Vibrational transitions operator

Figure 9.24 shows part of the laser Stark spectrum of the bent triatomic molecule FNO obtained with a CO infrared laser operating at 1837.430 cm All the transitions shown are Stark components of the rotational line of the Ig vibrational transition, where Vj is the N-F stretching vibration. The rotational symbolism is that for a symmetric rotor (to which FNO approximates) for which q implies that AA = 0, P implies that A/ = — 1 and the numbers indicate that K" = 7 and J" = 8 (see Section 6.2.4.2). In an electric field each J level is split into (J + 1) components (see Section 5.2.3), each specified by its value of Mj. The selection mle when the radiation is polarized perpendicular to the field (as here) is AMj = 1. Eight of the resulting Stark components are shown. 

As described above, the ground state vibrational wavefunction is totally symmetric for most common molecules. Therefore, the product, -(1)0 must at least contain a totally symmetric component. The direct product of two irreducible representations contains the totally symmetric representation only if the two irreducible representations are identical. Therefore, transitions can occur from a symmetrical initial state only to those states that have the same symmetry properties as the transition operator, 0. 

The photolysis of Cr(CO)6 also provides evidence for the formation of both CO (69) and Cr(CO) species (91,92) in vibrationally excited states. Since CO lasers operate on vibrational transitions of CO, they are particularly sensitive method for detecting vibrationally excited CO. It is still not clear in detail how these vibrationally excited molecules are formed during uv photolysis. For Cr(CO)6 (69,92), more CO appeared to be formed in the ground state than in the first vibrational excited state, and excited CO continued to be formed after the end of the uv laser pulse. Similarly, Fe(CO) and Cr(CO) fragments were initially generated with IR absorptions that were shifted to long wavelength (75,91). This shift was apparently due to rotationally-vibrationally excited molecules which relaxed at a rate dependent on the pressure of added buffer gas. 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

In the case of in-plane stretching vibrations, the transition operator f, has... 

The general form of the energy of the harmonic oscillator indicates that the vibrational energy levels are equally spaced. Due to the vector character of the dipole transition operator, the transition between vibronic energy levels is allowed only if the following selection rule is satisfied ... 

In the case of direct vibrational excitation, the vibrational transition probability is given by p, where are the intermediate and ground vibrational states, respectively, and is the vibrational transition moment. The electronic transition probability out of the intermediate state is < n < e ng e > n>, where are the excited and ground electronic states, respectively, and is the electronic dipole moment operator and vibrational state in the upper electronic state. Applying the Born-Oppenheimer approximation, where the nuclear electronic motion are separated, S can be presented as... 

IR wavelengths are traditionally divided into three regions. The portion that adjoins the visible region is the near infrared = 0.8-2.5 jam the mid infrared extends from 2.5 to 50 /xm the far infrared extends from 50 to 1000 [im. Most commercial IR spectrometers operate in the mid IR. Pure rotational transitions of light molecules, and low-frequency vibrational transitions of heavy molecules occur in the far IR. 

Extensions from the preceding ideal, isolated systems to others that are coupled to an environment are quite demanding and nontrivial [32] because the IR femtosecond/picosecond laser pulse has to achieve the selective vibrational transition (9) in competition against nonselective processes such as dissipation. For simulations, we employ the equation of motion for the reduced-density operator... 

Here, instead of the usual delta function d(Eg, — Eg — ftv), which supplies only discontinuous spectral lines, a Lorentz function with a line width 29(normalized to one) is introduced for all vibrational levels labeled by n and m of electronic states j and k which contribute to the transition rate of Eq. (9) [58]. M is the transition operator, in general, pertinent to an electric dipole transition. The nuclear wavefunctions are approximated by products of harmonic oscillator functions... 

In accordance with Placzek s theory (1934) we can write the real part of the complex transition polarizability as the dynamic vibrational polarizability operator (which is a function of a static configuration Q of nuclei) acting on the vibrational state functions and... 

So far the analysis has lead to the concept of a carrier space which links the degeneracy to a doubly transitive orbit of cosets of maximal subgroups. Interactions in this space are expressed as transition operators between the cosets. The final part of the treatment should bring in the vibrational degrees of freedom which are responsible for the Jahn-Teller activity. 

In the previous section we noticed that all the scattering information can be obtained from the transition operator T which satisfies Lippmann-Schwinger equation (7). In order to obtain cross sections for vibrationally inelastic processes we project the equation (7) onto electronic space spanned by plane waves and a two-dimensional space for nuclear dynamics defined by a vibrational ground state and its first excited state /j. The resulting equation follows ... 

The intensity of a spectral transition is calculated from matrix elements involving the initial and final state wavefunctions and the transition operator. This leads to selection rules, as for multielectron atoms, in terms of restrictions upon the values of S, L and / in the transition (Table 1). Considering the selection rules for vibrational spectra, the k=0 phonons of the initial and terminal states transform as T, and Tf, respectively, of the point group... 

To proceed further, three major approximations to the theory are made [44] First, that the transition operator can be written as a pairwise summation of elements where the index I denotes surface cells and k counts units of the basis within each cell second, that the element is independent of the vibrational displacement and, third, that the vibrations can all be treated within the harmonic approximation. These assumptions yield a form for w(kf, k ) which is equivalent to the use of the Bom approximation with a pairwise potential between the probe and the atoms of the surface, as above. However, implicit in these three approximations, and therefore also contained within the Bom approximation, is the physical constraint that the lattice vibrations do not distort the cell, which is probably tme only for long-wavelength and low-energy phonons. 

Throughout this paper, we have seen that algebraic techniques often provide extremely simple numerical results with small computational effort. This is particularly true in the preliminary phases of one-dimensional calculations, where almost trivial relations can be found for the initial guesses for the algebraic parameters, as shown in Sections II.C.l and III.C.2. However, it is also true that as soon as real calculations of more complex vibrational spectra are requested, the problem of adapting the various algebraic Hamiltonian and transition operators to suitable computer routines must be resolved. The construction of a computer interface between theoretical models and numerical results is absolutely necessary. Nonetheless, it is rather atypical to discuss these problems explicitly in a theoretical paper such as this one. However, the novelty of these methods itself justifies further explanation and comment on the computational procedures required in practical applications. In this section we present only a brief outline of the development of algebraic software in the last few years, as well as the most peculiar situations one expects to encounter. 

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