Mechanisms of the Formation of the ν₃ Band Contour in Absorption and Raman Scattering Spectra of Tetrafluoromethane in Condensed Phases

Abstract

Within the framework of this investigation, the mechanisms of the formation of complex contours of ν₃ and ν₁ + ν₃ bands of tetrafluoromethane (CF₄) are studied by methods of absorption and Raman scattering spectroscopy in condensed low-temperature systems—fluids, plastic crystals, and low-temperature solutions in liquid argon. In this work, an experimental Raman spectrum of CF₄ in the liquid phase is obtained. Using the spectrum, parameters of the rotational collapse under conditions of a fluid are determined and quantitative data about band contours of Raman spectrum in the fundamental spectral region are obtained. The main aim of the investigation is to determine contributions of different mechanisms (such as resonant dipole–dipole interaction, Fermi resonance, and hindered rotation) to the formation of contours of the considered bands. To solve the posed problem, the mathematical apparatus of spectral moments is used. As a result, it is shown that the resonant dipole–dipole interaction the effect of which is also calculated in this work is the dominating mechanism in the formation of these contours.

APPENDIX

In the Appendix, we consider the solution sequence for the secular equation of order 3N² which describes the RDD interaction of N CF₄ molecules in the spectral region of the ν₁ + ν₃ combination vibration, where the ν₁ vibration is inactive in absorption and the ν₃ vibration is strong in dipole absorption.

The dipole moment function of each molecule can be expanded in a Taylor series in dimensionless vibrational coordinates q: ${{P}^{{(n)}}}$ = $P_{i}^{{(n)}}{\kern 1pt} '{\kern 1pt} q_{i}^{{(n)}}$ + $P_{j}^{{(n)}}{\kern 1pt} '{\kern 1pt} q_{j}^{{(n)}}$ + $P_{{ij}}^{{(n)}}{\kern 1pt} ''{\kern 1pt} q_{i}^{{(n)}}q_{j}^{{(n)}}$ + …, where the subscripts are numbers of normal vibrations and the superscripts are numbers of molecules. This expansion includes a descent with respect to the parameter $P_{{ij}}^{{''}}{\text{/}}P_{i}^{'}$.

The perturbation related to the RDD interaction of a system (see expression (3)) consisting of N molecules in a fluid can be represented as a sum of pairwise interactions:

$$\begin{gathered} V = \frac{1}{{hcR_{0}^{3}}}\sum\limits_{n \ne m}^N {[P_{i}^{{\left( n \right)}}{\kern 1pt} '{\kern 1pt} q_{i}^{{\left( n \right)}} + P_{{ij}}^{{\left( n \right)}}{\kern 1pt} ''{\kern 1pt} q_{i}^{{\left( n \right)}}q_{j}^{{\left( n \right)}}]} \\ \times \;[P_{i}^{{\left( m \right)}}{\kern 1pt} '{\kern 1pt} q_{i}^{{\left( m \right)}} + P_{{ij}}^{{\left( m \right)}}{\kern 1pt} ''{\kern 1pt} q_{i}^{{\left( m \right)}}q_{j}^{{\left( m \right)}}]{{\left( {\frac{{{{R}_{0}}}}{{{{R}_{{nm}}}}}} \right)}^{3}}f\left( {{{\theta }_{{nm}}}} \right). \\ \end{gathered} $$

(A.1)

In the zero approximation of the stationary perturbation theory, the wave function of the whole molecular system L is represented as a product of wave functions of all molecules of the system, $\Psi _{L}^{0}$ = $\prod\nolimits_{i = 1}^N {\Psi _{V}^{{\left( i \right)}}}$, each of them depends on its own set of vibrational quantum numbers; $E_{L}^{0}$ = $\sum\nolimits_{i = 1}^N {E_{V}^{{\left( i \right)}}}$ is the vibration energy of the molecular system obtained in the approximation that takes into account the anharmonicity in the second order of the perturbation theory.

To simplify the formulations, we introduce the following notation: the c-level (combinational vibration level) is the level corresponding to the combination vibration (ν₁ + ν₃)⁽ⁿ⁾ of a single molecule n and the s‑level (simultaneous vibration level) is the level corresponding to the simultaneous vibrational transition (ν₁)⁽ⁿ⁾ + (ν₃)^(m) on a pair of molecules n and m. The transitions corresponding to these levels are denoted as the c-transition and the s-transition.

Let us consider the form of the secular equation for the case of the RDD interaction of fundamental -nondegenerate vibrations ν_i in a system consisting of N molecules:

$$\begin{array}{*{20}{c}} {}&1&2&3&{...}&N \\ {{{{\left( {{{\nu }_{i}}} \right)}}^{{(1)}}}}&{\left\langle {1;} \right.}&{0;}&{0;}&{...}&{\left. 0 \right|} \\ {{{{\left( {{{\nu }_{i}}} \right)}}^{{(2)}}}}&{\left\langle 0 \right.;}&{1;}&{0;}&{...}&{\left. 0 \right|} \\ {{{{\left( {{{\nu }_{i}}} \right)}}^{{(3)}}}}&{\left\langle 0 \right.;}&{0;}&{0;}&{...}&{\left. 0 \right|} \\ {...}&{...}&{...}& \ldots &{...}&{...} \\ {{{{\left( {{{\nu }_{i}}} \right)}}^{{(N)}}}}&{\left\langle {0;} \right.}&{0;}&{0;}&{...}&{\left. 1 \right|} \end{array}\left| {\begin{array}{*{20}{c}} 1&2&3&{...}&N \\ { - \lambda }&{{{a}_{{12}}}f\left( {{{\vartheta }_{{12}}}} \right)}&{{{a}_{{13}}}f\left( {{{\vartheta }_{{13}}}} \right)}&{...}&{{{a}_{{1N}}}f\left( {{{\vartheta }_{{1N}}}} \right)} \\ {{{a}_{{21}}}f\left( {{{\vartheta }_{{21}}}} \right)}&{ - \lambda }&{{{a}_{{23}}}f\left( {{{\vartheta }_{{23}}}} \right)}&{...}&{{{a}_{{2N}}}f\left( {{{\vartheta }_{{2N}}}} \right)} \\ {{{a}_{{31}}}f\left( {{{\vartheta }_{{31}}}} \right)}&{{{a}_{{32}}}f\left( {{{\vartheta }_{{32}}}} \right)}&{ - \lambda }&{...}&{{{a}_{{3N}}}f\left( {{{\vartheta }_{{3N}}}} \right)} \\ {...}&{...}&{...}&{...}&{...} \\ {{{a}_{{N1}}}f\left( {{{\vartheta }_{{N1}}}} \right)}&{{{a}_{{N2}}}f\left( {{{\vartheta }_{{N2}}}} \right)}&{{{a}_{{N3}}}f\left( {{{\vartheta }_{{N3}}}} \right)}&{...}&{ - \lambda } \end{array}} \right|.$$

In this equation, the matrix element of the interaction is determined by values of the parameters a and f.

Let us introduce the following simplifications.

1. In the expression for the perturbation of system (A.1), the cross products are not taken into account. The perturbation is divided into two parts which are essentially different in the magnitude and correlate with the decrease in expansion parameters of the dipole moment function of the molecule. One can distinguish two types of matrix elements of the interaction.

The matrix element of the interaction

$$W = \frac{1}{{hc{{R}_{0}}^{3}}}{{\left( {\frac{{{{R}_{0}}}}{{{{R}_{{nm}}}}}} \right)}^{3}}P_{i}^{{\left( n \right)}}{\kern 1pt} '{\kern 1pt} P_{i}^{{\left( m \right)}}{\kern 1pt} '{\kern 1pt} f({{\theta }_{{nm}}}) = a{{\left( {\frac{{{{R}_{0}}}}{{{{R}_{{nm}}}}}} \right)}^{3}}f({{\theta }_{{nm}}})$$

corresponds to two types of the interaction: (i) of (ν₁)⁽ⁿ⁾ + (ν₃)^(m) and (ν₁)⁽ⁿ⁾ + (ν₃)^(l) s-levels, where $n \ne m \ne l$; and (ii) of (ν₁ + ν₃)⁽ⁿ⁾ c-level and (ν₁)⁽ⁿ⁾ + (ν₃)^(m) s-level, where n ≠ m and n is fixed.

The matrix element of the interaction

$$w = \frac{1}{{hc{{R}_{0}}^{3}}}{{\left( {\frac{{{{R}_{0}}}}{{{{R}_{{nm}}}}}} \right)}^{3}}P_{{ij}}^{{\left( n \right)}}{\kern 1pt} ''{\kern 1pt} P_{{ij}}^{{\left( m \right)}}{\kern 1pt} ''{\kern 1pt} f({{\theta }_{{nm}}}) = b{{\left( {\frac{{{{R}_{0}}}}{{{{R}_{{nm}}}}}} \right)}^{3}}f({{\theta }_{{nm}}})$$

corresponds to two types of the interaction: (i) of (ν₁ + ν₃)⁽ⁿ⁾ and (ν₁ + ν₃)^(m) c-levels, where n ≠ m; and (ii) of (ν₁)⁽ⁿ⁾ + (ν₃)^(m) and (ν₁)^(m) + (ν₃)⁽ⁿ⁾ s-levels, where n ≠ m.

When composing the secular equation, we omit indices at the parameter a because only the order of its magnitude is essential in this consideration. In addition, we omit the factor (R₀/R_nm)³f(θ_nm) in the secular equation because it has no effect on the order of magnitude of the matrix element of the interaction.

2. When solving the secular equation, we neglect the Fermi resonance because it has no effect on the contour broadening.

3. The ν₃ vibration is triply degenerate; for simplicity, however, we consider the case of a one-dimensional oscillator (for example, the x-component of the dipole moment) with the corresponding state 〈v₁v₃|.

Taking into account the simplifications above, we represent the secular equation in the following form: the equation consists of (N + 1) blocks; the first block takes into account the interaction between c-states; and the next blocks take into account the interaction between s-states:

$$\left| {\begin{array}{*{20}{c}} {}&1&2&3&{ \ldots N}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{} \\ {{{{({{\nu }_{1}} + {{\nu }_{3}})}}^{{(1)}}}}&{\left\langle {11;} \right.}&{00;}&{00;}&{\left. { \ldots 00} \right|}&{ - \lambda }&b&b&{ \ldots b}&a&a&{a \ldots }&a&0&0&{0 \ldots }&0 \\ {{{{({{\nu }_{1}} + {{\nu }_{3}})}}^{{(2)}}}}&{\left\langle {00;} \right.}&{11;}&{00;}&{\left. { \ldots 00} \right|}&b&{ - \lambda }&{b \ldots }&b&0&0&{0 \ldots }&0&0&0&{0 \ldots }&0 \\ {{{{({{\nu }_{1}} + {{\nu }_{3}})}}^{{(3)}}}}&{\left\langle {00;} \right.}&{00;}&{11}&{\left. { \ldots 00} \right|}&b&b&{ - \lambda \ldots }&b&0&0&{0 \ldots }&0&0&0&{0 \ldots }&0 \\ {...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...} \\ {{{{({{\nu }_{1}} + {{\nu }_{3}})}}^{{(N)}}}}&{\left\langle {00;} \right.}&{00;}&{00}&{\left. { \ldots 01} \right|}&b&b&{b \ldots }&{ - \lambda }&0&0&{00}&0&a&a&{a \ldots }&a \\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{} \\ {{{{({{\nu }_{1}})}}^{{(1)}}} + {{{({{\nu }_{3}})}}^{{(2)}}}}&{\left\langle {10;} \right.}&{01;}&{00;}&{\left. { \ldots 00} \right|}&a&0&{0 \ldots }&0&{\Delta - \lambda }&a&{a \ldots }&a&0&0&{0 \ldots }&0 \\ {{{{({{\nu }_{1}})}}^{{(1)}}} + {{{({{\nu }_{3}})}}^{{(3)}}}}&{\left\langle {10;} \right.}&{00;}&{01;}&{\left. { \ldots 00} \right|}&a&0&{0 \ldots }&0&a&{\Delta - \lambda }&{a \ldots }&a&0&0&{0 \ldots }&0 \\ {...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}& \ldots &{...} \\ {{{{({{\nu }_{1}})}}^{{(1)}}} + {{{({{\nu }_{3}})}}^{{(N)}}}}&{\left\langle {10;} \right.}&{00;}&{00;}&{\left. { \ldots 01} \right|}&a&0&{0 \ldots }&0&a&a&{a \ldots }&{\Delta - \lambda }&b&0&{0 \ldots }&0 \\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{} \\ {{{{({{\nu }_{1}})}}^{{(2)}}} + {{{({{\nu }_{3}})}}^{{(1)}}}}&{\left\langle {01;} \right.}&{10;}&{00}&{\left. { \ldots 00} \right|}&0&a&{0 \ldots }&0&b&0&{0 \ldots }&0&0&0&{0 \ldots }&0 \\ {{{{({{\nu }_{1}})}}^{{(2)}}} + {{{({{\nu }_{3}})}}^{{(3)}}}}&{\left\langle {00;} \right.}&{10;}&{01;}&{\left. { \ldots 00} \right|}&0&a&{0 \ldots }&0&0&0&{0 \ldots }&0&0&0&{0 \ldots }&0 \\ {...}&{}&{}&{}&{}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...} \\ {{{{({{\nu }_{1}})}}^{{(2)}}} + {{{({{\nu }_{3}})}}^{{(N)}}}}&{\left\langle {00;} \right.}&{10;}&{00;}&{\left. { \ldots 01} \right|}&0&a&{0 \ldots }&0&0&0&{0 \ldots }&0&0&b&{0 \ldots }&0 \\ {}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{}&{} \\ {{{{({{\nu }_{1}})}}^{{(N)}}} + {{{({{\nu }_{3}})}}^{{(1)}}}}&{\left\langle {01;} \right.}&{00;}&{00;}&{\left. {10} \right|}&0&0&{0 \ldots }&a&0&0&{0 \ldots }&b&{\Delta - \lambda }&a&{a \ldots }&a \\ {{{{({{\nu }_{1}})}}^{{(N)}}} + {{{({{\nu }_{3}})}}^{{(2)}}}}&{\left\langle {00;} \right.}&{01;}&{00;}&{\left. {10} \right|}&0&0&{0 \ldots }&a&0&0&{0 \ldots }&0&a&{\Delta - \lambda }&{a \ldots }&a \\ \ldots & \ldots & \ldots & \ldots & \ldots &{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...}&{...} \\ {{{{({{\nu }_{1}})}}^{{(N)}}} + {{{({{\nu }_{3}})}}^{{(N - 1)}}}}&{\left\langle {00;} \right.}&{00;}&{ \ldots 01;}&{\left. {10} \right|}&0&0&{0 \ldots }&a&0&0&{00}&0&a&a&{a \ldots }&{\Delta - \lambda } \end{array}} \right| = 0.$$

(A.2)

It is seen from the secular equation (A.2) that the first block of the equation corresponds to the interaction between (ν₁ + ν₃)⁽ⁿ⁾ c-states, where n = 1 : N; this block has the same structure as in the fundamental region of the spectrum. The only difference is that the interaction occurs through the small parameter b. The second spectral moment is determined as $M{{(2)}_{{{\text{Comb}}}}}$ = $\sum\nolimits_{i,k = 1}^N {b_{{ik}}^{2}}$. In the absorption spectrum, such contour depends mainly on contributions of rotation and anharmonicity effects. In our case, however, the presence of s-transitions in the secular equations leads to an extension of the exciton zone due to the large value of the parameter a. Let us write the same secular equation in a different order. The equation will consist of N blocks and each block will include the interaction of one (ν₁ + ν₃)⁽ⁿ⁾ c-state with (N – 1) (ν₁)⁽ⁿ⁾ + (ν₃)^(m) s‑states:

(A.3)

It is seen from Eq. (A.3) that the interaction inside each diagonal subblock occurs through the parameter a, which leads to the value of the second spectral moment $M{{(2)}_{{{\text{sim}}\,{\text{ult}}}}}$ = $\sum\nolimits_{i,k = 1}^N {a_{{ik}}^{2}}$.

The total second spectral moment of the OPDS contour of the ν₁ + ν₃ band is a sum of second spectral moments of two mechanisms:

$$\begin{gathered} M{{(2)}_{{{\text{OPDS}}}}}\left( {{{{{\nu }}}_{{\text{1}}}}\;{\text{ + }}\;{{{{\nu }}}_{{\text{3}}}}} \right) = M{{(2)}_{{{\text{comb}}}}} + M{{(2)}_{{{\text{sim}}\,{\text{ult}}}}} \\ = \sum\limits_{i,k = 1}^N {b_{{ik}}^{2}} + \sum\limits_{i,k = 1}^N {a_{{ik}}^{2}} = \sum\limits_{i,k = 1}^N {\left( {b_{{ik}}^{2} + a_{{ik}}^{2}} \right)} . \\ \end{gathered} $$

Thus, the characteristic half-width of the OPDS contour is $2\sqrt {M{{{(2)}}_{{{\text{OPDS}}}}}}$ = $\sqrt {\sum\nolimits_{i,k = 1}^N {(b_{{ik}}^{2} + a_{{ik}}^{2})} }$ and is determined mainly by the interaction of s-states, i.e., by simultaneous transitions.