, regardless of whether or not it is homogeneous.

It is interesting to ask what happens if the n _j ’s are not required to be zero or positive integers. If all the n _j ’s are real numbers greater than −1, then the right-hand side of (194) can still be evaluated and it has the form

$$ {\prod\limits_{{j = 1}}^d {\int\limits_{{ - \infty }}^{\infty } {\!\!\!{\text{d}}{x_j}\,\,x_j^{{{n_j}}}{{\text{e}}^{{ - x_j^2}}} = } \prod\limits_{{j = 1}}^d {\frac{1}{2}(1 + {{\text{e}}^{{i\pi {n_j}}}})} \,\Gamma \left( {\frac{{{n_j} + 1}}{2}} \right)}}. $$

(201)

Thus, (192) becomes

$$ I(n) = \frac{2}{{\Gamma \left( {\frac{{d + n}}{2}} \right)}}\prod\limits_{{j = 1}}^d {\frac{1}{2}} (1 + {e^{{i\pi {n_j}}}})\;\Gamma \left( {\frac{{{n_j} + 1}}{2}} \right)\quad {n_j} > - 1,\quad j = 1, \ldots, d. $$

(202)

This more general equation reduces to (192) in the special case where the n _j ’s are required to be either zero or positive integers.