What Does the Principal Quantum Number Determine

Chemical Foundations of Physiology I

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

Atomic Orbitals Explain the Periodicity of Chemical Reactivities

There are eight main "shells," referring to the principal quantum number, n=(1,2,3,4,5,6,7,8) that describes atomic orbitals. There are four major subshells: s, p, d, and f, whose names derive from spectroscopic descriptions of sharp, principal, diffuse, and fundamental. These orbitals are described by the azimuthal quantum number, l=(0,1,2,3) for (s,p,d,f), respectively. Each subshell has a structure and a capacity for electrons that is described by the magnetic quantum number, m, and the spin quantum number, s. The s subshell is spherically symmetrical and holds only 2 electrons; each set of p orbitals holds 6 electrons, the d orbitals hold 10, and the f orbitals hold 14. The sequential filling of these orbitals accounts for the periodic chemical behavior of the elements with their atomic number. This order of filling is shown in Figure 1.4.2. 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 quantum number, s, that is represented as "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," analogous to the filling of a bus where double seats tend to fill with single individuals before double occupancy occurs.

Figure 1.4.2. Order of filling of atomic orbitals. Electronic orbits are characterized by a principal quantum number that determines the main shell, an azimuthal quantum 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. There are four subshells: s, p, d, and f. These have 1, 3, 5, and 7 orbitals that each can hold up to two electrons of opposite spin. The order of filling with increasing number of electrons follows the blue diagonal arrows in the diagram: 1s fills first, followed by 2s and 2p; next is 3s followed by 3p and 4s, followed by 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 full orbitals are chemically unreactive. These correspond 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 can react with other atoms, in order to become more stable by attempting to fill their orbitals. They do this by sharing electrons, a process that constitutes chemical bonding. This sharing can be equal or very unequal, corresponding to the extremes of covalent bonding and ionic bonding.

Figure 1.4.3. Electronic structure of the inert gases. These inert gases are chemically unreactive because their orbitals are already filled. Helium, with n=2 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 downward. Neon (n=10) fills the 2s and 2p orbitals with a total of 8 electrons. Each orbital in the subshells carries at most two electrons. The order of filling of the orbitals corresponds to that shown in Figure 1.4.2.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128008836000057

Atoms and Atomic Arrangements

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

1.1.2 Nomenclature for the electronic states

The energy of an electron is mainly determined by the values of the principal and orbital quantum numbers. The principal quantum number is simply expressed by giving that number, but the orbital quantum number is denoted by a letter. These letters, which are derived from the early days of spectroscopy, are s, p, d and f, which signify that the orbital quantum numbers l are 0, 1, 2 and 3, respectively. ¹ When the principal quantum number n=1, l must be equal to zero, and an electron in this state would be designated by the symbol 1s. Such a state can only have a single value of the inner quantum number m=0 but can have values of $+\frac{1}{2}$ or $-\frac{1}{2}$ for the spin quantum number s. It follows, therefore, that there are only two electrons in any one atom which can be in a 1s-state, and that these electrons will spin in opposite directions. Thus when n=1, only s-states can exist and these can be occupied by only two electrons. Once the two 1s-states have been filled, the next lowest energy state must have n=2. Here l may take the value 0 or 1, and therefore electrons can be in either a 2s- or a 2p-state. The energy of an electron in the 2s-state is lower than in a 2p-state, and hence the 2s-states will be filled first. Once more there are only two electrons in the 2s-state, and indeed this is always true of s-states, irrespective of the value of the principal quantum number. The electrons in the p-state can have values of m=+1, 0, −1, and electrons having each of these values for m can have two values of the spin quantum number, leading therefore to the possibility of six electrons being in any one p-state. These relationships are shown more clearly in Table 1.1, and Figure 1.1 shows the shapes of the s, p and d orbitals.

Table 1.1. Allocation of States in the First Three Quantum Shells

Shell	n	l	m	s	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 2s-states
L			+1	± $\frac{1}{2}$		8
	2	1	0	± $\frac{1}{2}$	Six 2p-states
			−1	± $\frac{1}{2}$
		0	0	± $\frac{1}{2}$	Two 3s-states
M			+1	± $\frac{1}{2}$
		1	0	± $\frac{1}{2}$	Six 3p-states
			−1	± $\frac{1}{2}$
	3					18
			+2	± $\frac{1}{2}$
			+1	± $\frac{1}{2}$
		2	0	± $\frac{1}{2}$	Ten 3d-states
			−1	± $\frac{1}{2}$
			−2	± $\frac{1}{2}$

No further electrons can be added to the state for n=2 after two 2s- and six 2p-state are filled, and the next electron must go into the state for which n=3, which is at a higher energy. Here the possibility arises for l to have the values 0, 1 and 2 and hence, besides s- and p-states, d-states for which l=2 can now occur. When l=2, m may have the values +2, +1, 0, −1, −2, and each may be occupied by two electrons of opposite spin, leading to a total of ten d-states. Finally, when n=4, l will have the possible values from 0 to 4, and when l=4 the reader may verify that there are fourteen 4f-states.

Table 1.1 shows that the maximum number of electrons in a given shell is 2n ². It is an accepted practice to retain an earlier spectroscopic notation and to label the states for which n=1, 2, 3, 4, 5, 6 as K-, L-, M- N-, O- and P-shells, respectively.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780080982045000018

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

Konrad Patkowski , in Annual Reports in Computational Chemistry, 2017

2.1.1 CBS Extrapolation

In view of the analytical results that describe the convergence of atomic correlation energies with the maximum angular momentum and the maximum principal quantum number present in the basis set, it has been commonly assumed that molecular correlation energies, and the correlation parts of noncovalent interaction energies, follow the resulting X ⁻³ convergence pattern in the cc-pVXZ and aug-cc-pVXZ sequences (60, 61) . If that assumption is exactly true, the CBS-limit correlation energy $E_{\infty }^{\text{corr}}$ can be computed from the results in two consecutive-X basis sets E _X−1 ^corr, E _X ^corr as

(4) $\begin{array}{l}\hfill E_{\infty }^{\text{corr}}=E_X^{\text{corr}}+\frac{{(1-1/X)}^n}{1-{(1-1/X)}^n}\left(E_X^{\text{corr}}-E_{X-1}^{\text{corr}}\right)\end{array}$

with n = 3. The use of the X ⁻³ extrapolation defined by Eq. (4), popularized by Helgaker and coworkers (60, 61) , has become a routine step 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 applied to the correlation contribution only. 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 E _X−2 ^HF, E _X−1 ^HF, E _X ^HF assuming the exponential formula

(5) $\begin{array}{l}\hfill E_{\infty }^{\text{HF}}=E_X^{\text{HF}}+Aexp(-BX)\end{array}$

with some constants A, B.

While the X ⁻³ extrapolation formula of correlation energy has been the most popular by far, it is not the only one possible. In particular, various three-parameter extrapolations have been proposed to leverage the results computed in three consecutive bases E _X−2 ^corr, E _X−1 ^corr, E _X ^corr. Martin (80) generalized the X ⁻³ formula to account for an offset between the maximum angular momentum and the basis set cardinal number:

(6) $\begin{array}{l}\hfill E_{\infty }^{\text{corr}}=E_X^{\text{corr}}+\frac{A}{{(X+B)}^3}.\end{array}$

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-pVXZ basis sets, proposed an extrapolation formula that takes into account the different (X ⁻³ and X ⁻⁵, respectively) convergence of the singlet and triplet pair energies. Truhlar (83) set out to optimize the exponent n in the X ⁻ⁿ extrapolation (Eq. 4) in the specific, most economical case of the cc-pVDZ and cc-pVTZ bases, using basis set limit energies for three systems. He found the optimal exponents to be 2.2 for the MP2 correlation energy and 2.4 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 form of Eq. (4)) to the determination of a single linear parameter F _X in

(7) $\begin{array}{l}\hfill E_{\infty }=E_X+F_X\left(E_X-E_{X-1}\right),\end{array}$

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

The general conclusion from the investigations on the optimal CBS extrapolations is that the conventional X ⁻³ formula, while not quite the optimal choice for all systems at all theory levels, is never a bad choice for any correlated interaction energy contribution. At the bare minimum, if the E _(X−1,X) extrapolated value and the |E _(X−1,X) − E _X | difference are taken, respectively, as the best estimate of $E_{\infty }$ and its uncertainty, such an error estimate is very conservative: unless the convergence of E _X is not monotonic (in which case no extrapolation would work), the true value of $E_{\infty }$ is practically guaranteed to be in this confidence 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, but rather reducing the errors in the computed values (the input data for extrapolation) by any of the techniques discussed in this section. 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 be combined. Such an approach was employed in the determination of the helium pair potential to millikelvin accuracy by Jeziorska and coworkers (85–87) using, in addition to the X ⁻³ expression, several extrapolations where the basis set convergence of E _X was assumed to follow the same pattern as the basis set 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 basis sets warrants additional discussion. In 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 true for the δ _MP2 ^CCSD(T) correction within the composite approach, Eq. (3). In fact, Sherrill and coworkers (88) found that the values of the δ _MP2 ^CCSD(T) interaction energy term computed in the aug-cc-pVXZ sequence often have a turning point at X =T or even Q. As a result, the (aug-cc-pVDZ,aug-cc-pVTZ) extrapolation of this term moves the aug-cc-pVTZ result in the wrong direction. Therefore, it is recommended (88) to use 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.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S1574140017300075

Materials and Processes for Next Generation Lithography

D. Frank Ogletree , in Frontiers of Nanoscience, 2016

2.2.1 Atomic Photoemission

The electronic structure of an isolated atom can be described as a series of 1-electron orbitals or energy levels, defined by the principal quantum number n (=1,2,3…) and the orbital angular momentum l (s  = 0, p  = 1, d  = 2, f  = 3), each with its own binding energy (BE). The binding 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 BE plus the sample work function (ϕ) is the difference between the incoming photon energy (hν) and the outgoing 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 X-ray absorption cross section, the inelastic and elastic electron-molecule scattering cross sections are important for EUV radiation chemistry, as are the molecular ionization and fragmentation cross sections for both X-rays and electrons. If the cross section is small compared to the physical area of the target, the interaction probability is small. If the number-density of targets is given by n, then the interaction mean free path λ  =   1/nσ and the probability of photon absorption in a film 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  m², the typical scattering cross section of an atomic nucleus, so 100   Mb, or megabarn, corresponds to an area of 1   Ų (10⁻²⁰  m²).

Each orbital has its own energy-dependent X-ray absorption cross section σ _n,l (hν,Z), sometimes called a "subshell cross section," where Z is the atomic number. Subshell cross sections decrease 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 will usually have a higher cross section than a filled s-level with only two electrons for similar BE. Calculated subshell cross sections for all the elements are tabulated by Yeh and Lindau. ¹⁹

The photoemission cross section is proportional to a dipole matrix element between the wave functions of the initial bound electron orbital and the final outgoing free electron wave. This matrix element involves an integral over the overlap of the two wave functions. ¹⁸ The phase of the outgoing wave oscillates with a wavelength proportional to KE^1/2. 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 cross section is faster for weakly bound valence orbitals than for core orbitals since their radial wave functions are more extended. For large KE, the cross section will drop off ¹⁸ as KE⁻³.

The radial wavefuction of a bound orbital has a radial node, where the phase of the orbital changes sign, when the orbital angular momentum l is greater that the principal quantum number n. This phase change also affects the matrix element overlap integral, and causes a so-called Cooper minimum in the subshell cross section around 10–20   eV above threshold. ²⁰ Cooper minima occur for 3p, 4d, and 5f orbitals, as shown by Yeh and Lindau. ¹⁹

Cross sections for high angular momentum d and f orbitals can show a "delayed maxima." ²⁰ Unlike s and p levels, where the maximum cross section is very close to the threshold photon energy, the maximum for higher angular momentum states can occur 30 or 40   eV above threshold. This is due to the so-called "centrifugal repulsion term" in the effective radial potential for a high angular momentum electron wave. ²⁰ This effect can be important at EUV energies, as seen in Fig. 2, where many of the 4d elements have large cross sections.

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

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780081003541000028

Tribochemistry of Lubricating Oils

In Tribology and Interface Engineering Series, 2003

Nomenclature.

The spectroscopic nomenclature is directly equivalent to that used for spectroscopy X-ray, and is related to the various quantum numbers such as the principal quantum number n, the electronic quantum number 1, the total angular momentum quantum number j, and the spin quantum number s, which can take either of the values ±½. Where: n = 1, 2, 3, 4, …, are designated K, L, M, N, … respectively; 1 = 0, 1, 2, 3, … and j = 1 + s (can take the values of ½, 3/2, 5/2, 7/2), … are given conventional suffixes, 1, 2, 3, 4, … according to the listing in Table 4.8 (Briggs and Seah, 1990). This description of the summation is known as j-j coupling.

Table 4.8. X-ray and spectroscopic notation

Quantum numbers			X-ray suffix	X-ray level	Spectroscopic level
n	1	j
1	0	½	1	K	1s1/2
2	0	½	1	L1	2s1/2
2	1	½	2	L2	2p1/2
2	1	3/2	3	L3	2p3/2
3	0	½	1	M1	3s1/2
3	1	½	2	M2	3p1/2
3	1	3/2	3	M3	3p3/2
3	2	3/2	4	M4	3d3/2
3	2	5/2	5	M5	3d5/2
	etc.		etc.	etc.	etc.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0167892203800196

Self-Assembly Processes at Interfaces

Adam West , in Interface Science and Technology, 2018

3.2.5.2 Auger Electron Spectroscopy and Energy Dispersive X-Ray Analysis

As in XPS a monochromatic beam of X-rays allows for the ejection of core shell 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 n  =   2), will fall to the vacant energy level. This electronic transition is accompanied either by emission of a photon having an energy corresponding to the energy difference between the L and K levels or by transfer of the excess energy to another electron of the L level. This electron is then ejected from the atom, and its energy is measured in Auger electron spectroscopy (AES). AES has been the first developed surface-sensitive analysis method in the beginning of the 1960s. The energy of the emitted photons can also be measured in energy dispersive X-ray (EDX) analysis. 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 sensitivity for elements with a low atomic number, whereas EDX is more appropriate for heavier elements.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128019702000033

Rufus Ritchie, A Gentleman and A Scholar

Károly Tőkési , in Advances in Quantum Chemistry, 2019

2.1.3 Classical quantum numbers

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

(18) $n_c=Z_T{Z}_e{\left(\frac{\mu T_e}{2U}\right)}^{1/2}.$

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

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

The classical orbital angular momentum is defined by

(20) $l_c=\sqrt{m_e\left[{\left(x\dot{y}-y\dot{x}\right)}^2+{\left(x\dot{z}-z\dot{x}\right)}^2+{\left(y\dot{z}-z\dot{y}\right)}^2\right]},$

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

(21) $l\le \frac{n}{n_c}{l}_c\le l+1,$

where l is the quantum-mechanical orbital-angular-momentum.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0065327619300383

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 animal kingdoms as well as in the modern synthetic dye and pigment industry. Organic dyes and pigments, whether natural or synthetic, are very intense in colour. 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 absorbed energy may be re-emitted in the form of fluorescence. Graebe and Liebermann in 1867 undertook the first study of the relationship between colour and chemical structure. They found that reduction of some known dyes destroyed the colour instantly. They concluded that the dyes are chemically unsaturated compounds.

Most organic compounds are complex unsaturated compounds having certain substituent groups. It 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 composed of a basic chromophore or colour-bearing group, to which can be attached a variety of subsidiary groups called auxochrome or colour intensifier, which lead to the production of colour. Chromophores include carbon–carbon double bond specially conjugated systems having alternate single and double bond. Witt also claimed that the auxochromes confer dyeing properties on the molecule, but it is now established that colour and dyeing properties are not directly related. However, Witt's theory, in general, is still acceptable to colourists (Giles, 1974).

Three 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 absorption peak towards longer wavelength). Though Nietzki's rule initially proved useful, its utility decreased as many exceptions were subsequently discovered.

Armstrong proposed the quinoid theory in 1887, stating that only the compounds which can be written in a quinoid form are coloured. The theory soon 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 between the two tautomeric forms (see Fig. 2.11a and 2.11b), the chlorine atom flopping rapidly from one amino group to the other.

2.11. The two tautomeric forms of Doebner's violet.

Hewitt and Mitchell in 1907 first realised the importance of conjugation, i.e. the presence of alternate single and double bonds. From a study of azo dyes, they established Hewitt's rule stating that the longer the conjugated chain, the more bathochromic shift will be 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 basic group, the auxochrome connected to an electron-withdrawing acidic group by a system of conjugated double bonds. The greater the nucleophilic and electrophilic character 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 colour complementary to the light absorbed, or more specifically to the wavelength of maximum absorption (λ _max). The relation between the colour absorbed and colour perceived are shown in Table 2.3. Certain colours require more than one absorption band – green requires absorption of red and blue-violet. This is difficult to achieve, and the number of green dyes are comparatively less. Black requires a combination of several broad overlapping bands of similar extinction coefficients. The brown, olive green and other dull colours also require 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	Colour perceived
400–500	Blue	Yellow
400–440	Violet	Greenish-yellow
460–500	Greenish blue	Orange
400–620	Bluish green	Red
480–520	Green	Magenta
560–700	Orange	Cyan
600–700	Red	Bluish green

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

The organic colourants may be broadly classified into three groups (Nassau, 1983). Benzenoids are the most important group of synthetic colourants. This includes various aromatic compounds – in textile the most important chromophores are azobenzene, triphenylmethane and anthraquinone (Figs 2.12, 2.13 and 2.14 respectively). Azo groups (−   N = N–), which are not available in nature, when incorporated in benzenoids, form the basis of the majority of synthetic dyes. Benzenoids are also occasionally accompanied by thio (> C = S), nitroso (−   N = O) and many other groups.

2.12. Azobenzene.

2.13. Triphenylmethane.

2.14. Anthraquinone

Polyenes consist of non-benzenoid long conjugated double-bond systems, which are the basis for many biological colourants. When such a conjugated system is large enough, it can absorb visible light and become coloured. Carotenoids are typical non-cyclic natural colourants. An important member of this group, β-carotene (Fig. 2.15), is the orange colourant in carrots and many other vegetables. This is used for colouration of cosmetics and food products. When it is split in half, a portion having structure similar to Fig. 2.16 is vitamin A₁. A similar carotenoid, crocein, which is the principal colouring component of the natural colourant yellow dye saffron, is used for food colouring. Rhodopsin, the visual pigment of eye, is similar to β-carotene and vitamin A₁. Carotenoids are responsible for various colours in bird feathers.

2.15. β-Carotene.

2.16. A half of β-carotine resembling Vitamin A1.

Cyclic polyene non-benzenoid conjugated systems include porphyrins; the most important members belonging to this class are α-chlorophyll (Fig. 2.17) and similar heme. While the green-coloured chlorophyll is responsible photosynthesis of plants, red-coloured heme transports oxygen in blood. Both cyclic 18-member conjugated systems have a central metal ion, Mg^{2   +} in the former and Fe^{3   +} in the latter. Synthetic pigment blue and green phthalocyanines are also cyclic polyenes like porphyrins, but they have additionally benzenoid groups, e.g. copper phthalocyanine (Fig. 2.18).

2.17. α-Chlorophyll.

2.18. Copper phthalocyanine.

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

Primary, 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's 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 not the oscillation of atoms, which produces colour. Atomic vibrations give rise to the absorption of infrared radiation, whereas the oscillation of electrons causes the absorption of ultraviolet or visible radiation resulting in colour sensation.

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

It was proposed that π-bonding electrons involved in the second 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 can move through the molecule. It is, therefore, possible to write various electronic configurations of the molecule, called canonical forms or resonance hybrids. This does not imply any actual vibration or oscillation among these forms, but merely signifies that the structure is an intermediate one. 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 can move more readily along the molecule. The natural frequency of vibration is decreased, resulting in absorption 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 (2.19(a)–2.19(e)), and the uncharged form (2.19(c)) is the most stable. When two auxochromic groups are attached to the azobenzene, the configuration (2.19(g)) is more stable than structures (2.19(a)) to (2.19(e)), as the charges are now firmly held on oxygen or nitrogen atoms. The compound (2.19(f)) is, therefore, of more intense colour than azobenzene.

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

With the increase in stability of the alternating structures, its electrons may be considered to move more 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 string in which the longer the string, the lower is the frequency; hence, the longer is the wavelength of the note it emits when plucked. This applies to adsorption as well as emission of energy, because any oscillator absorbs energy most readily at the wavelength of its natural frequency of vibration.

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

1.

Angular momentum quantum number, l = n – 1. Each l value represents a specific orbit named after the description of the hydrogen spectrum such s for sharp (l = 0), p for principal (l = 1), d for diffuse (l = 2), f for fundamental (l = 3) etc.

2.

Magnetic quantum number, m = +   l, +   l–1, +   l–2, … 0, 1, 2, ..l

3.

Spin quantum number, +½, –½

The detailed orbital designation and the number of electrons in each orbit up to the fourth shell are listed in Table 2.4. When two atoms are close to each other, the respective atomic orbital forms various MOs by overlap interactions. The molecular orbital can accommodate exactly the same number of electrons as the atomic orbital from which they are formed. The formation of various molecular orbital by linear and non-linear combinations of the interacting atomic orbital has been studied by various research workers. Molecular orbital techniques without any approximation are possible at present. But 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 two atomic orbitals interact strongly by direct overlap and there is symmetry with respect to rotation about the axis joining the two 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. By absorption of energy, transition may occur from σ-bonding to σ* antibonding state (σ → σ*). When overlapping occurs only for the outer region of less electron density, the energy of the bond is less than that in an σ bond, and a π orbital resulting in π bonds being formed. A σ bond possesses zero angular momentum around the bond axis, whereas a π orbital possesses one unit of angular momentum. Again, π-bonds can be excited into a high energy π* antibonding state (π → π*). There may also be a transition from non-bonding to antibonding state (n → σ* or n → π*). In a molecule, there may be several of each type of orbital of varying energy levels formed by interaction of various pairs of atoms. For a molecule, the smallest amount of energy absorbed is the energy required for transition from the highest occupied molecular orbital (HOMO) to the lowest occupied molecular orbital (LOMO). The most important 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 various orbits

Shell No.	No. of Orbital	Orbital Designation	No. of Electrons
1	1	1   s	2
2	2	2   s	2
		2p	6
3	3	3   s	2
		3p	6
4		3d	10
	4	4   s	2
4d		4p	6
4f	10

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780857092298500024

State- and Property-Specific Quantum Chemistry

Cleanthes A. Nicolaides , in Advances in Quantum Chemistry, 2011

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

It is clear that, assuming the model of orbital configurations, one can construct, formally, an infinity of MES for each atom or molecule, at least as a superposition of such configurations for each symmetry. For example, the two-electron ionization threshold (TEIT) of He is at 79.0 eV. No bound or quasi-bound DES exist above this 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-bound DES (and a few stable ones due to nonrelativistic symmetry restrictions) of many symmetries for even and odd parity. Obviously, for the photoabsorption process in the nonrelativistic context, only the DES of ¹ P^o symmetry is excited. Assuming the basis of orbital configurations, these can be labeled, at least formally, by superposition of configurations such as 2s2p, (2s3p, 2p3s, 2p3d), (4s4p, 4p4d, 4d4f), etc.

One important question is whether one can identify classes of such states exhibiting some type of regularity as a function of excitation energy, just like the Rydberg levels do as they reach the ionization threshold. Indeed, for MES without a complex 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 Be, 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 work on the double-electron excitation from the 2s ² subshell of Neon to newly established DES embedded in one- and two-electron continua, although they obey an effective Rydberg-like energy formula, show 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 example, Refs. [60a, 60b, 61, 62]. Accordingly, the zero-order wavefunction was chosen as the direct, state-specific MCHF solution with only the intrashell configurations for each hydrogenic shell, having the lowest energy. These multiconfigurational SCF wavefunctions account for most of angular correlation and part of radial correlation. This is sufficient for the quantitative understanding 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 issue was how to choose and compute the zero-order multiconfigurational (Fermi-sea) wavefunctions, so as to be able to recognize without ambiguity possible regular series that tend to the Wannier state at E = 0, with 〈r ₁〉 = 〈r ₂〉 and ϑ = 180°. Up to n = 10, intrashell states were computed and analyzed from first 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 series of such DES were computed for the first time. The regular opening, as a function of excitation, of the angle between the two electrons with maximum density at 〈r ₁〉 = 〈r ₂〉 was established from conditional probability densities. These quantities were computed quantum mechanically and demonstrated clearly that, for such cases, it is strong angular correlations that dominate the nature of the wavefunctions [58–64].

Later work produced the first ab initio results on the degree and mechanism of their stability by computing the partial and the total autoionization widths of Wannier two-electron ionization ladders (TEILs) for the 1snℓ ^{2 2} S and ⁴ P states of He⁻ [63, 64] and the 1s ² nℓ ² TEIL states of Li⁻ [61]. The same general approach was implemented in order to establish and to analyze novel regular series 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 series of intrashell states lead to symmetric fragmentation thresholds [58, 62].

I give two examples from the aforementioned published results:

1.

Let us consider the quadruply excited states in Be of ⁵ S^o symmetry having as zero-order wavefunction the MCHF intrashell superposition [58] . For the lowest principal quantum number, n = 2, the reference wavefunction is the single configuration (2s2p ³). As n increases, angular correlation (hydrogenic near-degeneracy) dominates. Thus, for n = 6, the state-specific zero-order MCHF solution with the lowest energy is 0.77(6s6p ³) + 0.48(6s6p6d ²) + 0.23(6s6f6d ²) − 0.28(6d6f6p ²) + 0.13(6p6d6f ²) − 0.11(6f6d ³). As n increases and angular correlation contributes more, the average angle between the four electrons opens up, tending to that of a tetrahedron [58].

2

Table 2.5 presents two sets of results for the DES TEIL states in H⁻ (′nℓ ^{2′ 1} S), He⁻ (′1snℓ ^{2′ 2} S), and Li⁻ (′1s ² nℓ ^{2′ 1} S), n = 3, 4, 5 …, which were expected to have similar characteristics, produced by the strong electron pair correlations. One set contains the computed average values (in a.u.) of the radii of the two electrons, 〈r ₁〉 _n ≈ 〈r ₂〉 _n ≡ r_n . The second set contains the energy distance from the fragmentation threshold (in eV). It turns out that these energies fit the Rydberg-like analytic formula $E_n=A\frac{n(n-1)}{r_n{}^2}$ , where A is a slowly varying proportionality constant [61].

Table 2.5. Average radii, r_n , (in a.u.) and energies from threshold, ΔE, (in eV) for the TEIL states, H⁻ (nℓ ^{2 1} S), He⁻ (1snℓ ^{2 2} S), and Li⁻ (1s ² nℓ ^{2 1} S), n = 3, 4, 5 … ¹

	H−		He−		Li−
n	rn	ΔE	rn	ΔE	rn	ΔE
3	16.3	1.885	14.2	2.156	13.3	2.285
4	28.7	1.088	26.5	1.180	25.8	1.213
5	44.9	0.706	42.6	0.745	41.8	0.755
6	66.5	0.493	62.9	0.511	62.0	0.517
7	90.9	0.366	87.0	0.375	86.9	0.376
8	120.1	0.282	114.7	0.287	114.7	0.287
9	152.0	0.224	144.1	0.227	144.0	0.227

1

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

In order to acquire more definitive knowledge as to the properties of various types of DES in He, (effective Coulomb attractive potential) and in H⁻, the SPSA computations have dealt with intrashell and intershell DES up to the hydrogenic threshold N = 25 and have been accompanied by analysis and a brief commentary concerning other approaches [65–67].

This became possible not only by the state-specific nature of the computations but also by the realization that the natural orbitals produced from hydrogenic basis sets were the same as the MCHF orbitals that are computable for the intrashell states up to about N = 10 − 12. Therefore, for DES with very high N, instead of obtaining the multiconfigurational zero-order wavefunction from the solution of the SPSA MCHF equations (which are very hard 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 use wavefunctions of different degrees of accuracy regarding the contribution of electron correlation, we explored the degree of validity of the Herrick–Sinanoğlu (K, 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 series of DES are best represented by the (F, T) scheme compared with the (K, T) one [65–67].

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123864772000085

Atomic Physics

Francis M. Pipkin , Mark D. Lindsay , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IV Rydberg and Exotic Atoms

Certain highly excited atoms, with one or more electrons at very high energy, just below the ionization potential energy needed to tear the electron off, are called Rydberg atoms . The Rydberg electron is bound to the atom, but just barely so. The principal quantum number n is a measure of the excitation of the Rydberg atom; atoms with n up to 600 or more have been observed. (In principle, n can go up to infinity before the electron becomes unbound or ionized; however, then the Rydberg atom becomes extraordinarily fragile and very difficult to measure.)

Rydberg atoms have a number of interesting properties. Since the electron is nearly unbound (the binding energy drops as n ⁻²), it moves quite far away from the nucleus, as shown in Fig. 8. The size of a Rydberg atom goes as n ², and its cross-section goes as n ⁴. For very high n values, the electron orbital radius can be several microns, almost macroscopic in size. The distance and weak interaction of the Rydberg electron with the nucleus mean that all Rydberg atoms are very similar to H atoms. Even with an extended and more complex core such as Na⁺, or even with a molecular core such as H₂ ⁺, the distant electron "sees" a point source of positive charge to a good approximation, just as the electron in an hydrogen atom does. Thus, most of the quantum mechanics mathematical apparatus and notation developed for the hydrogen atom can be used. The radiative lifetime of a Rydberg electron is calculated according to the usual hydrogen atom electric dipole matrix elements, and varies as n ³. That is, as n 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 center of the nucleus, so it acts more and more like an isolated free electron, which does not radiate. The polarizability of the "floppy" Rydberg atom can be very large, and goes as n ⁷.

FIGURE 8. Probability density ∣Ψ∣² r ³  =   ∣R _nl (r/a₀ )∣² r ³ for Rydberg H atom, n  =   20, l  =   10. (Factor of r ³ takes into account the larger three-dimensional volume at larger radius, to show the probability of finding the electron at a particular r.) Compare with Fig. 6; recall the size of the H atom with n  =   1 is about r/a ₀  =   1 (a ₀ is the Bohr radius, 5.29   ×   10⁻¹¹ m).

Since the Rydberg electron interacts weakly with the rest of the atom, perturbation theory methods can be used to calculate various properties that would be 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 core.

The weak interaction between the Rydberg electron and its core can allow a relatively slow transfer of energy between the two. When an excited core transfers energy to the Rydberg electron, usually giving it enough to be ionized, the process is called 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 mechanism in plasmas whereby free electrons and ions combine to form a normal hot gas. These two process are the time 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 large number of Rydberg states are available, all at nearly the same energy close to but just below the ionization potential. (Autoionizing states, counting the core energy, actually lie above the ionization potential.) There are so many states close in energy that usually they overlap in energy and interfere with each other in a quantum mechanical way, leading to very complex situations and to a so-called quasi-continuum of states.

An "atom" of positronium is formed by an electron and its antiparticle, a positron. Although the two eventually annihilate each other (via the overlapping of their wave functions), they can live for up to 10⁻⁷ sec, orbiting each other very much like in a hydrogen atom, except the reduced mass is half that of a normal hydrogen atom's electron. While positronium exists, it can absorb and emit photons with a spectrum similar to atomic hydrogen, except all wavelengths are doubled relative to atomic hydrogen. The lifetime of the positron is sensitive to the details of the wave function and so can probe the inside solid-state systems of the wave function. The quantum mechanical state labels of He apply to positronium.

A muonic atom 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 function and transitions just as the electron does, but the much higher mass means that the energies are higher and the wave functions are "tighter" (occupy less space). In fact, a significant fraction of the muonic wave function exists inside the nucleus of the muonic atom, thus muonic atoms are used to probe the exact spatial distribution of mass and charge of the nucleus, especially near the edge of the nucleus. Muonic molecules exist, with the muon pulling two nuclei very close together in a chemical bond. Unfortunately, thermalized muons are difficult to produce, and muons are unstable and only live 2   μsec before decaying, so muon catalyzed fusion of hydrogen nuclei has been observed but 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 great deal of work is ongoing to slow down the antihydrogen to thermal or less energies and even 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 plain antihydrogen. From the measured Rydberg constant of antiprotonic helium, the antiproton mass has been measured to be the same as that of a normal proton to within 3   ppm.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B0122274105000417

What Does the Principal Quantum Number Determine

Source: https://www.sciencedirect.com/topics/engineering/principal-quantum-number

Share this post