Advanced Electronic Structure Theory Density functional theory. Dr Fred Manby

Transcription

1 Advanced Electronic Structure Theory Density functional theory Dr Fred Manby

2 6 Strengths of DFT DFT is one of many theories used by (computational) chemists Advantages: Accurate relative term, of course Fast 99

3 Hierarchy of (some of the) available computation tools method description accuracy size MM atomistic, empirical potentials low > 10 4 AM1,PM3 HF with semi-empirical integrals. > 100 HF Slater-determinant. 100 MP2 simplest ab initio correlation. 50 DFT density based. > 100 CCSD(T) harder ab initio correlation. 20 MRCI multi-reference, OK if HF bad high <

4 6.1 Molecular structures Some important bond-lengths in Å [Koch and Holthausen] molecule bond SVWN BLYP BPW91 expt H 2 H H H 3 C CH 3 C C C H H 2 C=CH 2 C=C C H HC CH C C C H RMS error

5 Even SVWN is pretty good within 2% Gradient corrections improve matters 1% Gradient corrections sometimes over-compensate (eg C C) Errors largest for bonds to hydrogen How do hybrid functionals perform? Mean errors in bondlengths of diatomics in Å HF MP2 BLYP B3LYP 12 first row diatomics second row diatomics

6 B3LYP on G2 test-set gives error 0.008Å G2 consists of 55 experimentally well characterized molecules Small first- and second-row species [Curtiss et al. J Chem Phys (1991)] Bond angles usually come out to within 0.5 This is amazingly accurate! Only CCSD(T) in a large basis does better 103

7 6.2 Vibrational frequencies Obtained from second derivative of energy H ij = 2 q i q j E[ρ] Force constants are eigenvalues of the Hessian H This gives harmonic frequencies Comparison with anharmonic experimental frequencies difficult Scaling by uniform factors very common 104

8 Performance for vibrational frequencies of 122 molecules method scaling RMS/cm 1 >10% HF MP BLYP B3LYP Percentage of frequencies in error by more than 10% [From Koch and Holthausen, p134; from Scott and Radom, JCP (1996)] 105

9 6.3 Thermochemistry Have already seen some errors for atomization energies Chemical accuracy: errors within 2 kcal mol 1 B3LYP approaches this: 3 kcal mol 1 This can also be achieved by CCSD(T) but very expensive Hartree-Fock miles off: kcal mol 1 MP2 also quite erroneous: 20 kcal mol 1 errors 106

10 6.4 Reaction profiles Reaction profiles can be computed using DFT Structures often excellent Performance on reaction energies and barrier heights variable For many reactions, accuracy reasonable 107

11 6.4.1 Ring opening of cyclobutene 1 cyclobutene gauche-1,3-butadiene 2 trans-1,3-butadiene Barrier heights in kcal/mol barrier expt. HF SVWN BLYP B3LYP

12 6.4.2 Diels-Alder reaction (ethene and butadiene) Activation barriers for forward and reverse reactions (kcal mol 1 ) barrier expt. HF SVWN BLYP B3LYP forward reverse

13 6.5 Properties How properties are computed in DFT Properties in wavefunction methods computed like property = Ψ ˆp i Ψ i KS determinant is not the wavefunction Just there to serve up ρ and T s In DFT properties should arise as functionals property = P [ρ] Implies one has to think of a functional for every property 110

14 Measurement process disturbs the situation described by Ĥ Eg dipole moment: apply static electric field Energy of molecule in a weak electric field along z-axis ( ) E E(f z ) = E 0 + f z + f z Dipole moment (in z direction) ( ) E µ z = f z 0 Dipole moment functional ( ) E[ρ] µ z [ρ] = f z

15 Other properties from derivatives wrt static electric field Polarizabilities (second derivatives) Hyperpolarizabilities (third and higher derivatives) IR intensities Related to derivatives of dipole moments wrt normal mode coordinates Thus need second derivatives of energy Magnetic properties NMR shielding constants and ESR g-factors 112

16 6.5.2 Performance of DFT for properties DFT dipole moments in excellent agreement with expt. Dipole moments in Debye molecule HF MP2 BLYP B3LYP expt CO H 2 O HCl SO Polarizabilities less well described MP2 polarizabilities significantly better than DFT 113

17 6.6 Hydrogen bonds: the water dimer (H 2 O) 2 is the classic hydrogen bonded example H R OO O H O R OH H H property expt HF MP2 BLYP B3LYP R OO /Å R OH /Å µ/d α/å

18 6.7 Excited states Thus far we have only discussed ground states Excited states cannot be found in DFT by usual methods Excited states are reached when the molecule interacts with an oscillating electric field namely light Response of dipole to time-dependent electric field: frequency dependent polarizability α(ω) Amazingly, excitation energies appear as poles in α(ω) Requires time-dependent DFT (TDDFT) 115

19 Transition energies (in ev) in ethene transition B3LYP HCTH(AC) CASPT2 expt π π ( 3 B 1u ) π 3s( 3 B 1u ) π 3s( 1 B 1u ) mean abs. error HTCH(AC) is a functional designed for excitation energies HTCH(AC) and CASPT2 equally accurate; DFT much quicker Have to think about CASPT2, DFT can just be run 116

20 7 Summary of entire course DFT is used very widely in chemistry The density has the same info as the wavefunction DFT is possible: Hohenburg-Kohn DFT can be done in practice: Kohn-Sham Functionals: SVWN, BLYP, B3LYP Some problems: dispersion, multi-reference, self-interaction Works amazingly well for huge range of problems 117