Wave function nuclear spin

Symmetry properties of the nuclear wavefunction are different in the diabatic and adiabatic representations. The pair of adiabatic electronic states (see (1)) belong to the Al and A2 irreducible representations of the double group of S3. The diabatic states obtained from the adiabatic ones by applying the U matrix form a basis for the two dimensional irreducible representation E of S3. For quartet nuclear spin states, the electronuclear wavefunction, nuclear spin part excluded, must belong to the A2 irreducible representation. This requires the nuclear wavefunction (without nuclear spin) to be of the same E symmetry as the electronic one, because of the identity E x E = Ai + A2 + E. For doublet spin states, the E electronuclear wave-function (nuclear spin excluded) is obtained with an Ai or A2 nuclear wavefunction, combined with the E electronic ones. 

In the transition state region, the spin-pairing change mnst take place. At this nuclear configuration, the electronic wave function may be written as... 

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... 

As pointed out in the previous paragraph, the total wave function of a molecule consists of an electronic and a nuclear parts. The electrons have a different intrinsic nature from nuclei, and hence can be treated separately when one considers the issue of permutational symmetry. First, let us consider the case of electrons. These are fermions with spin and hence the subsystem of electrons obeys the Fermi-Dirac statistics the total electronic wave function... 

Let us discuss further the pemrutational symmetry properties of the nuclei subsystem. Since the elechonic spatial wave function t / (r,s Ro) depends parameti ically on the nuclear coordinates, and the electronic spacial and spin coordinates are defined in the BF, it follows that one must take into account the effects of the nuclei under the permutations of the identical nuclei. Of course. 

Since the total wave function must have the correct symmetry under the permutation of identical nuclei, we can determine the symmetiy of the rovi-bronic wave function from consideration of the corresponding symmetry of the nuclear spin function. We begin by looking at the case of a fermionic system for which the total wave function must be antisynmiebic under permutation of any two identical particles. If the nuclear spin function is symmetric then the rovibronic wave function must be antisymmetric conversely, if the nuclear spin function is antisymmebic, the rovibronic wave function must be symmetric under permutation of any two fermions. Similar considerations apply to bosonic systems The rovibronic wave function must be symmetric when the nuclear spin function is symmetric, and the rovibronic wave function must be antisymmetiic when the nuclear spin function is antisymmetric. This warrants... 

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. 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

In this section, we extend the above discussion to the isotopomers of X3 systems, where X stands for an alkali metal atom. For the lowest two electronic states, the permutational properties of the electronic wave functions are similar to those of Lij. Their potential energy surfaces show that the baniers for pseudorotation are very low [80], and we must regard the concerned particles as identical. The Na atom has a nuclear spin " K, and K have nuclear... 

As discussed in preceding sections, FI and have nuclear spin 5, which may have drastic consequences on the vibrational spectra of the corresponding trimeric species. In fact, the nuclear spin functions can only have A, (quartet state) and E (doublet) symmetries. Since the total wave function must be antisymmetric, Ai rovibronic states are therefore not allowed. Thus, for 7 = 0, only resonance states of A2 and E symmetries exist, with calculated states of Ai symmetry being purely mathematical states. Similarly, only -symmetric pseudobound states are allowed for 7 = 0. Indeed, even when vibronic coupling is taken into account, only A and E vibronic states have physical significance. Table XVII-XIX summarize the symmetry properties of the wave functions for H3 and its isotopomers. 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

If any atoms have nuclear spin this part of the total wave function can be factorized and the energy treated additively. ft is for these reasons that we can treat electronic, vibrational, rotational and NMR spectroscopy separately. 

When nuclear spin is included the total wave function p for a molecule is modified from that of Equation (1.58) to... 

Equation (1.48) shows that, for I =, space quantization of nuclear spin angular momentum results in the quantum number Mj taking the values 5 or — 5. The nuclear spin wave function J/ is usually written as a or /i, corresponding to Mj equal to 5 or —5,... 

In general, for a homonuclear diatomic molecule there are (21+ )(/+1) symmetric and (21+ 1)/antisymmetric nuclear spin wave functions therefore... 

If / = 1 for each nucleus, as in H2 and N2, the total wave function must be symmetric to nuclear exchange. There are nine nuclear spin wave functions of which six are symmetric and three antisymmetric to exchange. Figure 5. f 8 illustrates the fact that ortho- ll2 (or N2)... 

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