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Composite systems Physical Chemistry A quantum system such as the helium atom is a composite system Lecture 19 Angular Momentum of Composite Systems Overall properties of composite systems are derivable from the properties of components Adding angular momenta Angular momentum – a vector property Finding the sum of two angular momenta requires finding all possible vectors that can be produced Must keep in mind quantum nature of angular momentum Parts are identifiable Parts may consist of distinguishable or indistinguishable particles Total electron density at a point is the sum of the electron densities due to the various electrons Angular momentum of the total system is derivable from the angular momenta of the parts Angular momentum of composite system Must obey rules of 2 2 quantum mechanical L Ψ = L(L + 1)h Ψ angular momentum Lz Ψ = mh Ψ Operators for total angular momentum L = L1 + L 2 related to angular L2 = L • L momentum operators of the = L12 + L22 + 2 L1 • L 2 parts of the system L = L + L Can add only certain orientations Retain uncertainty about orientation z Combination of two angular momenta Only certain values of the total angular momentum quantum number, L Determined by the values of the angular momentum quantum numbers of the subparts of the system Can determine the total orbital angular momentum for any number of electrons combined | L1 − L2 | ≤ L ≤ L1 + L2 1z 2z Orbital terms The orbital wave functions for multielectron systems are simultaneous eigenfunctions of L2 and Lz of the total system Multi-electron states labeled by the total angular momentum Use capital Roman letters L Symbol 0 S 1 P 2 D 3 F 4 G 5 H 1 Labeling spin states of composite systems Spin of multi-electron systems Like orbital angular momentum Combine spins of subparts to determine the possible values of the total spin Operators of the total system are related to the operators of the subparts Can add more than two Repeat combining operations for additional spins S 2 Ψspin = S (S + 1)h 2 Ψspin S z Ψspin = mS h Ψspin S = S1 + S2 S2 = S•S = S + S = S1 z + S2 z 2 1 Sz Combined spin states are eigenfunctions of S2 and Sz for the total system of spins Superscript labels the total spin state of the term 2 2 + 2 S1 • S 2 Pauli’s exclusion principle β⎠ ⎝ Limit on quantum numbers requires increment of principal quantum number All electrons indistinguishable. Requires antisymmetrization. Configuration S L s1 1/2 0 s2 0 0 1 1 ss 0 or 1 0 s1p1 0 or 1 1 Term 2 S 1 S 1 S, 3S 1 P, 3P 1 1 1 pp 0 or 1 0, 1, or 2 D, 1P, 1S 3 D, 3P, 3S p2 0 or 1 0, 1, or 2 1 S, 3P, 1D ½ 1 3/2 2 Label 1 2 3 4 5 Name singlet doublet triplet quartet quintet The spatial part of the wave function has a degeneracy, gL The spin part of the wave function has a degeneracy. gS The total degeneracy is determined by the product of these two degeneracies g State g Terms arising from various configurations 0 Degeneracies of terms In principle, one could create a configuration in which all electrons are in the lowestΨspatial (r1 , r2 , r3 ) = Ψ1s (r1 )Ψ1s (r2 ) Ψ1s (r3 ) energy state Example: lithium atom Cannot happen Pauli’s principle: there can never be two equivalent electrons in an atom for which the values of all the quantum α⎞ ⎛ ΨLi , gs = A ⎜⎜ Ψ1sαΨ1s βΨ2 s ⎟⎟ numbers are the same. S = gL × gS 1S 3S 1P 3P 2S 2D 1 3 3 9 2 10 Unsőld’s theorem An important simplification when considering a complex system with many electrons All closed shells can have only zero spin and zero orbital angular momentum. 2 Ground terms of atoms Atom Ground configuration Ground term H 1s1 2S He 1s2 1S Li 1s22s1 2S Be 1s22s2 1S B 1s22s22p1 2P C 1s22s22p2 3P N 1s22s22p3 4S O 1s22s22p4 3P F 1s22s22p5 2P Ne 1s22s22p6 1S Summary The labeling of states (terms) is determined by the total angular momenta of the spin and space parts of the wave function Atomic terms are specified by straightforward labels Must focus on the unfilled orbtials of the atom to determine the term symbol (application of Unsöld’s theorem) 3