The Diatomic Molecule

In Sect.5,4.2, we demonstrate the basic physical principles with the help of the diatomic molecule. This is done in a rather intuitive way without any detailed derivations. In Sect.5.4.3, we discuss the SCHA for a Bravais crystal and in Sect.5.4.4, we treat the self-consistent Einstein model. 

Consider a diatomic molecule with a Morse-type potential of the form of (5.1), namely, 

ag is the distance at the minimum of cp(R), D is the dissociation energy, X is an adjustable parameter, 

The result of the SCHA (see Appendix 0) is physically appealing the first and second derivatives of tp(R) are simply replaced by their thermal averages with respect to the effective harmonic Hamiltonian H. The equation which determines r(T) at zero pressure is therefore 

Results similar to the relations (5.103,105) have been obtained for the self- 

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We will now treat the internal motion on the PES in cases of progressively increasing molecular complexity. We start with the simplest case of all, the diatomic molecule, where the notions of the Bom-Oppenlieimer PES and internal motion are particularly simple. 

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Suppose that W(r,Q) describes the radial (r) and angular (0) motion of a diatomic molecule constrained to move on a planar surface. If an experiment were performed to measure the component of the rotational angular momentum of the diatomic molecule perpendicular to the surface (Lz= -ih d/dQ), only values equal to mh (m=0,1,-1,2,-2,3,-3,...) could be observed, because these are the eigenvalues of ... 

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

Here the total moment of inertia I of the molecule takes the place of iRe2 in the diatomic molecule case... 

The Hamiltonian in this problem contains only the kinetic energy of rotation no potential energy is present because the molecule is undergoing unhindered "free rotation". The angles 0 and (j) describe the orientation of the diatomic molecule s axis relative to a laboratory-fixed coordinate system, and p is the reduced mass of the diatomic molecule p=mim2/(mi+m2). 

The advantages of INDO over CNDO involve situations where the spin state and other aspects of electron spin are particularly important. For example, in the diatomic molecule NH, the last two electrons go into a degenerate p-orbital centered solely on the Nitrogen. Two well-defined spectroscopic states, S" and D, result. Since the p-orbital is strictly one-center, CNDO results in these two states having exactly the same energy. The INDO method correctly makes the triplet state lower in energy in association with the exchange interaction included in INDO. 

D is the chemical energy of dissociation which cair be obtained from thermodynamic data, aird is the reduced mass of the diatomic molecule... 

If we were to calculate the potential energy V of the diatomic molecule AB as a function of the distance tab between the centers of the atoms, the result would be a curve having a shape like that seen in Fig. 5-1. This is a bond dissociation curve, the path from the minimum (the equilibrium intemuclear distance in the diatomic molecule) to increasing values of tab describing the dissociation of the molecule. It is conventional to take as the zero of energy the infinitely separated species. 

In Chapter 1,1 discussed the concept of mutual potential energy and demonstrated its relationship to that of force. So, for example, the mutual potential energy of the diatomic molecule discussed in Section 1.1.2 is... 

To illustrate molecular orbital theory, we apply it to the diatomic molecules of the elements in the first two periods of the periodic table. 

Among the diatomic molecules of the second period elements are three familiar ones, N2,02, and F2. The molecules Li2, B2, and C2 are less common but have been observed and studied in the gas phase. In contrast, the molecules Be2 and Ne2 are either highly unstable or nonexistent. Let us see what molecular orbital theory predicts about the structure and stability of these molecules. We start by considering how the atomic orbitals containing the valence electrons (2s and 2p) are used to form molecular orbitals. 

Apparently the diatomic molecules of the halogens already have achieved some of the stability characteristic of the inert gas electron arrangement. How is this possible How could one chlorine atom satisfy its need for one more electron (so it can reach the argon stability) by... 

The diatomic molecule of fluorine does not form higher compounds (such as F3, F4, - ) because each fluorine atom has only one partially filled valence orbital. Each nucleus in Fs is close to a number of electrons sufficient to fill the valence orbitals. Under these circumstances, the diatomic molecule behaves like an inert gas atom toward other such molecules. The forces that cause molecular fluorine to condense at 85°K are, then, the same as those that cause the inert gases to condense. These forces are named van der Waals forces, after the Dutch scientist who studied them. 

Most values were taken from M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, "JANAF Thermochemical Tables, Third Edition, J. Phys. Chem. Ref. Data, 14, Supplement No. 1, 1985. For the diatomic molecules, a few (in parentheses) came from G. Hertzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, and II. Infrared and Raman Spectra of Polyatomic Molecules. Van Nostrand Reinhold Co., New York, 1950 and 1945. 

Under most circumstances the equations given in Table 10.4 accurately calculate the thermodynamic properties of the ideal gas. The most serious approximations involve the replacement of the summation with an integral [equations (10.94) and (10.95)] in calculating the partition function for the rigid rotator, and the approximation that the rotational and vibrational partition functions for a gas can be represented by those for a rigid rotator and harmonic oscillator. In general, the errors introduced by these approximations are most serious for the diatomic molecule." Fortunately, it is for the diatomic molecule that corrections are most easily calculated. It is also for these molecules that spectroscopic information is often available to make the corrections for anharmonicity and nonrigid rotator effects. We will summarize the relationships... 

E10.6 For the diatomic molecule Na2, 5 = 230.476 J-K-1-mol" at T= 300 K, and 256.876 J-K-,-mol-1 at T= 600 K. Assume the rigid rotator and harmonic oscillator approximations and calculate u, the fundamental vibrational frequency and r, the interatomic separation between the atoms in the molecule. For a diatomic molecule, the moment of inertia is given by l pr2, where p is the reduced mass given by... 

Consider the collision of Ar+ with HD. If one assumes that one of the partners in the diatomic molecule does not participate in the collision but merely acts as a spectator, then conservation of energy and momentum permit the following equation for head-on collisions... 

The values in Table VI were obtained in the following way. Values for C, Si, Ge, and Sn are the same as in Table III, for the tetrahedral configuration is the normal one for these atoms. Radii for F, Cl, Br, and I were taken as one-half the band-spectral values for the equilibrium separation in the diatomic molecules of these substances. Inasmuch as these radii for F and Cl are numerically the same as the tetrahedral radii for these atoms, the values for N, 0, P, and S given in Table III were also accepted as normal-valence radii for these atoms. The differences of 0.03 A between the normal-valence radius and the tetrahedral radius for Br and... 

This qualitative description of the interactions in the metal is compatible with quantum mechanical treatments which have been given the problem,6 and it leads to an understanding of such properties as the ratio of about 1.5 of crystal energy of alkali metals to bond energy of their diatomic molecules (the increase being the contribution of the resonance energy), and the increase in interatomic distance by about 15 percent from the diatomic molecule to the crystal. 

The energy values and the derived quantities for the diatomic molecules of the alkali metals are given in table 1. It is seen that the amount of p character is calculated to he between 5 and 14 %. 

The absorption spectra of Zr atoms isolated in a variety of matrices have been reported. In addition, the diatomic molecule ZrN, prepared using a hollow cathode source and Na, was observed. Other work involving Na included the identification of ThN and Th(Na), and TaN in various matrices. 

First, by taking the diatomic molecule LiH, we try to excite a displaced wave packet on the ground-state at / = 6.0 a.u. to the B II excited state (see... 

Hence, we find two energy levels for the diatomic molecule where the electron can reside, one bonding and the other antibonding. This simplified approach does not describe the situation quantitatively too well, but in a qualitative sense it captures all the important effects. In the following we consider a few illustrative cases in the limit where the overlap S is small (this is the usual approximation for elementary work). In this limit Eq. (15) reduces to ... 

The application of the Bom-Oppenheimer and the adiabatic approximations to separate nuclear and electronic motions is best illustrated by treating the simplest example, a diatomic molecule in its electronic ground state. The diatomic molecule is sufficiently simple that we can also introduce center-of-mass coordinates and show explicitly how the translational motion of the molecule as a whole is separated from the internal motion of the nuclei and electrons. 

The first step in the solution of equation (10.28b) is to hold the two nuclei fixed in space, so that the operator drops out. Equation (10.28b) then takes the form of (10.6). Since the diatomic molecule has axial symmetry, the eigenfunctions and eigenvalues of He in equation (10.6) depend only on the fixed value R of the intemuclear distance, so that we may write them as tpKiy, K) and Sk(R). If equation (10.6) is solved repeatedly to obtain the ground-state energy eo(K) for many values of the parameter R, then a curve of the general form... 

The first derivative U HRe) vanishes because the potential is a minimum at the distance R. The second derivative U XR ) is called the force constant for the diatomic molecule (see Section 4.1) and is given the symbol k. We also introduce the relative distance variable q, defined as... 

The quantity I (= /jiR ) is the moment of inertia for the diatomic molecule with the intemuclear distance fixed at Re and Be is known as the rotational constant (see Section 5.4). 

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