Quantum number function

Fig. 1. An example of the quantum number function n2(q1,n1), here for a collision of He and H2 at a total energy E = 0hoj and with = 1. The ordinate is the final value of the vibrational quantum number as a function of the initial phase qx of the oscillator, along a classical trajectory with the initial conditions in (28). The dashed line at n2 = 2 indicates the graphical solution for the two roots of the equation n2(ql, 1) = 2.

Fig. 4 shows another example of the collinear reactive quantum number function n2(9i) from the paper of Wu and Levine,32 this one also for n, = 0 and for a total energy of 13 7 kcal. Here the function is considerably more complicated than those of Figs. 1 or 3, there being four trajectories that contribute to the 0 - 0 reaction. At first glance one might expect no semiclassical treatment to be possible Connor,38 however, has developed more general uniform semiclassical formulae which take account of four terms, and it would clearly be desirable to see if these expressions give more accurate results for this application. 

In highly quantum-like situations such as these, therefore, it is necessary to use the appropriate uniform semiclassical expressions to obtain quantitative results for transitions between individual quantum states. These will, too, undoubtedly be cases for which the quantum number function is too highly structured for any semiclassical treatment to be quantitatively useful. 

To see how Feshbach resonances appear in classical S-matrix theory, consider the collinear H + Cl2 collision as studied by Rankin and Miller.47 Fig. 8 shows the quantum number function n2(q,) for one region of ql the function is smooth, these trajectories being direct . The remaining interval of ql leads to complex trajectories, those which spend a number of additional vibrational periods in the interaction region for this region of qY values the final vibrational quantum number changes dramatically with small changes in q,. The S-matrix for the particular transition indicated in Fig. 8 thus has the form... 

If the excited state potential surface is repulsive, so that the dissociative trajectories are direct , the dipole matrix element will be a smooth function of 2, i.e. the absorption spectrum is continuous . If, on the other hand, V2 t) is attractive so that the complex ABC lives a long time before dissociating into A and BC, the quantum number function nfN Q ) will be highly structured (cf, Fig, 8) and thus a large number of terms will contribute to(125). Analogous to the semiclassical discussion of resonances in Section III.C, these many terms will interfere destructively at all but certain specific values of E2 at which the interference is constructive and the matrix element extremely large. In such cases, therefore, there will be a line spectrum , with the width of the absorption lines related to the time the excited state lives before dissociating,... 

Here h(x) is the Heaviside step function with h(x > 0) = 1 and h(x > 0) = 0 (not to be confused with Planck s constant). The limit a(J.. . ) indicates that the sunnnation is restricted to channel potentials witir a given set of good quantum numbers (J.. . ). 

In the following, it shall always be assumed that the zeroth-order solution is known, that is, we have a complete set of eigenvalues and wave functions, labeled by the electronic quantum number n, which satisfy... 

Apparently, the most natural choice for the electronic basis functions consist of the adiabatic functions / and tli defined in the molecule-bound frame. By making use of the assumption that A" is a good quantum number, we can write the complete vibronic basis in the form... 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

Vo + V2 and = Vo — 2 (actually, effective operators acting onto functions of p and < )), conesponding to the zeroth-order vibronic functions of the form cos(0 —4>) and sin(0 —(()), respectively. PL-H computed the vibronic spectrum of NH2 by carrying out some additional transformations (they found it to be convenient to take the unperturbed situation to be one in which the bending potential coincided with that of the upper electi onic state, which was supposed to be linear) and simplifications (the potential curve for the lower adiabatic electi onic state was assumed to be of quartic order in p, the vibronic wave functions for the upper electronic state were assumed to be represented by sums and differences of pairs of the basis functions with the same quantum number u and / = A) to keep the problem tiactable by means of simple perturbation... 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, E, P, Uj, It, Uc, and Ic-The terms of H involving the parameters aj, ac, and bo aie diagonal with respect to both the Ij and Ic quantum numbers, while the f>2 term connects with one another the basis functions with I j = Ij 2, 4- 2. The c terms... 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs of the 3D rotational group SO(3), where a is the rotational vector, written in the operator form as [36]... 

Eqs. (D.5)-(D.7). However, when perturbations occur due to anharmonicity, the wave functions in Eqs. (D.11)-(D.13) will provide the conect zeroth-order ones. The quantum numbers and v h are therefore not physically significant, while V2 arid or V2 and I2 = m, are. It should also be pointed out that the degeneracy in the vibrational levels will be split due to anharmonicity [28]. 

The next step in iin proving a basis set could be to go to triple zeta, quadruple zeta, etc. Ifone goes in this direction rather than adding functions of higher angular quantum number, the basis set would not be well balanced. With a large number of s and p functions only, one finds, for example, that the equilibrium geometry of am monia actually becomes planar. The next step beyond double z.ela n sit ally in voices addin g polarization fn n ciion s, i.e.. addin g d-... 

The hydrogen atom is a three-dimensional problem in which the attractive force of the nucleus has spherical symmetr7. Therefore, it is advantageous to set up and solve the problem in spherical polar coordinates r, 0, and three parts, one a function of r only, one a function of 0 only, and one a function of [Pg.171]

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