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Materials 100A: Orbitals, bonding, etc.Orbitals, and the Periodic TableRam Seshadri MRL 2031, x6129, [email protected] notes closely follow P. W. Atkins, Physical ChemistryThe 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 Schrödinder 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 13. The magnetic quantum number corresponding to the z component of the angular momentum ml , whichtakes on the values 0, 1, 2, . . . l4. The spin quantum number ms which takes on the values 12The energy of an electron in an orbital with quantum number n for an atom with atomic number Z is given by:En Z 2 µe432π 2 20 2 n2Where e is the charge on the electron, 0 is the vacuum permittivity, and µ is the reduced mass of the system.Shells, subshells . . . :n 1K2L3M4NThe different quantum numbers define the shell, subshells . . .andl 0s1p2d3f4g.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, 357Electrons2262610261014Rules 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 12 and the other, ms 21 ).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:1s2s2p3s3p3d4s4p4d4f5s5p5d5f6s6p6d2

Materials 100A: Orbitals, bonding, etc.s block1n 1energy, filling, ZHe4LiBe511n 3Na19n rCa2131n 5d block2H3n 2p 43Tc44Ru45Rh46Pd47Ag48CdXeFrom 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 12 C 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 12 C 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 12 C 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

OTASSIUM19SODIUMNa ROGENHY88.906ActinideAc-LrEditor: Aditya Vardhan ([email protected])However three such elements (Th, Pa, and U)do have a characteristic terrestrial isotopiccomposition, and for these an atomic weight istabulated.Relative atomic mass is shown with fivesignificant figures. For elements have no stablenuclides, the value enclosed in bracketsindicates the mass number of the longest-livedisotope of the element.91.224178.4992.906Cr NIDE89 (227) u IUM LEVIUM174.97(262)Lr103LUTETIUMLu71NOBELIUM LAWRENCIUMFm Md NE53BROMINEBr35CHLORINECl1720.180Ne10HELIUMHe18 VIIIA2 4.0026Copyright 1998-2002 EniG. MSe34SULPHUR16FLUORINEF18.998VIIAEr Tm OVIA 1715.999 9OXYGENVA 1614.007 8NITROGENIVA 1512.011 7CARBONIIIA 1410.811 6BORON13135BERKELIUM CALIFORNIUM EINSTEINIUMPu Am Cm Bk94157.25Gd64MEITNERIUM UNUNNILIUM UNUNUNIUMNEPTUNIUM PLUTONIUMNp93107.87Ag47COPPER65.39IIBZnIB 1263.546 30- synthetic- solidCu1129FeTcMt Uun Uuu Uub109IRIDIUM77RHODIUM46NICKELNi58.693Nd Pm Sm RASEODYMIUM NEODYMIUM PROMETHIUM 262)Tc(98)Noble gasHalogens element- liquid- gasVIIIB91027 58.933 28NeGaFeMOLYBDENUM TECHNETIUM RUTHENIUMDUBNIUM105TANTALUMTa73NIOBIUM43CHROMIUM MANGANESE55.845ActinideLanthanideTransition metalsNonmetalChalcogens elementSTANDARD STATE (100 C; 101 kPa)SemimetalAlkaline earth metalAlkali metalVIB 7 VIIB 851.996 25 54.938 26Nb Mo41VANADIUMVVB 650.942 24LANTHANIDE57 138.91 58 140.12 NB10.811IVB 547.867 23TITANIUM89-103 104LanthanideLa-Lu57-71YTTRIUM39SCANDIUM135GROUP CASELEMENT NAMETiIIIB 444.956 22Sc321SYMBOLATOMIC NUMBERGROUP TIVE ATOMIC MASS (1)PERIODIC TABLE OF THE ELEMENTSIA1.00794(1) Pure Appl. Chem., 73, No. 4, 667-683 (2001)PERIODGROUP11Materials 100A: Orbitals, bonding, etc.

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 Schrödingerequation (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).1 Hydrogen 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.nucleuselectron cloud 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:AtomNeArKrXeTM (K)2484116161TB (K)2787120165 Other columns of elements in the periodic table don’t follow this simple trend. For example:AtomCuAgAuTM (K)135312351333TB (K)283324333133Ionic 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 S2.6Se2.6Te2.1Po2.0F4.0Cl3.2Br3.0I2.7At2.2 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 up6

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 e24π 0 r 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. LBrnwhere 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 e2LB n4π 0 rrCovalent 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:Energyantibonding molecular orbitalatomic orbitalbonding molecular orbital Why is covalent bonding strongly directional ? The example of sp3 hybrids in diamond and Si:7

Materials 100A: Orbitals, bonding, etc.spxpypzHybrid 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 bondingThis is a special case of covalent bonding where all the sta