Diatomic molecules expectation values

One of the most important characteristics of molecular systems is their behavior as a function of the nuclear coordinates. The most important molecular property is total energy of the system which, as a function of the nuclear coordinates, is called the potential energy (hyper)surface, an obvious generalization of the potential energy curve in diatomics. Other expectation values as functions of the nuclear coordinates are frequently called property surfaces. The notion of the total molecular energy in a given electronic state, which depends only parametrically on the nuclear coordinates, is based on the fixed-nuclei approximation. In most cases (e.g. closed-shell molecules in the ground electronic state and in a low vibrational state) this is an excellent approximation. Even when it breaks down, the most convenient treatment is based on the fixed-nuclei picture, i.e. on the assumption that the nuclear mass is infinite compared with the electronic mass. 

In diatomic molecules, T2 = 0, and thus the expectation value of C vanishes. This is the reason why this operator was not considered in Chapter 2. However, for linear triatomic molecules, t2 = / / 0, and the expectation value of C does not vanish. We note, however, that D J is a pseudoscalar operator. Since the Hamiltonian is a scalar, one must take either the absolute value of C [i.e., IC(0(4 2))I or its square IC(0(412))I2. We consider here its square, and add to either the local or the normal Hamiltonians (4.51) or (4.56) a term /412IC(0(412))I2. We thus consider, for the local-mode limit,... 

Proceeding in the spirit above it seems reasonable to inquire why s is equal to the number of equivalent rotations, rather than to the total number of symmetry operations for the molecule of interest. Rotational partition functions of the diatomic molecule were discussed immediately above. It was pointed out that symmetry requirements mandate that homonuclear diatomics occupy rotational states with either even or odd values of the rotational quantum number J depending on the nuclear spin quantum number I. Heteronuclear diatomics populate both even and odd J states. Similar behaviors are expected for polyatomic molecules but the analysis of polyatomic rotational wave functions is far more complex than it is for diatomics. Moreover the spacing between polyatomic rotational energy levels is small compared to kT and classical analysis is appropriate. These factors appreciated there is little motivation to study the quantum rules applying to individual rotational states of polyatomic molecules. 

The adiabatic correction to the Born-Oppenheimer potential energy for a diatomic molecule A-B is simply given by the sum of the expectation values of the nuclear... 

Let s calculate the entropy of a tiny solid made up of four diatomic molecules of a binary compound such as carbon monoxide, CO. Suppose the four molecules have formed a perfectly ordered crystal in which all molecules are aligned with their C atoms on the left. Because T = 0, all motion has ceased (Fig. 7.5). We expect the sample to have zero entropy, because there is no disorder in either location or energy. This value is confirmed by the Boltzmann formula because there is only one way of arranging the molecules in the perfect crystal, W = l and S = k In 1 =0. Now suppose thar the molecules can lie with their C atoms on either side yet still have the same total energy (Fig. 7.6). The total number of ways of arranging the four molecules is... 

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... 

This Report deals with the calculation of spectroscopic constants both for diatomic and for polyatomic molecules, while concentrating on the former. The spectroscopic constants included are restricted to those measured in high-resolution gas-phase work. We cover the whole range of complexity in computation since this is determined not by the method used in computing the expectation value, but by the quality of the wavefunction used. Generally the wavefunctions used are of the ab initio type, but their quality will depend on the size and type of basis set employed as well as the method. 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

Table 2 is an extended and re-calculated version of table VI in ref. 5. The values of the anharmonic constants of diatomic molecules as tabulated here are often used to estimate the expected bond-stretching anharmonicity associated with bonds between corresponding pairs of atoms in polyatomic molecules, as discussed in Section 5. 

In the concluding remark of Section 5.2 we asked the question whether the transition from confined chaos to global chaos K = Kc can be seen in an experiment with diatomic molecules. The technical feasibility of such an experiment is discussed in Section 5.4. Here we ask the more modest question whether, and if so, how, the transition to global chaos manifests itself within the framework of the quantum kicked rotor. Since the transition to global chaos is primarily a classical phenomenon, we expect that we have the best chance of seeing any manifestation of this transition in the quantum kicked rotor the more classical we prepare its initial state and control parameters. Thus, we choose a small value... 

The relatively small value of lo = 10 was chosen with an eye on the experimental verification of the quantum threshold. It is not expected that any diatomic molecule is a good rotor for angular momenta exceeding 100. Therefore, Iq has to be chosen large enough to enter the semi-... 

The case of LiH is a very particular one, because of its very large value of an. The corresponding C Keesom coefficient is hardly reliable, the complete series expansion showing that a reduction of over 57% is needed for C (8.475 x 103 Ef,aq with n = 17) and about 5% for Cs (5.169 x 104 Ei,a with n = 7). The two-term asymptotic formula (4.89) given by Battezzati and Magnasco (2004) yields 8.436 x 103 E a, which is within 0.5% of the complete series expansion result. So, the simple two-term formula (4.89) is expected to work well for other fluorides and chlorides of the alkaline metals (considered as gaseous diatomic molecules), all of which have even larger values of the dipole-dipole constant a. ... 

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. 

This works well for the fundamental vibrational frequency of simple diatomic molecules, and is not too far from the average value of a two-atom stretch within a polyatomic molecule. However, this approximation only gives the average or center frequency of the diatomic bond. In addition, one might expect that since the reduced masses of, for example, CH, OH, and NH are 0.85, 0.89, and 0.87, respectively (these constitute the major absorption bands in the near-infrared spectrum), the ideal frequencies of all these pairs would be quite similar. 

The auxiliary data calculated from estimated molecular parameters for a diatomic molecule are expected to be quite accurate. The uncertainty in the recalculation to the standard temperature of the result for this iso-molecular reaction is anticipated to be moderate compared to the experimental uncertainty. The value for the enthalpy of formation of BSe(g) is therefore adopted... 

The derived values of the heat of formation of AuSe(g) agree reasonably well. Since the entropy and heat capacity of a diatomic molecule calculated from estimated molecular parameters are expected to be quite accurate, the review adopts the mean of the two results and obtains ... 

---