Notation, molecular states

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

In Tables 5 and 6 we compare the spectral intensities, in the form of oscillator strengths, for 3s-3p and 3p-3d transitions (in the united atom notation) between states of the same and different molecular symmetries within each corresponding molecular point group, in the Rydberg radicals CH5, H3O, H2F, and NeH, all ofwhich have sodium as the united atom limit. 

Platt s notation for molecular states is N M lt,pl,c lyStateSymmelry, where N labels the first, second, third, etc., states of the same type, the multiplicity is (25 + 1), A and B states are non-degenerate and degenerate, respectively, and g and u identify symmetric (gerade) and antisymmetric (ungerade) states, respectively. 

In describing states of the small molecule (as well as the solid) the first step is to enumerate each of the electronic states in the atom that will be used in the mathematical expansion of the electron states in (he molecule. These become our basis states. We let the index a = 1, 2, 3,..., n run from one up to the number of states that are used. Then the molecular state may be written (with the notation discussed in Section 1-A) as... 

It should be noted that instead of dressing the molecular states g and s by 1 and 0 photons, respectively, we could use any photon numbers p and p — 1. The corresponding matrix elements are than proportional to p. In processes pertaining to linear spectroscopy it is convenient to stick with photon populations 1 and 0, keeping in mind that all observed fluxes should be proportional to the incident photon numberp or, more physically, to the incident field intensity fo P- With this in mind we will henceforth use the notation g, k) (or g, m if the incident direction is not important for the discussion) as a substitute for g, Ik)-... 

In order that a consistent notation be used we shall denote total molecular states with capital letters, e. g. I and J. The I > state will always be considered to be lower than the corresponding Jth states of the same multiplicity. Generally the Ith and J h states will refer to the ground and first excited molecular states. The small letters, i. e. i, i, and i", represent the orbital, vibrational, and spin substates of the total molecular states. 

The simplest paramagnetic molecular state with A = 0 is The rotational quantum number N is equal to R from the above discussion, and from here forward we will use N for the discussion of nuclear rotation, in accordance with standard notation for E states. The lone valence electron contributes 1 p.B of magnetic moment, making E molecules suitable for trapping. The rotational ground state of E is a pure N = 0) state, which eliminates helium-induced Zeeman relaxation from first-order effects, in accordance with Equation 13.6. In order to change the spin projection Ms, there must be a mechanism by which the spin can be coupled to the rotational wavefunction, which can be altered by the electrostatic interaction. 

In the BO approximation, the total molecular state i can be written, in Dirac notation, as T,) = iXr(i))> where hereafter an underhned subscript indicates an electronic state (i) while a vibrational state r associated to the electronic state 1 is indicated with r f). Moreover, the dipole moment is separated into its electronic and nuclear components, respectively, and therefore, from Eq. 8.17, we have... 

Even with the high laser intensities employed for the study of multiphoton processes, the electric fields they produce are seldom comparable to intramolecular coulombic fields. As such, the interaction operator [9] can be treated as a perturbation on the molecular states. The result of these perturbations is to modify the form of the Schrodinger equation so that its usual eigenfunctions no longer represent stationary states, and hence transitions occur. The quantum amplitude for a transition between a given initial state i and a final state f is given by the following result from time-dependent perturbation theory, expressed in Dirac notation ... 

There is another commonly used notation known as second quantization. In this language the wave function is written as a series of creation operators acting on the vacuum state. A creation operator aj working on the vacuum generates an (occupied) molecular orbital i. 

Operational definitions of molecular structure are needed to clarify experimental significance. In addition, some statistical notation is needed to clarify physical meaning. All statistical definitions hinge on the minimum of potential energy in a bound electronic state, which defines the equilibrium geometry or r,-intemuclear distance type. 

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