Pattern forming quantum number

In this form of HROT, the pattern-forming quantum number is N rather than N+ and N 1 is problematic because it gives rise to nonzero off-diagonal matrix elements within a 2Z(2S++1A+) Rydberg complex,... 

The complication in analysing the spectrum is that the energy levels, and hence the spectral lines, can no longer be fitted by a power series in the quantum numbers that converges sufficiently rapidly to form a simple pattern of energy levels and spectral lines, so that simple analytical formulae cannot be used, and assignment becomes a problem. Usually the spectrum can only be analysed in conjunction with an appropriate theoretical treatment of the hamiltonian for the interacting levels. 

Notice that in both case (d) and case (e) there is no molecular projection quantum number. An example of case (e) coupling, probably the first, has been observed [60] for vibration rotation levels of the HeKr+ ion which lie very close to the dissociation limit. The Kr+ atomic ion has L = 1 and S= 1/2, so that. Ja is 3/2 or 1 /2, and the spin orbit interaction is strong. When a very weak bond is formed with a He atom,. Ja remains a good quantum number, at least for the most weakly bound levels, but there are nevertheless series ofrotation levels, with rotational energy BR(R + 1). The details are described in chapter 10, where we show that case (e) coupling is identified, both by the observed pattern of the rotational levels, and by the measured Zeeman effects and effective g factors for individual rotational levels. 

Atomic wave functions with magnetic quantum number m/ = 0 are real functions and their corresponding orbitals can be mapped in the form of well-defined geometrical shapes. Wave functions of electrons with mj 0 are complex functions and do not generate orbitals in real space. But, if by some procedure, these complex functions could be transformed into real orbitals in three-dimensional space, it would in principle be possible to use these spatially directed orbitals to predict the three-dimensional shape of molecules according to the pattern of overlap. The well-known scheme of hybridization by linear combination of atomic orbitals represents such an attempt. 

Each Hund s case is associated with a pattern-forming rotational quantum number, a problematic term in HROT that generates off-diagonal matrix elements within a systematically near degerate group of basis states, and terms in H 0- that lift the degeneracy of the basis states coupled by this problematic term in HROT. For case (a) the pattern-forming rotational quantum number is... 

In case (b) the pattern-forming rotational quantum number is N and the /-uncoupling term,... 

Assignments Based on Pattern-Forming Rotational Quantum Numbers... 

When the electronic transition is between a known rotational level (N", e/f determined in a double resonance scheme) of a case (b) initial state and an unknown rotational level of a case (d) final state, the existence of pattern-forming rotational quantum numbers [[B"N"(N" + 1) for case (b) and B+N+ (N+ +1) for case (d)] provides a basis for N+ — N" rotational assignments. 

However, when the PES transition is between molecular and ion states which belong to different Hund s cases, then the rotational branches are often labeled according to the numerical value of the change in pattern-forming rotational quantum number (respectively J,N,J,N+, and J+ for Hund s cases (a), (b), (c), (d), and (e)), which can have either integer or half-integer value. 

For solids and liquids, electronic absorption bands are usually broad and essentially featureless, but more information is obtainable from electronic spectra of gas-phase molecules. Transitions between two levels with long lifetimes are the most informative. Such an electronic transition for a gas-phase sample has various possible changes in vibrational and rotational quantum numbers associated with it, so that the spectrum, however it is obtained, consists of a number of vibration bands, each with rotational fine structure, together forming an electronic system of bands. The selection rules governing the changes in vibrational and rotational quantum numbers depend on the nature of the electronic transition, and they can be ascertained by analyzing the pattern and structures of the bands. 

The central point within our consciousness, our "spirit" in the hermetic sense, can now be seen as an entity that can work to control quantum probabilities. To our "spirits" our brain is a quantum sea providing a rich realm in which it can incarnate and manifest patterns down into the electrical/chemical impulses of the nervous system. (It has been calculated that the number of interconnections existing in our brains far exceeds the number of atoms in the whole universe - so in this sense the microcosm truly mirrors the macrocosm ). Our "spirit" can through quantum borrowing momentarily press a certain order into this sea and this manifests as a thought, emotion, etc. Such an ordered state can only exist momentarily, before our spirit or point of consciousness is forced to jump and move to other regions of the brain, where at that moment the pattern of probability waves for the particles in these nerve cells, can reflect the form that our spirit is trying to work with. 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

A number of other, but minor primary photoproducts was also found, among them the products expected from a radical (photo-Claisen) rearrangement and from photohydrolysis of the ortho chlorine 2- and 4-chlorophenol were detected too, but their formation remained unexplained. The photodegradation quantum yield of dichlorprop did not depend on pH and was 50 times smaller than that of the anionic form of the related monohalo-genated compound mecoprop (see above) [77]. This is another example of the marked influence of the pattern of ring halogen substitution on the course and on the efficiency of photodegradation. 

The persistent correlation that recurs between number patterns and physical structures indicates a similarity between the structure of space-time and number. Like numbers and chiral growth, matter has a symmetry-related conjugate counterpart. The mystery about this antimatter is its whereabouts in the universe. By analogy with numbers, the two chiral forms of fermionic matter may be located on opposite sides of an achiral bosonic interface. In the case of numbers this interface is the complex plane, in the physical world it is the vacuum. An equivalent mapping has classical worlds located in the two surfaces and the quantum world, which requires complex formulation, in the interface. 

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