Computational Spectroscopy in Solution: Methods and Models for Investigating Complex Systems

Abstract

In this contribution, some issues related to the interpretation, simulation and modelling of solvent effects on the absorption and emission spectra of organic dyes are presented and discussed. First, a brief analysis of the physical basis of solvent effects on the electronic transitions is reported, in order to introduce the most important phenomena and quantities tuning the so-called solvatochromic shifts. This is followed by a general discussion of the most common models employed for the interpretation, simulation and prediction of such effects. A general and effective multilayer scheme is analyzed in some detail, which has been developed in the past years and is known to provide—in most cases—quantitative predictions of the spectral features of solvated molecules. Afterwards, starting from this general model, some approximations are introduced, leading to simplified and cost effective analytical schemes. In order to sketch a more complete perspective of the models still used by spectroscopists, phenomenological methods are critically discussed. Finally, broadening of spectral lines by both symmetric (solvent relaxation) and possibly asymmetric (vibronic) contributions is shortly analysed. In all cases, the theoretical bases of the methods, as well as practical applications and test cases are given, in order to clarify the most interesting aspects of all the discussed models.

Appendices

Appendix A: Spherical Cavity Embedded By One or More Concentric Dielectric Shells Surrounded By the Bulk

In the following, we show a natural extension of the spherical model for concentric dielectric continua, and in particular the case of a cavity embedded by one concentric dielectric shell surrounded by the bulk (see Fig. A.1a), whose solution is due to Beveridge and Schnuelle [87], and the case of two or more concentric dielectric shells surrounded by the bulk (see Fig. A.1b, c), which we propose here.

Fig. A.1

Spherical models of concentric dielectric media

17.1.1 A.1 Spherical Cavity Embedded by One Concentric Dielectric Shell Surrounded by the Bulk

We consider now the charge distribution inside a spherical cavity of radius $r = a_{C} = {\text{constant}}$ and dielectric constant $\varepsilon_{C}$, with a first shell of polarisable dielectric, extending from $r = a_{C}$ to $r = a_{1} = {\text{constant}}$, characterised by dielectric constant $\varepsilon_{1}$. Beyond this shell, there is the bulk region, from $r = a_{1}$ to infinity, with dielectric constant $\varepsilon_{B}$. The solution of Laplace’s Eq. (3.10) is the same of the single spherical cavity case, but it is needed to introduce coefficients so that the expressions for the potential in the three regions (viz. cavity, shell, and bulk) are:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {A_{nm}^{C} + A_{nm}^{\left( 1 \right)} } \right)r^{n} + B_{nm}^{C} r^{{ - \left( {n + 1} \right)}} } \right]P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad r < a_{C} , $$

(A.1)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {A_{nm}^{\left( 1 \right)} r^{n} + B_{nm}^{\left( 1 \right)} r^{{ - \left( {n + 1} \right)}} } \right]P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad a_{C} < r < a_{1} , $$

(A.2)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {B_{nm}^{B} r^{{ - \left( {n + 1} \right)}} P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad r > a_{1} . $$

(A.3)

The Reaction Potential,

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {A_{nm}^{C} + A_{nm}^{\left( 1 \right)} } \right)r^{n} P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad r < a_{C} , $$

(A.4)

contains now two contributions, due to the polarisation of the local dielectric due to the cavity ($A_{nm}^{C}$) and the polarisation of the bulk dielectric due to the shell ($A_{nm}^{\left( 1 \right)}$). Concerning the boundary conditions, we have two per each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{r = a_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{r = a_{C}^{ + } }} , $$

(A.5a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial r}} \right)_{{r = a_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial r}} \right)_{{r = a_{C}^{ + } }} , $$

(A.5b)

$$ \left( {\varPhi_{1} } \right)_{{r = a_{1}^{ - } }} = \left( {\varPhi_{B} } \right)_{{r = a_{1}^{ + } }} , $$

(A.5c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial r}} \right)_{{r = a_{1}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial r}} \right)_{{r = a_{1}^{ + } }} , $$

(A.5d)

or, explicitly:

$$ \varepsilon_{C}^{ - 1} \left[ {\left( {A_{nm}^{C} + A_{nm}^{\left( 1 \right)} } \right)a_{C}^{n} + B_{nm}^{C} a_{C}^{{ - \left( {n + 1} \right)}} } \right] = \varepsilon_{1}^{ - 1} \left[ {A_{nm}^{\left( 1 \right)} a_{C}^{n} + B_{nm}^{\left( 1 \right)} a_{C}^{{ - \left( {n + 1} \right)}} } \right], $$

(A.6a)

$$ nA_{nm}^{C} a_{C}^{n - 1} - \left( {n + 1} \right)B_{nm}^{C} a_{C}^{{ - \left( {n + 2} \right)}} = - \left( {n + 1} \right)B_{nm}^{\left( 1 \right)} a_{C}^{{ - \left( {n + 2} \right)}} , $$

(A.6b)

$$ \varepsilon_{1}^{ - 1} \left[ {A_{nm}^{\left( 1 \right)} a_{1}^{n} + B_{nm}^{\left( 1 \right)} a_{1}^{{ - \left( {n + 1} \right)}} } \right] = \varepsilon_{B}^{ - 1} B_{nm}^{B} a_{1}^{{ - \left( {n + 1} \right)}} , $$

(A.6c)

$$ nA_{nm}^{\left( 1 \right)} a_{1}^{n - 1} - \left( {n + 1} \right)B_{nm}^{\left( 1 \right)} a_{1}^{{ - \left( {n + 2} \right)}} = - \left( {n + 1} \right)B_{nm}^{B} a_{1}^{{ - \left( {n + 2} \right)}} . $$

(A.6d)

Eliminating $B_{nm}^{\left( 1 \right)}$ from (A.6a) and (A.6b), and $B_{nm}^{B}$ from (A.6c) and (A.6d), we obtain:

$$ A_{nm}^{C} = \frac{{\left( {n + 1} \right)\left( {1 - \varepsilon^{\prime}_{{C^{*} }} } \right)}}{{\left( {n + 1} \right)\varepsilon^{\prime}_{{C^{*} }} + n}}\frac{{B_{nm}^{C} }}{{a_{C}^{2n + 1} }}, $$

(A.7)

$$ A_{nm}^{\left( 1 \right)} = \frac{{\left( {n + 1} \right)\left( {1 - \varepsilon_{{1^{*} }} } \right)}}{{\left( {n + 1} \right)\varepsilon_{{1^{*} }} + n}}\frac{{B_{nm}^{\left( 1 \right)} }}{{a_{1}^{2n + 1} }}, $$

(A.8)

$$ \begin{aligned} B_{nm}^{\left( 1 \right)} & = \varepsilon^{\prime}_{{C^{*} }} \left[ {A_{nm}^{\left( 1 \right)} a_{C}^{2n + 1} + B_{nm}^{C} } \right] \\ & \quad = \left[ {1 - \frac{n}{n + 1}g^{\prime}_{n} \left( {\varepsilon^{\prime}_{{C^{*} }} } \right)} \right]B_{nm}^{C} , \\ \end{aligned} $$

(A.9)

where:

$$ \varepsilon^{\prime}_{{C^{*} }} = \varepsilon_{{C^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{C^{*} }} } \right)g^{\prime}_{n} \left( {\varepsilon_{{1^{*} }} } \right)\left( {\frac{{a_{C} }}{{a_{1} }}} \right)^{2n + 1} } \right]^{ - 1}, $$

(A.10)

$$ g^{\prime}_{n} \left( \varepsilon \right) \equiv \frac{1 - \varepsilon }{{\frac{n}{n + 1} + \varepsilon }}, $$

(A.11)

and $\varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C}$, $\varepsilon_{{1^{*} }} \equiv \varepsilon_{B} /\varepsilon_{1}$. Finally, Helmholtz’s free energy reads:

$$ A = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\frac{{Q^{\prime}_{n} }}{{a_{C}^{2n + 1} }}} , $$

(A.12)

$$ Q^{\prime}_{n} \equiv \left\{ {g^{\prime}_{n} \left( {\varepsilon^{\prime}_{{C^{*} }} } \right) + \left[ {1 - \frac{n}{n + 1}g^{\prime}_{n} \left( {\varepsilon^{\prime}_{{C^{*} }} } \right)} \right]g^{\prime}_{n} \left( {\varepsilon_{{1^{*} }} } \right)\left( {\frac{{a_{C} }}{{a_{1} }}} \right)^{2n + 1} } \right\}Q_{n} . $$

(A.13)

17.1.2 A.2 Spherical Cavity Embedded by Two Concentric Dielectric Shells Surrounded by the Bulk

If we consider two spherical concentric dielectric shells, defined by the conditions:

$$ r = a_{C} = {\text{constant,}} $$

(A.14a)

$$ r = a_{1} = {\text{constant}} > a_{C} , $$

(A.14b)

$$ r = a_{2} = {\text{constant}} > a_{1} , $$

(A.14c)

we have now four different dielectric regions, viz. the cavity (with dielectric constant $\varepsilon_{C}$), the inner shell (with dielectric constant $\varepsilon_{1}$), the outer shell (with dielectric constant $\varepsilon_{2}$), and the bulk (with dielectric constant $\varepsilon_{B}$). The electric potential in the four regions is expressed by:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {A_{nm}^{C} + A_{nm}^{\left( 1 \right)} + A_{nm}^{\left( 2 \right)} } \right)r^{n} + B_{nm}^{C} r^{{ - \left( {n + 1} \right)}} } \right]P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad r < a_{C} , $$

(A.15)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {A_{nm}^{\left( 1 \right)} + A_{nm}^{\left( 2 \right)} } \right)r^{n} + B_{nm}^{\left( 1 \right)} r^{{ - \left( {n + 1} \right)}} } \right]P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad a_{C} < r < a_{1} , $$

(A.16)

$$ \varPhi_{2} = \varepsilon_{2}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {A_{nm}^{\left( 2 \right)} r^{n} + B_{nm}^{\left( 2 \right)} r^{{ - \left( {n + 1} \right)}} } \right]P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad a_{1} < r < a_{2} , $$

(A.17)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {B_{nm}^{B} r^{{ - \left( {n + 1} \right)}} P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } .\quad r > a_{2} . $$

(A.18)

The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {A_{nm}^{C} + A_{nm}^{\left( 1 \right)} + A_{nm}^{\left( 2 \right)} } \right)r^{n} P_{n}^{m} \left( {\cos \theta } \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad r < a_{C} , $$

(A.19)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{r = a_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{r = a_{C}^{ + } }} , $$

(A.20a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial r}} \right)_{{r = a_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial r}} \right)_{{r = a_{C}^{ + } }} , $$

(A.20b)

$$ \left( {\varPhi_{1} } \right)_{{r = a_{1}^{ - } }} = \left( {\varPhi_{2} } \right)_{{r = a_{1}^{ + } }} , $$

(A.20c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial r}} \right)_{{r = a_{1}^{ - } }} = \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial r}} \right)_{{r = a_{1}^{ + } }} , $$

(A.20d)

$$ \left( {\varPhi_{2} } \right)_{{r = a_{2}^{ - } }} = \left( {\varPhi_{B} } \right)_{{r = a_{2}^{ + } }} , $$

(A.20e)

$$ \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial r}} \right)_{{r = a_{2}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial r}} \right)_{{r = a_{2}^{ + } }} , $$

(A.20f)

which enable to reach the final formulae for Helmholtz’s free energy:

$$ A = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {g_{n} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} } \right)\frac{{Q^{\prime\prime}_{n} }}{{a_{C}^{2n + 1} }}} , $$

(A.21)

where:

$$ Q^{\prime\prime}_{n} \equiv \sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} r_{k}^{n} r_{l}^{n} \sum\limits_{m} {g^{\prime\prime}_{nm} \frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}P_{n}^{m} \left( {\cos \theta_{k} } \right)P_{n}^{m} \left( {\cos \theta_{l} } \right){\text{e}}^{{{\text{i}}m\left( {\phi_{l} - \phi_{k} } \right)}} } } } , $$

(A.22)

$$ \left\{ \begin{aligned} \varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C} \hfill \\ \varepsilon_{{1^{*} }} \equiv \varepsilon_{2} /\varepsilon_{1} \hfill \\ \varepsilon_{{2^{*} }} \equiv \varepsilon_{B} /\varepsilon_{2} \hfill \\ \end{aligned} \right., $$

(A.23)

$$ \left\{ \begin{aligned} \varepsilon^{\prime}_{{C^{*} }} \equiv \varepsilon_{{C^{*} }} \left[ {1 + g_{n} \left( {\varepsilon^{\prime}_{{1^{*} }} } \right)\frac{{1 - \varepsilon_{{C^{*} }} }}{{\left( {n + m} \right)!}}\left( {\frac{{a_{C} }}{{a_{1} }}} \right)^{2n + 1} } \right]^{ - 1} \hfill \\ \varepsilon^{\prime}_{{1^{*} }} \equiv \varepsilon_{{1^{*} }} \left[ {1 + g_{n} \left( {\varepsilon_{{2^{*} }} } \right)\frac{{1 - \varepsilon_{{1^{*} }} }}{{\left( {n + m} \right)!}}\left( {\frac{{a_{1} }}{{a_{2} }}} \right)^{2n + 1} } \right]^{ - 1} \hfill \\ \end{aligned} \right., $$

(A.24)

$$ \varepsilon^{\prime\prime}_{{C^{*} }} \equiv \varepsilon^{\prime}_{{C^{*} }} \left\{ {1 - \frac{{\varepsilon^{\prime}_{{C^{*} }} }}{{\varepsilon_{{C^{*} }} }}\left[ {\left( {n + m} \right)! - \left( {1 + \frac{{\varepsilon^{\prime}_{{1^{*} }} }}{{n\left( {1 + \varepsilon^{\prime}_{{1^{*} }} } \right)}}} \right)^{ - 1} } \right]g_{n} \left( {\varepsilon_{{2^{*} }} } \right)\frac{{1 - \varepsilon_{{C^{*} }} }}{{\left[ {\left( {n + m} \right)!} \right]^{2} }}\left( {\frac{{a_{C} }}{{a_{2} }}} \right)^{2n + 1} } \right\}^{ - 1} , $$

(A.25)

$$ g^{\prime\prime}_{nm} \equiv \frac{1}{{\left( {n + m} \right)!}}\left\{ {\begin{array}{*{20}l} {1 + \frac{{g_{n} \left( {\varepsilon^{\prime}_{{1^{*} }} } \right)}}{{g_{n} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} } \right)}}\left( {\frac{{a_{1} }}{{a_{C} }}} \right)^{2n + 1} \left[ {1 - g_{n} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} } \right)\frac{{\frac{n}{n + 1}}}{{\left( {n + m} \right)!}}\left( {\frac{{a_{1} }}{{a_{C} }}} \right)^{2n + 1} } \right]} \hfill \\ { + \frac{{g_{n} \left( {\varepsilon_{{2^{*} }} } \right)}}{{g_{n} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} } \right)}}\left( {\frac{{a_{2} }}{{a_{C} }}} \right)^{2n + 1} \left[ {1 - g_{n} \left( {\varepsilon^{\prime}_{{1^{*} }} } \right)\frac{{\frac{n}{n + 1}}}{{\left( {n + m} \right)!}}} \right]\left[ {1 - g_{n} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} } \right)\frac{{\frac{n}{n + 1}}}{{\left( {n + m} \right)!}}} \right]} \hfill \\ \end{array} } \right\}, $$

(A.26)

$$ g_{n} \left( \varepsilon \right) \equiv \frac{1 + \varepsilon }{{\frac{n}{n + 1} + \varepsilon }}. $$

(A.27)

Appendix B Spheroidal Cavity Embedded By One or More Concentric Dielectric Shells Surrounded By the Bulk

17.2.1 B.1 Spheroidal Cavity Embedded by One Concentric Dielectric Shell Surrounded by the Bulk

As we have seen for the spherical model, also the spheroidal model can be extended, considering concentring shells (Fig. B.1).

Fig. B.1

Spheroidal models of concentric dielectric media

We can investigate first the charge distribution inside a prolate spheroidal cavity, defined by the condition $\lambda = \lambda_{C} = a_{C} /d = {\text{constant}}$ and dielectric constant $\varepsilon_{C}$, with a first shell of polarisable dielectric, extending from $\lambda = \lambda_{C}$ to $\lambda = \lambda_{1} = a_{1} /d = {\text{constant}} > \lambda_{C}$, characterised by dielectric constant $\varepsilon_{1}$. Beyond this shell, there is the bulk region, from $\lambda = \lambda_{1}$ to infinity, with dielectric constant $\varepsilon_{B}$. The solution of Laplace’s Equation (3.10) is the same of the single spheroidal cavity case, but it is needed to introduce coefficients so that the expressions for the potential in the three regions (viz. cavity, shell, and bulk) are:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} } \right)d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{C} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda < \lambda_{C} , $$

(B.1)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\text{A}_{nm}^{\left( 1 \right)} d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda_{C} < \lambda < \lambda_{1} , $$

(B.2)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\text{B}_{nm}^{B} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda > \lambda_{1} . $$

(B.3)

The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} } \right)d^{n} P_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda < \lambda_{C} , $$

(B.4)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{\lambda = \lambda_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{\lambda = \lambda_{C}^{ + } }} , $$

(B.5a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{C}^{ + } }} , $$

(B.5b)

$$ \left( {\varPhi_{1} } \right)_{{\lambda = \lambda_{1}^{ - } }} = \left( {\varPhi_{B} } \right)_{{\lambda = \lambda_{1}^{ + } }} , $$

(B.5c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{1}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{1}^{ + } }} , $$

(B.5d)

or, explicitly:

$$ \varepsilon_{C}^{ - 1} \left[ {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} } \right)d^{n} P_{n}^{m} \left( {\lambda_{C} } \right) + \text{B}_{nm}^{C} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( {\lambda_{C} } \right)} \right] = \varepsilon_{1}^{ - 1} \left[ {\text{A}_{nm}^{\left( 1 \right)} d^{n} P_{n}^{m} \left( {\lambda_{C} } \right) + \text{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( {\lambda_{C} } \right)} \right] $$

(B.6a)

$$ \varvec{A}_{nm}^{C} d^{n} \dot{P}_{n}^{m} \left( {\lambda_{C} } \right) + \varvec{B}_{nm}^{C} d^{{ - \left( {n + 1} \right)}} \dot{Q}_{n}^{m} \left( {\lambda_{C} } \right) = \varvec{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} \dot{Q}_{n}^{m} \left( {\lambda_{C} } \right), $$

(B.6b)

$$ \varepsilon_{1}^{ - 1} \left[ {\varvec{A}_{nm}^{\left( 1 \right)} d^{n} P_{n}^{m} \left( {\lambda_{1} } \right) + \varvec{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( {\lambda_{1} } \right)} \right] = \varepsilon_{B}^{ - 1} \varvec{B}_{nm}^{B} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( {\lambda_{1} } \right), $$

(B.6c)

$$ \varvec{A}_{nm}^{\left( 1 \right)} d^{n} \dot{P}_{n}^{m} \left( {\lambda_{1} } \right) + \varvec{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} \dot{Q}_{n}^{m} \left( {\lambda_{1} } \right) = \varvec{B}_{nm}^{B} d^{{ - \left( {n + 1} \right)}} \dot{Q}_{n}^{m} \left( {\lambda_{1} } \right). $$

(B.6d)

Eliminating $\text{B}_{nm}^{\left( 1 \right)}$ from (B.6a) and (B.6b), and $\text{B}_{nm}^{B}$ from (B.6c) and (B.6d), we obtain:

$$ \text{A}_{nm}^{C} = \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\lambda_{C} } \right)\frac{{\text{B}_{nm}^{C} }}{{d^{2n + 1} }}, $$

(B.7)

$$ \begin{aligned} \varvec{A}_{nm}^{\left( 1 \right)} & = \gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\lambda_{1} } \right)\frac{{\varvec{B}_{nm}^{\left( 1 \right)} }}{{d^{2n + 1} }} \\ & \quad = \left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\lambda_{C} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{C} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{C} } \right)}}} \right]\gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\lambda_{1} } \right)\frac{{\varvec{B}_{nm}^{C} }}{{d^{2n + 1} }}, \\ \end{aligned} $$

(B.8)

where:

$$ \text{B}_{nm}^{C} = \left( { - 1} \right)^{m} \left( {2n + 1} \right)\left[ {\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \right]^{2} d^{n} \sum\limits_{k} {q_{k} P_{n}^{m} \left( {\lambda_{k} } \right)P_{n}^{m} \left( {\mu_{k} } \right){\text{e}}^{{ - {\text{i}}m\phi_{k} }} } $$

(B.9)

$$ \varepsilon^{\prime}_{{C^{*} }} = \varepsilon_{{C^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{C^{*} }} } \right)\gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\lambda_{1} } \right)\frac{{P_{n}^{m} \left( {\lambda_{C} } \right)}}{{Q_{n}^{m} \left( {\lambda_{C} } \right)}}} \right]^{ - 1} , $$

(B.10)

and $\varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C}$, $\varepsilon_{{1^{*} }} \equiv \varepsilon_{B} /\varepsilon_{1}$. Finally, Helmholtz’s free energy reads:

$$ \begin{aligned} A & = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\frac{2n + 1}{d}\sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} \sum\limits_{m} {\left( { - 1} \right)^{m} \gamma^{\prime}_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\varepsilon_{{1^{*} }} ,\lambda_{C} ,\lambda_{1} } \right)\left[ {\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \right]^{2} } } } } \\ & \quad \times P_{n}^{m} \left( {\lambda_{k} } \right)P_{n}^{m} \left( {\lambda_{l} } \right)P_{n}^{m} \left( {\mu_{k} } \right)P_{n}^{m} \left( {\mu_{l} } \right){\text{e}}^{{{\text{i}}m\left( {\phi_{l} - \phi_{k} } \right)}} , \\ \end{aligned} $$

(B.11)

where:

$$ \gamma^{\prime}_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\varepsilon_{{1^{*} }} ,\lambda_{C} ,\lambda_{1} } \right) \equiv \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\lambda_{C} } \right) + \gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\lambda_{1} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\lambda_{C} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{C} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{C} } \right)}}} \right]. $$

(B.12)

The final expressions for the oblate spheroid problem can be straightforwardly obtained by those of the prolate case, by making the substitutions (4.47).

17.2.2 B.2 Spheroidal Cavity Embedded by Two or More Concentric Dielectric Shells Surrounded by the Bulk

If we consider two prolate spheroidal concentric dielectric shells, defined by the conditions:

$$ \lambda = \lambda_{C} = a_{C} /d = {\text{constant,}} $$

(B.13a)

$$ \lambda = \lambda_{1} = a_{1} /d = {\text{constant}} > \lambda_{C} , $$

(B.13b)

$$ \lambda = \lambda_{2} = a_{2} /d = {\text{constant}} > \lambda_{1} , $$

(B.13c)

we have now four different dielectric regions, viz. the cavity (characterised by dielectric constant $\varepsilon_{C}$), the inner shell (characterised by dielectric constant $\varepsilon_{1}$), the outer shell (characterised by dielectric constant $\varepsilon_{2}$), and the bulk (characterised by dielectric constant $\varepsilon_{B}$). The electric potential in the four regions are expressed by:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{C} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda < \lambda_{C} , $$

(B.14)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda_{C} < \lambda < \lambda_{1} , $$

(B.15)

$$ \varPhi_{2} = \varepsilon_{2}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\text{A}_{nm}^{\left( 2 \right)} d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{\left( 2 \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda_{1} < \lambda < \lambda_{2} , $$

(B.16)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\text{B}_{nm}^{B} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } .\quad \lambda > \lambda_{2} . $$

(B.17)

The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)d^{n} P_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda < \lambda_{C} , $$

(B.18)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{\lambda = \lambda_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{\lambda = \lambda_{C}^{ + } }} , $$

(B.19a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{C}^{ + } }} , $$

(B.19b)

$$ \left( {\varPhi_{1} } \right)_{{\lambda = \lambda_{1}^{ - } }} = \left( {\varPhi_{2} } \right)_{{\lambda = \lambda_{1}^{ + } }} , $$

(B.19c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{1}^{ - } }} = \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{1}^{ + } }} , $$

(B.19d)

$$ \left( {\varPhi_{2} } \right)_{{\lambda = \lambda_{2}^{ - } }} = \left( {\varPhi_{B} } \right)_{{\lambda = \lambda_{2}^{ + } }} , $$

(B.19e)

$$ \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{2}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{2}^{ + } }} , $$

(B.19f)

which enable to reach the final formulae for the reaction potential:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\lambda_{C} ,\lambda_{1} ,\lambda_{2} } \right)\text{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} P_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } , $$

(B.20)

and Helmholtz’s free energy:

$$ \begin{aligned} A & = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\frac{2n + 1}{d}\sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} \sum\limits_{m} {\left( { - 1} \right)^{m} \gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\lambda_{C} ,\lambda_{1} ,\lambda_{2} } \right)\left[ {\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \right]^{2} } } } } \\ & \quad \times P_{n}^{m} \left( {\lambda_{k} } \right)P_{n}^{m} \left( {\lambda_{l} } \right)P_{n}^{m} \left( {\mu_{k} } \right)P_{n}^{m} \left( {\mu_{l} } \right){\text{e}}^{{{\text{i}}m\left( {\phi_{l} - \phi_{k} } \right)}} , \\ \end{aligned} $$

(B.21)

where:

$$ \left\{ \begin{aligned} \varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C} \hfill \\ \varepsilon_{{1^{*} }} \equiv \varepsilon_{2} /\varepsilon_{1} \hfill \\ \varepsilon_{{2^{*} }} \equiv \varepsilon_{B} /\varepsilon_{2} \hfill \\ \end{aligned} \right., $$

(B.22)

$$ \left\{ \begin{aligned} \varepsilon^{\prime}_{{C^{*} }} \equiv \varepsilon_{{C^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{C^{*} }} } \right)\gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\lambda_{1} } \right)\frac{{P_{n}^{m} \left( {\lambda_{C} } \right)}}{{Q_{n}^{m} \left( {\lambda_{C} } \right)}}} \right]^{ - 1} \hfill \\ \varepsilon^{\prime}_{{1^{*} }} \equiv \varepsilon_{{1^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{1^{*} }} } \right)\gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\lambda_{2} } \right)\frac{{P_{n}^{m} \left( {\lambda_{1} } \right)}}{{Q_{n}^{m} \left( {\lambda_{1} } \right)}}} \right]^{ - 1} \hfill \\ \end{aligned} \right., $$

(B.23)

$$ \varepsilon^{\prime\prime}_{{C^{*} }} \equiv \varepsilon^{\prime}_{{C^{*} }} \left\{ {1 + \varepsilon^{\prime}_{{C^{*} }} \left( {1 - \frac{1}{{\varepsilon_{{C^{*} }} }}} \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\lambda_{1} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{1} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{1} } \right)}}} \right]\gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\lambda_{2} } \right)\frac{{P_{n}^{m} \left( {\lambda_{C} } \right)}}{{Q_{n}^{m} \left( {\lambda_{C} } \right)}}} \right\}^{ - 1} , $$

(B.24)

$$ \begin{aligned} & \gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\lambda_{C} ,\lambda_{1} ,\lambda_{2} } \right) \equiv \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\lambda_{C} } \right) \\ & \quad + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\lambda_{1} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\lambda_{C} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{1} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{1} } \right)}}} \right] \\ & \quad + \gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\lambda_{2} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\lambda_{1} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{1} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{1} } \right)}}} \right]\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\lambda_{C} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{C} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{C} } \right)}}} \right]. \\ \end{aligned} $$

(B.25)

The expressions for the oblate spheroidal cavity can be obtained with the substitutions (4.47).

This approach can be generalised for a system accounting for L prolate spheroidal concentric dielectric shells, where the S-th shell is defined by the condition $\lambda = \lambda_{S} = a_{S} /d = {\text{constant}}$, and characterised by a dielectric constant $\varepsilon_{S}$. The potentials can be expressed as:

$$ \varPhi_{S} = \varepsilon_{S}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\sum\limits_{\sigma = S}^{L} {\text{A}_{nm}^{\left( \sigma \right)} } } \right)d^{n} P_{n}^{m} \left( \lambda \right) + \text{B}_{nm}^{\left( S \right)} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)} \right]P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad S = 0,1, \ldots L, $$

(B.26)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\text{B}_{nm}^{B} d^{{ - \left( {n + 1} \right)}} Q_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda > \lambda_{L} . $$

(B.27)

where S = 0 = C correspond to the inner cavity. The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {\sum\limits_{\sigma = 0}^{L} {\text{A}_{nm}^{\left( \sigma \right)} } } \right)d^{n} P_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } ,\quad \lambda < \lambda_{C} , $$

(B.28)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{S} } \right)_{{\lambda = \lambda_{S}^{ - } }} = \left( {\varPhi_{S + 1} } \right)_{{\lambda = \lambda_{S}^{ + } }} , $$

(B.29a)

$$ \varepsilon_{S} \left( {\frac{{\partial \varPhi_{S} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{S}^{ - } }} = \varepsilon_{S + 1} \left( {\frac{{\partial \varPhi_{S + 1} }}{\partial \lambda }} \right)_{{\lambda = \lambda_{S}^{ + } }} , $$

(B.29b)

where S = 0, 1,… L (S = 0 = C corresponds to the inner cavity, S = L + 1 = B corresponds to the outer bulk). which enable to reach the final formulae for the reaction potential:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\gamma_{nm}^{\left( L \right)} \left( {\left\{ {\varepsilon_{{\sigma^{*} }}^{{\left( {L - \sigma } \right)}} ,\lambda_{\sigma } } \right\}_{\sigma = 0,1, \ldots L} } \right)\text{B}_{nm}^{\left( 1 \right)} d^{{ - \left( {n + 1} \right)}} P_{n}^{m} \left( \lambda \right)P_{n}^{m} \left( \mu \right){\text{e}}^{{{\text{i}}m\phi }} } } , $$

(B.30)

and Helmholtz’s free energy:

$$ \begin{aligned} A & = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\frac{2n + 1}{d}\sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} \sum\limits_{m} {\left( { - 1} \right)^{m} \gamma_{nm}^{\left( L \right)} \left( {\left\{ {\varepsilon_{{\sigma^{*} }}^{{\left( {L - \sigma } \right)}} ,\lambda_{\sigma } } \right\}_{\sigma = 0,1, \ldots L} } \right)\left[ {\frac{{\left( {n - m} \right)!}}{{\left( {n + m} \right)!}}} \right]^{2} } } } } \\ & \quad \times P_{n}^{m} \left( {\lambda_{k} } \right)P_{n}^{m} \left( {\lambda_{l} } \right)P_{n}^{m} \left( {\mu_{k} } \right)P_{n}^{m} \left( {\mu_{l} } \right){\text{e}}^{{{\text{i}}m\left( {\phi_{l} - \phi_{k} } \right)}} , \\ \end{aligned} $$

(B.31)

where:

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon_{{\sigma^{*} }}} \equiv \varepsilon_{{\sigma + 1 }}/ \varepsilon_{{\sigma}} \\ {\varepsilon^{\prime}_{{\sigma^{*} }} \equiv \varepsilon_{{\sigma^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{\sigma^{*} }} } \right)\gamma_{nm} \left( {\varepsilon^{\prime}_{{\left( {\sigma + 1} \right)^{*} }} ,\lambda_{\sigma + 1} } \right)\frac{{P_{n}^{m} \left( {\lambda_{\sigma } } \right)}}{{Q_{n}^{m} \left( {\lambda_{\sigma } } \right)}}} \right]^{ - 1} } \hfill \\ {\varepsilon^{\prime\prime}_{{\sigma^{*} }} \equiv \varepsilon^{\prime}_{{\sigma^{*} }} \left\{ {1 + \varepsilon^{\prime}_{{\sigma^{*} }} \left( {1 - \frac{1}{{\varepsilon_{{\sigma^{*} }} }}} \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{\left( {\sigma + 1} \right)^{*} }} ,\lambda_{\sigma + 1} } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{\sigma + 1} } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{\sigma + 1} } \right)}}} \right]\gamma_{nm} \left( {\varepsilon_{{\left( {\sigma + 2} \right)^{*} }} ,\lambda_{\sigma + 2} } \right)\frac{{P_{n}^{m} \left( {\lambda_{\sigma } } \right)}}{{Q_{n}^{m} \left( {\lambda_{\sigma } } \right)}}} \right\}^{ - 1} } \hfill \\ \vdots \hfill \\ \end{array} ,} \right. $$

(B.32)

$$ \gamma_{nm}^{\left( L \right)} \left( {\left\{ {\varepsilon_{{\sigma^{*} }}^{{\left( {L - \sigma } \right)}} ,\lambda_{\sigma } } \right\}_{\sigma = 0,1, \ldots L} } \right) = \left\{ {\sum\limits_{\sigma = 0}^{L} {\gamma_{nm} \left( {\varepsilon_{{\sigma^{*} }}^{{\left( {L - \sigma } \right)}} ,\lambda_{\sigma } } \right)\prod\limits_{\tau = 0}^{\sigma } {\left[ {1 + \gamma_{nm} \left( {\varepsilon_{{\sigma^{*} }}^{{\left( {L - \sigma } \right)}} ,\lambda_{\sigma } } \right)\frac{{\dot{P}_{n}^{m} \left( {\lambda_{\sigma } } \right)}}{{\dot{Q}_{n}^{m} \left( {\lambda_{\sigma } } \right)}}} \right]} } } \right\}. $$

(B.33)

Appendix C Ellipsoidal Cavity Embedded By One or More Concentric Dielectric Shells Surrounded By the Bulk

17.3.1 C.1 Ellipsoidal Cavity Imbedded by One Concentric Dielectric Shell Surrounded by the Bulk

The formalism described above lends itself to be easily extended to the case of cavity imbedded by a concentric dielectric shell surrounded by the bulk. We aim here to investigate the charge distribution inside an ellipsoidal cavity, defined by the condition $\rho = \rho_{C} = {\text{constant}}$ and dielectric constant $\varepsilon_{C}$, with a first shell of polarisable dielectric, extending from $\rho = \rho_{C}$ to $\rho = \rho_{1} = {\text{constant}} > \rho_{C}$, characterised by dielectric constant $\varepsilon_{1}$. Beyond this shell, there is the bulk region, from $\rho = \rho_{1}$ to infinity, with dielectric constant $\varepsilon_{B}$. The solution of Laplace’s Equation (3.10) is the same of the single ellipsoidal cavity case, but it is needed to introduce coefficients so that the expressions for the potential in the three regions (viz. cavity, shell, and bulk) are:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\varvec{A}_{nm}^{C} + \varvec{A}_{nm}^{\left( 1 \right)} } \right)E_{n}^{m} \left( \rho \right) + \varvec{B}_{nm}^{C} F_{n}^{m} \left( \rho \right)} \right]E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho < \rho_{C} , $$

(C.1)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\text{A}_{nm}^{\left( 1 \right)} E_{n}^{m} \left( \rho \right) + \text{B}_{nm}^{\left( 1 \right)} F_{n}^{m} \left( \rho \right)} \right]E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho_{C} < \rho < \rho_{1} , $$

(C.2)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\text{B}_{nm}^{B} F_{n}^{m} \left( \rho \right)E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho > \rho_{1} . $$

(C.3)

The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} } \right)E_{n}^{m} \left( \rho \right)E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho < \rho_{C} , $$

(C.4)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{\rho = \rho_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{\rho = \rho_{C}^{ + } }} , $$

(C.5a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial \rho }} \right)_{{\rho = \rho_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \rho }} \right)_{{\rho = \rho_{C}^{ + } }} , $$

(C.5b)

$$ \left( {\varPhi_{1} } \right)_{{\rho = \rho_{1}^{ - } }} = \left( {\varPhi_{B} } \right)_{{\rho = \rho_{1}^{ + } }} , $$

(C.5c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \rho }} \right)_{{\rho = \rho_{1}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial \rho }} \right)_{{\rho = \rho_{1}^{ + } }} , $$

(C.5d)

or, explicitly:

$$ \varepsilon_{C}^{ - 1} \left[ {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} } \right)E_{n}^{m} \left( {\rho_{C} } \right) + \text{B}_{nm}^{C} F_{n}^{m} \left( {\rho_{C} } \right)} \right] = \varepsilon_{1}^{ - 1} \left[ {\text{A}_{nm}^{\left( 1 \right)} E_{n}^{m} \left( {\rho_{C} } \right) + \text{B}_{nm}^{\left( 1 \right)} F_{n}^{m} \left( {\rho_{C} } \right)} \right], $$

(C.6a)

$$ \varvec{A}_{nm}^{C} \dot{E}_{n}^{m} \left( {\rho_{C} } \right) + \varvec{B}_{nm}^{C} \dot{F}_{n}^{m} \left( {\rho_{C} } \right) = \varvec{B}_{nm}^{\left( 1 \right)} \dot{F}_{n}^{m} \left( {\rho_{C} } \right), $$

(C.6b)

$$ \varepsilon_{1}^{ - 1} \left[ {\varvec{A}_{nm}^{\left( 1 \right)} E_{n}^{m} \left( {\rho_{1} } \right) + \varvec{B}_{nm}^{\left( 1 \right)} F_{n}^{m} \left( {\rho_{1} } \right)} \right] = \varepsilon_{B}^{ - 1} \varvec{B}_{nm}^{B} F_{n}^{m} \left( {\rho_{1} } \right), $$

(C.6c)

$$ \varvec{A}_{nm}^{\left( 1 \right)} \dot{E}_{n}^{m} \left( {\rho_{1} } \right) + \varvec{B}_{nm}^{\left( 1 \right)} \dot{F}_{n}^{m} \left( {\rho_{1} } \right) = \varvec{B}_{nm}^{B} \dot{F}_{n}^{m} \left( {\rho_{1} } \right). $$

(C.6d)

Eliminating $\text{B}_{nm}^{\left( 1 \right)}$ from (C.6a) and (C.6b), and $\text{B}_{nm}^{B}$ from (C.6c) and (C.6d), we obtain:

$$ \text{A}_{nm}^{C} = \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\rho_{C} } \right)\text{B}_{nm}^{C} , $$

(C.7)

$$ \begin{aligned} \varvec{A}_{nm}^{\left( 1 \right)} & = \gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\rho_{1} } \right)\varvec{B}_{nm}^{\left( 1 \right)} \\ & = \left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\rho_{C} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{C} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{C} } \right)}}} \right]\gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\rho_{1} } \right)\varvec{B}_{nm}^{C} , \\ \end{aligned} $$

(C.8)

where:

$$ \text{B}_{nm}^{C} = N_{nm} Q_{nm} , $$

(C.9)

$$ \varepsilon^{\prime}_{{C^{*} }} = \varepsilon_{{C^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{C^{*} }} } \right)\gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,r_{1} } \right)\frac{{E_{n}^{m} \left( {\rho_{C} } \right)}}{{F_{n}^{m} \left( {\rho_{C} } \right)}}} \right]^{ - 1} , $$

(C.10)

and $\varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C}$, $\varepsilon_{{1^{*} }} \equiv \varepsilon_{B} /\varepsilon_{1}$. Finally, Helmholtz’s free energy reads:

$$ \begin{aligned} A & = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\sum\limits_{m} {\gamma^{\prime}_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\varepsilon_{{1^{*} }} ,\rho_{C} ,\rho_{1} } \right)N_{nm} \sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} } } } } \\ & \quad \times E_{n}^{m} \left( {\rho_{k} } \right)E_{n}^{m} \left( {\rho_{l} } \right)E_{n}^{m} \left( {\mu_{k} } \right)E_{n}^{m} \left( {\mu_{l} } \right)E_{n}^{m} \left( {\nu_{k} } \right)E_{n}^{m} \left( {\nu_{l} } \right), \\ \end{aligned} $$

(C.11)

where:

$$ \gamma^{\prime}_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\varepsilon_{{1^{*} }} ,\rho_{C} ,\rho_{1} } \right) \equiv \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\rho_{C} } \right) + \gamma_{nm} \left( {\varepsilon_{{1^{*} }} ,\rho_{1} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{C^{*} }} ,\rho_{C} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{C} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{C} } \right)}}} \right]. $$

(C.12)

17.3.2 C.2 Ellipsoidal Cavity Embedded by Two Concentric Dielectric Shells Surrounded by the Bulk

If we consider two ellipsoidal concentric dielectric shells, defined by the conditions:

$$ \rho = \rho_{C} = {\text{constant,}} $$

(C.13a)

$$ \rho = \rho_{1} = {\text{constant}} > \rho_{C} , $$

(C.13b)

$$ \rho = \rho_{2} = {\text{constant}} > \rho_{1} , $$

(C.13c)

we have now four different dielectric regions, viz. the cavity (characterised by dielectric constant $\varepsilon_{C}$), the inner shell (characterised by dielectric constant $\varepsilon_{1}$), the outer shell (characterised by dielectric constant $\varepsilon_{2}$), and the bulk (characterised by dielectric constant $\varepsilon_{B}$). The electric potential in the four regions are expressed by:

$$ \varPhi_{C} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)E_{n}^{m} \left( \rho \right) + \text{B}_{nm}^{C} F_{n}^{m} \left( \rho \right)} \right]E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho < \rho_{C} , $$

(C.14)

$$ \varPhi_{1} = \varepsilon_{1}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\left( {\text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)E_{n}^{m} \left( \rho \right) + \text{B}_{nm}^{\left( 1 \right)} F_{n}^{m} \left( \rho \right)} \right]E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho_{C} < \rho < \rho_{1} , $$

(C.15)

$$ \varPhi_{2} = \varepsilon_{2}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left[ {\text{A}_{nm}^{\left( 2 \right)} E_{n}^{m} \left( \rho \right) + \text{B}_{nm}^{\left( 2 \right)} F_{n}^{m} \left( \rho \right)} \right]E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho_{1} < \rho < \rho_{2} , $$

(C.16)

$$ \varPhi_{B} = \varepsilon_{B}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\text{B}_{nm}^{B} F_{n}^{m} \left( \rho \right)E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } .\quad \rho > \rho_{2} . $$

(C.17)

The reaction potential is given by:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\left( {\text{A}_{nm}^{C} + \text{A}_{nm}^{\left( 1 \right)} + \text{A}_{nm}^{\left( 2 \right)} } \right)E_{n}^{m} \left( \rho \right)E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } ,\quad \rho < \rho_{C} , $$

(C.18)

and the boundary conditions are imposed for each surface, i.e.:

$$ \left( {\varPhi_{C} } \right)_{{\rho = \rho_{C}^{ - } }} = \left( {\varPhi_{1} } \right)_{{\rho = \rho_{C}^{ + } }} , $$

(C.19a)

$$ \varepsilon_{C} \left( {\frac{{\partial \varPhi_{C} }}{\partial \rho }} \right)_{{\rho = \rho_{C}^{ - } }} = \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \rho }} \right)_{{\rho = \rho_{C}^{ + } }} , $$

(C.19b)

$$ \left( {\varPhi_{1} } \right)_{{\rho = \rho_{1}^{ - } }} = \left( {\varPhi_{2} } \right)_{{\rho = \rho_{1}^{ + } }} , $$

(C.19c)

$$ \varepsilon_{1} \left( {\frac{{\partial \varPhi_{1} }}{\partial \rho }} \right)_{{\rho = \rho_{1}^{ - } }} = \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial \rho }} \right)_{{\rho = \rho_{1}^{ + } }} , $$

(C.19d)

$$ \left( {\varPhi_{2} } \right)_{{\rho = \rho_{2}^{ - } }} = \left( {\varPhi_{B} } \right)_{{\rho = \rho_{2}^{ + } }} , $$

(C.19e)

$$ \varepsilon_{2} \left( {\frac{{\partial \varPhi_{2} }}{\partial \rho }} \right)_{{\rho = \rho_{2}^{ - } }} = \varepsilon_{B} \left( {\frac{{\partial \varPhi_{B} }}{\partial \rho }} \right)_{{\rho = \rho_{2}^{ + } }} , $$

(C.19f)

which enable to reach the final formulae for the reaction potential:

$$ \varPhi_{\text{R}} = \varepsilon_{C}^{ - 1} \sum\limits_{n} {\sum\limits_{m} {\gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\rho_{C} ,\rho_{1} ,\rho_{2} } \right)N_{nm} Q_{nm} E_{n}^{m} \left( \rho \right)E_{n}^{m} \left( \mu \right)E_{n}^{m} \left( \nu \right)} } , $$

(C.20)

and Helmholtz’s free energy:

$$ \begin{aligned} A & = \frac{1}{{2\varepsilon_{C} }}\sum\limits_{n} {\sum\limits_{m} {\gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\rho_{C} ,\rho_{1} ,\rho_{2} } \right)N_{nm} \sum\limits_{k} {\sum\limits_{l} {q_{k} q_{l} } } } } \\ & \quad \times E_{n}^{m} \left( {\rho_{k} } \right)E_{n}^{m} \left( {\rho_{l} } \right)E_{n}^{m} \left( {\mu_{k} } \right)E_{n}^{m} \left( {\mu_{l} } \right)E_{n}^{m} \left( {\nu_{k} } \right)E_{n}^{m} \left( {\nu_{l} } \right), \\ \end{aligned} $$

(C.21)

where:

$$ \left\{ \begin{aligned} \varepsilon_{{C^{*} }} \equiv \varepsilon_{1} /\varepsilon_{C} \hfill \\ \varepsilon_{{1^{*} }} \equiv \varepsilon_{2} /\varepsilon_{1} \hfill \\ \varepsilon_{{2^{*} }} \equiv \varepsilon_{B} /\varepsilon_{2} \hfill \\ \end{aligned} \right., $$

(C.22)

$$ \left\{ \begin{aligned} \varepsilon^{\prime}_{{C^{*} }} \equiv \varepsilon_{{C^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{C^{*} }} } \right)\gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\rho_{1} } \right)\frac{{E_{n}^{m} \left( {\rho_{C} } \right)}}{{F_{n}^{m} \left( {\rho_{C} } \right)}}} \right]^{ - 1} \hfill \\ \varepsilon^{\prime}_{{1^{*} }} \equiv \varepsilon_{{1^{*} }} \left[ {1 + \left( {1 - \varepsilon_{{1^{*} }} } \right)\gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\rho_{2} } \right)\frac{{E_{n}^{m} \left( {\rho_{1} } \right)}}{{F_{n}^{m} \left( {\rho_{1} } \right)}}} \right]^{ - 1} \hfill \\ \end{aligned} \right., $$

(C.23)

$$ \varepsilon^{\prime\prime}_{{C^{*} }} \equiv \varepsilon^{\prime}_{{C^{*} }} \left\{ {1 + \varepsilon^{\prime}_{{C^{*} }} \left( {1 - \frac{1}{{\varepsilon_{{C^{*} }} }}} \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\rho_{1} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{1} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{1} } \right)}}} \right]\gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\rho_{2} } \right)\frac{{E_{n}^{m} \left( {\rho_{C} } \right)}}{{F_{n}^{m} \left( {\rho_{C} } \right)}}} \right\}^{ - 1} , $$

(C.24)

$$ \begin{aligned} & \gamma^{\prime\prime}_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\varepsilon^{\prime}_{{1^{*} }} ,\varepsilon_{{2^{*} }} ,\rho_{C} ,\rho_{1} ,\rho_{2} } \right) \equiv \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\rho_{C} } \right) \\ & \quad + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\rho_{1} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\rho_{C} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{1} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{1} } \right)}}} \right] \\ & \quad + \gamma_{nm} \left( {\varepsilon_{{2^{*} }} ,\rho_{2} } \right)\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime}_{{1^{*} }} ,\rho_{1} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{1} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{1} } \right)}}} \right]\left[ {1 + \gamma_{nm} \left( {\varepsilon^{\prime\prime}_{{C^{*} }} ,\rho_{C} } \right)\frac{{\dot{E}_{n}^{m} \left( {\rho_{C} } \right)}}{{\dot{F}_{n}^{m} \left( {\rho_{C} } \right)}}} \right]. \\ \end{aligned} $$

(C.25)

Appendix D: Different equations for Stokes shifts

$$ y_{m,s} = y_{m,g}^{\left( 0 \right)} + a_{m} \left[ {x_{s}^{\left( 1 \right)} + x_{s}^{\left( 2 \right)} } \right], $$

(D.1)

Equation	Refs.	$y_{m,s}$	$a_{m}^{\left( i \right)}$	$x_{s}^{\left( 1 \right)}$	$x_{s}^{\left( 2 \right)}$
(D.1.1)	[52, 54–56, 59, 74]	$\tilde{\nu }^{a}$	$a_{m}^{\left( 1 \right)}$	$F_{1} \left( {\varepsilon ,n} \right)$	$f_{1} \left( {n^{2} } \right)$
(D.1.2)	[49–51]		$a_{m}^{\left( 1 \right)}$	$F_{2} \left( {\varepsilon ,n} \right)$	$f_{2} \left( {n^{2} } \right)$
(D.1.3)	[64, 76, 84]		$a_{m}^{\left( 1 \right)}$	$F_{3} \left( {\varepsilon ,n} \right)$	$f_{1} \left( {n^{2} } \right)$
(D.1.4)	[60]		$a_{m}^{\left( 1 \right)}$	$F_{3} \left( {\varepsilon ,n} \right)$	$f_{3} \left( {n^{2} } \right)$
(D.1.5)	[77, 88–90]		$a_{m}^{\left( 1 \right)}$	$f_{2} \left( \varepsilon \right)$	$f_{2} \left( {n^{2} } \right)$
(D.1.6)	[80, 85, 91]		$2a_{m}^{\left( 2 \right)}$	$f_{2} \left( \varepsilon \right)$
(D.1.7)	[80, 85, 91]		$a_{m}^{\left( 2 \right)}$	$F_{6} \left( {\varepsilon ,n} \right)$
(D.1.8)	[60, 168]	$\tilde{\nu }^{f}$	$a_{m}^{\left( 3 \right)}$	$F_{3} \left( {\varepsilon ,n} \right)$	$f_{3} \left( {n^{2} } \right)$
(D.1.9)	[66, 69, 70, 78, 81, 86, 98]		$a_{m}^{\left( 4 \right)}$	$F_{5} \left( {\varepsilon ,n} \right)$
(D.1.10)	[92]	$\tilde{\nu }^{a} - \tilde{\nu }^{f}$	$a_{m}^{\left( 5 \right)}$	$F_{1} \left( {\varepsilon ,n} \right)$
(D.1.11)	[57, 61]		$a_{m}^{\left( 5 \right)}$	$F_{2} \left( {\varepsilon ,n} \right)$
(D.1.12)	[63, 75, 82, 93]		$a_{m}^{\left( 5 \right)}$	$F_{3} \left( {\varepsilon ,n} \right)$
(D.1.13)	[62, 65, 67]		$a_{m}^{\left( 5 \right)}$	$F_{3} \left( {\varepsilon ,n} \right)$	$f_{1} \left( {n^{2} } \right)$
(D.1.14)	[58]	$\tilde{\nu }^{a} + \tilde{\nu }^{f}$	$2a_{m}^{\left( 2 \right)}$	$F_{2} \left( {\varepsilon ,n} \right)$	$f_{2} \left( {n^{2} } \right)$
(D.1.15)	[62, 65, 67, 71]		$2a_{m}^{\left( 2 \right)}$	$F_{4} \left( {\varepsilon ,n} \right)$	$f_{1} \left( {n^{2} } \right)$
(D.1.16)	[72, 73, 83]		$4a_{m}^{\left( 2 \right)}$	$F_{7} \left( {\varepsilon ,n} \right)$

where:

$$ \begin{aligned} f_{1} \left( \xi \right) & = \frac{\xi - 1}{\xi + 2},\quad f_{2} \left( \xi \right) = \frac{\xi - 1}{2\xi + 1},\quad f_{3} \left( \xi \right) = \frac{3}{2}\frac{{\xi^{2} - 1}}{{\left( {\xi + 2} \right)^{2} }}, \\ F_{1} \left( {\varepsilon ,n} \right) & = f_{1} \left( \varepsilon \right) - f_{1} \left( {n^{2} } \right),\quad F_{2} \left( {\varepsilon ,n} \right) = f_{2} \left( \varepsilon \right) - f_{2} \left( {n^{2} } \right), \\ F_{3} \left( {\varepsilon ,n} \right) & = F_{1} \left( {\varepsilon ,n} \right)\frac{{f_{1} \left( {n^{2} } \right)}}{{f_{2} \left( {n^{2} } \right)}},\quad F_{4} \left( {\varepsilon ,n} \right) = f_{1} \left( \varepsilon \right)\frac{{f_{1} \left( {n^{2} } \right)}}{{f_{2} \left( {n^{2} } \right)}}, \\ F_{5} \left( {\varepsilon ,n} \right) & = f_{2} \left( \varepsilon \right) - \frac{1}{2}f_{2} \left( {n^{2} } \right),\quad F_{6} \left( {\varepsilon ,n} \right) = 2f_{1} \left( \varepsilon \right) - f_{1} \left( {n^{2} } \right), \\ F_{7} \left( {\varepsilon ,n} \right) & = \frac{1}{2}F_{3} \left( {\varepsilon ,n} \right) + f_{3} \left( {n^{2} } \right), \\ a_{m}^{\left( 1 \right)} & = - \frac{2}{{hca_{0}^{3} }}{\varvec{\upmu}}_{g} \cdot \left( {{\varvec{\upmu}}_{e} - {\varvec{\upmu}}_{g} } \right),\quad a_{m}^{\left( 2 \right)} = - \frac{1}{{hca_{0}^{3} }}\left( {\mu_{e}^{2} - \mu_{g}^{2} } \right),\quad a_{m}^{\left( 3 \right)} = - \frac{2}{{hca_{0}^{3} }}{\varvec{\upmu}}_{e} \cdot \left( {{\varvec{\upmu}}_{e} - {\varvec{\upmu}}_{g} } \right), \\ a_{m}^{\left( 4 \right)} & = - \frac{2}{{hca_{0}^{3} }}\mu_{e}^{2} ,\quad a_{m}^{\left( 5 \right)} = + \frac{2}{{hca_{0}^{3} }}\left( {{\varvec{\upmu}}_{e} - {\varvec{\upmu}}_{g} } \right)^{2} . \\ \end{aligned} $$

Appendix E: Derivation of Ravi’s Equation

The starting point for the derivation of Ravi’s formula is the following equation [193, 211, 255–258],

$$ \tilde{\nu }^{a} - \tilde{\nu }^{f} = \frac{2}{{hca_{0}^{3} }}\left( {{\varvec{\upmu}}_{e} - {\varvec{\upmu}}_{g} } \right)^{2} \left[ {f_{1} \left( \varepsilon \right) - f_{1} \left( {n^{2} } \right)} \right], $$

(E.1)

which is the best analytical model for the description of the solvatochromism (see Sect. 4.1); the equation is written—according to the Authors’ notation—as follows:

$$ Y\left( {m,s} \right) = B\left( m \right)X\left( s \right) + Y^\circ \left( {m,g} \right), $$

(E.2)

where:

$$ \left\{ {\begin{array}{*{20}l} {Y\left( {m,s} \right) \equiv \tilde{\nu }_{a} - \tilde{\nu }_{f} } \hfill \\ {B\left( m \right) \equiv \frac{{2\left( {\Delta {\varvec{\upmu}}} \right)^{2} }}{{hca^{3} }} = \frac{{2\left( {{\varvec{\upmu}}_{e} - {\varvec{\upmu}}_{g} } \right)^{2} }}{{hca^{3} }}} \hfill \\ {X\left( s \right) \equiv \left( {\frac{\varepsilon - 1}{\varepsilon + 2} - \frac{{n^{2} - 1}}{{n^{2} + 2}}} \right)\frac{{2n^{2} + 1}}{{n^{2} + 2}} \equiv F_{3} \left( {\varepsilon ,n^{2} } \right)} \hfill \\ \end{array} ,} \right. $$

(E.3)

and $Y^\circ \left( {m,g} \right)$ is a constant, independent by the solvent, which represents Stokes shift of the molecule m in gas phase. The Authors actually do not use the squared difference of the two vectors when expressing $B\left( m \right)$ (as it should be), but the squared difference of their moduli, viz. $\left( {\Delta \mu } \right)^{2} = \left( {\mu_{e} - \mu_{g} } \right)^{2}$: in other words, they implicitly assume that ${\varvec{\upmu}}_{e}$ and ${\varvec{\upmu}}_{g}$ must have (always) the same direction and versus. (Since this is not explicitly stated in the paper, their equation, that will be shown soon, has been often improperly evoked in the experimental Literature.)

They apply Eq. (E.2) to the special case of the so-called Reichardt’s betaine–30, m = D, which the Eq. (17.1) was built on (Chart 5.1, molecule A). Also at this point there arise some issues. First of all, they use (approximated) experimental values of the dipole moments which actually refer to another betaine (i.e. Chart 5.1, molecule C), viz. $\mu_{e} = 6{\text{ D}}$ and $\mu_{g} = 15{\text{ D}}$, and thus $\Delta \mu_{\text{C}} = 9 {\text{D}}$, although this is not explicitly stated. In order to evaluate whether the two molecules have different values of electric dipole moments, we have calculated the values of $\mu_{g}$ and $\mu_{e}$ (in the Franck-Condon S ₁) for the dyes A and C at PBE0 [6]/SNSD [219] level of theory, both in vacuo and in the reference solvent (i.e. 1,4-dioxane); in the latter, $\Delta \mu_{\text{A}}$ = 6.120 D and $\Delta \mu_{\text{C}}$ = 9.018 D. These QM calculations confirm that ${\varvec{\upmu}}_{e}$ and ${\varvec{\upmu}}_{g}$ are parallel and directed along the x-axis [259], and seems to suggest that actually the two dyes show a different dipole moment change when passing from S ₀ to S ₁. So, according to our calculations, $\Delta \mu_{\text{A}}$ is smaller than $\Delta \mu_{\text{C}}$. Therefore, by employing the improper value of $\Delta \mu_{\text{C}}$ leads to an under-estimation of the electric dipole moment change of the system in study.

For the reference compound, they assume $Y\left( {D,s} \right) \equiv \tilde{\nu }_{a}^{D} - {\text{constant}}$, since the dye is written not to fluoresce at all and hence one can only follow the shifts of absorption maximum with change in solvent polarity. This is actually another contradictory point. As explained in Sect. 4.1, Eq. (D.1) gives good results, but to be employed from the experimental point of view the knowledge of the emission data is required and thus the compound must be well-fluorescing. They put $\tilde{\nu }_{f}^{D} \equiv {\text{constant}}$, since they consider $\mu_{e}^{D} = \mu_{g}^{D}$, and then they feel confident of making the “safe” assumption that the solvatochromic shift of the florescence peak, if any, will be negligible compared to the shift of the absorption peak, i.e. $\tilde{\nu }_{f}^{D} \left( s \right) = {\text{constant}}$ for any solvent s, but this is not necessarily true. (In particular in the case of the dye A, according to our PBE0 [6]/SNSD [219] calculations, the energy of the relaxed excited state is affected by the polarity of the medium; vide infra) Anyway, they write Eq. (D.2) for the dye m = D, as:

$$ \begin{array}{*{20}l} {Y\left( {D,s} \right) = B\left( D \right)X\left( s \right) + Y^{ \circ } \left( {D,g} \right)} \hfill \\ {\tilde{\nu }_{a}^{D,s} - {\text{constant}} = \frac{{2\left( {\Delta \mu_{D} } \right)^{2} }}{{hca_{D}^{3} }}X\left( s \right) + Y^{ \circ } \left( {D,g} \right)} \hfill \\ {\tilde{\nu }_{a}^{D,s} = \frac{{2\left( {\Delta \mu_{D} } \right)^{2} }}{{hca_{D}^{3} }}X\left( s \right) + Y^{{ \circ {\prime }}} ,} \hfill \\ \end{array} $$

where $Y^{{ \circ {\prime }}} \equiv Y^{ \circ } \left( {D,g} \right) + {\text{constant}}\,{ = }\left[ {\tilde{\nu }_{a}^{D,g} - \tilde{\nu }_{f}^{D,g} } \right] + \tilde{\nu }_{f}^{D,s}$, and thus:

$$ X\left( s \right) = \frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }}\tilde{\nu }_{a}^{D,s} - Y^{{ \circ {\prime\prime}}} , $$

(E.4)

where $Y^{{ \circ {\prime }}} \equiv \frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }}Y^{{ \circ {\prime\prime}}}$ [259]. (The Authors state that the nature of the function $X\left( s \right)$ can be left unspecified, but the mathematical expression for the polarity function is well-defined once the theoretical framework is chosen.) From Eq. (E.1), we have:

$$ \tilde{\nu }_{a}^{D,s} = E_{T} \left( {30} \right)/2.8591 \times 10^{ - 3} = 349.7604E_{T} \left( {30} \right), $$

(E.5)

whereas from Eq. (D.9), we have:

$$ E_{T} \left( {30} \right) = 32.4E_{T}^{N} + 30.7, $$

(E.6)

so that Eq. (D.6) becomes:

$$ \tilde{\nu }_{a}^{D,s} = 349.7604\left( {32.4E_{T}^{N} + 30.7} \right) = 1 1 3 3 2. 2E_{T}^{N} + 1 0 7 3 7. 6. $$

(E.6’)

(Please, note that the Authors actually report slightly different values of the numerical constants, in particular: 11307.6 instead of 11332.2. In the algebraic derivation, we shall continue in using the value of the numerical constants that we have found, but during the numerical evaluations, we shall adopt their equation as is, vide infra.) Substituting into Eq. (E.4), we have:

$$ \begin{aligned} X\left( s \right) & = \frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }}\left( { 1 1 3 3 2. 2E_{T}^{N} + 1 0 7 3 7. 6} \right) - Y^{{ \circ {\prime }}} \\ & = 1 1 3 3 2. 2\frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }}E_{T}^{N} + Y^{{ \circ {\prime \prime }}} , \\ \end{aligned} $$

(E.7)

where $Y^{{ \circ {\prime }}} \equiv \left( { 1 0 7 3 7. 6- Y^{{ \circ {\prime }}} } \right)\frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }} = \left\{ { 1 0 7 3 7. 6- \left[ {\left( {\tilde{\nu }_{a}^{D,g} - \tilde{\nu }_{f}^{D,g} } \right) + \tilde{\nu }_{f}^{D,s} } \right]} \right\}\frac{{hca_{D}^{3} }}{{2\left( {\Delta \mu_{D} } \right)^{2} }}$. Finally, substituting Eq. (D.7) into Eq. (D.2) instead of the correct $X\left( s \right)$ function, we have:

$$ \tilde{\nu }_{a}^{m,s} - \tilde{\nu }_{f}^{m,s} = 1 1 3 3 2. 2\frac{{\left( {\Delta {\varvec{\upmu}}} \right)^{2} /a^{3} }}{{\left( {\Delta \mu_{D} } \right)^{2} /a_{D}^{3} }}E_{T}^{N} + Y*\left( {m,g} \right), $$

(E.8)

where $Y*\left( {m,g} \right) \equiv Y^{ \circ } \left( {m,g} \right) + \frac{{2\left( {\Delta {\varvec{\upmu}}} \right)^{2} }}{{hca^{3} }}Y^{{ \circ {\prime \prime }}} .$