Wavefunctions general form

Using the approach already described for combination generation, one can formulate in a short but completely general form the Cl wavefunctions [16]. 

The radial parts of the wavefunctions for the hydrogen atom can be constructed from the general form of the associated Laguerre polynomials, as developed in Section 5.5.3. However, in applications in physics and chemistry it is often the probability density that is more important (see Section 5.4.1). This quantity in this case represents the probability of finding the electron in the appropriate three-dimensional volume element. 

In its most general form, a quantum graph is defined in terms of a (finite) graph G together with a unitary propagator U it describes the dynamics of wavefunctions

[Pg.79]

Obviously, the functions /,( ) diverge at large u, which are not appropriate to represent tip wavefunctions. The functions kiu) are regular at large u, which satisfies the desired boundary condition. Therefore, a component of tip wavefunction with quantum numbers I and m has the general form... 

The understanding of electronic states in atoms is to a great extent based on Schrodinger s solution of the hydrogen-atom problem. These wavefunctions have the general form (Landau and Lifshitz, 1977) ... 

In parabolic coordinates the volume element dr = +rj)d drjd

general form of the wavefunction given in Eq. (6.21) and carrying out the angular integration, leads to... 

As in the one-dimensional case, the general form of the spectrum is of Gaussian-type and can be regarded as a reflection of the two-dimensional bound-state wavefunction at the upper-state PES. It is noteworthy that ... 

The intermolecular term has the same general form as the absorption cross section in the case of direct photodissociation, namely the overlap of a set of continuum wavefunctions with outgoing free waves in channel j, a bound-state wavefunction, and a coupling term. For absorption cross sections, the coupling between the two electronic states is given by the transition dipole moment function fi (R,r, 7) whereas in the present case the coupling between the different vibrational states n and n is provided by V (R, 7) = dVi(R, r, 7)/dr evaluated at the equilibrium separation r = re. 

For this purpose it would seem profitable to review briefly in the next section some of the, perhaps, less well-known properties of the exact non-relativistic wavefunction, but which are, nevertheless, important when discussing VB theory. Also in this section a short description is given of the construction and manipulation of antisymmetric wavefunctions of more general form than a simple Slater determinant. This is then followed by a brief survey of some of the more commonly used spin functions. 

The particle will have an associated wavefunction, 4, which must have a value of 0 when x = 0 and when x = a this is known as a boundary condition. The general form of the wavefunction 4 (x) is written as ... 

Consider a model system of four electrons moving in an arbitrary electrostatic field generated by the nuclei in a molecule. For our purposes, it is not necessary to specify the number of these nuclei, their types, or positions only the general form of the electronic wavefunction is of interest. It is convenient to describe the motions of each electron separately by assigning them to one-electron functions, (l),(xi), where Xi is a vector of the coordinates (including spin) of electron 1. In addition, electrons are fermions, so the electronic wave-function must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. A traditional and very useful starting point for such a four-electron wavefunction is the so-called Slater determinant... 

In the event that the Cl cluster operator, C, is not truncated, however, it is possible to write the resulting full Cl wavefunction as a product of wavefunctions for each separated fragment, since the linear operator may be transformed into an exponential using a generalized form of Eq. [34]. 

The general form of the wavefunctions is still given by Eq. (2.50), which is more conveniently written (see Problem 2.10) as... 

Momentum-space concepts are not, in general, familiar to the chemist and so we outline first the calculation of momentum-space electron densities, p p), from ab initio wavefunctions. The form of pip) for different molecules is discussed, using as examples (i) the ground state of H2, (ii) bond formation in BH", and (iii) the n-orbitals in large conjugated polyenes. 

The usual treatment of confinement is where the charge p = 37 (r) 2 is essentially zero on the sphere (or hypersphere) r = R and the system may therefore be isolated. For example, if we envisage a large repulsive potential in r > R, then we may model this as an infinite barrier on r = R and consequently F(TJ) = 0. This model isolates the wavefunction completely in r < R and, for the examples considered here, leads to the general form... 

The wavefunction for two well-separated particles has the general form ... 

The general form of state-specific wavefunctions for discrete and for resonance sfafes is Refs. [1,42] and references fherein. 

We first establish a general form of the wavefunction I satisfying Equation 10.11, with the boundary condition 10.12 and symmetry relations 10.10. In our case the total energy E = —Ih fma < 0, and the Green function of Equation 10.11 reads... 

A spin-orbit wavefunction of this form is called a Slater determinant, after the pioneering physicist who developed the concept. Any can be written as a Slater determinant using the general form shown, creating a total wavefunction that obeys the Pauli principle. Note that we have now collapsed the spatial and spin parts into a single symbolism, such that y/ (m) contains both space and spin components. 

Actually, this derivation of Eq. 2.17 includes one other assumption, which was not mentioned because it doesn t affect K. We used the unnormalized function sin(/cx + 0) as the general form of a wavefunction with well-defined kinetic energy. There s a hidden assumption here that the wavefunction has no complex values, that it is a pure real function. 

Returning now to the larger problem of the angular Hamiltonian, we can replace the second derivative of cp in by the eigenvalue —mj. Now we have to find the 0-dependent part of the wavefunction, 0(0). This is more difficult, and we take a shortcut by assuming a reasonable, general form for the wavefunction. We need a function that can suffer differentiation with respect to 0, suffer multiplication and division by sin0, and still look the way it did when we started. 

The Hamiltonian includes terms with different powers of r, and in solving the Schrodinger equation we will need all those powers of r to cancel, leaving us with only the energy times our wavefunction. That suggests that in addition to the exponential term e, we will also need a polynomial in r, with the coefficients set so that any unwanted terms vanish. The general form for our radial wave-function R(r)is therefore ... 

Neglecting some relatively small effects, we can solve the Schrodinger equation of the hydrogen atom to get algebraic expressions for the energies and wavefunctions. Once we add a second electron to an atom, that becomes impossible. Although many numerical methods work well, the exact solution cannot be written in general form. We will go as far as we can in the space of one chapter, which is pretty far. 

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