Pseudo-quantum number

We are on our way testing if similar approaches can be applied to other, larger molecules such as pyrrole, thus with the hope to find regular patterns, supported by pseudo quantum numbers, allowing to frame the whole rovibrational structure, and therefore the IVR. 

In diatomic spectra, one distinguishes between individual bands each corresponding to a definite pair of quantum numbers v, v", and band systems, each composed of an ensemble of bands associated with a particular electronic transition. In polyatomic spectra, often (a), the individual bands of an electronic transition are so numerous and strongly overlapping that it is difficult or impossible to distinguish them individually, or (b), the electronic transition gives rise only to continuous absorption in both these situations the entire spectrum of an electronic transition is commonly called a band. IT IS RECOMMENDED (REC. 39) that the word band be reserved for definite individual bands, and that electronic transition or transition be used for the entire spectrum, whether discrete, pseudo-continuous, or strictly continuous, associated with an electronic transition or band system if the spectrum consists of discrete bands. ... 

Fig. 2. Correlation diagram of the doublet states between weak (left) and strong (right) spin-orbit coupling. In the strong coupling limit the splitting pattern is determined by the pseudo-./ quantum number...

Pseudo-potential energy curves are extracted from the perturbative Hamiltonian of Eq. (17) by retaining only the terms with P = N = 0—that is, the terms without differential operator. One obtains a one-dimensional pseudopotential curve Fvi.vsCQ) for each pair of quantum numbers v (H-CN stretch) and V3 (C-N stretch)... 

Figure 3. Plot of the lowest 11 pseudo-potential energy curves Vvi.v, (0) obtained by applying sixth-order CPT to the HCN <- CNH surface of Refs. 7 and 8. The stretch quantum numbers vi (H-CN stretch) and Vs(C-N stretch) are indicated for each curve as (vijVs).

Exponents of a set of primitive Gaussian function have been optimized to yield the lowest pseudo atom energies for all first- and second-row elements with an atomic DFT code employing the appropriate GTH potential for each element. A family basis set scheme has been adopted using the same set of exponents for each angular momentum quantum number of the occupied valence... 

The analytical forms of the modern PPs used today have little in common with the formulas we obtain by a strict derivation of the theory (Dolg 2000). Formally, the pseudo-orbital transformation leads to nodeless pseudovalence orbitals for the lowest atomic valence orbitals of a given angular quantum number l (one-component) or Ij (two-component). The simplest and historically the first choice is the local ansatz for A VCy in Equation (3.4). However, this ansatz turned out to be too inaccurate and therefore was soon replaced by a so-called semilocal form, which in two-component form may be written as... 

The HF equations (5.47) (in matrix form Eqs (5.44) and (5.46)) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction to generate an energy value e , times. Pseudo g ma uQ because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the HF equations F depends on xf/ because (Eq. (5.36)) the operator contains J and K, which in turn depend (Eqs (5.29) and (5.30)) on xjr. Each of the equations in the set (5.47) is for a single electron ( electron one is indicated, but any ordinal number could be used), so the HF operator F is a one-electron operator, and each spatial molecular orbital is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the full description of each of these electrons requires a spin/unction a or (section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital do not have all four quantum numbers the same (for an atomic 1j orbital, e.g. oneelectron has quantum numbers = ,/= 0, m= 0 and 5 = 5, while the other has n= 1, / = 0, m = 0 and 5 = — 5), and so the Pauli exclusion principle is not violated. 

In the A -operator formalism the relations (35), (37) and (38) are built in to the excitation operators. A full definition is given by Jensen et al. [38] who introduce the pseudo quantum number Mk to play the role of the non-relativistic Ms quantum number. This number is defined as Mk = (M - M 2 with M and Mp the number of occupied unbarred and barred spinors, respectively. The analogue of the non-relativistic spin-restricted excitation operators are now those that preserve Mk, either by disallowing spin flips... 

In the B state, the three sodium atoms perform a nearly free pseudorotational motion in the moat of a pseudo Jahn-Teller potential that is characterized by a vibronic angular momentum quantum number j. 

Notice that two quantum numbers specify the exciton eigenstates, eqn (6.13) or eqn (6.16) the principle quantum number, n, and the (pseudo) momentum quantum number, K (or fUj). For every n there are a family of excitons with different centre-of-mass momenta, and hence different centre-of-mass kinetic energy. Odd and even values of n correspond to the relative wavefunction, tl>n r), being even or odd under a reversal of the relative coordinate, respectively. We refer to even and odd parity excitons as excitons whose relative wavefunction is even or odd imder a reversal of the relative coordinate. This does not mean that the overall parity of the eigenstate (eqn (6.12)), determined by both the centre-of-mass and relative wavefiictions, is even or odd. The number of nodes in the exciton wavefunction, V rt( ), is n— 1. Figure 6.2 illustrates the wavefunctions and energies of excitons in the effective-particle model. 

The external peaks in Fig. 1, labeled by the pseudo spin quantum numbers =0,2, are symmetric and anti-symmetric combinations of states with spin quantum numbers 1=0,2. The reorientation process does not affect states with zero spin, whereas states with 1=2 may make a transition to one of the five different hf components causing a broadening. 

---