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Advanced Physical Chemistry

Advanced Physical Chemistry. G. H. CHEN Department of Chemistry University of Hong Kong. Quantum Chemistry. G. H. Chen Department of Chemistry University of Hong Kong. Emphasis Hartree-Fock method Density-functional theory Concepts Hands-on experience. Text Book

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Advanced Physical Chemistry

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  1. Advanced Physical Chemistry G. H. CHEN Department of Chemistry University of Hong Kong

  2. Quantum Chemistry G. H. Chen Department of Chemistry University of Hong Kong

  3. Emphasis Hartree-Fock method Density-functional theory Concepts Hands-on experience Text Book “Quantum Chemistry”, 4th Ed. Ira N. Levine “Density-functional theory of atoms and molecules”, R.G. Parr and W. T. Yang http://yangtze.hku.hk/teaching.php

  4. Beginning of Computational Chemistry In 1929, Dirac declared, “The underlying physical laws necessary for the mathematical theory of ...the whole of chemistry are thus completely know, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Dirac

  5. Quantum Chemistry Methods • Ab initio molecular orbital methods • Semiempirical molecular orbital methods • Density functional method

  6. SchrÖdinger Equation Hy = Ey Wavefunction Hamiltonian H = (-h2/2ma)2 - (h2/2me)ii2 +  ZaZbe2/rab - i Zae2/ria + ije2/rij Energy Contents 1. Variation Method 2. Hartree-Fock Self-Consistent Field Method

  7. The Variation Method The variation theorem Consider a system whose Hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E1. If f is any normalized, well- behaved function that satisfies the boundary conditions of the problem, then  f* Hf dt >E1

  8. Proof: Expand f in the basis set { yk} f = kakyk where {ak} are coefficients Hyk = Ekyk then f* Hf dt = kjak* aj Ej dkj = k | ak|2Ek> E 1k | ak|2 = E1 Since is normalized, f*f dt = k | ak|2 = 1

  9. i. f : trial function is used to evaluate the upper limit of ground state energy E1 ii. f= ground state wave function,  f* Hf dt = E1 iii. optimize paramemters in f by minimizing  f* Hf dt / f* f dt

  10. Application to a particle in a box of infinite depth l 0 Requirements for the trial wave function: i. zero at boundary; ii. smoothness  a maximum in the center. Trial wave function: f = x (l - x)

  11. * H  dx = -(h2/82m)  (lx-x2) d2(lx-x2)/dx2 dx = h2/(42m)  (x2 - lx)dx = h2l3/(242m) * dx =  x2 (l-x)2 dx = l5/30 E = 5h2/(42l2m)  h2/(8ml2) = E1

  12. Variational Method (1) Construct a wave function (c1,c2,,cm) (2) Calculate the energy of : E E(c1,c2,,cm) (3) Choose {cj*} (i=1,2,,m) so that Eis minimum

  13. Example: one-dimensional harmonic oscillator Potential: V(x) = (1/2) kx2 = (1/2) m2x2 = 22m2x2 Trial wave function for the ground state: (x) = exp(-cx2) * H  dx = -(h2/82m)  exp(-cx2) d2[exp(-cx2)]/dx2 dx + 22m2  x2 exp(-2cx2) dx = (h2/42m) (c/8)1/2 + 2m2 (/8c3)1/2 * dx =  exp(-2cx2) dx = (/2)1/2 c-1/2 E= W = (h2/82m)c + (2/2)m2/c

  14. To minimize W, 0 = dW/dc = h2/82m - (2/2)m2c-2 c = 22m/h W= (1/2) h

  15. Extension of Variation Method    . . . E3y3 E2y2 E1y1 For a wave function f which is orthogonal to the ground state wave function y1, i.e. dtf*y1 = 0 Ef = dtf*Hf / dtf*f>E2 the first excited state energy

  16. The trial wave function f: dtf*y1 = 0 f = k=1 akyk dtf*y1 = |a1|2 = 0 Ef = dtf*Hf / dtf*f = k=2|ak|2Ek / k=2|ak|2 >k=2|ak|2E2 / k=2|ak|2 = E2

  17. Application to H2+ e + + y1 y2 f = c1y1 + c2y2 W = f*H f dt / f*f dt = (c12H11 + 2c1 c2H12+ c22H22 ) / (c12 + 2c1 c2S + c22 ) W (c12 + 2c1 c2S + c22) = c12H11 + 2c1 c2H12+ c22H22

  18. Partial derivative with respect to c1(W/c1 = 0) : W (c1 + S c2) = c1H11 + c2H12 Partial derivative with respect to c2(W/c2 = 0) : W (S c1 + c2) = c1H12 + c2H22 (H11 - W) c1 + (H12 - S W) c2 = 0 (H12 - S W) c1 + (H22 -W) c2 = 0

  19. To have nontrivial solution: H11 - W H12 - S W H12 - S W H22 -W For H2+,H11 = H22; H12 < 0. Ground State: Eg = W1 = (H11+H12) / (1+S) f1= (y1+y2) / 2(1+S)1/2 Excited State: Ee = W2 = (H11-H12) / (1-S) f2= (y1-y2) / 2(1-S)1/2 = 0 bonding orbital Anti-bonding orbital

  20. Results: De = 1.76 eV, Re = 1.32 A Exact: De = 2.79 eV, Re = 1.06 A 1 eV = 23.0605 kcal / mol

  21. 2p 1s Further Improvements H p-1/2exp(-r) He+ 23/2p-1/2exp(-2r) Optimization of 1s orbitals Trial wave function: k3/2p-1/2exp(-kr)  Eg = W1(k,R) at each R, choose kso thatW1/k = 0 Results: De = 2.36 eV, Re = 1.06 A Resutls: De = 2.73 eV, Re = 1.06 A Inclusion of other atomic orbitals

  22. Molecular Orbital (MO):  = c11 + c22   ( H11 - W ) c1 + ( H12 - SW ) c2 = 0 S11=1 ( H21 - SW ) c1 + ( H22 - W ) c2 = 0 S22=1 Generally : i a set of atomic orbitals, basis set LCAO-MO  = c11 + c22 + ...... + cnn linear combination of atomic orbitals n  ( Hik - SikW ) ck = 0 i = 1, 2, ......, n k=1 Hikdti* H k Sikdti*k Skk = 1

  23. The Born-Oppenheimer Approximation Hamiltonian H =(-h2/2ma)2 - (h2/2me)ii2 +  ZaZbe2/rab - i Zae2/ria + ije2/rij H y(ri;ra) = E y(ri;ra)

  24. The Born-Oppenheimer Approximation: • (1) y(ri;ra) = yel(ri;ra) yN(ra) • (2) Hel(ra )= - (h2/2me)ii2- i Zae2/ria • + ije2/rij • VNN = b ZaZbe2/rab • Hel(ra)yel(ri;ra) = Eel(ra)yel(ri;ra) • (3) HN =(-h2/2ma)2 +U(ra) • U(ra) = Eel(ra) + VNN • HN(ra)yN(ra) = E yN(ra)

  25. Hydrogen Molecule H2 e + + e two electrons cannot be in the same state. The Pauli principle

  26. Wave function: f(1,2) = ja(1)jb(2) + c1 ja(2)jb(1) f(2,1) = ja(2)jb(1) + c1 ja(1)jb(2) Since two wave functions that correspond to the same state can differ at most by a constant factor f(1,2) = c2f(2,1) ja(1)jb(2) + c1ja(2)jb(1) =c2ja(2)jb(1) +c2c1ja(1)jb(2) c1 = c2 c2c1 = 1 Therefore:c1 = c2 = 1 According to the Pauli principle, c1 = c2 =- 1

  27. Wave function f of H2 : y(1,2) = 1/2! [f(1)a(1)f(2)b(2) - f(2)a(2)f(1)b(1)] f(1)a(1) f(2)a(2) = 1/2! f(1)b(1) f(2)b(2) The Pauli principle (different version) the wave function of a system of electrons must be antisymmetric with respect to interchanging of any two electrons. Slater Determinant

  28. Energy: E • E=2dt1 f*(1) (Te+VeN) f(1) + VNN • + dt1 dt2 |f2(1)| e2/r12 |f2(2)| • = i=1,2 fii + J12 + VNN • To minimize Eunder the constraintdt |f2|= 1, • useLagrange’s method: • L = E - 2 e [dt1 |f2(1)|- 1] • dL = dE - 4 e dt1 f*(1)df(1) = 4dt1 df*(1)(Te+VeN)f(1) • +4dt1 dt2 f*(1)f*(2) e2/r12 f(2)df(1) • - 4 e dt1 f*(1)df(1) • = 0

  29. [ Te+VeN +dt2 f*(2) e2/r12 f(2) ] f(1) = e f(1) Average Hamiltonian Hartree-Fock equation ( f + J ) f = e f f(1) = Te(1)+VeN(1) one electron operator J(1) =dt2 f*(2) e2/r12 f(2) two electron Coulomb operator

  30. f(1) is the Hamiltonian of electron 1 in the absence of electron 2; J(1) is the mean Coulomb repulsion exerted on electron 1 by 2; eis the energy of orbital f. LCAO-MO: f = c1y1 + c2y2 Multiple y1 from the left and then integrate : c1F11 + c2F12 = e (c1 + S c2)

  31. Multiple y2from the left and then integrate : c1F12 + c2F22 = e (S c1 + c2) where, Fij = dt yi*( f + J) yj = Hij + dt yi*Jyj S = dt y1y2 (F11 - e) c1 + (F12 - S e) c2 = 0 (F12 - S e) c1 + (F22 -e) c2 = 0

  32. Secular Equation: F11 - eF12 - S e = 0 F12 - SeF22 -e bonding orbital: e1 = (F11+F12) / (1+S) f1= (y1+y2) / 2(1+S)1/2 antibonding orbital: e2 = (F11-F12) / (1-S ) f2= (y1-y2) / 2(1-S)1/2

  33. Molecular Orbital Configurations of Homo nuclear Diatomic Molecules H2, Li2, O, He2, etc Moecule Bond order De/eV H2+ 2.79 H2 1 4.75 He2+  1.08 He2 0 0.0009 Li2 1 1.07 Be2 0 0.10 C2 2 6.3 N2+ 8.85 N2 3 9.91 O2+ 2 6.78 O2 2 5.21 The more the Bond Order is, the stronger the chemical bond is.

  34. Bond Order: one-half the difference between the number of bonding and antibonding electrons

  35. f1(1)a(1) f2(1)a(1) y(1,2) = 1 /2 f1(2)a(2) f2(2)a(2) = 1/2 [f1(1) f2(2) - f2(1)f1(2)] a(1) a(2) ---------------- f1 ---------------- f2

  36. Ey = dt1dt2 y* H y = dt1dt2 y* (T1+V1N+T2+V2N+V12+VNN) y = <f1(1)| T1+V1N|f1(1)> + <f2(2)| T2+V2N|f2(2)> + <f1(1) f2(2)| V12 | f1(1) f2(2)> - <f1(2) f2(1)| V12 | f1(1) f2(2)> + VNN = i<fi(1)| T1+V1N |fi(1)> + <f1(1) f2(2)| V12 | f1(1) f2(2)> - <f1(2) f2(1)| V12 | f1(1) f2(2)> +VNN = i=1,2 fii + J12 -K12 + VNN

  37. Average Hamiltonian Particle One: f(1) + J2(1)-K2(1) Particle Two: f(2) + J1(2)-K1(2) f(j) -(h2/2me)j2 - Za/rja Jj(1) q(1)  q(1)  dr2 fj*(2) e2/r12 fj(2) Kj(1) q(1)  fj(1) dr2 fj*(2) e2/r12 q(2)

  38. Hartree-Fock Equation: [ f(1)+ J2(1) -K2(1)] f1(1) = e1 f1(1) [ f(2)+ J1(2) -K1(2)] f2(2) = e2 f2(2) Fock Operator: F(1) f(1)+ J2(1) -K2(1) Fock operator for 1 F(2) f(2)+ J1(2) -K1(2) Fock operator for 2

  39. Hartree-Fock Method 1. Many-Body Wave Function is approximated by Slater Determinant 2. Hartree-Fock Equation Ffi = ei fi FFock operator fi the i-th Hartree-Fock orbital ei the energy of the i-th Hartree-Fock orbital

  40. 3. Roothaan Method (introduction of Basis functions) fi= k ckiyk LCAO-MO { yk }is a set of atomic orbitals (or basis functions) 4. Hartree-Fock-Roothaanequation j ( Fij - ei Sij ) cji = 0 Fij  < i| F | j > Sij  < i| j > 5. Solve the Hartree-Fock-Roothaan equation self-consistently

  41. Assignment one 8.40, 10.5, 10.6, 10.7, 10.8, 11.37, 13.37

  42. The Condon-Slater Rules <fa(1)fb(2)fc(3)...fd(n) | f(1) | fe(1)ff(2)fg(3)...fh(n)> = <fa(1) | f(1) | fe(1)> < fb(2)fc(3)...fd(n) | ff(2)fg(3)...fh(n)> = <fa(1) | f(1) | fe(1)> ifb=f, c=g, ..., d=h; 0, otherwise <fa(1)fb(2)fc(3)...fd(n) | V12 | fe(1)ff(2)fg(3)...fh(n)> = <fa(1) fb(2) | V12 | fe(1) ff(2)> < fc(3)...fd(n) | fg(3)...fh(n)> = <fa(1) fb(2) | V12 | fe(1) ff(2)> ifc=g, ..., d=h; 0, otherwise

  43. ------- the lowest unoccupied molecular orbital  ------- the highest occupied molecular orbital  ------- ------- LUMO HOMO Koopman’s Theorem The energy required to remove an electron from a closed-shell atom or molecules is well approximated by minus the orbital energy e of the AO or MO from which the electron is removed.

  44. # HF/6-31G(d) Route section water energy Title 0 1 Molecule Specification O -0.464 0.177 0.0 (in Cartesian coordinates H -0.464 1.137 0.0 H 0.441 -0.143 0.0

  45. Slater-type orbitals (STO) nlm = Nrn-1exp(-r/a0) Ylm(,) x the orbitalexponent * is used instead of  in the textbook Basis Set i = p cip p Gaussian type functions gijk = N xi yj zk exp(-ar2) (primitive Gaussian function) p = u dupgu (contracted Gaussian-type function, CGTF) u = {ijk} p = {nlm}

  46. Basis set of GTFs STO-3G, 3-21G, 4-31G, 6-31G, 6-31G*, 6-31G** ------------------------------------------------------------------------------------- complexity & accuracy Minimal basis set: one STO for each atomic orbital (AO) STO-3G: 3 GTFs for each atomic orbital 3-21G: 3 GTFs for each inner shell AO 2 CGTFs (w/ 2 & 1 GTFs) for each valence AO 6-31G: 6 GTFs for each inner shell AO 2 CGTFs (w/ 3 & 1 GTFs) for each valence AO 6-31G*: adds a set of d orbitals to atoms in 2nd & 3rd rows 6-31G**: adds a set of d orbitals to atoms in 2nd & 3rd rows and a set of p functions to hydrogen Polarization Function

  47. Diffuse Basis Sets: For excited states and in anions where electronic density is more spread out, additional basis functions are needed. Diffuse functions to 6-31G basis set as follows: 6-31G* - adds a set of diffuse s & p orbitals to atoms in 1st & 2nd rows (Li - Cl). 6-31G** - adds a set of diffuse s and p orbitals to atoms in 1st & 2nd rows (Li- Cl) and a set of diffuse s functions to H Diffuse functions + polarisation functions: 6-31+G*, 6-31++G*, 6-31+G** and 6-31++G** basis sets. Double-zeta (DZ) basis set: two STO for each AO

  48. 6-31G for a carbon atom: (10s4p)  [3s2p] 1s 2s 2pi (i=x,y,z) 6GTFs 3GTFs 1GTF 3GTFs 1GTF 1CGTF 1CGTF 1CGTF 1CGTF 1CGTF (s) (s) (s) (p) (p)

  49. Minimal basis set: One STO for each inner-shell and valence-shell AO of each atom example: C2H2 (2S1P/1S) C: 1S, 2S, 2Px,2Py,2Pz H: 1S total 12 STOs as Basis set Double-Zeta (DZ) basis set: two STOs for each and valence-shell AO of each atom example: C2H2 (4S2P/2S) C: two 1S, two 2S, two 2Px, two 2Py,two 2Pz H: two 1S (STOs) total 24 STOs as Basis set

  50. Split -Valence (SV) basis set Two STOs for each inner-shell and valence-shell AO One STO for each inner-shell AO Double-zeta plus polarization set(DZ+P, or DZP) Additional STO w/l quantum number larger than the lmax of the valence - shell  ( 2Px, 2Py ,2Pz ) to H  Five 3d Aos to Li - Ne , Na -Ar  C2H5 O Si H3 : (6s4p1d/4s2p1d/2s1p) Si C,O H

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