What Feature of an Orbital Is Most Directly Related to the Principal Quantum Number (N)?

Chemic Foundations of Physiology I

Joseph Feher , in Quantitative Human Physiology (Second Edition), 2017

Atomic Orbitals Explain the Periodicity of Chemical Reactivities

At that place are eight main "shells," referring to the principal breakthrough number, n=(i,2,3,4,five,6,seven,eight) that describes atomic orbitals. There are four major subshells: southward, p, d, and f, whose names derive from spectroscopic descriptions of sharp, principal, diffuse, and primal. These orbitals are described past the azimuthal quantum number, l=(0,ane,ii,3) for (s,p,d,f), respectively. Each subshell has a structure and a chapters for electrons that is described by the magnetic quantum number, m, and the spin quantum number, s. The southward subshell is spherically symmetrical and holds only ii electrons; each set of p orbitals holds 6 electrons, the d orbitals concord x, and the f orbitals hold 14. The sequential filling of these orbitals accounts for the periodic chemic beliefs of the elements with their atomic number. This order of filling is shown in Effigy 1.4.ii. Each subshell (s, p, d, f) is typically filled with the requisite number of electrons before filling the remaining subshells. Each electron has a spin breakthrough number, south, that is represented every bit "up" or "down." The orbitals in the subshells are typically filled singly with electrons of parallel spin before double occupancy begins. This is the so-called "bus seat rule," coordinating to the filling of a charabanc where double seats tend to fill with single individuals before double occupancy occurs.

Figure one.4.two. Lodge of filling of atomic orbitals. Electronic orbits are characterized by a master quantum number that determines the chief vanquish, an azimuthal breakthrough number that determines the subshell, a magnetic quantum number that determines the orbital, and the spin quantum number that determines the spin of the electron. In that location are four subshells: s, p, d, and f. These have 1, three, v, and seven orbitals that each can hold upward to 2 electrons of reverse spin. The club of filling with increasing number of electrons follows the blue diagonal arrows in the diagram: 1s fills starting time, followed by 2s and 2p; next is 3s followed by 3p and 4s, followed past 3d, 4p, and 5s; next is 4d, 5p, and 6s; then 4f, 5d, 6p, and 7s.

Full orbitals are inherently stable, because they have low energy, and atoms having total orbitals are chemically unreactive. These stand for to the noble gases, helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). The electronic structure of some of these stable atoms is shown in Figure 1.4.3. All of the other elements tin react with other atoms, in social club to become more than stable by attempting to make full their orbitals. They practice this past sharing electrons, a process that constitutes chemic bonding. This sharing tin can be equal or very unequal, corresponding to the extremes of covalent bonding and ionic bonding.

Effigy 1.four.3. Electronic structure of the inert gases. These inert gases are chemically unreactive because their orbitals are already filled. Helium, with n=two protons in its nucleus, fills the 1s orbital with 2 electrons of opposite spin. Spin is indicated in the drawing by an arrow pointed upward or downwardly. Neon (n=x) fills the 2s and 2p orbitals with a total of 8 electrons. Each orbital in the subshells carries at well-nigh 2 electrons. The gild of filling of the orbitals corresponds to that shown in Figure 1.four.two.

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Atoms and Atomic Arrangements

R.E. Smallman , A.H.West. Ngan , in Mod Physical Metallurgy (Eighth Edition), 2014

one.one.2 Nomenclature for the electronic states

The free energy of an electron is mainly determined by the values of the chief and orbital quantum numbers. The principal quantum number is simply expressed past giving that number, but the orbital quantum number is denoted by a letter. These messages, which are derived from the early days of spectroscopy, are south, p, d and f, which signify that the orbital quantum numbers l are 0, 1, ii and 3, respectively. ^ane When the principal quantum number n=1, l must be equal to nada, and an electron in this state would be designated by the symbol 1s. Such a country can simply have a unmarried value of the inner quantum number m=0 merely can accept values of $+\frac{1}{2}$ or $-\frac{1}{2}$ for the spin quantum number due south. It follows, therefore, that there are just 2 electrons in any one atom which can be in a 1s-state, and that these electrons will spin in contrary directions. Thus when northward=one, only s-states can exist and these can be occupied past only two electrons. Once the two anes-states take been filled, the adjacent lowest free energy country must take north=two. Hither l may have the value 0 or i, and therefore electrons can exist in either a 2s- or a 2p-country. The energy of an electron in the iisouth-state is lower than in a 2p-country, and hence the 2s-states will be filled starting time. Once again there are merely ii electrons in the iidue south-state, and indeed this is always truthful of s-states, irrespective of the value of the principal quantum number. The electrons in the p-state tin accept values of chiliad=+1, 0, −ane, and electrons having each of these values for m can accept two values of the spin quantum number, leading therefore to the possibility of half-dozen electrons being in any ane p-state. These relationships are shown more than conspicuously in Table 1.one, and Figure ane.1 shows the shapes of the s, p and d orbitals.

Table i.1. Allocation of States in the First 3 Quantum Shells

Shell	north	l	m	south	Number of States	Maximum Number of Electrons in Shell
K	1	0	0	± $\frac{1}{2}$	Two 1s-states	2
		0	0	± $\frac{1}{2}$	Two 2southward-states
L			+1	± $\frac{\mathrm{one}}{2}$		8
	2	1	0	± $\frac{\mathrm{i}}{2}$	Six 2p-states
			−1	± $\frac{\mathrm{i}}{2}$
		0	0	± $\frac{1}{2}$	Two 3southward-states
Chiliad			+1	± $\frac{1}{2}$
		1	0	± $\frac{\mathrm{one}}{2}$	Six threep-states
			−1	± $\frac{\mathrm{ane}}{2}$
	three					eighteen
			+2	± $\frac{\mathrm{ane}}{2}$
			+ane	± $\frac{1}{2}$
		two	0	± $\frac{1}{\mathrm{two}}$	Ten 3d-states
			−one	± $\frac{1}{2}$
			−2	± $\frac{1}{2}$

No farther electrons can exist added to the state for n=2 afterwards two 2s- and six iip-state are filled, and the side by side electron must go into the country for which n=3, which is at a higher energy. Here the possibility arises for fifty to accept the values 0, 1 and 2 and hence, as well south- and p-states, d-states for which l=2 tin can now occur. When l=two, m may have the values +2, +1, 0, −1, −two, and each may be occupied by two electrons of opposite spin, leading to a total of ten d-states. Finally, when n=4, l volition have the possible values from 0 to 4, and when l=four the reader may verify that in that location are 14 fourf-states.

Table 1.one shows that the maximum number of electrons in a given shell is 2n ². Information technology is an accustomed practice to retain an earlier spectroscopic notation and to label the states for which due north=1, ii, iii, 4, 5, 6 as Grand-, 50-, M- N-, O- and P-shells, respectively.

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Benchmark Databases of Intermolecular Interaction Energies: Design, Construction, and Significance

Konrad Patkowski , in Annual Reports in Computational Chemistry, 2017

two.1.one CBS Extrapolation

In view of the analytical results that depict the convergence of atomic correlation energies with the maximum angular momentum and the maximum principal breakthrough number nowadays in the footing set, it has been commonly causeless that molecular correlation energies, and the correlation parts of noncovalent interaction energies, follow the resulting Ten ⁻³ convergence design in the cc-pVXZ and aug-cc-pV10Z sequences (60, 61) . If that assumption is exactly truthful, the CBS-limit correlation energy ${\mathrm{Eastward}}_{\infty }^{\text{corr}}$ can exist computed from the results in two consecutive-Ten footing sets E _X−ane ^corr, E _X ^corr as

(4) $\begin{array}{l}\hfill E_{\infty }^{\text{corr}}={\mathrm{East}}_{\mathrm{10}}^{\text{corr}}+\frac{{(1-1/X)}^{\mathrm{due\; north}}}{1-{(1-\mathrm{one}/X)}^n}\left(E_{\mathrm{10}}^{\text{corr}}-E_{X-1}^{\text{corr}}\right)\end{array}$

with n = 3. The utilise of the X ⁻ⁱⁱⁱ extrapolation defined by Eq. (4), popularized by Helgaker and coworkers (60, 61) , has go a routine pace in obtaining near-CBS interaction energies. It should be stressed that the X ⁻³ formula stems from the properties of correlation energy and should be practical to the correlation contribution just. The remaining Hartree–Fock part of interaction energy E _HF ^int converges much faster with the basis set, and this term can be either taken from the largest-basis calculation without extrapolation, or extrapolated from the results in three consecutive bases East _X−2 ^HF, East _X−one ^HF, E _Ten ^HF assuming the exponential formula

(5) $\begin{array}{l}\hfill E_{\infty }^{\text{HF}}={\mathrm{East}}_{\mathrm{Ten}}^{\text{HF}}+Aexp(-B\mathrm{Ten})\end{array}$

with some constants A, B.

While the Ten ⁻ⁱⁱⁱ extrapolation formula of correlation energy has been the about popular by far, information technology is not the only one possible. In particular, various three-parameter extrapolations accept been proposed to leverage the results computed in three consecutive bases E _X−2 ^corr, E _10−one ^corr, Eastward _X ^corr. Martin (fourscore) generalized the X ^−three formula to business relationship for an offset between the maximum angular momentum and the basis fix central number:

(half-dozen) $\begin{array}{l}\hfill {\mathrm{Eastward}}_{\infty }^{\text{corr}}=E_X^{\text{corr}}+\frac{A}{{(X+B)}^3}.\end{array}$

Klopper (81) carried out reference explicitly correlated CCSD-R12 calculations (82) for seven small molecules and, looking at the convergence of conventional CCSD correlation energies in the cc-pV10Z basis sets, proposed an extrapolation formula that takes into account the different (X ^−three and X ⁻⁵, respectively) convergence of the singlet and triplet pair energies. Truhlar (83) set up out to optimize the exponent n in the X ⁻ⁿ extrapolation (Eq. iv) in the specific, nearly economical instance of the cc-pVDZ and cc-pVTZ bases, using footing set limit energies for iii systems. He found the optimal exponents to exist two.2 for the MP2 correlation energy and 2.four for the CCSD and CCSD(T) correlation energies, indicating quite a bit slower convergence than that implied by the X ⁻³ formula. Schwenke (84) noticed that the determination of an optimal exponent for the X ⁻ⁿ extrapolation is equivalent (as evident by the course of Eq. (4)) to the decision of a single linear parameter F _X in

(7) $\begin{array}{l}\hfill E_{\infty }=E_X+F_{\mathrm{Ten}}\left({\mathrm{Due\; east}}_X-E_{X-\mathrm{one}}\right),\end{array}$

where E can be whatever component of correlation free energy (singlet CCSD pairs, triplet CCSD pairs, total CCSD, or (T)). Schwenke went on to compute the optimal F _X values based on Klopper'due south benchmark CCSD-R12 energies (81) and his ain estimates of the CBS limit of the (T) correction, improving the accuracy of extrapolated energies relative to the (Ten ⁻³/X ⁻⁵ for singlet/triplet pairs) extrapolation for CCSD and the X ^−three one for (T).

The full general conclusion from the investigations on the optimal CBS extrapolations is that the conventional X ^−three formula, while not quite the optimal selection for all systems at all theory levels, is never a bad choice for whatsoever correlated interaction free energy contribution. At the bare minimum, if the Due east _(X−i,X) extrapolated value and the |E _(X−1,Ten) − E _X | difference are taken, respectively, as the best estimate of ${\mathrm{Eastward}}_{\infty }$ and its dubiety, such an error gauge is very conservative: unless the convergence of East _X is not monotonic (in which case no extrapolation would work), the true value of ${\mathrm{Eastward}}_{\infty }$ is practically guaranteed to be in this conviction interval even if the exponent n = 3 is far from optimal. More generally, the best strategy for improving the accuracy of CBS extrapolations is not a reoptimization of the extrapolation form, merely rather reducing the errors in the computed values (the input data for extrapolation) by any of the techniques discussed in this department. In the specific cases when extraordinarily accurate interaction energies are required, several CBS extrapolations utilizing different basis set families and/or different extrapolation forms might exist combined. Such an approach was employed in the conclusion of the helium pair potential to millikelvin accurateness by Jeziorska and coworkers (85–87) using, in addition to the X ⁻³ expression, several extrapolations where the footing set up convergence of Eastward _X was causeless to follow the aforementioned blueprint every bit the basis prepare convergence of some easy to compute contribution $E_X^{\text{easy}}$ , for example, the atomic MP2 correlation energy.

The correlation energy extrapolation from double- and triple-zeta footing sets warrants additional discussion. In full general, no extrapolation formula can work if the convergence is not monotonic. While molecular correlation energies tend to converge to the CBS limit monotonically from above (61) , the same is not necessarily truthful for the δ _MP2 ^CCSD(T) correction within the composite approach, Eq. (three). In fact, Sherrill and coworkers (88) found that the values of the δ _MP2 ^CCSD(T) interaction free energy term computed in the aug-cc-pVXZ sequence oftentimes have a turning point at X =T or fifty-fifty Q. As a result, the (aug-cc-pVDZ,aug-cc-pVTZ) extrapolation of this term moves the aug-cc-pVTZ outcome in the wrong direction. Therefore, information technology is recommended (88) to utilize nonextrapolated δ _MP2 ^CCSD(T)/aug-cc-pVTZ values instead of their (aug-cc-pVDZ,aug-cc-pVTZ) extrapolated counterparts; in a sense, the double-zeta basis is so unconverged that its admixture to the triple-zeta results does more harm than good.

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Materials and Processes for Side by side Generation Lithography

D. Frank Ogletree , in Frontiers of Nanoscience, 2016

2.2.1 Diminutive Photoemission

The electronic structure of an isolated cantlet can exist described as a series of 1-electron orbitals or free energy levels, defined by the principal quantum number n (=1,two,three…) and the orbital angular momentum l (s  = 0, p  = 1, d  = 2, f  = 3), each with its own binding energy (Be). The bounden energies of the occupied states are directly measured through photoemission experiments on solid materials ¹² where BE _Fermi   + ϕ  = hν  − KE _{photoelectron} represents energy conservation—the Exist plus the sample work function (ϕ) is the difference betwixt the incoming photon energy (hν) and the approachable electron KE. For solids the Be is referred to the Fermi level, while for atomic and molecular systems ¹⁷ it is more useful to refer to the vacuum level, Be _vacuum   = hν  − KE _{photoelectron} .

The "cross section" σ, with units of area, is way of describing the probability of molecular, atomic, and nuclear interactions with radiation. In addition to the 10-ray absorption cross section, the inelastic and elastic electron-molecule scattering cross sections are important for EUV radiations chemistry, as are the molecular ionization and fragmentation cross sections for both 10-rays and electrons. If the cross department is small compared to the concrete expanse of the target, the interaction probability is minor. If the number-density of targets is given past n, and then the interaction hateful free path λ  =   1/nσ and the probability of photon absorption in a movie of thickness t is 1   − e ^−nσt . In the nuclear and atomic physics literature, cross sections are sometimes expressed in units of "barns" or 10⁻²⁸  mⁱⁱ, the typical scattering cross section of an atomic nucleus, so 100   Mb, or megabarn, corresponds to an area of one   Ų (10^−twenty  m²).

Each orbital has its own energy-dependent X-ray absorption cross section σ _northward,l (hν,Z), sometimes chosen a "subshell cross department," where Z is the diminutive number. Subshell cantankerous sections subtract rapidly with increasing photoelectron KE, although the energy dependence can be more complex close to threshold. ¹⁸ The cross section also depends on the occupation of the orbital, so a filled d-level with 10 electrons volition usually accept a higher cross section than a filled southward-level with merely two electrons for like Exist. Calculated subshell cantankerous sections for all the elements are tabulated by Yeh and Lindau. ¹⁹

The photoemission cross section is proportional to a dipole matrix chemical element between the moving ridge functions of the initial bound electron orbital and the final outgoing gratuitous electron wave. This matrix element involves an integral over the overlap of the ii wave functions. ^eighteen The phase of the outgoing wave oscillates with a wavelength proportional to KE^i/2. As the wavelength becomes smaller than the radial size of the bound electron orbital, the positive and negative phase contributions to the overlap integral tend to cancel out, resulting in a rapid decrease of the orbital cross section with energy. The decrease in cantankerous department is faster for weakly leap valence orbitals than for core orbitals since their radial wave functions are more extended. For large KE, the cantankerous section will drop off ^xviii as KE⁻ⁱⁱⁱ.

The radial wavefuction of a bound orbital has a radial node, where the phase of the orbital changes sign, when the orbital athwart momentum l is greater that the principal quantum number northward. This phase change likewise affects the matrix element overlap integral, and causes a then-called Cooper minimum in the subshell cross department around 10–xx   eV above threshold. ^xx Cooper minima occur for 3p, 4d, and 5f orbitals, equally shown by Yeh and Lindau. ^nineteen

Cross sections for high angular momentum d and f orbitals tin can show a "delayed maxima." ²⁰ Unlike s and p levels, where the maximum cross department is very close to the threshold photon energy, the maximum for higher angular momentum states can occur 30 or xl   eV above threshold. This is due to the so-called "centrifugal repulsion term" in the constructive radial potential for a high angular momentum electron moving ridge. ²⁰ This effect can be important at EUV energies, as seen in Fig. ii, where many of the 4d elements take large cross sections.

Figure 2. Calculated atomic photoabsorption cross sections at 92   eV for selected elements ⁶⁵ in Mb (megabarn). Centre for X-ray optics "X-ray interactions with matter" Website, http://www.cxro.lbl.gov.

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Tribochemistry of Lubricating Oils

In Tribology and Interface Engineering Serial, 2003

Nomenclature.

The spectroscopic nomenclature is directly equivalent to that used for spectroscopy X-ray, and is related to the various quantum numbers such every bit the principal breakthrough number northward, the electronic quantum number ane, the full angular momentum quantum number j, and the spin quantum number s, which tin accept either of the values ±½. Where: north = 1, two, 3, iv, …, are designated K, L, M, Northward, … respectively; 1 = 0, 1, two, 3, … and j = i + south (can accept the values of ½, iii/2, 5/2, 7/2), … are given conventional suffixes, one, two, 3, 4, … co-ordinate to the listing in Table iv.8 (Briggs and Seah, 1990). This description of the summation is known as j-j coupling.

Table 4.viii. X-ray and spectroscopic notation

Quantum numbers			Ten-ray suffix	X-ray level	Spectroscopic level
northward	i	j
1	0	½	1	Thousand	1s1/ii
ii	0	½	i	Li	2s1/2
2	1	½	2	502	2p1/2
2	1	iii/2	three	Lthree	2p3/2
3	0	½	1	Mane	3s1/2
3	1	½	2	M2	3p1/ii
3	1	iii/2	3	Miii	3p3/2
3	2	three/2	4	Mfour	3d3/2
iii	ii	5/2	5	Grand5	3d5/2
	etc.		etc.	etc.	etc.

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Self-Assembly Processes at Interfaces

Adam Westward , in Interface Scientific discipline and Engineering science, 2018

three.2.v.2 Auger Electron Spectroscopy and Free energy Dispersive X-Ray Assay

Equally in XPS a monochromatic beam of Ten-rays allows for the ejection of core vanquish electron from an atom, say, from the K level (principal quantum number n  =   1). An electron from a higher orbital, say, from the L level (principal quantum number due north  =   2), will autumn to the vacant energy level. This electronic transition is accompanied either past emission of a photon having an free energy corresponding to the energy deviation between the L and K levels or past transfer of the excess energy to another electron of the 50 level. This electron is then ejected from the atom, and its free energy is measured in Auger electron spectroscopy (AES). AES has been the beginning developed surface-sensitive assay method in the beginning of the 1960s. The energy of the emitted photons can too be measured in energy dispersive X-ray (EDX) assay. Both processes occur simultaneously and their efficiency depends on the atomic number of the atoms present on the surface of the investigated material. AES affords more than sensitivity for elements with a low atomic number, whereas EDX is more advisable for heavier elements.

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Rufus Ritchie, A Admirer and A Scholar

Károly Tőkési , in Advances in Quantum Chemical science, 2019

two.one.3 Classical quantum numbers

In the CTMC calculations, the energy level E of an electron after the excitation is determined just past calculating its binding energy U   =   −   E . A classical chief quantum number is assigned according to

(18) ${\mathrm{northward}}_c=Z_T{Z}_e{\left(\frac{\mu T_e}{2U}\right)}^{\mathrm{ane}/\mathrm{ii}}.$

The classical values of n _c are "quantized" to a specific level n ⁴² if they satisfy the relation:

(19) ${\left[\left(n-\mathrm{one}\right)\left(n-1/2\right)n\right]}^{1/3}\le n_c\le {\left[n\left(n+1/2\right)\left(n+\mathrm{i}\right)\right]}^{\mathrm{i}/3}.$

The classical orbital athwart momentum is defined by

(twenty) ${\mathrm{fifty}}_c=\sqrt{m_e\left[{\left(x\dot{y}-y\dot{x}\right)}^2+{\left(x\dot{z}-z\dot{x}\right)}^{\mathrm{two}}+{\left(y\dot{z}-z\dot{y}\right)}^2\right]},$

where ten, y, z are the Cartesian coordinates of the electron relative to the nucleus. Since 50 _c is uniformly distributed for a given n level, ⁴² the quantal statistical weights are reproduced by choosing bin sizes such that

(21) $l\le \frac{n}{{\mathrm{northward}}_c}{l}_c\le l+\mathrm{i},$

where 50 is the breakthrough-mechanical orbital-angular-momentum.

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Object appearance and colour

Asim Kumar Roy Choudhury , in Principles of Colour and Appearance Measurement, 2014

Conjugated bonds

Organic dyes occur widely in the plant and brute kingdoms besides as in the modernistic constructed dye and pigment industry. Organic dyes and pigments, whether natural or synthetic, are very intense in color. The colour is so intense that a small quantity of such material is capable of colouration of large quantity of various substances such as textiles, paper, leather, etc. Just as with ligand field energy levels, some of the captivated energy may be re-emitted in the form of fluorescence. Graebe and Liebermann in 1867 undertook the start study of the relationship between color and chemical structure. They found that reduction of some known dyes destroyed the color instantly. They concluded that the dyes are chemically unsaturated compounds.

Most organic compounds are complex unsaturated compounds having certain substituent groups. Information technology was from a study of compounds such as azobenzene and p-benzoquinone that O. N. Witt was led in 1876 to formulate his celebrated theory. Witt proposed that a dye contains a colour-producing chromogen, which is equanimous of a basic chromophore or color-bearing group, to which can be attached a variety of subsidiary groups chosen auxochrome or colour intensifier, which atomic number 82 to the production of colour. Chromophores include carbon–carbon double bond especially conjugated systems having alternate unmarried and double bond. Witt also claimed that the auxochromes confer dyeing properties on the molecule, but it is now established that color and dyeing properties are non directly related. All the same, Witt'south theory, in general, is yet acceptable to colourists (Giles, 1974).

Iii years later, Nietzki stated that increasing the molecular weight of a dye by the introduction of substituents, such as methyl, ethyl, phenyl, ethoxy or bromo, produced a bathochromic shift (i.e. shift of assimilation acme towards longer wavelength). Though Nietzki's rule initially proved useful, its utility decreased as many exceptions were after discovered.

Armstrong proposed the quinoid theory in 1887, stating that simply the compounds which tin can be written in a quinoid form are coloured. The theory presently proved to be erratic. Gomberg first discovered in 1900 the coloured free radical triphenylmethane, which is devoid of keto, azo chromophores and auxochrome. On the basis of the new chromophore, Baeyer proposed the theory of halochromy whereby a colourless compound is rendered coloured on salt formation. Halochromism is still used to denote a colour change of a dye on the addition of acid or alkali. Baeyer further proposed in 1907 the possibility of tautomerism to account for the colour of dyes. For example, in Doebner's violet there is a rapid oscillation betwixt the two tautomeric forms (see Fig. 2.11a and ii.11b), the chlorine atom flopping speedily from one amino group to the other.

2.11. The 2 tautomeric forms of Doebner'southward violet.

Hewitt and Mitchell in 1907 first realised the importance of conjugation, i.e. the presence of alternating single and double bonds. From a study of azo dyes, they established Hewitt'southward dominion stating that the longer the conjugated chain, the more bathochromic shift volition exist in the colour of the dye. Dithey and Wizinger in 1928 refined Witt's theory and proposed that a dye consists of an electron-releasing bones group, the auxochrome connected to an electron-withdrawing acidic group by a system of conjugated double bonds. The greater the nucleophilic and electrophilic graphic symbol respectively of the two groups and/or the longer the unsaturated chain joining them, the greater is the resulting bathochromic shift.

When light absorption takes place in the visible range, the compound attains a color complementary to the light captivated, or more specifically to the wavelength of maximum absorption (λ _max). The relation between the colour absorbed and color perceived are shown in Tabular array 2.3. Certain colours require more than than ane absorption band – light-green requires absorption of red and blue-violet. This is difficult to achieve, and the number of greenish dyes are comparatively less. Black requires a combination of several wide overlapping bands of similar extinction coefficients. The brown, olive green and other irksome colours too crave bands covering the whole visible spectrum, but of different extinction coefficients (McLaren, 1983).

Table 2.3. Colour absorbed and colour perceived

Wavelength of absorption (nm)	Colour absorbed	Color perceived
400–500	Blue	Yellowish
400–440	Violet	Green-yellow
460–500	Dark-green bluish	Orange
400–620	Blueish dark-green	Red
480–520	Green	Magenta
560–700	Orange	Cyan
600–700	Blood-red	Bluish green

The early dye-chemist regarded the colour changes (by introduction of auxochromes) from yellow through green to red every bit deepening of colour; the shift was, therefore, termed every bit bathochromic and the alter in reverse direction as hypsochromic. In spectroscopy, these terms presently mean ruby shift and blueish shift respectively.

The organic colourants may be broadly classified into three groups (Nassau, 1983). Benzenoids are the most important grouping of synthetic colourants. This includes diverse aromatic compounds – in cloth the near important chromophores are azobenzene, triphenylmethane and anthraquinone (Figs 2.12, 2.thirteen and 2.14 respectively). Azo groups (−   N = Northward–), which are not available in nature, when incorporated in benzenoids, form the basis of the majority of synthetic dyes. Benzenoids are as well occasionally accompanied by thio (> C = S), nitroso (−   Northward = O) and many other groups.

2.12. Azobenzene.

2.xiii. Triphenylmethane.

2.xiv. Anthraquinone

Polyenes consist of not-benzenoid long conjugated double-bond systems, which are the footing for many biological colourants. When such a conjugated system is large enough, it can absorb visible light and become coloured. Carotenoids are typical not-circadian natural colourants. An important member of this group, β-carotene (Fig. 2.15), is the orangish colourant in carrots and many other vegetables. This is used for colouration of cosmetics and food products. When information technology is split in one-half, a portion having structure similar to Fig. 2.16 is vitamin A₁. A like carotenoid, crocein, which is the principal colouring component of the natural colourant xanthous dye saffron, is used for food colouring. Rhodopsin, the visual paint of eye, is like to β-carotene and vitamin A₁. Carotenoids are responsible for various colours in bird feathers.

2.15. β-Carotene.

two.sixteen. A half of β-carotine resembling Vitamin A1.

Circadian polyene non-benzenoid conjugated systems include porphyrins; the most of import members belonging to this class are α-chlorophyll (Fig. 2.17) and like heme. While the light-green-coloured chlorophyll is responsible photosynthesis of plants, cerise-coloured heme transports oxygen in claret. Both circadian 18-member conjugated systems have a key metallic ion, Mg^{2   +} in the erstwhile and Fe^{3   +} in the latter. Synthetic pigment bluish and greenish phthalocyanines are also cyclic polyenes like porphyrins, merely they take additionally benzenoid groups, e.one thousand. copper phthalocyanine (Fig. 2.18).

2.17. α-Chlorophyll.

ii.18. Copper phthalocyanine.

Auxochromes are electron donor or acceptor substituent groups, which shift light absorption inside the visible range. Typical electron donors are:

Chief, secondary and tertiary amines, alkoxide, hydroxide and acetate groups.

Typical electron acceptors are:

Nitrate, cyanide, alkyl sulphite, carboxylate, nitrite and carbonate.

As various limitations of Witt'south approach came to light, the resonance theory was put forward. Adam and Rosenstein in 1914 first proposed that it is the oscillation of electrons, and non the oscillation of atoms, which produces color. Atomic vibrations give rise to the assimilation of infrared radiation, whereas the oscillation of electrons causes the absorption of ultraviolet or visible radiation resulting in colour awareness.

Bury in 1935 highlighted the human relationship between resonance and the colour of a dye. He realised that it is only the electrons that movement and not the atoms. The intense assimilation of light, which characterises dyes, is due to resonance in the molecule. The resonance is enhanced past the auxochrome. The greater the number of limiting structures of similar free energy, the more than will be bathochromic shift in the dye.

It was proposed that π-bonding electrons involved in the 2nd bond of the double bonds are not localised but belong to the whole conjugated system of alternate single and double bonds. One or more mobile π-electrons of the system tin move through the molecule. It is, therefore, possible to write diverse electronic configurations of the molecule, called canonical forms or resonance hybrids. This does not imply whatsoever actual vibration or oscillation amongst these forms, but but signifies that the structure is an intermediate ane. When a donor-auxochrome is introduced in such a molecule, additional electrons are pumped into the conjugated system, while an acceptor-auxochrome pumps electron out from the system. Consequently, the structure becomes stable. The electrons tin move more readily along the molecule. The natural frequency of vibration is decreased, resulting in assimilation at longer wavelengths. In other words, the absorption range moves from ultraviolet to visible light, consequently simultaneous hyperchromic and bathochromic shift.

Azobenzene may exist in five resonance forms (ii.xix(a)–two.xix(eastward)), and the uncharged form (ii.xix(c)) is the most stable. When two auxochromic groups are attached to the azobenzene, the configuration (two.19(grand)) is more stable than structures (2.19(a)) to (two.19(east)), as the charges are now firmly held on oxygen or nitrogen atoms. The chemical compound (2.nineteen(f)) is, therefore, of more intense colour than azobenzene.

2.19. Different resonance structures of azobenzene (a-e) and with auxochromes (f and m)

With the increase in stability of the alternating structures, its electrons may exist considered to move more than readily along the chromophore. The natural frequency of its vibration is decreased. Consequently, the absorption occurs at longer wavelength. This is analogous to a violin cord in which the longer the cord, the lower is the frequency; hence, the longer is the wavelength of the note it emits when plucked. This applies to adsorption as well every bit emission of energy, because any oscillator absorbs energy about readily at the wavelength of its natural frequency of vibration.

Every bit the resonance theory cannot provide a completely satisfactory business relationship of colour generation in organic molecules, the molecular orbital theory has been proposed. The electrons exist in various layers called shells (denoted equally principal quantum number, n = 1, 2, three or whatsoever integer) around the atomic nucleus. The shells are further divided into various orbits. The number of orbits in a shell is decided by 3 factors:

1.

Angular momentum quantum number, 50 = n – 1. Each fifty value represents a specific orbit named subsequently the description of the hydrogen spectrum such south for sharp (fifty = 0), p for main (l = 1), d for diffuse (50 = two), f for central (l = 3) etc.

2.

Magnetic quantum number, thousand = +   l, +   fifty–1, +   50–2, … 0, i, two, ..l

iii.

Spin quantum number, +½, –½

The detailed orbital designation and the number of electrons in each orbit upwards to the 4th trounce are listed in Table 2.4. When ii atoms are close to each other, the respective atomic orbital forms various MOs by overlap interactions. The molecular orbital tin accommodate exactly the aforementioned number of electrons as the diminutive orbital from which they are formed. The germination of various molecular orbital by linear and non-linear combinations of the interacting atomic orbital has been studied by diverse enquiry workers. Molecular orbital techniques without any approximation are possible at present. Simply the necessity of very complex and high levels of computation restricts its application to large dye molecules. However, in a simplified approach, the molecular orbital can be classified into low energy bonding orbital, intermediate energy non-bonding orbital and high energy antibonding orbital. When ii diminutive orbitals collaborate strongly past direct overlap and there is symmetry with respect to rotation near the axis joining the 2 atoms, a sigma (σ) orbital results in a σ bond. The sigma orbital will have less energy than the individual atomic orbital from which it is formed, because some energy is utilised for bonding. Past absorption of free energy, transition may occur from σ-bonding to σ* antibonding country (σ → σ*). When overlapping occurs only for the outer region of less electron density, the energy of the bail is less than that in an σ bond, and a π orbital resulting in π bonds existence formed. A σ bond possesses zip athwart momentum around the bond axis, whereas a π orbital possesses one unit of measurement of angular momentum. Again, π-bonds tin can be excited into a loftier free energy π* antibonding state (π → π*). In that location may also be a transition from non-bonding to antibonding state (n → σ* or northward → π*). In a molecule, there may exist several of each type of orbital of varying energy levels formed past interaction of diverse pairs of atoms. For a molecule, the smallest amount of free energy absorbed is the free energy required for transition from the highest occupied molecular orbital (Man) to the everyman occupied molecular orbital (LOMO). The most of import transitions in respect of minimum absorption resulting generation of colour are (n → π*) and (π→ π*).

Table 2.4. Orbital designation and number of electrons in diverse orbits

Beat out No.	No. of Orbital	Orbital Designation	No. of Electrons
1	1	1   s	ii
2	2	2   due south	2
		2p	6
3	3	3   south	ii
		3p	6
4		3d	10
	4	4   due south	2
4d		4p	six
4f	10

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State- and Belongings-Specific Quantum Chemistry

Cleanthes A. Nicolaides , in Advances in Quantum Chemical science, 2011

four.5.ii Regular Ladders of Doubly, Triply, and Quadruply Excited States Tending to Fragmentation Thresholds where the Electronic Geometry is Symmetric

Information technology is clear that, assuming the model of orbital configurations, one can construct, formally, an infinity of MES for each atom or molecule, at least equally a superposition of such configurations for each symmetry. For example, the 2-electron ionization threshold (TEIT) of He is at 79.0 eV. No bound or quasi-leap DES exist above this free energy. However, below this TEIT, the spectrum contains one-electron continua for each hydrogenic threshold, n = 1, 2, …, and a very large number of unstable quasi-leap DES (and a few stable ones due to nonrelativistic symmetry restrictions) of many symmetries for even and odd parity. Evidently, for the photoabsorption process in the nonrelativistic context, only the DES of ⁱ P^o symmetry is excited. Assuming the basis of orbital configurations, these tin exist labeled, at to the lowest degree formally, by superposition of configurations such equally 2southward2p, (twos3p, 2pthrees, twop3d), (4s4p, 4p4d, 4d4f), etc.

One important question is whether one tin identify classes of such states exhibiting some blazon of regularity as a function of excitation energy, but like the Rydberg levels do as they reach the ionization threshold. Indeed, for MES without a circuitous electronic core, that is, for DES in H⁻, He, Li⁺, and Li⁻, for triply excited states in He⁻and Li, and for quadruply excited states in Exist, such regularities have been identified in terms of a Rydberg-like energy formula and in terms of the symmetric geometries of the electrons as they reach the fragmentation thresholds—Ref. [58] and references below.

On the other hand, recent piece of work on the double-electron excitation from the 2southward ² subshell of Neon to newly established DES embedded in i- and two-electron continua, although they obey an effective Rydberg-like energy formula, testify no tendency toward a symmetric geometry of fragmentation due to valence-core electron interactions [59].

These calculations and findings were made possible in the framework of the analyses that were published in the 1980s, for instance, Refs. [60a, 60b, 61, 62]. Accordingly, the zero-order wavefunction was chosen as the directly, land-specific MCHF solution with merely the intrashell configurations for each hydrogenic shell, having the lowest energy. These multiconfigurational SCF wavefunctions account for most of angular correlation and office of radial correlation. This is sufficient for the quantitative agreement of the properties of interest, such as geometric arrangements of the correlated electrons or absorption oscillator strengths.

The first domain of investigations was that of classes of DES of H⁻, He, Li⁺, and Li⁻. The crucial outcome was how to cull and compute the naught-order multiconfigurational (Fermi-ocean) wavefunctions, and so as to be able to recognize without ambiguity possible regular series that tend to the Wannier land at E = 0, with 〈r ₁〉 = 〈r _ii〉 and ϑ = 180°. Up to n = ten, intrashell states were computed and analyzed from starting time principles [60a, 60b]. When needed, additional radial and angular electron correlations were calculated variationally. In this way, accurate energies and oscillator strengths to the whole serial of such DES were computed for the offset time. The regular opening, as a function of excitation, of the bending between the ii electrons with maximum density at 〈r ₁〉 = 〈r _ii〉 was established from provisional probability densities. These quantities were computed quantum mechanically and demonstrated clearly that, for such cases, it is potent athwart correlations that boss the nature of the wavefunctions [58–64].

Later work produced the get-go ab initio results on the caste and mechanism of their stability past computing the partial and the total autoionization widths of Wannier two-electron ionization ladders (TEILs) for the isnℓ ^{two 2} S and ⁴ P states of He⁻ [63, 64] and the 1s ² nℓ ² TEIL states of Li⁻ [61]. The same full general approach was implemented in order to found and to analyze novel regular serial of excited unstable states labeled by triply and quadruply excited configurations. Specifically, by combining notions of angular momentum and spin symmetry and of electronic structure, we determined from first principles that identifiable serial of intrashell states lead to symmetric fragmentation thresholds [58, 62].

I give two examples from the aforementioned published results:

1.

Allow us consider the quadruply excited states in Exist of ⁵ Southward^o symmetry having as zero-guild wavefunction the MCHF intrashell superposition [58] . For the lowest primary quantum number, n = 2, the reference wavefunction is the unmarried configuration (2south2p ³). As due north increases, athwart correlation (hydrogenic near-degeneracy) dominates. Thus, for n = 6, the state-specific zero-society MCHF solution with the lowest energy is 0.77(visouth6p ³) + 0.48(6s6p6d ^two) + 0.23(sixs6fsixd ^two) − 0.28(6dsixfsixp ²) + 0.13(6phalf-dozendvif ^two) − 0.eleven(6f6d ³). As n increases and athwart correlation contributes more, the average angle between the iv electrons opens up, tending to that of a tetrahedron [58].

ii

Table 2.v presents 2 sets of results for the DES TEIL states in H⁻ (′nℓ ^{ii′ one} S), He⁻ (′1snℓ ^{ii′ ii} S), and Li⁻ (′is ² nℓ ^{two′ 1} S), n = 3, 4, 5 …, which were expected to have like characteristics, produced by the stiff electron pair correlations. One prepare contains the computed average values (in a.u.) of the radii of the two electrons, 〈r ₁〉 _n ≈ 〈r _two〉 _north ≡ r_n . The 2d fix contains the energy distance from the fragmentation threshold (in eV). It turns out that these energies fit the Rydberg-similar analytic formula $E_n=A\frac{n(n-\mathrm{one})}{r_{\mathrm{northward}}{}^{\mathrm{two}}}$ , where A is a slowly varying proportionality constant [61].

Table ii.5. Boilerplate radii, r_northward , (in a.u.) and energies from threshold, ΔE, (in eV) for the TEIL states, H⁻ (nℓ ^{2 1} S), He⁻ (1snℓ ^{2 ii} S), and Li⁻ (1s ² nℓ ^{two 1} S), n = 3, 4, five … ¹

	H−		He−		Li−
north	rnorth	ΔE	rnorthward	ΔE	rn	ΔE
3	16.iii	1.885	14.2	two.156	thirteen.3	2.285
4	28.7	1.088	26.5	i.180	25.viii	ane.213
5	44.9	0.706	42.6	0.745	41.8	0.755
6	66.5	0.493	62.nine	0.511	62.0	0.517
seven	90.9	0.366	87.0	0.375	86.nine	0.376
viii	120.one	0.282	114.7	0.287	114.7	0.287
9	152.0	0.224	144.ane	0.227	144.0	0.227

one

From Ref. [61]. Notation that the results are very similar, reflecting the similar beliefs of the excited pair of electrons in specific states.

In order to larn more than definitive noesis equally to the backdrop of various types of DES in He, (effective Coulomb attractive potential) and in H⁻, the SPSA computations take dealt with intrashell and intershell DES upward to the hydrogenic threshold N = 25 and have been accompanied past analysis and a brief commentary concerning other approaches [65–67].

This became possible not only past the state-specific nature of the computations simply also past the realization that the natural orbitals produced from hydrogenic basis sets were the same every bit the MCHF orbitals that are computable for the intrashell states upwards to about N = 10 − 12. Therefore, for DES with very loftier N, instead of obtaining the multiconfigurational nil-order wavefunction from the solution of the SPSA MCHF equations (which are very difficult to converge numerically if at all), we replaced the MCHF orbitals by natural orbitals obtained from the diagonalization of the appropriate density matrices with hydrogenic orbitals.

In fact, by being able to obtain and apply wavefunctions of different degrees of accuracy regarding the contribution of electron correlation, we explored the degree of validity of the Herrick–Sinanoğlu (G, T) quantum numbers [68] and of new ones, namely the Komninos et al. (F, T) classification scheme that was introduced in 1993 [65–67]. It was demonstrated that the accurate wavefunctions of the serial of DES are best represented by the (F, T) scheme compared with the (K, T) i [65–67].

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Atomic Physics

Francis One thousand. Pipkin (deceased) , Mark D. Lindsay , in Encyclopedia of Physical Scientific discipline and Technology (Third Edition), 2003

Iv Rydberg and Exotic Atoms

Certain highly excited atoms, with 1 or more than electrons at very high free energy, but below the ionization potential free energy needed to tear the electron off, are chosen Rydberg atoms . The Rydberg electron is bound to the cantlet, but only barely so. The principal breakthrough number n is a measure of the excitation of the Rydberg cantlet; atoms with n up to 600 or more take been observed. (In principle, north can go up to infinity earlier the electron becomes unbound or ionized; even so, then the Rydberg cantlet becomes extraordinarily delicate and very difficult to measure.)

Rydberg atoms have a number of interesting properties. Since the electron is well-nigh unbound (the binding energy drops every bit n ⁻²), it moves quite far away from the nucleus, as shown in Fig. eight. The size of a Rydberg atom goes equally north ^two, and its cross-section goes equally n ⁴. For very high due north values, the electron orbital radius can be several microns, nigh macroscopic in size. The distance and weak interaction of the Rydberg electron with the nucleus hateful that all Rydberg atoms are very similar to H atoms. Fifty-fifty with an extended and more than complex core such equally Na⁺, or even with a molecular core such equally H₂ ⁺, the distant electron "sees" a betoken source of positive charge to a good approximation, just as the electron in an hydrogen atom does. Thus, virtually of the quantum mechanics mathematical apparatus and annotation developed for the hydrogen atom can exist used. The radiative lifetime of a Rydberg electron is calculated according to the usual hydrogen atom electric dipole matrix elements, and varies every bit northward ³. That is, every bit northward goes up, the Rydberg electron becomes less and less likely to radiate. This is explained physically as an isolation of the Rydberg electron from the charge centre of the nucleus, so it acts more and more similar an isolated free electron, which does not radiate. The polarizability of the "floppy" Rydberg atom tin be very large, and goes equally due north ⁷.

Figure viii. Probability density ∣Ψ∣^two r ³  =   ∣R _nl (r/a₀ )∣² r ³ for Rydberg H atom, north  =   twenty, l  =   10. (Gene of r ^three takes into account the larger three-dimensional volume at larger radius, to evidence the probability of finding the electron at a particular r.) Compare with Fig. 6; recall the size of the H atom with n  =   one is virtually r/a ₀  =   1 (a ₀ is the Bohr radius, 5.29   ×   10^−eleven thou).

Since the Rydberg electron interacts weakly with the remainder of the atom, perturbation theory methods tin can be used to calculate diverse backdrop that would exist impossible to calculate for a low-lying, strongly interacting electron. The Rydberg electron can act as a sensitive probe of various core properties such as polarizability and quadrupole moment of the ion cadre.

The weak interaction between the Rydberg electron and its core tin allow a relatively wearisome transfer of energy between the 2. When an excited core transfers energy to the Rydberg electron, normally giving it enough to be ionized, the procedure is chosen autoionization. In this, the core loses the energy the Rydberg electron gains. When a Rydberg electron loses energy, which is transferred to the core, the process is called dielectronic recombination. This is the most important machinery in plasmas whereby costless electrons and ions combine to form a normal hot gas. These two process are the fourth dimension reversal of each other, but are described by the same mathematics.

Since n is very high for Rydberg states, and l and m can take on a large number of values, typically a very big number of Rydberg states are available, all at most the same energy close to but just below the ionization potential. (Autoionizing states, counting the cadre energy, actually lie to a higher place the ionization potential.) At that place are so many states close in energy that commonly they overlap in energy and interfere with each other in a breakthrough mechanical way, leading to very circuitous situations and to a so-called quasi-continuum of states.

An "atom" of positronium is formed past an electron and its antiparticle, a positron. Although the two somewhen annihilate each other (via the overlapping of their wave functions), they can alive for up to 10^−seven sec, orbiting each other very much like in a hydrogen cantlet, except the reduced mass is one-half that of a normal hydrogen atom'south electron. While positronium exists, information technology can absorb and emit photons with a spectrum similar to diminutive hydrogen, except all wavelengths are doubled relative to atomic hydrogen. The lifetime of the positron is sensitive to the details of the wave function then tin can probe the inside solid-country systems of the wave function. The quantum mechanical state labels of He apply to positronium.

A muonic cantlet is formed by a normal atom with one electron replaced by a negative muon, which is very similar to an electron but weighs 207 times as much. The muon in an atom has a wave office and transitions just as the electron does, but the much higher mass means that the energies are college and the wave functions are "tighter" (occupy less space). In fact, a meaning fraction of the muonic wave role exists within the nucleus of the muonic cantlet, thus muonic atoms are used to probe the exact spatial distribution of mass and charge of the nucleus, peculiarly near the edge of the nucleus. Muonic molecules be, with the muon pulling two nuclei very close together in a chemical bond. Unfortunately, thermalized muons are hard to produce, and muons are unstable and only live two   μsec before decomposable, so muon catalyzed fusion of hydrogen nuclei has been observed just is not efficient.

An antiproton and a positron form the exotic atom of antihydrogen. This atom has been formed and detected at the high energies commensurate with the formation of the antiproton. A corking deal of work is ongoing to tedious down the antihydrogen to thermal or less energies and fifty-fifty to laser trap them. An atom consisting of a normal He⁺⁺ nucleus, an antiproton, and an electron (antiprotonic helium) has been spectroscopically measured, as it is easier to form than manifestly antihydrogen. From the measured Rydberg constant of antiprotonic helium, the antiproton mass has been measured to exist the same equally that of a normal proton to inside 3   ppm.

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