Diatomic molecules polarizability

Figure Bl.2.2. Schematic representation of the polarizability of a diatomic molecule as a fimction of vibrational coordinate. Because the polarizability changes during vibration, Raman scatter will occur in addition to Rayleigh scattering.

Figure 6.2 Variation of the mean polarizability c/. with intemuclear distance r in a diatomic molecule...

Let us take the problem of a diatomic molecule, such as Cl2, with the two atoms as the two polarizable parts. Each atom has a polarizability ax parallel to and a2 perpendicular to the interatomic axis, respectively. The Silberstein theory yields for the effective polarizability of the molecule the values bx and b2, respectively, parallel and perpendicular to the axis. 

Because an applied field in the y direction Ev can induce a dipole M with a component in the x direction Mx as well as the component in the y direction My, it is necessary that we specify the components of the polarizability tensor by two subscripts (Fig. 3). If the bond A—B of a diatomic molecule stretches during a vibrational mode, Mx and Mv will vary and therefore the corresponding polarizability tensor components will vary. 

Let us suppose that the system of interest does not possess a dipole moment as in the case of a homonuclear diatomic molecule. In this case, the leading term in the electric field-molecule interaction involves the polarizability, a, and the Hamiltonian is of the form ... 

Another problem comes in examining the polarizability. In the physical picture, the spherically symmetric molecule, just like an atom, has isotropic polarizability. In the chemical picture, for a diatomic molecule we have two unique polarizabilities (1) and in the internal coordinate system or (2) dzz = 5 (o xc + (isotropic polarizability) and Aa = — [polar-... 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

The article is organized as follows. The main features of the linear response theory methods at different levels of correlation are presented in Section 2. Section 3 describes the calculation of the dipole and quadmpole polarizabilities of two small diatomic molecules LiH and HF. Different computational aspects are discussed for each of them. The LiH molecule permits very accurate MCSCF studies employing large basis sets and CASs. This gives us the opportunity to benchmark the results from the other linear response methods with respect to both the shape of the polarizability radial functions and their values in the vibrational ground states. The second molecule, HF, is undoubtedly one of the most studied molecules. We use it here in order to examine the dependence of the dipole and quadmpole polarizabilities on the size of the active space in the CAS and RASSCF approaches. The conclusions of this study will be important for our future studies of dipole and quadmpole polarizabilities of heavier diatomic molecules. 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

In the recent work reviewed in this chapter, we have shown the creation and properties of molecular superrotors, illustrated by application of the optical centrifuge to simple diatomic molecules. The technique is certainly not limited to diatomics, and to date we have created superrotor wave packets in a range of molecules. All that is required is an anisotropic polarizability such that the molecule can be... 

We saw that homonuclear diatomic molecules exhibit no pure-rotation or vibration-rotation spectra, because they have zero electric dipole moment for all internuclear separations. The Raman effect depends on the polarizability and not the electric dipole moment homonuclear diatomic molecules do have a nonzero polarizability which varies with varying internuclear separation. Hence they exhibit pure-rotation and vibration-rotation Raman spectra. Raman spectroscopy provides information on the vibrational and rotational constants of homonuclear diatomic molecules. 

Consideration of the matrix elements m a n of the polarizability shows that the selection rule for a pure-rotational Raman transition of a l2 diatomic molecule is (see Wilson, Decius, and Cross)... 

The magnitude of the surface dipole. For the system hydrogen-nickel, formula (13) leads to a value of 0.66 D. The experimentally determined value 0.022 D. is therefore a factor of about 30 smaller. This is quite conceivable because (13) has been derived for a diatomic molecule. In our case one of the partners of the bond, the metal, has a very high polarizability, and hence the surface dipole will be quenched to a large extent. The value of the dipole moment calculated from (13), though larger than the experimental value, is still far smaller than that to be expected for a pure ionic bond (fora bond distance of 2 A. fj, = 10 D.). This is one of the reasons for us to think that the contribution of the ionic type M+X to the total bond... 

Inelastic or Raman scattering of light can be understood classically as arising from modulation of the electron distribution, and hence the molecular polarizability, because of vibrations of the nuclei. For example, for a diatomic molecule, a can be represented adequately by the first two terms of a power series in the vibrational coordinate Q ... 

To understand the complete role of vibration in determining electrical properties, it is useful to consider a diatomic molecule in the harmonic oscillator approximation, where the stretching potential is taken to be quadratic in the displacement coordinate. The doubly harmonic model takes the various electrical properties to be linear functions of the coordinate. This turns out to be most reasonable in the vicinity of an equilibrium structure, but it breaks down at long separations. Letting x be a coordinate giving the displacement from equilibrium of a one-dimensional harmonic oscillator, the dipole moment, dipole polarizability, and dipole hyperpolarizability, within the doubly harmonic (dh) model, may be written in the following way ... 

Lastly, we turn to consideration of the Rg—HF heterodimers (the atom—diatomic molecule system of the right-hand side of Figure 5.2), where a crucial role is played by the induction interaction occurring between the higher multipole moments of HF and the induced dipoles originating the polarizability of the rare gas (Magnasco et al., 1989a). 

Atom-Atom Interactions. - The methods applied, usually to interactions in the inert gases, are a natural extension of diatomic molecule calculations. From the interaction potentials observable quantities, especially the virial coefficients can be calculated. Maroulis et al.31 have applied the ab initio finite field method to calculate the interaction polarizability of two xenon atoms. A sequence of new basis sets for Xe, especially designed for interaction studies have been employed. It has been verified that values obtained from a standard DFT method are qualitatively correct in describing the interaction polarizability curves. Haskopoulos et al.32 have applied similar methods to calculate the interaction polarizability of the Kr-Xe pair. The second virial coefficients of neon gas have been computed by Hattig et al.,33 using an accurate CCSD(T) potential for the Ne-Ne van der Waals potential and interaction-induced electric dipole polarizabilities and hyperpolarizabilities also obtained by CCSD calculations. The refractivity, electric-field induced SHG coefficients and the virial coefficients were evaluated. The authors claim that the results are expected to be more reliable than current experimental data. 

There is presently a large volume of literature devoted to the properties of isoelectronic diatomic molecules. Interest in these molecules comes from diverse areas of physics and chemistry. However, most of such literature reports calculations of lifetimes, transition probabilities, and energy levels. To the lesser extent, there have also been calculations and experiments on polarizabilities, NMR chemical shifts, harmonic and higher order force constants, and so on. 

For diatomic molecules, expressions such as (44) can be evaluated very accurately and this has been done for H2, where the vibrational hyper-polarizability yv (Pv is zero) is ... 

A decade ago Laaksonen et al. published a paper giving an outline of the finite difference (FD) (or numerical) Hartree-Fock (HF) method for diatomic molecules and several examples of its application to a series of molecules (1). A summary of the FD HF calculations performed until 1987 can be found in (2). The work of Laaksonen et al. can be considered a second attempt to solve numerically the HF equations for diatomic molecules exactly. The earlier attempt was due to McCullough who in the mid 1970s tried to tackle the problem using the partial wave expansion method (3). This approach had been extended to study correlation effects, polarizabilities and hyper-fine constants and was extensively used by McCullough and his coworkers (4-6). Heinemann et al. (7-9) and Sundholm et al. (10,11) have shown that the finite element method could also be used to solve numerically the HF equations for diatomic molecules. 

It is generally believed that the polarizabilities of monatomic ions and molecules are independent of field direction. For undistorted quasi-spherical molecules (e.g. CH4, CC14, etc.) the same is usually assumed. When two such atoms are held together, as in a diatomic molecule, the new system is not isotropically polarizable. The model discussed by Silberstein (1917) makes this understandable. If a unit field acts along the line of centres A-B it will induce primary moments parallel to itself in both A and B, and likewise if it acts at 90° to A-B. Each primary moment will induce a secondary moment in its neighbour in the first case the secondary moments will add to the primary moments, but in the second they will subtract. Hence b along the line of centres exceeds that across it, and the polarizability of A-B is an anisotropic property. A similar situation is to be expected with the majority of polyatomic ions or molecules (see Table 21). 

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