Electronic wave function fundamental property

For the purpose of quantum mechanical calculations of electronic wave functions, it is convenient to regard certain fundamental constants (and combinations of such constants) as though they were units. They are customarily called atomic units (abbreviated au), and they may be regarded as forming a coherent system of units for the calculation of electronic properties in theoretical chemistry, although there is no authority from... 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

Symmetry considerations have long been known to be of fundamental importance for an understanding of molecular spectra, and generally molecular dynamics [28-30]. Since electrons and nuclei have distinct statistical properties, the total molecular wave function must satisfy appropriate symmehy... 

The HF method determines the best one-determinant trial wave function (within the given basis set). It is therefore clear that in order to improve on HF results, the starting point must be a trial wave function which contains more than one Slater Determinant (SD). This also means that the mental picture of electrons residing in orbitals has to be abandoned, and the more fundamental property, the electron density, should be considered. As the HF solution usually gives 99% of the correct answer, electron correlation methods normally use the HF wave function as a starting point for improvements. 

The method of superposition of configurations as well as the method of different orbitals for different spins belong within the framework of the one-electron scheme, but, as soon as one introduces the interelectronic distance rijt a two-electron element has been accepted in the theory. In treating the covalent chemical bond and other properties related to electron pairs, it may actually seem more natural to consider two-electron functions as the fundamental building stones of the total wave function, and such a two-electron scheme has also been successfully developed (Hurley, Lennard-Jones, and Pople 1953, Schmid 1953). 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

Alternatively, we can base our analysis on the electron density, which as we have seen, is readily obtained from the wave function. The advantage of analyzing the electron density is that, unlike the wave function, the electron density is a real observable property of a molecule that, as we will see in Chapter 6, can be obtained from X-ray crystallographic studies. At the present time however, it is usually simpler to obtain the electron density of a molecule from an ab initio calculations rather than determine it experimentally. Because this analysis is based on a real physically observable property of a molecule, this approach appears to be the more fundamental. It is the approach taken by the atoms in molecules (AIM) theory, which we discuss in Chapters 6 and 7, on which we base part of the discussion in Chapters 8 and 9. 

There is a fundamental difference between the properties of the molecular bond length in the STIRAP and APLIP processes that is overlooked by the average bond distance. In APLIP the wave function is always localized around the average internuclear distance, since it is constrained by the electronic forces exerted by Uup(x,t) x(t) is also stable since it corresponds to the minimum of a potential, and it is quasi-static, since the rate of change of Uup(x,t) can be arbitrarily controlled. Therefore, it truly corresponds to the classical notion of a bond length. 

The cross section a is a fundamental property of the molecule and as such is related to the molecular wave functions for the two states between which a transition is induced. Hence it is desirable to separate the contributions to a that arise from purely kinematic quantities such as the impact energy of the electron beam from those that depend solely on the properties of the molecule. To this end, a dimensionless quantity, the oscillator strength, is introduced in optical absorption spectroscopy, defined by the relation22... 

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