Molecular properties in quantum chemistry

Transcription

1 Molecular properties in quantum chemistry Monika Musiał Department of Theoretical Chemistry

2 Outline 1 Main computational schemes for correlated calculations 2 Developmentoftheabinitiomethodsforthe calculation of molecular properties. Monika Musiał Molecular properties 2/ 40

3 Literature L. Piela, Idee Chemii Kwantowej, PWN, Warszawa, R.J.Bartlett,M.Musial,Rev.Mod.Phys.,79, (2007). M. Musiał, Mol. Phys., 108, 2921(2010). M. Jaworska, M. Musiał, T. Pluta, Wybrane zagadnienia chemii kwantowej, Uniwersytet Śląski(2009). J.F.Stanton,R.J.Bartlett,J.Chem.Phys., 98, 7029(1993). J.F.Stanton,J.Chem.Phys.,99,8840(1993). Monika Musiał Molecular properties 3/ 40

4 Computational methods of quantum chemistry basedonthewavefunction Ψ Born-Oppenheimer approximation Electronic Schrödinger equation: HΨ = EΨ Monika Musiał Molecular properties 4/ 40

5 Computational strategy in the wave function based calculations Two-step approach to the solution of the Schrödinger equation: 1 construction of the wave function within the 1-electron approximation(independent particle model): Hartree-Fock method 2 post-hartree-fock approach: determination of electronic correlation Monika Musiał Molecular properties 5/ 40

6 Computational strategy in the wave function based calculations Hartree-Fock (alsoknownasameanfieldapproach) Determination of(spin)orbitals Self Consistent Field(SCF) method E HF 99%oftotalenergy Monika Musiał Molecular properties 6/ 40

7 RHFvs.UHF RHF(Restricted Hartree-Fock)- closed shell molecules, N 2 orbitals(doublyoccupied) UHF(Unrestricted Hartree-Fock)- open shell molecules, l spinorbitals with the α spin andkspinorbitalswiththeβspin (DODS- Different Orbitals for Different Spins) Monika Musiał Molecular properties 7/ 40

8 RHFvs.UHF RHF UHF α β Monika Musiał Molecular properties 8/ 40

9 UHF UHF cons: incorrect spin problems with convergence UHF pros: size-consistent(extensive) RHF RHF cons: incorrect dissociation limit(in standard formulation) RHF pros: well defined spin good convergence for HF equations Monika Musiał Molecular properties 9/ 40

10 RHFvs.UHF E (a.u.) RHF UHF FCI R (a.u.) Monika Musiał Molecular properties 10/ 40

11 Computational strategy in the wave function based calculations Determination of correlation energy Perturbation theory(mpn) Configuration Interaction(CI) Coupled Cluster(CC) Monika Musiał Molecular properties 11/ 40

12 Correlation energy HΨ=EΨ ExpressingthewavefunctionΨasalinear combination of(ground and excited) state configurations: Ψ = Φ o +Σ a,i c a i Φa i +Σ a,b,i,j c ab ij Φab ij +Σ a,b,c,i,j,k c abc ijk Φabc ijk +... Monika Musiał Molecular properties 12/ 40

13 Configurations... c. b. a. i. j. k Φ o Φ a i Φ ab ij Φ abc ijk ˆT 1 ˆT 2 ˆT 3 Ĉ 1 Ĉ 2 Ĉ 3 CCD,CCSD,CCSDT,CCSDTQ,...,FULLCC CID,CISD,CISDT,CISDTQ,...,FULLCI Monika Musiał Molecular properties 13/ 40

14 Expansion coefficients are obtained using: 1 perturbation theory Moeller-Plesset corrections: MP2, MP3,... 1 (J q K q ) r ij q V= i>j 2 linear parametrization of the wave function Ψ=(1+C)Φ o Configuration Interaction method 3 exponential parametrization of the wave function Ψ=exp(T)Φ o Coupled Cluster method Monika Musiał Molecular properties 14/ 40

15 Coupled cluster method exponential parametrization of the wave function Ψ o =e T Φ o T = T 1 +T 2 +T 3...+T N wheret n n-tupleexcitationoperator T n = (n!) 2 Σ ab... Σ ij... t ab... ij...a b...ji andt ab... ij...a b...ji-coefficients(amplitudes)tobedetermined Monika Musiał Molecular properties 15/ 40

16 Coupled cluster method amplitude equation: Φ ab... ij... e T He T Φ o = Φ ab... ij... H Φ o =0 energy expression: E = Φ o e T He T Φ o = Φ o H Φ o where H similarity transformed Hamiltonian. Monika Musiał Molecular properties 16/ 40

17 Coupled cluster method SR CCD (Bartlett, Purvis, 1978; Pople et al. 1978) CCSD (Purvis, Bartlett, 1982) CCSDT (Noga, Bartlett, 1987; Watts, Bartlett, 1989) CCSDTQ (Kucharski Bartlett, 1992) CCSDTQP (Musiał, Kucharski, Bartlett, 2002) S Singles D Doubles T Triples Q Quadruples P Pentuples Monika Musiał Molecular properties 17/ 40

18 Coupled cluster method size-extensivity Correctscalingoftheenergywiththesizeof the system = correct separation of the non-interacting fragments. FortheABmoleculecomposedofthe non-interacting fragments A and B assuming that Φ AB =Φ A Φ B weget: Ψ AB =exp(t AB ) Φ AB =exp(t A ) Φ A exp(t B ) Φ B =Ψ A Ψ B E AB CC=E A CC+E B CC Monika Musiał Molecular properties 18/ 40

19 Important quantity in the CC theory: H- similarity transformed Hamiltonian H=e T He T =(He T ) c H operator includes one-body, two-body and also higher(three, four,...)-body elements. Monika Musiał Molecular properties 19/ 40

20 Howdoweobtain Hoperator? Get T amplitudes from the CC amplitude equations: Φ ab... ij... e T He T Φ o =0 With T amplitudes construct H: H=(He T ) c Monika Musiał Molecular properties 20/ 40

21 EOM-CC method Equation-Of-Motion CC (EOM-CC) method Monika Musiał Molecular properties 21/ 40

22 EOM-CC method EOM-CC method CI-like calculations for the H operator Reminder: Standard CI are based on the regular Hamiltonian (H). Monika Musiał Molecular properties 22/ 40

23 EOM-CC method Wave function: Ψ k = R(k) Ψ o k=1,2,... Schrödinger equation: HR(k) Ψ o =E k R(k) Ψ o R(k)H Ψ o =E o R(k) Ψ o Equation-of-motion: [H,R(k)] Ψ o =ω k R(k) Ψ o Monika Musiał Molecular properties 23/ 40

24 EOM-CC method Since Ψ o =e T Φ o weobtain-aftersimple algebra- eigenvalue equation: ( HR(k)) c Φ o =ω k R(k) Φ o orinmatrixform: HR(k)=ω k R(k) Monika Musiał Molecular properties 24/ 40

25 EOM-CC method R EE (k) = r o (k)+σ a Σ i r a i(k)a i+ 1 4 Σ abσ ij r ab ij(k)a b ji Σ abcσ ijl r abc ijl (k)a b c lji+... R IP (k) = Σ i r i (k)i+ 1 2 Σ aσ ij r a ij (k)a ji Σ abσ ijl r ab ijl (k)a b lji+... R EA (k) = Σ a r a (k)a Σ abσ i r ab i (k)a b i Σ abcσ ij r abc ij (k)a b c ji+... Monika Musiał Molecular properties 25/ 40

26 EOM-CC method R DIP (k) = 1 r ij (k)ji+ 1 r a 2 ij 6 ijl(k)a lji+... a ijl R DEA (k) = 1 r ab (k)a b + 1 r abc i (k)a b c i ab 6abc i Monika Musiał Molecular properties 26/ 40

27 EOM-CC method EOM-CCSD model H= S H S S H D D H S D H D S Φ j D Φ a ij Ionizationpotential(IP) S Φ b D Φ ab i Electron affinity(ea) S Φ a i D Φ ab ij Excitation energy(ee) Monika Musiał Molecular properties 27/ 40

28 EOM-CC method EOM-CCSDT model H= IP: T Φ ab ijl EA:T Φ abc ij EE:T Φ abc ijl S H S S H D S H T D H S D H D D H T T H S T H D T H T Monika Musiał Molecular properties 28/ 40

29 EOM-CC method EOM-CCSDforDIPandDEA H= [ D H D ] D Φ ij Doubleionizationpotential(DIP) D Φ ab Doubleelectronaffinity(DEA) Monika Musiał Molecular properties 29/ 40

30 EOM-CC method EOM-CCSDT for DIP and DEA H= D H D D H T T H D T H T DIP: T Φ a ijk DEA:T Φ abc i Monika Musiał Molecular properties 30/ 40

31 EOM-CC method Efficient technique used to diagonalize large matrices: Davidson method generalized for non-hermitian matrices. Monika Musiał Molecular properties 31/ 40

32 EOM-CC method A crucial step in the Davidson procedure is taking productoftheamplitudevectorrandthematrixtobediagonalized- H x k = HR k Monika Musiał Molecular properties 32/ 40

33 EOM-CC method Theproduct HR k cannotbetakendirectlyas a matrix product. Why? Rankof H goesintomilions. Weknow Hoperatorbutwedonotconstruct H matrix. Thusourequationscanbeseenas: Monika Musiał Molecular properties 33/ 40

34 EOM-CC method EE-EOM-CCSDT equations R(k) = R o (k)+r 1 (k)+r 2 (k)+r 3 (k) x a i(k)= Φ a i ( H N R(k)) c Φ o x ab ij (k)= Φab ij ( H N R(k)) c Φ o x abc ijl (k)= Φabc ijl ( H N R(k)) c Φ o x a i,xab ij,xabc ijl elementsofthex k vector Monika Musiał Molecular properties 34/ 40

35 EOM-CC method The diagrammatic representation of the elements of H N used in derivation of the EE-EOM-CCSDT equations. one-body I 1 : I I1 2 a b j i i a I a b I i j I i a a i b c I ab ci j k a i I ia jk two-body I 2 : I I2 2 + I2 3 + I2 4 a b k l a i a i c d i j b j b c I ab cd I ij kl I aj bi I ai bc k j i a i a j b I ij ka I ij ab i b j c a d i I abc dij a l k j I ajk ilb b i k a l b j i I iab jkl a b j c I abj icd d three-body I 3 : I I I 3 3 m a k l a j k b i j c i I ija klm I aib cjk a b i c d e I abc dei i j bk c l d a I ibcd ajkl four-body I 4 : I I4 3 j c k d l b m c b i k a e i a j I ibcd aejk I ijbc aklm Monika Musiał Molecular properties 35/ 40

36 EOM-CC method S D T I 1 1 I 1 2 I 2 4 S I 2 2 I 2 3 I1 2 I1 I2 1 D I2 3 I2 I3 2 I3 I1 3 I1 2 I1 T I2 4 I2 3 I2 I3 4 I3 Monika Musiał Molecular properties 36/ 40

37 EOM-CC method Diagrammatic form of the standard EE-EOM-CCSDT equations. = = a + b +c Monika Musiał Molecular properties 37/ 40

38 EOM-CC method = d + + e + f + + g h + i + j + k Monika Musiał Molecular properties 38/ 40

39 EOM-CC method H operator 1 H is non-hermitian 2 It includes three-, four- and higher-body elements H=e T He T =(He T ) c Expanding H N intoone-body,two-body, three-body,... etc. contributions we get: H N =I o + 2 k=0 I 1 k + 4 k=0 I 2 k + 3 k=0 I 3 k + 3 k=0 I 4 k + Monika Musiał Molecular properties 39/ 40

40 EOM-CC Computational strategy Steps applied in the standard EOM-CC method: solve the standard CCSD(or CCSDT) equations for the ground state problem usingtheconvergedt 1,T 2 (ort 3 )amplitudes constructall Helements apply a generalized Davidson diagonalization procedure to obtain eigenvalues of the H matrix. Monika Musiał Molecular properties 40/ 40