Molecular properties in quantum chemistry Monika Musiał Department of Theoretical Chemistry
Outline 1 Main computational schemes for correlated calculations 2 Developmentoftheabinitiomethodsforthe calculation of molecular properties. Monika Musiał Molecular properties 2/ 40
Literature L. Piela, Idee Chemii Kwantowej, PWN, Warszawa, 2003. R.J.Bartlett,M.Musial,Rev.Mod.Phys.,79, 291-352(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
Computational methods of quantum chemistry basedonthewavefunction Ψ Born-Oppenheimer approximation Electronic Schrödinger equation: HΨ = EΨ Monika Musiał Molecular properties 4/ 40
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
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
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
RHFvs.UHF RHF UHF α β Monika Musiał Molecular properties 8/ 40
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
RHFvs.UHF -14.800-14.820-14.840-14.860-14.880 E (a.u.) -14.900-14.920-14.940-14.960 RHF -14.980 UHF FCI -15.000 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0 R (a.u.) Monika Musiał Molecular properties 10/ 40
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
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
Configurations... c. b. a. i. j. k 2. 1. 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
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
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
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
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
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
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
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
EOM-CC method Equation-Of-Motion CC (EOM-CC) method Monika Musiał Molecular properties 21/ 40
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
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
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
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 + 1 36 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 + 1 12 Σ abσ ijl r ab ijl (k)a b lji+... R EA (k) = Σ a r a (k)a + 1 2 Σ abσ i r ab i (k)a b i + 1 12 Σ abcσ ij r abc ij (k)a b c ji+... Monika Musiał Molecular properties 25/ 40
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+... 2 ab 6abc i Monika Musiał Molecular properties 26/ 40
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
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
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
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
EOM-CC method Efficient technique used to diagonalize large matrices: Davidson method generalized for non-hermitian matrices. Monika Musiał Molecular properties 31/ 40
EOM-CC method A crucial step in the Davidson procedure is taking productoftheamplitudevectorrandthematrixtobediagonalized- H x k = HR k Monika Musiał Molecular properties 32/ 40
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
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
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 1 1 + 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 2 1 + 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 3 1 + I 3 2 + 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 4 2 + 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
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
EOM-CC method Diagrammatic form of the standard EE-EOM-CCSDT equations. = + + + + + + = + + + + + + + a + b +c + + + Monika Musiał Molecular properties 37/ 40
EOM-CC method = d + + e + f + + g + + + h + i + j + k + + + + + Monika Musiał Molecular properties 38/ 40
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
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