OVERVIEW OF QUANTUM CHEMISTRY METHODS

Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density functional methods Principle Functionals

Outline I Generalities Correlation, Basis set Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods points which will be important III Density functional methods for the rest of the course Principle Functionals Non exhaustive my goal is just to highlight

Outline I Generalities Correlation, basis set Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density functional methods Principle Functionals

Generalities - Hamiltonian H i = E i i H = T N + T e + V NN + V Ne + V ee H = T N + V NN + H el T e = X i r 2 i 2 X V ne = - ia r i Z A R A V ee = X ij 1 r ij

Generalities - Resolution H i = E i i ii = X I c i I Ii A set of n-e- functions Become an eigenvalue equation Each eigenvector is a state Each eigenvalue is its energy Can also be optimized using the variational principle E exact appleh test H testi

Generalities - Resolution is a n-electron function The easiest is to define it as a Hartree Product: a (tensor) product of 1-e - functions Ii = 1(1) 2 (2)... n(n)i The (1-e - ) basis set

Generalities - Correlation Let us build some intuition on correlation Example: Helium atom Basis set: «Physical intuition» of a localized electron i = X µ c µ r 1 = r µ,r 2 = r i Configuration interaction

Generalities - Correlation i = X µ c µ r 1 = r µ,r 2 = r i P (r 1 = r µ,r 2 = r )=c 2 µ Let us now assume that the 2 electrons are independent i = X µ c µ c r 1 = r µ,r 2 = r i P (r 1 = r µ,r 2 = r )=c 2 µc 2 P (r µ,r µ )=c 4 µ

Generalities - Correlation i = X µ c µ c r 1 = r µ,r 2 = r i µi = X c µ r = r µ i the 1s orbital i = µ i One Hartree Product is uncorrelated

Generalities - Correlation i = X µ c µ c p 2 ( r µ,r i r,r µ i) P (r µ,r µ )= c4 µ 2c 4 µ + c 4 µ 2 =0 Same spin electrons P (r µ,r µ )= c4 µ + c 4 µ 2 = c 4 µ Opposite spin electrons One Slater determinant has correlation for same-spin electrons (which is by far the largest part)

Generalities - Correlation In practice : We still call a Slater Determinant «uncorrelated» One Slater Determinant is usually a good approximation to the exact wave function Adding Slater determinant adds correlation Determinants other than the main one (in the ground state) often called «excitation» which makes things very tricky

Generalities - Basis sets 1-e - basis can be anything in principle «localized electron» sine/cosine The closer to the solution the better (since need less functions) Slater-type atomic orbitals = P (x, y, z)e r Gaussian-type atomic orbitals = P (x, y, z)e r 2

Generalities - Basis sets Basis set requirements may vary In uncorrelated He, one s function is enough In correlated He, other functions are needed We sometimes call them 2s, 2p, 3s but they really are just functions!

Generalities - Basis sets Basis set requirements may vary 0 HF (s-only) HF (full) MP2 (s-only) MP2 (full) Energy difference (kcal/mol) -7,5-15 -22,5-30 1 2 3 4 Zeta

Generalities - Spin In QC, spin is an ad hoc property s z "i = 1 2 "i s z #i = 1 2 #i s + "i =0 s + #i = "i s "i = #i s #i =0 s 2 = s + s s z + s 2 z = s s + + s z + s 2 z

Generalities - Spin Any Slater determinant is eigenvalue of ŝ z ŝ z i = m s i m s = N N not necessarily of ŝ 2 2 ŝ 2 i = s(s + 1) i singlet triplet aāi a bi+ bāi abi a bi bāi p p 2 2 ā bi

Generalities - Spin Components of a same spin multiplicity are degenerate In UHF, spin up and down can be in different orbitals a bi Can lead to spin contamination (mixing with other multiplicities) Can give a correct energy but not correct wavefunction (and properties)

Wfn methods - Hartree-Fock Hartree-Fock is the optimization of a single Slater determinant (usually closed-shell) E = 2 X i h i + X ij 2J ij K ij J ij =(ii jj) Z (ij kl) = i(1) j (1) 1 K ij =(ij ji) r 12 k(2) l (2)dr 1 dr 2 For i = j, the Coulomb and Exchange cancel out, preventing «self-interaction»

Wfn methods - Hartree-Fock However, one first needs to optimize the orbitals The derivative of the energy is linked to the Fock matrix ˆK j ˆF = ĥ + X j 2Ĵj non local! hi ˆF ii = h i + X j 2J ij K ij depends on orbitals Then one solves F ii = X j ij ji

Wfn methods - Hartree-Fock Optimization done by «rotating the orbitals» E = 2 X i h i + X ij 2J ij K ij No dependence on unoccupied orbitals ĩi = c i ii + c j ji ji = c i ji c j ii hĩ h ĩi + h j h ji = hi h ii + hj h ji No dependence on rotations among occupied orbitals

Wfn methods - Hartree-Fock Thus, there is an infinite number of valid orbital sets with the same energy! A common choice is the canonical orbitals F ii = X j ij ji F ii = i ii The Lagrange multiplier then becomes an orbital property: the orbital energy However, very delocalized orbitals!

Wfn methods - Hartree-Fock i = h i + X j 2J ij K ij Koopman s theorem: good approximation to the first ionization energy (error cancellation) IE -HOMO IE -HOMO Li 5.4 5.4 Be 9.3 9.4 B 8.3 5.7 C 11.3 10.7 N 14.5 12.9 O 13.6 15.9 F 17.4 18.6 Ne 21.6 21.6 Na 5.1 5.2 Mg 7.6 7.6 Al 6.0 6.0 Si 8.2 7.8 P 10.5 9.8 S 10.4 11.7 Cl 13.0 13.7 Ar 15.8 15.8

Wfn methods - Hartree-Fock Hartree-Fock recovers most of the electronic energy Often insufficient for quantitative analysis Typical trend: Too short bonds Too low dissociation energy Favors higher spin multiplicities Traditional codes have N 4 scaling 300 225 150 75 0-75 singlet triplet UHF singlet -150 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0

Wfn methods - CI If one adds all possible Slater determinant one gets the exact N-electron description in the 1-e - basis set fullci Easier said than done! the number of determinants grows factorially with number of electrons/orbitals Done for benchmarking on very small systems

Wfn methods - CI But usually all Slater determinants do not contribute equally Singly-excited determinants do not interact directly with the Hartree-Fock ground state (Brillouin s theorem) Higher than double excitations do not interact directly Truncated CI! they do interact, through the double!!

Wfn methods - CI Common approach CI(S)D Most of the correlation for small systems Not size-consistent! Ground state double excitations disconnected quadruple excitations

Wfn methods - CI Adding higher level only delays the problem (and is costly) Some possible corrections (Davidson for example) Some derived method (QCISD, CPF, ) Truncated CI abandoned in favor of CC Still used in the multi-reference case

Wfn methods - Coupled cluster Exponential instead of linear: CCSD i = e ˆ t1+ ˆ t2 0i e x =1+x + x2 2 + x3 6 +... ˆt 1 = X ia t a i ê a i i =(1+ˆt 1 + ˆt 2 + ˆt 1 2 ˆt 2 = 1 4 X ijab t ab ij ê ab ij 2 + ˆt ˆt ˆt ˆt 1 2 1 2 + 6 + 3 2 2 +...) 0i All excitations of fullci, but parameters of CISD

Wfn methods - Coupled cluster Size-consistent Very high quality wavefunction Can be extended to CCSDT, CCSDTQ Converge faster than the truncated CI series With perturbative triple = CCSD(T), «gold standard» of QC Only weakness: (very) strong correlation (diagnostic)

Wfn methods - Coupled cluster Strong correlation Relative energy (kcal/mol) 100 50 0-50 -100 O-O dissocation Reference UHF triplet CCSD triplet CCSD(T) triplet T1 diagnostic 0.5 1.5 2.5 3.5 4.5 5.5 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01-150 O-O bond length (Å) 0

Wfn methods - Perturbative methods Radically different methods H = H 0 + V In principle different possible splittings Standard: Møller-Plesset H = F + (H F ) E MP1 = E HF

Wfn methods - Perturbative methods X (ij ab) 2 E (2) = 1 4 ijab i + j a b Always negative Cheapest correlation scheme Size-consistent Problem for (quasi-)degenerate states Can go to higher order, but doesn t necessarily converge

Wfn methods - summary General scaling rules for «standard» methods : N 4 for HF and single excitations Multiply by N 2 for any additional excitation order or by N for perturbative treatment MP2 N 5 MP3, CISD, CCSD N 6 MP4, CCSD(T) N 7 MP5, CISDT, CCSDT N 8

DFT - Principle If you don t like the answer change the question!

DFT - Principle Solving the Hamiltonian brings a factorial complexity Instead, one may find another equation which we know how to solve Remove explicit use of wavefunction Better candidate : the electronic density

DFT - Principle Sound physical quantity Only a 3-D variable Z X V Ne = N (r)z N dr r R N J = Z (r) (r 0 ) r r 0 drdr 0

DFT - Principle First attempts : Thomas Fermi model (1927) Kinetic energy of a uniform electron gas with the same density Molecules are not stable

DFT - Principle Hohenberg & Kohn theorem The electronic density of the GS uniquely defines the external potential and thus the whole Hamiltonian! All properties of the GS can be expressed as universal functional of the density In practice, we don t know such functional Nothing says it would be «simple» to compute

DFT - Principle The kinetic energy was especially troublesome Kohn-Sham formalism Fictitious system with same density as the real one H f = T + V f + V ee A Slater determinant is the exact solution One step back to the wavefunction world One step forward to a usable DFT

DFT - Principle Put as many known ingredients as possible Put the rest in v xc v f = v ne + J +v xc This is the functional you choose when you do DFT Contains exchange and correlation Probably contains also some kinetic energy

DFT - Principle Very similar equations to HF potential of F = T e + V Ne + X j 2J j + v xc ( ) exchange and correlation Can scale better than HF because of lack of exchange Obtained orbitals are those of the fictitious system! Same (low) basis set requirements as HF Be careful for any interpretation/properties Often those orbitals are close to the expected ones

DFT - Functionals A hierarchy Locality? Coulomb term local Correlation term (mostly) local Exchange term non local Z LDA E xc =.v xc ( )dr

DFT - Functionals A hierarchy Hybrid meta-gga Adds some HF exchange Z GGA E xc = Z.v xc (, r )dr LDA E xc =.v xc ( )dr

DFT - Functionals In HF the exchange cancels the Coulomb term for same electron In DFT, the local exchange may not Self interaction Exact (HF) High exchange for transition states Exchange amount critical for spin DFT energetics Pure functional = low spin Hybrid functional = high spin

DFT - Functionals A hierarchy Double-hybrid Adds some wfn correlation Hybrid meta-gga Adds some HF exchange Z GGA E xc = Z.v xc (, r )dr LDA E xc =.v xc ( )dr

DFT - Functionals And a jungle! From ab-initio to heavily parametrized Different theoretical constraints Different fitting sets

Summary 80 74.5 G2-1 Data Set 70 mean absolute error 60 50 40 39.6 30 20 10 0 7.4 4.9 2.2 HF MP2 LSDA BLYP B3LYP