Diatomic potential

Figure Al.2.1. Potential V(R) of a diatomic molecule as a fiinction of the intemuclear separation i . The equilibrium distance Rq is at the potential minimum.

Hutson J M and Howard B J 1980 Spectroscopic properties and potential surfaces for atom-diatom van der Waals molecules Mol. Phys. 41 1123... 

Figure A3.9.8. An elbow potential energy surface representing the dissociation of a diatomic in two dimensions-the molecular bond lengdi and tlie distance from the molecule to the surface.

To compare the relative populations of vibrational levels, the intensities of vibrational transitions out of these levels are compared. Figure B2.3.10 displays typical potential energy curves of the ground and an excited electronic state of a diatomic molecule. The intensity of a (v, v ) vibrational transition can be written as... 

Tellinghuisen J A 1974 A fast quadrature method for computing diatomic RKR potential energy curves Comput. Phys. Commun. 6 221-8... 

Figure B3.4.7. Schematic example of potential energy curves for photo-absorption for a ID problem (i.e. for diatomics). On the lower surface the nuclear wavepacket is in the ground state. Once this wavepacket has been excited to the upper surface, which has a different shape, it will propagate. The photoabsorption cross section is obtained by the Fourier transfonn of the correlation function of the initial wavefimction on tlie excited surface with the propagated wavepacket.

To see physically the problem of motion of wavepackets in a non-diagonal diabatic potential, we plot in figure B3.4.17 a set of two adiabatic potentials and their diabatic counterparts for a ID problem, for example, vibrations in a diatom (as in metal-metal complexes). As figure B3.4.17 shows, if a wavepacket is started away from the crossing point, it would slide towards this crossing point (where where it would... 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

In the following, we shall demonstrate techniques for calculating the electronic potential energy terms up to the second order. For simplicity, we shall study the case of H2 molecule, the simplest multi-electron diatomic molecule. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

The second summation is over all the orbitals of the system. This equation is used in IlyperChem ah imiio calculations to generate contour plots of electrostatic potential, [fwe choose the approximation whereby we n eglect the effects of the diatomic differen tial overlap (NDDO). then the electrostatic potential can be rewritten... 

This Schrodinger equation relates to the rotation of diatomic and linear polyatomic molecules. It also arises when treating the angular motions of electrons in any spherically symmetric potential... 

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 ... 

The fact that the separated-atom and united-atom limits involve several crossings in the OCD can be used to explain barriers in the potential energy curves of such diatomic molecules which occur at short intemuclear distances. It should be noted that the Silicon... 

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). 

DIM (diatomics-in-molecules) a semiempirical method used for representing potential energy surfaces... 

This difference is shown in the next illustration which presents the qualitative form of a potential curve for a diatomic molecule for both a molecular mechanics method (like AMBER) or a semi-empirical method (like AMI). At large internuclear distances, the differences between the two methods are obvious. With AMI, the molecule properly dissociates into atoms, while the AMBERpoten-tial continues to rise. However, in explorations of the potential curve only around the minimum, results from the two methods might be rather similar. Indeed, it is quite possible that AMBER will give more accurate structural results than AMI. This is due to the closer link between experimental data and computed results of molecular mechanics calculations. 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

For each excited electronic state of a diatomic molecule there is a potential energy curve and, for most states, the curve appears qualitatively similar to that in Figure 6.4. 

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