Quantum number fractional populations

Figure la. Fractional populations as a function of rotational quantum number F2(3) is pumped. A2X state (O),Ft levels, (9),F, levels. 

Equation [9] shows how the fraction of total population in a given rovibrational energy level varies with temperature and rotational and vibrational quantum numbers. For most gas-phase diatomic molecules, if the population in a known rovibrational level is measured. Equation [9] allows the total population (and hence total pressure) of the gas to be calculated. 

Fig. 3.5.1 Fractional populations N(v, /)/A TcyrAL of the v = 0 state of rotational levels in carbon monoxide at 100,200, and 300 K. The lower scale is the rotational quantum number J. The upper scale is the energy in cm for the rotational levels.

The fluorescent components are denoted by I (intensity) followed by a capitalized subscript (D, A or s, for respectively Donors, Acceptors, or Donor/ Acceptor FRET pairs) to indicate the particular population of molecules responsible for emission of/and a lower-case superscript (d or, s) that indicates the detection channel (or filter cube). For example, / denotes the intensity of the donors as detected in the donor channel and reads as Intensity of donors in the donor channel, etc. Similarly, properties of molecules (number of molecules, N quantum yield, Q) are specified with capitalized subscript and properties of channels (laser intensity, gain, g) are specified with lowercase superscript. Factors that depend on both molecular species and on detection channel (excitation efficiency, s fraction of the emission spectrum detected in a channel, F) are indexed with both. Note that for all factorized symbols it is assumed that we work in the linear (excitation-fluorescence) regime with negligible donor or acceptor saturation or triplet states. In case such conditions are not met, the FRET estimation will not be correct. See Chap. 12 (FRET calculator) for more details. 

Sensitometric tests at variable light intensity I under conditions of photography, achieved on emulsions doped at the relative concentration of 10 mol HCO2 per mol Ag, confirmed the photo-induced bielectronic transfer (Fig. 15, bottom) [200]. The emulsion is completely stable in the dark. The number of photons required to induce development of the same grain population fraction is 5 times less (after immediate development) or 10 times less (development delayed by 20 min after exposure) in doped than in undoped emulsions where < eff=0.20. The quantum yield is thus close to the theoretical limit atom/photon in immediately developed doped emulsion (7 = 0),... 

The gain, from Eq. (3), is governed by a product of the stimulated emission cross section and the population inversion (N3-N2). The latter is dependent upon the absorption spectrum and its spectral match with the pump source, the lifetime of the metastable level 3 which determines the pumping rate required, and the quantum efficiency. The last quantity includes the fluorescence conversion efficiency (the number of ions excited to the fluorescing level per incident pump photon) and the quantum efficiency of the fluorescing state (the fractional number of photons emitted per excited ion in the upper laser level). 

The time scale for electron dynamics studied above is very short, typically shorter than 1 femtosecond, during which the nuclei are supposed to stay still. In the above process, the electron loss has been taken into account through the change of the population of natural orbitals and subsequently the change of the coefficients of the Slater determinants. Therefore the same set of self-consistent field molecular orbitals (SCF MO) may be used for the short time period. However, for a longer time scale, it would be better to redetermine the molecular orbitals under the gradual loss of electronic population. This demands an efficient way to determine the molecular orbitals with variable and fractional occupation numbers, although this is not a new problem for quantum chemistry. 

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