Variational Methods in the Quantum Mechanical Three-Body Problem with a Coulomb Interaction

Abstract

The variational Rayleigh–Ritz method for bound states in nonrelativistic quantum mechanics is formulated and the mathematical foundations of the method are discussed. A review of the most frequently used methods for constructing the Ritz variational basis is given on the example of the helium atom. Numerous applications of variational methods to solving the quantum mechanical three-body bound state problem are considered, and a comparison of the most accurate calculations for various physical systems is given. As a generalization of the variational Rayleigh–Ritz method to “quasi-bound” states (resonances), the complex coordinate rotation (CCR) method and its application to calculating various characteristics of resonances are considered.

Appendices

APPENDIX A

ANALYTICAL CALCULATION OF MATRIX ELEMENTS FOR HYLLERAAS VARIATIONAL EXPANSION

The presentation follows [77].

A1. Basic Recurrence Relation

To calculate integrals of radial variables, we should be able to calculate

$${{\Gamma }_{{lm}}}(\alpha ,\beta ) = {{\left( { - \frac{\partial }{{\partial \alpha }}} \right)}^{l}}{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{m}}\frac{{f(\alpha ,\beta )}}{{\alpha + \beta }}.$$

(A.1)

By Leibniz’s theorem, we have

$$\begin{gathered} {{\Gamma }_{{lm}}}(\alpha ,\beta ) = \sum\limits_{\begin{subarray}{l} \lambda + \lambda {\kern 1pt} ' = l \\ \mu + \mu \,' = m \end{subarray}} {\frac{{l!}}{{\lambda !\lambda {\kern 1pt} '!}}} \frac{{m!}}{{\mu !\mu {\kern 1pt} '!}}\frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} ')!}}{{{{{(\alpha + \beta )}}^{{\lambda {\kern 1pt} '\, + \mu {\kern 1pt} '\,\, + 1}}}}} \\ \times \,\,{{\left( { - \frac{\partial }{{\partial \alpha }}} \right)}^{\lambda }}{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{\mu }}f(\alpha ,\beta ). \\ \end{gathered} $$

Further, using the recurrence relation for binomial coefficients,

$$\begin{gathered} \frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} ')!}}{{\lambda {\kern 1pt} '!\mu {\kern 1pt} '!}} = C_{{\lambda {\kern 1pt} '\, + \,\mu {\kern 1pt} '\,\, - 1}}^{{\lambda {\kern 1pt} '}} \\ + \,\,C_{{\lambda {\kern 1pt} '\, + \,\mu {\kern 1pt} '\, - 1}}^{{\lambda {\kern 1pt} '\,\, - 1}} = \frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} '\,\, - 1)!}}{{\lambda {\kern 1pt} '!(\mu {\kern 1pt} '\,\, - 1)!}} + \frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} '\,\, - 1)!}}{{(\lambda {\kern 1pt} '\,\, - 1)!\mu {\kern 1pt} '!}}, \\ \end{gathered} $$

we obtain

$$\begin{gathered} {{\Gamma }_{{lm}}} = \sum\limits_{\begin{subarray}{l} \lambda + \lambda {\kern 1pt} ' = l \\ \mu + \mu {\kern 1pt} ' = m \end{subarray}} {\left[ {\frac{{l!}}{{\lambda !(\lambda {\kern 1pt} '\,\, - 1)!}}\frac{{m!}}{{\mu !\mu {\kern 1pt} '!}}\frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} '\,\, - 1)!}}{{{{{(\alpha + \beta )}}^{{\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} ' + 1}}}}}} \right.} \\ \left. { + \,\,\frac{{l!}}{{\lambda !\lambda {\kern 1pt} '!}}\frac{{m!}}{{\mu !(\mu {\kern 1pt} '\,\, - 1)!}}\frac{{(\lambda {\kern 1pt} '\,\, + \mu {\kern 1pt} '\,\, - 1)!}}{{{{{(\alpha + \beta )}}^{{\lambda {\kern 1pt} '\,\, + \,\,\mu {\kern 1pt} ' + 1}}}}}} \right] \\ \times \,\,{{\left( { - \frac{\partial }{{\partial \alpha }}} \right)}^{\lambda }}{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{\mu }}f(\alpha ,\beta ), \\ \end{gathered} $$

whence, applying Leibniz’s theorem again, we arrive at the final expression:

$$\begin{gathered} {{\Gamma }_{{lm}}}(\alpha ,\beta ) = \frac{1}{{\alpha + \beta }} \\ \times \,\,\left[ {l{{\Gamma }_{{l - 1,m}}} + m{{\Gamma }_{{l,m - 1}}} + {{{\left( { - \frac{\partial }{{\partial \alpha }}} \right)}}^{l}}{{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}}^{m}}f(\alpha ,\beta )} \right]. \\ \end{gathered} $$

(A.2)

A2. Singular Integrals of the Form $( - 1,m,n)$

A generating function similar to (17) is obtained by direct integration analytically:

$$\begin{gathered} {{\Gamma }_{{ - 1,00}}}\alpha ,\beta ,\gamma ) = \int {\int {\frac{1}{{{{r}_{1}}}}} } {{e}^{{ - \alpha {{r}_{1}} - \beta {{r}_{2}} - \gamma {{r}_{{12}}}}}}d{{r}_{1}}d{{r}_{2}}d{{r}_{{12}}} \\ = \int\limits_0^\infty {d{{r}_{1}}} \left\{ {\left( {\int\limits_0^{{{r}_{1}}} {d{{r}_{2}}} \int\limits_{{{r}_{1}} - {{r}_{2}}}^{{{r}_{1}} + {{r}_{2}}} {d{{r}_{{12}}}} + \int\limits_{{{r}_{1}}}^\infty {d{{r}_{2}}} \int\limits_{{{r}_{2}} - {{r}_{1}}}^{{{r}_{1}} + {{r}_{2}}} {d{{r}_{{12}}}} } \right)\frac{{{{e}^{{ - \alpha {{r}_{1}} - \beta {{r}_{2}} - \gamma {{r}_{{12}}}}}}}}{{{{r}_{1}}}}} \right\} \\ = \frac{{\ln(\alpha + \beta ) - \ln(\alpha + \gamma )}}{{(\beta - \gamma )(\beta + \gamma )}}. \\ \end{gathered} $$

(A.3)

Using (A.3), other integrals of the form $( - 1,m,n)$ can be obtained by differentiation:

$$\begin{gathered} {{\Gamma }_{{ - 1,mn}}} = {{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{m}}{{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{n}}{{\Gamma }_{{ - 1,00}}} \\ = {{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{m}}{{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{n}}\frac{{\ln(\alpha + \beta ) - \ln(\alpha + \gamma )}}{{(\beta - \gamma )(\beta + \gamma )}}. \\ \end{gathered} $$

(A.4)

Unfortunately, the method used to generate regular functions ${{\Gamma }_{{lmn}}}(\alpha ,\beta ,\gamma )$ cannot be used directly in this case. The factor ${1 \mathord{\left/ {\vphantom {1 {(\beta - \gamma )}}} \right. \kern-0em} {(\beta - \gamma )}}$ makes the recursion unstable.

To overcome this difficulty, we rewrite expression (A.4), assuming that $\beta > \gamma$, in the form

$$\begin{gathered} {{\Gamma }_{{ - 1,mn}}} = {{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{m}}{{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{n}} \\ \times \,\,\left\{ {\frac{1}{{(\alpha + \gamma )(\beta + \gamma )}}\left[ {{{{\left( {\frac{{\beta - \gamma }}{{\alpha + \gamma }}} \right)}}^{{ - 1}}}\ln\left( {1 + \frac{{\beta - \gamma }}{{\alpha + \gamma }}} \right)} \right]} \right\}, \\ \end{gathered} $$

(A.5)

and introduce a new variable $u = {{(\beta - \gamma )} \mathord{\left/ {\vphantom {{(\beta - \gamma )} {(\alpha + \gamma )}}} \right. \kern-0em} {(\alpha + \gamma )}}$; then, the following recurrent scheme can be suggested:

$$\begin{array}{*{20}{c}} {{{\Gamma }_{{ - 1,mn}}} = \frac{1}{{\beta + \gamma }}\left[ {m{{\Gamma }_{{ - 1,m - 1,n}}} + n{{\Gamma }_{{ - 1,m,n - 1}}} + {{D}_{{mn}}}} \right],} \\ {{{D}_{{mn}}} = {{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}}^{m}}{{{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}}^{n}}\frac{1}{{(\alpha + \gamma )}}\left[ {{{u}^{{ - 1}}}\ln(1 + u)} \right],} \\ {{{D}_{{mn}}} = \frac{1}{{\alpha + \gamma }}\left[ {n{{D}_{{m,n - 1}}} + {{C}_{{mn}}}} \right],} \\ {{{C}_{{mn}}} = {{{\left( { - \frac{\partial }{{\partial \beta }}} \right)}}^{m}}{{{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}}^{n}}\left[ {{{u}^{{ - 1}}}\ln(1 + u)} \right].} \end{array}$$

(A.6)

To generate the array ${{C}_{{mn}}}$, it is convenient to first differentiate with respect to $\gamma$, which is given by the recurrence relation,

$${{C}_{{mn}}} = \frac{1}{{\alpha + \gamma }}[(m + n - 1){{C}_{{m,n - 1}}} - (\alpha + \beta ){{C}_{{m + 1,n - 1}}}]{\kern 1pt} ,$$

which requires filling a triangular array $(m,n)$ with $m,n = 1, \ldots ,M + N$ and $m + n \leqslant M + N$. The first layer is specified as

$${{C}_{{m0}}} = {{\left( { - \frac{\partial }{{\partial \beta }}} \right)}^{m}}\left[ {{{u}^{{ - 1}}}\ln(1 + u)} \right] = \frac{{{{g}_{m}}(u)}}{{{{{(\alpha + \gamma )}}^{m}}}}.$$

Finally, we will define the recursion required to compute a one-dimensional array of functions ${{g}_{m}}$:

$$\begin{gathered} {{g}_{m}}(u) = {{\left( { - \frac{d}{{du}}} \right)}^{m}}\left[ {{{u}^{{ - 1}}}\ln(1 + u)} \right], \\ {{g}_{m}} = \frac{m}{u}{{g}_{{m - 1}}} + \frac{1}{u}{{f}_{m}}{\kern 1pt} , \\ {{f}_{m}}(u) = {{\left( { - \frac{d}{{du}}} \right)}^{m}}\ln(1 + u) = - \frac{{(m - 1)!}}{{{{{(1 + u)}}^{m}}}}, \\ \end{gathered} $$

(A.7)

which completes the construction of a recurrent scheme for the analytical calculation of integrals of the form $( - 1,m,n)$. However, when $u$ is small, forward recursion (A.7) becomes unstable. In this case, it is better to use recursion in the opposite direction:

$${{g}_{{m - 1}}} = \frac{1}{m}\left( {u{{g}_{m}} - {{f}_{m}}} \right),\,\,\,m = M,M - 1, \ldots ,$$

(A.7a)

where $M$ is chosen in such a way that, if ${{g}_{M}}$ is set to zero, then the error introduced by the value of ${{g}_{M}}$ at this step has decreased to machine epsilon during the backward recursion, by the time when the required value of $m$ is reached. In particular, when $u = 0$ (or $\beta = \gamma$, which is typical of physical systems containing identical particles), we obtain

$${{\Gamma }_{{ - 1,00}}}(\alpha ,\beta ,\gamma ) = \frac{1}{{(\alpha + \beta )\beta }} = \frac{1}{{(\alpha + \gamma )\gamma }},$$

which allows us to get rid of the indeterminate forms ${0 \mathord{\left/ {\vphantom {0 0}} \right. \kern-0em} 0}$ during the calculation.

A3. Singular Integrals of the Form $( - 1, - 1,n)$

The next step is to calculate singular integrals of the form ${{\Gamma }_{{ - 1, - 1,n}}}(\alpha ,\beta ,\gamma )$, which are required, e.g., for obtaining the mean value of the operator ${{{\mathbf{p}}}^{4}}$. Let us start again with the generating function

$$\begin{gathered} \begin{array}{*{20}{c}} {{{\Gamma }_{{ - 1, - 1,0}}}(\alpha ,\beta ,\gamma ) = \frac{1}{{4\gamma }}{{{\left\{ {\left[ {\ln \left( {\frac{{\alpha + \gamma }}{{\beta + \gamma }}} \right)} \right]} \right.}}^{2}}} \end{array} \\ + \,\,2\left. {\left[ {{\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\alpha + \gamma }}} \right) + {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right)} \right] + \frac{{{{\pi }^{2}}}}{3}} \right\}. \\ \end{gathered} $$

(A.8)

Here, dilog(x) is the dilogarithm function as defined in the Abramowitz and Stegun reference book [78] (see Appendix B). Next, the computational scheme is defined as follows:

$$\begin{gathered} {{\Gamma }_{{ - 1, - 1,n}}}(\alpha ,\beta ,\gamma ) = {{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{n}}{{\Gamma }_{{ - 1, - 1,0}}}(\alpha ,\beta ,\gamma ) \\ = {{\gamma }^{{ - 1}}}\left[ {n{{\Gamma }_{{ - 1, - 1,n - 1}}} - {{B}_{n}}} \right], \\ \end{gathered} $$

(A.9)

where

$$\begin{gathered} {{B}_{n}} = - {{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{n}}\frac{1}{4}\left\{ {{{{\left[ {\ln\left( {\frac{{\alpha + \gamma }}{{\beta + \gamma }}} \right)} \right]}}^{2}}} \right. \\ + \,\,2\left. {\left[ {{\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\alpha + \gamma }}} \right) + {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right)} \right] + \frac{{{{\pi }^{2}}}}{3}} \right\} \\ = {{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{{n - 1}}}\left[ {\frac{\alpha }{{{{\alpha }^{2}} - {{\gamma }^{2}}}}\ln\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right)} \right. \\ + \,\,\left. {\frac{\beta }{{{{\beta }^{2}} - {{\gamma }^{2}}}}\ln\left( {\frac{{\alpha + \beta }}{{\alpha + \gamma }}} \right)} \right] \\ = {{\left( { - \frac{\partial }{{\partial \gamma }}} \right)}^{{n - 1}}}\left\{ {\frac{\alpha }{{(\alpha + \gamma )(\beta + \gamma )}}[{{v}^{{ - 1}}}\ln(1 + v)]} \right. \\ + \,\,\left. {\frac{\beta }{{(\alpha + \gamma )(\beta + \gamma )}}[{{u}^{{ - 1}}}\ln(1 + u)]} \right\}, \\ \end{gathered} $$

(A.9a)

where $u = {{(\beta - \gamma )} \mathord{\left/ {\vphantom {{(\beta - \gamma )} {(\alpha + \gamma )}}} \right. \kern-0em} {(\alpha + \gamma )}}$ and $v = {{(\alpha - \gamma )} \mathord{\left/ {\vphantom {{(\alpha - \gamma )} {(\beta + \gamma )}}} \right. \kern-0em} {(\beta + \gamma )}}$. The final expression, written in the last line of Eq. (A.9), gives a stable recurrent relation if only the function ${{g}_{0}}(u) = {{u}^{{ - 1}}}\ln(1 + u)$ and its derivatives are calculated in the same way as in the previous section.

A4. Other Singular Integrals

Singular integrals of the form ${{\Gamma }_{{ - 2,m,n}}}(\alpha ,\beta ,\gamma )$ are not integrable, but they usually only occur in combinations such as ${{\Gamma }_{{ - 2,m + 2,n}}} - {{\Gamma }_{{ - 2,m,n + 2}}}$, which have finite values. In this situation, it is required to find such a generating function that would give the correct value for the combinations representing the integrands. In this section, we present expressions only for generating functions ${{\Gamma }_{{ - 2,m,n}}}(\alpha ,\beta ,\gamma )$ and other singular integrals in a form suitable for calculations:

$$\begin{gathered} {{\Gamma }_{{ - 2,00}}} = - \frac{{(\alpha + \beta )\ln(\alpha + \beta ) - (\alpha + \gamma )\ln(\alpha + \gamma )}}{{{{\beta }^{2}} - {{\gamma }^{2}}}} \\ = - \left[ {\frac{\alpha }{{\beta + \gamma }} + \frac{1}{2}} \right]\frac{{\ln(\alpha + \beta ) - \ln(\alpha + \gamma )}}{{\beta - \gamma }} \\ - \,\,\frac{{\ln(\alpha + \beta ) + \ln(\alpha + \gamma )}}{{2(\beta + \gamma )}}. \\ \end{gathered} $$

(A.10a)

$$\begin{gathered} {{\Gamma }_{{ - 3,00}}} = - \frac{\alpha }{{2(\beta + \gamma )}} + \left[ {\frac{{{{\alpha }^{2}}}}{{2(\beta + \gamma )}} + \frac{\alpha }{2} + \frac{{{{\beta }^{2}} + {{\gamma }^{2}}}}{{4(\beta + \gamma )}}} \right] \\ \times \,\,\frac{{\ln(\alpha + \beta ) - \ln(\alpha + \gamma )}}{{\beta - \gamma }} + \left[ {\frac{\alpha }{{2(\beta + \gamma )}} + \frac{1}{4}} \right] \\ \times \,\,\left[ {\ln(\alpha + \beta ) + \ln(\alpha + \gamma )} \right]. \\ \end{gathered} $$

(A.10b)

$$\begin{gathered} {{\Gamma }_{{ - 4,00}}} = \frac{{5{{\alpha }^{2}}}}{{12(\beta + \gamma )}} + \frac{1}{6}\alpha - \frac{1}{{12}}\left[ {\frac{{2{{\alpha }^{3}}}}{{\beta + \gamma }}} \right. \\ + \left. {3{{\alpha }^{2}} + 3\alpha \frac{{{{\beta }^{2}} + {{\gamma }^{2}}}}{{\beta + \gamma }} + ({{\beta }^{2}} - \beta \gamma + {{\gamma }^{2}})} \right] \\ \times \,\,\frac{{\ln(\alpha + \beta ) - \ln(\alpha + \gamma )}}{{\beta - \gamma }} - \frac{1}{{12}} \\ \times \,\,\left[ {\frac{{3{{\alpha }^{2}}}}{{\beta + \gamma }} + 3\alpha + \frac{{{{\beta }^{2}} + \beta \gamma + {{\gamma }^{2}}}}{{\beta + \gamma }}} \right] \\ \times \,\,\left[ {\ln(\alpha + \beta ) + \ln(\alpha + \gamma )} \right]. \\ \end{gathered} $$

(A.10c)

$$\begin{array}{*{20}{c}} \begin{gathered} {{\Gamma }_{{ - 2, - 1,0}}} = \frac{1}{2}l{{n}^{2}}(\beta + \gamma ) + {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right) \hfill \\ - \,\,\frac{{\left( {\alpha + \gamma } \right)}}{{4\gamma }}\left\{ {{{{\left[ {\ln\left( {\frac{{\alpha + \gamma }}{{\beta + \gamma }}} \right)} \right]}}^{2}}} \right. \hfill \\ \end{gathered} \\ { + \,\,2\left. {\left[ {{\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\alpha + \gamma }}} \right) + {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right)} \right] + \frac{{{{\pi }^{2}}}}{3}} \right\}.} \end{array}$$

(A.11a)

$$\begin{gathered} {{\Gamma }_{{ - 3, - 1,0}}} = \frac{{(\alpha + \beta )[1 - \ln(\alpha + \beta )]}}{2} \\ + \,\,\frac{{\alpha \ln(\beta + \gamma )[1 - \ln(\beta + \gamma )]}}{2} \\ - \,\,\alpha {\kern 1pt} {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right) + \frac{{{{{\left( {\alpha + \gamma } \right)}}^{2}}}}{{8\gamma }}\left\{ {\mathop {\left[ {\ln\left( {\frac{{\alpha + \gamma }}{{\beta + \gamma }}} \right)} \right]}\nolimits^2 } \right. \\ + \,\,2\left[ {{\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\alpha + \gamma }}} \right) + {\text{dilog}}\left( {\frac{{\alpha + \beta }}{{\beta + \gamma }}} \right)} \right] + \left. {\frac{{{{\pi }^{2}}}}{3}} \right\}. \\ \end{gathered} $$

(A.11b)

The last two equations are used in the calculation of the Bethe logarithm when calculating the kinetic energy operator for a basis that includes singular terms of the form $r_{i}^{{ - 1}}{{e}^{{ - {{\alpha }_{i}}{{r}_{1}} - {{\beta }_{i}}{{r}_{2}} - {{\gamma }_{i}}{{r}_{{12}}}}}}$ in expansion (22).

To get rid of polynomials in $\alpha$, $\beta$, and $\gamma$ in the numerator of complex expressions, one can use the following recursion:

$$\begin{gathered} {{( - {{\partial }_{\gamma }})}^{n}}[(\alpha + \gamma )f(\gamma )] \\ = - n{{( - {{\partial }_{\gamma }})}^{{n - 1}}}f(\gamma ) + (\alpha + \gamma ){{( - {{\partial }_{\gamma }})}^{n}}f(\gamma ). \\ \end{gathered} $$

When tabulating ${{\Gamma }_{{lmn}}}$ in the required range of integer parameters $(l,m,n)$, one can then use the general scheme described in [27, 28] to calculate the matrix elements for states with an arbitrary total angular momentum.

APPENDIX B

DILOGARITHM

In this appendix, a convenient way to calculate the dilogarithm function with high precision is presented.

For the dilogarithm function, we use the definition from the Abramowitz and Stegun reference book [78]:

$$\begin{gathered} {\text{dilog}}(x) = \int\limits_1^x {\frac{{\ln(t)}}{{1 - t}}} dt = \sum\limits_{k = 1}^\infty {\frac{{{{{(1 - x)}}^{k}}}}{{{{k}^{2}}}}} , \\ {\text{dilog}}{\kern 1pt} {\text{'}}{\kern 1pt} (x) = \frac{{\ln x}}{{1 - x}}. \\ \end{gathered} $$

(B.1)

The dilogarithm satisfies the functional relations

$$\begin{array}{*{20}{c}} {{\text{dilog}}(x) + {\text{dilog}}(1 - x) = - \ln x\ln(1 - x) + \frac{{{{\pi }^{2}}}}{6},} \\ {{\text{dilog}}(x) + {\text{dilog}}({{x}^{{ - 1}}}) = - {{{{{(\ln x)}}^{2}}} \mathord{\left/ {\vphantom {{{{{(\ln x)}}^{2}}} 2}} \right. \kern-0em} 2},} \end{array}$$

(B.2)

which allow one to define the function on the entire complex plane, while it is already defined on the set $D = \{ x:\left| {x - 1} \right| \leqslant 1,\,\,\operatorname{Re} (x) \geqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}\}$. Next, we use the relation of the dilogarithm with the Debye function:

$${\text{dilog}}({{e}^{{ - z}}}) = \int\limits_0^z {\frac{{tdt}}{{{{e}^{t}} - 1}}} ,$$

(B.3)

when the set $D$ is mapped into $\left| z \right| < 2\pi$. Finally, the Debye function can be calculated on $\left| z \right| < 2\pi$ using a rapidly converging power series:

$$\int\limits_0^z {\frac{{{{t}^{n}}dt}}{{{{e}^{t}} - 1}}} = {{z}^{n}}\left[ {\frac{1}{n} - \frac{z}{{2(n + 1)}} + \sum\limits_{k = 1}^\infty \,\frac{{{{B}_{{2k}}}{{z}^{{2k}}}}}{{(2k + n)(2k)!}}} \right],$$

(B.4)

where ${{B}_{{2k}}}$ are the Bernoulli numbers.