V, vibrational quantum number

V vibrational quantum number of the state with the measured value of li in column 3... 

V vibrational quantum number of the /-th normal vibration J= 1,. ..,3A-6 (for linear moleculesJ = 1,..., 3N-5) vibrational ground state ... 

B = rotational constant = ionization limit h = planck constant J = rotational quantum number n = principal quantum number 0 = multiphoton transition operator R = Rydberg constant Tq 0) = qth component of the tensor of rank k representing O d = quantum defect 8 = electric vector v = frequency V = vibrational quantum number v = wave-number. 

B = constant for methyl-group rotation / = vibronic angular momentum J = total angular momentum with a projection of K in the molecular frame m = rotational quantum number n = principal quantum number N = total angular momentum excluding spin a = torsional angle v = vibrational quantum number (o = corresponding vibrational constant. 

Ei = energy of ith mode F= quadratic force constant matrix G = inverse kinetic energy matrix h = Planck s constant icg = Boltzmann s constant 1 = pathlength M= molecular mass Q = normal mode i = internal coordinate v = vibrational quantum number T = integrated absorption intensity jji = dipole moment v = frequency p = radiation density 0) = wavenumber. 

Each such nonual mode can be assigned a synuuetry in the point group of the molecule. The wavefrmctions for non-degenerate modes have the following simple synuuetry properties the wavefrmctions with an odd vibrational quantum number v. have the same synuuetry as their nonual mode 2the ones with an even v. are totally symmetric. The synuuetry of the total vibrational wavefrmction (Q) is tlien the direct product of the synuuetries of its constituent nonual coordinate frmctions (p, (2,). In particular, the lowest vibrational state. 

Figure B2.3.10. Potential energy eiirves [42] of the ground X and exeited A eleetronie states of the hydroxyl radieal. Several vibrational levels are explieitly drawn in eaeh eleetronie state. One vibrational transition is explieitly indieated, and the upper and lower vibrational wavefiinetions are plotted. The upper and lower state vibrational quantum numbers are denoted V and v", respeetively. Also shown is one of the three repulsive potential energy eurves whieh eorrelate with the ground 0( P) + H dissoeiation asymptote. These eause predissoeiation of the higher rotational and vibrational levels of the A state.

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

Over the next few years, both the mid-infrared and the far-infrared spectra for Ar-HF and Ar-HCl were extended to numerous other bands and to other isotopic species (most importantly those containing deuterium). In 1992, Hutson [18, 39] combined all the available spectroscopic data to produce definitive potential energy surfaces that included both the angle dependence and the dependence on the HF/HCl monomer vibrational quantum number v... 

Figure C3.5.5. Vibronic relaxation time constants for B- and C-state emitting sites of XeF in solid Ar for different vibrational quantum numbers v, from [25]. Vibronic energy relaxation is complicated by electronic crossings caused by energy transfer between sites.

We can only determine and up to now. Later, we shall demonstrate that this equation is just the equations of motion of haimonic nucleai vibrations. The set of eigenstates of Eq. (43) can be written as IXBr). symbolizing that they are the vibrational modes of the nth electronic level, where v = (ui, 112,..., v ) if Q is N dimensional, and vi is the vibrational quantum number of the I th mode. 

The intensities are plotted vs. v, the final vibrational quantum number of the transition. The CSP results (which for this property are almost identical with CI-CSP) are compared with experimental results for h in a low-temperature Ar matrix. The agreement is excellent. Also shown is the comparison with gas-phase, isolated I. The solvent effect on the Raman intensities is clearly very large and qualitative. These show that CSP calculations for short timescales can be extremely useful, although for later times the method breaks down, and CTCSP should be used. 

Experimental. The vibrational spectrum of an ideal harmonic oscillator would consist of one line at frequency v corresponding to A = hv, where A is the distance between levels on the vertical energy axis in Fig. 10-la. In the harmonic oscillator, AE is the same for a transition from one energy level to an adjacent level. A selection rule An = 1, where n is the vibrational quantum number, requires that the transition be to an adjacent level. 

Symmetry coordinates Vibrational quantum numbers FiJ V, k (varies) Fy = d V/dS,dSj... 

In the case of H2O it is easy to see from the form of the normal modes, shown in Figure 4.15, that all the vibrations Vj, V2 and V3 involve a change of dipole moment and are infrared active, that is w=l-0 transitions in each vibration are allowed. The transitions may be labelled Ig, 2q and 3q according to a useful, but not universal, convention for polyatomic molecules in which N, refers to a transition with lower and upper state vibrational quantum numbers v" and v, respectively, in vibration N. 

The general symbolism for indicating a vibronic transition between an upper and lower level with vibrational quantum numbers v and v", respectively, is i/ — v", consistent with the general spectroscopic convention. Thus the electronic transition is labelled 0-0. 

Although it is less often done, I have used an analogous symbolism for pure vibrational transitions for the sake of consistency. Here N refers to a vibrational (infrared or Raman) transition from a lower state with vibrational quantum number v" to an upper state v in the vibration numbered N. 

First of all, the vibrational energy is quantized, and we write the single quantum number v. This quantum number can take values 0, 1, 2,... 

If the interaction V does not depend significantly on vibrational quantum numbers as in Eq. (4.38) then dephasing is negligible and... 

Figure 1. Eigenvalues of the scaled champagne bottle Hamiltonian (Eq. (2)) for p = 0.00625, in the energy, , and angular momentum, k map. The eigenvalues, represented by points, are joined (a) by lines of constant bent vibrational quantum number, vt, and (b) by lines of constant linear quantum number, v = 2vt + k. ...

---