“Simple-harmonic motion” (SHM) electronic potentials
The locally maximum FCFs in a Deslandres table can be connected like dots in a child's puzzle. The result is a curve called the Condon locus, which more or less[5-7] resembles a parabola (Fig. 1).
Figure 1. A Deslandres table showing some locally maximal FCFs q(v′,v′′). The simple harmonic approximation Condon locus is shown. The locus symmetry axis appears to lie at 45° from v″. Adapted from Herzberg, pg. 197.
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Classically, it is clear that the most likely transitions are at internuclear separations at the extrema of the motion, when one upper-state extremum is directly above a lower-state extremum (Fig. 2). The FCF for that band will be large compared to others, and the totality of all such factors is the Condon locus. Quantum mechanically the strongest transitions will occur when the wave function antinodes are more or less one above the other near the classical extrema. The wave functions also allow relatively strong transitions when the antinodes are above one another but not at the classical extrema; such transitions will form secondary Condon loci.[5-7]
Figure 2. Schematic of two SHM potential-energy curves of a molecule, with energy on the vertical axis and internuclear separation on the horizontal axis. The uppermost feature represents the probability of finding a classical oscillator at points in its domain.
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The transitions that occur when the molecule is at an internuclear separation matching the end-points of vibrational states, in both the upper and lower electronic levels, will have the highest FCFs. The probability of finding the molecule at a given internuclear separation is shown schematically in Figure 2 for the upper state. These transitions take place almost instantaneously, in a time that is very short compared with the period of molecular vibration (the Born–Oppenheimer approximation); they take place with virtually no change in internuclear distance. For that reason, transitions can be indicated in energy level diagrams by means of vertical lines.
If we take it as given that the two electronic potentials are parabolas, it is a good exercise in conic sections to show that the Condon locus that follows the strongest FCFs is also a parabola. The derivation appears in full in Ref. ; here, we review the results. The slope of the axis of the Condon parabola with respect to v″ is given simply by
where ωe are in cm−1, the k are force constants, and the single and double primes indicate the upper and lower electronic potentials. The length of the latus rectum is given by
with l being dimensionless and L having units of cm, and where
The latera recta are taken to have the same length scale as the quantum numbers v′ and v and m is the reduced mass. The behaviors of the lower-state vibration frequencies are well-studied and smooth; those of the upper-state vibration frequencies are irregular, and these irregularities account for the deviations of data point in the graphs to be shown in a later section.
The derivation implies the following:
- If θ = 45°, then the force constants in the two potentials are equal. If θ is smaller, then the lower electronic state is more tightly bound than the upper state.
- A Condon parabola has its vertical and horizontal tangents at v = −1/2.
- If the internuclear distances are equal, the parabola degenerates to a straight line, and the strongest bands have coordinates (v′,v″) = (0,0), (1,1), (2,2), and so on. Otherwise, the parabola is open.
Morse and other anharmonic potential curves
The Condon loci for Morse potentials can be calculated in much the same way as the simple harmonic case, except that additional anharmonic spectroscopic constants are necessary. The same applies also to all anharmonic potentials, such as the Lennard–Jones potential or indeed any potential function, even if the potential is given only as a table of numerical values.
The derivation which follows is classical, based on the premise that, at any instant of time, the molecule is most likely to be at its vibrational energy level position of greatest compression or greatest separation. For a quantum mechanical calculation, one can assume that, at any instant of time, the internuclear separation is most likely to be in both levels where there are maximum antinodes of the wave function. For large values of the vibrational quantum number, these maxima are close to the classical extrema; the classical approximation will become less accurate at very low quantum numbers.
Suppose that the potential energy functions of the two electronic states are
Both De are the depths of the potential wells. Ve′ and Ve″ do not come into the calculation of the FCFs, so we might as well take both of them to be zero. Thus, the potential functions are
The a′ and a″ are given in terms of the spectroscopic constants by
The ωexe is the anharmonicity constant in m−1 and m is the reduced mass in amu. De′ and De″are given in terms of the spectroscopic constants by
One begins by taking some value of V′ (between 0 and De′) and we calculate the extreme values of r, which we call rL and rR (for left and right). These are given by
Now, one goes to the lower potential curve and to eq. (6b). One chooses r = rL and r = rR to find the two corresponding values of V″. This process gives two points of the Condon locus in the (V′, V″) plane.
To go from the (V′,V″) plane to the (v′,v″) plane, one uses
One inverts these quadratic formulae in (v + ½), eq. (10), to obtain
Thus, one has two points of the Condon locus in the (v′, v″) plane, one on each branch of the locus. Repeat for many values of Ve′ between 0 and De′ to delineate the Condon locus. Note that v = ωe/(2ωexe) corresponds to a vibrational level whose energy is equal to the dissociation limit. In principle, vibrational levels with larger values of v (with energies below the dissociation limit), and transitions from them, are possible. These are unlikely to be observed, and so we restrict our attention to solutions with a negative sign before the square roots in eqs. (11).
The derivation, and computer experiments that were done afterward, imply the following:
- The upper branch of the Condon locus will pertain to the horizontally asymptotic portion of the Morse curve of the upper electronic potential curve. (At a given instant of time, the molecule is more likely to be at its condition of greatest extension than of greatest compression.)
- If the anharmonicity constants for the two electronic states are equal but not very large, there is little effect on the Condon locus.
- If the anharmonicity constants are equal but very large, however, the Morse locus deviates with upward curvature from the SHM locus.
- If the anharmonicity constants are unequal, with ωexe′ > ωexe″, the Morse locus is inside the SHM locus and crosses itself.
- If the anharmonicity constants are unequal, with ωexe′ < ωexe″, then both arms of the Morse locus diverge outward from SHM locus.