Wave function of state

The square of the molecular wave function 2, defines the molecular charge distribution. The wave function of state i, ( ), can be calculated in the presence or absence of an electric field. Details of the zero field occupied and unoccupied states determine the size of the hyperpolarizability. Summing the expectation value of the electronic position over occupied states (1 to M) gives the polarization [11]... 

In terms of the quantum-mechanical dipole matrix element Vjk 2 which determines the strength of an allowed spectral line from the wave-functions of states j > and k >, the oscillator strength is defined as... 

In this expression, < Pi M P2> = ptfiMT dr, where Pj and T2 are the wave-functions of states 1 and 2, respectively, dr is over the whole configuration space of the 3N coordinates, M is the dipole moment operator (M = Seq, where q is the vector joining the electron j to the origin of a coordinate system linked to the molecule). 

Values of (r" ) are based primarily on theoretical calculations of the 4f wave functions, for example, those of Freeman and Watson (1962) and of Judd (1963). In table 18.5 are listed the values of (/ N /), (r ), and Hfs for the tripositive rare-earth ions. Strictly speaking, correction factors differing from unity by a few percent should be applied to these (/ M /) values [Bleaney (1972)] to take account of intermediate coupling effects which arise from the admixture into the ground state (L, S, J) wave function of states of different L, S, but the same /, by the spin-orbit interaction. A table of these values is included in Bleaney (1972). 

The first stoand of Ham s argument [11] is that V(c()) supports continuous bands of Floquet states, with wave functions of the form... 

We shall assume, for simplifying the notation, that the k values are positive. For a phase-inverting reaction, the wave function of the transition state is therefore written as... 

It is clear from Figure 5 that the phase of the electronic wave function of the ground state is constant when moving along the Qp coordinate, until a certain... 

This situation arises when the electronic wave function of the transition state is described by the out-of-phase combination of the two base functions. If the electronic wave function of the transition state is described by the in-phase coinbination. no curve crossing occurs. 

Adopting the view that any theory of aromaticity is also a theory of pericyclic reactions [19], we are now in a position to discuss pericyclic reactions in terms of phase change. Two reaction types are distinguished those that preserve the phase of the total electi onic wave-function - these are phase preserving reactions (p-type), and those in which the phase is inverted - these are phase inverting reactions (i-type). The fomier have an aromatic transition state, and the latter an antiaromatic one. The results of [28] may be applied to these systems. In distinction with the cyclic polyenes, the two basis wave functions need not be equivalent. The wave function of the reactants R) and the products P), respectively, can be used. The electronic wave function of the transition state may be represented by a linear combination of the electronic wave functions of the reactant and the product. Of the two possible combinations, the in-phase one [Eq. (11)] is phase preserving (p-type), while the out-of-phase one [Eq. (12)], is i-type (phase inverting), compare Eqs. (6) and (7). Normalization constants are assumed in both equations ... 

In this chapter, we resfiict the discussion to elementary chemical reactions, which we define as reactions having a single energy bamer in both dhections. As discussed in Section I, the wave function R) of any system undergoing an elementary reaction from a reactant A to a product B on the ground-state surface, is written as a linear combination of the wave functions of the reactant, A), and the product, B) [47,54] ... 

B. The phase changes near the transition state lying along this coordinate. It must therefore be positive close to that locality. The electronic wave function of... 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

The task is now to calculate the structure and energy of the system in the transition state between A and B. Its wave function is assumed to be constmcted from a linear combination of the two. It is convenient to use VB terminology for this purpose. Let the wave function of A be denoted by a VB function A) and that of B by B). 

According to Eq. (A.4), if < 0, the ground state will be the in-phase combination, and the out-of-phase one, an excited state. On the other hand, if > 0, the ground state will be the out-of-phase combination, while the in-phase one is an excited state. This conclusion is far reaching, since it means that the electronic wave function of the ground state is nonsymmetric in this case, in contrast with common chemical intuition. We show that when an even number of electron pairs is exchanged, this is indeed the case, so that the transition state is the out-of-phase combination. 

Now we can calculate the ground-state energy of H2. Here, we only use one basis function, the Is atomic orbital of hydrogen. By symmetry consideration, we know that the wave function of the H2 ground state is... 

The Symmetry Properties of Wave Functions of Li3 Electronically Ground State in [Pg.582]

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