Diatomic molecules wavefunctions

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 ... 

According to the argument presented above, any molecule must be described by wavefunctions that are antisymmetric with respect to the exchange of any two identical particles. For a homonuclear diatomic molecule, for example, thepossibility of permutation of the two identical nuclei must be considered. Although both the translational and vibrational wavefunctions are symmetric under such a permutation, die parity of the rotational wavefunction depends on the value of 7, the rotational quantum number. It can be shown that the wave-function is symmetric if J is even and antisymmetric if J is odd The overall... 

In order to examine molecular orbitals at their simplest, we shall consider the case of diatomic molecules. The interaction of the wave-functions of two identical atomic orbitals gives rise to wavefunctions of two distinct molecular orbitals (Figure 1.4). 

The methods described above are all based on the Born-Oppenheimer approximation. Therefore, they can be used to calculate polarizabilities of diatomic molecules for a given internuclear distance R. However, if one is interested in values of the polarizability tensors, and C", for a particular vibrational state /i )), one has to average the polarizability radial functions a(R) and C(R) with the vibrational wavefunction i.e., one has to... 

For diatomic molecules the vibrational wavefunctions can be obtained numerically as solution of the one-dimensional Schrddinger equation... 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Treatment of the rotational motion at the zeroth-order level described above introduces the so-called rigid rotor energy levels and wavefunctions Ej = h2 J(J+l)/(2 4,Re2) and Yjm (0,( )) these same quantities arise when the diatomic molecule is treated as a rigid rod of length Re. The spacings between successive rotational levels within this approximation are... 

Figure 3.6 shows the Morse potential energy curves for two hypothetical electronic states of a diatomic molecule, the vibrational energy levels for each, and the shape of the vibrational wavefunctions (i//) within... 

For a determination of the permanent multipole moments, Eq. 2.42 must be averaged over the ground state nuclear vibrational wavefunction transition elements are also often of interest which are matrix elements between initial and final rotovibrational states. For example, for a diatomic molecule with rotovibrational states vJM), the transition matrix elements (v J M Q(m vJM) will be of interest a prime designates final states. 

The time-dependent wavepacket constructed in Section 4.1 is not the wavepacket that a laser with finite duration creates in the excited electronic state. It represents the wavepacket created by a pulse with infinitely narrow width in time. In order to construct the real wavefunction of the molecular system we must go back to Section 2.1. For simplicity of presentation, let us consider a diatomic molecule with internuclear separation R. We assume that the excitation takes place from the electronic ground state (index 0) to a bound upper state (index 1). The extension to a dissociative state, several coupled excited states, or several degrees of freedom is formally straightforward. 

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. 

To summarize, therefore, it is reasonable to say that ab initio calculations of spin-orbit coupling constants may be successfully performed on atoms (although relativistic wavefunctions will be necessary for the heavier ones) and diatomic molecules (especially hydrides). For larger molecules, such methods may be too time-consuming and resort to semi-empirical techniques will be necessary. The atoms-in-molecules approach has proved extremely successful, but it should be possible to use semi-empirical wavefunctions with the full hamiltonian before long. This will be probably more useful with very large molecules. 

Unfortunately the calculation of transition probabilities has not been attempted in many instances and the results are not encouraging. Apart from a valiant early attempt by Davies173 for Na2 in a calculation where not all electrons were specifically considered, the first serious attempt to compute transition probabilities from ab initio wavefunctions was due to La Paglia.174 In this important paper La Paglia considered the dipole strengths of various 1 -iE transitions for some first-row diatomic molecules. 

Relationship between Valence Bond and Spin Valence Theories.—We consider first for simplicity a diatomic molecule AB. The basic physical idea behind all the variants of VB theory is that the wavefunction for the molecule, Pab, should in some way be written as a product of the wavefunctions Pa, Pb for particular states of the participating atoms. Thus... 

Atoms in Molecules.—In this approach, which was first proposed by Moffitt,105 a wavefunction for a particular electronic state of a molecule is constructed from products of atomic wavefunctions, these, moreover, being taken to be exact eigenfunctions of their respective atomic hamiltonians. We confine our attention to the case of diatomic molecules AB so that, according to this procedure, the wavefunction is written as... 

Direct printout of Mathematica commands to calculate and plot wavefunctions and electronic transition intensity factors for a diatomic molecule (I2) using harmonic and Morse oscillator wavefunctions. See text for discussion. A Maple version of this calculation can be found on the Maple applications website. 

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