Energy curve, diatomic molecule

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... 

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... 

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. 

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. 

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. 

Figure S-1. Form of a potential energy curve for diatomic molecule AB. VfrAa) is the potential energy, Tab is the intemuclear distance, is the equilibrium intemuclear distance, and D is the bond dissociation energy. (The zero point energy is neglected in the figure.)...

FIGURE 7.2 Energy curves for a diatomic molecule. Two possible transitions are shown. When an electron has been excited to the point marked A, the molecule may cleave (p. 312). 

Figure 6.34 shows potential energy curves for a hypothetical diatomic molecule X2, which approaches a surface, coming from the right-hand side of the diagram. First... 

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... 

Considering again the case of a structureless continuum, we have that 8j3 arises from excitation of a superposition of continuum states, hence from coupling within PHmP [69]. The simplest model of this class of problems, depicted schematically in Fig. 5b, is that of dissociation of a diatomic molecule subject to two coupled electronic dissociative potential energy curves. Here the channel phase can be expressed as... 

This time period is too short for a change in geometry to occur (molecular vibrations are much slower). Hence the initially formed excited state must have the same geometry as the ground state. This is illustrated in Figure 1.2 for a simple diatomic molecule. The curves shown in this figure are called Morse curves and represent the relative energy of the diatomic system as a... 

To help visualize this process, let us consider a diatomic molecule with the energy curves shown in Figure 2.5(a). In this example the ground and excited states have the same equilibrium intemuclear distance ra. Since in solution at room temperature almost all the molecules will be in the lowest vibrational level of the ground state vt° (subscripts refer to the electronic... 

The relationships between bond length, stretching force constant, and bond dissociation energy are made clear by the potential energy curve for a diatomic molecule, the plot of the change in the internal energy AU of the molecule A2 as the internuclear separation is increased until the molecule dissociates into two A atoms ... 

The surface molecule model has been used to study chemisorption of hydrogen 47) and nitrogen 48) on tungsten (100). The parameters used in these calculations are collected in Table IV. Preliminary calculations on the diatomic molecules WH and WW showed that inclusion of tungsten 5 p orbitals is essential to produce a minimum in the energy/ distance curves. However, the repulsion due to inner electrons could be calculated by the empirical relationship ... 

Before we do this, though, we point out that for a simple diatomic molecule, assuming ideal conditions, one can in principle calculate the rate of the uni-molecular process. This is so because the lower excited states of the ion are (relatively) few and well separated. If the potential curves are then given, the value of the rate can be provided. For a polyatomic molecule, two great complications immediately arise (1) the number of lower excited states increases tremendously and (2) multidimensional potential energy surfaces make trajectory calculations intractable. 

The first-row homonuclear diatomic molecules A2 of main-group elements (A = B, C, N, O, F) exhibit a well-known diversity of ground-state multiplicities, bond lengths, and bond energies. Calculated potential-energy curves for low-lying singlet and triplet states of these species are pictured in Fig. 3.27 and summarized in Table 3.13 (with comparison experimental values). Because these homonuclear... 

Figure 19.1 Potential energy curve for a diatomic molecule.

Figure 19.2 Two potential energy curves of a diatomic molecule.

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Another way of looking at the Morse curve in Figure 9.12 is to say it represents the energy E (as y ) of the two atoms of X as a function of their bond length r (as V). The two atoms of X form a simple diatomic molecule in its ground state, i.e. before it absorbs a photon of light. 

The potential energy of a diatomic molecule depends on only the distance between two bonded atoms. The potential energy of a diatomic molecule can be plotted in two dimensions by plotting PE as a function of the bond length. The curve is known as potential energy curve (Fig. 9.9). 

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