Modification of Vibrational Parameters of a D_∞h-Symmetric Triatomic Molecule in a Laser Plasma

Abstract

We estimate theoretically the strong-field-modified vibrational parameters of three-atomic $ABA$ molecules with $D_{\infty h}$ symmetry in a laser-induced plasma. The linear CO${}_2$ and CS${}_2$ molecules in the $X{}^1{\mathsf{\Sigma }}_g$ state are considered as examples. We show that double degeneracy of the ${\mathsf{\Pi }}_u$ mode is removed due to reduction of the $ABA$ molecule symmetry by the laser field. The linear form of $ABA$ molecule is preserved, the bond length being elongated.

1. Introduction

The interaction of laser radiation with polyatomic molecules is studied actively both theoretically and experimentally. For instance, Ref. [1] studied experimentally a rearrangement of atoms due to two-electron dissociative ionization of CO${}_2$, OCS, and D${}_2$O triatomics by laser radiation in near-infrared (NIR) range. In the resulting laser plasma, the atomic ${\mathrm{O}}_2^{+}+{\mathrm{C}}^{+}$, ${\mathrm{SO}}^{+}+{\mathrm{C}}^{+}$, ${\mathrm{D}}_2^{+}+{\mathrm{O}}^{+}$ ions were detected. Refs. [2,3] report an experimental study of two-electron dissociative ionization of water isotopologues by femtosecond laser pulses. Experimental and theoretical study of Coulomb explosion resulting from ionization of OCS${}^{+}$ using a strong IR-laser pump and probe technique was presented in Ref. [4].

It should be noted that laser radiation influences not only the dissociation fragments, but also on the nuclear motion in the neutral molecule. The mechanism of this influence can be comparatively simply explained theoretically using the concept of “laser dressing”: the laser field polarizes the electron shells, and this results in a deformation of the potential energy surfaces (PES). This approach was proposed in Ref. [5] for the DC field, and it was developed in Ref. [6] for calculation of non-adiabatic susceptibilities of molecules. Under the DC field, PES are modified by some corrections proportional to some powers of the field strength. In optical fields, the odd-order corrections vanish [7], and the modified PES are expanded in powers of the radiation intensity, I.

The role of the laser-modified nuclear motion in a molecule is also important for the material science. In particular, recent paper [8] presents the experimental study of a vibroelectronic quantum dots–molecule coupling based on dipolar interaction. Namely, the amplitude of the vibrational response of the phenyltriethoxysilane ligands (over the infrared range) was modulated by the linear optical susceptibility of the Cd Te${}_{0.25}$S${}_{0.75}$ quantum dots (over the visible range). These phenomena are employed for the design of optical biosensors with a low detection threshold used in nonlinear spectroscopy.

The laser-field modification of vibrational parameters (frequency and bond length) of the diatomics having $C_{\infty v}$ or $D_{\infty h}$ symmetry was first studied theoretically by Zon in [9]. In Ref. [10], it was shown that taking into account the laser-field modification of the vibrational parameters can change the tunnelling ionization rate (and, consequently, the rate of laser plasma generation) by a factor of up to 2.5.

Using the normal mode formalism, the above approach was generalized on arbitrary polyatomic molecules. In Ref. [11], the laser-induced deformation of nonlinear symmetric triatomics $ABA$ ($C_{2v}$ symmetry) was theoretically studied for H${}_2$O and SO${}_2$. These molecules have three non-degenerate normal modes. Two of them are of $A_1$ symmetry and correspond to (i) symmetric vibrations of the A atoms with respect to the B atom and (ii) scissoring vibrations; the third $B_1$ mode corresponding to antisymmetric vibrations of A with respect to B. Such molecules demonstrate only quantitative modification of their vibrational parameters in a laser field. The vibration frequencies (as well as the normal modes) undergo some shifts; bond length $\left(AB\right)$ and bending angle $\angle \left(ABA\right)$ are changed. In the lowest order in intensity, all these changes are proportional to I. The $C_{2v}$ symmetry is preserved in the laser field as well. However, the tunnelling ionization rate, e.g., for the H${}_2$S molecule, can be changed by a factor of 20 due to the modification of vibrational parameters and the Franck–Condon factors (FCFs).

We note the importance of FCFs in the theoretical study of optical phenomena not only in molecules in a gas (or laser plasma) but also in material science. Many publications are devoted to calculation of the FCFs in a condensed matter. In particular, Ref. [12] considers the solvent-modified FCFs in diatomics. The authors of [13] calculated the molecular spectra of metal-based tetrapyrrole derivatives in pigments. The authors of [14,15] investigated the vibrational spectra of 2D structures. Refs. [16,17] deal with the FCFs in fullerenes. For photoluminescence in solids, the FCFs were involved, e.g., in [18].

In the present work, we study the influence of the laser field on the vibrational parameters of linear $ABA$ triatomics that possess the higher symmetry, $D_{\infty h}$. The degeneracy of vibrational modes, which is inherent to such molecules, can be removed by the laser radiation.

As the examples, we consider CO${}_2$ and CS${}_2$ molecules that have the same electronic terms, $X{}^1{\mathsf{\Sigma }}_g$. The carbon dioxide, CO${}_2$, is abundant in the Earth’s atmosphere and therefore can influence the propagation of ionizing laser radiation and laser plasma filamentation [19]. The carbon disulphide, CS${}_2$, is an important reagent in organic synthesis [20], so the influence of a laser radiation on its vibrational parameters can be of interest for laser chemistry [21].

The theoretical model is presented in Section 2. Section 2.1 contains the general formulas which determine the laser-field modification of the vibrational parameters. This formalism is applied to linear $ABA$ molecules in Section 2.2.

The numerical results are obtained and discussed in Section 3. The main conclusions are given in Section 4.

The atomic units ($\hslash =e=m_e=1$) are used throughout this work.

2. Methods

2.1. General Formalism of Vibrational-Parameters Modification in a Laser Field

For simplicity, we consider the modification of vibrational parameters of N-atomic molecules in a monochromatic linearly polarized laser field whose electric vector has the form:

$$\mathbf{F}\left(t\right)=\mathbf{u}Fcos\omega t.$$

(1)

Here, F is the laser wave’s electric field amplitude, $\omega$ is its frequency, $\mathbf{u}$ is the unit polarization vector.

In the laser field (1), the molecule’s energy acquires the dynamic Stark shift:

$$\Delta E_{\mathrm{Stark}}=-\frac{1}{2}\alpha (\omega ,\mathbf{u})F^2.$$

(2)

Here,

$$\alpha (\omega ,\mathbf{u})={\mathbf{u}}^T\widehat{\alpha }\left(\omega \right)\mathbf{u},$$

(3)

${\mathbf{u}}^T$ is the row-vector, and $\widehat{\alpha }\left(\omega \right)$ is the symmetric tensor of dynamic (i.e., frequency-dependent) electric dipole polarizability. In the Cartesian coordinates, one has

$$\alpha (\omega ,\mathbf{u})=\sum _{i,j=x,y,z}{\alpha }_{ij}\left(\omega \right)u_i{u}_j,{\alpha }_{ij}\left(\omega \right)={\alpha }_{ji}\left(\omega \right).$$

(4)

In the NIR range, the laser radiation frequency, $\omega$, is substantially low as compared to the resonance frequencies of the molecular electron shells, so we can use the static polarizability instead of the dynamic one: $\widehat{\alpha }\left(\omega \right)\approx \widehat{\alpha }$.

In the Born–Oppenheimer approximation, the Stark shift (2) results in a deformation of molecular PES. In particular, the molecular geometry can be deformed due to the shift of equilibrium positions of the nuclei, and the vibrational frequencies can be modified due to the changes in PES curvature.

The small vibrations of polyatomic molecules are convenient to study in the normal mode formalism. In the classical approach, instead of real vibrations of N nuclei, one can consider independent harmonic vibrations of f fictitious particles. The latter are the normal modes which can be expressed as linear combinations of the real vibrations, f being the number of the vibrational degrees of freedom. For the linear molecules (having $C_{\infty v}$ or $D_{\infty h}$ symmetry), one has $f=3N-5$. For the other molecules, $f=3N-6$.

The vibrational Lagrangian in the normal coordinates takes the form:

$$\mathcal{L}\left(Q\right)=\sum _{k=1}^f{\mathcal{L}}_k\left(Q_k\right),{\mathcal{L}}_k\left(Q_k\right)=\frac{1}{2}{M}_k{\dot{Q}}_k^2-\frac{1}{2}{M}_k{\mathsf{\Omega }}_{0k}^2{(Q_k-Q_{ek})}^2.$$

(5)

Here, $Q\equiv \{Q_1,\dots ,Q_k\}$; $Q_k$, $M_k$ and ${\mathsf{\Omega }}_{0k}$ are the coordinate, reduced mass and frequency of the k-th normal mode, respectively; $Q_{ek}$ is the equilibrium position of the k-th normal coordinate. The observable quantities are the frequencies, ${\mathsf{\Omega }}_{0k}$, which are supposed to be known for each particular molecule. The reduced masses and normal coordinates are determined to within the simultaneous substitutions,

$$M_k\to {\beta }_k{M}_k,Q_k\to Q_k/\sqrt{{\beta }_k},Q_{0k}\to Q_{0k}/\sqrt{{\beta }_k},{\beta }_k=\mathrm{const},$$

(6)

which leave the Lagrangian (5) invariant.

The deformation of the molecule is determined by the changes of its normal coordinates, Q, so the polarizability tensor, $\widehat{\alpha }$, depends on Q. The normal modes, ${\mathsf{\Omega }}_{0k}$, and the equilibrium normal coordinates, $Q_e$, are modified by the laser field (1) via the deformation of PES. The correspondent formulas are derived in Ref. [11] through the Hessian matrix. Up to the terms of the order of $\sim F^2$, the non-degenerate normal modes are modified according to the formula

$${\mathsf{\Omega }}_k\approx {\mathsf{\Omega }}_{0k}-\frac{{\alpha }^{\left(2kk\right)}\left(\mathbf{u}\right)}{8M_k{\mathsf{\Omega }}_{0k}}{F}^2,$$

(7)

and the equilibrium normal coordinates are modified according to

$$Q_{Ek}=Q_{ek}+\frac{{\alpha }^{\left(1k\right)}\left(\mathbf{u}\right)}{4M_k{\mathsf{\Omega }}_{0k}^2}{F}^2.$$

(8)

Here, ${\mathsf{\Omega }}_k$ and $Q_{Ek}$ are the laser-modified normal frequencies and the equilibrium normal coordinates, respectively, and the derivatives

$${\alpha }^{\left(1k\right)}\left(\mathbf{u}\right)={\left.\frac{\partial }{\partial Q_k}\alpha \left(\mathbf{u}\right)\right|}_{Q_e},{\alpha }^{\left(2k{k}^{\prime }\right)}\left(\mathbf{u}\right)={\left.\frac{{\partial }^2}{\partial Q_k\partial Q_{k^{\prime }}}\alpha \left(\mathbf{u}\right)\right|}_{Q_e}$$

(9)

are calculated numerically at the equilibrium geometry using the quantum chemistry methods.

The molecular deformations (changes in the bond length, as well as in bending, dihedral, and torsion angles) are calculated from $Q_{Ek}$ using an algorithm which is specific for the symmetry of a particular molecule. This algorithm is based on a method of normal-to-Cartesian coordinate transition.

If some vibrational modes (say, k and $k^{\prime }$) are degenerate (i.e., $M_k=M_{k^{\prime }}$ and ${\mathsf{\Omega }}_{0k}={\mathsf{\Omega }}_{0k^{\prime }}$), this degeneracy can be removed by the laser field. In the general case, the field shift of the frequencies is determined by the following expression [11]:

$$\begin{array}{c}{\mathsf{\Omega }}_k^{(\pm )}\approx {\mathsf{\Omega }}_{0k}-\frac{F^2}{16M_k{\mathsf{\Omega }}_{0k}}\left\{{\alpha }^{\left(2kk\right)}\left(\mathbf{u}\right)+{\alpha }^{\left(2k^{\prime }{k}^{\prime }\right)}\left(\mathbf{u}\right)\phantom{\left(\frac{f_f}{f_f}\right)}\right.\hfill \\ \hfill \pm \left.{\left[{\left({\alpha }^{\left(2kk\right)}\left(\mathbf{u}\right)-{\alpha }^{\left(2k^{\prime }{k}^{\prime }\right)}\left(\mathbf{u}\right)\right)}^2+4{\left({\alpha }^{\left(2k{k}^{\prime }\right)}\left(\mathbf{u}\right)\right)}^2\right]}^{1/2}\right\}.\end{array}$$

(10)

Note that the corresponding modes can be mixed in the general case.

Once all the ${\mathsf{\Omega }}_k$ and $Q_{Ek}$ are determined, a standard quantization procedure can be applied to the molecular vibrations.

2.2. Normal Modes in a $D_{\infty h}$-Symmetric Triatomic Molecule

In a symmetric linear triatomic $ABA$ molecule with the equilibrium bond length, $\left(AB\right)=l_e$, there are four vibrational degrees of freedom which are shown in Figure 1 (see also [22], Section 24).

The classical interpretation of the ${\mathsf{\Sigma }}_g$ mode is the in-phase vibrations of the A atoms along the resting molecular axis, z (symmetric stretching). The bond lengths $\left(AB\right)$ change symmetrically; therefore, we will refer to such vibrations as symmetric and label them with the “s” subscript, for instance, the frequency $\mathsf{\Omega }\left({\mathsf{\Sigma }}_g\right)$ will be denoted as ${\mathsf{\Omega }}_s$.

The ${\mathsf{\Sigma }}_u$ mode, in classical interpretation, is due to out-of-phase vibrations of the A atoms along the resting molecular axis, z (antisymmetric stretching). The B is also moving along the z-axis to ensure that the center of mass is resting. The bond lengths $\left(AB\right)$ change antisymmetrically; we will refer to such vibrations as antisymmetric and label them with the “a” subscript.

The ${\mathsf{\Sigma }}_s$ and ${\mathsf{\Sigma }}_u$ modes can be referred to as longitudinal modes.

The ${\mathsf{\Pi }}_u$ mode is due to molecular bending with equal bond lengths $\left(AB\right)$. This mode is labelled with the “b” subscript. This mode is doubly degenerate because of axial symmetry of the non-deformed molecule. Indeed, this bending motion can take place in two mutually orthogonal directions, e.g., along the x- or y-axis. These transverse modes are labelled with the “$bx$” and “$by$” subscripts.

The normal coordinates in the above listed modes are related to the Cartesian coordinates of the nuclei according to Figure 2. We enumerate the atoms as follows: (1) “upper” A atom, (2) B atom, and (3) “lower” A atom. Their masses are denoted as $m_A$ and $m_B$, respectively.

The longitudinal normal coordinates, $Q_s$ and $Q_a$, are determined as follows [22]:

$$z_1=(Q_a+Q_s)/2,z_2=-\frac{m_A}{m_B}{Q}_a,z_3=(Q_a-Q_s)/2.$$

(11)

The bending angles, ${\delta }_{x,y}$, can be chosen as transverse normal coordinates (see Figure 2). However, to preserve the length dimension for all normal coordinates, we will use $Q_{bx}=l_e{\delta }_x$ and $Q_{by}=l_e{\delta }_y$ instead. The relationship between the normal coordinates corresponding to the motion along the x-axis with the Cartesian coordinates of the nuclei is given by the equations,

$$x_1=x_3=\frac{m_B}{2M}{Q}_{bx},x_2=-\frac{m_A}{M}{Q}_{bx},M=2m_A+m_B.$$

(12)

and the similar relationships for the motion along the y-axis. The transformations (11) and (12) eliminate the motion of the center of mass as well the overall rotation of the molecule.

The vibrational Lagrangian of a $ABA$ molecule in normal coordinates is separated into a sum of independent linear harmonic oscillator Lagrangians:

$$\mathcal{L}={\mathcal{L}}_s+{\mathcal{L}}_a+{\mathcal{L}}_{bx}+{\mathcal{L}}_{by},$$

where each term has the form (5). The reduced masses, $M_k$, of the normal mode oscillators are expressed in terms of the masses of the nuclei:

$$M_s=m_A/2,M_a=\frac{m_{A}M}{2m_B},M_{bx}=M_{by}=M_b=\frac{m_A{m}_B}{2M}.$$

(13)

The equilibrium positions of the nuclei are

$$z_{e1}=-z_{e3}=l_e,z_{e2}=x_{ei}=y_{ei}=0,i=1,2,3.$$

Hence, the equilibrium positions of the normal coordinates in Equation (5), according to Equation (12), are

$$Q_{es}=2l_e,Q_{ea}=Q_{ebx}=Q_{eby}=0.$$

(14)

The $D_{\infty h}$ symmetry is axial. Therefore, the polarizability tensor in principal axes (of those, the z-axis is the molecular rotation axis) has the form $\widehat{\alpha }=\mathrm{diag}({\alpha }_{xx},{\alpha }_{xx},{\alpha }_{zz})$. The interaction of the molecule with the linearly polarized laser radiation (1) is determined only by the angle, $\theta$, between the polarization vector, $\mathbf{u}$, and the z-axis. From the Cartesian components of $\mathbf{u}=(sin\theta ,0,cos\theta )$, one can see that the derivatives (9), with the account for symmetry of the polarizability tensor, take a comparatively simple form:

$${\alpha }^{\left(q\right)}\left(\mathbf{u}\right)={\alpha }_{xx}^{\left(q\right)}{sin}^2\theta +{\alpha }_{zz}^{\left(q\right)}{cos}^2\theta +{\alpha }_{xz}^{\left(q\right)}sin2\theta ,$$

(15)

where $q=1k$ or $2k{k}^{\prime }$, and the Cartesian components of the ${\alpha }_{ij}^{\left(q\right)}$ tensor (i, $j=x$, y, z) are defined according to Equation (9).

Further analysis of the molecular symmetry shows that

$${\alpha }^{\left(1a\right)}\left(\mathbf{u}\right)={\alpha }^{\left(1bx\right)}\left(\mathbf{u}\right)={\alpha }^{\left(1by\right)}\left(\mathbf{u}\right)=0,{\widehat{\alpha }}^{\left(2bxby\right)}=\left(\begin{array}{ccc}0& \xi & 0\\ \xi & 0& 0\\ 0& 0& 0\end{array}\right),\xi \ne 0.$$

Therefore, ${\alpha }^{\left(2bxby\right)}\left(\mathbf{u}\right)=0$, and, as a result of Equations (7) and (10), the frequencies of all the normal modes will be modified by the laser radiation in the same way (this is not correct in the case of elliptic polarization when ${\alpha }^{\left(2bxby\right)}\left(\mathbf{u}\right)\ne 0$). The relative modifications of the normal modes can be calculated by the general formula:

$$\Delta \left({\mathsf{\Omega }}_k\right)=\frac{\Delta {\mathsf{\Omega }}_k}{{\mathsf{\Omega }}_{0k}}=-\frac{{\alpha }^{\left(2kk\right)}\left(\mathbf{u}\right)}{8M_k{\mathsf{\Omega }}_{0k}^2}{F}^2.$$

(16)

Here, $k=s$, a, $bx$, $by$. Due to the degeneracy, one has $M_{bx}=M_{by}=M_b$ and ${\mathsf{\Omega }}_{0bx}={\mathsf{\Omega }}_{0by}={\mathsf{\Omega }}_{0b}$. The reduced masses are calculated according to Equation (13). The doubly degenerate ${\mathsf{\Pi }}_u$ mode (“b” in our notation) is split by two modes. In the classical interpretation, the ${\mathsf{\Omega }}_{bx}$ frequency corresponds to vibrations in the plane formed by the linear polarization vector, $\mathbf{u}$ and the molecular z-axis. The ${\mathsf{\Omega }}_{by}$ frequency corresponds to vibrations in the perpendicular plane. The laser field does not mix these degenerate modes. A similar situation takes place, for instance, in Zeeman effect when the degeneracy can be removed without combining non-perturbed states of the atom.

The accounting for the terms $\sim F^2$ does not reveal modification of the equilibrium quantities of $Q_{ea}$, $Q_{ebx}$, and $Q_{eby}$. Therefore, the $ABA$ molecule retains its linear and symmetric form in a laser field. Since the $ABA$ molecule has an inversion center, this conclusion is valid for an arbitrary intensity of monochromatic radiation, and not only within the perturbation theory. However, in the case of a few-cycle laser pulse or different isotopes of A atoms (e.g., ${}^{18}$OC${}^{16}$O), as well as non-symmetric linear molecules (HCN, OCS, etc.), this problem requires additional study even using the perturbative approach.

Thus, only the ($AB$) bond length is modified due to the modification of $Q_{es}$ (see Equation (8)). The relative change of the ($AB$) bond length is

$$\Delta \left(l\right)=\frac{\Delta l}{l_e}=\frac{{\alpha }^{\left(1s\right)}\left(\mathbf{u}\right)}{8M_s{l}_e{\mathsf{\Omega }}_{0s}^2}{F}^2.$$

(17)

The expressions (16) and (17) are the main formulas in the present work.

3. Numerical Results and Discussion

In this section, we present the results of calculation of the vibrational parameters of CO${}_2$ and CS${}_2$ molecules modified by the laser radiation (1). We assume that these molecules are composed of the most abundant isotopes, ${}^{12}$C, ${}^{16}$O and ${}^{32}$S. Both of the molecules belong to the same symmetry group, $D_{\infty h}$, and we assume that they are in the same ground electronic state, $X{}^1{\mathsf{\Sigma }}_g$. The ionization potentials, $I_p$, bond lengths, $l_e$, isotropic static electric dipole polarizabilities, $\overline{\alpha }={\displaystyle \frac{1}{3}}\mathrm{Sp}\widehat{\alpha }$, and the normal mode frequencies of these molecules are given in

Table 1. Experimental electronic and vibrational parameters of CO${}_2$ and CS${}_2$ molecules in $X{}^1{\mathsf{\Sigma }}_g$ state, according to [23]. The theoretical $\overline{\alpha }$ values were obtained in this work with the CCSD(T) method and (a) aug-cc-pV$\zeta$Z ($\zeta =6$ for CO${}_2$ and 5 for CS${}_2$) and (b) 6-311++G(3df,3pd) basis sets.

Molecule	${\mathit{I}}_{\mathit{p}}$, eV	$\overline{\mathit{\alpha }}$, Å${}^3$			${\mathit{l}}_{\mathit{e}}$, Å	${\mathbf{\Omega }}_0$, cm${}^{-1}$
		Expt.	Theory (a)	Theory (b)		${\mathsf{\Sigma }}_{\mathbf{g}}$	${\mathsf{\Sigma }}_{\mathbf{u}}$	${\mathsf{\Pi }}_{\mathbf{u}}$
CO${}_2$	13.777	2.507	2.584	2.556	1.1621	1333	2349	677
CS${}_2$	10.073	8.749	8.343	8.432	1.5540	658	1535	397

.

The derivatives of the polarizability tensor with respect to normal coordinates in the equilibrium geometry were obtained by five-point numerical differentiation with the same step, 0.005 Å, for all the normal coordinates. The mutual position of nuclei in the deformed molecule at the given normal coordinate positions was defined according to Equations (11) and (12). The polarizability tensor of the deformed molecule was calculated using the CCSD(T) method which proved its efficiency in calculation of static [24] and dynamic [25] polarizabilities of simple molecules. These calculations [24,25] used Dunning’s correlated-consistent basis sets of high quality ($\zeta =5$ or 6). However, these basis sets are shown to have limited capabilities for calculations involving multiple bonds. In deformed molecules, these basis sets do not always ensure smooth dependence of the polarizability tensor $\widehat{\alpha }\left(Q\right)$ on the normal coordinates, and even the symmetry of this tensor. This problem can be solved using alternative (Pople’s 6-311++G(3df,3pd)) valence-split basis sets of a comparable quality (see also Table 1). The calculations were performed with the help of the NWChem package [26], which calculates polarizabilities by solving $\mathsf{\Lambda }$-equations.

We note that the derivative of the static CO${}_2$ polarizability with respect to the C–O bond length was calculated by Maroulis [27] using the CCSD(T) method. Instead of $\mathsf{\Lambda }$-equations, he calculated the polarizability as a result of applying a finite electric field (with the strength of 0.005 a.u.). The basis sets DU1, DU2, DU3, Q1, and Q2 developed by Maroulis [27] to maximize the mean value of quadrupole polarizability. Unfortunately, we are not able to compare our results with those of Ref. [27], since (i) Ref. [27] considered symmetric stretching deformation only (in order to preserve the $D_{\infty h}$ symmetry); (ii) Ref. [27] used the C–O bond length $l_e=2.192$ Å, which differs from the most recent experimental results listed by NIST [23].

CCSD(T) is the most reliable method not only for the polarizabilities (which are determined mainly by deformation of HOMO in electric field), but also for other molecular properties, for instance, spin–orbital interaction in deep shells of molecules containing heavy atoms [28].

For our calculations, it is sufficient to only use the deformations preserving ${\alpha }_{xz}=0$. Thus, according to Equation (15), the relative change of the vibrational parameters (16) and (17) can be parametrized by a very simple expression:

$$\Delta \left(G\right)=[c\left(G\right){cos}^2\theta +s\left(G\right){sin}^2\theta ]I/I_a.$$

(18)

Here, $G={\mathsf{\Omega }}_s$, ${\mathsf{\Omega }}_a$, ${\mathsf{\Omega }}_{bx}$, ${\mathsf{\Omega }}_{by}$, l; I is the radiation intensity, $I_a=3.509\times {10}^{16}$ W/cm${}^2$ is atomic unit of the radiation intensity. The constants, $c\left(G\right)$ and $s\left(G\right)$, obtained in the present work from calculation of the derivatives of the polarizability tensor, are given in

Table 2. The parameters of Formula (18) for $X{}^1{\mathsf{\Sigma }}_g$ term. Here, $I_0=1.00\times {10}^{14}$ W/cm${}^2$ is the intensity typical for modern NIR Ti:Sapphire lasers.

Molecule	CO${}_2$		CS${}_2$
Quantity	$\mathit{c}\left(\mathit{G}\right)$	$\mathit{s}\left(\mathit{G}\right)$	$\mathit{c}\left(\mathit{G}\right)$	$\mathit{s}\left(\mathit{G}\right)$
$\Delta \left({\mathsf{\Omega }}_s\right)$	$-1.5300$	$0.029682$	$-5.80435$	$0.081479$
$\Delta \left({\mathsf{\Omega }}_a\right)$	$1.3569$	$-0.51852$	$-1.7797$	$-1.1387$
$\Delta \left({\mathsf{\Omega }}_{bx}\right)$	$-10.209$	$-8.4021$	$-30.332$	$-14.178$
$\Delta \left({\mathsf{\Omega }}_{by}\right)$	$-10.209$	$-3.6133$	$-30.332$	$-6.6173$
$\Delta \left(l\right)$	$1.5217$	$0.33435$	$4.6619$	$0.67071$
$\Delta \left({\mathsf{\Omega }}_{bx}\right)/\Delta \left({\mathsf{\Omega }}_{by}\right){|}_{\theta ={90}^{\circ }}$	$2.3253$		$2.1426$
${\Delta \left(l\right)|}_{I=I_0,\theta =0}$, %	$0.43361$		$0.095274$
${\Delta \left(l\right)|}_{I=I_0,\theta ={90}^{\circ }}$, %	$1.3284$		$0.19112$

. It is easy to observe that the absolute values of these constants for CS${}_2$ are approximately ∼3 times greater than those for CO${}_2$. This can be explained by the same differences in the isotropic polarizabilities of these molecules (see also Table 1).

It can be seen from Table 2 that the vibrational parameters of the both molecules depend monotonically on the orientation angle, $\theta$. In particular, the influence of the laser field results in an increase of the molecular length. The maximal (minimal) elongation is achieved when the polarization vector, $\mathbf{u}$, is parallel (perpendicular) to the molecular axis. The dependence of the relative elongation of CO${}_2$ and CS${}_2$ molecules upon $\theta$ is shown in Figure 3 for intensity, $I_0=1.00\times {10}^{14}$ W/cm${}^2$, which is typical for the focal volumes of modern NIR Ti:Sapphire lasers. The dependence shown in Figure 3 can be easily explained by the fact that the longitudinal component of the polarizability tensor of linear molecules is substantially greater than the transverse component. Remember that the laser field does not change the geometric (linear and symmetric) form of the molecule.

Unlike the bond lengths, the frequencies of the longitudinal normal modes can increase as well as decrease. For example, the frequency of ${\mathsf{\Sigma }}_u$ mode in CO${}_2$ increases for the orientation angle domain, $\theta <{58}^{\circ }{17}^{\prime }$, and decreases in the $\theta >{58}^{\circ }{17}^{\prime }$ domain. At the angle value $\theta ={58}^{\circ }{17}^{\prime }$, this frequency is not modified (up to the terms of the order of ∼I). In the small $\theta$ domain, the ${\mathsf{\Sigma }}_u$ mode frequency of CS${}_2$ molecule is decreased compared to that of CO${}_2$. These peculiarities are probably due to the electronic structure of the above molecules.

While the laser-field modification of the geometry and the frequencies of the longitudinal modes is merely quantitative, the transverse ${\mathsf{\Pi }}_u$ modes demonstrate qualitative changes. Namely, the double degeneracy is removed (see Figure 4) due to the appearance of a physically preferential plane (formed by the molecular z-axis and the polarization vector, $\mathbf{u}$). The degeneracy is therefore removed due to breaking of the axial symmetry. Indeed, if the vibrational x axis lies in the above physically preferred plane (see Figure 1) and the corresponding mode is labelled by the “x” subscript, then the vibrational “y” axis lies in the $\left(yz\right)$ plane which is perpendicular to the physically preferred plane (see also Figure 1) and the corresponding mode is labeled by the “y” subscript. If the molecule is oriented along the polarization vector, $\mathbf{u}$ (i.e., when $\theta =0$), then the physically preferred plane disappears, and the degeneracy arises again. If the molecule is oriented perpendicularly to the polarization vector ($\theta ={90}^{\circ }$), then the field-independent relative difference between the frequencies of the ${\mathsf{\Pi }}_u$ modes will be maximal, and this difference is approximately the same for both of the molecules (see Table 2).

Thus, the main difference of the behavior of linear symmetric ($ABA$) and nonlinear [29] molecules in a laser field consists of reduction of their symmetry of the former from $D_{\infty h}$ to $D_{2h}$ due to removal of degeneracy of the ${\mathsf{\Pi }}_u$ modes.

4. Conclusions

In the first nonvanishing order in the intensity, I, a linearly polarized monochromatic laser radiation changes the length of the linear symmetric $ABA$ molecule, but preserves the bending angle $\angle \left(ABA\right)={180}^{\circ }$. The normal modes also undergo shifts ∼I, and the double degeneracy of the ${\mathsf{\Pi }}_u$ mode is removed. The main result of this work is theoretical demonstration of the reduction of symmetry of a linear $ABA$ molecule from $D_{\infty h}$ to $D_{2h}$.

The first-order (in I) laser-field modifications of the vibrational parameters are parametrized by simple and convenient expression (18) depending only on the angle, $\theta$, between the radiation polarization vector, $\mathbf{u}$, and the molecular z-axis. The parameters of (18) are calculated for CO${}_2$ and CS${}_2$ molecules using quantum chemistry methods.

The obtained results can be used in calculation of Franck–Condon factors which are very sensitive to the vibrational parameters [30]. In turn, the FCFs are needed in calculation of a number of phenomena in interaction of molecular systems with high-intensity NIR laser radiation, for instance, ionization of molecules [29] or photodetachment from molecular anions [31].