Lecture 5: More about one- Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory.

Transcription

1 Lecture 5: More about one- determinant wave functions Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory.

2 Items from Lecture 4 Could the Koopmans theorem be employed to the other direction, i.e. adding electrons to the N- electron system as N 1 < HF? aa t Derive the Roothan-Hall equations Study the Chapter 3 Do the preparatory exercise for Friday at

3 Hartree-Fock theory recap In the Hartree-Fock approximation, the electronic wave function is approximated by a single Slater determinant The optimal determinant may be found by solving a set of effective one-electron Schrödinger equations for spinorbitals the Hartree-Fock equations The associated effective Hamiltonian is the Fock operator f< h V V < (2 g, g ) E ÂÂ pq iœocc pqii piiq pq

4 Hartree-Fock theory recap Since the Fock matrix is defined in terms of its own eigenvectors, an iterative procedure the selfconsistent field (SCF) method is needed In the HF picture an electron experiences the Coulomb potential generated by the nuclear framework and by the charge distribution of N-1 remaining electrons Fermi correlation The HF closed-shell wave function corresponds to N/2 lowest orbitals doubly occupied HF Ê < Á aa ˆ i a i b ÁË i vac

5 Hartree-Fock theory recap The HF wave function is an eigenstate of the Fock operator with an eigenvalue equal to the sum of occupied orbital energies f HF < 2  e i HF i The occupied orbital energies can be associated with the ionization potential (Koopmans theorem) N IP < HF a Ha HF, E <, f i is is HF ii The Hartree-Fock energy for a closed-shell system 1 E < HF H HF < HF ÂD h d g pq pq  2 pq <  Â, pqrs E 2 h (2 g g ) HF i ii iijj ijji ij pqrs pqrs

6 Roothaan-Hall equations Expand MOs as a linear combination of AOs (LCAO), using the expansion coefficients C as variational parameters The variational conditions for optimized energy become AO f C < SCe  1 ( ) AO AO f < h D g, g mn mn rs mnrs msrn rs 2 AO D < 2ÂCC rs ri si i These are solved by applying the SCF method

7 Solution of the Roothan-Hall equations Initial orbitals initial D AO Construct the Fock matrix f AO from D AO New C by solving f AO C=SCe Transform f to MO basis f MO = C T f AO C Form an error vector eas the occupiedvirtual block of f MO AO f C < SCe  1 ( ) AO AO f < h D g, g mn mn rs mnrs msrn rs 2 AO D < 2ÂCC rs ri si i No e <t? Yes SCF>

8 Concerning SCF convergence A straightforward implementation of the RH SCF may fail to converge or converges slowly A number of means to improve the convergence has been established Direct Inversion of the Iterative Subspace (DIIS) scheme: Employ also n older iterations in addition to last one to construct the improved density

9 Solution of the RH equations: DIIS acceleration Initial orbitals initial D AO Construct the weighted Fock matrix Transform f to MO basis f MO = C T f AO C Form an error vector e as the occupiedvirtual block of f MO New C by solving f AO C=SCe Determine No weights w i e <t? Yes SCF>

10 Concerning SCF convergence r OH =r OH e r OH =2r OH e HF SCF iterations of a water molecule

11 Concerning SCF convergence Second-order methods Newtonian trust-region methods (Dalgaard & Jørgensen, JCP 69, 3833 (1978) Augmented Roothan-Hall [Høst et al., JCP 129, (2008)] Other convergence accelerators C2-DIIS [H. Sellers, IJQC 45, 31, (1993)] Energy DIIS (EDIIS) [Kudin et al., JCP 116, 8255 (2002)] ADIIS [Hu & Yang, JCP 132, (2010)]

12 Computational considerations The computational cost of the RH SCF scheme with N AOs (basis set functions) consists of the following components AO f C < SCe AO AO 1 f < h ÂD ( g, g ) mn mn rs mnrs msrn rs 2 AO D < CC Â 2 rs ri si i Computing one-electron integrals and construction of the f AO are O(N 2 ) procedures Diagonalization is a O(N 3 ) process (with a small prefactor) Calculation of two-electron integrals: we need O(N 4 )of them

13 Computational considerations It turns out that most of the two-electron integrals are very small It can be shown that g g g mnrs mnmn rsrs Thus we can easily identify in advance which integrals to evaluate and which not There are O(N 2 ) non-zero two-electron integrals The exchange integrals decay rapidly as a function of the separation between two AO centres Prescreen the exchange contribution with the AO overlap matrix It is quite straightforward to avoid computation of all but O(N) exchange integrals

14 Computational considerations Coulomb part on the other hand decays only as 1/r Non-vanishing contributions also over large distances We can however evaluate only a O(N) part of them exactly we can approximate the rest with simpler formulae These are known as (continuous) fast multipole methods With O(N) scaling two-electron integrals, we need to overcome the diagonalization bottleneck Do everything in the AO basis => matrices are sparse for large systems => O(N) scaling Overall, the cost of RH SCF scales as O(N )

15 Møller-Plesset perturbation theory Standard machinery of Rayleigh-Schrödinger perturbation theory H < H U Ξ < E < 0  k k  E Ξ () k () k (0) (0) (0) E < Ξ H Ξ 0 ( n) (0) ( n, 1) E < Ξ U Ξ, n = 0 Ê Ξ <,, Ξ, Ξ ÁË n ( n) (0) ( n 1) ( k) ( n k) ( H E ),, U E 0 Á  n< 1 ˆ

16 Møller-Plesset perturbation theory Define the fluctuation potential Ε as the difference between the true two-electron operator and the Fock potential: Ε< g, V H < f Ε Using these in the context of the RSPT is known as the Møller-Plesset perturbation theory E (0) < HF f HF < Â ( n) ( n, 1) E < HF Ε Ξ, n = 0 Ê Ξ <,, Ε Ξ, Ξ ÁË I n ( n) ( n 1) ( k) ( n k) ( f Âe ),, E IÁ Â I n< 1 e I ˆ

17 Møller-Plesset perturbation theory To the first order in perturbation we obtain E < HF Ε HF < HF H HF, HF f HF (1) MP E < E E 2 (0) (1) HF MP MP MP1 <, m e m H HF < T HF m  m 2, 1 (1) 2 m A B < < a aa a HF, A= B, I = J A I B J I J e < e e, e, e ABIJ A B I J

18 Møller-Plesset perturbation theory And to the second order in perturbation MP2 E Â <, m e m Ε, E m, 1 (1) m MP (2) (2) (2) (2) T T T T ( MP1 < HF < HF Ε MP1 < HF HT HF (2) (1) MP 2 And by similar argumentation E < HF [( T ),[ Ε, T ]] HF (3) (1) (1) MP 2 2 The MP2 correction can be expressed in closed form, for higher orders the amplitudes in the operators T n have to be solved iteratively

19 Møller-Plesset perturbation theory MPn wave functions in description of dissociation of a water molecule (angle fixed). Full line: with RHF reference state; dashed line: with UHF reference state.

20 About the accuracy: Bond lengths Comparison of models by the deviation from experimental molecular geometries of 29 small main-group element species Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002) HF MP2 MP3 cc-pvdz cc-pvtz cc-pvqz MP4

21 About the accuracy: Bond lengths Relationship between the calculated bond distances for the standard models (in pm) Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

22 About the accuracy: Reaction enthalpies Error in the reaction enthalpies (kj/mol) for 14 reactions involving small main-group element molecules Helgaker, Jorgensen, Olsen: Molecular Electronic-Structure Theory (Wiley 2002)

23 Concluding remarks Hartree-Fock theory: the foundation of ab initio quantum chemistry Usually not good enough model chemistry as is Møller-Plesset perturbation theory: an approach for inclusion of dynamical correlation with a reasonable computational cost The series is inherently divergent; the second-order in perturbation (i.e. the MP2 model) the only useful level of theory Fails outside equilibrium geometries

24 Things to think about & homework Read the following review article on modern Hartree- Fock theory: Echenique & Alonso, A mathematical and computational review of Hartree Fock SCF methods in quantum chemistry, Molecular Physics 105, (2007) Study Chapter 3, have a look at the Chapter 4 Derive (find) the expression for MP2 energy correction No exercise session on Friday But we re going to continue with the lectures on Wednesday I am waiting for more plans for the project works!