Wavefunction well-behaved

It is important to stress that the behavior of the wave packet outside the interaction region portrayed in Figures 1.4, 1.7 and correspondingly Eqs. (7, 23) does not extend over the whole space and the wavefunction eventually decays (in space). This is to be expected because we are dealing with a well-behaved square integrable function. This important point will be addressed in Section 4. 

The rationale behind this approach is the variational principle. This principle states that for an arbitrary, well-behaved function of the coordinates of the system (e.g., the coordinates of all electrons in case of the electronic Schrodinger equation) that is in accord with its boundary conditions (e.g., molecular dimension, time-independent state, etc.), the expectation value of its energy is an upper bound to the respective energy of the true (but possibly unkown) wavefunction. As such, the variational principle provides a simple and powerful criterion for evaluating the quality of trial wavefunctions the lower the energetic expectation value, the better the associated wavefunction. 

Q Which of the wavefunctions shown in Figure 1.26 are well behaved Give reasons for your answer. 

A Only the wavefunction shown in (c) is well behaved. The one shown in (a) is tending towards infinity, and therefore violates rule (3). The one shown in (b) has multiple values of v for a given value of X, and the one shown in (d) has discontinuities in the gradient, so that dv dx cannot be defined at certain points. 

First one can build up other effective Hamiltonians based on hierarchized orthogonalization procedures. The Gram-Schmidt procedure is recommended if one starts from the best projected wavefunctions of the bottom of the spectrum. Thus one can obtain a quite reliable effective Hamiltonian with well behaved wavefunctions and good transferability properties (see Section III.D.2). The main drawback of this approach is that the Gram-Schmidt method, which involves triangular matrices, does not lead to simple analytical expressions for perturbation expansions. A partial solution to these limitations is brought about by the new concept of intermediate Hamiltonian,... 

To describe a physical system, the wavefunction (if/) must also be well-behaved that is, it must satisfy the following conditions ... 

Any wavefunction for a real system obeys the following rules, common to well-behaved functions that describe physical phenomena ... 

This is nothing new the classical solution in this region is the same. Conveniently, though, this establishes boundary conditions for our solution inside the box. The wavefunction must be a well-behaved (in this case meaning continuous) function, and therefore if/ must be zero at x = 0 and at x = a. 

In this expression, j x) is an arbitrary, well-behaved fimction that gives the overall shape of the wavepacket and that varies slowly enough so that several oscillations of the cosine wave are possible within the wavepacket. If we operate with p at one point along the wavefunction, we will define the local squared momentum at that point only to be... 

This is not a problem (as inspection of the cosine/sine form of the wavefunction shows). They must be continuous and differentiable. Again, exponential functions of this sort are mathematically well behaved. 

A complete set of functions of a given variable has the property that a linear combination of its members can be used to construct any well-behaved function of that variable, even if the target function is not a member of the set. Well-behaved in this context means a function that is finite everywhere in a defined interval and has finite first and second derivatives everywhere in this region. Fourier analysis uses this property of the set of sine and cosine functions (Appendix A.3). In quantum mechanics, wavefunctions of complex systems often are approximated by combinations of the wavefunctions of simpler or ideahzed systems. Wave-functions of time-dependent systems can be described similarly by combining wavefunctions of stationary systems in proportions that vary with time. We ll return to this point in Sects. 2.3.6 and 2.5. 

The Schrodinger equation has not been solved exactly for electrons in molecules larger than the H2 ion the interactions of multiple electrons become too complex to handle. However, the eigenfunctions of the Hamiltonian operator provide a complete set of functions, and as mentioned in Sect. 2.2.1, a linear combination of such functions can be used to construct any well-behaved function of the same coordinates. This suggests the possibility of representing a molecular electronic wavefunction by a linear combination of hydrogen atomic orbitals centered at the nuclear positions. In principle, we should include the entire set of atomic orbitals... 

Acceptable radial wavefunctions must as usual be well behaved (single-valued, continuous, etc.), and, since the electron is radially constrained by the electrostatic potential, quantum conditions apply with a radial quantum number, n. A few of these radial wavefunctions are shown in Table 3.3. 

For a systematic study of many related systems, a practical but nevertheless reliable approach had to be identified. It is clear that the reactions of the type depicted in Fig. 8.5 inherently have some diradical and hence multi-determinantal character. Therefore, semiempirical, HF and MP2 methods are unsuitable approaches because a single-reference will not be able to describe all the species involved equally well. DFT may be an option but it is also single-reference however, as it is not the wavefunction that is the operand but the electron density, DFT may still do a reasonable job in describing these reactions. The simple take on this is that the density may be well-behaved even if the wavefunction is not. For instance, DFT has been proven to work exceptionally well for carbenes although singlet carbenes are... 

The individual partial cross sections are even more structured than the total cross sections and a simple explanation of the energy dependences is probably impossible, except at the low-energy tail of the spectrum. For total energies below the barrier of the A-state PES (E = —2.644 eV in this normalization), the dissociative wavefunction is mainly confined to the two H + OH channels with little amplitude in the intermediate region. It therefore overlaps only that part of the ground-state wavefunction which extends well into the two exit channels. There, however, the 40-) and the 31-) states behave completely differently (see Figure 13.4),... 

Namely, the circular orbit (l = n — 1), which is a rotating state with a nodeless radial wavefunction, corresponds to a vibrational quantum number v = 0, and the next-to-circular single-node state (l = n — 2) corresponds to v = 1,.... This theoretical possibility of large-/ circular orbits behaving like bound states in a Morse potential seems to have no other natural manifestation than in the present case of metastable exotic helium. This situation is presented in Fig. 2, where the potential as well as the wavefunctions are shown. 

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