Wavefunction normalization

Consider a state described by the wavefunction normalized to unity. Then we define the fe-particle (reduced) density matrices as expectation values of... 

Nk being the normalization constant to be determined. The equivalent wavefunction normalized per unit energy is... 

Using Eqs. (2.36) and (2.37) we calculate iVw, yielding the wavefunction normalized per unit energy. Explicitly,... 

To do this we must change the i wavefunctions from normalization per unit energy to normalization per state, i.e. Eq. (20.20). Using the derivative dtT,/dr, = 1/iy5 we may convert the squared wavefunctions 0, 2 from energy to state normalization by multiplying by 1/vj3. Equivalently, a bound 0, wavefunction which is normalized per unit energy has a normalization integral of v2. Since the wavefunction T = is composed of bound wavefunctions normalized per... 

The flux is defined up to the constant A. Taking A = (a single particle wavefunction normalized in the volume Q) implies that the relevant observable is 2VJ(r), that is, is the particle flux for a system with a total of N particles with N Sometimes it is convenient to normalize the wavefunction to unit flux, J = 1 by choosing A =... 

Table B2.2.1 Continuum wavefunction normalization, density of states and cross section factors.

Wavefunctions normally have to meet the three conditions listed below. 

In many instances, it is necessary to find an extremal of a functional, subject to some constraint. These constraints can be introduced via the method of Lagrange multipliers, which can be global or have a local (pointwise) dependence. For instance, in WFT, one minimizes the energy of a molecular system, ( H ), keeping the wavefunction,normalized to unity, namely. 

In other words, the energy of the exact wavefunction serves as a lower bound to the energies calculated by any other normalized antisymmetric function. Thus, the problem becomes one of finding the set of coefficients that minimize the energy of the resultant wavefunction. 

The normalized vibrational wavefunctions are given by the general expression... 

The matrix S is of course symmetric. The normalization condition for a single LCAO wavefunction can be written in a compact notation as... 

The first and third terms on the right-hand side are equal for a real wavefunction. (hfferentiating the normalization condition gives... 

If we further assume that the vibrational wavefunctions associated with normal mode i are the usual harmonic oscillator ones, and r = u + 1, then the integrated intensity of the infrared absorption band becomes... 

In standard quantum-mechanical molecular structure calculations, we normally work with a set of nuclear-centred atomic orbitals Xi< Xi CTOs are a good choice for the if only because of the ease of integral evaluation. Procedures such as HF-LCAO then express the molecular electronic wavefunction in terms of these basis functions and at first sight the resulting HF-LCAO orbitals are delocalized over regions of molecules. It is often thought desirable to have a simple ab initio method that can correlate with chemical concepts such as bonds, lone pairs and inner shells. A theorem due to Fock (1930) enables one to transform the HF-LCAOs into localized orbitals that often have the desired spatial properties. 

There are a few interesting points about the treatment. First of all, there is no variational HF-LCAO calculation (because every available x is doubly occupied) and so the energy evaluation is straightforward. For a wavefunction comprising m doubly occupied orthonormal x s the normalizing factor N is... 

Appendix Normal Coordinates, Vibrational Wavefunctions, and Spectral Activities. 339... 

In general, the first excited state (i.e. the final state for a fundamental transition) is described by a wavefunction pt which has the same symmetry as the normal coordinate (Appendix). The normal coordinate is a mathematical description of the normal mode of vibration. 

The vibrational wavefunctions may be expressed as functions of the jth normal coordinate ... 

Cjl.115 Wavefunctions are normalized to 1. This term means that the total probability of finding an electron in the system is 1. Verify this statement for a particle-in-the-box wavefunction (Eq. 10). 

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