HARMONY

In common sense, harmony refers to agreement and well fitting together of parts of any set that are forming the whole construction. In particular, harmony means 

The ancient Greek philosophers searched harmony in the universe as well as in the human arts. Certain geometric constmctions are addressed as platonic bodies. The school named after Pythagoras tried to measure harmony in numbers. As simpler a ration of numbers that are characterizing a system, more harmonic the system is itself. The starting point was problems of music, because here harmony is very easy to demonstrate. 

Alberti was an excellent mathematician and geometrician. He wrote 10 books on architecture [1]. He experimented with proportions in space and he stated 

We shall therefore borrow all our rules for the finishing our proportions, from the musicians, who are the greatest masters of this sort of numbers, and from those things wherein nature shows herself most excellent and compleat. 

Thompson stated in his famous book On Growth and Form in 1917 [2]  

As long as there is at least uniaxial symmetry and the fiber axis is in the detector plane, the scattering pattern can be split into four quadrants which should carry each identical information. This means that there is some harmony in the scattering pattern, from which missing data can be reconstructed.  

After this reconstruction the scattering pattern should be smooth. If, on the other hand, seams are observed at the edges of the former invalid regions this shows that penumbra was not detected and the mask of the valid pixels was chosen too large. Solution erode the old mask and return to the start of pre-evaluation. 

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The new instrument Harmonie 2000 features additional advantages which are specifically linked to the digital architecture. 

The usual situation, true for the first three cases, is that in which the reactant and product solids are mutually insoluble. Langmuir [146] pointed out that such reactions undoubtedly occur at the linear interface between the two solid phases. The rate of reaction will thus be small when either solid phase is practically absent. Moreover, since both forward and reverse rates will depend on the amount of this common solid-solid interface, its extent cancels out at equilibrium, in harmony with the thermodynamic conclusion that for the reactions such as Eqs. VII-24 to VII-27 the equilibrium constant is given simply by the gas pressure and does not involve the amounts of the two solid phases. 

Fig. 9.24 A restraining potential that does not penalise struetures in which the distance lies between the leaver and upper distances di and and uses harmonie functions outside this range (left). The harmonic potentials may also he replaeed by linear restraints further from this region (right).

A slender vertieal filament of negligible mass supports a 0.200-g mass at one end and is fixed at the other end. A foree of 0.0800 N displaees the mass 0.0200 m. The mass exeeutes simple harmonie motion as the filament bends. What is the bending eonstant KB of the filament What is the frequeney v of the motion in Hz What is the period t of oseillation ... 

The Seetion entitled The BasiC ToolS Of Quantum Mechanics treats the fundamental postulates of quantum meehanies and several applieations to exaetly soluble model problems. These problems inelude the eonventional partiele-in-a-box (in one and more dimensions), rigid-rotor, harmonie oseillator, and one-eleetron hydrogenie atomie orbitals. The eoneept of the Bom-Oppenheimer separation of eleetronie and vibration-rotation motions is introdueed here. Moreover, the vibrational and rotational energies, states, and wavefunetions of diatomie, linear polyatomie and non-linear polyatomie moleeules are diseussed here at an introduetory level. This seetion also introduees the variational method and perturbation theory as tools that are used to deal with problems that ean not be solved exaetly. 

The harmonie oseillator energies and wavefunetions eomprise the simplest reasonable model for vibrational motion. Vibrations of a polyatomie moleeule are often eharaeterized in terms of individual bond-stretehing and angle-bending motions eaeh of whieh is, in turn, approximated harmonieally. This results in a total vibrational wavefunetion that is written as a produet of funetions one for eaeh of the vibrational eoordinates. 

Two of the most severe limitations of the harmonie oseillator model, the laek of anharmonieity (i.e., non-uniform energy level spaeings) and laek of bond dissoeiation, result from the quadratie nature of its potential. By introdueing model potentials that allow for proper bond dissoeiation (i.e., that do not inerease without bound as x=>°o), the major shorteomings of the harmonie oseillator pieture ean be overeome. The so-ealled Morse potential (see the figure below)... 

Here, Dg is the bond dissoeiation energy, rg is the equilibrium bond length, and a is a eonstant that eharaeterizes the steepness of the potential and determines the vibrational frequeneies. The advantage of using the Morse potential to improve upon harmonie-oseillator-level predietions is that its energy levels and wavefunetions are also known exaetly. The energies are given in terms of the parameters of the potential as follows ... 

Thus far, exaetly soluble model problems that represent one or more aspeets of an atom or moleeule s quantum-state strueture have been introdueed and solved. For example, eleetronie motion in polyenes was modeled by a partiele-in-a-box. The harmonie oseillator and rigid rotor were introdueed to model vibrational and rotational motion of a diatomie moleeule. 

Caleulate the expeetation value of the operator for the first two states of the harmonie oseillator. Use the v=0 and v=l harmonie oseillator wavefunetions given below... 

Lets use these ideas to solve some problems foeusing our attention on the harmonie... 

Assume that the vibrational motion of CO is purely harmonie and use the redueed mass p = 6.857 amu. 

Note that this identity enables you to utilize the orthonormality of the spherieal harmonies. 

The angular part of the atomie orbitals is deseribed in terms of the spherieal harmonies Yi m that is, eaeh atomie orbital (j) ean be expressed as... 

The eigenfunetions belonging to these energy levels are the spherieal harmonies Yl,m(0 (1>) whieh are normalized aeeording to... 

The simple harmonie motion of a diatomie moleeule was treated in Chapter 1, and will not be repeated here. Instead, emphasis is plaeed on polyatomie moleeules whose eleetronie energy s dependenee on the 3N Cartesian eoordinates of its N atoms ean be written (approximately) in terms of a Taylor series expansion about a stable loeal minimum. We therefore assume that the moleeule of interest exists in an eleetronie state for whieh the geometry being eonsidered is stable (i.e., not subjeet to spontaneous geometrieal distortion). 

The Harmonie Vibrational Energies and Normal Mode Eigenveetors... 

Within this harmonie treatment of vibrational motion, the total vibrational energy of the moleeule is given as... 

The hi, b 1 and a2 bloeks are formed in a similar manner. The eigenvalues of eaeh of these bloeks provide the squares of the harmonie vibrational frequeneies, the eigenveetors provide the normal mode displaeements as linear eombinations of the symmetry adapted... 

The eleetronie energy of a moleeule, ion, or radieal at geometries near a stable strueture ean be expanded in a Taylor series in powers of displaeement eoordinates as was done in the preeeding seetion of this Chapter. This expansion leads to a pieture of uneoupled harmonie vibrational energy levels... 

At larger bond lengths, the true potential is "softer" than the harmonie potential, and eventually reaehes its asymptote whieh lies at the dissoeiation energy Dg above its minimum. This negative deviation of the true V(R) from 1/2 k(R-Rg)2 eauses the true vibrational energy levels to lie below the harmonie predietions. 

The first term is the harmonie expression. The next is termed the first anharmonieity it (usually) produees a negative eontribution to E(vj) that varies as (vj + 1/2)2. spaeings... 

A plot of the spaeing between neighboring energy levels versus vj should be linear for values of vj where the harmonie and first overtone terms dominate. The slope of sueh a plot is expeeted to be -2h(cox)j and the small -vj intereept should be h[cOj - 2(cox)j]. Sueh a plot of experimental data, whieh elearly ean be used to determine the coj and (cox)j parameter of the vibrational mode of study, is shown in the figure below. 

Explain how the eonelusion is "obvious", how for J = 0, k = k, and A = 0, we obtain the usual harmonie oseillator energy levels. Deseribe how the energy levels would be expeeted to vary as J inereases from zero and explain how these ehanges arise from ehanges in k and... 

B. Seleetion Rules on the Vibrational Quantum Number in the Harmonie Approximation... 

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