Homonuclear diatomic molecules electronic wave functions

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

Next, we address some simple cases, begining with homonuclear diatomic molecules in 1S electronic states. The rotational wave functions are in this case the well-known spherical harmonics for even J values, yr(R) is symmetric under permutation of the identical nuclei for odd J values, y,.(R) is antisymmetric under the same permutation. A similar statement applies for any D.yjh type molecule. 

Atoms do not all have the same ability to attract electrons. When two different types of atoms form a covalent bond by sharing a pair of electrons, the shared pair of electrons will spend more time in the vicinity of the atom that has the greater ability to attract them. In other words, the electron pair is shared, but it is not shared equally. The ability of an atom in a molecule to attract electrons to it is expressed as the electronegativity of the atom. Earlier, for a homonuclear diatomic molecule we wrote the combination of two atomic wave functions as... 

MO wave functions in the above form give equal importance to covalent and ionic structures, which is unrealistic in homonuclear diatomic molecules like H2. This should be contrasted with (/>Vb> which in its simple form neglects the ionic contributions. Both and i//MO are inadequate in their simplest forms while in the VB theory the electron correlation is overemphasized, simple MO theory totally neglects it giving equal importance to covalent and ionic structures. Therefore neither of them is able to predict binding energies closer to experiment. The MO theory could be... 

In Section 1.19 we classified the electronic wave functions of homonuclear diatomic molecules as g or u, according to whether they were even or odd with respect to inversion g and u refer to inversion of the electronic coordinates with respect to the molecule-fixed axes. This is to be distinguished from the inversion of electronic and nuclear coordinates with respect to space-fixed axes, which was discussed in this section. The electronic Hamiltonian for a diatomic molecule is... 

Interchanging the nuclear coordinates does not affect R, but it does affect the electronic spatial coordinates since they are defined with respect to the molecule-fixed xyz axes, which are rigidly attached to the nuclei. To find the effect on el of interchanging the nuclear coordinates, we will first invert the space-fixed coordinates of the nuclei and the electrons, and then carry out a second inversion of the space-fixed electronic coordinates only the net effect will be the interchange of the space-fixed coordinates of the two nuclei. We found in the last section that inversion of the space-fixed coordinates of all particles left //e, unchanged for 2+,n+,... electronic states, but multiplied it by —1 for 2, II ,... states. Consider now the effect of the second step, reinversion of the electronic space-fixed coordinates. Since the nuclei are unaffected by this step, the molecule-fixed axes remain fixed for this inversion, so that inversion of the space-fixed coordinates of the electrons also inverts their molecule-fixed coordinates. But we noted in Section 1.19 that the electronic wave functions of homonuclear diatomics could be classified as g or m, according to whether inversion of molecule-fixed electronic coordinates multiplies ptl by + 1 or -1. We conclude that for 2+,2,7,11, IV,... electronic states, i//el is symmetric with respect to interchange of nuclear coordinates, whereas for... 

For homonuclear diatomic molecules, the electronic wave functions have definite parity (g or w), and since del is of odd parity, we must have a change in parity of f/el (corresponding to the Laporte rule in atoms) ... 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

As in section 6.9.3, we start out with the case (a) functions and consider the effect of l 2 on each of its three factors in turn. For a homonuclear diatomic molecule, we write the electronic orbital wave function as a linear combination of spherical harmonics ... 

As was shown in the preceding discussion (see also Sections VIII and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and electronic states antisymmetric for odd J values in and electronic states symmetric for odd J values in S7 and electronic... 

We now use the H2 MOs developed in the last section to discuss many-electron homonuclear diatomic molecules. If we ignore the interelectronic repulsions, the zeroth-order wave function is a Slater determinant of H -like one-electron spin-orbitals. We approximate the spatial part of the spin-orbitals by the LCAO-MOs of the last section. Treatments that go beyond this crude first approximation will be discussed later. 

The MO approximation puts the electrons of a molecule in molecular orbitals, which extend over the whole molecule. As an approximation to the molecular orbitals, we usually use linear combinations of atomic orbitals. The VB method puts the electrons of a molecule in atomic orbitals and constructs the molecular wave function by allowing for exchange of the valence electron pairs between the atomic orbitals of the bonding atoms. We compared the two methods for H2. We now consider other homonuclear diatomic molecules. 

We have seen that the electronic wave functions of homonuclear diatomic molecules can be classified as g or m, according to their parity. Hence a homonuclear diatomic molecule has a zero permanent electric dipole moment, a not too astonishing result. The same holds true for any molecule with a center of symmetry. The electric dipole-moment operator for a molecule includes summation over both the electronic and nuclear charges ... 

Bader, Henneker, and Cade have taken the electron probability densities for homonuclear diatomic molecules at as calculated from Hartree-Fock functions and subtracted off the probability densities for the corresponding separated atoms, as calculated from atomic Hartree-Fock wave functions [R. F. W. Bader, W. H. Henneker,... 

For a homonuclear diatomic molecule composed of even(odd) mass-ntmber nuclei, the total wave function, which we assume to be a product of electronic, vibrational, rotational, and nuclear-spin functions, must be symmetric (antisymmetric). If the electronic wave function is symmetric, and if the nuclear spin is zero, as in the ground state of 0a, only even values of J, the rotational quantum number, are allowed. If the nuclear spin is not zero, both even and odd values of J (l.e., symmetric and antisymmetric rotational wave functions) are allowed, but with different statistical weights. These may be determined from the nuclear-spin part of the wave function. 

The electronic wave functions of homonuclear diatomic molecules can be approximated as products of LCAOMOs. 

The stretching or compression of a bond changes the electronic wave function and therefore changes the polarizability. The vibration of a diatomic molecule is Raman active, whether the molecule is homonuclear or heteronuclear. A vibrational normal... 

The complete, nonrelativistic Hamiltonian for a diatomic molecule is given by (1.272). If one inverts the Cartesian coordinates of all particles (nuclei and electrons), then H in (1.272) is unchanged, since all interparticle distances are unchanged. Thus the parity operator IT commutes with this Hamiltonian, and we can characterize the overall wave function of a diatomic molecule by its parity. (This statement applies to both homonuclear and heteronuclear diatomics.)... 

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