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Lecture 5: Methods and Terminology, Part III Coupled Cluster Theory, Hellmann-Feynman Theorem, Densities, Density Functional Theory, Exchange-Correlation Functionals

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Lecture 5: Methods and Terminology, Part III Coupled Cluster Theory, Hellmann-Feynman Theorem, Densities, Density Functional Theory, Exchange-Correlation Functionals Evans boldly put 50 atm of ethylene in a cell with 25 atm of oxygen. The apparatus subsequently blew up, but luckily not before he had obtained the spectra shown in Figure 8. A.J. Mehrer and R.S. Mulliken, Chem. Rev. 69 (1969) 639-656 AUTHORS AND REVIEWERS Lecture 5: methods and terminology, part III coupled cluster theory, Hellmann-Feynman theorem, densities, density functional theory, exchange-correlation functionals Dr Ilya Kuprov, University of Southampton, 2012 (for all lecture notes and video records see http://spindynamics.org) Coupled-cluster theory Configuration interaction, introduced in the previous lecture, was summing over all single, double, etc. excitations: Coupled-cluster theory uses a sum over all excitations of a specified type. That is, whereas each determinant is parameterized in CI theory, it is the excitation process that is parameterized in the CC theory. CC includes higher excitations, but their coefficients are products of lower order excitation coefficients. CC theory is size- Method Configurations Scaling with basis size consistent, that is, CCD linked double excitations O(n6) for two systems: linked single and double CCSD O(n6) excitations ECC (1 2) linked single and double CCSD(T) O(n7) EECC(1) CC (2) excitations with MP4 triples R.J. Bartlett, M. Musal, http://dx.doi.org/10.1103/RevModPhys.79.291 Coupled-cluster theory Coupled-cluster theory is based on the observation that there must exist a unitary operator that would take the Hartree-Fock reference to the exact solution: ˆˆexpTTTTT : ˆ; exp ˆˆˆˆ 1 23 2 6 ... 0 After a lengthy derivation, the cluster operator T may be shown to introduce single, double, etc. excitations into the wavefunction: ˆˆˆˆ ˆa a ˆ ab ab TTTT123...; T 10 ti i, T 20 t ij ij , ... ia,, i ja b If we truncate the cluster operator at T2,its exponential would still contain triple and quad- ˆ 2 ab cd abcd ruple excitations, but their coefficients are pro- Ttt20 ij kl ijkl ijkl, ducts of single and double excitation coefficients, abcd, this makes them easier to compute. TTˆˆ tacd t acd Some excitations are missing from truncated CC 12 0 iklikl iak,, lc , d compared to the full CI, but it may be shown to account for the most important ones. J. Cizek, http://dx.doi.org/10.1063/1.1727484 Coupled-cluster theory Method RMS bond length error, Method ECC-EFCI for water in (Angstrom from XRD) cc-pVDZ, Hartree HF/6-31G(d,p) 0.021 RHF 0.217822 HF/6-311G(d,p) 0.022 CCSD 0.003744 MP2/6-31G(d,p) 0.014 CCSDT 0.000493 MP2/6-311G(d,p) 0.014 QCISD/6-311G(d,p) 0.013 CCSDTQ 0.000019 CCSD(T)/6-311G(d,p) 0.013 CCSDTQP 0.000003 Method RMS atomization energy error, Convergence with excitation (kcal/mol from experiment) level may be much slower in multi-reference systems (such HF/6-31G(d) 85.9 as twisted double bonds, stret- HF/6-311G(2df,p) 82.0 ched single bonds, etc.)–just MP2/6-31G(d) 22.4 as with CI, if a single deter- minant is not a good first ap- MP2/6-311G(d,p) 23.7 proximation to the ground sta- QCISD/6-311G(d) 28.8 te, CASSCF / CASPT methods CCSD(T)/6-311G(2df,p) 11.5 should be used instead. F. Jensen, Introduction to Computational Chemistry, Wiley, 1998. Summary of post-HF methods N.B. Single reference Method Ecorr(Be) Method Ecorr(Be) Method Ecorr(Be) methods are very accu- MP2 67.85% rate if and only if the HF reference state is a good MP3 86.90% CISD 96.05% CCSD 99.75% starting point. MP4 94.58% CCSD(T) 99.99% In systems with homoly- tic bond cleavage or de- MP5 98.15% CISDT 96.29% CCSDT 99.99% generacies, MPn, CI and MP6 99.64% CC methods can fail dra- matically. MP7 100.15% CISDTQ 100% CCSDTQ 100% S. Niu, M.B. Hall, http://dx.doi.org/10.1021/jp961558p Summary of post-HF methods Methods Scaling with the Hartree-Fock: (without basis number of basis • Up to a few hundred atoms. screening) functions • First and second analytical derivatives readily 4 HF O(n ) available with respect to common perturbations. MP2 O(n5) MP2, MP4: MP3, CISD, • Efficient implementations scale to ~100 atoms CCSD, O(n6) for MP2 and to ~20 atoms for MP4. QCISD • First and second analytical derivatives readily MP4, available with respect to common perturbations CCSD(T), O(n7) for MP2, numerical derivatives only for MP4. QCISD(T) CISD, CCSD: MP5, CISDT, O(n8) CCSDT • Efficient implementations scale to ~20 atoms. • First analytical derivatives usually available, MP6 O(n9) second derivatives mostly numerical for now. MP7, FCI: CISDTQ, O(n10) CCSDTQ • 3-4 simple atoms at most. FCI O(n!) • All derivatives numerical. F. Jensen, Introduction to Computational Chemistry, Wiley, 1998. Summary of post-HF methods Dissociation Distribution of curve for N deviations (pm) 2 molecule as from X-ray a function geometries for Distribution of of method. 31 molecules as deviations (pm) a function of from X-ray method. geometries for 31 molecules as a function of basis set size. For properties other than energy, a higher level method does not necessarily mean a more accurate answer – beware of error compensation situations. R.J. Bartlett, M. Musal, http://dx.doi.org/10.1103/RevModPhys.79.291 Summary of post-HF methods Fast approximate methods (but capturing the essentials): HF/6-31G(d,p) or HF/cc-pVDZ Slow accurate methods: MP2/6-311G(2d,2p) or MP2/cc-pVTZ Very slow, very accurate methods: CCSD(T)/cc-pVQZ and higher Specialized high-accuracy compound methods: MP2-R12, G1-G3, G2MP2, G3MP2 etc. Energies (but not usually other parameters) can be extrapolated to the basis set limit – effectively to the complete basis set. Dunning-Feller extrapolation scheme is given below, in which n is the zeta index of an (aug-)cc-pVnZ basis set: n En EeCBS M.P. de Lara-Castells et al., http://dx.doi.org/10.1063/1.1415078 Hellmann-Feynman theorem Many properties of molecules (most notably forces, vibrational frequencies and magnetic resonance parameters) may be expressed as derivatives of the ground state energy with respect to some parameters of the Hamiltonian. Hellmann- Feynman theorem states that: dE dHˆ dd Where the Hamiltonian is assumed to have a continuous dependence on the parameter and the ground state wavefunction (being an eigenfunction of the Hamiltonian) also implicitly depends on it. The second derivative version also exists: dE22 dHˆˆˆ d dH dHd d 22 d dd dd dd where the wavefunction is assumed to be differentiable and have a derivative with a bounded norm: dd R.P. Feynman, http://dx.doi.org/10.1103/PhysRev.56.340 Densities and distributions Density refers to a continuous spatial distribution of a certain property. Commonly encountered densities of interest are: 2 Probability density px,,y zx ,,y z 33 3 2 Electron density ()r N dr dr dr (, rr , , r ) 23NN 2 Spin density ()rrr () () 33 3 2 Charge density ()rNedrdrdrrrr (, , , ) C232 NN Electron isodensity map- surface pedpotential. electrostatic with Charge density map of a modified silicon surface. Spin density oxygen an around in cerium oxide. vacancy F. Jensen, Introduction to Computational Chemistry, Wiley, 1998. Density functional theory The basic lemma of DFT: the ground state density of a bound system of interacting electrons in some external potential determines this potential uniquely. ground Hamiltonian All properties of external state operator (we know the system (via potential density the kinetic terms) standard QM) It should therefore be possible to reformulate quantum chemistry in terms of electron density alone (just three dimensions!) – a major simplification. Hohenberg-Kohn variational principle: there exists a well-defined (though not explicitly known) functional of electron density, which attains a global minimum at the ground state density. nuclear electron energy inter-electron attraction repulsion functional repulsion energy functional ˆˆˆ 3 minETVU [ ] 00ETU[] [] [] Vrrdr ()() nuclear attraction kinetic energy ground state kinetic energy functional functional wavefunction terms (known) F. Jensen, Introduction to Computational Chemistry, Wiley, 1998. Density functional theory Let us choose a basis and set up a collection of one-electron “orbitals” defined so as to generate some trial density for us to optimize against E: N electron 2 basis density ()rrkk () () rarkn n () functions k 1 n one-electron “orbitals” Hartree-Fock style equations for single electrons in these “orbitals” would be: 2 2 k Vrs ()kkk () r () r 2m kinetic energy potential energy term (known) energy term eigenvalue The potential is only known partially, all unknowns are dumped into the last term: s ()r 3 VrsC() Vr () d r VXs [ ()] r rr exchange and correlation potential nuclear attraction (unknown) Coulomb repulsion F. Jensen, Introduction to Computational Chemistry, Wiley, 1998. Exchange-correlation functionals in DFT The exact form of the exchange-correlation functional is unknown, but several practically usable approximations exist. Local spin density approximation (LSDA): it is assumed that the exchange- correlation energy only depends on the density itself, and not on its derivatives. LDA 3 ErdrXC [,] ()(,)xc exchange-correlation energy density Homogeneous electron gas (HEG): this is a special case of LSDA. Exchange functional is known exactly and correlation functionals are known in several limits, which are usually interpolated. Generalized gradient approximation (GGA): are local, but also take into account density gradient at each point. ErGGA[,] ()(,, , ) dr 3 XC XC Meta-GGAs also include second and higher derivatives of the density. This effectively amounts to building more and more accurate Taylor type approxi- mations to the true XC functional.
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