Wave functions diatomic molecules

McLean and Yoshimine 58, 59) published an extensive hst of Tables of Linear Molecule Wave Functions which include the diatomic molecules e.g. the hydrogen halides, the oxides (e.g. BeO, MgO), the alkali halides, PN, as well as HCN, HFi, C02, N2O, SCN, C2H2, each for a number of intemuclear configurations. These Tables represent the best ... 

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. 

Next, we address some simple eases, begining with honronuclear diatomic molecules in E electronic states. The rotational wave functions are in this case the well-known spherical haimonics for even J values, Xr( ) symmetric under permutation of the identical nuclei for odd J values, Xr(R) is antisymmetric under the same pemrutation. A similar statement applies for any type molecule. 

In general, for a homonuclear diatomic molecule there are (21+ )(/+1) symmetric and (21+ 1)/antisymmetric nuclear spin wave functions therefore... 

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... 

For a homonuclear diatomic molecule with nuclei labelled 1 and 2 the LCAO method gives the MO wave function... 

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. 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

Hurley, A. C., Proc. Roy. Soc. [London) A216, 424, The molecular orbital theory of chemical valency. XIII. Orbital wave functions for excited states of a homonuclear diatomic molecule."... 

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... 

Although we have dealt with a diatomic molecule consisting of two hydrogen atoms, the procedure is exactly the same if the molecule is Li2, except that the atomic wave functions are 2s wave functions and the energies involved are those appropriate to lithium atoms. The VSIP for lithium is only 513 kj mol-1 rather than 1312kj mol-1 as it is for hydrogen. 

The simplest diatomic molecule consists of two nuclei and a single electron. That species, H2+, has properties some of which are well known. For example, in H2+ the internuclear distance is 104 pm and the bond energy is 268kJ/mol. Proceeding as illustrated in the previous section, the wave function for the bonding molecular orbital can be written as... 

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... 

Only those problems that can be reduced to one-dimensional one-particle problems can be solved in closed form by the methods of wave mechanics, which excludes all systems of chemical interest. As shown before, several chemical systems can be approximated by one-dimensional model systems, such as a rotating diatomic molecule modelled in terms of a rotating particle in a fixed orbit. The trick is to find a one-dimensional potential function, V that provides an approximate model of the interaction of interest, in the Schrodinger formulation... 

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