Application of the Uniformly Charged Sphere Stabilization for Calculating the Lowest ¹ S Resonances of H ⁻

Abstract

The uniformly charged sphere stabilization method has been used to calculate the lowest ¹ S resonances of H ⁻. It was shown that this method is sensitive to the choice of basis set and parameters of the stabilization potential. The conclusion on the suitability of this method for calculating resonance energies and widths is based on the analysis of our computational results.

Appendix: Calculation of Matrix Elements for One- and Two-Electron Operators

The matrix elements of the one-electron operators can be written as

$$\begin{aligned} \bigl\langle n'l'm'\bigr| \frac{1}{r}|nlm\rangle =&\frac {1}{b^2}Q_{n'l'nl}^{l'+l+1}(0)N_{n'l'}N_{nl} \delta_{l'l}\delta_{m'm} , \end{aligned}$$

(5.18)

$$\begin{aligned} \bigl\langle n'l'm'\bigr|- \frac{1}{2}\varDelta|nlm\rangle =& -\frac{1}{2b} \biggl\{ \frac {1}{4}Q_{n'l'nl}^{l'+l+2}(0)-(n+l+1)Q_{n'l'nl}^{l'+l+1}(0) \\ &{}-nQ_{n'l'nl}^{l'+l}(0)+(n+2l+2)Q_{n'l'n-1l}^{l'+l}(0) \biggr\} N_{n'l'}N_{nl} \delta_{l'l} \delta_{m'm}, \end{aligned}$$

(5.19)

$$\begin{aligned} \bigl\langle n'l'm'\bigr|V_{st}(r)|nlm \rangle =&\frac{V_0}{b^3} \bigl\{ Q_{n'l'nl}^{l'+l+2}(bR) -bRQ_{n'l'nl}^{l'+l+1}(bR) \bigr\}N_{n'l'}N_{nl} \delta_{l'l}\delta_{m'm}, \end{aligned}$$

(5.20)

where $Q_{n'l'nl}^{m}(x)$ is the auxiliary integral

$$ Q_{n'l'nl}^m(x)=\int_x^{\infty} \mathrm{e}^{-z}z^m L_{n'}^{2l'+2}(z) L_{n}^{2l+2}(z) dz . $$

(5.21)

For the calculation of integrals (5.18)–(5.20) the method based on the recurrence relation for product of Laguerre polynomials has been used [51]. The main relation of this method can be obtained by substituting the recursion $(n+1)L_{n+1}^{k}(z)-(2n+k+1-z)L_{n}^{k}(z)+(n+k)L_{n-1}^{k}(z)=0$ [40] for $L_{n}^{2l+2}$ or $L_{n'}^{2l'+2}$ polynomial in (5.21) which gives two different three-term recursions for integrals over the indexes n and n′. Elimination of the integral $Q_{n'l'nl}^{m+1}(x)$ from the both recursions leads to the new relation

$$\begin{aligned} nQ_{n'l'nl}^{m}(x) =&\bigl(n'+1 \bigr)Q_{n'+1l'n-1l}^{m}(x)+\bigl(n'+2l'+2 \bigr)Q_{n'-1l'n-1l}^{m}(x) \\ &{}- (n+2l+1)Q_{n'l'n-2l}^{m}(x)+2\bigl(n+l-n'-l'-1 \bigr)Q_{n'l'n-1l}^{m}(x). \end{aligned}$$

(5.22)

This relation is applied for construction of the ascending recursion which starts from the elements $Q_{kl0l}^{m}(x)$ with n′−n+1≤k≤n′+n−1 and finishes at $Q_{n'lnl}^{m}(x)$ (assume that n≤n′). The integrals with m=2l+2 and x=0 can be evaluated directly using the orthogonality relation

$$ Q_{n'lnl}^{2l+2}(0)=\frac{(n+2l+2)!}{n!} \delta_{n'n} . $$

(5.23)

For integrals with m<2l+2 and x=0 the recursion (5.22) is initialized by the elements

$$\begin{aligned} Q_{0l0l}^{m}(0) =&m! , \end{aligned}$$

(5.24)

$$\begin{aligned} Q_{1l0l}^{m}(0) =&Q_{0l1l}^{m}(0)=m!(2l+2-m), \end{aligned}$$

(5.25)

$$\begin{aligned} Q_{1l1l}^{m}(0) =&m! \bigl( (m+1) (m+2)-2(m+1) (2l+3)+(2l+3)^2 \bigr) . \end{aligned}$$

(5.26)

All elements $Q_{kl0l}^{m}(0)$ needed for the evaluation of integrals (5.18) and (5.19) are calculated by means of the special three-term recursion $(n'+1)Q_{n'+1l0l}^{m}(x)-(2n'+2l+1)Q_{n'l0l}^{m}(x)+(2n'+2l+1)Q_{n'l0l}^{m+1}(x) +(n'+2l+2)Q_{n'-1l0l}^{m}(x)=0$.

The case of the integrals (5.20) is more complicate. The elements $Q_{n'l0l}^{m}(x)$ and $Q_{n'l1l}^{m}(x)$ (m=2l+2 or m=2l+1) needed for the initializing (5.21) can be taken analytically

$$ Q_{n'lnl}^{m}(x)=\sum _{t=0}^{n'+n}\varGamma(m+t+1,x)W_t \bigl(n',l',n,l\bigr) , $$

(5.27)

where $\varGamma(a,x)=\int_{x}^{\infty}z^{a-1}\mathrm{e}^{-z}dz$—incomplete gamma function, W _t (n′,l,n,l)—coefficient of the term z ^t in the product of two associated Laguerre polynomials. This coefficient is calculated by the formula

$$ W_t \bigl(n',l',n,l\bigr)=\sum _{u=0}^{u\le n, t-u\le n'}\frac{(-1)^t(n'+v)!(n+v)!}{ u!(t-u)!(n'-t+u)!(n-u)!(v'+t-u)!(v+u)!} . $$

(5.28)

In this formula v=2l+2 and v′=2l′+2. The function Γ(a,x) from expression (5.27) can be derived analytically at the integer values of parameter a. For this purpose the function is represented as the product Γ(a,x)=e^−x x ^a g(a,x), where g(a,x)—the auxiliary function defined for the a=1 as g(1,x)=1/x. This representation allows to use the standard recurrence relation for incomplete gamma function Γ(a+1,x)=aΓ(a,x)+e^−x x ^a by $g(a+1,x)=\frac{a}{x}g(a,x)+\frac{1}{x}$ with the minimal numerical error [52]. The analytical method of calculation is suitable for integrals with n′≤23, that one can see by the comparison of the values obtained from the formula (5.27) with ones obtained by Gauss-Laguerre integration. For the integrals with n′>23 the quadrature method was applied.

During the calculation of electron-electron repulsion integrals the operator 1/r ₁₂ can be represented by the series

$$ \frac{1}{r_{12}}=\sum_{l=0}^{\infty} \sum_{m=-l}^{l}\frac{4\pi }{2l+1}Y_{lm}( \hat{r}_1) Y_{lm}^{*}(\hat{r}_2) \frac{r_{<}^l}{r_{>}^{l+1}} , $$

(5.29)

where r _<=min(r ₁,r ₂) and r _>=max(r ₁,r ₂) [53–57]. With this expression the two-electron integral is

$$\begin{aligned} &\bigl\langle n'_1 l'_1 m'_1 n'_2 l'_2 m'_2\bigr| \frac{1}{r_{12}}|n_1l_1m_1n_2l_2m_2 \rangle \\ &\quad= N_{n'_1l'_1}N_{n'_2l'_2}N_{n_1l_1}N_{n_2l_2}b^{-5} \sum_{L=0}^{\infty }R_L\sum _{M=-L}^{L}D_{LM} , \end{aligned}$$

(5.30)

where R _L —radial integral in the form

$$ R_L=\int_0^{\infty} \int_0^{\infty}dxdy f_1(x)f_2(y) \frac{x_{<}^L}{x_{>}^{L+1}} , $$

(5.31)

in which variables r ₁, r ₂ and r _</> are replaced by x, y and x _</>, and $f_{i}(x)=\mathrm{e}^{-x}x^{l'_{i}+l_{i}+2}L_{n'_{i}}^{2l'_{i}+2}(x)L_{n_{i}}^{2l_{i}+2}(x)$ (i=1,2). The multiplier D _LM from the formula (5.30) is

$$\begin{aligned} D_{LM} =&(-1)^{m+m'_1+m'_2}\sqrt {{\frac{(2l'_1+1)(2l'_2+1)(2l_1+1)(2l_2+1)}{(2L+1)^2}} } \\ &{}\times C^{L0}_{l'_1 0,l_1 0} C^{L0}_{l'_2 0,l_2 0}C^{L-M}_{l'_1-m'_1,l_1 m_1} C^{LM}_{l'_2-m'_2,l_2 m_2}, \end{aligned}$$

(5.32)

where $C^{lm}_{l_{1}m_{1},l_{2}m_{2}}$—Clebsch-Gordan coefficients.

Substitution x _</> by x and y the integral (5.31) can be represented as

$$ R_L=\int_{ x>y}\int \mathrm{d}x\mathrm{d}yf_1(x)f_2(y)\frac{y^L}{x^{L+1}} + \int _{ x<y}\int \mathrm{d}x\mathrm{d}yf_1(x)\mathrm{d}yf_2(y) \frac{x^L}{y^{L+1}} , $$

(5.33)

where the integration over variables x,y is realized on the region [0,∞); integrals from the right part correspond to x>y and x<y cases, respectively. As the integration order in this relation is arbitrary, each of integrals from the right part can be taken in two variants [53–57]. For the first integral there is the expression

$$ \int_0^{\infty} \mathrm{d}xf_1(x)\int_0^{x}\mathrm{d}yf_2(y)\frac{y^L}{x^{L+1}}=\int_0^{\infty} \mathrm{d}xf_2(x)\int_x^{\infty} \mathrm{d}yf_1(y)\frac{x^L}{y^{L+1}} . $$

(5.34)

The second integral from the right part of (5.33) can be obtained from (5.34) by substitution f ₂ for f ₁ (and vice versa). Using the expressions with the internal integration over the interval x≤y≤∞, the integrals from (5.33) can be modified as

$$ R_L=\int_0^{\infty} \mathrm{d}xf_2(x)\int_x^{\infty} \mathrm{d}yf_1(y)\frac {x^L}{y^{L+1}} + \int_0^{\infty} \mathrm{d}xf_1(x)\int_x^{\infty} \mathrm{d}yf_2(y)\frac {x^L}{y^{L+1}} . $$

(5.35)

The expansion of Laguerre polynomials from (5.35) over powers of variables x and y permits to present R _L as

$$ R_L=\sum_{t=0}^{n_1+n'_1} \sum_{u=0}^{n_2+n'_2}W_t \bigl(n'_1,l'_1,n_1,l_1 \bigr) W_u\bigl(n'_2,l'_2,n_2,l_2 \bigr) [ H_{\alpha+t,\beta+u-\gamma}+H_{\beta+u,\alpha+t-\gamma} ] , $$

(5.36)

where $\alpha=l'_{1}+l_{1}+L+2$, $\beta=l'_{2}+l_{2}+L+2$, γ=2L+1 and H _a,b —integral in the form

$$ H_{a,b}=\int_0^{\infty}x^a \mathrm{e}^{-x}\varGamma(b+1,x)dx=\frac{\varGamma (a+b+2)}{(a+1)2^{a+b+2}}{}_2F_1(1,a+b+2,a+2;1/2) . $$

(5.37)

By means the integral $\int_{0}^{\infty}x^{a}\mathrm{e}^{-x}\gamma(b+1,x)\mathrm{d}x=\frac {\varGamma(a+b+2)}{(b+1)2^{a+b+2}}{}_{2}F_{1}(1,a+b+2,b+2;1/2)=H_{b,a}$ it can be shown that H _a,b =Γ(a+1)Γ(b+1)−H _b,a and H _a,a =(Γ(a+1))²/2. Putting the recurrence relation Γ(b+1,x)=bΓ(b,x)+x ^be^−x into integral (5.37), one obtains

$$ H_{a,b}=bH_{a,b-1}+\frac{\varGamma(a+b+1)}{2^{a+b+1}} . $$

(5.38)

To evaluate the H _a,b it is useful to introduce the new formula $h_{a,b}=H_{a,b}\frac{2^{a+b+1}}{\varGamma(a+b+1)}$ (h _0,0=1), which allows to rewrite recursion (5.38) as $h_{a,b}=\frac{2b}{a+b}h_{a,b-1}+1$.

The numerical tests demonstrated, that the evaluation of the radial integral by formulas (5.36)–(5.38) gives the accuracy up to 12th significant digits for the basis functions with $\max(n'_{1} ,n_{1} ,n'_{2} ,n_{2}) \le7$, if the 64th bits representation of real numbers is used. The 128th bit representation allows to preserve the same accuracy level for the basis functions with $\max(n'_{1} ,n_{1} ,n'_{2} ,n_{2}) \le10$. The check of the two-indexes recursion, which is analogous to the (5.22) [51], showed that this one provides a good precision only for integrals with $\max (n'_{1},n_{1},n'_{2},n_{2}) \le13$ when 128th bit representation of real numbers is used. This is in accord with the remark in [51] about the possibility of using this recursion for integral calculation with a good accuracy $\max(n'_{1},n_{1},n'_{2},n_{2}) \le14$. As it was not known beforehand if in constructing a wave function one can select polynomials with n≤13 only, the combined scheme was used. Calculations have been done with the formulas (5.36)–(5.38), if $\max(n'_{1},n_{1},n'_{2},n_{2}) < 11$ and the quadrature integration being used in the opposite case. Using the replacement of u=x and v=y−x, similar to [57], the integral (5.35) can be represented as

$$\begin{aligned} R_L =&\int_0^{\infty} \mathrm{d}uf_1(u)\int_0^{\infty} \mathrm{d}vf_2(u+v)\frac {u^L}{(u+v)^{L+1}} \\ &{}+ \int_0^{\infty} \mathrm{d}uf_2(u)\int _0^{\infty} \mathrm{d}vf_1(u+v)\frac{u^L}{(u+v)^{L+1}} . \end{aligned}$$

(5.39)

Thus, it is possible to use Gauss-Laguerre integration for calculation (5.39). Note, that the relation as $\sum_{m=0}^{n}L_{m}^{\alpha}(x)L_{n-m}^{\beta}(y)=L_{n}^{\alpha+\beta +1}(x+y)$ [41] allows to factorize the two-dimensional integral completely.