Relative wavefunction

V n( ) is the relative wavefunction that describes the internal structure of the exciton. Owing to particle-hole symmetry it satisfies... 

As shown in Appendix D, the relative wavefunction, V n(r), satisfies the following Schrodinger difference equation. 

Fig. 6.2. The effective-particle model of excitons on a linear chain. The total exciton wavefunction, = ipn r) j R), where ipn r) is the relative wavefunction...

Notice that two quantum numbers specify the exciton eigenstates, eqn (6.13) or eqn (6.16) the principle quantum number, n, and the (pseudo) momentum quantum number, K (or fUj). For every n there are a family of excitons with different centre-of-mass momenta, and hence different centre-of-mass kinetic energy. Odd and even values of n correspond to the relative wavefunction, tl>n r), being even or odd under a reversal of the relative coordinate, respectively. We refer to even and odd parity excitons as excitons whose relative wavefunction is even or odd imder a reversal of the relative coordinate. This does not mean that the overall parity of the eigenstate (eqn (6.12)), determined by both the centre-of-mass and relative wavefiictions, is even or odd. The number of nodes in the exciton wavefunction, V rt( ), is n— 1. Figure 6.2 illustrates the wavefunctions and energies of excitons in the effective-particle model. 

Substituting eqn (D.14) into eqn (D.12), we obtain the following Schrodinger difference equation for the relative wavefunction,... 

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

One fomis a reference wavefunction T (this can be of the SCF, MPn, CC, etc variety) tlie energy differences are computed relative to the energy of this fiinction. 

The function/( C) may have a very simple form, as is the case for the calculation of the molecular weight from the relative atomic masses. In most cases, however,/( Cj will be very complicated when it comes to describe the structure by quantum mechanical means and the property may be derived directly from the wavefunction for example, the dipole moment may be obtained by applying the dipole operator. 

Assume that an experiment has been carried out on an atom to measure its total angular momentum L. According to quantum mechanics, only values equal to L(L+1) h will be observed. Further assume, for the particular experimental sample subjected to observation, that values of equal to 2 and 04f were detected in relative amounts of 64 % and 36%, respectively. This means that the atom s original wavefunction / could be represented as ... 

I la2g la2y 2a2g 2a2 and all single and double excitations relative to this (dominant) CSF, which is a very common type of Cl procedure to follow, the Bc2 wavefunction would not have contained the particular CSFs ls2 2p2 ls2 2p2 b because these CSFs are four-fold excited relative to the la2g la2y 2a2g 2a2 reference CSF. 

The first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

The dispersion coefficients for the mixed-symmetry component 7 5 which describes the deviation from Kleinman symmetry are for methane more than an order of magnitude smaller than coefficients of the same order in the frequencies for 7. Their varations with basis sets and wavefunction models are, however, of comparable absolute size and give rise to very large relative changes for the mixed-symmetry dispersion coefficients. 

This view somehow seems dubious in the case of heavier elements like 6 row metals. The high energy separation, as well as the very different spatial distribution of the 6s/6p wavefunctions, which are found for these elements because of the strong influence of relativity, stand against an efficient s-p hybridization. The first excited state of Th (in the gas phase), s p lies 7.4 eV above the... 

Since the HPHF wavefunction for singlet states does not contain any triplet contamination, this model was seen to produce relatively good results for singlet ground states, very close to those of the fully projected one [1-10],... 

The selection rules illustrated above are general, as they depend only on the symmetry properties of the functions involved. However, more limiting, selection rules depend on the form of the wavefunctions involved. A relatively simple example of the development of specific selection rules is provided by the harmonic oscillator. The solution of this problem in quantum mechanics,... 

In view of the fact that recent parameterisations make use of reference data from high-level calculations, the corresponding error functions used to develop these methods can in principle involve any given property that can be calculated. Thus, in addition to structural information, the error function can involve atomic charges and spin densities, the value for the wavefunction, ionisation potentials and the relative energies of different structures within the reference database [26, 32], Detailed information concerning the actual wavefunction can be extremely useful for... 

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