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William A. Goddard, III, wag3@kaist.ac.kr

Lecture 4, September 10, 2009. Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy. Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday.

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William A. Goddard, III, wag3@kaist.ac.kr

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  1. Lecture 4, September 10, 2009 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday William A. Goddard, III, wag3@kaist.ac.kr WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Senior Assistant: Dr. Hyungjun Kim: linus16@kaist.ac.kr Teaching Assistant: Ms. Ga In Lee: leeandgain@kaist.ac.kr EEWS-90.502-Goddard-L04

  2. Last time: The nodal Theorem Φ00 Φ10 + Φ01 Φ20 + - + - - + + - + Φ11 Φ21 + - + + - - + - The ground state of a system is nodeless (more properly, the ground state never changes sign). Useful in reasoning about wavefunctions. Implies that ground state wavefunctions for H2+ are g not u) For one dimensional finite systems, we can order all eigenstates by the number of nodes E0 < E1 < E2 .... En < En+1 The nodal Theorem sometimes orders excited states in 2D, 3D E00 < E10 < E20< E21 and E00 < E01 < E11 < E21 but nodal argument does not indicate the relative energies of E10 and E20 versus E01 EEWS-90.502-Goddard-L04

  3. 4th postulate of QM Consider the exact eigenstate of a system HΦ = EΦ and multiply the Schrödinger equation by some CONSTANT phase factor (independent of position and time) exp(ia) = eia eia HΦ = H (eiaΦ) = E (eiaΦ) Thus Φ and (eiaΦ) lead to identical properties and we consider them to describe exactly the same state. EEWS-90.502-Goddard-L04

  4. The Hamiltonian for H2+ Coordinates of H atom r 1 1 1 For H atom the Hamiltonian is Ĥ = - (Ћ2/2me)2– e2/r or Ĥ = - ½ 2– 1/r (in atomic units) For H2+ molecule the Hamiltonian (in atomic units) is Ĥ = - ½ 2+ V(r) where Coordinates of H2+ We will rewrite this as EEWS-90.502-Goddard-L04

  5. The Schrödinger Equation for H2+ The exact (electronic wavefunction of H2+ is obtained by solving Here we can ignore the 1/R term (not depend on electron coordinates) to write where e is the electronic energy Then the total energy E becomes R E= e + 1/R Since v(r) depends on R, the wavefunction φ depends on R. E(R) Thus for each R we solve for φ and e and add to 1/R to get the total energy E(R) EEWS-90.502-Goddard-L04

  6. Inversion Symmetry Applying the inversion twice, leads to the identity x  -x  x; y  -y  y z  -z  z Nothing changes! The identity or do-nothing operator, is called einheit. The inversion operator is of order 2 = ( )2 = Thus ^ ^ e e ^ ^ ^ ^ I I I I The operation of inversion (denoted as ) through the origin of a coordinate system changes the coordinates as x  -x y  -y z  -z Taking the origin of the coordinate system as the bond midpoint, inversion changes the electronic coordinates as illustrated. EEWS-90.502-Goddard-L04

  7. Symmetry Theorem φg(r) = + φg(r) φu(r) = - φu(r) φg(r) φu(r) ^ ^ I I If h(r) is invariant under inversion: h(-r) = h(r) Then for all eigenfunctions φ(r)of h h(r)φ(r) = eφ(r) g for gerade or even u for ungerade or odd EEWS-90.502-Goddard-L04

  8. Now consider symmetry for H2 molecule For multielectron systems, inversion, , inverts all electron coordinates simultaneously ^ I For H2 the Hamiltonian is 1/r12 interaction between 2 electrons all terms depending only on electron i H(1,2)Φ(1,2) = E Φ(1,2) If H(1,2) is invariant under inversion: H(-r1,-r2) = H(r1,r2) Then for all eigenfunctionsΦ(1,2) of H EEWS-90.502-Goddard-L04

  9. inversion symmetry for H2 wavefunctions g u u g EEWS-90.502-Goddard-L04

  10. Permutation Symmetry Transposing the two electrons in H(1,2) must leave the Hamiltonian invariant since the electrons are identical H(2,1) = h(2) + h(1) + 1/r12 + 1/R = H(1,2) We denote transposition as t where tΦ(1,2) = Φ(2,1) Applying t twice leads to the identity t2 = e, t2Φ(1,2) = Φ(1,2) Thus the previous arguments on inversion apply equally to transposition EEWS-90.502-Goddard-L04

  11. permutational symmetry for H2 wavefunctions symmetric antisymmetric EEWS-90.502-Goddard-L04

  12. Electron spin Consider application of a magnetic field Our hamiltonian has not no terms dependent on the magnetic field. Hence no effect. But experimentally there is a huge effect. Namely The ground state of H atom splits into two states This leads to the 5th postulate of QM In addition to the 3 spatial coordinates x,y,z each electron has internal or spin coordinates which leads to a magnetic dipole that is either aligned with the external magnetic field or it is opposite. We label these as a for spin up and b for spin down. Thus the ground states of H atom are φ(xyz)a(spin) and φ(xyz)b(spin) EEWS-90.502-Goddard-L04

  13. Spin states for 1 electron systems Our Hamiltonian does not involve any terms dependent on the spin, so without a magnetic field we have 2 degenerate states for H atom. φ(r)a, with up-spin, ms = +1/2 φ(r)b, with down-spin, ms = -1/2 The electron is said to have a spin anglular momentum of S=1/2 with projections along a polar axis (say the external magnetic moment) of +1/2 (spin up) or -1/2 (down spin). This explains the observed splitting of the H atom into two states in a magnetic field Similarly for H2+ the ground state becomes φg(r)a and φg(r)b While the excited state becomes φu(r)a and φu(r)b EEWS-90.502-Goddard-L04

  14. Electron spin b B=0 Increasing B a So far we have considered the electron as a point particle with mass, me, and charge, -e. In fact the electron has internal coordinates, that we refer to as spin, with two possible angular momenta. +½ or a or up-spin and -½ or b or down-spin But the only external manifestation is that this spin leads to a magnetic moment that interacts with an external magnetic field to splt into two states, one more stable and the other less stable by an equal amount. DE = -gBzsz Now the wavefunction of an atom is written as ψ(r,s) where r refers to the vector of 3 spatial coordinates, x,y,z whiles to the internal spin coordinates EEWS-90.502-Goddard-L04

  15. Spinorbitals The Hamiltonian does not depend on spin the spatial and spin coordinates are independent. Hence the wavefunction can be written as a product of a spatial wavefunction, φ(s), called an orbital, and a spin function, х(s) = a or b. ψ(r,s) = φ(s) х(s) where r refers to the vector of 3 spatial coordinates, x,y,z whiles to the internal spin coordinates. EEWS-90.502-Goddard-L04

  16. spinorbitals for two-electron systems Thus for a two-electron system with independent electrons, the wavefunction becomes Ψ(1,2) = Ψ(r1,s1,r2,s2) = ψa(r1,s1) ψb(r2,s2) = φa(r1) хa(s1) φb(r2) хb(s2) =[φa(r1) φb(r2)][хa(s1) хb(s2)] Where the last term factors the total wavefunction into space and spin parts EEWS-90.502-Goddard-L04

  17. Spin states for 2-electron systems Since each electron can have up or down spin, any two-electron system, such as H2 molecule will lead to 4 possible spin states each with the same energy Φ(1,2) a(1) a(2) Φ(1,2) a(1) b(2) Φ(1,2) b(1) a(2) Φ(1,2) b(1) b(2) This immediately raises an issue with permutational symmetry Since the Hamiltonian is invariant under interchange of the spin for electron 1 and the spin for electron 2, the two-electron spin functions must be symmetric or antisymmetric with respect to interchange of the spin coordinates, s1and s2 Symmetric spin Neither symmetric nor antisymmetric Symmetric spin EEWS-90.502-Goddard-L04

  18. Spin states for 2 electron systems Combining the two-electron spin functions to form symmetric and antisymmetric combinations leads to Φ(1,2) a(1) a(2) Φ(1,2) [a(1) b(2) + b(1) a(2)] Φ(1,2) b(1) b(2) Φ(1,2) [a(1) b(2) - b(1) a(2)] Adding the spin quantum numbers, ms, to obtain the total spin projection, MS = ms1 + ms2 leads to the numbers above. The three symmetric spin states are considered to have spin S=1 with components +1.0,-1, which are referred to as a triplet state (since it leads to 3 levels in a magnetic field) The antisymmetric state is considered to have spin S=0 with just one component, 0. It is called a singlet state. MS +1 0 -1 0 Symmetric spin Antisymmetric spin EEWS-90.502-Goddard-L04

  19. Permutational symmetry Our Hamiltonian for H2, H(1,2) =h(1) + h(2) + 1/r12 + 1/R Does not involve spin This it is invariant under 3 kinds of permutaions Space only: r1  r2 Spin only: s1 s2 Space and spin simultaneously: (r1,s1)  (r2,s2) Since doing any of these interchanges twice leads to the identity, we know from previous arguments that Ψ(2,1) =  Ψ(1,2) symmetry for transposing spin and space coord Φ(2,1) = Φ(1,2) symmetry for transposing space coord Χ(2,1) =  Χ(1,2) symmetry for transposing spin coord EEWS-90.502-Goddard-L04

  20. Permutational symmetries for H2 and He H2 He the only states observed are those for which the wavefunction changes sign upon transposing all coordinates of electron 1 and 2 Leads to the 6th postulate of QM EEWS-90.502-Goddard-L04

  21. The 6th postulate of QM: the Pauli Principle For every state of an electronic system H(1,2,…i…j…N)Ψ(1,2,…i…j…N) = EΨ(1,2,…i…j…N) The electronic wavefunction Ψ(1,2,…i…j…N) changes sign upon transposing the total (space and spin) coordinates of any two electrons Ψ(1,2,…j…i…N) = - Ψ(1,2,…i…j…N) We can write this as tijΨ = - Ψ for all I and j EEWS-90.502-Goddard-L04

  22. Implications of the Pauli Principle Consider two independent electrons, 1 on the earth described by ψe(1) and 2 on the moon described by ψm(2) Ψ(1,2)= ψe(1) ψm(2) And test whether this satisfies the Pauli Principle Ψ(2,1)= ψm(1) ψe(2) ≠ - ψe(1) ψm(2) Thus the Pauli Principle does NOT allow the simple product wavefunction for two independent electrons EEWS-90.502-Goddard-L04

  23. Quick fix to satisfy the Pauli Principle Combine the product wavefunctions to form a symmetric combination Ψs(1,2)= ψe(1) ψm(2) + ψm(1) ψe(2) And an antisymmetric combination Ψa(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2) We see that t12Ψs(1,2) = Ψs(2,1) = Ψs(1,2) (boson symmetry) t12Ψa(1,2) = Ψa(2,1) = -Ψa(1,2) (Fermion symmetry) Thus the Pauli Principle only allows the antisymmetric combination for two independent electrons EEWS-90.502-Goddard-L04

  24. Consider some simple cases: identical spinorbitals Ψ(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2) Identical spinorbitals: assume that ψm = ψe Then Ψ(1,2)= ψe(1) ψe(2) - ψe(1) ψe(2) = 0 Thus two electrons cannot be in identical spinorbitals Note that if ψm = eia ψe where a is a constant phase factor, we still get zero EEWS-90.502-Goddard-L04

  25. Consider some simple cases: orthogonality Consider the wavefuntion Ψold(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2) where the spinorbitals ψm and ψe areorthogonal hence <ψm|ψe> = 0 Define a new spinorbital θm = ψm + lψe (ignore normalization) That is not orthogonal to ψe. Then Ψnew(1,2)= ψe(1) θm(2) - θm(1) ψe(2) = ψe(1) θm(2) + l ψe(1) ψe(2) - θm(1) ψe(2) - l ψe(1) ψe(2) = ψe(1) ψm(2) - ψm(1) ψe(2) =Ψold(1,2) Thus the Pauli Principle leads to orthogonality of spinorbitals for different electrons, <ψi|ψj> = dij = 1 if i=j =0 if i≠j EEWS-90.502-Goddard-L04

  26. Consider some simple cases: nonuniqueness Starting with the wavefuntion Ψold(1,2)= ψe(1) ψm(2) - ψm(1) ψe(2) Consider the new spinorbitals θm and θe where θm = (cosa) ψm + (sina) ψe θe = (cosa) ψe - (sina) ψm Note that <θi|θj> = dij Then Ψnew(1,2)= θe(1) θm(2) - θm(1) θe(2) = +(cosa)2ψe(1)ψm(2) +(cosa)(sina) ψe(1)ψe(2) -(sina)(cosa) ψm(1) ψm(2) - (sina)2ψm(1) ψe(2) -(cosa)2ψm(1) ψe(2) +(cosa)(sina) ψm(1) ψm(2) -(sina)(cosa) ψe(1) ψe(2) +(sina)2ψe(1) ψm(2) [(cosa)2+(sina)2] [ψe(1)ψm(2) - ψm(1) ψe(2)] =Ψold(1,2) Thus linear combinations of the spinorbitals do not change Ψ(1,2) EEWS-90.502-Goddard-L04

  27. Determinants The determinant of a matrix is defined as The determinant is zero if any two columns (or rows) are identical Adding some amount of any one column to any other column leaves the determinant unchanged. Thus each column can be made orthogonal to all other columns.(and the same for rows) The above properities are just those of the Pauli Principle Thus we will take determinants of our wavefunctions. EEWS-90.502-Goddard-L04

  28. The antisymmetrized wavefunction Now put the spinorbitals into the matrix and take the deteminant Where the antisymmetrizer can be thought of as the determinant operator. Similarly starting with the 3!=6 product wavefunctions of the form The only combination satisfying the Pauil Principle is EEWS-90.502-Goddard-L04

  29. Example: Interchanging electrons 1 and 3 leads to From the properties of determinants we know that interchanging any two columns (or rows) that is interchanging any two spinorbitals, merely changes the sign of the wavefunction Guaranteeing that the Pauli Principle is always satisfied EEWS-90.502-Goddard-L04

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