Simplified Wave Functions

The simplest many-electron wave function that satisfies the Exclusion Principle is a product of N different one-electron functions that have been antisymmetrized, or written as a determinant. Here, N is the number of electrons (or valence electrons) in the molecule. HyperChem uses this form of the wave function for most semi-empirical and ab initio calculations. Exceptions involve using the Configuration Interaction option (see page 119). HyperChem computes one-electron functions, termed molecular spin orbitals, by relatively simple integration and summation calculations. The many-electron wave function, which has N terms (the number of terms in the determinant), never needs to be evaluated. 

To avoid having the wave function zero everywhere (an unacceptable solution), the spin orbitals must be fundamentally different from one another. Eor example, they cannot be related by a constant factor. You can write each spin orbital as a product of a space function which depends only on the x, y, and z coordinates of the electron—and a spin function. The space function isusually called the molecular orbital. While an infinite number of space functions are possible, only two spin functions are possible alpha and beta. 

To use HyperChem for calculations, you specify the total molecular charge and spin multiplicity (see Charge, Spin, and Excited State on page 119). The calculation selects the appropriate many-electron wave function with the correct number of alpha or beta electrons. You don t need to specify the spin function of each orbital. 

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Purely electrostatic interactions are taken into account by another even more simplified wave function, expressed as the simple product... 

The drastically simplified wave function reads, therefore, as... 

Originally, chemists have built their imderstanding of orbitals on constructs that resulted from WFT-analyzes, and such an orbital is usually referred to as molecular orbital (MO). One important aspect of an MO-analysis is the investigation of orbital-overlap. A simplified wave function theory that emphasizes this particular featare of orbital-interaction, the Extended Hiickel-Theory (EHT), has revolutionized the general perception of molecular structure and reaction mechanisms. 

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... 

Equality between the 1, 2 wave function and the modulus of the 2, 1 wave function, v /(j2, i), shows that they have the same curve shape in space after exchange as they did before, which is necessary if their probable locations are to be the same. The phase factor orients one wave function relative to the other in the complex plane, but Eq. (9-17) is simplified by one more condition that is always true for particle exchange. When exchange is canied out twice on the same particle pair, the operation must produce the original configuration of particles... 

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Many problems simplify significantly by choosing a suitable coordinate system. At the heart of these transfonnations is the separability theorem. If a Hamilton operator depending on N coordinates can be written as a sum of operators which only depend on one coordinate, the corresponding N coordinate wave function can be written as a product of one-coordinate functions, and the total energy as a sum of energies. 

The study of the two-electron systems was greatly simplified by the fact that the total wave function could be factorized into a space part and a spin part according to Eq. III. I. ForiV = 3, 4,. . , such a separation of space and spin is no longer possible, and an explicit treatment of the spin is actually needed in considering correlation effects. This question of the connection between space and spin in an antisymmetric spin function is a rather complicated problem, which has been brought to a simple solution first during the last few years. 

Because of the success of the r12 method in the applications, one had almost universally in the literature adopted the idea of the necessity of introducing the interelectronic distances r j explicitly in the total wave function (see, e.g., Coulson 1938). It was there-fore essential for the development that Slater,39 Boys, and some other authors at about 1950 started emphasizing the fact that a wave function of any desired accuracy could be obtained by superposition of configurations, i.e., by summing a series of Slater determinants (Eq. 11.38) built up from a complete basic one-electron set. Numerical applications on atoms and molecules were started by means of the new modern electronic computers, and the results have been very encouraging. It is true that a wave function delivered by the machine may be the sum of a very large number of determinants, but the result may afterwards be mathematically simplified and physically interpreted by means of natural orbitals.22,17... 

A great deal of work has later been carried out in order to simplify and refine the wave function for the H2 molecule, and, for a more detailed survey, we will refer to a special table in the bibliography. There is little doubt that, even as to the H2 molecule, one can in due time expect a similar development as is now going on concerning the He atom, and, since the former is being complicated also by the nuclear motion involved, several new interesting problems will probably appear. 

Putting Wigner s theory in the simplified form (Eq. III. 130), Krisement was able to compare it with the results of the plasma model where, in a first approximation, Pines wave function may be written in the generalized form (Eq. III. 129) with... 

In Section II.B, we have used the density matrices to simplify the calculations, but the wave functions W are still the fundamental quantities. Relation II. 11 shows,however, that the expectation value of the energy p)Av depends only on the second-order density matrix, and we can rewrite it in the form22... 

The simplified theory allows the time-dependent wave function to be calculated rapidly for any specified laser field. However, controlling the dynamics of the charge carriers requires the answer to an inverse question [18-22]. That is, given a specific target or objective, what is the laser field that best drives the system to that objective Several methods have been developed to address this question. This section sketches one method, valid in the weak response (perturbative) regime in which most experiments on semiconductors are performed. 

It is important to distinguish between mmetiy properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations. 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

The last important contribution to intermolecular energies that will be mentioned here, the dispersion energy (dEnis). is not accessible in H. F. calculations. In our simplified picture of second-order effects in the perturbation theory (Fig. 2), d mS was obtained by correlated double excitations in both subsystems A and B, for which a variational wave function consisting of a single Slater determinant cannot account. An empirical estimate of the dispersion energy in Li+...OH2 based upon the well-known London formula (see e.g. 107)) gave a... 

To date, the only applications of these methods to the solution/metal interface have been reported by Price and Halley, who presented a simplified treatment of the water/metal interface. Briefly, their model involves the calculation of the metal s valence electrons wave function, assuming that the water molecules electronic density and the metal core electrons are fixed. The calculation is based on a one-electron effective potential, which is determined from the electronic density in the metal and the atomic distribution of the liquid. After solving the Schrddinger equation for the wave function and the electronic density for one configuration of the liquid atoms, the force on each atom is ciculated and the new positions are determined using standard molecular dynamics techniques. For more details about the specific implementation of these general ideas, the reader is referred to the original article. ... 

Erom a general experience with wave-packet motion in periodic potentials [237], it may be expected that the complexity of the dynamics is partially caused by the symmetric excitation of the system (i.e., at wave function right from the beginning. To simplify the analysis, it is therefore helpful to invoke an initial preparation that results in a preferred direction of motion of the system. With this end in mind, we next assume that the initial wave packet contains a dimensionless average momentum of po = 23.24, corresponding to an... 

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