Complete wave function

Although we will not write the complete wave functions as we did for the case of an octahedral complex, the molecular orbitals give rise to the energy level diagram shown in Figure 17.20. 

To complete our description of the HF method, we have to define how the solutions of the single-electron equations above are expressed and how these solutions are combined to give the /V-clcctron wave function. The HF approach assumes that the complete wave function can be approximated using a single Slater determinant. This means that the N lowest energy spin orbitals of the... 

B) Delocalization It is customary to represent the delocalization effect as due to the existence, in the complete wave function, of terms corresponding to some or all of the structures (i) (v). 

In this case wiy (i = 1,2,3) are antisymmetrical wave functions formed from atomic one-electron orbitals. The complete wave function was written as a linear combination of the ipi... 

The complete wave function of a molecule is called the rovibronic wave function. In the simplest approximation, the rovibronic function is a product of rotational, vibrational, and electronic functions. For certain applications, the rotational motion is first neglected, and the vibrational and electronic motions are treated together. The rotational motion is then taken into account. The wave function for electronic and vibrational motion is called the vibronic wave function. Just as we separately classified the electronic and vibrational wave functions according to their symmetries, we can do the same for the vibronic functions. In the simplest approximation, the vibronic wave function is a product of electronic and vibrational wave functions, and we can thus readily determine its symmetry. For example, if the electronic state is an e2 state and the vibrational state is a state, then the vibronic wave function is... 

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... 

It is then assumed that none of these factors of the complete wave function are interdependent, so that instead of having to solve one large wave equation... 

In order to plot the complete wave functions, one would in general require a four-dimensional graph with coordinates for each of the three spatial dimensions (.x. y,or r, 6, (J>) and a fourth value, the wave function. 

Nature requires—as we have seen—that the complete wave function (space and spin) be antisymmetric. Consequently, for two electrons in one orbital tp, ... 

The fundamental approximation used for describing the electron and nuclear motion in molecules and in condensed media is the well-known adiabatic approximation. Let us recall its essence. It is based upon the large difference in the masses of electrons and nuclei. Due to this difference the electron motion is fast in comparison with the nuclear motion, and thus electrons have time to adjust themselves to the nuclear motion and at every moment they can be in a state very close to the one they would be in if nuclei were immobile. Within this picture, as the first step in the construction of the complete wave function of the system, it proves useful to find wave functions describing electron motion with fixed positions of the nuclei, i.e. to resolve the Schrodinger equation... 

If one adopts the correct point of view that the complete wave function of any state of a diatomic molecule has contributions from all other states of that molecule, one can understand that all degrees of perturbation and hence probabilities of crossover may be met in practice. If the perturbation by the repulsive or dissociating state is very small, the mean life of the excited molecule before dissociation may be sufficiently long to permit the absorption spectrum to be truly discrete. Dissociation may nevertheless occur before the mean radiative lifetime has been reached so that fluorescence will not be observed. Predissociation spectra may therefore show all gradations from continua through those with remnants of vibrational transitions to discrete spectra difficult to distinguish from those with no predissociation. In a certain sense photochemical data may contribute markedly to the interpretation of spectra. 

The complete wave function of a molecule will often change in the presence of foreign fields. Foreign fields, either electrostatic or electromagnetic, may under certain conditions change wave functions sufficiently to permit perturbations which would otherwise be unimportant. Perturbations may also be induced by collisions with other molecules, particularly with molecules which are themselves paramagnetic. These effects give rise to what is often called collision-induced predissociation. 

Triplet states are, however, formed to larger extents and play more important roles than would have been thought to be true twenty or thirty years ago. The classical work of Lewis and Kasha129, whichshowed certain emissions to come from triplet states, opened the way for the rationalization of many phenomena which would otherwise prove to be quite incomprehensible. Perhaps, as Matsen etal.130 have pointed out, an undue emphasis has been placed on electron spin and on the multiplicity of states. The symmetry of the entire wave function is really the important point, and the contributions to it of all states of the molecule must be considered. Viewed in this light the triplet component , to use rather crude language, will depend on the vibrational quantum numbers in the excited state. If other isomers can exist, their contributions to the complete wave function must also be considered. 

If contributions from all states, including those of different multiplicity, must be considered in developing the complete wave function, it is not possible in one sense to say that collisions or foreign fields induce crossovers from singlet to triplet states. If, however, a molecule is to be stabilized in a triplet state enough energy must somehow be removed so that the wave function can be mainly triplet. Thus, to say that collisions play no role in the crossover process would be fallacious. 

Since the complete wave function will have variable contributions from other energy states and even from isomers and will depend on the vibrational energy level, it would be very desirable to use light of such monochromaticity that one and only one vibrational level could be formed at a time. The importance of doing this may vary greatly from one compound to another but we will illustrate the nature of the problem with the following simplified mechanism... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

The Bohm interpretation assumes that the interaction between measuring apparatus and observed system breaks the wave function into a series of classically separated wave packets, corresponding to the possible outcomes of a measurement. The particle enters one of the packets and remains in that packet. All other packets can now be neglected and the complete wave function is replaced by a simplified one corresponding to the actual result of the measurement. The fluctuations in the fluid are never large enough to move the particle from one packet to another. 

In the above treatment of the hydrogen molecule ion and the hydrogen molecule, the effect of electronic spin has been excluded, but the complete wave function of an electron must include not only the orbital motion, with which we have been concerned so far, but also a contribution for the spin. With single atoms it was possible to introduce a fourth quantum number s in addition to the three quantum numbers , I and m in order to account for the spin of the electron, and for polyelectronic molecules it is possible to proceed in an analogous manner. The complete wave function of an electron is considered to be the product of the orbital wave function, i,e. the wave function that we have been considering so far, and a wave function representing the orientation of the spin axis of the electron. 

These conditions mean that the wave function a refers to the state of the electron in which the spin quantum number s, has the value + /2> consequence the probability of it having the value — V2 zero. Similarly the function j8 describes those states corresponding to a value for of — and consequently the probability of it having the value -h V2 zero. The complete wave function of the electron is then represented by... 

For two electrons, the complete wave function of the first electron will be (l)a(l) which denotes that the first electron is attached to the nucleus a and the solution is given by... 

Table XV. Complete Wave Functions for the Two Electron Problem...

The complete wave function is different, however, and it will appear that only an antisymmetric complete function, which reverses its sign on transposition of the electrons, is acceptable on the basis of Pauli s principle, which states in wave mechanical terms Only that complete wave function may be selected which does not permit the two electrons to exist in the same state Let us consider a symmetrical complete wave function and assume that the two electrons have the same values of the quantum numbers n, /, m and s when the electrons are interchanged the wave function will be identical with the original. Thus by the use of a symmetric function it is possible to describe an electronic state which is not permitted by the Pauli principle. It may therefore be concluded that the complete wave function cannot be symmetric. 

Now let us consider an antisymmetric complete wave function which is also prohibited by Pauli s principle. Again it is assumed that there are two electrons having the same values of , /, m and s. Transposition of the electrons should not alter the function since both electrons are in identical states, but since the function is antisymmetric, the sign must be reversed these two statements can be compatible only when the function is zero. Thus when the complete wave function is antisymmetric it becomes zero when in disagreement with Pauli s principle. Thus not all the linear combinations of the functions in Table XV arc possible, but only... 

Thus on the basis of the above conclusions we see that for the hydrogen molecule there are four possible complete wave functions ... 

The complete wave function of the hydrogen molecule must describe the electronic orbital motion, the electronic spin orientation, the vibration of the nuclei, the rotation of the nuclei and the nuclear spin orientation. As a first approximation the various forms of motion must be considered as being independent of each other and the complete wave function may thus be represented as the product of five separate functions ... 

The complete wave function, , of the molecule is thus antisymmetric as required by Pauli s principle. 

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