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The algebraic way to molecular structure Introductory Lecture given at National Institute of Chemical Physics and Biophysics , Tallinn, Estonia (Feb 2017) Lorenzo Fortunato – Padova Univ. (Italy) & I.N.F.N. H2 University of Padova, Italy (founded 1222) Dept. of Physics and Astronomy + National Insitute of Nuclear Physics Summary Online at: www.pd.infn.it/~fortunat/lectures.html • Why Lie algebras allow to write meaningful model hamiltonians • Dynamical symmetries and spectrum generating algebra • Examples (useful for Molecular Physics): Angular momentum Rigid rotor SO(3) : CO molecule The U(4) vibron model for diatomic molecules • Seminar: The algebraic approach to endohedral molecules Two equivalent visions of quantum mechanics Schroedinger wave eq.: Heisenberg matrix eq. : • coordinates, momenta & energies • operators (differential matrices, etc.) • interactions have often a clear physical interpretation • interactions are more general, but sometimes difficult to intepret Non • equations are algebrized • differential equation Commutativity ! E.Schroedinger, Ann.Physik 79, 361 (1926) W.Heisenberg, Z.Physik 33, 879 (1925) Equivalence E.Schroedinger, Ann.Physik 79, 734 (1926) 2° quant. : V.Fock, Z.Physik 98, 145 (1935) L. Fortunato Example: Angular Momentum so(3) L =r p [Lx , Ly ] = iLz Lx , Ly , Lz are generators of so(3) +cyclic permutations over indices C[so(3)] = L2= Lx2 + Ly2 + Lz2 is the quadratic Casimir operator (rank 1) [C , Li ] = 0 , Li SO(3) so(3) so(2) L M | Branching problem Branching rules: -L M L L. Fortunato … continues ... One must specify the action of the generators on a given orthonormal basis |LM L±|LM = [(LM)(L±M+1)]½|LM±1 iLy L± = Lx ± Lz|LM = M|LM L2|LM = L(L+1)|LM Application in the rigid rotor: H=k L² E=k L(L+1) with k=(h²/2I) Notice it is the Casimir operator of SO(3) L. Fortunato … quantum rotors H2 HCl, CO, etc. EJ= B J(J+1) EJ= 2B(J+1) L. Fortunato Intensità /unità arbitrarie Spettro a Microonde (Lontano IR ) CO 0 20 40 60 80 100 Numeri d’onda cm-1 EJ= B J(J+1) EJ= 2B(J+1) L. Fortunato Spectrum Generating Algebra (SGA) When, in general, one can write an hamiltonian H= E0 + ckXk + cklXk Xl +… with Xk G as a polynomial in the elements of an algebra then G is called spectrum generating algebra (SGA) for H, because it is always possible to diagonalize (numerically) H in the ONC basis labelled by all the quantum numbers of a Complete Set of Commuting Operators (CSCO) of any of the possible chains of subalgebras of G G’ G’’ … Once the action of the Xk on | is given, then one can calculate the matrix elements |H|’ L. Fortunato Example of SGA The non-compact algebra SO(2,1) can be realized with the following differential operators: Z1=p2 Z2=r2 Z3= k(r·p + p·r) These operators close under commutation with the structure constants typical of the SO(2,1) algebra. Once the action of the Zk on some | is given, then one can calculate the matrix elements |H|’ of hamiltonians, for example the harmonic oscillator : H = ½p2 + ½r2 (a part from some constants) L. Fortunato Dynamical Symmetry In some cases H= E0 + ckXk + cklXk Xl +… with Xk G we have only some terms that correspond to invariant operators of the algebras in the chain: G G’ G’’ … C C’ C’’ chain of subalgebras (one or more for each subalgebra) H= E0 + aC + a’C’ +a’’C’’ +… in these cases we speak of a dynamical symmetry and we have immediately (NO NEED FOR CALCULATIONS) E= E0 + aC + a’ C’ +a’’ C’’ +… L. Fortunato Subalgebra chains: just as an example Use in the IBM in nuclear physics The number of possible chains grows with the dimensions of the algebra. It might get really complicated... L. Fortunato Consequences of a DS -1 1) All the states are soluble and we have analytic expressions for energy and other observables E= E0 + aC + a’ C’ +a’’ C’’ +… 2) All the states are characterized by quantum numbers that “label” the irreducible representations (IRREPS) of the chain of subalgebras |12...n 3) The structure of the wavefunctions is dictated by symmetry and it’s independent from the details of the hamiltonian L. Fortunato Consequences of DS -2 Assume that H commutes with a set of operators that form a given Lie algebra: If | is eigenstate of H, then also gi| is an eigenstate of H and we have degeneration. At the very origin of degeneration there is a conserved quantity, an invariant, that is the Casimir operator of some group. L. Fortunato Multiplets, degeneration and splitting. = { | lm } ONC basis states l m The dynamical symmetry splits, but do not admix the states of a basis! L. Fortunato Magnetic Degeneration M= +Jz J2 M= -Jz J1 O(3) O(2) States with a definite total angular momentum contain a multiplet of substates with different third component: these are called magnetic substates because can be separated with a magnetic field (Zeeman effect) Said another way, the magnetic interaction -mB breaks the symmetry of the hamiltonian (Zeeman) L. Fortunato Not easy to digest…let’s go through it once again D o th e ir c o m m u ta to rs c lo s e ? [X i,X j]= X k X i i= 1 ,...,n o p e ra to rs C a n y o u s o lv e th e b ra n c h in g p ro b le m ? C h a in o f n e s te d L ie a lg e b ra s If y e s L ie A lg e b ra o f so m e r a n k (g e n e ra to rs) so ( 3 ) m u st b e co n ta in e d a s a su b a lg e b r a a ll cla ssifie d kn o w n p r o p e r tie s: - in va r ia n t o p . - m a tr ix e le m e n ts C a n H c a n b e e x p re s s e d a s a lin .c o m b . o f C a s im ir o p e ra to rs o f th e c h a in ? C a n H c a n b e e x p re s s e d in te rm s o f th e g e n e ra to rs ? If y e s T h e m a trix e le m e n ts o f th e h a m ilto n a in c a n b e c a lc u la te d a n d th e n H c a n b e d ia g o n a liz e d S p e c tru m g e n e ra tin g a lg e b ra If y e s H is a lre a d y d ia g o n a l in th e O .N . b a s is d e fin e d b y th e c h a in . S p e c tru m c a n b e re a d o ff d ire c tly D y n a m ic a l S y m m e tr y L. Fortunato Example: hydrogen atom so(4) The states of the hamiltonian of the H atom are clearly invariant with respect to SO(3), but… L. Fortunato Further degeneration The spectrum shows a further degeneration in . Where does it come from ? Degeneration Conserved Quantity Runge-Lenz vector: [H,A]=0 there’s a larger symm. group that contains both L and A so(4) so(3) so(3) L. Fortunato L. Fortunato Other important dynamical Symmetries From R.Bijker L. Fortunato The vibron model , U(4) diatomic molecules Diatomic Molecules • Dipolar interaction N = 3 • Spectrum Generating Algebra: U(4) • Vibron Model F. Iachello. Chem. Phys. Lett. 78 581 (1981) F. Iachello and R.D.Levine, book, “Algebraic theory of molecules” L. Fortunato Definitions: “Elementary” bosons (IBM building blocks): b , b † with =1,…,4 b1= s b2,…,4= pm One constructs the u(4) algebra by taking bilinear operators: Such that they close into u(4): [Gr , Gs] = ct Gt L. Fortunato Connection with important physical operators Dipole From the book : F.Iachello & R.D.Levine «Algebraic theory of molecules» Oxford Univ. Press 1995 L. Fortunato Dynamical symmetries: Chains I & II I. Non-rigid molecule (or soft) II. Rigid molecule L. Fortunato Dynamical symmetries: textbook examples of spectra Chain I U(3) Chain II SO(4) L. Fortunato Application to H2 molecular states Figures from A.Frank and P.van Isacker «Symmetry methods in molecules and nuclei» Little discrepancies are explained because this has been fitted with pure SO(4) dyn.symm. and no Dunham Ideally a mixture with U(3) should be used: H = H[U(3)] + (1-) H[SO(4)] Needs powerpoint to see animation L. Fortunato Message to be taken with you • Some simple models are “naturally” written in terms of creation and annihilation operators. • To them we can always associate an algebra that brings with itself a dynamical symmetry. • By knowing how to deal mathematically with the algebra one can get analytic solutions that can be compared with experimental data (not only: you can get also new, unexpected solutions!) • The algebra naturally entails quantum numbers (=classification), selection rules (= explain some weird observations). • The algebra gives a conceptual frame and might give hints on new physics!! L. Fortunato References • F. Iachello. Chem. Phys. Lett. 78 581 (1981) • F. Iachello and R.D.Levine, « Algebraic theory of molecules » • A.Frank and P.van Isacker «Symmetry methods in molecules and nuclei» SyG editores • S. Oss, Advances in chemical physics 93, 455-647 • F. Iachello, S.Oss, Chemical physics letters 205 (2-3), 285-289 • F. Pérez-Bernal and L.F., Physics Letters A 376 (2012) 236–244 • L.F. and F. Pérez-Bernal, Phys.Rev. A 94 (2016) 032508 L. Fortunato Back-up slides L. Fortunato Casimir Operators and Rank For each algebra one can costruct a set of operators, called Casimir operators or invariants, C, such that [C, Xi]=0 , Xi G Hendrik Casimir (1909-2000) dutch physicist The number of independent invariants is called rank of the algebra, namely #C. The order is the number of operators (generators) that form the algebra, namely #Xi G. L. Fortunato Intensities in U(4) vibron model From A.Frank and P.van Isacker book: «Symmetry methods in Molecules and Nuclei» L. Fortunato Fisica Molecolare: acetilene C2H2 F. Pérez-Bernal and L.F. Physics Letters A 376 (2012) 236–244 L. Fortunato

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