Exchange approximation

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

The Spectral Density of Pure Fermi Resonance Beyond the Exchange Approximation... 

D. Discussion on the Exchange Approximation in the Absence of a Hydrogen Bond... 

In this section we shall give the connections between the nonadiabatic and damped treatments of Fermi resonances [53,73] within the strong anharmonic coupling framework and the former theory of Witkowski and Wojcik [74] which is adiabatic and undamped, involving implicitly the exchange approximation (approximation later defined in Section IV.C). 

At last, the Fermi coupling potential may be expressed beyond the exchange approximation by introducing the dimensionless Fermi coupling parameter A ... 

More explicitly, the exchange approximation needs to keep in the total Hamiltonian the simple coupling term... 

Following the steps shown in section IV.B which remain valid within the exchange approximation, one may easily obtain the expression of the resulting spectral density /Sf ex ... 

We shall compare later the spectral density /sf ex and the adiabatic one If [given later by Eq. (103)] that supposes implicitly the exchange approximation. We may expect no difference between their lineshape in cases for which the adiabatic approximation is valid. 

A physical picture of the Fermi coupling within the exchange approximation is given in Fig. 7. 

Figure 7. The Fermi resonance mechanism within the adiabatic and exchange approximations. F, fast mode S, slow mode B, bending mode.

Figure 8. Comparison between the adiabatic spectral density and the standard one (with or without the exchange approximation), (a) and (c) display the spectral density Isf from Eq. (81), using dashed lines, (b) and (d) display the spectral density /sf ex from Eq. (88), within the exchange approximation, using dashed lines. Comparison is made with the adiabatic spectral density If (thin plain lines) obtained from (96). Spectra (a) and (b) are computed with Oo = 1.2, whereas spectra (c) and (d) are computed with a0 = 0.6. Common parameters A = 120cm-1, (n0 = 3000cm-1, C05 = 1440 cm-1, co00 = 150 cm-1, y0 = y5 = 60 cm-1, and T = 300 K.

We kept the same structure in Figs. 8(a) and 8(b), but the spectra were computed for a greater value a0 = 1.2. As may be seen, some differences between the adiabatic and nonadiabatic spectral densities appear in all cases, whether applying the exchange approximation or not. Within the exchange approximation, Fig. 8(b), these discrepancies may be safely attributed to the... 

Now, comparing 7Sf and /Sf ex, we observe also some splitting modifications. We may attribute them to some additional changes in the frequency gap. Indeed, as shown in Section V.B, the full treatment of the Fermi coupling mechanism leads to a displacement of the potential of the fast mode (both in energy and position) that does not appear within the exchange approximation. The fast mode then involves an effective frequency that differs from oo0, which leads to an effective gap which differs from (oo0 — 2oog). 

Applying the exchange approximation and neglecting the zero-point energy terms, we may safely limit the representation of the Hamiltonian Hsf ex within the following reduced base which accounts for the ground states of each mode and for the first (second) overtone of the fast (bending) mode ... 

These behaviors may be observed in Fig. 10(b), which displays three spectra computed with an increasing parameter A and A = —120 cm-1. We shall see in the following that beyond the exchange approximation the spectral density behaves in an opposite way. 

We have seen that the exchange approximation consists mainly in ignoring the driven part of the fast mode motion, given by... 

The exchange approximation when dealing with Fermi resonance (A O)... 

Some sample calculations are displayed in Fig. 10. As may be seen, the spectral density (124) involves two sub-bands in the vs (X-H) frequency region, like it is observed within the exchange approximation. Note that other submaxima appear at overtone frequencies (near 2oo0, 3co0,. ..) with a much lower intensity (less than 0.1% of the doublet intensity) and will not be studied here. 

Moreover, because the driven term depends on A, a strengthening of the Fermi coupling implies an increase of the effective frequency gap between the coupled levels, and therefore an increase of the intensity gap, that is at the opposite of the behavior observed within the exchange approximation. This... 

Figure 10. Pure Fermi coupling within or beyond the exchange approximation. Left column spectra were obtained from expression (110) of the spectral density /sf (m, V. = 0) ex (a) Resonant case A — 0. A — 60 cm-1 (b) nonresonant case A — 120cm 1 with A — 60 cm-1 (dotted line),...

We may finally conclude that, with the purpose of comparison with experiments, one has to be careful and must remember the following (i) Two sub-bands of the same intensity may not be the consequence of a resonant situation 0, (ii) The frequency of each submaxima is governed by the three parameters co0, o>0. and A, and (iii) The frequency of the Evans hole, which appears between the two sub-bands, is given within the exchange approximation by the average frequency j ( 0 + 2o>0), but it is dependent on A within the full treatment of Fermi resonances. 

Figure 11. Pure Fermi coupling within the exchange approximation relative influence of the damping parameters. Common parameters

Figure 12. Hydrogen bond involving a Fermi resonance damping parameters switching the intensities. The lineshapes were computed within the adiabatic and exchange approximations. Intensities balancing between two sub-bands are observed when modifying the damping parameters (a) with y0 =0.1, and y5 = 0.8 (b) with ya = ys = 0.8 (c) with yB — 0.8 and ys — 0.1. Common parameters oto = 1, A = 150cm, 2g)5 = 2850cm-1, and T = 30 K.

As a consequence of the two above points, there is presently no satisfactory theory able to incorporate the specific relaxation of the fast mode and of the bending modes in a model working beyond both the adiabatic and exchange approximations. 

---