Quartic anharmonic oscillator

In order to better understand the structure of the scalet equation formalism, we consider the problem of the quartic anharmonic oscillator potential... 

For the quartic anharmonic oscillator case, the (approximate) form of a depends on whether c b, m) is positive or negative. In the positive case, the term (in the exponential of the second and third modes given above) dominates. If c(6, m) is negative, then the a terms (with a real factor) dominates. As such we have ... 

We examine the behavior of the above asymptotic expansions, at the real turning points (for simplicity), for the ground state of both the quar-tic anharmonic oscillator (e = m = = l), and the double well quartic anharmonic oscillator e = g = 1 and m —5). 

The (convergent) asymptotic series results agree with the direct integration of the scalet equation, where the missing moments and energies can be obtained by MRF analysis. Specifically, for the quartic anharmonic oscillator, Egr = 1.392351642, the nonzero missing moments are /x(0) =. 6426706223, and p(2) =. 3573293777. For the quartic double well potential, Egr = -3.410142761, p(0) = 0.3223013271, and p(2) = 0.6776986729. 

In this section we present a more intuitive proof of the results derived for the quartic anharmonic oscillator which argue that either all, or a large number, of scalet solutions converge to solutions of the Schrodinger equation. 

In addition, we develop a slightly more comprehensive analysis of the scaling transform and its derivatives, including the significance of the adopted definition for the quantization scale, as presented within our analysis of the quartic anharmonic oscillator. 

It is for this case, as argued earlier (for the quartic anharmonic oscillator), that we can take the quantization scale, ag(T), to be defined as the smallest, nontrivial scale-root satisfying... 

The Morse potential provides a useful model for the anharmonic stretching vibrations of a polyatomic molecule. It is superior to a harmonic oscillator perturbed by cubic and quartic anharmonicity terms in terms of both convergence (a V(r) that includes an r3 term cannot be bound, thus cannot have any rigorously bound vibrational levels) and the need for a smaller number of adjustable parameters to describe both the potential energy curve (a and De for Morse frr, frrri and frrrr for the cubic plus quartic perturbed harmonic oscillator) and the energy levels. 

D. G. Truhlar, Oscillators with quartic anharmonicity Approximate energy levels,/. Molec. Spec. 38 4151 (1971). 

Where wg is the corresponding frequency of vibration (in cm i). The remaining terms give the cubic, quartic, etc. contributions of the potential, with the cubic, quartic, etc. contributions of the potential, with the cubic, quartic, etc., force constants, respectively. The energy levels of an anharmonic oscillator follow from the term values... 

In Section A, we will describe several methods that only take into account poles and are therefore only appropriate at low orders. The low-order terms of dimensional expansions are relatively easy to derive, so these methods are useful for a quick qualitative analysis. This level of approximation corresponds to a simple physical model Eq is the energy of the infinite-P Lewis configuration, E corresponds to the harmonic Langmuir oscillations, and E2 represents cubic and quartic anharmonicities [13]. 

This approach gives very good results for the quartic anharmonic double well oscillator V x) = —5x + x (i.e. ground and first excited states), and the quartic potential V(x) = x. The first has all real turning points whereas the second has real and pure imaginary turning points. The results of this analysis are given in Tables 8 and 9, respectively (Handy and Brooks (2000)). 

Figure 3. Schematic depiction of first three terms in dimensional perturbation expansion of Eq.(12) for hydrogenic atom. For each the effective potential W(r) for the D oo limit is shown (solid curve). At left, the zeroth-order term corresponds to the electron at rest at the minimum of T (r). In the middle, the first-order term, proportional to 1/D, corresponds to harmonic oscillations (as if potential were replaced by the dashed parabola). At right, the second-order term corresponds to anharmonic vibrations (arising from cubic and quartic portions of the potential).

The most realistic vibrational potentials of molecules are not strictly harmonic. For a diatomic molecule, the stretching potential s dependence on the separation distance may turn out to be cubic, quartic, and so on. A potential is harmonic if it has only a quadratic dependence. Any higher-order dependence is an anharmonicity in the potential. Vibrational anharmonicity refers to the effects on energy levels and transitions of an otherwise harmonic oscillator arising from anharmonic potential terms. 

The first term is a harmonic element. For as < 1, the typical situation, the terms diminish in size beyond the harmonic term. Even for as 1, the terms diminish beyond the cubic term. The cubic and quartic terms are the leading sources of anharmonicity near the equilibrium of a Morse oscillator, and it is very often the case that a cubic potential element is the most important contributor to anharmonicity effects in real molecules. 

---