Molecular Quantum Numbers

Hamiltonian contains (h2/2mc r2) cfi/dy2, whereas the potential energy part is independent of y, the energies of the molecular orbitals depend on the square of the m quantum number. Thus, pairs of orbitals with m= 1 are energetically degenerate pairs with m= 2 are degenerate, and so on. The absolute value of m, which is what the energy depends on, is called the X quantum number. Molecular orbitals with X = 0 are called o orbitals those with X = 1 are Jt orbitals and those with X = 2 are 8 orbitals. 

The solutions can be labelled by their values of F and m.p. We say that F and m.p are good quantum. num.bers. With tiiis labelling, it is easier to keep track of the solutions and we can use the good quantum numbers to express selection rules for molecular interactions and transitions. In field-free space only states having the same values of F and m.p can interact, and an electric dipole transition between states with F = F and F" will take place if and only if... 

We have described here one particular type of molecular synnnetry, rotational symmetry. On one hand, this example is complicated because the appropriate symmetry group, K (spatial), has infinitely many elements. On the other hand, it is simple because each irreducible representation of K (spatial) corresponds to a particular value of the quantum number F which is associated with a physically observable quantity, the angular momentum. Below we describe other types of molecular synnnetry, some of which give rise to finite synnnetry groups. 

The rotational states are characterized by a quantum number J = 0, 1, 2,. .. are degenerate with degeneracy (2J + 1) and have energy t r = ) where 1 is the molecular moment of inertia. Thus... 

Most molecular vibrations are well described as hannonic oscillators with small anlrannonic perturbations [5]. Por an hannonic oscillator, all single-quantum transitions have the same frequency, and the intensity of single-quantum transitions increases linearly with quantum number v. Por the usual anhannonic oscillator, the single-quantum transition frequency decreases as v increases. Ultrashort pulses have a non-negligible frequency bandwidth. Por a 1... 

Now, we have besides the vibrational, the electronic angular momentum the latter is characterized by the quantum number A corresponding to the magnitude of its projection along the molecular axis, L. Here we shall consider A as a unsigned quantity, that is, for each A 7 0 state there will be two possible projections of the electronic angular momentum, one corresponding to A and the other to —A. The operator Lj can be written in the form... 

The trends in chemical and physical properties of the elements described beautifully in the periodic table and the ability of early spectroscopists to fit atomic line spectra by simple mathematical formulas and to interpret atomic electronic states in terms of empirical quantum numbers provide compelling evidence that some relatively simple framework must exist for understanding the electronic structures of all atoms. The great predictive power of the concept of atomic valence further suggests that molecular electronic structure should be understandable in terms of those of the constituent atoms. 

Much of quantum chemistry attempts to make more quantitative these aspects of chemists view of the periodic table and of atomic valence and structure. By starting from first principles and treating atomic and molecular states as solutions of a so-called Schrodinger equation, quantum chemistry seeks to determine what underlies the empirical quantum numbers, orbitals, the aufbau principle and the concept of valence used by spectroscopists and chemists, in some cases, even prior to the advent of quantum mechanics. 

Ab initio ECPs are derived from atomic all-electron calculations, and they are then used in valence-only molecular calculations where the atomic cores are chemically inactive. We start with the atomic HF equation for valence orbital Xi whose angular momentum quantum number is 1 ... 

The quasi-classical theory of spectral shape is justified for sufficiently high pressures, when the rotational structure is not resolved. For isotropic Raman spectra the corresponding criterion is given by inequality (3.2). At lower pressures the well-resolved rotational components are related to the quantum number j of quantized angular momentum. At very low pressure each of the components may be considered separately and its broadening is qualitatively the same as of any other isolated line in molecular or atomic spectroscopy. 

Initial conditions for the total molecular wavefunction with n = I (including electronic, vibrational and rotational quantum numbers) can be imposed by adding elementary solutions obtained for each set of initial nuclear variables, keeping in mind that the xi and 5 depend parametrically on the initial variables... 

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