Wavefunction left side

Since the wavefunction is not a stationary state, it evolves according to Equation 6.11. If there are only two stationary states in the superposition state, as in Figure 6.3, the probability distribution and all of the observables oscillate at a frequency a> = (If — E )/Ti (see Problem 6-10). If we have the left side wavefunction in Figure 6.3 at time / = 0, at later times we will have a right side wavefunction. At... 

Recall that the reason for the enhancement of the absorption cross section for small clusters is the enhanced overlap between the electron and hole wavefunctions, that is, U(0) 2 in Eq. (6). In an ideal cluster, the maxima of the electron and hole wavefunctions are located at the center of the cluster and good overlap between them is insured (left side of structure below). In reality, however, the hole wavefunction may be localized near the surface owing to the presence of defects of charged centers (right side of Structure 2). This reduces the electron hole overlap and therefore the exciton... 

At the left side, we have shown the energy levels of a single electron as the energy, E, increases. When lattice sites are imposed, we get a series of bands (shown horizontedly) as E increases. Note also the unoccupied zones, indicated by the hatched area. Occupation of the allowed bands is a function of the types and numbers of electrons present. At the bottom is the wavefunction associated, as a function of atom-position in the lattice. 

Figure 1.2 illustrates the resonance ) decaying into a quasi-continuum (left side) and into a true continuum (right side). The black rectangle is the useful part of the continuum implied in the dynamics. It corresponds to the wavefunction H (p) (a doorway state in spectroscopy), which is the second term in the method of moments (see Eq. (17)). The physical results are obtained at the limit 5 -> 0 while tF/5 remains constant. The transition rate r to the continuum is equal to... 

The upper left side shows the resulting shape of the corresponding excited complex, with the symmetric 4ai orbital and the now singly occupied Ib electron. Its structure is very well characterized, because both the electronic structure and the nuclear configuration are known. A very intettesting feature of the excited complex is its electronic symmetry wifh respect to a reflection at the nuclear plane. Because of the singly occupied ptt lobe perpendicular to the molecular plane (the Ibj electron), the electronic wavefunction is antisymmetric to the plane. 

Calculation of the reaction paths for both reactions (6) and (7) have been reported by Harding Wagner (38). These calculations employed a polarized double zeta basis set and a four orbital - four electron CASSCF wavefunction. This wavefunction allows for the correlation of both CH bond pairs of formaldehyde and thus provides a qualitatively correct description of both reaction pathways. The angular dependence of the potential surface for a hydrogen atom moving in the field of a frozen formyl radical is shown in Figure 6. Two reactive channels are clearly seen in this plot. One, on the left side of the plot, leads to the addition product, formaldehyde, while the other, on the right side of the plot, leads directly to abstraction products, H2 + CO. From this plot it can be seen that there is no barrier for either reaction. 

Our knowing that the particle is on the left side at f = 0 means that the wavefunction for this state is not one of the time-independent box eigenfunctions we saw in Chapter 2, because those all predict equal probabilities for finding the particle on the two sides of the box. If the state function is not stationary, it must be time dependent, and it must satisfy Schrddinger s time-dependent equation (6-1). We have, then, that the state function is time dependent, and that I s q/p is zero everywhere on the right side of the box when f = 0. (We have not yet been specific enough to describe p in detail on the left side of the box.)... 

Suppose, for instance, v e choose to describe the starting wavefunction (x, 0) as a normalized half sine wave in the left side of the box and zero at the right (Fig. 6-3). Then we can calculate the amount (c ) of each of the stationary-state functions ij/n present in this function as follows ... 

Symmetry is one of the most powerful tools that can be applied to quantum mechanics and wavefunctions. Most people are generally aware of the concept of symmetry An object is round or square, or the left side is the same as the right side, or maybe they are mirror images. All of these statements imply a recognition of symmetry, a spatial similarity due to the shape of an object. But more technically, symmetry is a powerful mathematical tool that can potentially simplify our study of quantum mechanics. 

Eq. (A.l 1) constitutes the quantum mechanical virial theorem for molecular solutes described within the PCM model, which involves, on the left side, the kinetic and the total potential energies for exact state-wavefunctions. The terms on right side of Eq.(A.ll) have a physical meaning which can be clarified with the aid of the Hellmann-Feynman theorem discussed in Chap. 2 (see Eq. 2.1 ). 

It is interesting to compare the Ce(CgH6)2 ground state wavefunction with a corresponding one of H2 in its S+ ground state at a stretched bond distance of 2 A (2.7 R<,), as shown in Figurelfi.lO on the left side. Here the two electrons are treated in an active space built by the ag and linear combinations of the Is orbitals on each of the two atoms. The orbitals a and b now represent any orthonormalized pair of orbitals built from these Is orbitals. The natural orbitals (rotation angle d> = 0°) correspond to the ag and orbitals, delocalized between the two atoms. A rotation by = 45° leads to the Is orbitals localized on the two... 

Once o(i) is diagonalized, the uncoupled translational wavefunctions can be propagated from the left side to the right side of sector i. This process is accomplished with the 2N x2N sector propagator. 

Dividing left- and right-hand sides by the product wavefunction gives... 

The term d2iji/dx 2 can be thought of as a measure of how sharply the wavefunction is curved. The left-hand side of the Schrodinger equation is commonly written Hv i, where H is called the hamiltonian for the system then the equation takes the deceptively simple form... 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

As shown in Fig. 7.2, at a shorter proton-proton separation (/ <16 a.u. or <8 A), the electron in the Is state in the vicinity of one proton has an appreciable probability of tunneling to the Is state in the vicinity of another proton. The tunneling matrix element can be evaluated using the perturbation theory we presented in Chapter 2. A schematic of this problem is shown in Fig. 7.3. By defining a pair of one-center potentials, Ul and Ur, we define the right-hand-.side states and the left-hand-side states. Because the potential Ul is different from the potential of a free proton, Uro, the wavefunction i ii, and the energy level Eo are different from the Is state of a free hydrogen atom. (The same is true for Ur and We will come back to the effect of such a distortion later in this section. 

Fig. 7.4. Wavefunctions of the hydrogen molecular ion. (a) The exact wavefunctions of the hydrogen molecular ion. The two lowest states are shown. The two exact solutions can be considered as symmetric and antisymmetric linear combinations of the solutions of the left-hand-side and right-hand-side problems, (b) and (c), defined by potential curves in Fig. 7.3. For brevity, the normalization constant is omitted. (Reproduced from Chen, 1991c, with permission.)...

Fig. 6.4. Schematic illustration of the multi-dimensional reflection principle in the adiabatic limit. The left-hand side shows the vibrationally adiabatic potential curves en(R). The independent part of the bound-state wavefunction in the ground electronic state is denoted by

Fig. 6.9. Left-hand side Vibrational excitation function N(ro) and weighting function W(ro) versus the initial oscillator coordinate ro for three values of the coupling parameter e. The equilibrium separation of the free BC molecule is f = 0.403 A and the equilibrium value within the parent molecule is re = 0.481 A. Right-hand side Final vibrational state distributions P(n) for fixed energy E the quantum mechanical and the classical distributions are normalized to the same height at the maxima. The classical distributions are obtained with the help of (6.32). The lowest part of the figure contains also the pure Franck-Condon (FC) distribution (

Cuts of the stationary wavefunctions along the transition line, shown on the left-hand side of Figure 10.15 and denoted by 4 ts(70)> clearly manifest that ... 

The excitation function together with the transition-state wavefunctions for FNO are shown on the left-hand side of Figure 10.15. This figure underlines very clearly the success of the simple picture in interpreting the quantum mechanically calculated final state distributions. The reflection principle is obvious in the light of Section 6.3 and needs no further explanation. Each maximum and each minimum in the distribution has its counterpart in the transition state wavefunction. 

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