Energy gaps calculations

Table 5 Stokes shift, i.e. difference between absorption and emission energy gaps calculated as total energy differences within the A-SCF approach. All values are in eV...

Verify the HOMO LUMO energy gaps calculated by PMO theory for the systems shown in Scheme 4.1. 

Figure C3.5.6 compares the result of this ansatz to the numerical result from the Wiener-Kliintchine theorem. They agree well and the ansatz exliibits the expected exponential energy-gap law (VER rate decreases exponentially with Q). The ansatz was used to detennine the VER rate with no quantum correction Q= 1), with the Bader-Beme hannonic correction [61] and with a correction based [83, M] on Egelstaff s method [62]. The Egelstaff corrected results were within a factor of five of experiment, whereas other corrections were off by orders of magnitude. This calculation represents the present state of the art in computing VER rates in such difficult systems, inasmuch as the authors used only a model potential and no adjustable parameters. However the ansatz procedure is clearly not extendible to polyatomic molecules or to diatomic molecules in polyatomic solvents.

Not all Iterative semi-empirical or ah iniiio calculations converge for all cases. For SCF calculation s of electronic stnictiire. system s with a small energy gap between the highest occupied orbital and the lowest unoccupied orbital may not converge or may converge slowly. (They are generally poorly described by the Ilartree-Foch method.)... 

Some researchers use molecule computations to estimate the band gap from the HOMO-LUMO energy separation. This energy separation becomes smaller as the molecule grows larger. Thus, it is possible to perform quantum mechanical calculations on several molecules of increasing size and then extrapolate the energy gap to predict a band gap for the inhnite system. This can be useful for polymers, which are often not crystalline. One-dimensional band structures are... 

The exponent in this formula is readily obtained by calculating the difference of quasiclassical actions between the turning and crossing points for each term. The most remarkable difference between (2.65) and (2.66) is that the electron-transfer rate constant grows with increasing AE, while the RLT rate constant decreases. This exponential dependence k AE) [Siebrand 1967] known as the energy gap law, is exemplified in fig. 14 for ST conversion. 

Fig. 3. Energy gap versus inverse nanotube diameter, for the nine nanotubes studied the dashed line is a regression through the points, the full line is a calculation for semiconducting zigzag nanolubes[7,13] (adapted from Oik et n/.(ll]).

The electronic properties of single-walled carbon nanotubes have been studied theoretically using different methods[4-12. It is found that if n — wr is a multiple of 3, the nanotube will be metallic otherwise, it wiU exhibit a semiconducting behavior. Calculations on a 2D array of identical armchair nanotubes with parallel tube axes within the local density approximation framework indicate that a crystal with a hexagonal packing of the tubes is most stable, and that intertubule interactions render the system semiconducting with a zero energy gap[35]. 

Table 2. Calculated energy gap due to an in-plane Kekul distortion for CNTs having chiral vector L/a = (m, 2m). The critical magnetic flux (p. and the corresponding magnetic field are also shown. The coupling constant is A, = 1.62.

Unusual photophysical properties of polyazaanthracenes and polyazapentacenes having low values of calculated singlet-triplet energy gap 99PAC295. 

The upper VB for ratile is composed of the 02p orbitals and is 5.52 eV wide. The lower 02s band has a width of 2.08 eV. These numbers agree well with the experimental values of 5.4 and 1.9 eV, respectively. 1 The calculated direct energy gap of 1.83 eV is in good agreement with other LDA results and is smaller than the experimental value of 3.0 eV. 1... 

Predictions obtained by using the frontier orbital approximation213 were unsuccessful, apparently due to inadequacies in these MO calculations mostly involving the energy gap between HO of the dipole and LU of the dipolarophile. 

FIGURE 3.5. The actual free-energy profile for the ground-state surface as a function of the energy gap As. The calculations are done for the CF + CH3C1- C1CH3 + CP exchange reaction (Ref. 11). 

Such an orbital phase picture in Fig. 14 is also applicable to rationalize the relative S-T gaps of hetero diradicals 19 and 20. hi comparison with their parent system, 1,3-dimethylenecyclobutadiene (DMCBD, 10), the introduction of oxygen atoms does destabilize the triplet state. The calculated energy gap between singlet and triplet states, AE deaeases in the order 10 (18.2 kcal moF ) > 19 (7.7 kcal moF ) > 20 (-20.7 kcal moF ) [64]. These results supported the orbital phase predictions. 

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