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Materials 100A: Orbitals, bonding, etc.
Orbitals, and the Periodic Table
Ram Seshadri MRL 2031, x6129, [email protected]
These notes closely follow P. W. Atkins, Physical Chemistry
The Hydrogen atom: To understand the quantum mechanics of the hydrogen atom, we recognize that weneed to set up the Hamiltonian H that describes the kinetic energy of the electron and recognizes the potentialenergy (Coulombic) arising from the negatively charged electron being in the vicinity of a positively chargednucleus:
H = K.E(electron) + K.E.(nucleus) + P.E.(electron-nucleus)
and the Schrodinder equation (S.E.) Hψ = Eψ can be written and solved. The best way to do this is to usepolar coordinates and the equation as well as the solution is written ψ(r, θ, φ) rather than ψ(x, y, z).
Quantum numbers: From solving the S.E. for hydrogen-like atoms, one finds that electrons in many-electronatoms are completely described by a set of four quantum numbers:
1. The principal quantum number n, that can take on values 1, 2, 3 . . .
2. The angular momentum quantum number l that takes on values 0, 1, 2 . . .n− 1
3. The magnetic quantum number corresponding to the z component of the angular momentum ml, whichtakes on the values 0, ±1, ±2, . . .±l
4. The spin quantum number ms which takes on the values ± 12
The energy of an electron in an orbital with quantum number n for an atom with atomic number Z is given by:
En = − Z2µe4
32π2ε20~2n2
Where e is the charge on the electron, ε0 is the vacuum permittivity, and µ is the reduced mass of the system.
Shells, subshells . . . : The different quantum numbers define the shell, subshells . . .
n = 1 2 3 4 . . .K L M N . . .
and
l = 0 1 2 3 4 . . .s p d f g . . .
The s, p, d, f and g are called atomic orbitals. Filling up these orbitals with electrons builds atoms, andthe way in which atoms are build up gives rise to the periodic table. There is only one s orbital (ml = 0),but there are three p orbitals (ml = −1, 0, 1), five d orbitals (ml = −2,−1, 0, 1, 2), and seven f orbitals(ml = −3,−2,−1, 0, 1, 2, 3).
1
Materials 100A: Orbitals, bonding, etc.
n Shell Subshells States Electrons1 K s 1 22 L s 1 2
p 3 63 M s 1 2
p 3 6d 5 10
4 N s 1 2p 3 6d 5 10f 7 14
Rules for filling in the electrons: Atoms have in the nucleus, protons and neutrons and outside the nucleus,electrons. The number of electrons = number of protons = Z, the atomic number.
1. The Pauli principle: No more than two electrons can occupy a given orbital. If there are two electrons inan orbital, their spins must be paired (one must have ms = 1
2 and the other, ms = − 12 ).
2. The aufbau (building-up) principle: When electrons are filled in to orbitals in an atom, the orbitals withlower energy are filled first. The order of filling is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s . . .
3. The Hund rule: Electrons will occupy different orbitals in a given subshell, before two electrons will occupya single orbital.
There is a simple way of remembering how electrons fill up orbitals, shown in the accompanying diagrams:
6d
1s
2s
3s
4s
5s
6s
2p
3p
4p
5p
6p
3d
4d
5d
4f
5f
2
Materials 100A: Orbitals, bonding, etc.
19 20
K Ca
1 2
H He
3 4
Li Be
11 12
Na Mg
Rb Sr
37 38
21 22 23 24 25 26 27 28 29 30
Sc Ti V Cr Mn Fe Co Ni Cu Zn
5 6 7 8 9 10
CB N O F Ne
13 14 15 16 17 18
Al Si P S Cl Ar
31 32 33 35 35 36
Ga Ge As Se Br Kr
49 50 51 52 53 54
In Sn Sb Te I Xe
39 40 41 42 43 44 45 46 47 48
Zr Nb Mo Tc Ru Rh Pd Ag CdY
ener
gy, f
illin
g, Z
n = 4
n = 2
n = 3
n = 1
n = 5
s block p block d block
From such diagrams, we are able to extract the electronic configurations of elements.
More about the atom: The atomic mass (which is numerically, a value close to the mass number) is theweighted average mass of a number of isotopes of the element, expressed in a system of units where the commonisotope of carbon 12C has an atomic mass of precisely 12.00000. The unit of atomic mass in g is equal to1.00000/(Avogadro Number) = 1.00000/6.0221367×1023 = 1.66054×10−24 g. This is sometimes called theLochschmidt number. One atom of 12C weighs 12 times this, 1.99265×1023 g. If instead of counting atom byatom, we count in bunches corresponding to the Avogadro number, we have moles of something, and 1 moleof 12C weights precisely 12.00000 g. One mole of a normal carbon sample (which is a mixture of isotopes withdifferent mass numbers) actually weighs 12.011 g.
The Periodic Table: The rules for filling up electrons in an atom result in the periodic table. Note thatelements in the periodic table are separated into various categories. You must learn to understand thesedifferent categories: Alkali metals, alkaline earth metals, transition metals, main group elements (consisting ofmetalloids and non-metals) and the noble gases. Also, there are the lanthanide and actinide elements.
3
Materials 100A: Orbitals, bonding, etc.
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51.9
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4
Materials 100A: Orbitals, bonding, etc.
Count the electrons in the noble gases. Note that they correspond to filled K shells (He), filled L shells(Ne), filled M shells (Ar) . . . . These are stable configurations and the noble gases are rather unreactive. It isalways useful to know how far an element is from the nearest noble gas. For instance, Br is just one electronaway from Kr, and as a result, will grab an electron whenever it gets the chance. K has just one electron morethan Ar and is always trying to get rid of it (the one electron). Another way of stating this is that Br has 7valence electrons (and tries to get one more to reach 8) while K has 1 valence electron and tries to get rid of itto reach zero.
In general, atoms that can adopt the configuration of the nearest noble gas by gaining electrons, have atendency to grab electrons from other atoms. This tendency is called electronegativity, and Pauling introduceda scale to describe this tendency. The scale runs to 4 (corresponding to F) which is an atom that always triesvery hard to grab electrons. Atoms that have can gain a noble gas configuration by giving up electrons areelectropositive, and their electronegativity values are small (usually below 1.5).
Bonding:
• There are four forces in nature. The strong and the weak interactions act between electrons, protons,neutrons and other elementary particles and do not concern us. We do not know of any normal materialwhose properties (melting point, for example) depend on the magnitude of these forces. The two otherforces are gravitational and electromagnetic.
• Gravitational forces account for large scale phenomena such as tides, and seasons, and together withintermolecular forces, decide the length of a giraffe’s neck. We shall not discuss gravitation.
• All interactions that are important for solids, should in principle, come out as solutions of the Schrodingerequation (SE). Unfortunately, solutions of the SE are hard to come by for many real systems, and evenif they were available, their utility would not be assured. We therefore continue to propagate the usefulfiction that cohesive interactions in materials can be classified as belonging to one of four categories – vander Waals, ionic, covalent or metallic.1 We keep in mind that these are not very easily distinguished fromone-another in many solids.
For a delightfully readable text on the nature of cohesion between molecules, and between molecules andsurfaces, look at J. N. Israelachvili, Intermolecular and Surface Forces.
van der Waals:
• The simplest solids are perhaps those obtained on cooling down a noble gas – He, Ne, Ar, Kr or Xe. Hedoes not form a solid at ambient pressure. All the other noble gases do.
• The interactions between noble gas atoms (which have closed shells of electrons) is of the van derWaals type (note: van der Waals, not van der Waal’s !) which means that the interaction is betweeninstantaneous dipoles formed because the atoms “breathe” and this breathing causes the centers ofpositive and negative charges to, from time to time, not coincide. The forces are therefore also referredto as induced dipole-induced dipole interactions, or London dispersion forces (after F. W. London).
1Hydrogen bonds are somewhere between being ionic and covalent and we do not see a good reason to place them in a class bythemselves.
5
Materials 100A: Orbitals, bonding, etc.
electron cloudnucleus
• If we were to believe the above scheme, it should come as no surprise that the largest noble gas atomshould be the most polarisable and therefore the most cohesive. The boiling points (often better indicatorsof cohesion than melting points) testify to this:
Atom TM (K) TB (K)Ne 24 27Ar 84 87Kr 116 120Xe 161 165
• Other columns of elements in the periodic table don’t follow this simple trend. For example:
Atom TM (K) TB (K)Cu 1353 2833Ag 1235 2433Au 1333 3133
Ionic
• As a good thumb rule, atoms at the two ends of the electronegativity scale either give up their valenceelectrons very easily to form stable cations (ions with small values of electronegativity) or take up electronsvery easily to form anions (ions with large electronegativities).
H . . .2.2 . . .Li Be . . . B C N O F1.0 1.6 . . . 2.0 2.6 3.0 3.4 4.0Na Mg . . . Al Si P S Cl0.9 1.3 . . . 1.6 1.9 2.2 2.6 3.2K Ca . . . Ga Ge As Se Br0.8 1.0 . . . 1.8 2.0 2.2 2.6 3.0Rb Sr . . . In Sn Sb Te I0.8 0.9 . . . 1.8 1.9 2.1 2.1 2.7Cs Ba . . . Tl Pb Bi Po At0.8 0.9 . . . 1.8 2.1 2.0 2.0 2.2
• The process of giving up electrons (in the case of cations) and of taking electrons (anions) permits the ionto achieve a stable electronic configuration such as that of
– a noble gas: For example, Na+ and F− have the Ne configuration
– the d10 configuration: Ga3+ takes this up
6
Materials 100A: Orbitals, bonding, etc.
– the s2 configuration: Pb2+ and Bi3+ take this up
• Once they have done this, they can pair up suitably to form ionic solids that are held together byCoulombic interactions
• For any ionic crystal, the attractive Coulombic part (per mole) is:
∆UAtt. = −(LA|z+||z−|e2
4πε0r
)L is the Avogadro number.
• The repulsive part arises because atoms and ions behave nearly like hard spheres. This is a consequenceof the Pauli exclusion principle which says that no two electrons in a system can have all four quantumnumbers the same.
• The repulsion can be approximated by the expression:
∆URep. =LB
rn
where B is called the repulsion coefficient and n is the Born exponent. n is normally around 8 or 9.
The two terms add:
∆U(0 K) = −LA|z+||z−|e2
4πε0r+LB
rn
Covalent bonding
• Covalent bonds are formed between non-metallic (usually) atoms of similar electronegativity. s or porbitals are used.
For example, the 1s orbitals on two hydrogen atoms combine to form the molecular orbitals σ(1s) whichis bonding and σ∗(1s), which is antibonding. The two electrons occupy the bonding level and leave theantibonding level empty. In the following depiction, the circles are the 1s orbitals:
Ene
rgy
bonding molecular orbital
atomic orbital
antibonding molecular orbital
• Why is covalent bonding strongly directional ? The example of sp3 hybrids in diamond and Si:
7
Materials 100A: Orbitals, bonding, etc.
s
pz
px py
Hybrid orbitals are obtained from linear combinations of atomic orbitals on the same atom. These hybridorbitals can then overlap with similar hybrid orbitals on neighboring atoms, just as the 1s orbitals do inthe hydrogen chain.
Metallic bonding
This is a special case of covalent bonding where all the states are not filled up, and the electrons floataround fixed nuclei in the solid.
8
