Primed quantum numbers

The primed quantum numbers represent the final state. The coordinate representation of the length operator Lu for the kth electron is... 

As the species looked at in chemical reactions are mostly molecules, the two electronic levels depicted in Figure 7.2 spht into sub-levels, according to the molecular vibrational and rotational energy quanta. The vibrational levels are customarily numbered with the quantum number v,(i = 0, 1, 2,...). The notation for the rotational levels is more complex and depends as well on the size of the molecule, but typically one associates the rotation with the quantum number Ji i = 0, 1, 2,...). In order to distinguish between states, double primes are used to mark the (lower) ground-state levels and single-primed quantum numbers mark the excited (upper) state levels. The main processes observed in molecule-laser photon interactions are shown in Figure 7.3. 

With the help of functions (25) and the introduction of the new quantum number M we represent the perturbation energy matrix elements P with respect to the coupling term —Qar we retain the notation (16). The elements read, when we use primed quantum numbers for the final state and unprimed quantum numbers for the initial state,... 

Fermi resonance occurs in C02. We note from (6.100) that 2vl e (= 1346 cm-1) is very close to e (= 1354 cm-1). Hence harmonic-oscillator levels of C02 for which 2v + v 2 = 2vx +1>2 (where the primed and unprimed quantum numbers refer to different vibrational levels) are quite close together and we have Fermi resonance. For example, (6.99) and (6.100) predict the levels (10°0), (02°0), and (0220) to lie 1335, 1339, and 1335 cm-1, respectively, above the ground vibrational level the observed spectrum (Table 6.2) shows these levels actually lie 1388, 1285, and 1335 cm-1, respectively, above the ground level (Problems 6.20 and 6.21). Clearly, the (10°0) and (02°0) levels have interacted with each other, thus shifting their energies. The (0220) level is unaffected because the matrix element H j is zero if states i and j have different values of / (Problem 6.22) Fermi resonance occurs only between states of the same symmetry. Fermi resonance between two levels increases the energy of the upper level and decreases the energy of the lower level the levels repel each other. 

Using primes to indicate quantum numbers of state j, we want to evaluate... 

In all cases a two-color multiphoton process is used to excite the molecules to the high-n Rydberg states. In the remainder of this text we use a primed notation to refer to the transitions from the ground state to the intermediate state (e.g., O branch, O branch, etc.), whereas an unprimed notation is used for transitions from the intermediate state J to the Rydberg states with core rotation quantum number for example, 5(2) implies the transition J = 2- N+ = 4, whereas S (2) implies J" = 2 — J = 4. 

We now calculate the matrix elements of each of the four main terms in H cir in turn for simplicity we will omit the primes when the matrix elements are diagonal in those quantum numbers. 

The case (b) basis functions are of the form rj, A N, S, J, I, F, MF) in the hyperfine-coupled basis set and the matrix elements of the effective Hamiltonian are given below. We include A in the basis set because although A = 0 for a state, there are terms in the effective Hamiltonian which can mix the ground state with excited states having A / 0. As before, the absence of primes on the right-hand side denotes that the matrix elements are diagonal in the relevant quantum numbers. 

Generally, constants and quantum numbers with single and double prime refer to the upper and the lower state, respectively. It is m = J -b 1 for the P-branch and m = -J for the P-branch. Wavenumber combinations of pairs of lines with common upper or lower rotation-vibration state can give the corresponding rotational constants of the ground or excited vibrational state ... 

The 77 coupling representation is generally useful in the reduction of the Lippmann—Schwinger equations since it applies to situations where spin—orbit coupling is not negligible. The quantum numbers used in the representation are defined in table 7.1. Primed and double-primed quantities are used to distinguish different angular-momentum states. 

Just as several STOs are needed to give an accurate representation of Hartree-Fock AOs (Section 11.1), one needs more than one STO of a given n and / in the linear combination of STOs that is to accurately represent the Hartree-Fock MO. The primed and double-primed AOs in the extended-basis-set function are STOs with different orbital exponents. The 3dquantum number m = 0, that is, the 3do and 4/o AOs. The total energies found are -197.877 and -198.768 hartrees for the minimal and extended calculations, respectively. The experimental energy of F2 at Rg is -199.670 hartrees, so the error for the minimal calculation is twice that of the extended calculation. The extended-basis-set calculation is believed to give a wave function quite close to the true Hartree-Fock wave function. Therefore, the correlation energy in F2 is about -0.90 hartrees = -24.5 eV. 

Reaction rate constants cannot be used to describe such detailed processes. Instead the differential reaction cross section, areact ( i, 2, 3, n l, 2 0 ployed, where n, are various quantum numbers and the primed quantities refer to reaction products. Such cross sections represent the effective collision area for reagents with given i, 2, , to give specific products. Rate constants represent the effective average of the product of the cross section with the approach velocity taken over the calculated distribution of reagent quantum states. 

The sequence Av = 0 is allowed when two different electronic states, E and E2, are involved, i.e. (Ei, v" = n) ( 2, v = n), where the double prime and single prime indicate the lower and upper state quantum numbers respectively. 

Here v denotes the vibrational quantum number while the prime stands for upper electronic state and the double prime for lower electronic state in optical spectroscopy (in infrared spectroscopy the electronic state is the same). 

In the following tables and corresponding diagrams, U and L stand for upper and lower electronic state, respectively U = A( E ), L = X( E ) the vibrational and rotational quantum numbers v and J are labeled with a prime or a double prime depending on their referring to the upper or lower state of the transition. We shall first display and comment some tables and diagrams for CuH, CuD, and CuD, and then consider the tables and diagrams for the and iso-... 

Here unprimed and primed quantities refer to the initial and final states, respectively, and refers to the distribution of initial rotational quantum numbers J. We separate the total energy into electronic-vibrational plus rotational terms, and obtain the following relations... 

Centrifugal distortion constants are not of prime importance in the determination of molecular stmcture, but it may be necessary to determine them in order to interpret an observed speetmm. To determine a quartic distortion constant it is generally necessary to have very precise data, preferably over a good range of the quantum numbers (7, and also K for a symmetric top). Otherwise, there is no partieular difficulty in obtaining reliable values for Dj and Djx for a symmetrie top, or D for a linear molecule. ) has very little influence on the form of the speetmm for a symmetric top, and is correspondingly difficult to determine. 

Here the primes refer to post-collision quantum numbers, the quantum numbers without primes are pre-collision, and f and i refer to final and initial "spectroscopic states (this is the same convention as is ordinarily used e,g, in line broadening ). The indices and KjijKj indicate the ranks of spherical tensors which produce coupling of orbital and rotor angular momenta, respectively. The index K is the total tensor index. The symbols (m) and... 

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