The Wavefunction

The rest of this chapter expands on the above ideas. A proper understanding of quantum mechanics requires an understanding of the principles that it uses. An adequate familiarity with these principles is essential, even irreplaceable. In your dealings with these principles, do not lose sight of that last statement in the above synopsis Quantum mechanics properly describes the behavior of matter, as determined by observation. 

The behavior of a wave can be expressed as a simple mathematical function. For example, 

Since de Broglie indicated that matter should have wave properties, why not describe the behavior of matter using an expression for a wave The first postulate of quantum mechanics is that the state of a system can be described by an expression called a wavefunction. Wavefunctions in quantum mechanics are typically given the symbol if/ or (the Greek letter psi). For various physical and mathematical reasons, these P s are limited, or constrained, to being functions that are  

Single-valued (that is, a wavefunction must have only one possible F x) value for each and every value of x.) 

Differentiable (that is, there must be no mathematical reasons why the derivative of F cannot exist.)  

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To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Electrons, protons and neutrons and all other particles that have s = are known as fennions. Other particles are restricted to s = 0 or 1 and are known as bosons. There are thus profound differences in the quantum-mechanical properties of fennions and bosons, which have important implications in fields ranging from statistical mechanics to spectroscopic selection mles. It can be shown that the spin quantum number S associated with an even number of fennions must be integral, while that for an odd number of them must be half-integral. The resulting composite particles behave collectively like bosons and fennions, respectively, so the wavefunction synnnetry properties associated with bosons can be relevant in chemical physics. One prominent example is the treatment of nuclei, which are typically considered as composite particles rather than interacting protons and neutrons. Nuclei with even atomic number tlierefore behave like individual bosons and those with odd atomic number as fennions, a distinction that plays an important role in rotational spectroscopy of polyatomic molecules. 

The object now will be to steer the wavefunction out of a specific exit chaimel on the ground electronic state, using the excited electronic state as an intennediate. Insofar as the control is achieved by transferring amplitude between two electronic states, all tire concepts regarding the central quantity introduced above will now come into play. 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

In many kinds of atomic and molecular collision problem the wavefunction has many oscillations because the... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

Note, irrespective of the method chosen to nomialize the wavefunctions, that... 

A partial acknowledgment of the influence of higher discrete and continuum states, not included within the wavefunction expansion, is to add, to the tmncated set of basis states, functions of the fomi T p(r)<6p(r) where dip is not an eigenfiinction of the internal Flamiltonian but is chosen so as to represent some appropriate average of bound and continuum states. These pseudostates can provide fiill polarization distortion to die target by incident electrons and allows flux to be transferred from the the open channels included in the tmncated set. 

Hamiltonian = t+ The additivity of implies that the mean-field energies il/are additive and the wavefunctions [Pg.2162]

Figure B3.4.16. A generic example of crossing 2D potential surfaces. Note that, upon rotating around the conic intersection point, the phase of the wavefunction need not return to its original value.

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

It seems that surface hopping (also called Molecular Dynamics with Quantum Transitions, MDQT) is a rather heavy tool to simulate proton dynamics. A recent and promising development is path integral centroid dynamics [123] that provides approximate dynamics of the centroid of the wavefunctions. Several improvements and applications have been published [123, 124, 125, 126, 127, 128). 

Knowledge of molecular orbitals, particularly of the HOMO Highest Occupied Molecular Orbital) and the LUMO Lowest Unoccupied Molecular Orbital), imparts a better understanding of reactions Figure 2-125b). Different colors e.g., red and blue) are used to distinguish between the parts of the orbital that have opposite signs of the wavefunction. 

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. 

The wavefunctions are commonly referred to as orbitals and are characterised by three Ljuantiim numbers n, m and 1. The quantum numbers can adopt values as follows ... 

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

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