Wave function two electrons

D. A. Mazziotti, Quantum chemistry without wave functions two-electron reduced density matrices. Acc. Chem. Res. 39, 207 (2006). 

The permutational symmetry of the rotational wave function is determined by the rotational angular momentum J, which is the resultant of the electronic spin S, elecbonic orbital L, and nuclear orbital N angular momenta. We will now examine the permutational symmetry of the rotational wave functions. Two important remarks should first be made. The first refers to the 7 = 0 rotational... 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

Any determinant changes sign when any two columns are interchanged. Moreover, no two of the product functions (columns) can be the same since that would cause the determinant to vanish. Thus, in all nonvanishing completely anti-symmetric wave functions, each electron must be in a different quantum state. This result is known as Pauli s exclusion principle, which states that no two electrons in a many-electron system can have all quantum numbers the same. In the case of atoms it is noted that since there are only two quantum states of the spin, no more than two electrons can have the same set of orbital quantum numbers. 

The Hamiltonian for two electrons in the field of two fixed protons is given by (39). For large values of rab the system reasonably corresponds to two H atoms. The wave functions of the degenerate system are ipi = ui5a(l)ui 6(2) and ip2 = UiSt(l)u1Sa(2), where ul5o(l) is the hydrogenic wave function for electron 1 about nucleus A, etc. For smaller values of rab a linear combination of the two product functions is a reasonable variational trial function, i.e. 1p = 1pl +... 

As mentioned in Sec. 1.3, the electrochemical potential of electrons in condensed phases corresponds to the Fermi level of electrons in the phases. There are two possible cases of electron ensembles in condensed phases one to which the band model is applicable (in the state of degenera< where the wave functions of electrons overlap), and the other to which the band model cannot apply (in the state of nondegeneracy where no overlap of electron wave functions occurs). In the former case electrons or holes are allowed to move in the bands, while in the latter case electrons are assumed to be individual particles rather than waves and move in accord with a thermal hopping mechanism between the a4jacent sites of localized electron levels. 

In an MC wave function, the HF wave function in Eq. 2 is augmented by additional configurations, only four of which are actually given in Eq. 5. As shown pic-torially in Figure 22.2, in these four additional configurations different pairs of electrons have been excited from two of the orbitals (vf/ and / ) that are doubly occupied in the HF wave function. The electrons have been excited into one of the many virtual orbitals (v[/y), which, if empty, are the HF wave function. It is the... 

The density functional theory of Hohenberg, Kohn and Sham [173,205] has become the standard formalism for first-principles calculations of the electronic structure of extended systems. Kohn and Sham postulate a model state described by a singledeterminant wave function whose electronic density function is identical to the ground-state density of an interacting /V-clcctron system. DFT theory is based on Hohenberg-Kohn theorems, which show that the external potential function v(r) of an //-electron system is determined by its ground-state electron density. The theory can be extended to nonzero temperatures by considering a statistical electron density defined by Fermi-Dirac occupation numbers [241], The theory is also easily extended to the spin-indexed density characteristic of UHF theory and of the two-fluid model of spin-polarized metals [414],... 

A semiempirical method can be developed for the arbitrary form of the trial wave function of electrons, which is predefined by the specific class of molecules to be described and by the physical properties and/or effects which have to be reproduced within its framework. Two characteristic examples will be considered in this section. One is the strictly local geminal (SLG) wave function the other is the somewhat less specified wave function of the GF form selected to describe transition metal complexes. 

The construction of the LD theory of the ligand influence evolves in terms of two key objects the electron-vibration (vibronic) interaction operator and the substitution operator. The vibronic interaction in the present context is the formal expression for the effect of the system Hamiltonian (Fockian) dependence on the molecular geometry taken in the lower - linear approximation with respect to geometry variations. It describes coupling between the electronic wave function (or electron density) and molecular geometry. 

The condition j + j > 1 for a matrix element of a first rank tensor operator implies, e.g., that there is no first-order SOC of singlet wave functions. Two doublet spin wave functions may interact via SOC, but the selection rule /+ / > 2 for i (2)(Eq. [171]) tells us that electronic spin-spin interaction does not contribute to their fine-structure splitting in first order. 

A specific wave function solution is called an orbital. The different orbitals define different energies and distributions for the different electrons. The name orbital goes back to earlier theories where the electron was thought to orbit the nucleus in the way that planets orbit the sun. It seems to apply more to an electron seen as a particle, and orbitals of electrons thought of as particles and wave functions of electrons thought of as waves are really two different ways of looking at the same thing. Each different orbital has its own individual quantum numbers, , , and mg. 

Such a separation is exact for atoms. For molecules, only the translational motion of the whole system can be rigorously separated, while their kinetic energy includes all kinds of motion, vibration and rotation as well as translation. First, as in the case of atoms, the translational motion of the molecule is isolated. Then a two-step approximation can be introduced. The first is the separation of the rotation of the molecule as a whole, and thus the remaining equation describes only the internal motion of the system. The second step is the application of the Born-Oppenheimer approximation, in order to separate the electronic and the nuclear motion. Since the relatively heavy nuclei move much more slowly than the electrons, the latter can be assumed to move about a fixed nuclear arrangement. Accordingly, not only the translation and rotation of the whole molecular system but also the internal motion of the nuclei is ignored. The molecular wave function is written as a product of the nuclear and electronic wave functions. The electronic wave function depends on the positions of both nuclei and electrons but it is solved for the motion of the electrons only. 

All of the discussion to this point has concerned the spatial wave function i/ (r) of an electron. An electron also has spin. For any t) there are two possible spin states. Thus, assertion (a) set forth earlier should be amended to say that an electron is described by its spatial wave function and its spin state. The term state is commonly used to refer to only the spatial wave function, when electron spin is not of interest. It is also frequently used to encompass both wave function and electron spin. 

For very large values of rAB we know that in its normal state the system consists of two normal hydrogen atoms. Its wave functions (the state having two-fold degeneracy) are then Ui,x(l) Ui,s(2) and i,B(l) Wi ,(2) or any two independent linear combinations of these two (the wave function ulwave function for electron 1 about nucleus A,... 

Just as atomic orbitals are solutions to the quantum mechanical treatment of atoms, molecular orbitals (MOs) are solutions to the molecular problem. Molecular orbitals have many of the same characteristics as atomic orbitals. Two of the most important are that they can hold two electrons with opposite spins and that the square of the molecular orbital wave function indicates electron probability. 

---