Vibration rotational wave function

Substitute the proposed potential function into the Schrodinger equation and solve it as exactly as possible in order to find the vibrational-rotational wave functions and the corresponding eigenenergies. 

This is integrated over the Q,Q2Q,-space. If the collision pair wave functions never overlap the vibration wave function Xiku(Qi>Q2>Q3 2Zu) of the QTS, there will be zero contribution to the cross section. In this case, the QTS defines the reaction domain. This is quantized by the corresponding vibration-rotation wave function. Therefore, from all possible collisions among the reactants, only those having a non-zero FC factor will contribute to the reaction rate. This is related to the steric factor, P, in elementary chemical kinetics theory. Selection rules for VR-transitions apply. The probability to find the system in one of the product channel states when starting from a QTS is controlled by the FC integral formed by the products of the type... 

Relations exist between the molecular parameters of eq. (2) and (3) which can be obtained from the molecular potential of the electronic state and the resulting vibrational-rotational wave function. Formulas can be found in [61 Seh]. 

For a given electronic state of the molecule the matrix elements refer to vibrational-rotational wave functions Fvr- In the approximation in which vibrational and rotational energies are separable we may write [4]... 

IT. Total Molecular Wave Functdon TIT. Group Theoretical Considerations TV. Permutational Symmetry of Total Wave Function V. Permutational Symmetry of Nuclear Spin Function VT. Permutational Symmetry of Electronic Wave Function VIT. Permutational Symmetry of Rovibronic and Vibronic Wave Functions VIIT. Permutational Symmetry of Rotational Wave Function IX. Permutational Symmetry of Vibrational Wave Function X. Case Studies Lis and Other Systems... 

Planar molecules, permutational symmetry electronic wave function, 681-682 rotational wave function, 685-687 vibrational wave function, 687-692... 

We must now combine the nuclear wave functions with the rest of the molecular wave function to generate a total wave function which is antisymmetric with respect to exchange of Fermions. For Bosons the total wave function must be symmetric. To do so we write r]r = i rans r]rviB rot r Nuc-spiN and recognize that both the vibrational and translational wave functions are symmetric. Rotational wave functions... 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

We have seen that the molecular electronic and vibrational wave functions el and vib each transform according to the irreducible representations of the molecular point group. We now consider the rotational wave function ptot. 

To a first approximation, and usually a rather good one, the complete wave function F for a molecule can be written as a product of an electronic wave function y/e, a vibrational wave function y/vn and a rotational wave function... 

Reid, B.P., Janda, K.C., and Halberstadt, N. (1988). Vibrational and rotational wave functions for the triatomic van der Waals molecules HeCl2, NeCl2, and ArCl2, J. Phys. Chem. 92, 587-593. 

The nuclear function %a(R) is usually expanded in terms of a wave function describing the vibrational motion of the nuclei, and a rotational wave function [36, 37]. Analysis of the vibrational part of the wave function usually assumes that the vibrational motion is harmonic, such that a normal mode analysis can be applied [36, 38]. The breakdown of this approximation leads to vibrational coupling, commonly termed intramolecular vibrational energy redistribution, IVR. The rotational basis is usually taken as the rigid rotor basis [36, 38 -0]. This separation between vibrational and rotational motions neglects centrifugal and Coriolis coupling of rotation and vibration [36, 38—401. Next, we will write the wave packet prepared by the pump laser in terms of the zeroth-order BO basis as... 

In chapter 2 we show how a separation of the vibrational and rotational wave functions can be achieved by using the product functions... 

In the same way that we dealt with the transformation of the vibrational, case (a) spin and rotational wave functions in section 6.9.3, it is easy to show that... 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

For transition states, the total parity is the product of the parities of the vibrational and rotational wave functions, and it depends on both J and K. The parity of the rotational wave function is (—l)7. For K = 0, the vibrational wave function has even parity (+1). 

Wc). The initial wave function ) =, ) r) t) where the is the quantum number associated with the harmonic oscillator wave function for the strong molecular bond ( =i in fig. 10.7), is the rotational quantum number associated with the rotational wave function (not included in the one-dimensional picture of fig. 10.7), and is the quantum number associated with the vibrational wave function for the van der Waals bond ( Pv in fig. 10.7). A Morse potential is assumed for this latter interaction potential. For the final state, the wave function is f) = 4f) 7) t)> where f) is the quantum number associated with the final state of the strong bond ( I>v"=o fig- 10-7), the rotational quantum number, J, represents the rotational wave function (now a free... 

The wave function for a given electro-nuclear state is the product of the universal electronic function, the relative vibrational motion of the nuclei, the rotational wave function and the center-of-mass free particle wavepacket. 

Now, we are amazed to see that Eq. (6.23) is identical (ef., p. 199) to that which appeared as a result of the transformation of the Schrddinger equation for a rigid rotator. Y denoting the corresponding wave function. As we know from p. 200, this equation has a solution only if A = —J J + 1), where / — 0,1, 2,... Since Y stands for the rigid rotator wave function, we now concentrate exclusively on the function Xk which describes vibrations (changes in the length of R). 

For a homonuclear diatomic molecule composed of even(odd) mass-ntmber nuclei, the total wave function, which we assume to be a product of electronic, vibrational, rotational, and nuclear-spin functions, must be symmetric (antisymmetric). If the electronic wave function is symmetric, and if the nuclear spin is zero, as in the ground state of 0a, only even values of J, the rotational quantum number, are allowed. If the nuclear spin is not zero, both even and odd values of J (l.e., symmetric and antisymmetric rotational wave functions) are allowed, but with different statistical weights. These may be determined from the nuclear-spin part of the wave function. 

The transition electric dipole moment in eqn [57] can be developed by invoking the Born-Oppenheimer approximation to express the total molecular wave function as a product of electronic and vibrational parts. (Rotational wave functions do not have to be included here since eqn [57] refers to an isotropic system. That is, the equation is a result of a rotational average which is equivalent to a summation over all the rotational states involved in the transition.) A general molecular state can now be expressed as the product of vibrational and electronic parts. Assuming that the initial and final electronic states are the ground state jcg). 

In the previous chapter it was indicated that when dassical mechanics is employed the rotation and vibration of a molecule can be treated separately. The proof of this will be given in Chap. 11, where it will likewise be show n that rotation and vibration are also approximately separable when Avave mechanics is used. To this degree of approximation the total wave function for the motions of the atoms can be written as a product of a vibrational wave function f v and a rotational wave function that is,... 

In propagation, the wave packet is expanded in terms of a body-fixed translational-vibrational-rotational basis functions as follows ... 

If the nucleus has an odd mass number, the overall wave function is anti-symmetrical with regard to the nuclei it is the product of all the translational, rotational, vibrational, electronic and nuclear wave functions. With all the translational, vibrational and electronic wave functions being symmetrical, we only have to consider the rotational and nuclear functions, one of which must be symmetrical and the other anti-symmetrical or vice versa. The g(g-l)/2 wave functions with anti-symmetrical nuclear spin must have corresponding symmetrical rotational wave functions, i.e. with even values of j the g(g+l)/2 wave functions with symmetrical nuclear spin must have corresponding rotational wave functions, i.e. with odd values of j. The combined nuclear-rotational partition function will therefore be ... 

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