Advanced Electronic Structure Theory Density functional theory Dr Fred Manby fred.manby@bris.ac.uk http://www.chm.bris.ac.uk/pt/manby/
6 Strengths of DFT DFT is one of many theories used by (computational) chemists Advantages: Accurate relative term, of course Fast 99
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 < 10 100
6.1 Molecular structures Some important bond-lengths in Å [Koch and Holthausen] molecule bond SVWN BLYP BPW91 expt H 2 H H 0.765 0.748 0.749 0.741 H 3 C CH 3 C C 1.510 1.542 1.533 1.526 C H 1.101 1.100 1.100 1.088 H 2 C=CH 2 C=C 1.327 1.339 1.336 1.339 C H 1.094 1.092 1.092 1.085 HC CH C C 1.203 1.209 1.209 1.203 C H 1.073 1.068 1.070 1.061 RMS error 0.020 0.009 0.008 101
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 0.024 0.011 0.012 0.004 12 second row diatomics 0.016 0.017 0.024 0.006 102
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 94 7221 (1991)] Bond angles usually come out to within 0.5 This is amazingly accurate! Only CCSD(T) in a large basis does better 103
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
Performance for vibrational frequencies of 122 molecules method scaling RMS/cm 1 >10% HF 0.8953 50 10 MP2 0.9434 63 10 BLYP 0.9945 45 10 B3LYP 0.9614 34 6 Percentage of frequencies in error by more than 10% [From Koch and Holthausen, p134; from Scott and Radom, JCP 100 16502 (1996)] 105
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: 70 80 kcal mol 1 MP2 also quite erroneous: 20 kcal mol 1 errors 106
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
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 1 33 43 33 27 33 2 3 2 4 4 3 108
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 27 51 5 26 28 reverse 38 30 59 22 29 109
6.5 Properties 6.5.1 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
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 0 0 111
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
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 0.25 0.31 0.19 0.10 0.11 H 2 O 1.98 1.85 1.80 1.86 1.85 HCl 1.21 1.14 1.08 1.12 1.11 SO 2 1.99 1.54 1.57 1.67 1.63 Polarizabilities less well described MP2 polarizabilities significantly better than DFT 113
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 2.952 0.09 0.044 0.004 0.034 R OH /Å 0.957 0.02 +0.004 +0.015 +0.005 µ/d 1.854 +0.08 +0.006 0.051 0.006 α/å 3 1.427 0.21 0.004 +0.143 +0.026 114
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
Transition energies (in ev) in ethene transition B3LYP HCTH(AC) CASPT2 expt π π ( 3 B 1u ) 4.07 4.33 4.39 4.36 π 3s( 3 B 1u ) 6.50 7.10 7.05 6.98 π 3s( 1 B 1u ) 6.57 7.16 7.17 7.11. mean abs. error 0.85 0.07 0.09 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
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