香港大学化学系：高级物理化学 Advanced Physical Chemistry（PPT课件讲稿）量子化学 Quantum Chemistry

Advanced Physical Chemistry G. H CHEN Department of chemistry University ofHong Kong 明

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

Quantum Chemistr G.H. Chen Department of chemistry University ofHong Kong 明

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

Emphasis Hartree-Fock method Concepts Hands-on experience Text book Quantum Chemistry, 4th ed Iran. levine http:/yangtze.hkuhk/lecture/chem3504-3.ppt

Emphasis Hartree-Fock method Concepts Hands-on experience Text Book “Quantum Chemistry”, 4th Ed. Ira N. Levine http://yangtze.hku.hk/lecture/chem3504-3.ppt

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

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.” Beginning of Computational Chemistry Dirac

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

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

Schrodinger Equation H≡E Wavefunction Hamiltonian H=2(h2/2mav 2-(h2/2me)2Vi e ∑∑a Celtic Energy +∑∑e2/r Contents 1. Variation method 2. Hartree-Fock Self-Consistent field method

H y = E y SchrÖdinger Equation Hamiltonian H =  (-h 2 /2m ) 2 - (h 2 /2me )ii 2 +  ZZe 2 /r - i  Ze 2 /ri + i j e 2 /rij Wavefunction Energy Contents 1. Variation Method 2. Hartree-Fock Self-Consistent Field Method

The variation method The variation theorem Consider a system whose hamiltonian operator H is time independent and whose lowest-energy eigenvalue is E. If o is any normalized, well behaved function that satisfies the boundary conditions of the problem, then φHφdτ>E

The Variation Method 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 * H f dt > E1 The variation theorem

00 Expand o in the basis set ( vr k kYk where fa, are coefficients k= Ek the jo o dt=2,E.osy - EkIaklek2exklal=er Since is normalized,∫φpdr=ΣαP=1

Proof: Expand f in the basis set { yk } f = k kyk where {k } are coefficients Hyk = Ekyk then  f * H f dt = k j k * j Ej dkj = k | k | 2 Ek > E 1 k | k | 2 = E1 Since is normalized,  f *f dt = k | k | 2 = 1

o: trial function is used to evaluate the upper limit of ground state energy El φ= ground state wave function, JφHψdτ=E1 iii. optimize paramemters in q by minimizing ∫φHφdτ/∫φφdτ

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

Application to a particle in a box of infinite depth Requirements for the trial wave function 1.zero at boundary ii. smoothness a maximum in the center Trial wave function: o=x(L-X

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