A Diatomic Molecule

Consider a diatomic molecule such as Within the Bom-Oppenheimer 

A more advanced calculation shows that the potential (defined by the data points) has a minimum at 127.34pm with a corresponding energy of t460.244 0222 h- 

Suppose for example that we want to test the validity of our data against the (larraonic model 

Alternatively, we might try to allow for anharmonicity and write a more general expression such as 

If for the minute we identify the constant b with in every case then we obtain Table 14.2. 

Consider a diatomic molecule such as H35C1. Within the Bom-Oppenheimer approximation, we focus attention on the electronic wavefimction and calculate enough data points to give a potential energy curve. Such a curve shows the variation of the electronic energy with intemuclear separation. The nuclei vibrate in this potential. 

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Although a diatomic molecule can produce only one vibration, this number increases with the number of atoms making up the molecule. For a molecule of N atoms, 3N-6 vibrations are possible. That corresponds to 3N degrees of freedom from which are subtracted 3 translational movements and 3 rotational movements for the overall molecule for which the energy is not quantified and corresponds to thermal energy. In reality, this number is most often reduced because of symmetry. Additionally, for a vibration to be active in the infrared, it must be accompanied by a variation in the molecule s dipole moment. 

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.

Figure Al.2.2. Internal nuclear motions of a diatomic molecule. Top the molecule in its equilibrium configuration. Middle vibration of the molecule. Bottom rotation of the molecule.

The interaction energy can be written as an expansion employing Wigner rotation matrices and spherical hamionics of the angles [28, 130], As a simple example, the interaction between an atom and a diatomic molecule can be expanded hr Legendre polynomials as... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

The energy of a diatomic molecule can be divided into translational and internal contributions = (/ik) /(2A7)... 

Consider the collision of an atom (denoted A) with a diatomic molecule (denoted BC), with motion of the atoms constrained to occur along a line. In this case there are two important degrees of freedom, the distance R between the atom and the centre of mass of the diatomic, and the diatomic intemuclear distance r. The Flamiltonian in tenns of these coordinates is given by ... 

Meier C and Engel V 1995 Pump-probe ionization spectroscopy of a diatomic molecule sodium molecule as a prototype example Femtosecond Chemistry Proc. Berlin Conf Femtosecond Chemistry (Berlin, March 1993) (Weinheim Verlag Chemie)... 

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.

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 reaction of an atom with a diatomic molecule is the prototype of a chemical reaction. As the dynamics of a number of atom-diatom reactions are being understood in detail, attention is now being turned to the study of the dynamics of reactions involving larger molecules. The reaction of Cl atoms with small aliphatic hydrocarbons is an example of the type of polyatomic reactions which are now being studied [M, 72, 73]. 

These electronic energies dependence on the positions of the atomic centres cause them to be referred to as electronic energy surfaces such as that depicted below in figure B3.T1 for a diatomic molecule. For nonlinear polyatomic molecules having atoms, the energy surfaces depend on 3N - 6 internal coordinates and thus can be very difficult to visualize. In figure B3.T2, a slice tln-oiigh such a surface is shown as a fimction of two of the 3N - 6 internal coordinates. 

Figure B3.2.12. Schematic illustration of geometries used in the simulation of the chemisorption of a diatomic molecule on a surface (the third dimension is suppressed). The molecule is shown on a surface simulated by (A) a semi-infinite crystal, (B) a slab and an embedding region, (C) a slab with two-dimensional periodicity, (D) a slab in a siipercell geometry and (E) a cluster.

Miller W H 1969 Coupled equations and the minimum principle for collisions of an atom and a diatomic molecule, including rearrangements J. Chem. Phys. 50 407... 

The simplest condensed phase VER system is a dilute solution of a diatomic in an atomic (e.g. Ar or Xe) liquid or crystal. Other simple systems include neat diatomic liquids or crystals, or a diatomic molecule bound to a surface. A major step up in complexity occurs with poly atomics, with several vibrations on the same molecule. This feature guarantees enonnous qualitative differences between diatomic and polyatomic VER, and casts doubt on the likelihood of understanding poly atomics by studying diatomics alone. 

A simple example would be in a study of a diatomic molecule that in a Hartree-Fock calculation has a bonded cr orbital as the highest occupied MO (HOMO) and a a lowest unoccupied MO (LUMO). A CASSCF calculation would then use the two a electrons and set up four CSFs with single and double excitations from the HOMO into the a orbital. This allows the bond dissociation to be described correctly, with different amounts of the neutral atoms, ion pair, and bonded pair controlled by the Cl coefficients, with the optimal shapes of the orbitals also being found. For more complicated systems... 

Consider a diatomic molecule as shown in Figure 1. The nuclear kinetic energy is expressed as... 

Although all of the nuclear coordinates participate in this kinetic energy operator, and in our previous discussions, all of the nuclear coordinates are expanded, with respect to an equivalent position, in power series of the parameter K, here in the specific case of a diatomic molecule, we found that only the R coordinate seems to have an equilibrium position in the molecular fixed coordinates. This means that actually we only have to, or we can only, expand the R coordinate, but not the other coordinates, in the way that... 

Figure 2. The space-fixed (XYZ) and body-fixed xyz) frames in a diatomic molecule AB. The nuclei are at A and B, and 1 represents the location of a typical electron. The results of inversions of their SF coordinates are A A, B B, and 1 1, respectively. After one executes only the reinversion of the electronic SF coordinates, one obtains 1 — 1. The net effect is then the exchange of the SF nuclear coordinates alone.

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. 

As our first model problem, we take the motion of a diatomic molecule under an external force field. For simplicity, it is assumed that (i) the motion is pla nar, (ii) the two atoms have equal mass m = 1, and (iii) the chemical bond is modeled by a stiff harmonic spring with equilibrium length ro = 1. Denoting the positions of the two atoms hy e 71, i = 1,2, the corresponding Hamiltonian function is of type... 

Figure 9-5 A Diatomic Molecule. The molecule can undergo translation without changing the distance X2 — Xi), or it can undergo vibration, in which (X2 — Xi) changes, or it can undergo translation while vibrating.

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 ... 

A diatomic molecule constrained to rotate on a flat surface can be modeled as a planar... 

The rotational Hamiltonian for a diatomic molecule as given in Chapter 3 is... 

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. 

For a diatomic molecule, for example, there is only one internal coordinate and the energy as a function of configuration (inter-nuclear distance) will look something like the following ... 

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 1.11(b) illustrates the ball-and-spring model which is adequate for an approximate treatment of the vibration of a diatomic molecule. For small displacements the stretching and compression of the bond, represented by the spring, obeys Hooke s law ... 

The same expression applies also to any linear polyatomic molecule but, because / is likely to be larger than for a diatomic molecule, the energy levels of Figure 1.12 tend to be more closely spaced. 

We have seen in Section 1.3.6 how the vibrational energy levels of a diatomic molecule, treated in the harmonic oscillator approximation, are given by... 

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