n 4 l 3 m 3 2 1 0 1 2 3 7 7 4 f orbitals but play lesser role and we don t do them here Size and shape of orbitals Drumming http en wikipedia org wiki Atomic orbital 0 1 1s 1 1 2p 0 2 2s 1 2 3p 0 3 3s 2 1 3d Chladni 1787 Patterns Sand on violin Electron Spin Fig 5 15 p 209 Electron spin quantum number ms Stern and Gerlach experiment H beam n 1 l 0 m 0 Beam split into 2 paths by magnet Electrons have intrinsic angular momentum characterized by quantum number relativistic effect Electron spin Chemistry we have the spin quantum number ms 1 2 or 1 2 up or down spin Electron is not actually spinning Spin doubles the number of allowed quantum states We now have 2n2 states rather than n2 n l m ms n l n 1 0 m l 0 l and ms 1 2 Overcoming the multielectron problem many bodied problem for multielectron atoms Each electron s motion can be described by a single particle function orbital which does not depend explicitly on the instantaneous motions of the other electrons Is the basis of Molecular Orbital Theory Orbitals are mathematical constructs which only approximate reality Hartree theory o For multi electron atom must consider Coulomb interactions between its Z electrons and its nucleus of charge Ze Largest effects due to large nuclear charge o Must also consider Coulomb interactions between each electron and all other electrons in atom Effect is weak o Assume electrons are moving independently in a spherically symmetric net potential o The net potential is the sum of the spherically symmetric attractive Coulomb potential due to the nucleus and a spherically symmetric repulsive Coulomb potential which represents the average effect of the electrons and its Z 1 colleagues o Hartree 1928 attempted to solve the time independent Schr dinger equation for Z electrons in a net potential o Total potential of the atom can be written as the sum of a set of Z identical net potentials V r each depending on r of the electron only Screening Hartree theory results in a shell model of atomic structure which includes the concept of screening For example alkali atom can be modelled as having a valence electron at a large distance from nucleus Moves in an electrostatic field of nucleus Ze which is screened by the Z 1 inner electrons This is described by the effective potential Veff r At r small Veff r Ze2 r Unscreened nuclear Coulomb potential At r large Veff r e2 r Nuclear charge is screened to one unit of charge e r Ze Z 1 e Shell Model for many electron atoms Hartree orbitals Self consistent field approximation SCF Generates set of approximate one electron orbitals with associated energy is represents set of quantum numbers Put Z electrons in these orbits according to rules later Hartree orbitals assumptions Approximating orbitals Electron moves as independent particle described by one electron orbital similar to H atom For boron 1s 2s 2p this is known as the Hartree Product Radial probability density vs r Hartree orbitals in Argon n 1 2 3 SCF 18 electrons 1s 2s 2p 3s 3p Notice similarities of all orbitals at a given n suggesting shell model Note the radius of the peaks of the orbitals 10 less than that of H n 2 n 3 Fig 5 16 p 212 Total radial charge distribution of Argon calculated using Hartree s method The sum of the summing up that of Ar 1s 2s 2px y z Notice shell structures at n 1 2 3 Subshell structure within each shell is defined as orbitals with same n and l values Fig 5 17 p 213 Shielding effect and energy degeneracy Assume that in each shell n electron moves in an effective field Vneff Zeff n is the effective nuclear charge in shell n Zeff n Z S Zeff n ranges from Z near nucleus to 1 where complete sheilding occurs Argon calculations show that Zeff 1 16 Zeff 2 8 Zeff 3 2 5 Hartree orbital energy n Zeff n 2 n2 Rydbergs Slater s rules empirical method to determine energies and radii of orbitals Determines the shielding S for ns or np orbitals Zeff n Z S n Zeff n 2 n2 Rydbergs 1 write electronic config in order and grouped as 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 2 electrons to right of group don t shield 3 each electron in a ns np group contributes 0 35 to S 3 each electron in a n 1 shell contributes 0 8 to S 4 each electron in a n 2 or lower fully shields S 1 Today these values are computed with advanced methods Table 5 3 p 215 Dependence of energy on l as well as n s orbitals penetrate nucleus p and d have nodes ns np nd l degeneracy is lifted Fig 5 18 p 215 Aufbau Principles Pauli exclusion principle no two electrons in atom can have same set of n l m ms each orbital n l m holds 2 electrons with ms 1 2 1 2 Hund s Rule Electrons fill each orbital in the subshell before pairing up with opposite spins minimizes e e repulsion Carbon 6 electrons 1s2 2s2 2p2 Fig 5 19 p 216 Diamagnetic and Paramagnetic atoms confirms Hund s Rule Atoms in a magnetic field paramagnetic if attracted into a magnetic field diamagnetic if its pushed out of a magnetic field smaller force H Li B C paramagnetic He Be diamagnetic All of the electrons are spin paired in diamagnetic elements Paramagnetic elements subshells are not completely filled with electrons For the atoms write out the electron configuration Note in molecules we have considerations covered later iron nickel cobalt and gadolinium metals are ferromagnetic Periodic table can be built up using this shell filling process Must apply 1 Pauli exclusion principle Only two electrons with opposite spin can occupy an atomic orbital i e no two electrons have the same 4 quantum numbers 2 Hunds rule Electrons fill each orbital in the subshell before pairing up with opposite spins shorthand electron configurations notation using noble gas symbol Silicon Si 1s2 2s2 2p6 3s2 3p2 in general we are interested in valence outer electrons write as Ne 3s2 3p2 Transition Metals d block and Beyond After 3p are filled with 6 electrons Ar 18 electrons 3d orbitals are due to be filled K 9 Zn 30 However 3d 4s for K Ca calculations show 3d 4s K is Ar 3d0 4s1 Ca is Ar 3d0 4s2 Sc and beyond 3d 4s Sc Ar 3d3 4s0 has more e e repulsion than Ar 3d1 4s2 expect that Sc Zn Ar 3d1 4s2 follow this energy minimization except Cr Ar 3d5 4s1 and Cu Ar 3d10 4s1 here half filled and fully filled d shell preferred similar considerations occur in 5th period and f block Anomalous electron configurations Fig 5 21 p 220 o Below are atomic shells listed in order of increasing energy Nshell 2 2l 1 …