Complex atoms, angular momenta electronic states

The familiar set of the three t2g orbitals in an octahedral complex constitutes a three-dimensional shell. Classical ligand field theory has drawn attention to the fact that the matrix representation of the angular momentum operator t in a p-orbital basis is equal to the matrix of — if in the basis of the three d-orbitals with t2g symmetry [2,3]. This correspondence implies that, under a d-only assumption, l2 g electrons can be treated as pseudo-p electrons, yielding an interesting isomorphism between (t2g)" states and atomic (p)" multiplets. We will discuss this relationship later on in more detail. 

Intermediate species in the gas phase may also be studied by electron resonance methods, although the spectra obtained are usually more complex than those observed with condensed phases. In small gaseous radicals and atoms with degenerate orbital states, the orbital angular momentum will not be quenched, and may make a contribution to the paramagnetism of the species. Even in orbitally non-degener-... 

The 3E and p-complex structures resemble each other because both consist of one unit of spin or electronic angular momentum (S or L) coupled to the nuclear rotation (R). However, since fj, operates exclusively on electron spatial coordinates, any resemblance between the rotational-branch intensity patterns for 3S —1E+ and p-complex —1E+ transitions would seem to be coincidental. A 3E —1E+ transition will look exactly like a p-complex —1E+ transition if, in addition to satisfying Eqs. (6.3.47), the cr-orbital of the 1E+ state is predominantly of scr united atom character. Then the transition moment ratio will be... 

In more complex atoms there may be a strong coupling between the motion of different electrons. The states are usually described in terms of the total orbital angular momentum L and the total spin angular momentum... 

A wavefunction, ip, is a solution to the Schrodinger equation. For atoms, wavefunctions describe the energy and probabihty of location of the electrons in any region around the proton nucleus. The simplest wavefunctions are found for the hydrogen atom. Each of the solutions contains three integer terms called quantum numbers. They are n, the principal quantum number, I, the orbital angular momentum quantum number and mi, the magnetic quantum number. These simplest wavefunctions do not include the electron spin quantum number, m, which is introduced in more complete descriptions of atoms. Quantum numbers define the state of a system. More complex wavefunctions arise when many-electron atoms or molecules are considered. 

Quantum conceptualizations on the eomposition of atoms and molecules make the foundation of modem natural seienee theories. Thus, the electronic angular momentum in stationary eondition equals the integral multiple from Planek s eonstant. This main quantum number and three other combined explieitly eharacterize the state of any atom. The repetition factors of atomic quantum characteristics are also expressed in spectral data for simple and complex stmctures. 

States of individual atoms are usually described by quantum numbers L, S, and for the electronic orbital, spin, and total angular momenta, respectively. However, in scattCTing and bound-state problems involving pairs of atoms or molecules it is common to use lower-case letters for quantum numbers of individual collision partners and reserve capital letters for quantities that refer to the collision system (or complex) as a whole. Thus, in this subsection we will use I and s for the quantum numbers of a single helium atom and reserve L and S for the end-over-end angular momentum of the atomic pair and the total spin, respectively. 

The radial part R i(r) of cancels out in Eqs. 2.6 and 2.7, because and operate only on 9 and 0.) This implies that the orbital angular momentum quantities and are constants of the motion in stationary state with values /(/ -I- l)h and mh, respectively. A common notation for one-electron orbitals combines the principal quantum number n with the letter s, p, d, or/ for orbitals with 1 = 0, 1, 2, and 3, respectively. (This notation is a vestige of the nomenclature sharp, principal, diffuse, and fundamental for the emission series observed in alkali atoms, as shown for K in Fig. 2.2.) An orbital with n = 2,1 = 0 is called a 2s orbital, one with n = 4, / = 3 a 4/ orbital, and so on. Numerical subscripts are occasionally added to indicate the pertinent m value the 2po orbital exhibits n = 2, I = i, and m = 0. Chemists frequently work with real (rather than complex) orbitals which transform as Cartesian vector (or tensor) components. A normalized 2p, orbital is the linear combination... 

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